Feynman V1 OCR

Feynman's Preface These are the lectures in physics that I gave last year and the year before to the freshman and sophomore classes at Caltech. The lectures are, of course, not verbatim—they have been edited, sometimes extensively and sometimes less so. The lectures form only part of the complete course. The whole group of 180 students gathered in a big lecture room twice a week to hear these lectures and then they broke up into small groups of 15 to 20 students in recitation sections under the guidance of a teaching assistant. In addition, there was a laboratory session once a week. The special problem we tried to get at with these lectures was to maintain the interest of the very enthusiastic and rather smart students coming out of the high schools and into Caltech. They have heard a lot about how interesting and excit- ing physics is—the theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many would be very dis- couraged because there were really very few grand, new, modern ideas presented to them. They were made to study inclined planes, electrostatics, and so forth, and after two years it was quite stultifying. The problem was whether or not we could make a course which would save the more advanced and excited student by maintaining his enthusiasm. The lectures here are not in any way meant to be a survey course, but are very serious. I thought to address them to the most intelligent in the class and to make sure, if possible, that even the most intelligent student was unable to completely encompass everything that was in the lectures—by putting in suggestions of appli- cations of the ideas and concepts in various directions outside the main line of attack. For this reason, though, I tried very hard to make all the statements as accurate as possible, to point out in every case where the equations and ideas fitted into the body of physics, and how—when they learned more—things would be modified. I also felt that for such students it is important to indicate what it is that they should—if they are sufficiently clever—be able to understand by deduc- tion from what has been said before, and what is being put in as something new. When new ideas came in, I would try either to deduce them if they were deducible, or to explain that it was a new idea which hadn't any basis in terms of things they had already learned and which was not supposed to be provable—but was just added in. At the start of these lectures, I assumed that the students knew something when they came out of high school—such things as geometrical optics, simple chemistry ideas, and so on. I also didn't see that there was any reason to make the lectures 37% in a definite order, in the sense that I would not be allowed to mention something until I was ready to discuss it in detail. There was a great deal of mention of things to come, without complete discussions. These more complete discussions would come later when the preparation became more advanced. Examples are the dis- cussions of inductance, and of energy levels, which are at first brought in in a very qualitative way and are later developed more completely. At the same time that I was aiming at the more active student, I also wanted to take care of the fellow for whom the extra fireworks and side applications are merely disquieting and who cannot be expected to learn most of the material in the lecture at all. For such students I wanted there to be at least a central core or backbone of material which he could get. Even if he didn't understand everything in a lecture, I hoped he wouldn't get nervous. I didn't expect him to understand everything, but only the central and most direct features. It takes, of course, a certain intelligence on his part to see which are the central theorems and central ideas, and which are the more advanced side issues and applications which he may understand only in later years. In giving these lectures there was one serious difficulty: in the way the course was given, there wasn't any feedback from the students to the lecturer to indicate how well the lectures were going over. This is indeed a very serious difficulty, and I don't know how good the lectures really are. The whole thing was essentially an experiment. And if I did it again I wouldn't do it the same way—I hope I don't have to do it again! I think, though, that things worked out—so far as the physics is concerned—quite satisfactorily in the first year. In the second year I was not so satisfied. In the first part of the course, dealing with electricity and magnetism, I couldn't think of any really unique or different way of doing it—of any way that would be particularly more exciting than the usual way of presenting it. So I don't think I did very much in the lectures on electricity and magnetism. At the end of the second year I had originally intended to go on, after the electricity and magnetism, by giving some more lectures on the properties of materials, but mainly to take up things like fundamental modes, solutions of the diffusion equation, vibrating systems, orthogonal functions,... developing the first stages of what are usually called "the mathematical methods of physics." In retrospect, I think that if I were doing it again I would go back to that original idea. But since it was not planned that I would be giving these lec- tures again, it was suggested that it might be a good idea to try to give an introduc- tion to the quantum mechanics—what you will find in Volume III. It is perfectly clear that students who will major in physics can wait until their third year for quantum mechanics. On the other hand, the argument was made that many of the students in our course study physics as a background for their primary interest in other fields. And the usual way of dealing with quantum mechanics makes that subject almost unavailable for the great majority of students because they have to take so long to learn it. Yet, in its real applications—espe- cially in its more complex applications, such as in electrical engineering and chem- istry—the full machinery of the differential equation approach is not actually used. So I tried to describe the principles of quantum mechanics in a way which wouldn't require that one first know the mathematics of partial differential equa- tions. Even for a physicist I think that is an interesting thing to try to do—to present quantum mechanics in this reverse fashion—for several reasons which may be apparent in the lectures themselves. However, I think that the experiment in the quantum mechanics part was not completely successful—in large part because I really did not have enough time at the end (I should, for instance, have had three or four more lectures in order to deal more completely with such matters as energy bands and the spatial dependence of amplitudes). Also, I had never presented the subject this way before, so the lack of feedback was particularly serious. I now believe the quantum mechanics should be given at a later time. Maybe I'll have a chance to do it again someday. Then I'll do it right. The reason there are no lectures on how to solve problems is because there were recitation sections. Although I did put in three lectures in the first year on how to solve problems, they are not included here. Also there was a lecture on inertial 4 don’! guidance which certainly belongs after the lecture on rotating systems, but which was, unfortunately, omitted. The fifth and sixth lectures are actually due to Matthew Sands, as I was out of town. The question, of course, is how well this experiment has succeeded. My own point of view—which, however, does not seem to be shared by most of the people who worked with the students—is pessimistic. I don't think I did very well by the students. When I look at the way the majority of the students handled the problems on the examinations, I think that the system is a failure. Of course, my friends point out to me that there were one or two dozen students who—very surprisingly —understood almost everything in all of the lectures, and who were quite active in working with the material and worrying about the many points in an excited and interested way. These people have now, I believe, a first-rate background in physics—and they are, after all, the ones I was trying to get at. But then, "The power of instruction is seldom of much efficacy except in those happy dispositions where it is almost superfluous." (Gibbon) Still, I didn't want to leave any student completely behind, as perhaps I did. I think one way we could help the students more would be by putting more hard work into developing a set of problems which would elucidate some of the ideas in the lectures. Problems give a good opportunity to fill out the material of the lectures and make more realistic, more complete, and more settled in the mind the ideas that have been exposed. I think, however, that there isn't any solution to this problem of education other than to realize that the best teaching can be done only when there is a direct individual relationship between a student and a good teacher—a situation in which the student discusses the ideas, thinks about the things, and talks about the things. It's impossible to learn very much by simply sitting in a lecture, or even by simply doing problems that are assigned. But in our modern times we have so many students to teach that we have to try to find some substitute for the ideal. Perhaps my lectures can make some contribution. Perhaps in some small place where there are individual teachers and students, they may get some inspiration or some ideas from the lectures. Perhaps they will have fun thinking them through—or going on to develop some of the ideas further. RICHARD P. FEYNMAN June, 1963 Foreword This book is based upon a course of lectures in introductory physics given by Prof. R. P. Feynman at the California Institute of Technology during the academic year 1961-62; it covers the first year of the two-year introductory course taken by all Caltech freshmen and sophomores, and was followed in 1962-63 by a similar series covering the second year. The lectures constitute a major part of a funda- mental revision of the introductory course, carried out over a four-year period. The need for a basic revision arose both from the rapid development of physics in recent decades and from the fact that entering freshmen have shown a steady increase in mathematical ability as a result of improvements in high school mathe- matics course content. We hoped to take advantage of this improved mathematical background, and also to introduce enough modern subject matter to make the course challenging, interesting, and more representative of present-day physics. In order to generate a variety of ideas on what material to include and how to present it, a substantial number of the physics faculty were encouraged to offer their ideas in the form of topical outlines for a revised course. Several of these were presented and were thoroughly and critically discussed. It was agreed almost at once that a basic revision of the course could not be accomplished either by merely adopting a different textbook, or even by writing one ab initio, but that the new course should be centered about a set of lectures, to be presented at the rate of two or three per week; the appropriate text material would then be produced as a secondary operation as the course developed, and suitable laboratory experi- ments would also be arranged to fit the lecture material. Accordingly, a rough outline of the course was established, but this was recognized as being incomplete, tentative, and subject to considerable modification by whoever was to bear the responsibility for actually preparing the lectures. Concerning the mechanism by which the course would finally be brought to life, several plans were considered. These plans were mostly rather similar, involv- ing a cooperative effort by N staff members who would share the total burden symmetrically and equally: each man would take responsibility for 1/N of the material, deliver the lectures, and write text material for his part. However, the unavailability of sufficient staff, and the difficulty of maintaining a uniform point of view because of differences in personality and philosophy of individual partici- pants, made such plans seem unworkable. The realization that we actually possessed the means to create not just a new and different physics course, but possibly a unique one, came as a happy inspira- tion to Professor Sands. He suggested that Professor R. P. Feynman prepare and deliver the lectures, and that these be tape-recorded. When transcribed and edited, they would then become the textbook for the new course. This is essentially the plan that was adopted. It was expected that the necessary editing would be minor, mainly consisting of supplying figures, and checking punctuation and grammar; it was to be done by one or two graduate students on a part-time basis. Unfortunately, this expectation was short-lived. It was, in fact, a major editorial operation to transform the ver- batim transcript into readable form, even without the reorganization or revision of The subject matter that was sometimes required. Furthermore, it was not a job for a technical editor or for a graduate student, but one that required the close attention of a professional physicist for from ten to twenty hours per lecture! 7 The difficulty of the editorial task, together with the need to place the material in the hands of the students as soon as possible, set a strict limit upon the amount of "polishing" of the material that could be accomplished, and thus we were forced to aim toward a preliminary but technically correct product that could be used immediately, rather than one that might be considered final or finished. Because of an urgent need for more copies for our students, and a heartening inter- est on the part of instructors and students at several other institutions, we decided to publish the material in its preliminary form rather than wait for a further major revision which might never occur. We have no illusions as to the completeness, smoothness, or logical organization of the material; in fact, we plan several minor modifications in the course in the immediate future, and we hope that it will not become static in form or content. In addition to the lectures, which constitute a centrally important part of the course, it was necessary also to provide suitable exercises to develop the students' experience and ability, and suitable experiments to provide first-hand contact with the lecture material in the laboratory. Neither of these aspects is in as ad- vanced a state as the lecture material, but considerable progress has been made. Some exercises were made up as the lectures progressed, and these were expanded and amplified for use in the following year. However, because we are not yet satisfied that the exercises provide sufficient variety and depth of application of the lecture material to make the student fully aware of the tremendous power being placed at his disposal, the exercises are published separately in a less perma- nent form in order to encourage frequent revision. A number of new experiments for the new course have been devised by Professor H. V. Neher. Among these are several which utilize the extremely low friction exhibited by a gas bearing: a novel linear air trough, with which quantitative measurements of one-dimensional motion, impacts, and harmonic motion can be made, and an air-supported, air-driven Maxwell top, with which accelerated rota- tional motion and gyroscopic precession and nutation can be studied. The develop- ment of new laboratory experiments is expected to continue for a considerable period of time. The revision program was under the direction of Professors R. B. Leighton, H. V. Neher, and M. Sands. Officially participating in the program were Professors R. P. Feynman, G. Neugebauer, R. M. Sutton, H. P. Stabler,* F. Strong, and R. Vogt, from the division of Physics, Mathematics and Astronomy, and Professors T. Caughey, M. Plesset, and C. H. Wilts from the division of Engineering Science. The valuable assistance of all those contributing to the revision program is grate- fully acknowledged. We are particularly indebted to the Ford Foundation, without whose financial assistance this program could not have been carried out. ROBERT B. LEIGHTON July, 1963 * 1961-62, while on leave from Williams College, Williamstown, Mass. Contents CHAPTER 1. ATOMS IN MOTION 1-1 Introduction 1-1 1-2 Matter is made of atoms 1-2 1-3 Atomic processes 1-5 1-4 Chemical reactions 1-6 CHAPTER 2. BASIC PHYSICS 2-1 Introduction 2-1 2-2 Physics before 1920 2-3 2-3 Quantum physics 2-6 2-4 Nuclei and particles 2-8 CHAPTER 3. THE RELATION OF PHYSICS TO OTHER SCIENCES 3-1 Introduction 3-1 3-2 Chemistry 3-1 3-3 Biology 3-2 3-4 Astronomy 3-6 3-5 Geology 3-7 3-6 Psychology 3-8 3-7 How did it get that way? 3-9 CHAPTER 4. CONSERVATION OF ENERGY 4-1 What is energy? 4-1 4-2 Gravitational potential energy 4-2 4-3 Kinetic energy 4-5 4-4 Other forms of energy 4-6 CHAPTER 5. TIME AND DISTANCE 5-1 Motion 5-1 5-2 Time 5-1 5-3 Short times 5-2 5-4 Long times 5-3 5-5 Units and standards of time 5-5 5-6 Large distances 5-5 5-7 Short distances 5-8 CHAPTER 6. PROBABILITY 6-1 Chance and likelihood 6-1 6-2 Fluctuations 6-3 6-3 The random walk 6-5 6-4 A probability distribution 6-7 6-5 The uncertainty principle 6-10 CHAPTER 7. THE THEORY OF GRAVITATION 7-1 Planetary motions 7-1 7-2 Kepler's laws 7-1 7-3 Development of dynamics 7-2 7-4 Newton's law of gravitation 7-3 7-5 Universal gravitation 7-5 7-6 Cavendish's experiment 7-9 7-7 What is gravity? 7-9 7-8 Gravity and relativity 7-11CHAPTER 8. MOTION 8-1 Description of motion 8-1 8-2 Speed 8-2 8-3 Speed as a derivative 8-5 8-4 Distance as an integral 8-7 8-5 Acceleration 8-8 CHAPTER 9. NEWTON'S LAWS OF DYNAMICS 9-1 Momentum and force 9-1 9-2 Speed and velocity 9-2 9-3 Components of velocity, acceleration, and force 9-3 9-4 What is the force? 9-3 9-5 Meaning of the dynamical equations 9-4 9-6 Numerical solution of the equations 9-5 9-7 Planetary motions 9-6 CHAPTER 10. CONSERVATION OF MOMENTUM 10-1 Newton's Third Law 10-1 10-2 Conservation of momentum 10-2 10-3 Momentum is conserved! 10-5 10-4 Momentum and energy 10-7 10-5 Relativistic momentum 10-8 VECTORS CHAPTER 11. 11-1 11-2 11-3 11-4 11-5 11-6 11-7Symmetry in physics 11-1 Translations 11-1 Rotations 11-3 Vectors 11-5 Vector algebra 11-6 Newton's laws in vector notation 11-7 Scalar product of vectors 11-8 CHAPTER 12. CHARACTERISTICS OF FORCE 12-1 What is a force? 12-1 12-2 Friction 12-3 12-3 Molecular forces 12-6 12-4 Fundamental forces. Fields 12-7 12-5 Pseudo forces 12-10 12-6 Nuclear forces 12-12 CHAPTER 13. WORK AND POTENTIAL ENERGY (A) 13-1 Energy of a falling body 13-1 13-2 Work done by gravity 13-3 13-3 Summation of energy 13-6 13-4 Gravitational field of large objects 13-8 CHAPTER 14. WORK AND POTENTIAL ENERGY (conclusion) 14-1 Work 14-1 14-2 Constrained motion 14-3 14-3 Conservative forces 14-3 14-4 Nonconservative forces 14-6 14-5 Potentials and fields 14-7 CHAPTER 15. THE SPECIAL THEORY OF RELATIVITY 15-1 The principle of relativity 15-1 15-2 The Lorentz transformation 15-3 15-3 The Michelson-Morley experiment 15-3 15-4 Transformation of time 15-5 15-5 The Lorentz contraction 15-7 15-6 Simultaneity 15-7 15-7 Four-vectors 15-8 15-8 Relativistic dynamics 15-9 15-9 Equivalence of mass and energy 15-10 CHAPTER 16. RELATIVISTIC ENERGY AND MOMENTUM 16-1 Relativity and the philosophers 16-1 16-2 The twin paradox 16-3 16-3 Transformation of velocities 16-4 16-4 Relativistic mass 16-6 16-5 Relativistic energy 16-8 CHAPTER 17. SPACE-TIME 17-1 The geometry of space-time 17-1 17-2 Space-time intervals 17-2 17-3 Past, present, and future 17-4 17-4 More about four-vectors 17-5 17-5 Four-vector algebra 17-7. CHAPTER 18. ROTATION IN Two DIMENSIONS 18-1 The center of mass 18-1 18-2 Rotation of a rigid body 18-2 18-3 Angular momentum 18-5 18-4 Conservation of angular momentum 18-6 CHAPTER 19. CENTER OF MASS; MOMENT OF INERTIA 19-1 Properties of the center of mass 19-1 19-2 Locating the center of mass 19-4 19-3 Finding the moment of inertia 19-5 19-4 Rotational kinetic energy 19-7 CHAPTER 20. ROTATION IN SPACE 20-1 Torques in three dimensions 20-1 20-2 The rotation equations using cross products 20-4 20-3 The gyroscope 20-5 20-4 Angular momentum of a solid body 20-8 CHAPTER 21. THE HARMONIC OSCILLATOR 21-1 Linear differential equations 21-1 21-2 The harmonic oscillator 21-1 21-3 Harmonic motion and circular motion 21-4 21-4 Initial conditions 21-4 21-5 Forced oscillations 21-5 ALGEBRA CHAPTER 22. 22-1 22-2 22-3 22-4 22-5 22-6Addition and multiplication 22-1 The inverse operations 22-2 Abstraction and generalization 22-3 Approximating irrational numbers 22-4 Complex numbers 22-7 Imaginary exponents 22-9 CHAPTER 23. RESONANCE 23-1 Complex numbers and harmonic motion 23-1 23-2 The forced oscillator with damping 23-323-3 Electrical resonance 23-5 23-4 Resonance in nature 23-7 CHAPTER 24. TRANSIENTS 24-1 The energy of an oscillator 24-1 24-2 Damped oscillations 24-2 24-3 Electrical transients 24-5 CHAPTER 25. LINEAR SYSTEMS AND REVIEW 25-1 Linear differential equations 25-1 25-2 Superposition of solutions 25-2 25-3 Oscillations in linear systems 25-5 25-4 Analogs in physics 25-6 25-5 Series and parallel impedances 25-8 CHAPTER 26. OPTICS: THE PRINCIPLE OF LEAST TIME 26-1 Light 26-1 26-2 Reflection and refraction 26-2 26-3 Fermat's principle of least time 26-3 26-4 Applications of Fermat's principle 26-5 26-5 A more precise statement of Fermat's principle 26-7 26-6 How it works 26-8 CHAPTER 27. GEOMETRICAL OPTICS 27-1 Introduction 27-1 27-2 The focal length of a spherical surface 27-1 27-3 The focal length of a lens 27-4 27-4 Magnification 27-5 27-5 Compound lenses 27-6 27-6 Aberrations 27-7 27-7 Resolving power 27-7 CHAPTER 28. ELECTROMAGNETIC RADIATION 28-1 Electromagnetism 28-1 28-2 Radiation 28-3 28-3 The dipole radiator 28-5 28-4 Interference 28-6 CHAPTER 29. INTERFERENCE 29-1 Electromagnetic waves 29-1 29-2 Energy of radiation 29-2 29-3 Sinusoidal waves 29-2 29-4 Two dipole radiators 29-3 29-5 The mathematics of interference 29-5 CHAPTER 30. DIFFRACTION 30-1 The resultant amplitude due to n equal oscillators 30-1 30-2 The diffraction grating 30-3 30-3 Resolving power of a grating 30-5 30-4 The parabolic antenna 30-6 30-5 Colored films; crystals 30-7 30-6 Diffraction by opaque screens 30-8 30-7 The field of a plane of oscillating charges 30-10 CHAPTER 31. THE ORIGIN OF THE REFRACTIVE INDEX 31-1 The index of refraction 31-1 31-2 The field due to the material 31-4 31-3 Dispersion 31-6 31-4 Absorption 31-8 31-5 The energy carried by an electric wave 31-9 31-6 Diffraction of light by a screen 31-10 10 9-4 20-4 CHAPTER 32. RADIATION DAMPING. LIGHT SCATTERING 32-1 Radiation resistance 32-1 32-2 The rate of radiation of energy 3.2-2 32-3 Radiation damping 32-3 32-4 Independent sources 32-5 32-5 Scattering of light 32-6 CHAPTER 33. POLARIZATION 33-1 The electric vector of light 33-1 33-2 Polarization of scattered light 33-3 33-3 Birefringence 33-3 33-4 Polarizers 33-5 33-5 Optical activity 33-6 33-6 The intensity of reflected light 33-7 33-7 Anomalous refraction 33-9 CHAPTER 34. RELATIVISTIC EFFECTS IN RADIATION 34-1 Moving sources 34-1 34-2 Finding the "apparent" motion 34-2 34-3 Synchrotron radiation 34-3 34-4 Cosmic synchrotron radiation 34-6 34-5 Bremsstrahlung 34-6 34-6 The Doppler effect 34-7 34-7 The w, k four-vector 34-9 34-8 Aberration 34-10 34-9 The momentum of light 34-10 CHAPTER 35. COLOR VISION 35-1 The human eye 35-1 35-2 Color depends on intensity 35-2 35-3 Measuring the color sensation 35-3 35-4 The chromaticity diagram 35-6 / 35-5 The mechanism of color vision 35-7 35-6 Physiochemistry of color vision 35-9 CHAPTER 36. MECHANISMS OF SEEING 36-1 The sensation of color 36-1 36-2 The physiology of the eye 36-3 36-3 The rod cells 36-6 36-4 The compound (insect) eye 36-6 36-5 Other eyes 36-9 36-6 Neurology of vision 36-9 CHAPTER 37. QUANTUM BEHAVIOR 37-1 Atomic mechanics 37-1 37-2 An experiment with bullets 37-2 37-3 An experiment with waves 37-3 37-4 An experiment with electrons 37-4 37-5 The interference of electron waves 37-5 37-6 Watching the electrons 37-7 37-7 First principles of quantum mechanics 37-10 37-8 The uncertainty principle 37-1138-5 Energy levels 38-7 38-6 Philosophical implications 38-8 CHAPTER 39. THE KINETIC THEORY OF GASES 39-1 Properties of matter 39-1 39-2 The pressure of a gas 39-2 39-3 Compressibility of radiation 39-6 39-4 Temperature and kinetic energy 39-6 39-5 The ideal gas law 39-10 CHAPTER 40. THE PRINCIPLES OF STATISTICAL MECHANICS 40-1 The exponential atmosphere 40-1 40-2 The Boltzmann law 40-2 40-3 Evaporation of a liquid 40-3 40-4 The distribution of molecular speeds 40-4 40-5 The specific heats of gases 40-7 40-6 The failure of classical physics 40-8 CHAPTER 41. THE BROWNIAN MOVEMENT 41-1 Equipartition of energy 41-1 41-2 Thermal equilibrium of radiation 41-3 41-3 Equipartition and the quantum oscillator 41-6 41-4 The random walk 41-8 CHAPTER 42. APPLICATIONS OP KINETIC THEORY 42-1 Evaporation 42-1 42-2 Thermionic emission 42-4 42-3 Thermal ionization 42-5 42-4 Chemical kinetics 42-7 42-5 Einstein's laws of radiation 42-8 CHAPTER 43. DIFFUSION 43-1 Collisions between molecules 43-1 43-2 The mean free path 43-3 43-3 The drift speed 43-4 43-4 Ionic conductivity 43-6 43-5 Molecular diffusion 43-7 43-6 Thermal conductivity 43-9 CHAPTER 44. THE LAWS OF THERMODYNAMICS 44-1 Heat engines; the first law 44-1 44-2 The second law 44-3 44-3 Reversible engines 44-4 44-4 The efficiency of an ideal engine 44-7 44-5 The thermodynamic temperature 44-9 44-6 Entropy 44-10 CHAPTER 45. ILLUSTRATIONS OF THERMODYNAMICS 45-1 Internal energy 45-1 45-2 Applications 45-4 45-3 The Clausius-Clapeyron equation 45-6 CHAPTER 38. THE RELATION OF WAVE AND PARTICLE VIEWPOINTS 38-1 Probability wave amplitudes 38-1 38-2 Measurement of position and momentum 38-2 38-3 Crystal diffraction 38-4 38-4 The size of an atom 38-5CHAPTER 46. RATCHET AND PAWL 46-1 How a ratchet works 46-1 46-2 The ratchet as an engine 46-2 46-3 Reversibility in mechanics 46-4 46-4 Irreversibility 46-5 46-5 Order and entropy 46-7 41-4 42-4 CHAPTER 47. SOUND. THE WAVE EQUATION 47-1 Waves 47-1 47-2 The propagation of sound 47-3 47-3 The wave equation 47-4 47-4 Solutions of the wave equation 47-6 47-5 The speed of sound 47-7 CHAPTER 48. BEATS 48-1 Adding two waves 48-1 48-2 Beat notes and modulation 48-3 48-3 Side bands 48-4 48-4 Localized wave trains 48-5 48-5 Probability amplitudes for particles 48-7 48-6 Waves in three dimensions 48-9 48-7 Normal modes 48-10 CHAPTER 49. MODES 49-1 The reflection of waves 49-1 49-2 Confined waves, with natural frequencies 49-2 49-3 Modes in two dimensions 49-3 49-4 Coupled pendulums 49-6 49-5 Linear systems 49-7 INDEXCHAPTER 50. HARMONICS 50-1 Musical tones 50-1 50-2 The Fourier series 50-2 50-3 Quality and consonance 50-3 50-4 The Fourier coefficients 50-5 50-5 The energy theorem 50-7 50-6 Nonlinear responses 50-8 CHAPTER 51. WAVES 51-1 Bow waves 51-1 51-2 Shock waves 51-2 51-3 Waves in solids 51-4 51-4 Surface waves 51-7 CHAPTER 52. SYMMETRY IN PHYSICAL LAWS 52-1 Symmetry operations 52-1 52-2 Symmetry in space and time 52-1 52-3 Symmetry and conservation laws 52-3 52-4 Mirror reflections 52-4 52-5 Polar and axial vectors 52-6 52-6 Which hand is right? 52-8 52-7 Parity is not conserved! 52-8 52-8 Antimatter 52-10 52-9 Broken symmetries 52-11 12 Atoms in Motion 1-1 Introductio n This two-year course in physics is presented from the poin t of view that you, the reader, are going to be a physicist. This is not necessarily the case of course, but that is what every professor in every subject assumes! If you are going to be a physicist, you will have a lot to study : two hundred years of the most rapidly developing field of knowledge that there is. So much knowledge, in fact, that you might think that you cannot learn all of it in four years, and truly you cannot; you will have to go to graduat e school too! Surprisingly enough, in spite of the tremendous amount of work that has been done for all this time it is possible to condense the enormous mass of results to a large extent—that is, to find laws which summarize all our knowledge. Even so, the laws are so hard to grasp that it is unfai r to you to start exploring this tremendous subject without some kind of map or outline of the relationship of one part of the subject of science to another. Following these preliminary remarks, the first three chapters will therefore outline the relation of physics to the rest of the sciences , the relations of the sciences to each other, and the meaning of science , to help us develop a "feel " for the subject. You might ask why we cannot teach physics by just giving the basic laws on page one and then showing how they work in all possible circumstances, as we do in Euclidean geometry, where we state the axioms and then mak e all sorts of de- ductions. (So , not satisfied to learn physics in four years, you want to learn it in four minutes?) We cannot do it in this way for two reasons. First, we do not yet know all the basic laws: there is an expanding frontie r of ignorance. Second, the correct statement of the laws of physics involves some very unfamiliar ideas which require advanced mathematics for their description. Therefore, one needs a considerable amount of preparatory trainin g even to learn what the words mean. No, it is not possible to do it that way. We can only do it piece by piece. Each piece, or part , of the whole of natur e is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, every- thing we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected. The principle of science , the definition , almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific "truth." But what is the source of knowledge? Where do the laws that are to be tested come from ? Experiment, itself, helps to produc e these laws, in the sense that it gives us hints. But also needed is imagination to create from these hint s the great generalizations—to guess at the wonderful, simple, but very strange patterns be- neath them all, and then to experiment to check again whether we have made the right guess. This imagining process is so difficul t that there is a division of labor in physics: there are theoretical physicists who imagine, deduce, and guess at new laws, but do not experiment; and then there are experimental physicists who ex- periment, imagine, deduce, and guess. We said that the laws of nature are approximate: that we first find the "wrong" ones, and then we find the "right" ones. Now, how can an experiment be "wrong" ? First, in a trivial way : if something is wrong with the apparatus that you did not notice. But these things are easily fixed , and checked back and forth . So without snatching at such minor things, how can the results of an experiment be wrong ? Only by being inaccurate. For example, the mass of an object never seems to 1-11-1 Introduction 1-2 Matter is made of atom s 1-3 Atomic processe s 1-4 Chemica l reaction s Atoms Motion WATER MAGNIFIED ONE BILLIO N TIMES Figur e 1-1change; a spinning top has the same weight as a still one. So a "law " was in- vented: mass is constant, independent of speed. That "law " is now foun d to be incorrect. Mass is foun d to increase with velocity, but appreciable increases require velocities near that of light. A true law is: if an object moves with a speed of less than one hundred miles a second the mass is constant to within one part in a million. In some such approximate form this is a correct law. So in practice one might think that the new law makes no significant difference. Well , yes and no. For ordinary speeds we can certainly forget it and use the simple constant- mass law as a good approximation. But for high speeds we are wrong, and the higher the speed, the more wrong we are. Finally, and most interesting, philosophically we are completely wrong with the approximat e law. Our entire picture of the world has to be altered even though the mass changes only by a little bit. This is a very peculiar thing about the philosophy, or the ideas, behind the laws. Even a very small effect sometimes requires profoun d changes in our ideas. Now, what should we teach first ? Should we teach the correct but unfamilia r law with its strange and difficul t conceptual ideas, for example the theor y of relativity, four-dimensional space-time, and so on? Or should we first teach the simple "constant-mass" law, which is only approximate, but does not involve such difficul t ideas? The first is more exciting, more wonderful , and more fun, but the second is easier to get at first , and is a first step to a real understandin g of the second idea. This point arises again and again in teaching physics. At differen t times we shall have to resolve it in differen t ways, but at each stage it is worth learning what is now known, how accurate it is, how it fits into everything else, and how it may be changed when we learn more. Let us now proceed with our outline, or general map, of our understandin g of scienc e today (in particular , physics, but also of other sciences on the periphery), so that when we later concentrate on some particula r point we will have some idea of the background, why that particular point is interesting, and how it fits into the big structure. So, what is our over-all pictur e of the world? 1-2 Matter is made of atoms If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, wha t statement would contain the most informatio n in the fewest words ? I believe it is the atomic hypothesis (or the atomic fact, or whateve r you wish to call it) that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied. To illustrate the power of the atomic idea, suppose that we have a drop of water a quarte r of an inch on the side. If we look at it very closely we see nothin g but water—smooth, continuous water. Even if we magnif y it with the best optical microscope available—roughly two thousand times—then the water drop will be roughly fort y feet across, about as big as a large room, and if we looked rathe r closely, we would still see relatively smooth water—but here and there small football-shaped things swimming back and forth . Very interesting. Thes e are paramecia. You may stop at this poin t and get so curious abou t the paramecia with thei r wiggling cilia and twisting bodies that you go no further , except per- haps to magnif y the paramecia still more and see inside. This, of course, is a subject for biology, but for the present we pass on and look still more closely at the water material itself, magnifyin g it two thousan d times again. Now the drop of water extends about fiftee n miles across, and if we look very closely at it we see a kind of teeming, something which no longer has a smooth appearance—it looks some- thing like a crowd at a football game as seen from a very great distance. In order to see what this teeming is about , we will magnif y it another two hundred and fifty times and we will see something similar to what is shown in Fig. 1-1. This is a picture of water magnified a billion times, but idealized in several ways. 1-2%Q9Crumw In the first place, the particles are drawn in a simple manner with sharp edges, which is inaccurate. Secondly, for simplicity, they are sketched almost schemati- cally in a two-dimensional arrangement, but of course they are moving aroun d in three dimensions. Notice that there are two kinds of "blobs" or circles to represent the atoms of oxygen (black) and hydrogen (white), and that each oxygen has two hydrogens tied to it. (Each little group of an oxygen with its two hydrogens is called a molecule.) The picture is idealized furthe r in that the real particles in nature are continually jiggling and bouncing, turnin g and twisting aroun d one another. You will have to imagine this as a dynamic rathe r than a static picture. Another thing that cannot be illustrated in a drawing is the fact that the particles are "stuck together"—that they attract each other, this one pulled by that one, etc. The whole group is "glue d together," so to speak. On the other hand, the particles do not squeeze through each other. If you try to squeeze two of them too close together, they repel. The atoms are 1 or 2 X 10-8 cm in radius. Now 10-8 cm is called an angstrom (just as another name), so we say they are 1 or 2 angstroms (Å) in radius. Another way to remember their size is this: if an apple is magnified to the size of the earth, then the atoms in the apple are approximately the size of the original apple. Now imagine this great drop of water with all of these jiggling particles stuck together and tagging along with each other. The water keeps its volume; it does not fall apart , because of the attraction of the molecules for each other. If the drop is on a slope, where it can move from one place to another , the water will flow, but it does not just disappear—things do not just fly apart—because of the molecular attraction. Now the jigglin g motion is what we represent as heat: when we increase the temperature, we increase the motion. If we heat the water, the jiggling increases and the volume between the atoms increases, and if the heating continues there comes a time when the pull between the molecules is not enough to hold them together and they do fly apar t and become separated from one another. Of course, this is how we manufactur e steam out of water—by increasing the temperature; the particles fly apart because of the increased motion. In Fig. 1-2 we have a picture of steam. This picture of steam fails in one respect: at ordinary atmospheric pressure there might be only a few molecules in a whole room, and there certainly would not be as many as three in this figure. Most squares this size would contain none—but we accidentally have two and a half or three in the picture (jus t so it would not be completely blank). Now in the case of steam we see the characteristic molecules more clearly than in the case of water. For simplicity, the molecules are drawn so that there is a 120° angle between them. In actual fact the angle is 105°3' , and the distance between the center of a hydrogen and the center of the oxygen is 0.957 Å, so we know this molecule very well. Let us see wha t some of the properties of steam vapor or any other gas are. The molecules, being separated from one another, will bounce against the walls. Imagine a room with a number of tennis balls (a hundred or so) bouncin g around in perpetual motion. When they bombard the wall, this pushes the wall away. (Of course we would have to push the wall back.) This means that the gas exerts a jitter y force which our coarse sense s (not being ourselves magnified a billion times) feels only as an average push. In order to confine a gas we mus t apply a pressure. Figure 1-3 shows a standard vesse l for holdin g gases (used in all textbooks), a cylinder with a piston in it. Now, it makes no differenc e wha t the shapes of water molecules are, so for simplicity we shall draw them as tenni s balls or little dots. These things are in perpetual motion in all directions. So many of them are hittin g the top piston all the time that to keep it from being patiently knocked out of the tank by this continuou s banging, we shall have to hold the piston down by a certai n force, which we call the pressure (really, the pressure times the area is the force). Clearly, the force is proportiona l to the area , for if we increase the area but keep the numbe r of molecules per cubic centimeter the same, we increase the numbe r of collisions with the piston in the same proportion as the area was increased . 1-3Figure 1-2 Figur e 1-3‘QSTElM ‘~ 1*44$14. °> *0’\x "AP Figure 1-4Now let us put twice as many molecules in this tank , so as to double the den- sity, and let them have the same speed, i.e., the same temperature. Then, to a close approximation , the number of collisions will be doubled, and since each will be just as "energetic" as before, the pressure is proportiona l to the density. If we consider the true natur e of the forces between the atoms, we would expect a slight decrease in pressure because of the attraction between the atoms, and a slight increase because of the finite volume they occupy. Nevertheless, to an excellent approximation, if the density is low enough that there are not many atoms , the pressure is proportional to the density. We can also see something else: If we increase the temperature without changing the density of the gas, i.e., if we increase the speed of the atoms, what is going to happen to the pressure? Well, the atoms hit harde r because they are moving faster , and in addition they hit more often , so the pressure increases. You see how simple the ideas of atomic theor y are. Let us consider anothe r situation. Suppose that the piston moves inward, so that the atoms are slowly compressed into a smaller space. What happens when an atom hits the moving piston ? Evidently it picks up speed from the collision. You can try it by bouncing a ping-pong ball from a forward-movin g paddle, for example, and you will find that it comes off with more speed than that with which it struck . (Special example: if an atom happens to be standing still and the piston hits it, it will certainly move.) So the atoms are "hotter" when they come away from the piston than they were before they struc k it. Therefore all the atoms which are in the vessel will have picked up speed. This means that when we compress a gas slowly, the temperature of the gas increases. So, under slow compression, a gas will increase in temperature, and under slow expansion it will decrease in temperature. We now return to our drop of water and look in another direction. Suppose that we decrease the temperatur e of our drop of water . Suppose that the jiggling of the molecules of the atoms in the water is steadily decreasing. We know that there are forces of attraction between the atoms, so that after a while they will not be able to jiggle so well. Wha t will happen at very low temperatures is in- dicated in Fig. 1-4: the molecules lock into a new patter n which is ice. This particula r schematic diagra m of ice is wrong because it is in two dimensions, but it is righ t qualitatively. The interesting point is that the material has a definite place for every atom, and you can easily appreciate that if somehow or othe r we were to hold all the atoms at one end of the drop in a certain arrangement , each atom in a certain place, then because of the structure of interconnections, which is rigid, the other end miles away (at our magnified scale) will have a definite location. So if we hold a needle of ice at one end, the other end resists our pushing it aside, unlike the case of water, in which the structure is broke n down because of the increased jigglin g so that the atoms all move aroun d in differen t ways. The differ - ence between solids and liquid s is, then, that in a solid the atoms are arrange d in some kind of an array , called a crystalline array, and they do not have a random position at long distances; the position of the atoms on one side of the crystal is determined by that of othe r atoms millions of atoms away on the othe r side of the crystal. Figure 1-4 is an invented arrangemen t for ice, and although it con- tains many of the correct features of ice, it is not the true arrangement . One of the correct feature s is that there is a part of the symmetry that is hexagonal. You can see that if we turn the pictur e around an axis by 120° , the picture returns to itself . So there is a symmetry in the ice which accounts for the six-sided appearanc e of snowflakes. Another thin g we can see from Fig. 1-4 is why ice shrinks when it melts. The particula r crystal patter n of ice shown here has many "holes " in it, as does the true ice structure . When the organizatio n break s down, these holes can be occupied by molecules. Most simple substances, with the exception of water and type metal, expand upon melting, because the atoms are closely packed in the solid crystal and upon melting need more room to jiggle around, but an open structure collapses, as in the case of water . Now although ice has a "rigid" crystalline form , its temperature can change— ice has heat. If we wish, we can change the amount of heat. What is the heat in 1-4 the case of ice? The atoms are not standing still. They are jigglin g and vibrating . So even though there is a definite order to the crystal—a definit e structure—all of the atoms are vibratin g "in place." As we increase the temperature, they vibrat e with greater and greater amplitude , unti l they shake themselves out of place. We call this melting. As we decrease the temperature, the vibration decreases and decreases until, at absolute zero, there is a minimu m amount of vibration that the atoms can have, but not zero. This minimum amoun t of motio n that atoms can have is not enough to melt a substance, with one exception: helium. Helium merely decreases the atomic motions as much as it can, but even at absolute zero there is still enough motion to keep it from freezing. Helium, even at absolute zero, does not freeze, unless the pressure is made so great as to make the atoms squash together. If we increase the pressure, we can make it solidify. 1-3 Atomic processe s So much for the description of solids, liquids, and gases from the atomic point of view. However, the atomic hypothesis also describes processes, and so we shall now look at a numbe r of processes from an atomic standpoint . The first process that we shall look at is associated with the surface of the water. What happens at the surface of the water ? We shall now make the picture more com- plicated—and more realistic—by imagining that the surface is in air. Figure 1-5 shows the surface of water in air. We see the water molecules as before, forming a body of liquid water, but now we also see the surface of the water. Above the surface we find a number of things: First of all there are water molecules, as in steam. This is water vapor, which is always foun d above liquid water. (There is an equilibrium between the steam vapor and the water which will be described later.) In addition we find some other molecules—here two oxygen atoms stuck together by themselves, formin g an oxygen molecule, there two nitrogen atoms also stuck together to make a nitrogen molecule. Air consists almost entirely of nitrogen, oxygen, some water vapor , and lesser amounts of carbon dioxide, argon, and other things. So above the water surface is the air, a gas, containin g some wate r vapor. Now wha t is happening in this picture ? The molecules in the water are always jiggling around . From time to time, one on the surface happens to be hit a little harder than usual, and gets knocked away. It is hard to see that happening in the picture because it is a still picture. But we can imagine that one molecule near the surface has just been hit and is flyin g out, or perhaps another one has been hit and is flyin g out. Thus, molecule by molecule, the water disappears— it evaporates. But if we close the vessel above, afte r a while we shall find a large number of molecules of wate r amongst the air molecules. From time to time, one of these vapor molecules comes flyin g down to the wate r and gets stuck again. So we see that what looks like a dead, uninterestin g thing— a glass of wate r with a cover, that has been sitting there for perhap s twenty years—really contains a dynamic and interesting phenomenon which is going on all the time. To our eyes , our crude eyes, nothing is changing, but if we could see it a billion times magni- fied, we would see that from its own point of view it is always changing: molecules are leaving the surface, molecules are coming back. Why do we see no change? Because just as many molecules are leaving as are coming back! In the long run "nothin g happens." If we then take the top of the vesse l off and blow the moist air away, replacing it with dry air, then the number of molecules leaving is just the same as it was before, because this depends on the jiggling of the water, but the numbe r coming back is greatly reduced be- cause there are so many fewer water molecules above the water. Therefore there are more going out than coming in, and the water evaporates. Hence , if you wish to evaporate water turn on the fan! Here is something else: Which molecules leave? When a molecule leaves it is due to an accidental, extra accumulation of a little bit more than ordinar y energy, which it needs if it is to break away from the attraction s of its neighbors. Therefore, since those that leave have more energy than the average, the ones that are left have less average motion than they had before. So the liquid gradually 1-5Figur e 1-5 molecule,0\\§\~ w"% -'@Q§b<>°‘ ‘W ro0.. . I WATER E\/VAPORATING (NAl.§_0A’.O\.I I o @OXYGEN HYDROGEN NITROGEN Figur e 1-6 Neares t neighbo r distanc e d=a/ 2 Figure 1-7cools if it evaporates. Of course, when a molecule of vapor comes from the air to the water below there is a sudden great attractio n as the molecule approaches the surface. This speeds up the incoming molecule and results in generation of heat. So when they leave they take away heat ; when they come back they generate heat. Of course when there is no net evaporatio n the result is nothing—th e water is not changing temperature . If we blow on the water so as to maintai n a continuou s preponderance in the number evaporating, then the water is cooled. Hence, blow on soup to cool it! Of course you should realize that the processes just described are more com- plicated than we have indicated. Not only does the water go into the air, but also, from time to time, one of the oxygen or nitrogen molecules will come in and "get lost" in the mass of water molecules, and work its way into the water. Thus the air dissolves in the water ; oxygen and nitrogen molecules will work their way into the water and the water will contain air. If we suddenly take the air away from the vessel , then the air molecules will leave more rapidly than they come in, and in doing so will mak e bubbles. This is very bad for divers, as you may know . Now we go on to another process. In Fig. 1-6 we see, from an atomic point of view, a solid dissolving in water. If we put a crystal of salt in the water, what will happen ? Salt is a solid, a crystal, an organized arrangemen t of "salt atoms." Figure 1-7 is an illustration of the three-dimensional structur e of common salt, sodium chloride. Strictly speaking, the crystal is not made of atoms, but of wha t we call ions. An ion is an atom which either has a few extra electrons or has lost a few electrons. In a salt crystal we find chlorine ions (chlorine atoms with an extra electron) and sodium ions (sodium atoms with one electron missing). The ions all stick together by electrical attraction in the solid salt, but when we put them in the water we find , because of the attractions of the negative oxygen and positive hydrogen for the ions, that some of the ions jiggle loose. In Fig. 1-6 we see a chlorine ion getting loose, and other atoms floatin g in the water in the form of ions. This picture was made with some care. Notice, for example, that the hydrogen ends of the water molecules are more likely to be near the chlorine ion, while near the sodium ion we are more likely to find the oxygen end, because the sodium is positive and the oxygen end of the water is negative, and they attract electrically. Can we tell from this pictur e whether the salt is dissolving in water or crystallizing out of water ? Of course we cannot tell, because while some of the atoms are leaving the crystal other atoms are rejoining it. The process is a dynamic one, just as in the case of evaporation, and it depends on whether there is more or less salt in the water than the amoun t needed for equilibrium. By equilibrium we mean that situation in which the rate at which atoms are leaving just matches the rate at which they are coming back. If there is almost no salt in the water, more atoms leave than return, and the salt dissolves. If, on the other hand , there are too many "salt atoms," more retur n than leave, and the salt is crystallizing. In passing, we mention that the concept of a molecule of a substance is only approximate and exists only for a certain class of substances. It is clear in the case of water that the three atoms are actually stuck together. It is not so clear in the case of sodium chloride in the solid. There is just an arrangement of sodium and chlorine ions in a cubic pattern . There is no natura l way to group them as "molecules of salt." Returnin g to our discussion of solution and precipitation, if we increase the temperature of the salt solution, then the rate at which atoms are taken away is increased, and so is the rate at which atoms are brought back. It turns out to be very difficult , in general, to predict which way it is going to go, whether more or less of the solid will dissolve. Most substances dissolve more, but some substances dissolve less, as the temperature increases . 1-4 Chemica l reactions In all of the processes which have been described so far, the atoms and the ions have not changed partners, but of course there are circumstances in which the atoms do change combinations, forming new molecules. This is illustrated in 1-63? ,1" UK.O: 0‘wio 6.0'1N>~,‘/\\SALT D|$$DLV|N6 1NWQTER 0 CHLORINE 0 somum [IE] NA Cl 5.64 K Cl6.28 A] Cl 5.54 My O4.20 Pb S 587 Pb Se 6.l4 P5 To 6.34 Fd 2B51hgayzvaygs.°Yi£i§'-Ql.Yin! Fig. 1-8. A process in which the rearrangement of the atomic partners occurs is what we call a chemical reaction. The other processes so far described are called physical processes, but there is no sharp distinction between the two. (Nature does not care what we call it, she just keeps on doing it.) This figure is supposed to represent carbo n burnin g in oxygen. In the case of oxygen, two oxygen atoms stick together very strongly. (Why do not three or even four stick together? That is one of the very peculiar characteristics of such atomic processes. Atoms are very special: they like certain particula r partners, certain particular directions, and so on. It is the job of physics to analyze why each one want s what it wants. At any rate, two oxygen atoms form , saturated and happy, a molecule.) The carbon atoms are supposed to be in a solid crystal (which could be graphite or diamond*). Now, for example, one of the oxygen molecules can come over to the carbon, and each atom can pick up a carbon atom and go flying off in a new combination—"carbon-oxygen"—which is a molecule of the gas calle d carbon monoxide. It is given the chemical name CO. It is very simple: the letters "CO " are practically a pictur e of that molecule. But carbon attracts oxygen much more than oxygen attract s oxygen or carbon attracts carbon. Therefore in this process the oxygen may arrive with only a little energy, but the oxygen and carbon will snap together with a tremendous vengeance and commotion, and everything near them will pick up the energy. A large amount of motion energy, kinetic energy, is thus generated. This of course is burning; we are getting heat from the com- bination of oxygen and carbon. The heat is ordinaril y in the form of the molecular motion of the hot gas, but in certain circumstances it can be so enormous that it generates light. That is how one gets flames. In addition, the carbon monoxide is not quite satisfied. It is possible for it to attach another oxygen, so that we might have a much more complicated reac- tion in which the oxygen is combining with the carbon, while at the same time there happens to be a collision with a carbon monoxide molecule. One oxygen atom could attach itself to the CO and ultimately form a molecule, composed of one carbon and two oxygens, which is designated CO 2 and called carbon dioxide. If we burn the carbon with very little oxygen in a very rapid reaction (for example, in an automobil e engine, where the explosion is so fast that there is not time for it to make carbon dioxide) a considerable amoun t of carbon monoxide is formed. In many such rearrangements, a very large amount of energy is released, forming explosions, flames, etc., depending on the reactions. Chemists have studied these arrangements of the atoms, and foun d that every substance is some type of arrange- ment of atoms. To illustrate this idea, let us consider another example. If we go into a field of small violets, we know what "that smell" is. It is some kind of molecule, or arrangement of atoms, that has worked its way into our noses. First of all, how did it work its way in? That is rathe r easy. If the smell is some kind of molecule in the air, jiggling around and being knocked every which way, it might have accidentally worked its way into the nose. Certainly it has no particula r desire to get into our nose. It is merely one helpless part of a jostling crowd of molecules, and in its aimless wanderings this particular chunk of matter happens to find itself in the nose. Now chemists can take special molecules like the odor of violets, and analyze them and tell us the exact arrangement of the atoms in space. We know that the carbon dioxide molecule is straight and symmetrical: O—C—O. (That can be deter- mined easily, too, by physical methods.) However, even for the vastly more com- plicated arrangements of atoms that there are in chemistry , one can, by a long, remarkable process of detective work , find the arrangements of the atoms. Figure 1-9 is a picture of the air in the neighborhood of a violet; again we find nitrogen and oxygen in the air, and water vapor. (Why is there water vapor ? Because the violet is wet. All plants transpire.) However, we also see a "monster" composed of carbon atoms, hydrogen atoms, and oxygen atoms, which have picked a certain particular pattern in which to be arranged. It is a much more complicated arrange-CARBON BURNIN G IN OXYGEN Figur e 1-8 ODOR OF VIOLET S Figure 1-9 *One can burn a diamond in air. 1-7 oi900 aaigi o9 @@@-l00° I°o Fig. 1-10 . Th e substance pictured is α-irone .ment than that of carbon dioxide; in fact, it is an enormously complicated arrange- ment. Unfortunately , we canno t pictur e all that is really known about it chemically, because the precise arrangement of all the atoms is actually known in three dimensions, while our picture is in only two dimensions. The six carbons which form a ring do not form a flat ring, but a kind of "puckered" ring . All of the angles and distances are known . So a chemical formula is merely a picture of such a molecule. When the chemist writes such a thing on the blackboard, he is trying to "draw," roughly speaking, in two dimensions. For example, we see a "ring" of six carbons, and a "chain" of carbons hanging on the end, with an oxygen second from the end, three hydrogens tied to that carbon, two carbons and three hydrogens sticking up here, etc. How does the chemist find what the arrangement is? He mixes bottles full of stuff together, and if it turn s red, it tells him that it consists of one hydrogen and two carbons tied on here; if it turn s blue, on the other hand , that is not the way it is at all. This is one of the most fantasti c pieces of detective work that has ever been done—organic chemistry. To discover the arrangement of the atoms in these enormously complicated array s the chemist looks at wha t happens when he mixes two differen t substances together. The physicist could never quite believe that the chemist knew what he was talkin g about when he described the arrangement of the atoms. For about twenty years it has been possible, in some cases, to look at such molecules (not quit e as complicated as this one, but some which contain parts of it) by a physical method, and it has been possible to locate every atom, not by looking at colors, but by measuring where they are. And lo and behold!, the chemists are almost always correct. It turns out, in fact, that in the odor of violets there are three slightly differen t molecules, which diffe r only in the arrangement of the hydrogen atoms. One problem of chemistry is to name a substance, so that we will know what it is. Find a name for this shape! Not only must the name tell the shape, but it must also tell that here is an oxygen atom, there a hydrogen—exactly what and where each atom is. So we can appreciate that the chemical names must be com- plex in order to be complete. You see that the name of this thin g in the more com- plete form that will tell you the structure of it is 4-(2, 2, 3, 6 tetramethyl-5- cyclohexanyl)-3-buten-2-one, and that tells you that this is the arrangement. We can appreciate the difficulties that the chemists have, and also appreciate the reason for such long names. It is not that they wish to be obscure, but they have an extremely difficul t problem in tryin g to describe the molecules in words! How do we know that there are atoms? By one of the tricks mentioned earlier: we make the hypothesis that there are atoms, and one afte r the other results come out the way we predict, as they ought to if thing s are made of atoms. There is also somewhat more direct evidence, a good example of which is the following: The atoms are so small that you cannot see them with a light microscope—in fact, not even with an electron microscope. (With a light microscope you can only see things which are much bigger.) Now if the atoms are always in motion, say in water, and we put a big ball of something in the water, a ball much bigger than the atoms, the ball will jiggle around—much as in a push ball game, where a great big ball is pushed around by a lot of people. The people are pushing in various directions, and the ball moves aroun d the field in an irregular fashion . So, in the same way, the "large ball" will move because of the inequalities of the collisions on one side to the other, from one moment to the next. Therefore, if we look at very tiny particles (colloids) in water throug h an excellent microscope, we see a perpetual jiggling of the particles, which is the result of the bombardment of the atoms. This is called the Brownian motion. We can see furthe r evidence for atoms in the structure of crystals. In many cases the structures deduced by x-ray analysis agree in their spatial "shapes" with the forms actually exhibited by crystals as they occur in nature . The angles be- tween the various "faces" of a crystal agree, within seconds of arc, with angles deduced on the assumption that a crystal is made of many "layers" of atoms. Everything is made of atoms. That is the key hypothesis. The most important hypothesis in all of biology, for example, is that everything that animals do, atoms 1-8 3HC_o“P_v CIH__u_H__H16C3/__my/H\C‘C¢3\C/XCIHNWH /H _W7H do. In other words, there is nothing that living things do that cannot be understood from the point of view that they are made of atoms acting according to the laws of physics. This was not known from the beginning: it took some experimenting and theorizing to suggest this hypothesis, but now it is accepted, and it is the most useful theory for producing new ideas in the field of biology. If a piece of steel or a piece of salt, consisting of atoms one next to the other, can have such interesting properties; if water—which is nothing but these little blobs, mile upon mile of the same thing over the earth—can form waves and foam, and make rushing noises and strange patterns as it runs over cement; if all of this, all the life of a stream of water, can be nothing but a pile of atoms, how much more is possible? If instead of arranging the atoms in some definite pattern, again and again repeated, on and on, or even forming little lumps of complexity like the odor of violets, we make an arrangement which is always different from place to place, with different kinds of atoms arranged in many ways, continually changing, not repeating, how much more marvelously is it possible that this thing might behave? Is it possible that that "thing" walking back and forth in front of you, talking to you, is a great glob of these atoms in a very complex arrangement, such that the sheer complexity of it staggers the imagination as to what it can do? When we say we are a pile of atoms, we do not mean we are merely a pile of atoms, because a pile of atoms which is not repeated from one to the other might well have the possibilities which you see before you in the mirror. 1-9 2 Basic Physics 2-1 Introduction In this chapter, we shall examine the most fundamental ideas that we have about physics—the nature of things as we see them at the present time. We shall not discuss the history of how we know that all these ideas are true; you will learn these details in due time. The things with which we concern ourselves in science appear in myriad forms, and with a multitude of attributes. For example, if we stand on the shore and look at the sea, we see the water, the waves breaking, the foam, the sloshing motion of the water, the sound, the air, the winds and the clouds, the sun and the blue sky, and light; there is sand and there are rocks of various hardness and perma- nence, color and texture. There are animals and seaweed, hunger and disease, and the observer on the beach; there may be even happiness and thought. Any other spot in nature has a similar variety of things and influences. It is always as com- plicated as that, no matter where it is. Curiosity demands that we ask questions, that we try to put things together and try to understand this multitude of aspects as perhaps resulting from the action of a relatively small number of elemental things and forces acting in an infinite variety of combinations. For example: Is the sand other than the rocks? That is, is the sand perhaps nothing but a great number of very tiny stones? Is the moon a great rock? If we understood rocks, would we also understand the sand and the moon? Is the wind a sloshing of the air analogous to the sloshing motion of the water in the sea? What common features do different movements have? What is common to dif- ferent kinds of sound? How many different colors are there? And so on. In this way we try gradually to analyze all things, to put together things which at first sight look different, with the hope that we may be able to reduce the number of different things and thereby understand them better. A few hundred years ago, a method was devised to find partial answers to such questions. Observation, reason, and experiment make up what we call the scientific method. We shall have to limit ourselves to a bare description of our basic view of what is sometimes called fundamental physics, or fundamental ideas which have arisen from the application of the scientific method. What do we mean by "understanding" something? We can imagine that this complicated array of moving things which constitutes "the world" is something like a great chess game being played by the gods, and we are observers of the game. We do not know what the rules of the game are; all we are allowed to do is to watch the playing. Of course, if we watch long enough, we may eventually catch on to a few of the rules. The rules of the game are what we mean by fundamental physics. Even if we knew every rule, however, we might not be able to under- stand why a particular move is made in the game, merely because it is too com- plicated and our minds are limited. If you play chess you must know that it is easy to learn all the rules, and yet it is often very hard to select the best move or to understand why a player moves as he does. So it is in nature, only much more so; but we may be able at least to find all the rules. Actually, we do not have all the rules now. (Every once in a while something like castling is going on that we still do not understand.) Aside from not knowing all of the rules, what we really can explain in terms of those rules is very limited, because almost all situations are so enormously complicated that we cannot follow the plays of the game using the rules, much less tell what is going to happen next. We must, therefore, limit our- selves to the more basic question of the rules of the game. If we know the rules, we consider that we "understand" the world. 2-12-1 Introduction 2-2 Physics before 1920 2-3 Quantum physics 2-4 Nuclei and particles How can we tell whether the rules which we "guess" at are really right if we cannot analyze the game very well? There are, roughly speaking, three ways. First, there may be situations where nature has arranged, or we arrange nature, to be simple and to have so few parts that we can predict exactly what will happen, and thus we can check how our rules work. (In one corner of the board there may be only a few chess pieces at work, and that we can figure out exactly.) A second good way to check rules is in terms of less specific rules derived from them. For example, the rule on the move of a bishop on a chessboard is that it moves only on the diagonal. One can deduce, no matter how many moves may be made, that a certain bishop will always be on a red square. So, without being able to follow the details, we can always check our idea about the bishop's motion by finding out whether it is always on a red square. Of course it will be, for a long time, until all of a sudden we find that it is on a black square (what happened of course, is that in the meantime it was captured, another pawn crossed for queening, and it turned into a bishop on a black square). That is the way it is in physics. For a long time we will have a rule that works excellently in an over-all way, even when we cannot follow the details, and then some time we may discover a new rule. From the point of view of basic physics, the most interesting phenomena are of course in the new places, the places where the rules do not work—not the places where they do work! That is the way in which we discover new rules. The third way to tell whether our ideas are right is relatively crude but prob- ably the most powerful of them all. That is, by rough approximation. While we may not be able to tell why Alekhine moves this particular piece, perhaps we can roughly understand that he is gathering his pieces around the king to protect it, more or less, since that is the sensible thing to do in the circumstances. In the same way, we can often understand nature, more or less, without being able to see what every little piece is doing, in terms of our understanding of the game. At first the phenomena of nature were roughly divided into classes, like heat, electricity, mechanics, magnetism, properties of substances, chemical phenomena, light or optics, x-rays, nuclear physics, gravitation, meson phenomena, etc. How- ever, the aim is to see complete nature as different aspects of one set of phenomena. That is the problem in basic theoretical physics, today—to find the laws behind experiment; to amalgamate these classes. Historically, we have always been able to amalgamate them, but as time goes on new things are found. We were amalga- mating very well, when all of a sudden x-rays were found. Then we amalgamated some more, and mesons were found. Therefore, at any stage of the game, it always looks rather messy. A great deal is amalgamated, but there are always many wires or threads hanging out in all directions. That is the situation today, which we shall try to describe. Some historic examples of amalgamation are the following. First, take heat and mechanics. When atoms are in motion, the more motion, the more heat the system contains, and so heat and all temperature effects can be represented by the laws of mechanics. Another tremendous amalgamation was the discovery of the relation between electricity, magnetism, and light, which were found to be dif- ferent aspects of the same thing, which we call today the electromagnetic field. Another amalgamation is the unification of chemical phenomena, the various properties of various substances, and the behavior of atomic particles, which is in the quantum mechanics of chemistry. The question is, of course, is it going to be possible to amalgamate everything, and merely discover that this world represents different aspects of one thing? Nobody knows. All we know is that as we go along, we find that we can amalga- mate pieces, and then we find some pieces that do not fit, and we keep trying to put the jigsaw puzzle together. Whether there are a finite number of pieces, and whether there is even a border to the puzzle, is of course unknown. It will never be known until we finish the picture, if ever. What we wish to do here is to see to what extent this amalgamation process has gone on, and what the situation is at present, in understanding basic phenomena in terms of the smallest set of principles. To express it in a simple manner, what are things made of and how few elements are there ? 2-2 2-2 Physics before 1920 It is a little difficult to begin at once with the present view, so we shall first see how things looked in about 1920 and then take a few things out of that picture. Before 1920, our world picture was something like this: The "stage" on which the universe goes is the three-dimensional space of geometry, as described by Euclid, and things change in a medium called time. The elements on the stage are particles, for example the atoms, which have some properties. First, the property of inertia: if a particle is moving it keeps on going in the same direction unless forces act upon it. The second element, then, is forces, which were then thought to be of two varieties: First, an enormously complicated, detailed kind of inter- action force which held the various atoms in different combinations in a com- plicated way, which determined whether salt would dissolve faster or slower when we raise the temperature. The other force that was known was a long-range interaction—a smooth and quiet attraction—which varied inversely as the square of the distance, and was called gravitation. This law was known and was very simple. Why things remain in motion when they are moving, or why there is a law of gravitation was, of course, not known. A description of nature is what we are concerned with here. From this point of view, then, a gas, and indeed all matter, is a myriad of moving particles. Thus many of the things we saw while standing at the seashore can immediately be connected. First the pressure: this comes from the collisions of the atoms with the walls or whatever; the drift of the atoms, if they are all moving in one direc- tion on the average, is wind; the random internal motions are the heat. There are waves of excess density, where too many particles have collected, and so as they Tush off they push up piles of particles farther out, and so on. This wave of excess density is sound. It is a tremendous achievement to be able to understand so much. Some of these things were described in the previous chapter. What kinds of particles are there? There were considered to be 92 at that time: 92 different kinds of atoms were ultimately discovered. They had different names associated with their chemical properties. The next part of the problem was, what are the short-range forces ? Why does carbon attract one oxygen or perhaps two oxygens, but not three oxygens? What is the machinery of interaction between atoms? Is it gravitation? The answer is no. Gravity is entirely too weak. But imagine a force analogous to gravity, varying inversely with the square of the distance, but enormously more powerful and having one difference. In gravity everything attracts everything else, but now imagine that there are two kinds of "things," and that this new force (which is the electrical force, of course) has the property that likes repel but unlikes attract. The "thing" that carries this strong interaction is called charge. Then what do we have? Suppose that we have two unlikes that attract each other, a plus and a minus, and that they stick very close together. Suppose we have another charge some distance away. Would it feel any attraction? It would feel practically none, because if the first two are equal in size, the attraction for the one and the repulsion for the other balance out. Therefore there is very little force at any appreciable distance. On the other hand, if we get very close with the extra charge, attraction arises, because the repulsion of likes and attraction of unlikes will tend to bring unlikes closer together and push likes farther apart. Then the repulsion will be less than the attraction. This is the reason why the atoms, which are constituted out of plus and minus electric charges, feel very little force when they are separated by appreciable distance (aside from gravity). When they come close together, they can "see inside" each other and rearrange their charges, with the result that they have a very strong interaction. The ultimate basis of an interaction between the atoms is electrical. Since this force is so enormous, all the plusses and all minuses will normally come together in as intimate a combina- tion as they can. All things, even ourselves, are made of fine-grained, enormously strongly interacting plus and minus parts, all neatly balanced out. Once in a while, by accident, we may rub off a few minuses or a few plusses (usually it is easier to rub off minuses), and in those circumstances we find the force of electricity unbalanced, and we can then see the effects of these electrical attractions. 2-3 To give an idea of how much stronger electricity is than gravitation, consider two grains of sand, a millimeter across, thirty meters apart. If the force between them were not balanced, if everything attracted everything else instead of likes repelling, so that there were no cancellation, how much force would there be? There would be a force of three million tons between the two! You see, there is very, very little excess or deficit of the number of negative or positive charges necessary to produce appreciable electrical effects. This is, of course, the reason why you cannot see the difference between an electrically charged or uncharged thing—so few particles are involved that they hardly make a difference in the weight or size of an object. With this picture the atoms were easier to understand. They were thought to have a "nucleus" at the center, which is positively electrically charged and very massive, and the nucleus is surrounded by a certain number of "electrons" which are very light and negatively charged. Now we go a little ahead in our story to remark that in the nucleus itself there were found two kinds of particles, protons and neutrons, almost of the same weight and very heavy. The protons are elec- trically charged and the neutrons are neutral. If we have an atom with six protons inside its nucleus, and this is surrounded by six electrons (the negative particles in the ordinary world of matter are all electrons, and these are very light compared with the protons and neutrons which make nuclei), this would be atom number six in the chemical table, and it is called carbon. Atom number eight is called oxygen, etc., because the chemical properties depend upon the electrons on the outside, and in fact only upon how many electrons there are. So the chemical properties of a substance depend only on a number, the number of electrons. (The whole list of elements of the chemists really could have been called 1, 2, 3, 4, 5, etc. Instead of saying "carbon," we could say "element six," meaning six electrons, but of course, when the elements were first discovered, it was not known that they could be numbered that way, and secondly, it would make everything look rather complicated. It is better to have names and symbols for these things, rather than to call everything by number.) More was discovered about the electrical force. The natural interpretation of electrical interaction is that two objects simply attract each other: plus against minus. However, this was discovered to be an inadequate idea to represent it. A more adequate representation of the situation is to say that the existence of the positive charge, in some sense, distorts, or creates a "condition" in space, so that when we put the negative charge in, it feels a force. This potentiality for produc- ing a force is called an electric field. When we put an electron in an electric field, we say it is "pulled." We then have two rules: (a) charges make a field, and (b) charges in fields have forces on them and move. The reason for this will be- come clear when we discuss the following phenomena: If we were to charge a body, say a comb, electrically, and then place a charged piece of paper at a distance and move the comb back and forth, the paper will respond by always pointing to the comb. If we shake it faster, it will be discovered that the paper is a little behind, there is a delay in the action. (At the first stage, when we move the comb rather slowly, we find a complication which is magnetism. Magnetic influences have to do with charges in relative motion, so magnetic forces and electric forces can really be attributed to one field, as two different aspects of exactly the same thing. A changing electric field cannot exist without magnetism.) If we move the charged paper farther out, the delay is greater. Then an interesting thing is observed. Although the forces between two charged objects should go inversely as the square of the distance, it is found, when we shake a charge, that the influence extends very much farther out than we would guess at first sight. That is, the effect falls off more slowly than the inverse square. Here is an analogy: If we are in a pool of water and there is a floating cork very close by, we can move it "directly" by pushing the water with another cork. If you looked only at the two corks, all you would see would be that one moved immediately in response to the motion of the other—there is some kind of "inter- action" between them. Of course, what we really do is to disturb the water; the water then disturbs the other cork. We could make up a "law" that if you pushed 2-4 the water a little bit, an object close by in the water would move. If it were farther away, of course, the second cork would scarcely move, for we move the water locally. On the other hand, if we jiggle the cork a new phenomenon is involved, in which the motion of the water moves the water there, etc., and waves travel away, so that by jiggling, there is an influence wry much farther out, an oscillatory influence, that cannot be understood from the direct interaction. Therefore the- idea of direct interaction must be replaced with the existence of the water, or-in- the electrical case, with what we call the electromagnetic field. The electromagnetic field can carry waves; some of these waves are light, others are used in radio broadcasts, but the general name is electromagnetic waves. These oscillatory waves can have various frequencies. The only thing that is really different from one wave to another is the frequency of oscillation. If we shake a charge back and forth more and more rapidly, and look at the effects, we get a whole series of different kinds of effects, which are all unified by specifying but one number, the number of oscillations per second. The usual "pickup" that we get from electric currents in the circuits in the walls of a building have a frequency of about one hundred cycles per second. If we increase the frequency to 500 or 1000 kilocycles (1 kilocycle = 1000 cycles) per second, we are "on the air," for this is the frequency range which is used for radio broadcasts. (Of course it has nothing to do with the air! We can have radio broadcasts without any air.) If we again increase the frequency, we come into the range that is used for FM and TV. Going still further, we use certain short waves, for example for radar. Still higher, and we do not need an instrument to "see" the stuff, we can see it with the human eye. In the range of frequency from 5 X 1014 to 5 X 1015 cycles per second our eyes would see the oscillation of the charged comb, if we could shake it that fast, as red, blue, or violet light, depending on the frequency. Frequencies below this range are called infrared, and above it, ultraviolet. The fact that we can see in a particular frequency range makes that part of the electromagnetic spectrum no more impressive than the other parts from a physicist's standpoint, but from a human standpoint, of course, it is more interesting. If we go up even higher in frequency, we get x-rays. X-rays are nothing but very high-frequency light. If we go still higher, we get gamma rays. These two terms, x-rays and gamma rays, are used almost synonymously. Usually electromagnetic rays coming from nuclei are called gamma rays, while those of high energy from atoms are called x-rays, but at the same frequency they are indistinguishable physically, no matter what their source. If we go to still higher frequencies, say to 1024 cycles per second, we find that we can make those waves artificially, for example with the synchrotron here at Caltech. We can find electromagnetic waves with stupendously high frequencies—with even a thousand times more rapid oscillation—in the waves found in cosmic rays. These waves cannot be controlled by us. Table 2-1 The Electromagnetic Spectrum 2-5Frequency in oscillations /sec NameRough behavior 102 5X105-106 108 1010 5X1014 10181021 1024 1027+1015Electrical disturbance Radio broadcast FM—TV Radar Light X-rays ‘Y-rays, nuclear ‘Y-rays, “artificial” ’Y_-rays, incosmic raysField Waves Particle 2-3 Quantum physics Having described the idea of the electromagnetic field, and that this field can carry waves, we soon learn that these waves actually behave in a strange way which seems very unwavelike. At higher frequencies they behave much more like particles! It is quantum mechanics, discovered just after 1920, which explains this strange behavior. In the years before 1920, the picture of space as a three-dimen- sional space, and of time as a separate thing, was changed by Einstein, first into a combination which we call space-time, and then still further into a curved space-time to represent gravitation. So the "stage" is changed into space-time, and gravitation is presumably a modification of space-time. Then it was also found that the rules for the motions of particles were incorrect. The mechanical rules of "inertia" and "forces" are wrong—Newton's laws are wrong—in the world of atoms. Instead, it was discovered that things on a small scale behave nothing like things on a large scale. That is what makes physics difficult—and very interesting. It is hard because the way things behave on a small scale is so "unnatural"; we have no direct experience with it. Here things behave like nothing we know of, so that it is impossible to describe this behavior in any other than analytic ways. It is difficult, and takes a lot of imagination. Quantum mechanics has many aspects. In the first place, the idea that a particle has a definite location and a definite speed is no longer allowed; that is wrong. To give an example of how wrong classical physics is, there is a rule in quantum mechanics that says that one cannot know both where something is and how fast it is moving. The uncertainty of the momentum and the uncertainty of the position are complementary, and the product of the two is constant. We can write the law like this: Dx Dp ³ h/2p, but we shall explain it in more detail later. This rule is the explanation of a very mysterious paradox: if the atoms are made out of plus and minus charges, why don't the minus charges simply sit on top of the plus charges (they attract each other) and get so close as to completely cancel them out? Why are atoms so big? Why is the nucleus at the center with the electrons around it? It was first thought that this was because the nucleus was so big; but no, the nucleus is very small. An atom has a diameter of about 10-8 cm. The nucleus has a diameter of about 10-13 cm. If we had an atom and wished to see the nucleus, we would have to magnify it until the whole atom was the size of a large room, and then the nucleus would be a bare speck which you could just about make out with the eye, but very nearly all the weight of the atom is in that infinitesimal nucleus. What keeps the electrons from simply falling in? This principle: If they were in the nucleus, we would know their position precisely, and the uncertainty principle would then require that they have a very large (but uncertain) momentum, i.e., a very large kinetic energy. With this energy they would break away from the nucleus. They make a compromise: they leave them- selves a little room for this uncertainty and then jiggle with a certain amount of minimum motion in accordance with this rule. (Remember that when a crystal is cooled to absolute zero, we said that the atoms do not stop moving, they still jiggle. Why? If they stopped moving, we would know where they were and that they had zero motion, and that is against the uncertainty principle. We cannot know where they are and how fast they are moving, so they must be continually wiggling in there!) Another most interesting change in the ideas and philosophy of science brought about by quantum mechanics is this: it is not possible to predict exactly what will happen in any circumstance. For example, it is possible to arrange an atom which is ready to emit light, and we can measure when it has emitted light by picking up a photon particle, which we shall describe shortly. We cannot, however, predict when it is going to emit the light or, with several atoms, which one is going to. You may say that this is because there are some internal "wheels" which we have not looked at closely enough. No, there are no internal wheels; nature, as we understand it today, behaves in such a way that it is fundamentally impossible to make a precise prediction of exactly what will happen in a given experiment. This is a horrible thing; in fact, philosophers have said before that one of the fundamental requisites of science is that whenever you set up the same 2-6 conditions, the same thing must happen. This is simply not true, it is not a funda- mental condition of science. The fact is that the same thing does not happen, that we can find only an average, statistically, as to what happens. Nevertheless, science has not completely collapsed. Philosophers, incidentally, say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong. For example, some philosopher or other said it is fundamental to the scientific effort that if an experiment is performed in, say, Stockholm, and then the same experiment is done in, say, Quito, the same results must occur. That is quite false. It is not necessary that science do that; it may be a fact of experience, but it is not necessary. For example, if one of the experiments is to look out at the sky and see the aurora borealis in Stockholm, you do not see it in Quito; that is a different phenomenon. "But," you say, "that is something that has to do with the outside; can you close yourself up in a box in Stockholm and pull down the shade and get any difference?" Surely. If we take a pendulum on a universal joint, and pull it out and let go, then the pendulum will swing almost in a plane, but not quite. Slowly the plane keeps changing in Stockholm, but not in Quito. The blinds are down, too. The fact that this happened does not bring on the destruction of science. What is the fundamental hypothesis of science, the fundamental philosophy? We stated it in the first chapter: the sole test of the validity of any idea is experiment. If it turns out that most experiments work out the same in Quito as they do in Stockholm, then those "most experi- ments" will be used to formulate some general law, and those experiments which do not come out the same we will say were a result of the environment near Stockholm. We will invent some way to summarize the results of the experiment, and we do not have to be told ahead of time what this way will look like. If we are told that the same experiment will always produce the same result, that is all very well, but if when we try it, it does not, then it does not. We just have to take what we see, and then formulate all the rest of our ideas in terms of our actual experience. Returning again to quantum mechanics and fundamental physics, we cannot go into details of the quantum-mechanical principles at this time, of course, be- cause these are rather difficult to understand. We shall assume that they are there, and go on to describe what some of the consequences are. One of the consequences is that things which we used to consider as waves also behave like particles, and particles behave like waves; in fact everything behaves the same way. There is no distinction between a wave and a particle. So quantum mechanics unifies the idea of the field and its waves, and the particles, all into one. Now it is true that when the frequency is low, the field aspect of the phenomenon is more evident, or more useful as an approximate description in terms of everyday experiences. But as the frequency increases, the particle aspects of the phenomenon become more evident with the equipment with which we usually make the measurements. In fact, al- though we mentioned many frequencies, no phenomenon directly involving a fre- quency has yet been detected above approximately 1012 cycles per second. We only deduce the higher frequencies from the energy of the particles, by a rule which assumes that the particle-wave idea of quantum mechanics is valid. Thus we have a new view of electromagnetic interaction. We have a new kind of particle to add to the electron, the proton, and the neutron. That new particle is called a photon. The new view of the interaction of electrons and protons that is electromagnetic theory, but with everything quantum-mechanically correct, is called quantum electrodynamics. This fundamental theory of the interaction of light and matter, or electric field and charges, is our greatest success so far in physics. In this one theory we have the basic rules for all ordinary phenomena except for gravitation and nuclear processes. For example, out of quantum electro- dynamics come all known electrical, mechanical, and chemical laws: the laws for the collision of billiard balls, the motions of wires in magnetic fields, the specific heat of carbon monoxide, the color of neon signs, the density of salt, and the reactions of hydrogen and oxygen to make water are all consequences of this one law. All these details can be worked out if the situation is simple enough for us to make an approximation, which is almost never, but often we can understand more 2-7 second or less what is happening. At the present time no exceptions are found to the quantum-electrodynamic laws outside the nucleus, and there we do not know whether there is an exception because we simply do not know what is going on in the nucleus. In principle, then, quantum electrodynamics is the theory of all chemistry, and of life, if life is ultimately reduced to chemistry and therefore just to physics because chemistry is already reduced (the part of physics which is involved in chemistry being already known). Furthermore, the same quantum electrodynamics, this great thing, predicts a lot of new things. In the first place, it tells the properties of very high-energy photons, gamma rays, etc. It predicted another very re- markable thing: besides the electron, there should be another particle of the same mass, but of opposite charge, called a positron, and these two, coming to- gether, could annihilate each other with the emission of light or gamma rays. (After all, light and gamma rays are all the same, they are just different points on a frequency scale.) The generalization of this, that for each particle there is an antiparticle, turns out to be true. In the case of electrons, the antiparticle has another name—it is called a positron, but for most other particles, it is called anti- so-and-so, like antiproton or antineutron. In quantum electrodynamics, two numbers are put in and most of the other numbers in the world are supposed to come out. The two numbers that are put in are called the mass of the electron and the charge of the electron. Actually, that is not quite true, for we have a whole set of numbers for chemistry which tells how heavy the nuclei are. That leads us to the next part. 2-4 Nuclei and particles What are the nuclei made of, and how are they held together? It is found that the nuclei are held together by enormous forces. When these are released, the energy released is tremendous compared with chemical energy, in the same ratio as the atomic bomb explosion is to a TNT explosion, because, of course, the atomic bomb has to do with changes inside the nucleus, while the explosion of TNT has to do with the changes of the electrons on the outside of the atoms. The question is, what are the forces which hold the protons and neutrons together in the nucleus? Just as the electrical interaction can be connected to a particle, a photon, Yukawa suggested that the forces between neutrons and protons also have a field of some kind, and that when this field jiggles it behaves like a particle. Thus there could be some other particles in the world besides protons and neutrons, and he was able to deduce the properties of these particles from the already known characteristics of nuclear forces. For example, he predicted they should have a mass of two or three hundred times that of an electron; and lo and behold, in cosmic rays there was discovered a particle of the right mass! But it later turned out to be the wrong particle. It was called a m-meson, or muon. However, a little while later, in 1947 or 1948, another particle was found, the p-meson, or pion, which satisfied Yukawa's criterion. Besides the proton and the neutron, then, in order to get nuclear forces we must add the pion. Now, you say, "Oh great!, with this theory we make quantum nucleodynamics using the pions just like Yukawa wanted to do, and see if it works, and everything will be explained." Bad luck. It turns out that the calculations that are involved in this theory are so difficult that no one has ever been able to figure out what the consequences of the theory are, or to check it against experiment, and this has been going on now for almost twenty years! So we are stuck with a theory, and we do not know whether it is right or wrong, but we do know that it is a little wrong, or at least incomplete. While we have been dawdling around theoretically, trying to calculate the consequences of this theory, the experimentalists have been discovering some things. For example, they had already discovered this m-meson or muon, and we do not yet know where it fits. Also, in cosmic rays, a large number of other "extra" particles were found. It turns out that today we have approximately thirty particles, and it is very difficult to understand the relationships of all these particles, and what nature 2-8 , wants them for, or what the connections are from one to another. We do not today understand these various particles as different aspects of the same thing, and the fact that we have so many unconnected particles is a representation of the fact that we have so much unconnected information without a good theory. After the great successes of quantum electrodynamics, there is a certain amount of knowledge of nuclear physics which is rough knowledge, sort of half experience and half theory, assuming a type of force between protons and neutrons and see- ing what will happen, but not really understanding where the force comes from. Aside from that, we have made very little progress. We have collected an enor- mous number of chemical elements. In the chemical case, there suddenly appeared a relationship among these elements which was unexpected, and which is embodied in the periodic table of Mendeleev. For example, sodium and potassium are about the same in their chemical properties and are found in the same column in the Mendeleev chart. We have been seeking a Mendeleev-type chart for the new particles. One such chart of the new particles was made independently by Gell-Mann in the U.S.A. and Nishijima in Japan. The basis of their classification is a new number, like the electric charge, which can be assigned to each particle, called its "strangeness," S. This number is conserved, like the electric charge, in reactions which take place by nuclear forces. In Table 2-2 are listed all the particles. We cannot discuss them much at this stage, but the table will at least show you how much we do not know. Under- neath each particle its mass is given in a certain unit, called the Mev. One Mev is equal to 1.782 X 10~27 gram. The reason this unit was chosen is historical, and we shall not go into it now. More massive particles are put higher up on the chart; we see that a neutron and a proton have almost the same mass. In vertical columns we have put the particles with the same electrical charge, all neutral objects in one column, all positively charged ones to the right of this one, and all negatively charged objects to the left. Particles are shown with a solid line and "resonances" with a dashed one. Several particles have been omitted from the table. These include the important zero-mass, zero-charge particles, the photon and the graviton, which do not fall into the baryon-meson-lepton classification scheme, and also some of the newer resonances (K*, <p, ri). The antiparticles of the mesons are listed in the table, but the antiparticles of the leptons and baryons would have to be listed in another table which would look exactly like this one reflected on the zero-charge column. Although all of the particles except the electron, neutrino, photon, graviton, and proton are unstable, decay products have been shown only for the resonances. Strangeness assignments are not applicable for leptons, since they do not interact strongly with nuclei. All particles which are together with the neutrons and protons are called baryons, and the following ones exist: There is a "lambda," with a mass of 1154 Mev, and three others, called sigmas, minus, neutral, and plus, with several masses almost the same. There are groups or multiplets with almost the same mass, within one or two percent. Each particle in a multiple! has the same strangeness. The first multiple! is the proton-neutron doublet, and then there is a singlet (the lambda) then the sigma triplet, and finally the xi doublet. Very recently, in 1961, even a few more particles were found. Or are they particles? They live so short a time, they disintegrate almost instantaneously, as soon as they are formed, that we do not know whether they should be considered as new particles, or some kind of "resonance" interaction of a certain definite energy between the A and T products into which they disintegrate. In addition to the baryons the other particles which are involved in the nuclear interaction are called mesons. There are first the pions, which come in three varie- ties, positive, negative, and neutral; they form another multiplet. We have also found some new things called A'-mesons, and they occur as a doublet, K+ and K°. Also, every particle has its antiparticle, unless a particle is its own antiparticle. For example, the ir~ and the 7T4' are antiparticles, but the TT" is its own antiparticle. The K~ and ^+ are antiparticles, and the K° and K°. In addition, in 1961 we also found some more mesons or maybe mesons which disintegrate almost immediately. 2-9Table 2-2. Elementary Particles MASS cums: cRouPanGev. -e O +e srnnuc |,3— L2- l.|- L0- .9— .8@- .7- .6- _5- .4*- .3— .2- .|— Q15Wb,_4.. 1n51f;»1r_' I.-Z-A‘l+z'I 15421.’; A’|3lI w% w’.‘ L938 <_-11-zwazr kn! 2';-1&2 pl-1&2 0-0 5<9Q Ts’§':r —3l-5£059--Is-‘I °s=0‘T335 _s-2' s=2 5=l S=l s-ol s-0' s-0 ,..A-- .451> f\ 1fill A thing called w which goes into three pions has a mass 780 on this scale, and somewhat less certain is an object which disintegrates into two pions. These parti- cles, called mesons and baryons, and the antiparticles of the mesons are on the same chart, but the antiparticles of the baryons must be put on another chart, "reflected" through the charge-zero column. Just as Mendeleev's chart was very good, except for the fact that there were a number of rare earth elements which were hanging out loose from it, so we have a number of things hanging out loose from this chart—particles which do not interact strongly in nuclei, have nothing to do with a nuclear interaction, and do not have a strong interaction (I mean the powerful kind of interaction of nuclear energy). These are called leptons, and they are the following: there is the electron, which has a very small mass on this scale, only 0.510 Mev. Then there is that other, the ^-meson, the muon, which has a mass much higher, 206 times as heavy as an electron. So far as we can tell, by all experiments so far, the difference between the electron and the muon is nothing but the mass. Everything works exactly the same for the muon as for the electron, except that one is heavier than the other. Why is there another one heavier; what is the use for it? We do not know. In addition, there is a lepton which is neutral, called a neutrino, and this particle has zero mass. In fact, it is now known that there are two different kinds of neutrinos, one related to electrons and the other related to muons. Finally, we have two other particles which do not interact strongly with the nuclear ones: one is a photon, and perhaps, if the field of gravity also has a quan- tum-mechanical analog (a quantum theory of gravitation has not yet been worked out), then there will be a particle, a graviton, which will have zero mass. What is this "zero mass"? The masses given here are the masses of the particles at rest. The fact that a particle has zero mass means, in a way, that it cannot be at rest. A photon is never at rest, it is always moving at 186,000 miles a second. We will understand more what mass means when we understand the theory of relativity, which will come in due time. Thus we are confronted with a large number of particles, which together seem to be the fundamental constituents of matter. Fortunately, these particles are not all different in their interactions with one another. In fact, there seem to be just four kinds of interaction between particles which, in the order of decreasing strength, are the nuclear force, electrical interactions, the beta-decay interaction, < and gravity. The photon is coupled to all charged particles and the strength of the interaction is measured by some number, which is 1/137. The detailed law of this coupling is known, that is quantum electrodynamics. • Gravity is coupled to all energy, but its coupling is extremely weak, much weaker than that of elec- tricity. This law is also known. Then there are the so-called weak decays— beta decay, which causes the neutron to disintegrate into proton, electron, and neutrino, relatively slowly. This law is only partly known. The so-called strong interaction, the meson-baryon interaction, has a strength of 1 in this scale, and the law is completely unknown, although there are a number of known rules, such as that the number of baryons does not change in any reaction. Table 2-3. Elementary Interactions Coupling Strength* Law Photon to charged particles ~10~2 Law known Gravity to all energy -^lO"40 Law known Weak decays ~10~5 Law partly known Mesons to baryons ~ 1 Law unknown (some rules known) * The "strength" is a dimensionless measure of the coupling constant involved in each interaction (~ means "approximately"). 2-10 /'L'I'l'1esOn 7 ~10—2 ~1()—40 ~l0—5 This then, is the horrible condition of our physics today. To summarize it, I would say this: outside the nucleus, we seem to know all; inside it, quantum mechanics is valid—the principles of quantum mechanics have not been found to fail. The stage on which we put all of our knowledge, we would say, is relativistic space-time; perhaps gravity is involved in space-time. We do not know how the universe got started, and we have never made experiments which check our ideas of space and time accurately, below some tiny distance, so we only know that our ideas work above that distance. We should also add that the rules of the game are the quantum-mechanical principles, and those principles apply, so far as we can tell, to the new particles as well as to the old. The origin of the forces in nuclei leads us to new particles, but unfortunately they appear in great profusion and we lack a complete understanding of their interrelationship, although we already know that there are some very surprising relationships among them. We seem gradually to be groping toward an understanding of the world of sub- atomic particles, but we really do not know how far we have yet to go in this task. 2-11 3 The Relation of Physics to Other Sciences 3-1 Introduction Physics is the most fundamental and all-inclusive of the sciences, and has had a profound effect on all scientific development. In fact, physics is the present- day equivalent of what used to be called natural philosophy, from which most of our modern sciences arose. Students of many fields find themselves studying physics because of the basic role it plays in all phenomena. In this chapter we shall try to explain what the fundamental problems in the other sciences are, but of course it is impossible in so small a space really to deal with the complex, subtle, beautiful matters in these other fields. Lack of space also prevents our discussing the relation of physics to engineering, industry, society, and war, or even the most remarkable relationship between mathematics and physics. (Mathe- matics is not a science from our point of view, in the sense that it is not a natural science. The test of its validity is not experiment.) We must, incidentally, make it clear from the beginning that if a thing is not a science, it is not necessarily bad. For example, love is not a science. So, if something is said not to be a science, it does not mean that there is something wrong with it; it just means that it is not a science. 3-2 Chemistry The science which is perhaps the most deeply affected by physics is chemistry. Historically, the early days of chemistry dealt almost entirely with what we now call inorganic chemistry, the chemistry of substances which are not associated with living things. Considerable analysis was required to discover the existence of the many elements and their relationships—how they make the various relatively simple compounds found in rocks, earth, etc. This early chemistry was very important for physics. The interaction between the two sciences was very great because the theory of atoms was substantiated to a large extent by experiments in chemistry. The theory of chemistry, i.e., of the reactions themselves, was summarized to a large extent in the periodic chart of Mendeleev, which brings out many strange relationships among the various elements, and it was the collection of rules as to which substance is combined with which, and how, that constituted inorganic chemistry. All these rules were ultimately explained in principle by quantum mechanics, so that theoretical chemistry is in fact physics. On the other hand, it must be emphasized that this explanation is in principle. We have already discussed the difference between knowing the rules of the game of chess, and being able to play. So it is that we may know the rules, but we cannot play very well. It turns out to be very difficult to predict precisely what will happen in a given chemical reaction; nevertheless, the deepest part of theoretical chemistry must end up in quantum mechanics. There is also a branch of physics and chemistry which was developed by both sciences together, and which is extremely important. This is the method of statistics applied in a situation in which there are mechanical laws, which is aptly called statistical mechanics. In any chemical situation a large number of atoms are involved, and we have seen that the atoms are all jiggling around in a very random and complicated way. If we could analyze each collision, and be able to follow in detail the motion of each molecule, we might hope to figure out what would happen, but the many numbers needed to keep track of all these molecules ex- ceeds so enormously the capacity of any computer, and certainly the capacity of 3-13-1 Introduction 3-2 Chemistry 3-3 Biology 3-4 Astronomy 3-5 Geology 3-6 Psychology 3-7 How did it get that way? the mind, that it was importan t to develop a method for dealing with such com- plicated situations . Statistical mechanics, then, is the science of the phenomena of heat, or thermodynamics. Inorganic chemistry is, as a science, now reduced essentially to wha t are called physical chemistry and quantu m chemistry; physical chemistry to study the rate s at which reactions occur and wha t is happenin g in detail (How do the molecules hit? Which pieces fly off first? , etc.), and quantum chemistry to help us understan d wha t happens in terms of the physical laws. The other branch of chemistry is organic chemistry, the chemistry of the substances which are associated with living things. For a time it was believed that the substances which are associated with living things were so marvelous that they could not be made by hand , from inorganic materials. This is not at all true—they are just the same as the substances made in inorganic chemistry, but more complicated arrangements of atoms are involved. Organic chemistry obviously has a very close relationship to the biology which supplies its substances, and to industry, and furthermore , much physical chemistry and quantu m mechanics can be applied to organic as well as to inorganic compounds. However, the main problems of organic chemistry are not in these aspects, but rathe r in the analysis and synthesis of the substances which are formed in biological systems, in living things . This leads imperceptibly, in steps, toward biochemistry, and then into biology itself, or molecular biology. 3-3 Biology Thus we come to the science of biology, which is the study of living things. In the early days of biology, the biologists had to deal with the purely descriptive problem of findin g out what living things there were, and so they just had to count such things as the hairs of the limbs of fleas. Afte r these matter s were worked out with a great deal of interest, the biologists wen t into the machinery inside the living bodies, first from a gross standpoint, naturally , because it takes some effor t to get into the finer details. There was an interesting early relationship between physics and biology in which biology helped physics in the discovery of the conservation of energy, which was first demonstrated by Mayer in connection with the amount of heat taken in and given out by a living creature. If we look at the processes of biology of living animals more closely, we see many physical phenomena: the circulation of blood, pumps , pressure, etc. There are nerves: we know what is happening when we step on a sharp stone, and that somehow or other the informatio n goes from the leg up. It is interesting how that happens. In their study of nerves, the biologists have come to the conclusion that nerves are very fine tubes with a complex wall which is very thin; through this wall the cell pumps ions, so that there are positive ions on the outside and nega- tive ions on the inside, like a capacitor. Now this membrane has an interestin g property; if it "discharges" in one place, i.e., if some of the ions were able to move through one place, so that the electric voltage is reduced there, that electrical influence makes itself felt on the ions in the neighborhood, and it affects the membrane in such a way that it lets the ions throug h at neighboring points also. This in turn affects it farthe r along, etc., and so there is a wave of "penetrability" of the membrane which runs down the fiber when it is "excited" at one end by stepping on the sharp stone. This wave is somewhat analogous to a long sequence of vertical dominoes ; if the end one is pushed over, that one pushes the next, etc. Of course this will transmit only one message unless the dominoes are set up again; and similarly in the nerve cell, there are processes which pump the ions slowly out again , to get the nerve ready for the next impulse. So it is that we know what we are doing (or at least where we are). Of course the electrical effects associated with this nerve impulse can be picked up with electrical instruments , and because there are electrical effects, obviously the physics of electrical effects has had a great deal of influence on understanding the phenomenon. The opposite effect is that, from somewhere in the brain, a message is sent out along a nerve. What happens at the end of the nerve? There the nerve branches 3-2 out into fine little things, connected to a structure near a muscle, called an end- plate. For reasons which are not exactly understood, when the impulse reaches the end of the nerve, little packets of a chemical called acetylcholine are shot off (five or ten molecules at a time) and they affect the muscle fiber and make it con- tract—how simple! Wha t makes a muscle contract ? A muscle is a very large num - ber of fibers close together, containing two differen t substances, myosin and actomyosin, but the machinery by which the chemical reaction induced by acetyl- choline can modif y the dimensions of the molecule is not yet known. Thus the fundamental processes in the muscle that make mechanical motions are not known. Biology is such an enormously wide field that there are hosts of other problems that we cannot mention at all—problems on how vision works (what the light does in the eye) , how hearing works, etc. (The way in which thinking works we shall discuss later under psychology.) Now, these things concerning biology which we have just discussed are, from a biological standpoint , really not fundamental, at the bottom of life, in the sens e that even if we understood them we still would not understand life itself. To illustrate : the men who study nerves feel their work is very important, because after all you cannot have animals without nerves. But you can have life without nerves. Plants have neither nerves nor muscles, but they are working, they are alive, just the same. So for the fundamenta l prob- lems of biology we must look deeper; when we do, we discover that all living things have a great many characteristics in common. The most common featur e is that they are made of cells, within each of which is complex machinery for doing things chemically. In plant cells , for example, there is machinery for picking up light and generating sucrose, which is consumed in the dark to keep the plan t alive. When the plant is eaten the sucrose itself generates in the animal a serie s of chemical reactions very closely related to photosynthesis (and its opposite effect in the dark) in plants. In the cells of living systems there are many elaborate chemical reactions, in which one compound is changed into another and another. To give some im- pression of the enormous effort s that have gone into the stud y of biochemistry, the chart in Fig. 3-1 summarizes our knowledge to date on just one small part of the many series of reactions which occur in cells, perhaps a percent or so of it. Here we see a whole serie s of molecules which change from one to another in a sequence or cycle of rathe r small steps. It is called the Krebs cycle, the respira- tory cycle . Each of the chemicals and each of the steps is fairl y simple, in terms of what change is made in the molecule, but—and this is a centrally importan t discovery in biochemistry—these changes are relatively difficult to accomplish in a laboratory. 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" Consider this analogy: If we wanted to take an object from one place to another, at the same level but on the other side of a hill, we could push it over the top, but to do so requires the addition of some energy. Thus most chemical reactions do not occur, because there is what is called an activa- tion energy in the way. In order to add an extra atom to our chemical requires that we get it close enough that some rearrangement can occur; then it will stick . But if we cannot give it enough energy to get it close enough, it will not go to com- pletion, it will just go part way up the "hill" and back down again. However, if we could literally take the molecules in our hands and push and pull the atoms around in such a way as to open a hole to let the new atom in, and then let it snap back, we would have foun d another way, around the hill, which would not require extra energy, and the reaction would go easily. Now there actually are, in the cells , very large molecules, much larger than the ones whose changes we have been de- scribing, which in some complicated way hold the smaller molecules just right, so that the reaction can occur easily. These very large and complicated things are called enzymes. (They were first called ferments, because they were originally discovered in the fermentation of sugar. In fact , some of the first reactions in the cycle were discovered there.) In the presence of an enzyme the reaction will go. An enzyme is made of another substance called protein. Enzymes are very big and complicated, and each one is different , each being built to control a certain special reaction. The names of the enzymes are written in Fig. 3-1 at each reaction. (Sometimes the same enzyme may control two reactions.) We emphasize that the enzymes themselves are not involved in the reaction directly. They do not change; they merely let an atom go from one place to another. Having done so, the enzyme is ready to do it to the next molecule, like a machine in a factory. Of course, there must be a supply of certain atoms and a way of disposing of other atoms. Take hydrogen, for example: there are enzymes which have special units on them which carry the hydrogen for all chemical reactions. For example, there are three or four hydrogen-reducing enzymes which are used all over our cycle in different places . It is interesting that the machinery which liberates som e hydrogen at one plac e will take that hydrogen and use it somewhere else. The most importan t feature of the cycl e of Fig. 3-1 is the transformation from GDP to GTP (guanadine-di-phosphate to guanadine-tri-phosphate) because the one substance has much more energy in it than the other. Just as there is a "box" in certain enzymes for carrying hydrogen atoms around, there are specia l energy-carrying "boxes " which involve the triphosphate group. So, GTP has more energy than GDP and if the cycle is going one way, we are producing molecules which have extra energy and which can go drive some other cycle which requires energy, for example the contraction of muscle. The muscle will not contract unless there is GTP. We can take muscle fiber , put it in water, and add GTP , and the fibers contract, changing GTP to GDP if the right enzymes are present. So the real system is in the GDP-GTP transformation ; in the dark the GTP which has been stored up during the day is used to run the whole cycle around the other way. An enzyme you see, does not care in which direction the reaction goes , for if it did it would violate one of the laws of physics. Physics is of great importance in biology and other science s for still another reason, that has to do with experimental techniques. In fact, if it were not for the great development of experimental physics, these biochemistry charts would not be known today. The reason is that the most useful tool of all for analyzing this fantastically complex system is to label the atoms which are used in the reactions. Thus, if we could introduce into the cycle some carbon dioxide which has a "gree n mark" on it, and then measure afte r three seconds where the green mark is, and again measure afte r ten seconds, etc., we could trace out the course of the reactions. What are the "green marks"? They are different isotopes. We recall that the chemical properties of atoms are determined by the number of electrons, not by the mass of the nucleus. But there can be, for example in carbon, six neutrons or seven neutrons, together with the six protons which all carbon nuclei have. Chemically, the two atoms C12 and C13 are the same, but they differ in weight and they have differen t nuclear properties, and so they are distinguishable. 3-4 electruzs, By using these isotopes of differen t weights, or even radioactive isotopes like C14, which provide a more sensitive means for tracing very small quantities, it is pos- sible to trace the reactions. Now, we retur n to the description of enzymes and proteins. All protein s are not enzymes, but all enzymes are proteins. There are many proteins, such as the proteins in muscle, the structura l proteins which are, for example, in cartilag e and hair, skin, etc., that are not themselves enzymes. However, proteins are a very characteristic substance of life: firs t of all they make up all the enzymes, and second, they make up much of the rest of living material. Proteins have a very interesting and simple structure. They are a series, or chain, of differen t ammo acids. There are twenty differen t amino acids, and they all can combine with each other to form chains in which the backbone is CO-NH, etc. Proteins are nothing but chains of various ones of these twenty amin o acids. Each of the amino acids probably serves some special purpose. Some, for example, have a sulphur atom at a certain place; when two sulphur atoms are in the same protein , they form a bond, that is, they tie the chain together at two points and form a loop. Another has extra oxygen atoms which make it an acidic substance, another has a basic characteristic. Some of them have big groups hanging out to one side, so - that they take up a lot of space. One of the amino acids, called prolene, is not really an amino acid, but imino acid. There is a slight difference, with the result that when prolene is in the chain, there is a kink in the chain. If we wished to manufactur e a particular protein , we would give these instructions: put one of those sulphur hooks here ; next, add something to take up space; then attach some- thing to put a kink in the chain. In this way, we will get a complicated-looking chain, hooked together and having some complex structure ; this is presumably just the manner in which all the various enzymes are made. One of the great tri- umphs in recent times (since 1960) , was at last to discover the exact spatial atomic arrangement of certain proteins, which involve some fifty-six or sixty amin o acids in a row. Over a thousand atoms (more nearly two thousand, if we count the hydrogen atoms) have been located in a complex pattern in two proteins. The first was hemoglobin. One of the sad aspects of this discovery is that we cannot see anything from the pattern ; we do not understand why it works the way it does . Of course, that is the next problem to be attacked. Another problem is how do the enzymes know what to be? A red-eyed fly makes a red-eyed fly baby, and so the information for the whole pattern of enzymes to make red pigment must be passed from one fly to the next. This is done by a substance in the nucleus of the cell, not a protein, called DNA (short for des- oxyribose nucleic acid). This is the key substance which is passed from one cell to another (for instance, sperm cells consist mostly of DNA) and carries the information as to how to make the enzymes. DNA is the "blueprint." What does the blueprint look like and how does it work ? First, the blueprint must be able to reproduce itself. Secondly, it must be able to instruct the protein. Concerning the reproduction, we might think that this proceeds like cell reproduction . Cell s simply grow bigger and then divide in half. Must it be thus with DNA molecules, then, that they too grow bigger and divide in half ? Every atom certainly does not grow bigger and divide in half! No, it is impossible to reproduce a molecule except by some more clever way. The structure of the substance DNA was studied for a long time, first chemi- cally to find the composition, and then with x-rays to find the pattern in space . The result was the following remarkable discovery: The DNA molecule is a pair of chains, twisted upon each other. The backbone of each of these chains , which are analogous to the chains of proteins but chemically quite different , is a serie s of sugar and phosphate groups, as shown in Fig. 3-2. Now we see how the chain can contain instructions, for if we could split this chain down the middle, we would have a series BAADC . . . and every living thing could have a different series . Thus perhaps, in some way, the specific instructions for the manufactur e of pro- teins are contained in the specific series of the DNA. Attached to each sugar along the line, and linking the two chains together, are certain pairs of cross-links. However, they are not all of the same kind ; there are 3-5| | Y ‘I’ msose msoseSUGAR O_8‘A—O' SUGAR <'>\oH RIBOSE RIBOSESUGAR l:>_A‘B'_O| SUGAR HO O OH meose lmeoszSUGAR O_A‘B_O| SUGAR (i)OH_f>1Z_ 'Y 5%:——’“e§"u/{ZO9 0-1:-o—Q—'u—o-0-00/\o/\o/\oRIBOSE _ _ maossSUGAR SUGAR -0;ass:|E>~<='I>—<Dl 252::O O I I Fig. 3-2. Schematic diagram ofDNA four kinds, called adenine, thymine, cytosine, and guanine, but let us call them A, B, C, and D. The interesting thing is that only certain pairs can sit opposite each other, for example A with B and C with D. These pairs are put on the two chains in such a way that they "fit together," and have a strong energy of interac- tion. However, C will not fit with A, and B will not fit with C; they will only fit in pairs, A against B and C against D. Therefore if one is C, the other must be D, etc. Whatever the letters may be in one chain, each one must have its specific complementary letter on the other chain. What then about reproduction? Suppose we split this chain in two. How can we make another one just like it? If, in the substances of the cells, there is a manufacturing department which brings up phosphate, sugar, and A, B, C, D units not connected in a chain, the only ones which will attach to our split chain will be the correct ones, the complements of BAADC . . ., namely, ABBCD ... Thus what happens is that the chain splits down the middle during cell division, one half ultimately to go with one cell, the other half to end up in the other cell; when separated, a new complementary chain is made by each half-chain. Next comes the question, precisely how does the order of the A, B, C, D units determine the arrangement of the amino acids in the protein? This is the central unsolved problem in biology today. The first clues, or pieces of information, however, are these: There are in the cell tiny particles called microsomes, and it is now known that that is the place where proteins are made. But the micro- somes are not in the nucleus, where the DNA and its instructions are. Something seems to be the matter. However, it is also known that little molecule pieces come off the DNA—not as long as the big DNA molecule that carries all the informa- tion itself, but like a small section of it. This is called RNA, but that is not essential. It is a kind of copy of the DNA, a short copy. The RNA, which somehow carries a message as to what kind of protein to make goes over to the microsome; that is known. When it gets there, protein is synthesized at the microsome. That is also known. However, the details of how the amino acids come in and are arranged in accordance with a code that is on the RNA are, as yet, still unknown. We do not know how to read it. If we knew, for example, the "lineup" A, B, C, C, A, we could not tell you what protein is to be made. Certainly no subject or field is making more progress on so many fronts at the present moment, than biology, and if we were to name the most powerful assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made of atoms, and that everything that living things do can be understood in terms of the jigglings and wigglings of atoms. 3-4 Astronomy In this rapid-fire explanation of the whole world, we must now turn to astronomy. Astronomy is older than physics. In fact, it got physics started by showing the beautiful simplicity of the motion of the stars and planets, the under- standing of which was the beginning of physics. But the most remarkable discovery in all of astronomy is that the stars are made of atoms of the same kind as those on the earth* How was this done? Atoms liberate light which has definite fre- * How I'm rushing through this! How much each sentence in this brief story contains. "The stars are made of the same atoms as the earth." I usually pick one small topic like this to give a lecture on. Poets say science takes away from the beauty of the stars—mere globs of gas atoms. Nothing is "mere." I too can see the stars on a desert night, and feel them. But do I see less or more ? The vastness of the heavens stretches my imagina- tion—stuck on this carousel my little eye can catch one-million-year-old light. A vast pattern—of which I am a part—perhaps my stuff was belched from some forgotten star, as one is belching there. Or see them with the greater eye of Palomar, rushing all apart from some common starting point when they were perhaps all together. What is the pattern, or the meaning, or the why ? It does not do harm to the mystery to know a little about it. For far more marvelous is the truth than any artists of the past imagined! Why do the poets of the present not speak of it ? What men are poets who can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent? 3-6 quencies, something like the timbre of a musical instrument, which has definite pitches or frequencies of sound. When we are listening to several different tones we can tell them apart, but when we look with our eyes at a mixture of colors we cannot tell the parts from which it was made, because the eye is nowhere near as discerning as the ear in this connection. However, with a spectroscope we can analyze the frequencies of the light waves and in this way we can see the very tunes of the atoms that are in the different stars. As a matter of fact, two of the chemical elements were discovered on a star before they were discovered on the earth. Helium was discovered on the sun, whence its name, and technetium was dis- covered in certain cool stars. This, of course, permits us to make headway in understanding the stars, because they are made of the same kinds of atoms which are on the earth. Now we know a great deal about the atoms, especially con- cerning their behavior under conditions of high temperature but not very great density, so that we can analyze by statistical mechanics the behavior of the stellar substance. Even though we cannot reproduce the conditions on the earth, using the basic physical laws we often can tell precisely, or very closely, what will happen. So it is that physics aids astronomy. Strange as it may seem, we understand the distribution of matter in the interior of the sun far better than we understand the interior of the earth. What goes on inside a star is better understood than one might guess from the difficulty of having to look at a little dot of light through a telescope, because we can calculate what the atoms in the stars should do in most circum- stances. One of the most impressive discoveries was the origin of the energy of the stars, that makes them continue to burn. One of the men who discovered this was out with his girl friend the night after he realized that nuclear reactions must be going on in the stars in order to make them shine. She said "Look at how pretty the stars shine!" He said "Yes, and right now I am the only man in the world who knows why they shine." She merely laughed at him. She was not impressed with being out with the only man who, at that moment, knew why stars shine. Well, it is sad to be alone, but that is the way it is in this world. It is the nuclear "burning" of hydrogen which supplies the energy of the sun; the hydrogen is converted into helium. Furthermore, ultimately, the manufacture of various chemical elements proceeds in the centers of the stars, from hydrogen. The stuff of which we are made, was "cooked" once, in a star, and spit out. How do we know? Because there is a clue. The proportion of the different isotopes— how much C12, how much C13, etc., is something which is never changed by chemical reactions, because the chemical reactions are so much the same for the two. The proportions are purely the result of nuclear reactions. By looking at the proportions of the isotopes in the cold, dead ember which we are, we can discover what the furnace was like in which the stuff of which we are made was formed. That furnace was like the stars, and so it is very likely that our elements were "made" in the stars and spit out in the explosions which we call novae and super- novae. Astronomy is so close to physics that we shall study many astronomical things as we go along. 3-5 Geology We turn now to what are called earth sciences, or geology. First, meteorology and the weather. Of course the instruments of meteorology are physical instru- ments, and the development of experimental physics made these instruments possible, as was explained before. However, the theory of meteorology has never been satisfactorily worked out by the physicist. "Well," you say, "there is nothing but air, and we know the equations of the motions of air." Yes we do. "So if we know the condition of air today, why can't we figure out the condition of the air tomorrow?" First, we do not really know what the condition is today, because the air is swirling and twisting everywhere. It turns out to be very sensitive, and even unstable. If you have ever seen water run smoothly over a dam, and then turn into a large number of blobs and drops as it falls, you will understand what I mean by unstable. You know the condition of the water before it goes over the 3-7 spillway; it is perfectly smooth ; but the moment it begins to fall, where do the drops begin ? Wha t determines how big the lumps are going to be and where they will be? That is not known , because the water is unstable. Even a smooth moving mass of air, in going over a mountai n turn s into complex whirlpools and eddies. In many fields we find this situatio n of turbulent flow that we cannot analyze today. Quickly we leave the subject of weather, and discuss geology! The question basic to geology is, what makes the earth the way it is? The most obvious processes are in fron t of you r very eyes, the erosion processes of the rivers, the winds, etc. It is easy enough to understand these, but for every bit of erosion there is an equal amoun t of something else going on. Mountain s are no lower today, on the average, than they were in the past. There must be moun- tsim-forming processes. You will find, if you stud y geology, that there are mountain-formin g processes and vulcanism, which nobody understand s but which is half of geology. The phenomenon of volcanoes is really not understood. Wha t makes an earthquak e is, ultimately, not understood. It is understood that if something is pushin g something else , it snaps and will slide—that is all right. But what pushes, and why ? The theory is that there are currents insid e the earth— circulating currents, due to the difference in temperature inside and outside— which, in their motion, push the surfac e slightly. Thus if there are two opposite circulations next to each other, the matter will collect in the region where they meet and make belts of mountains which are in unhappy stressed conditions, and so produce volcanoes and earthquakes. What abou t the inside of the earth ? A great deal is known about the spee d of earthquak e waves throug h the earth and the density of distributio n of the earth . However, physicists have been unable to get a good theory as to how dense a substance should be at the pressures that would be expected at the center of the earth. In other words, we cannot figur e out the properties of matter very well in these circumstances. We do much less well with the earth than we do with the conditions of matter in the stars. The mathematics involved seems a little too difficult , so far, but perhaps it will not be too long before someone realizes that it is an importan t problem, and really work it out. The other aspect, of course, is that even if we did kno w the density, we cannot figur e out the circulatin g currents . Nor can we really work out the properties of rocks at high pressure. We cannot tell how fast the rocks should "give" ; that must all be worked out by experiment. 3-6 Psychology Next, we consider the science of psychology. Incidentally, psychoanalysis is not a science: it is at best a medical process, and perhaps even more like witch- doctoring. It has a theory as to wha t causes disease—lots of differen t "spirits," etc. The witch doctor has a theory that a disease like malaria is caused by a spiri t which comes into the air; it is not cured by shakin g a snake over it, but quinin e does help malaria. So, if you are sick, I would advise that you go to the witch doctor because he is the man in the tribe who knows the most abou t the disease; on the other hand, his knowledge is not science. Psychoanalysis has not been checked carefully by experiment , and there is no way to find a list of the numbe r of cases in which it works , the numbe r of cases in whic h it does not work , etc. The other branches of psychology, which involve thing s like the physiology of sensation—what happens in the eye, and what happens in the brain—are , if you wish, less interesting . But some small but real progress has been made in studying them. One of the most interestin g technical problems may or may not be called psychology. The centra l problem of the mind , if you will, or the nervous system, is this : when an anima l learn s something , it can do somethin g differen t than it could before, and its brai n cell mus t have changed too, if it is mad e out of atoms. In what way is it different ? We do not kno w where to look , or wha t to look for, when something is memorized . We do not know wha t it means, or what change there is in the nervou s system, when a fact is learned . This is a very impor - tant problem which has not been solved at all. Assuming, however, that there is some kind of memory thing, the brain is such an enormous mass of interconnect- 3-8 ing wires and nerves that it probably cannot be analyzed in a straightforward manner. There is an analog of this to computing machines and computing ele- ments, in that they also have a lot of lines, and they have some kind of element, analogous, perhaps, to the synapse, or connection of one nerve to another. This is a very interesting subject which we have not the time to discuss further—the relationship between thinking and computing machines. It must be appreciated, of course, that this subject will tell us very little about the real complexities of ordinary human behavior. All human beings are so different. It will be a long time before we get there. We must start much further back. If we could even figure out how a dog works, we would have gone pretty far. Dogs are easier to under- stand, but nobody yet knows how dogs work. 3-7 How did it get that way? In order for physics to be useful to other sciences in a theoretical way, other than in the invention of instruments, the science in question must supply to the physicist a description of the object in a physicist's language. They can say "why does a frog jump?," and the physicist cannot answer. If they tell him what a frog is, that there are so many molecules, there is a nerve here, etc., that is different. If they will tell us, more or less, what the earth or the stars are like, then we can figure it out. In order for physical theory to be of any use, we must know where the atoms are located. In order to understand the chemistry, we must know exactly what atoms are present, for otherwise we cannot analyze it. That is but one limitation, of course. There is another kind of problem in the sister sciences which does not exist in physics; we might call it, for lack of a better term, the historical question. How did it get that way? If we understand all about biology, we will want to know how all the things which are on the earth got there. There is the theory of evolution, an important part of biology. In geology, we not only want to know how the mountains are forming, but how the entire earth was formed in the be- ginning, the origin of the solar system, etc. That, of course, leads us to want to know what kind of matter there was in the world. How did the stars evolve? What were the initial conditions? That is the problem of astronomical history. A great deal has been found out about the formation of stars, the formation of elements from which we were made, and even a little about the origin of the universe. There is no historical question being studied in physics at the present time. We do not have a question, "Here are the laws of physics, how did they get that way?" We do not imagine, at the moment, that the laws of physics are somehow changing with time, that they were different in the past than they are at present. Of course they may be, and the moment we find they are, the historical question of physics will be wrapped up with the rest of the history of the universe, and then the physicist will be talking about the same problems as astronomers, geologists, and biologists. Finally, there is a physical problem that is common to many fields, that is very old, and that has not been solved. It is not the problem of finding new funda- mental particles, but something left over from a long time ago—over a hundred years. Nobody in physics has really been able to analyze it mathematically satisfactorily in spite of its importance to the sister sciences. It is the analysis of circulating or turbulent fluids. If we watch the evolution of a star, there comes a point where we can deduce that it is going to start convection, and thereafter we can no longer deduce what should happen. A few million years later the star explodes, but we cannot figure out the reason. We cannot analyze the weather. We do not know the patterns of motions that there should be inside the earth. The simplest form of the problem is to take a pipe that is very long and push water through it at high speed. We ask: to push a given amount of water through that pipe, how much pressure is needed? No one can analyze it from first principles and the properties of water. If the water flows very slowly, or if we use a thick goo like honey, then we can do it nicely. You will find that in your textbook. 3-9 What we really cannot do is deal with actual, wet water running through a pipe. That is the central problem which we ought to solve some day, and we have not. A poet once said, "The whole universe is in a glass of wine." We will probably never know in what sense he meant that, for poets do not write to be understood. But it is true that if we look at a glass of wine closely enough we see the entire universe. There are the things of physics: the twisting liquid which evaporates depending on the wind and weather, the reflections in the glass, and our imagi- nation adds the atoms. The glass is a distillation of the earth's rocks, and in its composition we see the secrets of the universe's age, and the evolution of stars. What strange array of chemicals are in the wine? How did they come to be? There are the ferments, the enzymes, the substrates, and the products. There in wine is found the great generalization: all life is fermentation. Nobody can discover the chemistry of wine without discovering, as did Louis Pasteur, the cause of much disease. How vivid is the claret, pressing its existence into the conscious- ness that watches it! If our small minds, for some convenience, divide this glass of wine, this universe, into parts—physics, biology, geology, astronomy, psy- chology, and so on—remember that nature does not know it! So let us put it all back together, not forgetting ultimately what it is for. Let it give us one more final pleasure: drink it and forget it all! 3-10 4 Conservation of Energy 4-1 What is energy? In this chapter, we begin our more detailed study of the differen t aspects of physics, having finished our description of things in general. To illustrate the ideas and the kind of reasoning that might be used in theoretical physics, we shall now examine one of the most basic laws of physics, the conservation of energy. There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in the manifold changes which natur e undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantit y which does not change when something happens. It is not a description of a mechanism, or any- thing concrete; it is just a strange fact that we can calculate some number and when we finish watching natur e go through her tricks and calculate the number again, it is the same. (Something like the bishop on a red square, and after a number of moves—details unknown—it is still on some red square. It is a law of this nature.) Since it is an abstract idea, we shall illustrat e the meaning of it by an analogy. Imagine a child, perhaps "Dennis the Menace," who has blocks which are absolutely indestructible, and cannot be divided into pieces. Each is the same as the other. Let us suppose that he has 28 blocks. His mother puts him with his 28 blocks into a room at the beginning of the day. At the end of the day, being curious, she counts the blocks very carefully, and discovers a phenomenal law— no matter what he does with the blocks, there are always 28 remaining! This continues for a number of days, until one day there are only 27 blocks, but a little investigating shows that there is one under the rug—she must look everywhere to be sure that the number of blocks has not changed. One day, however, the number appears to change—there are only 26 blocks. Careful investigation in- dicates that the window was open, and upon looking outside, the other two blocks are found . Another day, careful count indicates that there are 30 blocks! Thi s causes considerable consternation, until it is realized that Bruce came to visit, bringing his blocks with him, and he left a few at Dennis' house. After she has disposed of the extra blocks, she closes the window, does not let Bruce in, and then everything is going along all right, unti l one time she counts and find s only 25 blocks. However, there is a box in the room, a toy box, and the mother goes to open the toy box, but the boy says "No, do not open my toy box," and screams. Mother is not allowed to open the toy box. Being extremely curious, and somewhat ingenious, she invents a scheme! She knows that a block weighs three ounces, so she weighs the box at a time when she sees 28 blocks, and it weighs 16 ounces. The next time she wishes to check, she weighs the box again, subtracts sixteen ounces and divides by three. She discovers the following:4-1 What is energy ? 4-2 Gravitational potential energ y 4-3 Kinetic energy 4-4 Other forms of energy There then appear to be some new deviations, but careful study indicates that the dirty water in the bathtu b is changing its level. The child is throwing blocks into the water, and she cannot see them because it is so dirty , but she can find out how many blocks are in the water by adding another term to her formula . Since the original height of the water was 6 inches and each block raises the water a quarter 4-1numbgf of +(weight ofbox) —16ounces =constant. 4 blocks seen 3ounces() of an inch , this new formula woul d be: In the gradual increase in the complexity of her world, she find s a whole serie s of terms representing ways of calculating how many blocks are in places where she is not allowed to look. As a result, she find s a complex formula, a quantity which has to be computed, which always stays the same in her situation. What is the analogy of this to the conservation of energy? Th e most re- markable aspect that must be abstracted from this picture is that there are no blocks. Take away the first terms in (4.1) and (4.2) and we find ourselves calculating more or less abstract things. The analogy has the following points. First, when we are calculating the energy, sometimes some of it leave s the system and goes away, or sometimes some comes in. In order to verif y the conservation of energy, we must be careful that we have not put any in or taken any out. Second, the energy has a large number of different forms, and there is a formula for each one. Thes e are: gravitational energy, kinetic energy, heat energy, elastic energy, electrical energy, chemical energy, radiant energy, nuclear energy, mass energy. If we total up the formulas for each of these contributions, it will not change except for energy going in and out. It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a pictur e that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity , and when we add it all together it gives "28"'—always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas. 4-2 Gravitational potential energ y Conservation of energy can be understood only if we have the formula for all of its forms. I wish to discuss the formula for gravitational energy near the surface of the Earth, and I wish to derive this formul a in a way which has nothin g to do with history but is simply a line of reasoning invented for this particular lecture to give you an illustration of the remarkable fact that a great deal about nature can be extracted from a few facts and close reasoning. It is an illustration of the kind of work theoretical physicists become involved in. It is patterned after a most excellent argument by Mr. Carnot on the efficiency of steam engines.* Consider weight-lifting machines—machines which have the property that they lift one weight by lowering another. Let us also make a hypothesis: that there is no such thing as perpetual motion with these weight-liftin g machines. (In fact , that there is no perpetual motion at all is a general statement of the law of conservation of energy.) We must be careful to define perpetual motion . First, let us do it for weight-lifting machines . If, when we have lifted and lowered a lot of weights and restored the machine to the original condition, we find that the net result is to have lifted a weight, then we have a perpetual motion machine because we can use that lifted weight to run something else. That is, provided the machine which lifted the weight is brough t back to its exact original condition, and furthermor e that it is completely self-contained— that it has not received the energy to lift that weight from some external source—like Bruce's blocks. A very simple weight-liftin g machine is shown in Fig. 4-1. This machine lifts weights three units "strong." We place three unit s on one balance pan, and one unit on the other. However, in order to get it actually to work , we must lift a little weight off the left pan. On the other hand, we could lift a one-unit weight * Our point here is not so much the result, (4.3), which in fact you may already know, as the possibility of arriving at it by theoretical reasoning. 4-2 g, Fig.4-1. Simple weight-lifting machine.4number of)+(weight ofbox) —16ounces blocks seen 3QuflQe$ a +(height ofwater) —6inches 1/4inch=constant. (4.2) “28"—-always by lowering the three-unit weight, if we cheat a little by lifting a little weight off the other pan. Of course, we realize that with any actual lifting machine, we must add a little extra to get it to run. This we disregard, temporarily. Ideal machines, although they do not exist, do not require anything extra. A machine that we actually use can be, in a sense, almost reversible: that is, if it will lift the weight of three by lowering a weight of one, then it will also lift nearly the weight of one the same amount by lowering the weight of three. We imagine that there are two classes of machines, those that are not re- versible, which includes all real machines, and those that are reversible, which of course are actually not attainable no matter how careful we may be in our design of bearings, levers, etc. We suppose, however, that there is such a thing—a reversible machine—which lowers one unit of weight (a pound or any other unit) by one unit of distance, and at the same time lifts a three-unit weight. Call this reversible machine, Machine A. Suppose this particular reversible machine lifts the three-unit weight a distance X. Then suppose we have another machine, Ma- chine B, which is not necessarily reversible, which also lowers a unit weight a unit distance, but which lifts three units a distance Y. We can now prove that Y is not higher than X; that is, it is impossible to build a machine that will lift a weight any higher than it will be lifted by a reversible machine. Let us see why. Let us suppose that Y were higher than X. We take a one-unit weight and lower it one unit height with Machine B, and that lifts the three-unit weight up a distance V. Then we could lower the weight from Y to X, obtaining free power, and use the reversible Machine A, running backwards, to lower the three-unit weight a distance X and lift the one-unit weight by one unit height. This will put the one-unit weight back where it was before, and leave both machines ready to be used again! We would therefore have perpetual motion if Y were higher than X, which we assumed was impossible. With those assumptions, we thus deduce that Y is not higher than X, so that of all machines that can be designed, the reversible machine is the best. We can also see that all reversible machines must lift to exactly the same height. Suppose that B were really reversible also. The argument that Y is not higher than X is, of course, just as good as it was before, but we can also make our argument the other way around, using the machines in the opposite order, and prove that X is not higher than Y. This, then, is a very remarkable observation because it permits us to analyze the height to which different machines are going to lift something without looking at the interior mechanism. We know at once that if somebody makes an enormously elaborate series of levers that lift three units a certain distance by lowering one unit by one unit distance, and we compare it with a simple lever which does the same thing and is fundamentally reversible, his machine will lift it no higher, but perhaps less high. If his machine is re- versible, we also know exactly how high it will lift. To summarize: every reversible machine, no matter how it operates, which drops one pound one foot and lifts a three-pound weight always lifts it the same distance, X. This is clearly a universal law of great utility. The next question is, of course, what is XI Suppose we have a reversible machine which is going to lift this distance X, three for one. We set up three balls in a rack which does not move, as shown in Fig. 4-2. One ball is held on a stage at a distance one foot above the ground. The machine can lift three balls, lowering one by a distance 1. Now, we have arranged that the platform which holds three balls has a floor and two shelves, exactly spaced at distance X, and further, that the rack which holds the balls is spaced at distance X, (a). First we roll the balls horizontally from the rack to the shelves, (b), and we suppose that this takes no energy because we do not change the height. The reversible machine then operates: it lowers the single ball to the floor, and it lifts the rack a distance X, (c). Now we have ingeniously arranged the rack so that these balls are again even with the platforms. Thus we unload the balls onto the rack, (d); having unloaded the balls, we can restore the machine to its original condition. Now we have three balls on the upper three shelves and one at the bottom. But the strange thing is that, in a certain way of speaking, we have not lifted two of them at all because, after all, there were balls on shelves 2 and/3 4-3 and,3.§+lO l<—.-1[QB .1_l :X (0)START (blLOAD BALLS __l Q __ Q _ Q (clllb.LIFTS 3lb.A DISTANCE X _<—— Q axid)UNLOAD BALLS QQ X L 1(e)REARRANGE inEND Fig.4-2. Areversible machine.lfi J before. The resulting effect has been to lift one ball a distance 3X. Now, if 3X exceeds one foot , then we can lower the ball to retur n the machine to the initial condition, (f), and we can run the apparatus again. Therefore 3 X cannot exceed one foot, for if 3 X exceeds one foot we can make perpetual motion. Likewise, we can prove that one foot cannot exceed 3X, by making the whole machine run the opposite way, since it is a reversible machine. Therefore 3X is neither greater nor less than a foot, and we discover then, by argument alone, the law that X = ^ foot . The generalization is clear: one poun d falls a certain distance in operating a reversible machine; then the machine can lift p pounds this distance divided by p. Another way of puttin g the result is that three pounds times the height lifted, which in our problem was X, is equal to one pound times the distance lowered, which is one foot in this case . If we take all the weights and multiply them by the heights at which they are now , above the floor, let the machine operate, and then multiply all the weights by all the heights again, there will be no change. (We have to generalize the example where we moved only one weight to the case where when we lower one we lift several differen t ones—but that is easy.) We call the sum of the weights times the heights gravitational potential energy— the energy which an object has because of its relationship in space, rela- tive to the earth . The formul a for gravitational energy, then, so long as we are not too far from the earth (the force weakens as we go higher) is It is a very beautiful line of reasoning. The only problem is that perhaps it is not true. (After all, natur e does not have to go along with our reasoning.) ,For example, perhaps perpetual motion is, in fact , possible. Some of the assumptions may be wrong, or we may have made a mistake in reasoning, so it is always necessary to check. /; turns out experimentally, in fact, to be true. The general name of energy which has to do with location relative to some- thing else is called potential energy. In this particular case, of course, we call it gravitational potential energy. If it is a question of electrical forces against which we are working, instead of gravitational forces, if we are "lifting" charges away from other charges with a lot of levers, then the energy content is called electrical potential energy. The general principle is that the change in the energy is the force times the distance that the force is pushed, and that this is a change in energy in general : We will return to many of these other kinds of energy as we continue the course. The principle of the conservation of energy is very usefu l for deducing what will happen in a numbe r of circumstances. In high school we learned a lot of laws about pulleys and levers used in differen t ways. We can now see that these "laws" are all the same thing, and that we did not have to memorize 75 rules to figure it out. A simple example is a smooth inclined plane which is, happily, a three-four-five triangle (Fig. 4-3) . We han g a one-pound weight on the inclined plan. e with a pulley, and on the other side of the pulley, a weight W. We want to know how heavy W must be to balance the one pound on the plane. How can we figure that out? If we say it is just balanced, it is reversible and so can move up and down, and we can consider the following situation. In the initial circumstance, (a), the one pound weight is at the botto m and weight W is at the top. When W has slipped down in a reversible way, we have a one-pound weight at the top and the weight W the slant distance, (b), or five feet, from the plane in which it was before. We lifted the one-pound weight only three feet and we lowered W pounds by five feet. Therefore W = f of a pound . Note that we deduced this from the conservation of energy, and not from force components. Cleverness, however, is relative. It can be deduced in a way which is even more brilliant, discovere d by 4-4 '5 l3 gravitational potential energy =(weight) X(height). 43 foroneobject change in_ distance force (energy >_(force) X(acts through (44) no (b)IW plane Fig.4-3. Inclined plane. 25 Stevinus and inscribed on his tombstone. Figure 4-4 explains that it has to be ^ of a pound, because the chain does not go around. It is evident that the lower part of the chain is balanced by itself, so that the pull of the five weights on one side must balance the pull of three weights on the other, or whatever the ratio of the legs. You see, by looking at this diagram, that W must be ^ of a pound. (If you get an epitaph like that on your gravestone, you are doing fine.) Let us now illustrate the energy principle with a more complicated problem, the screw jack shown in Fig. 4-5. A handle 20 inches long is used to turn the screw, which has 10 threads to the inch. We would like to know how much force would be needed at the handle to lift one ton (2000 pounds). If we want to lift the ton one inch, say, then we must turn the handle around ten times. When it goes around once it goes approximately 126 inches. The handle must thus travel 1260 inches, and if we used various pulleys, etc., we would be lifting our one ton with an un- known smaller weight W applied to the end of the handle. So we find out that W is about 1.6 pounds. This is a result of the conservation of energy. Take now the somewhat more complicated example shown in Fig. 4-6. A rod or bar, 8 feet long, is supported at one end. In the middle of the bar is a weight of 60 pounds, and at a distance of two feet from the support there is a weight of 100 pounds. How hard do we have to lift the end of the bar in order to keep it balanced, disregarding the weight of the bar? Suppose we put a pulley at one end and hang a weight on the pulley. How big would the weight W have to be in order for it to balance? We imagine that the weight falls any arbitrary dis- tance—to make it easy for ourselves suppose it goes down 4 inches—how high would-the two load weights rise? The center rises 2 inches, and the point a quarter of the way from the fixed end lifts 1 inch. Therefore, the principle that the sum of the heights times the weights does not change tells us that the weight W times 4 inches down, plus 60 pounds times 2 inches up, plus 100 pounds times 1 inch has to add up to nothing: Thus we must have a 55-pound weight to balance the bar. In this way we can work out the laws of "balance"—the statics of complicated bridge arrangements, and so on. This approach is called the principle of virtual work, because in order to apply this argument we had to imagine that the structure moves a little—even though it is not really moving or even movable. We use the very small imagined motion to apply the principle of conservation of energy. 4-3 Kinetic energy To illustrate another type of energy we consider a pendulum (Fig. 4-7). If we pull the mass aside and release it, it swings back and forth. In its motion, it loses height in going from either end to the center. Where does the potential energy go? Gravitational energy disappears when it is down at the bottom; nevertheless, it will climb up again. The gravitational energy must have gone into another form. Evidently it is by virtue of its motion that it is able to climb up again, so we have the conversion of gravitational energy into some other form when it reaches the bottom. We must get a formula for the energy of motion. Now, recalling our arguments about reversible machines, we can easily see that in the motion at the bottom must be a quantity of energy which permits it to rise a certain height, and which has nothing to do with the machinery by which it comes up or the path by which it comes up. So we have an equivalence formula something like the one we wrote for the child's blocks. We have another form to represent the energy. It is easy to say what it is. The kinetic energy at the bottom equals the weight times the height that it could go, corresponding to its velocity: K.E. = WH. What we need is the formula which tells us the height by some rule that has to do with the motion of objects. If we start something out with a certain velocity, say straight up, it will reach a certain height; we do not know what it is yet, but it depends on the velocity—there is a formula for that. Then to find the formula for kinetic energy 4-5 Q5 Q5 Q"'4';;'4'4‘ " '0 IIQU Fig.4-4. Theepitaph ofStevinus. IOTHREADSI INCH __ l_—4W +(2)(60) —l—(1)(100) =0, W='55lb. (4.5) Fig.4-5. Ascrew iack. Gravitationalniii Fig. 4-6. Weighted rod supported ononeend. \ l \ / \\ \ /// Fig.4-7. Pendulum. for an object moving with velocity V, we must calculate the height that it could reach, and multiply by the weight. We shall soon find that we can write it this way: Of course, the fact that motion has energy has nothing to do with the fact that we are in a gravitational field. It makes no difference where the motion came from . This is a general formul a for various velocities. Both (4.3) and (4.6) are approxi- mate formulas, the first because it is incorrect when the heights are great, i.e., when the heights are so high that gravity is weakening ; the second, because of the relativistic correction at high speeds. However, when we do finally get the exact formula for the energy, then the law of conservation of energy is correct. 4-4 Other forms of energy We can continue in this way to illustrate the existence of energy in other forms. First, consider elastic energy. If we pull down on a spring, we must do some work, for when we have it down, we can lift weights with it. Therefore in its stretched condition it has a possibility of doing some work. If we were to evaluate the sums of weights times heights, it would not check out—we must add something else to account for the fact that the spring is unde r tension. Elastic energy is the formula for a spring when it is stretched. How much energy is it? If we let go, the elastic energy, as the sprin g passes throug h the equilibrium point, is converted to kinetic energy and it goes back and fort h between compressing or stretching the spring and kinetic energy of motion. (There is also some gravitational energy going in and out, but we can do this experiment "sideways" if we like.) It keeps going unti l the losses—Aha! We have cheated all the way throug h by puttin g on little weights to move things or saying that the machines are reversible, or that they go on forever, but we can see that things do stop, eventually. Where is the energy when the spring has finished moving up and down? This brings in another form of energy: heat energy. Inside a spring or a lever there are crystals which are made up of lots of atoms, and with great care and delicacy in the arrangement of the part s one can try to adjust things so that as something rolls on something else, none of the atoms do any jiggling at all. But one must be very careful . Ordinarily when things roll, there is bumping and jiggling because of the irregularities of the material, and the atoms start to wiggle inside. So we lose track of that energy; we find the atoms are wiggling inside in a random and confused manner afte r the motion slows down. There is still kinetic energy, all right, but it is not associated with visible motion. What a dream! How do we know there is still kinetic energy? It turns out that with thermometers you can find out that, in fact, the spring or the lever is warmer, and that there is really an increase of kinetic energy by a definite amount. We call this form of energy heat energy, but we know that it is not really a new form , it is just kinetic energy—internal motion. (One of the difficulties with all these experiments with matter that we do on a large scale is that we cannot really demonstrate the conservation of energy and we canno t really make our reversible machines, because every time we move a large clump of stuff , the atoms do not remain absolutely undisturbed , and so a certai n amoun t of rando m motion goes into the atomic system. We cannot see it, but we can measure it with thermom- eters, etc.) There are man y other forms of energy, and of course we cannot describe them in any more detail just now. There is electrical energy, which has to do with push- ing and pulling by electric charges. There is radian t energy, the energy of light, which we know is a form of electrical energy because light can be represented as wigglings in the electromagnetic field . There is chemical energy, the energy which is released in chemical reactions. Actually, elastic energy is, to a certain extent, like chemical energy, because chemical energy is the energy of the attraction of the atoms, one for the other, and so is elastic energy. Our modern understanding is the following : chemical energy has two parts, kinetic energy of the electrons inside the atoms, so part of it is kinetic, and electrical energy of interaction of the 4-6K.E. =WV2/2g. (4.6) electron s and the protons—the rest of it, therefore, is electrical . Next we come to nuclear energy, the energy which is involved with the arrangement of particles inside the nucleus, and we have formulas for that, but we do not have the funda - mental laws. We know that it is not electrical, not gravitational, and not purely chemical, but we do not know what it is. It seems to be an additional form of energy. Finally, associated with the relativity theory, there is a modification of the laws of kinetic energy, or whatever you wish to call it, so that kinetic energy is combined with another thing called mass energy. An object has energy from its sheer existence. If I have a positron and an electron, standing still doing nothing —never mind gravity, never mind anything—and they come together and dis- appear, radiant energy will be liberated, in a definit e amount, and the amount can be calculated. All we need know is the mass of the object. It does not depend on what it is—we make two things disappear, and we get a certain amount of energy. The formula was first found by Einstein; it is E = mc2. It is obvious from our discussion that the law of conservation of energy is enormously useful in making analyses, as we have illustrated in a few examples without knowing all the formulas. If we had all the formulas for all kinds of energy, we could analyze how many processes should work without having to go into the details. Therefore conservation laws are very interesting. The question naturally arises as to what other conservation laws there are in physics. There are two other conservation laws which are analogous to the conservation of energy. One is called the conservation of linear momentum. The other is called the conservation of angular momentum. We will find out more about these later. In the last analysis, we do not-understan d the conservation laws deeply. We do not understand the conservation of energy. We do not understand energy as a certain number of little blobs. You may have heard that photons come out in blobs and that the energy of a photon is Planck's constant times the frequency. That is true, but since the frequency of light can be anything, there is no law that says that energy has to be a certain definite amount. Unlike Dennis' blocks, there can be any amount of energy, at least as presently understood. So we do not under- stand this energy as counting something at the moment, but just as a mathematical quantity, which is an abstract and rather peculiar circumstance. In quantum mechanics it turn s out that the conservation of energy is very closely related to another importan t property of the world, things do not depend on the absolute time. We can set up an experiment at a given moment and try it out, and then do the same experiment at a later moment, and it will behave in exactly the same way. Whether this is strictly true or not, we do not know. If we assume that it is true, and add the principles of quantum mechanics, then we can deduce Jhe principle of the conservation of energy. It is a rather subtle and interesting thing, and it is not easy to explain. The other conservation laws are also linked together. The conservation of momentum is associated in quantu m mechanics with the proposition that it makes no difference where you do the experiment, the results will always be the same. As independence in spac e has to do with the conserva- tion of momentum, independence of time has to do with the conservation of energy, and finally, if we turn our apparatus, this too makes no difference, and so the invariance of the world to angular orientation is related to the conservation of angular momentum. Beside s these, there are three other conservation laws, that are exact so far as we can tell today, which are much simpler to understand because they are in the nature of counting blocks. The first of the three is the conservation of charge, and that merely means that you count how many positive, minus how man y negative electrical charges you have, and the number is never changed. You may get rid of a positive with a negative, but you do not create any net exces s of positives over negatives. Two other laws are analogous to this one—one is called the conservation of baryons. There are a number of strange particles, a neutron and a proton are examples, which are calle d baryons. In any reaction whatever in nature, if we count how many baryons are coming into a process, the number of baryons* which come out * Counting antibaryons as —1 baryon. 4-7 quantum where conservation will be exactly the same. There is another law, the conservation of leptons. We can say that the group of particles calle d leptons are: electron, mu meson, and neutrino. There is an antielectron which is a positron, that is, a —1 lepton. Counting the total number of leptons in a reaction reveals that the number in and out never changes, at least so far as we know at present . These are the six conservation laws, three of them subtle, involving space and time, and three of them simple, in the sense of counting something. With regard to the conservation of energy, we should note that available energy is another matter—there is a lot of jiggling around in the atoms of the water of the sea, because the sea has a certain temperature, but it is impossible to get them herded into a definite motion without taking energy from somewhere else. That is, although we know for a fact that energy is conserved, the energy available for human utility is not conserved so easily. The laws which govern how much energy is available are called the laws of thermodynamics and involve a concept called entropy for irreversible thermodynamic processes. Finally, we remark on the question of where we can get our supplies of energy today. Our supplies of energy are from the sun, rain , coal, uranium , and hydrogen. The sun makes the rain, and the coal also, so that all these are from the sun. Although energy is conserved, natur e does not seem to be interested in it; she liberates a lot of energy from the sun, but only one part in two billion falls on the earth. Nature has conservation of energy, but does not really care; she spends a lot of it in all directions. We have already obtained energy from uranium ; we can also get energy from hydrogen, but at present only in an explosive and dangerous condition. If it can be controlled in thermonuclear reactions, it turn s out that the energy that can be obtained from 10 quart s of water per second is equal to all of the electrical power generated in the United States. With 150 gallons of running water a minute, you have enough fuel to supply all the energy which is used in the United States today! Therefore it is up to the physicist to figure out how to liberate us from the need for having energy . It can be done. 4-8 5 Time and Distance 5-1 Motion In this chapter we shall consider some aspects of the concepts of time and distance. It has been emphasized earlier that physics, as do all the sciences, de- pends on observation. One might also say that the development of the physical sciences to their present form has depended to a large extent on the emphasis which has been placed on the making of quantitative observations. Only with quantitative observations can one arrive at quantitative relationships, which are the heart of physics. Many people would like to place the beginnings of physics with the work done 350 years ago by Galileo, and to call him the first physicist. Until that time, the study of motion had been a philosophical one based on arguments that could be thought up in one's head. Most of the arguments had been presented by Aristotle and other Greek philosophers, and were taken as "proven." Galileo was skeptical, and did an experiment on motion which was essentially this: He allowed a ball to roll down an inclined trough and observed the motion. He did not, however, just look; he measured how far the ball went in how long a time. The way to measure a distance was well known long before Galileo, but there were no accurate ways of measuring time, particularly short times. Although he later devised more satisfactory clocks (though not like the ones we know), Galileo's first experiments on motion were done by using his pulse to count off equal in- tervals of time. Let us do the same. We may count off beats of a pulse as the ball rolls down the track: "one .. . two ... three .. . four .. . five ... six ... seven . . . eight..." We ask a friend to make a small mark at the location of the ball at each count; we can then measure the distance the ball travelled from the point of release in one, or two, or three, etc., equal intervals of time. Galileo expressed the result of his observations in this way: if the location of the ball is marked at 1, 2, 3, 4,... units of time from the instant of Its release, those marks are distant from the starting point in propor- tion to the numbers 1, 4, 9, 16, ... Today we would say the distance is propor- tional to the square of the time:5-1 Motion 5-2 Time 5-3 Short times 5-4 Long times 5-5 Units and standards of time 5-6 Large distances 5-7 Short distances Fig. 5-1. A ball rolls down an dined track. The study of motion, which is basic to all of physics, treats with the questions: where? and when? 5-2 Time Let us consider first what we mean by time. What is time? It would be nice if we could find a good definition of time. Webster defines "a time" as "a period," and the latter as "a time," which doesn't seem to be very useful. Perhaps we should say: "Time is what happens when nothing else happens." Which also doesn't get us very far. Maybe it is just as well if we face the fact that time is one of the things we probably cannot define (in the dictionary sense), and just say that it is what we already know it to be: it is how long we wait! What really matters anyway is not how we define time, but how we measure it. One way of measuring time is to utilize something which happens over and over again in a regular fashion—something which is periodic. For example, a day. A day seems to happen over and over again. But when you begin to think 5-1D<=<t2."START" ll ll ONE’..TwO.. sun’I2 ‘) I Io*56 ,~',THREE7 89I0 about it, you might well ask: "Are days periodic; are they regular? Are all days the same length?" One certainly has the impression that days in summer are longe r than days in winter. Of course, some of the days in winter seem to get awfully long if one is very bored. You have certainly heard someone say, "My , but this has been a long day!" It does seem , however, that days are about the same length on the average. Is there any way we can test whether the days are the same length—either from one day to the next, or at least on the average? One way is to make a comparison with some other periodic phenomenon. Let us see how such a comparison might be made with an hour glass. With an hour glass, we can "create" a periodic occurrence if we have someone standing by it day and night to turn it over when- ever the last grain of sand runs out. We could then coun t the turnings of the glass from each morning to the next. We would find , this time, that the number of "hours" (i.e., turnings of the glass ) was not the same each "day." We should distrust the sun, or the glass, or both. After some thought , it might occur to us to count the "hours" from noon to noon. (Noon is here defined not as 12:0 0 o'clock, but that instant when the sun is at its highest point.) We would find , this time, that the number of "hours" each day is the same. We now have some confidence that both the "hour" and the "day " have a regular periodicity, i.e., mark off successive equal intervals of time, although we have not proved that either one is "really" periodic. Someone might question whether there might not be some omnipotent being who would slow down the flow of sand every night and speed it up during the day. Our experiment does not, of course, give us an answer to this sort of question. All we can say is that we find that a regularity of one kind fits together with a regularity of another kind. We can just say that we base our definition of time on the repetition of some apparently periodic event. 5-3 Shor t times We should now notice that in the process of checking on the reproducibility of the day, we have received an importan t by-product. We have found a way of measuring, more accurately, fractions of a day. We have found a way of counting time in smaller pieces. Can we carry the process further, and learn to measure even smaller intervals of time ? Galileo decided that a given pendulum always swings back and forth in equal intervals of time so long as the size of the swing is kept small. A test comparing the number of swings of a pendulum in one "hour" shows that such is indeed the case. We can in this way mark fractions of an hour . If we use a mechanical device to count the swings—and to keep them going—we have the pendulum clock of our grandfathers. Let us agree that if our pendulum oscillates 3600 times in one hour (and if there are 24 such hours in a day), we shall call each period of the pendulum one "second. " We have then divided our original unit of time into approximately 105 parts. We can apply the same principles to divide the second into smaller and smalle r intervals. It is, you will realize, not practical to make mechanical pen- dulums which go arbitrarily fast, but we can now make electrical pendulums, called oscillators, which can provide a periodic occurrence with a very short period of swing. In these electronic oscillators it is an electrical current which swings to and fro, in a manner analogous to the swinging of the bob of the pendulum. We can make a series of such electronic oscillators, each with a period 10 times shorter than the previous one. We may "calibrate" each oscillator against the next slower one by counting the number of swings it makes for one swing of the slower oscillator. When the period of oscillation of our clock is shorter than a fraction of a second, we cannot count the oscillations withou t the help of some device which extends our powers of observation. One such device is the electron- beam oscilloscope, which acts as a sort of microscope for short times. This device plots on a fluorescent scree n a graph of electrical current (or voltage) versus time. 5-2 OCCllI‘IlHC€ By connecting the oscilloscope to two of our oscillators in sequence , so that it plots a graph first of the current in one of our oscillators and then of the current in the other, we get two graphs like those shown in Fig. 5-2. We can readily determine the number of periods of the faster oscillator in one period of the slower oscillator. With modern electronic techniques, oscillators have been built with periods as short as about 10~ 12 second, and they have been calibrated (by comparison methods such as we have described) in terms of our standard unit of time, the second. With the invention and perfection of the "laser," or light amplifier, in the past few years, it has become possible to make oscillators with even shorter periods than 10~ 12 second, but it has not yet been possible to calibrate them by the methods which have been described, although it will no doub t soon be possible. Times shorter than 10~ 12 second have been measured, but by a differen t tech- nique. In effect, a differen t definition of "time" has been used. One way has been to observe the distance between two happenings on a moving object. If, for example, the headlights of a moving automobile are turned on and then off, we can figure out how long the lights were on if we know where they were turned on and off and how fast the car was moving. The time is the distance over which the lights were on divided by the speed. Within the past few years, just such a technique was used to measure the lifetime of the πð°-meson. By observing in a microscope the minut e tracks left in a photographic emulsion in which π°-mesons had been created one saw that a π°-meson (known to be travelling at a certain speed nearly that of light) went a distance of about 10-7 meter, on the average, before disintegrating. It lived for only about 10~ 16 sec. It should be emphasized that we have here used a some- what differen t definition of "time" than before. So long as there are no inconsist- encie s in our understanding, however, we feel fairl y confident that our definitions are sufficiently equivalent. By extending our techniques—and if necessary our definitions—still furthe r we can infer the time duratio n of still faster physical events. We can speak of the period of a nuclear vibration. We can speak of the lifetime of the newly discovered strange resonances (particles) mentioned in Chapter 2. Their complete life occupies a time span of only 10-24 second, approximately the time it would take light (which moves at the fastes t know n speed) to cross the nucleus of hydrogen (the smallest known object). What about still smaller times? Does "time" exist on a still smaller scale? Does it make any sense to speak of smaller times if we canno t measure—o r perhaps even thin k sensibly about—somethin g which happens in a shorter time ? Perhaps not. These are some of the open questions which you will be askin g and perhaps answering in the next twenty or thirt y years.Fig. 5-2 . Two views of an oscilloscope screen. In (a) the oscilloscope is connected to one oscillator, in (b) it is connected to an oscillator with a period one-tenth as long . 5-4 Long times Let us now consider times longer than one day. Measurement of longer times is easy; we just count the days—so long as there is someone around to do the- counting. First we find that there is another natura l periodicity: the year, about 365 days. We have also discovered that natur e has sometimes provided a counter for the years, in the form of tree rings or river-bottom sediments. In some cases we can use these natura l time marker s to determine the time which has passed since some early event. When we cannot count the years for the measurement of long times, we must look for other ways to measure. One of the most successful is the use of radio- active material as a "clock." In this case we do not have a periodic occurrence, as for the day or the pendulum, but a new kind of "regularity." We find that the radioactivity of a particular sample of material decreases by the same fraction for successive equal increases in its age. If we plot a graph of the radioactivit y observed as a function of time (say in days), we obtain a curve like that shown in Fig. 5-3. We observe that if the radioactivity decreases to one-half in T days (called the "half-life"), then it decreases to one-quarter in another T days, and so 5-3Fig. 5-3. The decreas e with time of radioactivity. The activity decrease s by one-half in each "half-life," T. 10-12 10—12 10-1610-12 (C1) RADIOACTIVITY _1_______I-__ee ‘ii O "'—-Il I21 31 TIME> on. In an arbitrary time interval t there are t/T "half-lives," and the fraction left after this time t is ^') tlT. If we knew that a piece of material, say a piece of wood, had contained an amount A of radioactive material when it was formed, and we foun d out by a direct measurement that it now contains the amount B, we could compute the age of the object, t, by solving the equation There are, fortunately, cases in which we can know the amount of radioactivity that was in an object when it was formed. We know, for example, that the carbon dioxide in the air contains a certain small fraction of the radioactive carbon isotope C14 (replenished continuously by the action of cosmic rays). If we measure the total carbon content of an object, we know that a certain fraction of that amount was originally the radioactive C14; we know , therefore, the starting amount A to use in the formula above. Carbon-14 has a half-life of 5000 years. By carefu l measurements we can measure the amoun t left afte r 20 half-lives or so and can therefore "date" organic objects which grew as long as 100,00 0 years ago. We would like to know , and we think we do know, the life of still older things. Much of our knowledge is based on the measurements of other radioactive iso- topes which have differen t half-lives. If we make measurements with an isotope with a longer half-life, then we are able to measure longer times. Uranium , for example, has an isotope whose half-life is about 109 years, so that if some material was formed with uranium in it 109 years ago, only half the uraniu m would remain today. When the uraniu m disintegrates, it changes into lead. Consider a piece of rock which was formed a long time ago in some chemical process. Lead, being of a chemical natur e differen t from uranium , would appear in one part of the rock and uraniu m would appear in another part of the rock. The uranium and lead 5-4 YEARS SECONDSTIMES MEAN LIFE OF 109 106 103 I1018 1015 1012 109 106 I03 I l0'3 Io~° FIo-9 10-1 I0-1 Io-1 Io—2 lO"2 (%)”T-2 5 8 1 4???????? Age ofuniverse Age ofearth Earliest men Age ofpyramids Age ofU.S. Lifeofaman _ One day Light goes from suntoearth One heart beat Period ofsound wave Period ofradiowave Light travels onefoot Period ofmolecular rotation Period ofatomic vibration Light crosses anatom Period ofnuclear vibration Light crosses anucleus aassaasa (%)”T =3/4-U233 R0226 H3 Neutron Muon ‘Ki-meson 1r°-meson Strange particle woul d be separate. If we look at that piece of rock today , where ther e should only be uraniu m we will how find a certai n fractio n of uraniu m and a certai n fractio n of lead. By comparin g these fractions , we can tell wha t percent of the uraniu m disappeared and change d into lead . By this method , the age of certai n rocks has been determine d to be several billio n years . An extension of this method , not using particula r rock s but lookin g at the uraniu m and lead in the ocean s and usin g averages over the earth , has been used to determin e (withi n the past few years) that the age of the eart h itsel f is approximatel y 5.5 billio n years. It is encouragin g that the age of the eart h is foun d to be the same as the age of the meteorites which land on the earth , as determine d by the uraniu m method. It appears that the eart h was forme d out of rocks floatin g in space, and that the meteorites are, quit e likely , some of that materia l left over. At some time more than five billio n years ago, the universe started . It is now believed that at least our part of the univers e had its beginnin g abou t ten or twelve billio n years ago. We do not kno w wha t happened before then . In fact , we may well ask again : Does the question mak e any sense? Does an earlie r time have any meaning ? 5-5 Units and standards of time We have implie d that it is convenien t if we star t with some standar d unit of time, say a day or a second, and refe r all other time s to some multipl e or fractio n of this unit . What shal l we take as our basic standar d of time ? Shall we take the huma n pulse ? If we compare pulses, we find that they seem to vary a lot. On comparin g two clocks, one find s they do not vary so much . You migh t then say, well, let us take a clock. But whose clock ? There is a story of a Swiss boy who wanted all of the clocks in his town to ring noon at the same time. So he went aroun d tryin g to convince everyone of the value of this. Everyone though t it was a marvelous idea so long as all of the other clocks rang noon when his did! It is rather difficul t to decide whose clock we should take as a standard . Fortunately , we all share one clock—the earth . For a long time the rotationa l perio d of the earth has been take n as the basic standar d of time . As measurement s have been made mor e and more precise, however, it has been foun d that the rotatio n of the earth is not exactl y periodic, when measured in term s of the best clocks. These "best" clocks are those which we have reaso n to believe are accurat e because they agree with each other . We now believe that , for variou s reasons, some days are longer than others, some days are shorter , and on the average the period of the earth becomes a little longer as the centurie s pass. Until very recently we had foun d nothin g muc h better tha n the earth' s period , so all clocks have been related to the length of the day , and the second has been defined as 1/86400 of an average day . Recently we hav e been gainin g experience with some natura l oscillators which we now believe woul d provid e a more constan t time referenc e than the earth , and whic h are also based on a natura l phenomenon available to everyone. These are the so-called "atomic clocks." Their basic interna l period is that of an atomic vibratio n whic h is very insensitiv e to the temperatur e or any othe r external effects . These clocks keep tim e to an accuracy of one part in 10 9 or better. Withi n the past two year s an improve d atomi c clock whic h operate s on the vibratio n of the hydroge n atom has been de- signed and buil t by Professor Norma n Ramsey at Harvar d University . He believes that this clock might be 100 time s mor e accurat e still . Measurement s now in progress will show whether this is true or not. We may expect that since it has been possible to build clocks muc h more accurat e than astronomica l time , ther e will soon be an agreement amon g scientists to defin e the unit of time in term s of one of the atomi c clock standards . 5-6 Large distances Let us now turn to the question of distance. How far, or how big, are things ? Everybody know s that the way you measure distanc e is to star t with a stick and count . Or start with a thum b and count . You begin with a unit and count . How 5-5 Fig. 5-4. Th e height of a Sputnik is determined by triangulation Fig. 5-5. The distance of nearby stars can be measured by triangulation, using the diameter of the earth's orbit as a baseline.does one measure smaller things ? How does one subdivide distance? In the same way that we subdivided time: we take a smaller unit and count the number of such units it takes to make up the longer unit. So we can measure smaller and smalle r lengths. But we do not always mean by distance what one gets by counting off with a meter stick. It would be difficul t to measure the horizontal distance between two mountain tops using only a meter stick. We have foun d by experience that dis- tance can be measured in another fashion : by triangulation. Although this means that we are really using a different definition of distance, when they can both be used they agree with each other. Space is more or less what Euclid thought it was, so the two types of definitions of distance agree. Since they do agree on the earth it gives us some confidence in using triangulatio n for still larger distances. For example, we were able to use triangulation to measure the height of the first Sputnik. We foun d that it was roughly 5 X 105 meters high. By more careful measurements the distance to the moon can be measured in the same way. Two telescope s at different places on the earth can give us the two angles we need. It has been foun d in this way that the moon is 4 X 108 meters away. We cannot do the same with the sun, or at least no one has been able to yet. The accuracy with which one can focus on a given point on the sun and with which one can measure angles is not good enough to permit us to measure the distance to the sun. Then how can we measure the distance to the sun? We must invent an extension of the idea of triangulation. We measure the relative distances of all the planets by astronomical observations of where the planets appear to be, and we get a picture of the solar system with the proper relative distances of every- thing, but with no absolute distance. One absolute measurement is then required, which has been obtained in a number of ways. One of the ways, which was believed until recently to be the most accurate, was to measure the distance from the earth to Eros, one of the small planetoids which passes near the earth every now and then. By triangulation on this little object, one could get the one required scale measure- ment. Knowing the relative distances of the rest, we can then tell the distance, for example, from the earth to the sun, or from the earth to Pluto. Within the past year there has been a big improvement in our knowledge of the scale of the solar system. At the Jet Propulsion Laboratory the distance from the earth to Venus was measured quit e accurately by a direct radar observation. This, of course, is a still differen t type of inferred distance. We say we kno w the speed at which light travels (and therefore, at which radar waves travel), and we assume that it is the same speed everywhere between the earth and Venus. We send the radio wave out, and count the time unti l the reflected wave comes back. From the time we infe r a distance, assuming we know the speed . We have really another definition of a measurement of distance. How do we measure the distance to a star, which is much farthe r away? Fortunately, we can go back to our triangulation method, because the earth moving around the sun gives us a large baseline for measurements of objects outside the solar system. If we focus a telescope on a star in summe r and in winter, we might hope to determine these two angles accurately enough to be able to measure the distance to a star. What if the stars are too far away for us to use triangulation ? Astronomers are always inventing new ways of measuring distance. They find , for example, that they can estimate the size and brightness of a star by its color. The color and brightness of many nearby stars—whose distances are known by triangula - tion—have been measured, and it is foun d that there is a smooth relationship between the color and the intrinsi c brightness of stars (in most cases) . If one now measures the color of a distan t star, one may use the color-brightness relationship to determine the intrinsic brightness of the star. By measuring how bright the star appears to us at the earth (or perhaps we should say how dim it appears), we can compute how far away it is. (For a given intrinsi c brightness, the apparent bright- ness decreases with the square of the distance.) A nice confirmatio n of the correct- ness of this method of measuring stellar distances is given by the results obtained for groups of stars known as globular clusters. A photograph of such a grou p is 5-6 / \‘\\ // \\\/ \\ /‘Is its i”““““"I *3 \\\\\\\\\\/.2/-I /P /I / / / / / / /""'“‘\\SUN {/eel-nH_ ______EARfi\QNTER LQCATDN SUMMER LDCHTDN’ /\ /X *1‘?--.-Z?relative Fig. 5-6. A cluster of stars near the center of our galaxy . Thei r distance from the earth is 30,00 0 light-years, or about 3 X 10 20 meters. shown in Fig. 5-6. Just from looking at the photograph one is convinced that these stars are all together. The same result is obtained from distance measurements by the color-brightness method. A study of many globular clusters gives another importan t bit of informa - tion. It is found that there is a high concentration of such clusters in a certain part of the sky and that most of them are about the same distance from us. Cou- pling this information with other evidence, we conclude that this concentration of clusters marks the center of our galaxy. We then know the distance to the center of the galaxy—about 1020 meters. Knowing the size of our own galaxy, we have a key to the measurement of still larger distances—the distances to other galaxies. Figure 5-7 is a photograph of a galaxy, which has much the same shape as our own. Probably it is the same size, too. (Other evidence supports the idea that galaxies are all about the same size.) If it is the same size as ours, we can tell its distance. We measure the angle it subtends in the sky ; we kno w its diameter, and we compute its distance— triangulatio n again! Fig. 5-7. A spiral galax y like our own. Presuming that its diameter is similar to that of our own galaxy , we may compute its distance from its apparent size. It is 30 millio n light-years (3 X 1023 meters) from the earth. 5-7 30,000 102° galaxy, Fig. 5-8 . Th e most distant object, 3C295 in BOOTES (indicated by the arrow), measured by the 200-inch telescope to date (1960) . Photographs of exceedingly distant galaxies have recently been obtained with the giant Palomar telescope. One is shown in Fig. 5-8. It is now believed that some of these galaxies are abou t halfwa y to the limit of the universe—10 26 meters away—the largest distance we can contemplate! 5-7 Shor t distance s Now let's think about smaller distances. Subdividing the meter is easy . With- out much difficult y we can mar k off one thousand equal spaces which add up to one meter. With somewhat more difficulty , but in a similar way (using a good microscope), we can mark off a thousand equal subdivisions of the millimeter to make a scale of microns (millionths of a meter). It is difficul t to continue to smaller scales , because we cannot "see " objects smaller than the wavelength of visible light (about 5 X 10~ 7 meter). We need not stop, however, at what we can see. With an electron microscope, we can continue the process by making photographs on a still smaller scale , say down to 10~ 8 meter (Fig. 5-9). By indirect measurements—by a kind oftriangula - tion on a microscopic scale— we can continue to measure to smaller and smaller scales . First, from an observation of the way light of short wavelength (x-radiation) is reflected from a pattern of mark s of known separation, we determine the wave- Fig. 5-9. Electron micrograph of some virus molecules. The "large" sphere is for calibration and is known to have a diameter of 2 X 10~ 7 meter (2000 A). 5-8 I 10-810-’ Io—’GITOW l. DISTANCES LIGHT-YEARS METERS length of the light vibrations. Then, from the pattern of the scattering of the same light from a crystal, we can determine the relative location of the atoms in the crystal, obtaining results which agree with the atomic spacings also determined by chemical means. We find in this way that atoms have a diameter of about 10-10 meter. There is a large "gap " in physical sizes between the typical atomic dimension of about lO" 10 meter and the nuclear dimensions 10~ 15 meter, 10~ 5 times smaller. For nuclear sizes , a differen t way of measuring size becomes convenient. We meas- ure the apparent area, σ, calle d the effective cross section. If we wish the radius, we can obtain it from σ == rr 2, since nuclei are nearly spherical. Measurement of a nuclear cross section can be made by passing a beam of high-energy particles throug h a thin slab of material and observing the number of particles which do not get through . These high-energy particles will plow right through the thin cloud of electrons and will be stopped or deflected only if they hit the concentrated weight of a nucleus. Suppose we have a piece of material 1 centimeter thick. There will be about 108 atomic layers. But the nuclei are so small that there is little chance that any nucleus will lie behind another. We might imagine that a highly magnified view of the situation—looking along the particle beam—would look like Fig. 5-10 . The chance that a very small particle will hit a nucleus on the trip through is just the total area covered by the profiles of the nuclei divided by the total area in the picture. Suppose that we know that in an area A of our slab of material there are N atoms (each with one nucleus, of course). Then the total area "covered" by the nuclei is Nσ/A. Now let the number of particles of our beam which arriv e at the slab be n\ and the number which come out the other side be n^. The frac - tion which do not get throug h is (n1 — n^/n^, which should just equal the 5-9Fig. 5-10. Imagine d view throug h a block of carbon 1 cm thick if only the nucle i were observed. I027 1024 1021 I013 1015 I012 10° I011 I03 I lO“3 10-11 l0"9 10-12 Io-15 1rr2,?????3?-3 Edge ofuniverse Tonearest neighbor galaxy Tocenter ofourgalaxy Tonearest star Radius oforbit ofPluto Tothesun Tothemoon Height ofaSputnik Height ofaTVantenna tower Height ofachild Agrain ofsalt Avirus Radius ofanatom Radius ofanucleus 3?????3'2‘ 10*“ 10_’ H1 H2 n2)/"ls0 00'I‘. Q.I 0..0"0coo.on II9I:.'o.0.9 ‘Q0o.".'On'ID .qI'[.o..I we'veQ-.1."0 "0000'09.0‘.I..0...0.0'u.°'.I,I‘ l...OQ..gI.-. .OIOlQ.O.‘..,0‘0'c'oou''.00 .0000‘"0 u.0'.‘I .U...I'0.0'8' .Q0',»o"'000'5'IQ0 0\.¢"v0...!IU'I.I.'0'1.I.Q.0‘u‘I0 fraction of the area covered. We can obtain the radius of the nucleus from the equation* From such an experiment we find that the radii of the nuclei are from about 1 to 6 times 10~ 15 meter. The length unit lO" 15 meter is calle d fhe fermi, in honor of Enrico Fermi (1901-1958) . What do we find if we go to smaller distances? Can we measure smaller distances ? Such questions are not yet answerable. It has been suggested that the still unsolved mystery of nuclear forces may be unravelled only by some modifica- tion of our idea of space, or measurement, at such small distances. It might be thought that it would be a good idea to use some natural length as our unit of length—say the radius of the earth or some fraction of it. The meter was originally intended to be such a unit and was defined to be (π/2) X 10~ 7 times the earth's radius. It is neither convenient nor very accurate to determine the unit of length in this way. For a long time it has been agreed internationally that the meter would be defined as the distance between two scratches on a bar kept in a special laboratory in France. More recently, it has been realized that this definition is neither as precise as would be useful, nor as permanent or universal as one would like. It is currently being considered that a new definition be adopted, an agreed-upon (arbitrary) number of wavelengths of a chosen spectral line. Measurements of distance and of time give results which depend on the ob- server. Two observers moving with respect to each other will not measure the same distances and times when measuring what appear to be the same things. Distances and time intervals have different magnitudes, depending on the coordinate system (or "frame of reference") used for making the measurements. We shall study this subject in more detail in a later chapter. Perfectly precise measurements of distances or times are not permitted by the laws of nature. We have mentioned earlier that the errors in a measurement of the position of an object must be at least as large as where h is a small quantity called "Planck's constant" and Δp is the error in our knowledge of the momentum (mass times velocity) of the object whose posi- tion we are measuring. It was also mentioned that the uncertainty in position measurements is related to the wave nature of particles. The relativity of space and time implies that time measurements have also a minimum error, give n in fact by where AE is the error in our knowledge of the energy of the process whose time period we are measuring. If we wish to know more precisely when something happened we must kno w less about what happened, because our knowledge of the energy involved will be less. The time uncertainty is also related to the wave nature of matter. * This equation is right only if the area covered by the nuclei is a small fraction of the total, i.e., if (MI — ni)ln\ is much less than 1. Otherwise we must make a correction for the fact that some nuclei will be partly obscured by the nuclei in fron t of them. 5-10 10-15 AE (I11 H2)/I111|']'2=o'=£ N H1"1 lO_15 Ax=h/Ap, At=h/AE,-7 6 Probability “The truelogic ofthisworld isinthecalculus ofprobabilities.” -—James Clerk Maxwell 6-1Chance andlikelihood “Chance” isaword which isincommon useineveryday living. The radio reports speaking oftomorrow’s weather may say: “There isasixty percent chance ofrain.” You might say: “There isasmall chance that Ishall livetobeone hundred years old.” Scientists alsousetheword chance. Aseismologist may be interested inthequestion: “What isthechance thatthere willbeanearthquake ofacertain sizeinSouthern California nextyear?” Aphysicist might asktheques- tion: “What isthechance thataparticular geiger counter willregister twenty counts inthenexttenseconds?” Apolitician orstatesman might beinterested inthequestion: “What isthechance thatthere willbeanuclear warwithin the nexttenyears?” You maybeinterested inthechance thatyouwilllearn some- thing from thischapter. Bychance, wemean something likeaguess. Why dowemake guesses? Wemake guesses when wewish tomake ajudgment buthave incomplete infor- mation oruncertain knowledge. Wewant tomake aguess astowhat things are, orwhat things arelikely tohappen. Often wewish tomake aguess because we havetomake adecision. Forexample: Shall Itakemyraincoat withmetomorrow ? Forwhat earth movement should Idesign anewbuilding? Shall Ibuild myself afallout shelter? Shall Ichange mystand ininternational negotiations‘? Shall Igotoclass today? Sometimes wemake guesses because wewish, with ourlimited knowledge, tosayasmuch aswecanabout some situation. Really, anygeneralization isin thenature ofaguess. Anyphysical theory isakindofguesswork. There aregood guesses andthere arebadguesses. Thetheory ofprobability isasystem formaking better guesses. Thelanguage ofprobability allows ustospeak quantitatively about some situation which may behighly variable, butwhich does have some consistent average behavior. Letusconsider theflipping ofacoin. Ifthetoss—and thecoin—are “honest,” wehavenowayofknowing what toexpect fortheoutcome ofanyparticular toss. Yetwewould feelthatinalarge number oftosses there should beabout equal numbers ofheads andtails. Wesay:“The probability thatatosswilllandheads is0.5.” Wespeak ofprobability onlyforobservations thatwecontemplate being made inthefuture. Bythe“probability” ofaparticular outcome ofanobservation we mean ourestimate forthemost likely fraction ofanumber ofrepeated observa- tions thatwillyield thatparticular outcome. Ifweimagine repeating anobserva- tion——such aslooking atafreshly tossed coin—N times, andifwecallNAour estimate ofthemost likely number ofourobservations thatwillgivesome specified result A,saytheresult “heads,” then byP(A), theprobability ofobserving A, wemean P(A)=NA/N. (6.1) Ourdefinition requires several comments. First ofall,wemayspeak ofa probability ofsomething happening onlyiftheoccurrence isapossible outcome ofsome repeatable observation. Itisnotclear thatitwould make anysense to ask:“What istheprobability thatthere isaghost inthathouse?” 6-16-1Chance andlikelihood 6-2Fluctuations 6-3Therandom walk 6-4Aprobability distribution 6-5Theuncertainty principle Youmayobject thatnosituation isexactly repeatable. That isright. Every different observation must atleast beatadifferent time orplace. Allwecansay isthatthe“repeated” observations should, forourintended purposes, appear tobeequivalent. Weshould assume, atleast, thateach observation wasmade from anequivalently prepared situation, andespecially with thesame degree of ignorance atthestart. (Ifwesneak alook atanopponent's hand inacard game, ourestimate ofourchances ofwinning aredifferent than ifwedonot‘) Weshould emphasize thatNandNAinEq.(6.1) arenotintended torepresent numbers based onactual observations. NAisourbest estimate ofwhat would occur inNimagined observations. Probability depends, therefore, onourknowledge andonourability tomake estimates. Ineffect, onourcommon sense! Fortunately, there isacertain amount ofagreement inthecommon sense ofmany things, so thatdifferent people willmake thesame estimate. Probabilities need not,however, be“absolute” numbers. Since theydepend onourignorance, theymay become different ifourknowledge changes. You may have noticed another rather “subjective” aspect ofourdefinition ofprobability. Wehave referred toNAas“our estimate ofthemost likely num- ber...”Wedonotmean thatweexpect toobserve exactly NA,butthatweexpect anumber nearNA,andthatthenumber NAismore likely thananyother number inthevicinity. Ifwetossacoin, say,30times, weshould expect thatthenumber ofheads would notbeverylikely tobeexactly l5.butrather onlysome number near to15,say12,13,l4,15,l6,or17.However, ifwemust choose, wewould decide that 15heads ismore likely than anyother number. Wewould write P(heads) =0.5. Why didwechoose l5asmore likely than anyother number? Wemust have argued withourselves inthefollowing manner: Ifthemost likely number of heads isNHinatotal number oftosses N,thenthemost likely number oftails NTis(N—NH).(Weareassuming thatevery tossgives either heads ortails, andno“other” result!) Butifthecoinis“honest,” there isnopreference forheads ortails. Until wehavesome reason tothink thecoin(ortoss)isdishonest, wemust giveequal likelihoods forheads andtails. Sowemust setNT=NH.Itfollows thatNT=NH=N/2, orP(H) =p(T) =0.5. Wecangeneralize ourreasoning toanysituation inwhich there aremdif- ferent but“equivalent” (that is,equally likely) possible results ofanobservation. Ifanobservation canyield mdifferent results, andwehave reason tobelieve that anyoneofthem isaslikely asanyother, then theprobability ofaparticular outcome AisP(A) =1/m. Ifthere areseven different-colored balls inanopaque boxandwepickone out“atrandom” (that is,without looking), theprobability ofgetting aballofa particular color is%.Theprobability thata“blind draw” from ashufiled deck of52cards willshow thetenofhearts isgli.Theprobability ofthrowing adouble- onewithdiceis3%. InChapter 5wedescribed thesizeofanucleus interms ofitsapparent area, or “cross section.” When wedidsowewere really talking about probabilities. When we shoot ahigh-energy particle atathinslabofmaterial, there issome chance thatitwill passright through andsome chance thatitWlllhitanucleus. (Since thenucleus isso small thatwecannot seeit,wecannot aimright atanucleus. Wemust “shoot blind.") Ifthere arenatoms inourslabandthenucleus ofeach atom hasacross-sectional area Ir,then thetotal area“shadowed” bythenuclei isncr.Inalarge number Nofrandom shots, weexpect thatthenumber ofhitsNT;ofsome nucleus willbeintheratio toNas theshadowed areaistothetotal areaoftheslab: NC/N =no/A. (6.2) Wemay say,therefore, thattheprobability thatanyoneprojectile particle willsuffer acollision inpassing through theslabis PC=50', (6.3) where n/Aisthenumber ofatoms perunitareainourslab. 6-2 6-2Fluctuations Wewould likenowtouseourideas about probability toconsider insome greater detail thequestion: “How many heads doIreally expect togetifItoss acoin Ntimes?” Before answering thequestion, however, letuslook atwhat does happen insuch an“experiment.” Figure 6-1shows theresults obtained in thefirstthree “runs” ofsuch anexperiment inwhich N=30.Thesequences of “heads” and“tails” areshown justasthey were obtained. The firstgame gave llheads; thesecond also ll;thethird 16.Inthree trials wedidnotonce get15 heads. Should webegin tosuspect thecoin? Orwere wewrong inthinking that themost likely number of“heads” insuch agame is15?Ninety-seven more runs were made toobtain atotal of100experiments of30tosses each. The results oftheexperiment aregiven inTable 6-l.* Table 6-1H x xx T H x x xx T HXX IX! I XX X Xxxxxxxxxxx xxxx xx xxxii! xxxxxx xx xxxx xxxxx xxxxxx xxxx xxxxxx xx ‘I’ Fig.6—l. Observed sequences f heads and tails inthree games of30x xxxxx xx xxx tosses each. Number ofheads insuccessive trials of30tosses ofacoin. ll ll 16 I6 16 I4 I6 19 l5 14 I2 I8 IS l4 l7 17 I2 l3 l4 I7 9l6 l7 12 l2 10 l4 lll7 17 l5 ll l5 13 16l5 12 10 22 13 16 14l7 20 18 12 14 l5 l7l6 23 l7 20 l6 I9 I419 ll 13 12 15 21 ll18 16 15 l5 16 l4 l6 21 I3l5 l7 l4 16 l3 l2 l7 ll I613 14 15 12 18 l5 16 16 I3 NUIBE9 (F OAHES IN WHICH THE SCORE IIIAS OBTAINED 5 /1It / / 1,/ 0 5 I0I //1I \ 1 \1t\ 1, \ I I \ ' \90trials OBSERVED INTHIS EXPERIMENT ’ \\ II5 li=uuuszn orn.4-PROBABLE MUNICH I \ ’ \I \1 \ \\\\\\ \\\ D 25 EAD5xxxx Fig. 6-2. Summary oftheresults ofTOO games of30tosses each. Thevertical bars show thenumber ofgames inwhich ascore ofkheads was obtained. Thedashed curve shows theexpected numbers ofgames with thescore kobtained byaproba- bility COmpUtatiOn. Looking atthenumbers inTable 6-l,weseethatmost oftheresults are “near” 15,inthatthey arebetween 12and18.Wecangetabetter feeling forthe details ofthese results ifweplotagraph ofthedistribution oftheresults. We count thenumber ofgames inwhich ascore ofkwas obtained, andplot this number foreach k.Such agraph isshown inFig.6-2. Ascore ofl5heads was obtained inl3games. Ascore of14heads wasalso obtained 13times. Scores of 16and17were each obtained more than 13times. Arewetoconclude thatthere issome biastoward heads? Wasour“best estimate" notgood enough? Should *After thefirstthree games, theexperiment wasactually done byshaking 30pennies violently inaboxandthencounting thenumber ofheads thatshowed. 6-3 vuvs nvs vms soon: moo n I an |/a 2 szns/s //,.... mm "'rose sscouowas mumTOQ Fig.6-3. Adiagram forshowing the number ofways oscore ofO,l,2,or3 heads canbeobtained inagame of3 tosses. SCORE I G <§.;.;§;§;5I 0 Fig.6-4, Adiagram likethatofFig. 6-3, foragame of6tosses.weconclude nowthatthe“most likely” score forarunof30tosses isreally 16 heads? Butwait! Inallthegames taken together, there were 3000tosses. Andthe total number ofheads obtained was1492. Thefraction oftosses thatgave heads is0.497, verynearly, butslightly lessthan half. Weshould certainly notassume thattheprobability ofthrowing heads isgreater than0.5! Thefactthatonepar- ticular setofobservations gave 16heads most often, isafluctuation. Westillexpect thatthemost likely number ofheads is15. Wemayaskthequestion: “What istheprobability thatagame of30tosses willyield I5heads——or 16,oranyother number?” Wehave saidthatinagame ofonetoss, theprobability ofobtaining onehead is0.5,andtheprobability of obtaining nohead is0.5.Inagame oftwotosses there arefourpossible outcomes: HH, HT,TH,TT. Since each ofthese sequences isequally likely, weconclude that(a)theprobability ofascore oftwoheads isi,(b)theprobability ofascore ofonehead is%,(c)theprobability ofazeroscore isi.There aretwoways of obtaining onehead, butonlyoneofobtaining either zeroortwoheads. Consider nowagame of3tosses. Thethird tossisequally likely tobeheads ortails. There isonlyonewaytoobtain 3heads: wemust have obtained 2heads onthefirsttwotosses, andthenheads onthelast. There are,however, three ways ofobtaining 2heads. Wecould throw tailsafter having thrown twoheads (one way) orwecould throw heads after throwing onlyonehead inthefirsttwotosses (two ways). Soforscores of3-H, 2-H, 1-H, 0-Hwehave thatthenumber ofequally likely ways is1,3,3,1,withatotal of8different possible sequences. Theprob- abilities are%,%,%,%. Theargument wehave been making canbesummarized byadiagram likethat inFig.6-3. Itisclear how thediagram should becontinued forgames with alarger number oftosses. Figure 6-4shows such adiagram foragame of6tosses. The number of“ways” toanypoint onthediagram isjust thenumber ofdifferent “paths” (sequences ofheads andtails) which canbetaken from thestarting point. Thevertical position gives usthetotal number ofheads thrown. Thesetofnum- berswhich appears insuchadiagram isknown asPascal’s triangle. Thenumbers arealsoknown asthebinomial coefficients, because theyalsoappear intheex- pansion of(a+b)".Ifwecallnthenumber oftosses andkthenumber ofheads thrown, then thenumbers inthediagram areusually designated bythesymbol (Z).Wemayremark inpassing thatthebinomial coefficients canalsobecom- puted from = <6”) where n!,called “n-factorial,” represents theproduct (n)(n —l)(n—2)...(3) (2)(1)- Wearenow ready tocompute theprobability P(k, n)ofthrowing kheads in ntosses, using ourdefinition Eq.(6.1). Thetotal number ofpossible sequences is2"(since there are2outcomes foreach toss), andthenumber ofways ofobtain- ingkheads is(Z),allequally likely, sowehave P(k, n)=gg- (6.5) Since P(k,n)isthefraction ofgames which weexpect toyield kheads, then inI00games weshould expect tofindkheads 100-P(k,n)times. Thedashed curve inFig.6-2passes through thepoints computed from 100-P(k,30). We seethatweexpect toobtain ascore of15heads in14or15games, whereas this score wasobserved in13games. Weexpect ascore of16in13orl4games, but weobtained thatscore inl6games. Such fluctuations are“part ofthegame.” Themethod wehave justused canbeapplied tothemost general situation inwhich there areonly twopossible outcomes ofasingle observation. Letus designate thetwooutcomes byW(for“win”) andL(for“lose”). Inthegeneral case, theprobability ofWorLinasingle event need notbeequal. Letpbethe probability ofobtaining theresult W.Then q,theprobability ofL,isnecessarily 6-4 (I—p).Inasetofntrials, theprobability P(k,n)thatWwillbeobtained k times is P(k,") =('i)P'°¢1""‘- (6.6) Thisprobability function iscalled theBernoulli or,also, thebinomial probability. 6-3Therandom walk There isanother interesting problem inwhich theideaofprobability isre- quired. Itistheproblem ofthe“random walk.” Initssimplest version, weimagine a“game” inwhich a“player” starts atthepoint x=0andateach “move” isrequired totakeastepeither forward (toward +x)orbackward (toward —x). Thechoice istobemade randomly, determined, forexample, bythetossofacoin. How shall wedescribe theresulting motion? Initsgeneral form theproblem is related tothemotion ofatoms (orother particles) inagas—called Brownian motion-—and alsotothecombination oferrors inmeasurements. You willsee thattherandom-walk problem isclosely related tothecoin-tossing problem we havealready discussed. First, letuslook atafewexamples ofarandom walk. Wemaycharacterize thewalker’s progress bythenetdistance DNtraveled inNsteps. Weshow inthe graph ofFig.6-5three examples ofthepath ofarandom walker. (Wehave used fortherandom sequence ofchoices theresults ofthecointosses shown inFig. 6-1.) I 5_ D(N) (otsnwcz mom __.'°'._ _ _."'. ."~.__.-"-._' ..START) 0/\\\ J’I..u' '.“u' .5, J '0.-' V \\ /\‘\ 1/ \\ \ 1 \ _5_ \\ , ,"\ I/' \g\/ ' » \ _|o_ _ ta O\/\\/ \ V N(STEPS TAKEN) Fig. 6-5. The progress made inarandom walk. The horizontal coordinate N isthetotal number ofsteps taken; thevertical coordinate D(N) isthenetdistance moved from thestarting position. What canwesayabout such amotion? Wemight firstask:“How fardoes hegetontheaverage?” Wemust expect thathisaverage progress willbezero, since heisequally likely togoeither forward orbackward. Butwehave thefeel- ingthatasNincreases, heismore likely tohave strayed farther from thestarting point. Wemight, therefore, askwhat ishisaverage distance travelled inabsolute value, thatis,what istheaverage ofID]. Itis,however, more convenient todeal withanother measure of“progress,” thesquare ofthedistance: D2ispositive foreither positive ornegative motion, andistherefore areasonable measure of suchrandom wandering. Wecanshow thattheexpected value ofD;§isjustN,thenumber ofsteps taken. By“expected value” wemean theprobable value (ourbestguess), which wecanthink ofastheexpected average behavior inmany repeated sequences. Werepresent suchanexpected value by(Dfi), andmayrefertoitalsoasthe“mean square distance.” After onestep, D2isalways +1,sowehavecertainly (Di) =1. (Alldistances willbemeasured interms ofaunitofonestep. Weshall notcontinue towrite theunits ofdistance.) 6-5I . I |_no ao so Theexpected value ofD13forN>1canbeobtained from DN_1. If,after (N—l)steps, wehave DN_1, thenafter Nsteps wehave DN=DN_1 +Ior DN=DN_1 —1.Forthesquares, Dita+20~_1+1. oi,= or (6.7) D};_1 —2D1v-1 +l. Inanumber ofindependent sequences, weexpect toobtain each value one-half ofthetime, soouraverage expectation isjusttheaverage ofthetwopossible values. Theexpected value ofDNisthen Drir-1 -1-l.Ingeneral, weshould expect forD§_1 its“expected value” (D)(2;_1) (bydefinitionl). So (Dir) =(D§r_1> +1- (6-3) Wehave already shown that(Di) =l;itfollows thenthat D16=N, (6.9) aparticularly simple result! Ifwewishanumber likeadistance, rather thanadistance squared, torepre- sentthe“progress made away from theorigin” inarandom walk, wecanusethe “root-mean-square distance” Dms: D...=~/<71?=vs. (6.10) Wehave pointed outthattherandom walk isclosely similar initsmathe- matics tothecoin-tossing game weconsidered atthebeginning ofthechapter. Ifweimagine thedirection ofeachsteptobeincorrespondence withtheappearance ofheads ortailsinacointoss,thenDisjustNH—NT,thedifference inthenum- berofheads andtails. Since NH+NT=N,thetotalnumber ofsteps (andtosses), wehave D=2NH—N.Wehave derived earlier anexpression fortheexpected distribution ofNH(also called k)andobtained theresult ofEq.(6.5). Since Nisjustaconstant, wehave thecorresponding distribution forD.(Since for every head more thanN/2there isatail“missing,” wehave thefactor of2between NHandD.)Thegraphs ofFig.6—2represent thedistribution ofdistances wemight getin30random steps (where k=15istobereadD=0;k=16,D=2;etc.). Thevariation ofNHfrom itsexpected value N/2is N DNH—3— §- (6.11) Thermsdeviation is (NH— =%\/W. (6.12) According toourresult forD,m,, weexpect thatthe“typical” distance in 30steps ought tobe\/50 =5.5,oratypical kshould beabout 5.5/2 =2.8 units from 15.Weseethatthe“width” ofthecurve inFig.6—2,measured from thecenter, isjustabout 3units, inagreement withthisresult. Wearenowinaposition toconsider aquestion wehave avoided until now. How shall wetellwhether acoin is“honest” or“loaded”? Wecangivenow atleast apartial answer. Foranhonest coin, weexpect thefraction ofthetimes heads appears tobe0.5,thatis, 1%?=0.5. (6.13) Wealsoexpect anactual NHtodeviate from N/2byabout \/N/2, orthefraction todeviate by N_LT2\/Tv Thelarger Nis,thecloser weexpect thefraction NH/N tobetoone-half. 6-62»- ~s IO Fig. 6-6. Thefraction ofthetosses Fnsgljgu thatgave heads inaparticular sequence 0, /x1', ofNtosses ofapenny. O I I I I I I I I I I I I Z 4 8 I6 32 64 I28 256 512 I026 20484006 InFig.6-6wehave plotted thefraction NH/N forthecointosses reported earlier inthischapter. Weseethetendency forthefraction ofheads toapproach 0.5forlarge N.Unfortunately, foranygiven runorcombination ofrunsthere is noguarantee thattheobserved deviation willbeeven neartheexpected deviation. There isalways thefinite chance thatalarge fluctuation-—-a long string ofheads ortails—-will giveanarbitrarily large deviation. Allwecansayisthatifthe deviation isnear theexpected 1/2\/N (saywithin afactor of2or3),wehave no reason tosuspect thehonesty ofthecoin. Ifitismuch larger, wemaybesuspi- cious, butcannot prove, thatthecoin isloaded (orthatthetosser iscleverl). Wehave alsonotconsidered howweshould treat thecaseofa“coin” or some similar “chancy” object (sayastone thatalways lands ineither oftwoposi- tions) thatwehave good reason tobelieve should have adifferent probability for heads andtails. Wehave defined P(H) =(NH)/N. How shall weknow what to expect forNH?Insome cases, thebestwecandoistoobserve thenumber of heads obtained inlarge numbers oftosses. Forwant ofanything better, wemust set(N1;)=NH(observed). (How could weexpect anything else?) Wemust under- stand, however, thatinsuchacaseadifferent experiment, oradifferent observer, might conclude thatP(H) wasdifferent. Wewould expect, however, thatthevarious answers should agree within thedeviation 1/2\/N [ifP(H) isnear one-half]. An experimental physicist usually saysthatan“experimentally determined” probability hasan“error,” andwrites P(H)=-1%;b (6.14) There isanimplication insuch anexpression thatthere isa“true” or“correct” probability which could becomputed ifweknew enough, andthattheobservation maybein“error” duetoafluctuation. There is,however, nowaytomake such thinking logically consistent. Itisprobably better torealize thattheprobability concept isinasense subjective, thatitisalways based onuncertain knowledge, andthat itsquantitative evaluation issubject tochange asweobtain more information. 6-4Aprobability distribution Letusreturn now totherandom walk andconsider amodification ofit. Suppose thatinaddition toarandom choice ofthedirection (+or—)ofeach step,thelength ofeachstepalsovaried insome unpredictable way, theonlycondi- tionbeing thatontheaverage thesteplength wasoneunit. This caseismore representative ofsomething likethethermal motion ofamolecule inagas. If wecallthelength ofastepS,then Smayhave anyvalue atall,butmost often willbe“near” 1.Tobespecific, weshall let(S2) =1or,equivalently, Sm, =1. Ourderivation for(D2) would proceed asbefore except thatEq.(6.8) would be changed nowtoread <vt>=<vt_.>+<S’>=<v%&_.>+1. (6-15) Wehave, asbefore, that (0,6)=N. (6.16) 6-7 not) _"\\\\\\\\\\\\\\\\v;-.\\\\\\\\\\\\\\_v._]_\\\\\\\\\\\\\_\‘.\\\\\\\\\\\\_\.\\\\\\\\\\v.:->- 4 I2 X... Fig. 6-8. The probability that the distance Dtraveled in0random Walk is between x1and x;isthearea under the curve ofp(x) from x1toX2.i PIX) 'PROBABILIYY DENSITY N=|o,ooo srsos - 40,000 stews _ 1so,ooo stews A 1 _.1 /1 _J 1 | 1 1\\» 1... .t.,L“TOO -500 “5® _40O "300 ‘ZOO -I00 O IOO 200 300 400 500 500 100 D=DISTANCE FROM START Fig.6-7. Theprobability density forending upatthedistance Dfrom thestarting place inarandom walk ofNsteps. (Dismeasured inunits ofthermsstep length.) What would weexpect now forthedistribution ofdistances D?What is, forexample, theprobability thatD=0after 30steps? Theanswer iszero! Theprobability iszerothatDwillbeanyparticular value, since there isnochance atallthatthesumofthebackward steps (ofvarying lengths) would exactly equal thesumofforward steps. Wecannot plotagraph likethatofFig.6—2. Wecan,however, obtain arepresentation similar tothatofFig.6—2,ifwe ask,notWhat istheprobability ofobtaining Dexactly equal to0,l,or2,but instead what istheprobability ofobtaining Dnear0,l,or2.Letusdefine P(x,Ax) astheprobability thatDwilllieintheinterval Axlocated atx(sayfrom xto x+Ax). Weexpect thatforsmall Axthechance ofDlanding intheinterval isproportional toAx,thewidth oftheinterval. Sowecanwrite P(x,Ax)=p(x)Ax. (6.11) Thefunction p(x)iscalled theprobability density. Theform ofp(x) willdepend onN,thenumber ofsteps taken, andalsoon thedistribution ofindividual steplengths. Wecannot demonstrate theproofs here, butforlarge N,p(x)isthesame forallreasonable distributions inindividual steplengths, anddepends onlyonN.Weplotp(x)forthree values ofNinFig. 6-7. Youwillnotice thatthe“half-widths” (typical spread from x=0)ofthese curves is\/N, aswehave shown itshould be. Youmaynotice alsothatthevalue ofp(x)nearzeroisinversely proportional to\/N. Thiscomes about because thecurves areallofasimilar shape andtheir areas under thecurves must allbeequal. Since p(x)Axistheprobability offind- ingDinAxwhen Axissmall, wecandetermine thechance offinding Dsomewhere inside anarbitrary interval from x1tox2,bycutting theinterval inanumber of small increments Axandevaluating thesumoftheterms p(x)Axforeach incre- ment. Theprobability thatDlands somewhere between x,andx2,which wemay write P(x1 <D<x2),isequal totheshaded areainFig.6-8. Thesmaller we taketheincrements Ax,themore correct isourresult. Wecanwrite, therefore, P(x1<D<x2)=Zp(x)Ax=L"p(x)dx. (6.18) Theareaunder thewhole curve istheprobability thatDlands somewhere (that is,hassome value between x=—-so andx=+60). That probability is surely 1.Wemust have that /_*:°p(x)dx =1. (6.19) 6-8 Since thecurves inFig.6-7getwider inproportion to\/TV, their heights must be proportional to1/\/TV tomaintain thetotal areaequal to1. Theprobability density function wehave been describing isonethatisen- countered most commonly. Itisknown asthenormal orgaussian probability density. Ithasthemathematical form p(x)=-Ite"'2/2°i (6.20)ax/21r where 0'iscalled thestandard deviation andisgiven, inourcase, byer=\/1_V or,ifthermsstepsizeisdifferent from l,by0'=\/1—VS,,,,,,. Weremarked earlier thatthemotion ofamolecule, orofanyparticle, inagas islikearandom walk. Suppose weopen abottle ofanorganic compound andlet some ofitsvapor escape intotheair.Ifthere areaircurrents, sothattheairis circulating, thecurrents willalsocarry thevapor withthem. Buteveninperfectly stillair,thevapor willgradually spread out—will diffuse—until ithaspenetrated throughout theroom. Wemight detect itbyitscolor orodor. Theindividual molecules oftheorganic vapor spread outinstillairbecause ofthemolecular motions caused bycollisions withother molecules. Ifweknow theaverage “step” size,andthenumber ofsteps taken persecond, wecanfindtheprobability that one,orseveral, molecules willbefound atsome distance from theirstarting point afteranyparticular passage oftime. Astimepasses, more steps aretaken andthe gasspreads outasinthesuccessive curves ofFig.6-7. Inalaterchapter, weshall findouthowthestepsizesandstepfrequencies arerelated tothetemperature and pressure ofagas. Earlier, wesaidthatthepressure ofagasisduetothemolecules bouncing against thewalls ofthecontainer. When wecome later tomake amore quantita- tivedescription, wewillwish toknow how fastthemolecules aregoing when they bounce, since theimpact they make willdepend onthatspeed. Wecannot, how- ever, speak ofthespeed ofthemolecules. Itisnecessary touseaprobability description. Amolecule mayhave anyspeed, butsome speeds aremore likely thanothers. Wedescribe what isgoing onbysaying thattheprobability thatany particular molecule willhave aspeed between vandv+Avisp(v)Av,where p(v),aprobability density, isagiven function ofthespeed v.Weshall seelater howMaxwell, using common sense andtheideas ofprobability, wasable to findamathematical expression forp(v). Theform* ofthefunction p(v)isshown inFig.6—9. Velocities mayhave anyvalue, butaremost likely tobenear the most probable orexpected value (v). Weoften think ofthecurve ofFig.6-9inasomewhat different way. Ifwe consider themolecules inatypical container (with avolume of,say,oneliter), thenthere areaverylarge number Nofmolecules present (N~10”‘). Since p(v)Avistheprobability thatonemolecule willhave itsvelocity inAv,byour definition ofprobability wemean thattheexpected number (AN) tobefound with avelocity intheinterval Avisgiven by (AN)=Np(v)Av. (6.21) WecallNp(v)the“distribution invelocity.” Theareaunder thecurve between twovelocities v1andv2,forexample theshaded area inFig.6-9, represents [forthecurve Np(v)] theexpected number ofmolecules with velocities between v1andv2.Since withagasweareusually dealing withlarge numbers ofmolecules, weexpect thedeviations from theexpected numbers tobesmall (like l/\/IT’), so weoften neglect tosaythe“expected” number, andsayinstead: “The number of molecules with velocities between v1andv2istheareaunder thecurve.” We should remember, however, that such statements arealways about probable numbers. *Maxwell’s expression isp(v)=Cv2e"°”2, where aisaconstant related tothetem- perature andCischosen sothatthetotal probability isone. 6-99(1) or ND(V) Fig.6—9. Thedistribution ofvelocities ofthemolecules inagas. ,.o|(xl \(0) [ax] X: Hcam l I (b) __l__[AV] °<...V: Fig.6-10. Probability densities for observation oftheposition and velocity ofaparticle.6-5Theuncertainty principle Theideas ofprobability arecertainly useful indescribing thebehavior of the1022 orsomolecules inasample ofagas,foritisclearly impractical even toattempt towrite down theposition orvelocity ofeach molecule. When prob- ability wasfirstapplied tosuch problems, itwasconsidered tobeaconvenience- awayofdealing withverycomplex situations. Wenowbelieve thattheideas of probability areessential toadescription ofatomic happenings. According to quantum mechanics, themathematical theory ofparticles, there isalways some uncertainty inthespecification ofpositions andvelocities. Wecan,atbest, say thatthere isacertain probability thatanyparticle willhave aposition nearsome coordinate x. Wecangiveaprobability density p1(x),such thatp1(x) Axistheprobability thattheparticle willbefound between xandx+Ax.Iftheparticle isreasonably welllocalized, saynear x1,thefunction p1(x) might begiven bythegraph of Fig.6—10(a). Similarly, wemust specify thevelocity oftheparticle bymeans of aprobability density p2(v), withp2(v) Avtheprobability that thevelocity will befound between vandv+Av. Itisoneofthefundamental results ofquantum mechanics thatthetwofunc- tions p1(x) andp2(v)cannot bechosen independently and, inparticular, cannot both bemade arbitrarily narrow. Ifwecallthetypical “width” ofthep1(x) curve [Ax], andthatofthep2(v)curve [Av](asshown inthefigure), nature demands that theproduct ofthetwowidths beatleast asbigasthenumber h/m, where m isthemass oftheparticle andhisafundamental physical constant called Planck’s constant. Wemaywrite thisbasic relationship as [Ax] '[Av] Zh/m. (6.22) This equation isastatement oftheHeisenberg uncertainty principle that we mentioned earlier. Since theright-hand sideofEq.(6.22) isaconstant, thisequation saysthat ifwetryto“pindown” aparticle byforcing ittobeataparticular place, itends upbyhaving ahigh speed. Orifwetrytoforce ittogovery slowly, orata precise velocity, it“spreads out” sothatwedonotknow verywelljustwhere itis.Particles behave inafunny way! Theuncertainty principle describes aninherent fuzziness thatmust exist in anyattempt todescribe nature. Ourmost precise description ofnature must beinterms ofprobabilities. There aresome people whodonotlikethiswayof describing nature. They feelsomehow thatiftheycould only tellwhat isreally going onwithaparticle, theycould know itsspeed andposition simultaneously. Intheearly days ofthedevelopment ofquantum mechanics, Einstein wasquite worried about thisproblem. Heused toshake hishead andsay,“But, surely God does notthrow dice indetermining how electrons should go!” Heworried about that problem foralong time andheprobably never really reconciled himself to thefactthatthisisthebestdescription ofnature thatonecangive. There are stilloneortwophysicists whoareworking ontheproblem whohave anintuitive conviction thatitispossible somehow todescribe theworld inadifferent way andthatallofthisuncertainty about thewaythings arecanberemoved. Noone hasyetbeen successful. Thenecessary uncertainty inourspecification oftheposition ofaparticle becomes most important when wewish todescribe thestructure ofatoms. In thehydrogen atom, which hasanucleus ofoneproton withoneelectron outside ofthenucleus, theuncertainty intheposition oftheelectron isaslarge astheatom itself! Wecannot, therefore, properly speak oftheelectron moving insome “orbit” around theproton. Themost wecansayisthatthere isacertain chance p(r)AV, ofobserving theelectron inanelement ofvolume AVatthedistance rfrom the proton. Theprobability density p(r)isgiven byquantum mechanics. Foran undisturbed hydrogen atom p(r)=Ae—’2/"2, which isabell-shaped function like thatinFig.6-8.Thenumber aisthe“typical” radius, where thefunction isdecreas- ingrapidly. Since there isasmall probability offinding theelectron atdistances 6-10 Fig.6-1l.Away ofvisualizing ahy- drogen otom. Thedensity (whiteness) of the cloud represents the probability density forobserving theelectron. from thenucleus much greater than a,wemay think ofaas“the radius ofthe atom,” about l0‘1° meter. Wecanform animage ofthehydrogen atom byimagining a“cloud” whose density isproportional totheprobability density forobserving theelectron. Asample ofsuch acloud isshown inFig.6—l1. Thus ourbest“picture” ofa hydrogen atom isanucleus surrounded byan“electron cloud” (although wereally mean a“probability cloud”). Theelectron isthere somewhere, butnature per- mitsustoknow only thechance offinding itatanyparticular place. Initsefforts tolearn asmuch aspossible about nature, modern physics has found thatcertain things cannever be“known” with certainty. Much ofour knowledge must always remain uncertain. Themost wecanknow isinterms of probabilities. 6-ll 7 The Theory ofGravitation 7-1Planetary motions Inthischapter weshall discuss oneofthemost far-reaching generalizations ofthehuman mind. While weareadmiring thehuman mind, weshould takesome timeofftostand inaweofanature thatcould follow withsuch completeness and generality such anelegantly simple principle asthelawofgravitation. What is thislawofgravitation? Itisthat every object intheuniverse attracts every other object with aforce which foranytwobodies isproportional tothemass of eachandvaries inversely asthesquare ofthedistance between them. This state- ment canbeexpressed mathematically bytheequation F_ G_"L'I._ rz Iftothisweaddthefactthatanobject responds toaforce byaccelerating inthe direction oftheforce byanamount thatisinversely proportional tothemass of theobject, weshall have said everything required, forasufficiently talented mathematician could then deduce alltheconsequences ofthese twoprinciples. However, since youarenotassumed tobesufficiently talented yet,weshall dis- cusstheconsequences inmore detail. andnotjustleave youwith only these two bareprinciples. Weshall briefly relate thestory ofthediscovery ofthelawof gravitation anddiscuss some ofitsconsequences, itsefiects onhistory, themys- teries thatsuch alawentails, andsome refinements ofthelawmade byEinstein; weshall alsodiscuss therelationships ofthelawtotheother laws ofphysics. Allthiscannot bedone inonechapter, butthese subjects willbetreated indue timeinsubsequent chapters. Thestory begins withtheancients observing themotions ofplanets among the stars, andfinally deducing thattheywent around thesun,afactthatwasredis- covered later byCopernicus. Exactly how theplanets went around thesun, withexactly what motion, took alittle more work todiscover. Inthebeginning of thefifteenth century there were great debates astowhether theyreally went around thesunornot. Tycho Brahe hadanideathatwasdifferent from anything pro- posed bytheancients: hisidea wasthatthese debates about thenature ofthe motions oftheplanets would bestberesolved iftheactual positions oftheplanets intheskywere measured sufficiently accurately. Ifmeasurement showed exactly howtheplanets moved, then perhaps itwould bepossible toestablish oneor another viewpoint. This wasatremendous idea——that tofindsomething out,it isbetter toperform some careful experiments than tocarry ondeep philosophical arguments. Pursuing thisidea, Tycho Brahe studied thepositions oftheplanets formany years inhisobservatory ontheisland ofHven, nearCopenhagen. He made voluminous tables, which were then studied bythemathematician Kepler, after Tycho's death. Kepler discovered from thedata some very beautiful and remarkable, butsimple, laws regarding planetary motion. 7-2Kepler’s laws First ofall,Kepler found thateach planet goes around thesuninacurve called anellipse, with thesunatafocus oftheellipse. Anellipse isnotjustan oval, butisavery specific andprecise curve thatcanbeobtained byusing two tacks, oneateach focus, aloop ofstring, andapencil; more mathematically, it 7-17-1Planetary motions 7-2Kepler’s laws 7-3Development ofdynamics 7-4Newton’s lawofgravitation 7-5Universal gravitation 7-6Cavendisl1’s experiment 7-7What isgravity? 7-8Gravity andrelativity FigII‘ ,,, A\V-t-Y=20 4 Fig. 7—'l. Anellipse. ‘.""'lI'1"Iv, .7-2. Kepler's lawofareas.isthelocus ofallpoints thesumofwhose distances from twofixed points (thefoci) isaconstant. Or,ifyouwill,itisaforeshortened circle (Fig. 7-1). Kepler’s second observation wasthattheplanets donotgoaround thesun atauniform speed, butmove faster when they arenearer thesunandmore slowly when theyarefarther from thesun,inprecisely thisway: Suppose aplanet isobserved atanytwosuccessive times, letussayaweek apart, andthattheradius vector* isdrawn totheplanet foreach observed position. Theorbital arctraversed bytheplanet during theweek, andthetworadius vectors, bound acertain plane area, theshaded areashown inFig.7-2. Iftwosimilar observations aremade a week apart, atapartoftheorbit farther from thesun(where theplanet moves more slowly), thesimilarly bounded areaisexactly thesame asinthefirstcase. So,inaccordance withthesecond law,theorbital speed ofeachplanet issuchthat theradius “sweeps out” equal areas inequal times. Finally, athird lawwasdiscovered byKepler much later; thislawisofa different category from theother two,because itdeals notwithonlyasingle planet, butrelates oneplanet toanother. This lawsaysthatwhen theorbital period and orbit sizeofanytwoplanets arecompared, theperiods areproportional tothe 3/2power oftheorbit size. Inthisstatement theperiod isthetime interval it takes aplanet togocompletely around itsorbit, andthesizeismeasured bythe length ofthegreatest diameter oftheelliptical orbit, technically known asthe major axis. More simply, iftheplanets went incircles, asthey nearly do,the time required togoaround thecircle would beproportional tothe3/2power of thediameter (orradius). Thus Kepler’s three laws are: I.Each planet moves around thesuninanellipse, with thesunatonefocus. II.Theradius vector from thesuntotheplanet sweeps outequal areas in equal intervals oftime. III.Thesquares oftheperiods ofanytwoplanets areproportional tothe cubes ofthesemimajor axesoftheir respective orbits: T~a3’2. 7-3Development ofdynamics While Kepler wasdiscovering these laws, Galileo wasstudying thelaws of motion. Theproblem was, what makes theplanets goaround? (Inthose days, oneofthetheories proposed wasthattheplanets went around because behind them were invisible angels, beating their wings anddriving theplanets forward. You willseethatthistheory isnowmodified! Itturns outthatinorder tokeep theplanets going around, theinvisible angels must fiyinadifferent direction and they have nowings. Otherwise, itisasomewhat similar theory!) Galileo dis- covered avery remarkable factabout motion, which wasessential forunder- standing these laws. That istheprinciple ofinertia—if something ismoving, with nothing touching itandcompletely undisturbed, itwillgoonforever, coasting at auniform speed inastraight line. (Why does itkeep oncoasting? Wedonot know, butthatisthewayitis.) Newton modified thisidea, saying thattheonly waytochange themotion ofabody istouseforce. Ifthebody speeds up,aforce hasbeen applied inthe direction ofmotion. Ontheother hand, ifitsmotion ischanged toanewdirec- tion,aforce hasbeen applied sideways. Newton thusadded theideathataforce isneeded tochange thespeed orthedirection ofmotion ofabody. Forexample, ifastone isattached toastring andiswhirling around inacircle, ittakes aforce tokeep itinthecircle. Wehave topullonthestring. Infact,thelawisthatthe acceleration produced bytheforce isinversely proportional tothemass, orthe force isproportional tothemass times theacceleration. Themore massive a thing is,thestronger theforce required toproduce agiven acceleration. (The mass canbemeasured byputting other stones ontheendofthesame string and making them goaround thesame circle atthesame speed. Inthiswayitisfound thatmore orlessforce isrequired, themore massive object requiring more force.) *Aradius vector isalinedrawn from thesuntoanypoint inaplanet's orbit. 7-2 Thebrilliant idearesulting from these considerations isthatnotangential force isneeded tokeep aplanet initsorbit (theangels donothave toflytangentially) because theplanet would coast inthatdirection anyway. Ifthere were nothing atalltodisturb it,theplanet would goofl"inastraight line. Buttheactual motion deviates from thelineonwhich thebody would have gone ifthere were noforce, thedeviation being essentially atright angles tothemotion, notinthedirection ofthemotion. Inother words, because oftheprinciple ofinertia, theforce needed tocontrol themotion ofaplanet around thesunisnotaforce around thesun buttoward thesun. (Ifthere isaforce toward thesun,thesunmight betheangel, ofcourse!) 7-4Newton’s lawofgravitation From hisbetter understanding ofthetheory ofmotion, Newton appreciated thatthesuncould betheseatororganization offorces thatgovern themotion of theplanets. Newton proved tohimself (and perhaps weshall beabletoprove it soon) thatthevery factthatequal areas areswept outinequal times isaprecise signpostoftheproposition thatalldeviations areprecisely radial—-that thelawof areas isadirect consequence oftheideathatalloftheforces aredirected exactly toward thesun. Next, byanalyzing Kepler’s third lawitispossible toshow thatthefarther away theplanet, theweaker theforces. Iftwoplanets atdifferent distances from thesunarecompared, theanalysis shows thattheforces areinversely propor- tional tothesquares oftherespective distances. With thecombination ofthe twolaws, Newton concluded thatthere must beaforce, inversely asthesquare ofthedistance, directed inalinebetween thetwoobjects. Being aman ofconsiderable feeling forgeneralities, Newton supposed, of course, thatthisrelationship applied more generally thanjusttothesunholding theplanets. Itwasalready known, forexample, thattheplanet Jupiter hadmoons going around itasthemoon oftheearth goes around theearth, andNewton feltcertain thateach planet held itsmoons with aforce. Healready knew ofthe force holding usontheearth, soheproposed thatthiswasauniversal force- thateverything pulls everything else. Thenext problem waswhether thepulloftheearth onitspeople wasthe “same” asitspullonthemoon, i.e.,inversely asthesquare ofthedistance. Ifan object onthesurface oftheearth falls16feetinthefirstsecond after itisreleased from rest,how fardoes themoon fallinthesame time? Wemight saythatthe moon does notfallatall.Butifthere were noforce onthemoon, itwould gooil inastraight line,whereas itgoesinacircle instead, soitreally falls infrom where itwould have been ifthere were noforce atall.Wecancalculate from theradius ofthemoon’s orbit (which isabout 240,000 miles) andhow long ittakes togo around theearth (approximately 29days), how farthemoon moves initsorbit inlsecond, andcanthencalculate howfaritfallsinonesecond.* This distance turns outtoberoughly 1/20 ofaninchinasecond. That fitsverywellwith the inverse square law,because theearth’s radius is4000 miles, andifsomething which is4000 miles from thecenter oftheearth falls l6feetinasecond, something 240,000 miles, or60times asfaraway, should fallonly 1/3600 of16feet,which also isroughly l/20 ofaninch. Wishing toputthistheory ofgravitation toatestby similar calculations, Newton made hiscalculations very carefully andfound a discrepancy solarge thatheregarded thetheory ascontradicted byfacts, anddid notpublish hisresults. Sixyears later anewmeasurement ofthesizeoftheearth showed thattheastronomers hadbeen using anincorrect distance tothemoon. When Newton heard ofthis, hemade thecalculation again, with thecorrected figures, andobtained beautiful agreement. This ideathatthemoon “falls” issomewhat confusing, because, asyousee, itdoes notcome anycloser. Theidea issufiiciently interesting tomerit further *That is,how farthecircle ofthemoon's orbit fallsbelow thestraight linetangent toitatthepoint where themoon wasonesecond before. 7-3 fx ELECTROMAGNET .____mO\\°\, /’/ LLlSl0N' h I l"|=h2ha [ J Fig.7-3. Apparatus forshowing the independence ofvertical and horizontal motions. From X Plane Geoimtry X 211-S 2R‘S;.T-? I I‘R-radius of earth M000 miles 'x‘;d£at.nnce "travelled nort- zonull," Inone second '5‘duunee 'fal1en" uione second (16fee’) A4 Fig.7-4. Acceleration toward the center ofacircular path. From plane geometry, x/s=(2R—S)/x z2R/x, where Ristheradius oftheearth, 4000 miles; xisthedistance “travelled hori- zontally" inone second; and Sisthe distance "fallen" inone second (16feet).explanation: themoon fallsinthesense thatitfalls away from thestraight line thatitwould pursue ifthere werenoforces. Letustakeanexample onthesurface oftheearth. Anobject released near theearth’s surface willfall16feetinthefirst second. AnOb_]6Cl. shot outhorizontally willalsofall16feet; even though itis moving horizontally, itstillfalls thesame 16feetinthesame time. Figure 7-3 shows anapparatus which demonstrates this. Onthehorizontal track isaball which isgoing tobedriven forward alittle distance away. Atthesame height isaballwhich isgoing tofallvertically, andthere isanelectrical switch arranged sothatatthemoment thefirstballleaves thetrack, thesecond ballisreleased. That theycome tothesame depth atthesame time iswitnessed bythefactthat theycollide inmidair. Anobject likeabullet, shothorizontally, might goalong wayinonesecond—perhaps 2000 feet—but itwillstillfall16feetifitisaimed horizontally. What happens ifweshoot abullet faster andfaster? Donotforget thattheearth’s surface iscurved. Ifweshoot itfastenough, then when itfalls 16feetitmaybeatjustthesame height above theground asitwasbefore. How canthatbe?Itstillfalls, buttheearth curves away, soitfalls“around” theearth. Thequestion is,how fardoes ithave togoinonesecond sothattheearth is l6feetbelow thehorizon? InFig.7-4weseetheearth with its4000-mile radius, andthetangential, straightline path thatthebullet would take ifthere were no force. Now, ifweuseoneofthose wonderful theorems ingeometry, which says thatourtangent isthemean proportional between thetwoparts ofthediameter cutbyanequal chord, weseethatthehorizontal distance travelled isthemean proportional between thel6feetfallen andthe8000-mile diameter oftheearth. Thesquare root of(16/5280) X8000 comes outvery close to5miles. Thus weseethatifthebullet moves at5miles asecond, itthen willcontinue tofall toward theearth atthesame rateof16feeteach second, butwillnever getany closer because theearth keeps curving away from it.Thus itwasthatMr.Gagarin maintained himself inspace while going 25,000 miles around theearth atapproxi- mately 5miles persecond. (Hetook alittle longer because hewasalittle higher.) Anygreat discovery ofanewlawisuseful onlyifwecantake more outthan weputin.Now, Newton used thesecond andthird ofKepler’s laws todeduce hislawofgravitation. What didhepredict? First, hisanalysis ofthemoon’s motion wasaprediction because itconnected thefalling ofobjects ontheearth’s surface with thatofthemoon. Second, thequestion is,istheorbit anellipse? Weshall seeinalater chapter howitispossible tocalculate themotion exactly, andindeed onecanprove thatitshould beanellipse,* sonoextra factisneeded toexplain Kepler’s firstlaw. Thus Newton made hisfirstpowerful prediction. Thelawofgravitation explains many phenomena notpreviously understood. Forexample, thepullofthemoon ontheearth causes thetides, hitherto mysterious. Themoon pulls thewater upunder itandmakes thetides—people hadthought ofthatbefore, buttheywere notasclever asNewton, andsotheythought there ought tobeonlyonetideduring theday. Thereasoning wasthatthemoon pulls thewater upunder it,making ahightideandalowtide,andsince theearth spins underneath, thatmakes thetideatonestation goupanddown every 24hours. Actually thetidegoes upanddown in12hours. Another school ofthought claimed thatthehightideshould beontheother sideoftheearth because, sothey argued, themoon pulls theearth away from thewaterl Both ofthese theories arewrong. Itactually works likethis: thepullofthemoon fortheearth andfor thewater is“balanced” atthecenter. Butthewater which iscloser tothemoon is pulled more than theaverage andthewater which isfarther away from itispulled lessthan theaverage. Furthermore, thewater canflowwhile themore rigid earth cannot. Thetruepicture isacombination ofthese twothings. What dowemean by“balanced”? What balances? Ifthemoon pulls the whole earth toward it,whydoesn’t theearth fallright “up” tothemoon? Because theearth does thesame trick asthemoon, itgoesinacircle around apoint which isinside theearth butnotatitscenter. Themoon does notjustgoaround the *Theproof isnotgiven inthiscourse. 7-4 earth, theearth andthemoon both goaround acentral position, each falling toward thiscommon position, asshown inFig.7-5. This motion around the common center iswhat balances thefallofeach. Sotheearth isnotgoing ina straight lineeither; ittravels inacircle. Thewater onthefarsideis“unbalanced” because themoon‘s attraction there isweaker than itisatthecenter oftheearth, where itjust balances the“centrifugal force.” Theresult ofthisimbalance isthat thewater risesup,away from thecenter oftheearth. Onthenearside, theattrac- tionfrom themoon isstronger, andtheimbalance isintheopposite direction in space, butagain away from thecenter oftheearth. Thenetresult lSthatweget twotidal bulges. 7-5Universal gravitation What elsecanweunderstand when weunderstand gravity? Everyone knows theearth isround. Why istheearth round? That iseasy; itisduetogravitation. Theearth canbeunderstood toberound merely because everything attracts everything elseandsoithasattracted itself together asfarasitcan! Ifwegoeven further, theearth isnotexactly asphere because itisrotatings andthisbrings in centrifugal effects which tendtooppose gravity near theequator. Itturns outthat theearth should beelliptical, andweeven gettheright shape fortheellipse. Wecanthus deduce thatthesun, themoon, andtheearth should be(nearly) spheres, justfrom thelawofgravitation. What elsecanyoudowith thelawofgravitation? Ifwelook atthemoons ofJupiter wecanunderstand everything about thewaythey move around that planet. Incidentally, there wasonce acertain difficulty withthemoons ofJupiter thatisworth remarking on.These satellites were studied verycarefully byRoemer, whonoticed thatthemoons sometimes seemed tobeahead ofschedule, andsome- times behind. (One canfindtheir schedules bywaiting averylongtimeandfinding outhowlongittakes ontheaverage forthemoons togoaround.) Now they were ahead when Jupiter wasparticularly close totheearth andthey were behind when Jupiter wasfarther from theearth. Thiswould have been avery difficult thing to explain according tothelawofgravitation—it would have been, infact,thedeath ofthiswonderful theory iftherewerenoother explanation. Ifalawdoesnotwork even inoneplace where itought to,itisjustwrong. Butthereason forthisdis- crepancy wasverysimple andbeautiful: ittakes alittle while toseethemoons of Jupiter because ofthetime ittakes light totravel from Jupiter totheearth. When Jupiter iscloser totheearth thetime isalittle less,andwhen itisfarther from the earth, thetime ismore. This iswhymoons appear tobe,ontheaverage, alittle ahead oralittle behind, depending onwhether theyarecloser toorfarther from theearth. This phenomenon showed thatlight does nottravel instantaneously, andfurnished thefirstestimate ofthespeed oflight. Thiswasdone in1656. Ifalloftheplanets push andpulloneach other, theforce which controls, letussay,Jupiter ingoing around thesunisnotjusttheforce from thesun; there isalsoapullfrom, say,Saturn. Thisforce isnotreally strong, since thesun ismuch more massive than Saturn, butthere issome pull, sotheorbit ofJupiter should notbeaperfect ellipse, anditisnot;itisslightly off,and“wobbles” around thecorrect elliptical orbit. Such amotion isalittle more complicated. Attempts were made toanalyze themotions ofJupiter, Saturn, andUranus onthebasis ofthelawofgravitation. Theeffects ofeach ofthese planets oneach other were calculated toseewhether ornotthetinydeviations andirregularities inthese motions could becompletely understood from thisonelaw. Loandbehold, for Jupiter andSaturn, allwaswell, butUranus was“weird.” Itbehaved inavery peculiar manner. Itwasnottravelling inanexact ellipse, butthatwasunder- standable, because oftheattractions ofJupiter andSaturn. Buteven ifallowance were made forthese attractions, Uranus stillwasnotgoing right, sothelaws of gravitation were indanger ofbeing overturned, apossibility thatcould notbe ruled out.Two men, Adams andLeverrier, inEngland andFrance, independently, 7-5Bo//’Moo~/ /// H20 / / POINT AROUND WHICH EARTH BMOON ROTATE EARTH Fig.7-5. The earth-moon system, withtides. Fig.7-6. Adouble-star system. arrived atanother possibility: perhaps there isanother planet, dark andinvisible, which menhadnotseen. This planet, N,could pullonUranus. They calculated where such aplanet would have tobeinorder tocause theobserved perturba- tions. They sentmessages totherespective observatories, saying, “Gentlemen, point your telescope tosuch andsuch aplace, andyouwillseeanewplanet.” Itoften depends onwith whom youareworking astowhether theypayanyatten- tiontoyouornot. They didpayattention toLeverrier; they looked, andthere planet Nwas! Theother observatory then alsolooked very quickly inthenext fewdays andsawittoo. This discovery shows that Newton’s laws areabsolutely right inthesolar system; butdothey extend beyond therelatively small distances ofthenearest planets? Thefirsttestliesinthequestion, dostars attract each other aswellas planets? Wehave definite evidence thattheydointhedouble stars. Figure 7-6 shows adouble star—two stars very close together (there isalsoathird starin thepicture sothatwewillknow thatthephotograph wasnotturned). Thestars arealsoshown asthey appeared several years later. Weseethat, relative tothe “fixed” star, theaxisofthepairhasrotated, i.e.,thetwostars aregoing around each other. Dotheyrotate according toNewton’s laws? Careful measurements oftherelative positions ofonesuch double starsystem areshown inFig.7-7. There weseeabeautiful ellipse, themeasures starting in1862 andgoing allthe wayaround to1904 (bynow itmust have gone around once more). Everything coincides with Newton’s laws, except that thestarSirius Aisnotatthefocus. Why should thatbe? Because theplane oftheellipse isnotinthe“plane ofthe sky.” Wearenotlooking atright angles totheorbit plane, andwhen anellipse isviewed atatilt,itremains anellipse butthefocus isnolonger atthesame place. Thus wecananalyze double stars, moving about each other, according tothe requirements ofthegravitational law. 180” 0,,/,51» /,9Q-1 0,,QA‘,“Y.ea’-@ »%VI/. \ .,..0 Q. an0Q 0 0 3' 0 5'/tr‘.///»,/I////k,’ /////210° —__V__ 90° Li” w“ __,\ " 9Isa:,‘-\\ l \ \ 1% \ ii lb”O \ .0 Q4 ' 0»4 10 , mP q& a ?'.'...|....?" .4".‘f".l".1.~.WSCALE Fig.7-7. Orbit ofSirius Bwith respect toSirius A. 7-6 Fig.7-8. Aglobular slurcluster. That thelawofgravitation istrue ateven bigger distances isindicated in Fig.7—8. Ifonecannot seegravitation acting here, hehasnosoul. This figure shows oneofthemost beautiful things inthesky—a globular starcluster. Allof thedotsarestars. Although theylook asiftheyarepacked solid toward thecenter, thatisduetothefallibility ofourinstruments. Actually, thedistances between even thecentermost stars arevery great andthey very rarely collide. There are more stars intheinterior than farther out,andaswemove outward there are fewer andfewer. Itisobvious that there isanattraction among these stars. Itisclear thatgravitation exists atthese enormous dimensions, perhaps 100,000 times thesizeofthesolar system. Letusnow gofurther, andlook atanentire galaxy, shown inFig.7—9. Theshape ofthisgalaxy indicates anobvious tendency foritsmatter toagglomerate. Ofcourse wecannot prove thatthelawhere is precisely inverse square, only that there isstillanattraction, atthisenormous dimension, thatholds thewhole thing together. Onemay say,“Well, thatisall veryclever butwhyisitnotjustaball?” Because itisspinning andhasangular momentum which itcannot giveupasitcontracts; itmust contract mostly ina plane. (Incidentally, ifyouarelooking foragood problem, theexact details of how thearms areformed andwhat determines theshapes ofthese galaxies has notbeen worked out.) Itis,however, clear that theshape ofthegalaxy isdueto gravitation even though thecomplexities ofitsstructure have notyetallowed Fig.7-9. Agalaxy. 7-7 ustoanalyze itcompletely. Inagalaxy wehave ascale ofperhaps 50,000 to 100,000 light years. Theearth’s distance from thesunis8%light minutes, soyou canseehowlarge these dimensions are. Gravity appears toexist ateven bigger dimensions, asindicated byFig.7-10, which shows many “little” things clustered together. This isacluster ofgalaxies, justlikeastarcluster. Thus galaxies attract each other atsuch distances thatthey tooareagglomerated intoclusters. Perhaps gravitation exists even overdistances oftensofmillions oflight years; sofaraswenowknow, gravity seems togoout forever inversely asthesquare ofthedistance. Notonly canweunderstand thenebulae, butfrom thelawofgravitation we caneven getsome ideas about theorigin ofthestars. Ifwehave abigcloud ofdust andgas,asindicated inFig.7-11, thegravitational attractions ofthepieces of dustforoneanother might make them form little lumps. Barely visible inthefigure are“little” black spots which may bethebeginning oftheaccumulations ofdust andgases which, duetotheir gravitation, begin toform stars. Whether wehave everseenastarform ornotisstilldebatable. Figure 7-l2 shows theonepiece of evidence which suggests thatwehave. Attheleftisapicture ofaregion ofgas with some stars inittaken in1947, andattheright isanother picture, taken only 7years later, which shows twonewbright spots. Hasgasaccumulated, hasgravity acted hard enough andcollected itintoaballbigenough thatthestellar nuclear reaction starts intheinterior andturns itinto astar? Perhaps, andperhaps not. Itisunreasonable thatinonly seven years weshould besolucky astoseeastar change itself intovisible form; itismuch lessprobable thatweshould seetwo! Fig.7—lO. Acluster ofgalaxies. Fig7—llAninterstellar dustcloud. Fig.7-12. Theformation ofnew stars? 7-8 7-6Cavendish’s experiment Gravitation, therefore, extends over enormous distances. Butifthere isa force between anypairofobjects, weought tobeabletomeasure theforce between ourown objects. Instead ofhaving towatch thestars goaround each other, whycanwenottakeaballofleadandamarble andwatch themarble gotoward theballoflead? Thedifficulty ofthisexperiment when done insuch asimple manner istheveryweakness ordelicacy oftheforce. Itmust bedone withextreme care, which means covering theapparatus tokeep theairout,making sureitis notelectrically charged, andsoon;then theforce canbemeasured. Itwasfirst measured byCavendish with anapparatus which isschematically indicated in Fig.7-13. Thisfirstdemonstrated thedirect force between twolarge, fixed balls ofleadandtwosmaller balls ofleadontheends ofanarmsupported byavery finefiber, called atorsion fiber. Bymeasuring how much thefiber getstwisted, onecanmeasure thestrength oftheforce, verify thatitisinversely proportional tothesquare ofthedistance, anddetermine how strong itis.Thus, onemay accurately determine thecoefficient Gintheformula mm’F- G7- Allthemasses anddistances areknown. Yousay,“Weknew italready forthe earth.” Yes,butwedidnotknow themass oftheearth. Byknowing Gfrom this experiment andbyknowing how strongly theearth attracts, wecanindirectly learn howgreat isthemass oftheearth! Thisexperiment hasbeencalled “weighing theearth.” Cavendish claimed hewasweighing theearth, butwhat hewasmeas- uring wasthecoeflicient Gofthegravity law. This istheonly wayinwhich the mass oftheearth canbedetermined. Gturns outtobe 6.670 X10*‘ 1newton -m2/kg2. Itishard toexaggerate theimportance oftheeflect onthehistory ofscience produced bythisgreat success ofthetheory ofgravitation. Compare theconfu- sion, thelackofconfidence, theincomplete knowledge thatprevailed intheearlier ages, when there were endless debates andparadoxes, withtheclarity andsimplic- ityofthislaw—-this factthatallthemoons andplanets andstars have such a simple ruletogovern them, andfurther thatman could understand itanddeduce howtheplanets should move! This isthereason forthesuccess ofthesciences in following years, foritgave hope thattheother phenomena oftheworld might also have such beautifully simple laws. 7-7What isgravity? Butisthissuch asimple law? What about themachinery ofit?Allwehave done istodescribe howtheearth moves around thesun,butwehave notsaid what makes itgo.Newton made nohypotheses about this; hewassatisfied to findwhat itdidwithout getting intothemachinery ofit.Noonehassince given anymachinery. Itischaracteristic ofthephysical lawsthattheyhave thisabstract character. Thelawofconservation ofenergy isatheorem concerning quantities thathave tobecalculated andadded together, withnomention ofthemachinery, andlikewise thegreat laws ofmechanics arequantitative mathematical laws for which nomachinery isavailable. Why canweusemathematics todescribe nature without amechanism behind it?Nooneknows. Wehave tokeep going because wefindoutmore thatway. Many mechanisms forgravitation have been suggested. Itisinteresting tocon- sider oneofthese, which many people have thought offrom time totime. At first, oneisquite excited andhappy when he“discovers” it,buthesoon finds that itisnotcorrect. Itwasfirstdiscovered about 1750. Suppose there were many particles moving inspace ataveryhighspeed inalldirections andbeing onlyslightly absorbed ingoing through matter. When theyareabsorbed, theygiveanimpulse totheearth. However, since there areasmany going onewayasanother, the 7-9@ ‘oFig.7-13. Asimplified diagram of theapparatus used byCavendish to verify thelawofuniversal gravitation for small obiects andtomeasure thegravita- tional constant G. Gr-avifat." Attnd/Hi_-ii-—— 1-/ #2 Eltzlrrul Ra/Mlflflri /4'/7 ‘/0 I//4,/70,ooo, ooq 0090° _\°°o‘@0904°00°°°i°°O O09 O0‘? 000 090' - Fig.7-14. The relative strengths of electrical and gravitational interactions between twoelectrons.impulses allbalance. Butwhen thesunisnearby, theparticles coming toward the earth through thesunarepartially absorbed, sofewer ofthem arecoming from thesunthan arecoming from theother side. Therefore, theearth feels anetim- pulse toward thesunanditdoes nottakeonelong toseethatitisinversely asthe square ofthedistance—because ofthevariation ofthesolid angle thatthesun subtends aswevary thedistance. What iswrong with thatmachinery? Itin- volves some newconsequences which arenottrue. This particular idea hasthe following trouble: theearth, inmoving around thesun,would impinge onmore particles which arecoming from itsforward sidethan from itshind side(when yourunintherain, theraininyour faceisstronger than thatontheback ofyour headl). Therefore there would bemore impulse given theearth from thefront, andtheearth would feelaresistance tomotion andwould beslowing upinitsorbit. Onecancalculate howlong itwould take fortheearth tostopasaresult ofthis resistance, anditwould nottakelongenough fortheearth tostillbeinitsorbit, so thismechanism doesnotwork. Nomachinery haseverbeeninvented that“explains” gravity without alsopredicting some other phenomenon thatdoes notexist. Next weshall discuss thepossible relation ofgravitation toother forces. There isnoexplanation ofgravitation interms ofother forces atthepresent time. Itisnotanaspect ofelectricity oranything likethat, sowehave noexplanation. However, gravitation andother forces arevery similar, anditisinteresting to note analogies. Forexample, theforce ofelectricity between twocharged objects looks justlikethelawofgravitation: theforce ofelectricityis aconstant, withaminus sign, times theproduct ofthecharges, andvaries inversely asthesquare ofthe distance. Itisintheopposite direction———likes repel. Butisitstillnotveryremark- ablethatthetwolawsinvolve thesame function ofdistance? Perhaps gravitation andelectricity aremuch more closely related than wethink. Many attempts have been made tounify them; theso-called unified field theory isonly averyelegant attempt tocombine electricity andgravitation; but,incomparing gravitation and electricity, themost interesting thing istherelative strengths oftheforces. Any theory thatcontains them both must alsodeduce howstrong thegravity is. Ifwetake, insome natural units, therepulsion oftwoelectrons (nature’s universal charge) duetoelectricity, andtheattraction oftwoelectrons duetotheir masses, wecanmeasure theratio ofelectrical repulsion tothegravitational attraction. Theratio isindependent ofthedistance andisafundamental constant ofnature. Theratio isshown inFig.7-14. Thegravitational attraction relative totheelectrical repulsion between twoelectrons isldivided by4.17 X1042! Thequestion is,where does such alarge number come from? Itisnotaccidental, liketheratio ofthevolume oftheearth tothevolume ofaflea. Wehave considered twonatural aspects ofthesame thing, anelectron. This fantastic number isa natural constant, soitinvolves something deep innature. Where could such a tremendous number come from? Some saythatweshall onedayfindthe“universal equation,” andinit,oneoftheroots willbethisnumber. Itisvery difficult to findanequation forwhich such afantastic number isanatural root. Other pos- sibilities have been thought of;oneistorelate ittotheageoftheuniverse. Clearly, wehave tofindanother large number somewhere. Butdowemean theageofthe universe inyears ?No,because years arenot“natural”; theywere devised bymen. Asanexample ofsomething natural, letusconsider thetime ittakes light togo across aproton, l0_2“ second. Ifwecompare thistimewiththeageoftheuniverse, 2X101°years, theanswer is10*“. Ithasabout thesame number ofzeros going offit,soithasbeen proposed thatthegravitational constant isrelated totheage oftheuniverse. Ifthatwere thecase, thegravitational constant would change with time, because astheuniverse gotolder theratio oftheageoftheuniverse tothe time which ittakes forlight togoacross aproton would begradually increasing. Isitpossible thatthegravitational constant ischanging with time? Ofcourse thechanges would besosmall thatitisquite diflicult tobesure. Onetestwhich wecanthink ofistodetermine what would have been theeffect ofthechange during thepast 109years, which isapproximately theagefrom theearliest lifeontheearth tonow, andone-tenth oftheageoftheuniverse. Inthistime, thegravity constant would have increased byabout 10percent. It 7-10 turns outthatifweconsider thestructure ofthesun—-the balance between the weight ofitsmaterial andtherateatwhich radiant energy isgenerated inside it—— wecandeduce thatifthegravity were l0percent stronger, thesunwould bemuch more than 10percent brighter——by thesixth power ofthegravity constant! Ifwe calculate what happens totheorbit oftheearth when thegravity ischanging, we findthattheearth wasthen closer in.Altogether, theearth would beabout 100 degrees centigrade hotter, andallofthewater would nothave been inthesea,but vapor intheair,solifewould nothave started inthesea.S0wedonotnowbelieve thatthegravity constant ischanging with theageoftheuniverse. Butsuch argu- ments astheonewehave justgiven arenotvery convincing, andthesubject is notcompletely closed. Itisafactthattheforce ofgravitation isproportional tothemass, thequantity which isfundamentally ameasure ofinertia—of howhard itistohold something which isgoing around inacircle. Therefore twoobjects, oneheavy andonelight, going around alarger object inthesame circle atthesame speed because ofgravity, Wlllstaytogether because togoinacircle requires aforce which isstronger for abigger mass. That is,thegravity isstronger foragiven mass injusttheright proportion sothatthetwoobjects willgoaround together. Ifoneobject were inside theother itwould stayinside; itisaperfect balance. Therefore, Gagarin orTitov would findthings “weightless” inside aspace ship; ifthey happened toletgo ofapiece ofchalk, forexample, itwould goaround theearth inexactly thesame wayasthewhole space ship, andsoitwould appear toremain suspended before them inspace. Itisvery interesting thatthisforce isexactly proportional tothe mass with great precision, because ifitwere notexactly proportional there would besome effect bywhich inertia andweight would differ. Theabsence ofsuch an effect hasbeen checked with great accuracy byanexperiment done first by Eotvos in1909 andmore recently byDicke. Forallsubstances tried, themasses andweights areexactly proportional within lpartin1,000,000,000, orless. This isaremarkable experiment. 7-8Gravity andrelativity Another topic deserving discussion isEinstein’s modification ofNewton's lawofgravitation. Inspite ofalltheexcitement itcreated, Newton’s lawofgravi- tation isnotcorrect! Itwasmodified byEinstein totakeintoaccount thetheory ofrelativity. According toNewton, thegravitational effect isinstantaneous, that is,ifwewere tomove amass, wewould atonce feelanewforce because ofthe newposition ofthatmass; bysuch means wecould send signals atinfinite speed. Einstein advanced arguments which suggest that wecannot send signals faster thanthespeed oflight, sothelawofgravitation must bewrong. Bycorrecting it totakethedelays intoaccount, wehave anewlaw,called Einstein’s lawofgravi- tation. One feature ofthisnewlawwhich isquite easy tounderstand isthis: IntheEinstein relativity theory, anything which hasenergy hasmass—mass in thesense thatitisattracted gravitationally. Even light, which hasanenergy, hasa“mass.” When alight beam, which hasenergy init,comes pastthesunthere isanattraction onitbythesun. Thus thelight does notgostraight, butisde- flected. During theeclipse ofthesun,forexample, thestars which arearound the sunshould appear displaced from where theywould beifthesunwere notthere, andthishasbeen observed. Finally, letuscompare gravitation with other theories. Inrecent years we have discovered thatallmass ismade oftinyparticles andthatthere areseveral kinds ofinteractions, suchasnuclear forces, etc.None ofthese nuclear orelectrical forces hasyetbeen found toexplain gravitation. Thequantum-mechanical aspects ofnature have notyetbeen carried over togravitation. When thescale issosmall thatweneed thequantum effects, thegravitational effects aresoweak thatthe need foraquantum theory ofgravitation hasnotyetdeveloped. Ontheother hand, forconsistency inourphysical theories itwould beimportant toseewhether Newton’s lawmodified toEinstein’s lawcanbefurther modified tobeconsistent with theuncertainty principle. This lastmodification hasnotyetbeen completed. 7—ll 8 Motion 8-1Description ofmotion Inorder tofindthelaws governing thevarious changes thattake place in bodies astimegoeson,wemust beabletodescribe thechanges andhave some way torecord them. Thesimplest change toobserve inabody istheapparent change initsposition withtime,-which wecallmotion. Letusconsider some solidobject withapermanent mark, which weshall callapoint, thatwecanobserve. We shalldiscuss themotion ofthelittle marker, which might betheradiator capofan automobile orthecenter ofafalling ball,andshalltrytodescribe thefactthatit moves andhowitmoves. These examples maysound trivial, butmany subtleties enter intothedescrip- tionofchange. Some changes aremore difficult todescribe than themotion of apoint onasolid object, forexample thespeed ofdrift ofacloud thatisdrifting veryslowly, butrapidly forming orevaporating, orthechange ofawoman’s mind. Wedonotknow asimple waytoanalyze achange ofmind, butsince the cloud canberepresented ordescribed bymany molecules, perhaps wecandescribe themotion ofthecloud inprinciple bydescribing themotion ofallitsindividual molecules. Likewise, perhaps even thechanges inthemind may have aparallel inchanges oftheatoms inside thebrain, butwehave nosuch knowledge yet. Atanyrate, thatiswhywebegin withthemotion ofpoints; perhaps weshould think ofthem asatoms, butitisprobably better tobemore rough inthebegin- ningandsimply tothink ofsome kindofsmall objects——-small, thatis,compared withthedistance moved. Forinstance, indescribing themotion ofacarthatis going ahundred miles, wedonothavetodistinguish between thefront andthe backofthecar.Tobesure,there areslight differences, butforrough purposes we say“the car,” andlikewise itdoes notmatter thatourpoints arenotabsolute points; forourpresent purposes itisnotnecessary tobeextremely precise. Also, while wetake afirstlook atthissubject wearegoing toforget about thethree dimensions oftheworld. Weshall justconcentrate onmoving inonedirection, asinacarononeroad. Weshall return tothree dimensions after weseehow to describe motion inonedimension. Now, youmay say,“This isallsome kind of trivia,” andindeed itis.How canwedescribe such aone-dimensional motion—- letussay,ofacar? Nothing could besimpler. Among many possible ways, one would bethefollowing. Todetermine theposition ofthecaratdifferent times, wemeasure itsdistance from thestarting point andrecord alltheobservations. InTable 8-1,srepresents thedistance ofthecar,infeet,from thestarting point, andtrepresents thetime inminutes. Thefirstlineinthetable represents zero distance andzerotime—the carhasnotstarted yet.After oneminute ithasstarted andhasgone 1200 feet. Then intwominutes, itgoesfarther—'notice thatitpicked upmore distance inthesecond minute—it hasaccelerated; butsomething hap- pened between 3and4andeven more soat5—it stopped atalight perhaps? Then itspeeds upagain andgoes 13,000 feetbytheendof6minutes, 18,000 feetatthe endof7minutes, and23,500 feetin8minutes; at9minutes ithasadvanced to only24,000 feet,because inthelastminute itwasstopped byacop. That isonewaytodescribe themotion. Another wayisbymeans ofagraph. Ifweplotthetimehorizontally andthedistance vertically, weobtain acurve some- thing likethatshown inFig.8-l. Asthetime increases, thedistance increases, atfirstvery slowly andthen more rapidly, andveryslowly again foralittle while at4minutes; then itincreases again forafewminutes andfinally, at9minutes, appears tohave stopped increasing. These observations canbemade from the 8-18-1Description ofmotion 8-2Speed 8-3Speed asaderivative 8-4Distance asanintegral 8-5Acceleration Table 8-1 t(min) \O®\lO\U|-BUJIOF-*@ NWO FEET DSTANQTRAVELEDN"3§ 2 4 6 8 IOs(ft) 0 1200 4000 9000 9500 9600 13000 18000 23500 24000 TIME INMINUTES Fig. 8—l. Graph ofdistance versus time forthecar. Table 8-2 t(sec) s(ft) 0 16 64 144 256 400 576 400 F I I I . l 2 3 4 5 T|vE |NSECOND$ Fig.8-2. Graph ofdistance versus rime forafalling body.O\U\->l.AI\)>-‘Q DSTANCEFALLENNFEET6825‘OOOgraph, without atable. Obviously, foracomplete description onewould have to know where thecarisatthehalf-minute marks, too,butwesuppose thatthegraph means something, thatlhecarhassome position atalltheintermediate times. Themotion ofacariscomplicated. Foranother example wetakesomething that moves inasimpler manner, folldwing more simple laws: afalling ball. Table 8-2gives thetime inseconds andthedistance infeetforafalling body. Atzeroseconds theballstarts outatzerofeet,andattheendoflsecond ithas fallen 16feet. Attheendof2seconds, ithasfallen 64feet, attheendof3 seconds, 144feet, andsoon;ifthetabulated numbers areplotted, wegetthe nice parabolic curve shown inFig. 8-2. Theformula forthiscurve canbewritten as s=16t2. (8.1) This formula enables ustocalculate thedistances atanytime. You might say there ought tobeaformula forthefirstgraph too. Actually, onemaywrite such aformula abstractly, as S=f(l). (3-2) meaning thatsissome quantity depending ontor,inmathematical phraseology, sisafunction oft.Since wedonotknow what thefunction is,there isnowaywe canwrite itindefinite algebraic form. Wehave nowseen twoexamples ofmotion, adequately described with very simple ideas, nosubtleties. However, there aresubtleties—-several ofthem. In thefirstplace, what dowemean bytimeandspace? Itturns outthatthese deep philosophical questions have tobeanalyzed very carefully inphysics, andthis isnotsoeasytodo.Thetheory ofrelativity shows thatourideas ofspace and time arenotassimple asonemight think atfirstsight. However, forourpresent purposes, fortheaccuracy thatweneed atfirst, weneed notbeverycareful about defining things precisely. Perhaps yousay,“That’s aterrible thing—-I learned that inscience wehave todefine everything precisely.” Wecannot define anything precisely! Ifweattempt to,wegetintothatparalysis ofthought thatcomes to philosophers, who sitopposite each other, onesaying totheother, “You don’t know what youaretalking aboutl” Thesecond onesays, “What doyoumean byknow ?What doyoumean bytalking ?What doyoumean byyou?,”andsoon. Inorder tobeabletotalkconstructively, wejusthave toagree thatwearetalking about roughly thesame thing. Youknow asmuch about timeasweneed forthe present, butremember thatthere aresome subtleties thathave tobediscussed; weshall discuss them later. Another subtlety involved, andalready mentioned, isthatitshould bepossible toimagine thatthemoving point weareobserving isalways located somewhere. (Ofcourse when wearelooking atit,there itis,butmaybe when welook away itisn’tthere.) Itturns outthatinthemotion ofatoms, thatideaalsdisfalse- wecannot findamarker onanatom andwatch itmove. That subtlety weshall have togetaround inquantum mechanics. Butwearefirstgoing tolearn what the problems arebefore introducing thecomplications, andthenweshall beinabetter position tomake corrections, inthelight ofthemore recent knowledge ofthe subject. Weshall, therefore, take asimple point ofview about time andspace. Weknow what these concepts areinarough way, andthose whohave driven a carknow what speed means. 8-2Speed Even though weknow roughly what “speed” means, there arestillsome rather deepsubtleties; consider thatthelearned Greeks werenever abletoadequately describe problems involving velocity. Thesubtlety comes when wetrytocompre- hend exactly what ismeant by“speed.” TheGreeks gotveryconfused about this, andanewbranch ofmathematics hadtobediscovered beyond thegeometry and algebra oftheGreeks, Arabs, and Babylonians. Asanillustration ofthediffi- culty, trytosolve thisproblem bysheer algebra: Aballoon isbeing inflated so 8-2 thatthevolume oftheballoon isincreasing attherateof100cm?’ persecond; atwhat speed istheradius increasing when thevolume is1000 cm3? TheGreeks weresomewhat confused bysuch problems, being helped, ofcourse, bysome very confusing Greeks. Toshow thatthere were difliculties inreasoning about speed atthetime, Zeno produced alarge number ofparadoxes, ofwhich weshall men- tiononetoillustrate hispoint thatthere areobvious difficulties inthinking about motion. “Listen,” hesays, “tothefollowing argument: Achilles runs 10times as fastasatortoise, nevertheless hecannever catch thetortoise. For, suppose that theystart inaracewhere thetortoise is100meters ahead ofAchilles; then when Achilles hasrunthe100meters totheplace where thetortoise was,thetortoise has proceeded 10meters, having runone-tenth asfast. Now, Achilles hastorun another 10meters tocatch upwith thetortoise, butonarriving attheendofthat run,hefinds thatthetortoise isstill1meter ahead ofhim; running another meter, hefinds thetortoise 10centimeters ahead, andsoon,adinfinitum. Therefore, at anymoment thetortoise isalways ahead ofAchilles andAchilles cannever catch upwiththetortoise.” What iswrong withthat? Itisthatafinite amount oftime canbedivided intoaninfinite number ofpieces, justasalength oflinecanbe divided intoaninfinite number ofpieces bydividing repeatedly bytwo. And so, although there areaninfinite number ofsteps (intheargument) tothepoint at which Achilles reaches thetortoise, itdoesn’t mean thatthere isaninfinite amount oftime. Wecanseefrom thisexample thatthere areindeed some subtleties in reasoning about speed. Inorder togettothesubtleties inaclearer fashion, weremind youofajoke which yousurely must have heard. Atthepoint where theladyinthecariscaught byacop, thecopcomes uptoherandsays, “Lady, youwere going 60miles an hour!” Shesays, “That’s impossible, sir,Iwastravelling foronly seven minutes. Itisridiculous—how canIgo60miles anhour when Iwasn’t going anhour?” How would youanswer herifyouwere thecop? Ofcourse, ifyouwere really the cop,then nosubtleties areinvolved; itisverysimple: yousay,“Tell thattothe judge!” Butletussuppose thatwedonothave thatescape andwemake amore honest, intellectual attack ontheproblem, andtrytoexplain tothislady what wemean bytheideathatshewasgoing 60miles anhour. Justwhat dowemean? Wesay,“What wemean, lady, isthis: ifyoukept ongoing thesame wayasyou aregoing now, inthenexthour youwould go60miles.” Shecould say,“Well, myfootwasofftheaccelerator andthecarwasslowing down, soifIkeptongoing thatwayitwould notgo60miles.” Orconsider thefalling ballandsuppose we want toknow itsspeed atthetime three seconds iftheballkeptongoing theway itisgoing. What does thatmean—kept onaccelerating, going faster? No-kept ongoing withthesame velocity. Butthatiswhat wearetrying todefine! Forif theballkeeps ongoing thewayitisgoing, itwilljustkeep ongoing thewayitis going. Thus weneed todefine thevelocity better. What hastobekeptthesame? Theladycanalsoargue thisway: “IfIkept ongoing thewayI’mgoing forone more hour, Iwould runintothatwallattheendofthestreet!” Itisnotsoeasyto saywhat wemean. Many physicists think thatmeasurement istheonly definition ofanything. Obviously, then, weshould usetheinstrument that measures thespeed—the speedometer-—and say,“Look, lady, your speedometer reads 60.” Soshesays, “My speedometer isbroken anddidn’t read atall.” Does thatmean thecaris standing still? Webelieve that there issomething tomeasure before webuild thespeedometer. Only then canwesay,forexample, “The speedometer isn’t working right,” or“the speedometer isbroken.” That would beameaningless sentence ifthevelocity hadnomeaning independent ofthespeedometer. Sowe have inourminds, obviously, anidea that 1Sindependent ofthespeedometer, andthespeedometer ismeant onlytomeasure thisidea. Soletusseeifwecanget abetter definition oftheidea. Wesay,“Yes, ofcourse, before youwent anhour, youwould hitthatwall, butifyouwent onesecond, youwould go88feet; lady, youwere going 88feetpersecond, andifyoukept ongoing, thenext second it would be88feet,andthewalldown there isfarther away than that.” Shesays, “Yes, butthere’s nolawagainst going 88feetpersecoilidl There isonly alaw 8-3 against going 60miles anhour.” “But,” wereply, “it’sthesame thing.” Ifitis thesame thing, itshould notbenecessary togointothiscircumlocution about 88feetpersecond. Infact,thefalling ballcould notkeepgoing thesame way even onesecond because itwould bechanging speed, andweshall have todefine speed somehow. Now weseem tobegetting ontheright track; itgoes something likethis: Ifthelady kept ongoing foranother 1/1000 ofanhour, shewould go1/1000 of 60miles. Inother words, shedoes nothave tokeep ongoing forthewhole hour; thepoint isthatforamoment sheisgoing atthatspeed. Now what thatmeans isthatifshewent justalittle bitmore intime, theextra distance shegoes would bethesame asthatofacarthatgoesatasteady speed of60miles anhour. Per- haps theideaofthe88feetpersecond isright; weseehowfarshewent inthelast second, divide by88feet,andifitcomes out1thespeed was60miles anhour. Inother words, wecanfindthespeed inthisway: Weask,howfardowegoina very short time? Wedivide thatdistance bythetime, andthatgives thespeed. Butthetime should bemade asshort aspossible, theshorter thebetter, because some change could take place during thattime. Ifwetake thetime ofafalling body asanhour, theidea isridiculous. Ifwetake itasasecond, theresult is pretty good foracar,because there isnotmuch change inspeed, butnotfora falling body; soinorder togetthespeed more andmore accurately, weshould takeasmaller andsmaller time interval. What weshould doistakeamillionth ofasecond, anddivide thatdistance byamillionth ofasecond. Theresult gives thedistance persecond, which iswhat wemean bythevelocity, sowecandefine itthatway. Thatisasuccessful answer forthelady,orrather, thatisthedefinition thatwearegoing touse. Theforegoing definition involves anewidea, anideathatwasnotavailable totheGreeks inageneral form. That ideawastotake aninfinitesimal distance andthecorresponding infinitesimal time, form theratio, andwatch what happens tothatratio asthetimethatweusegetssmaller andsmaller andsmaller. Inother words, take alimit ofthedistance travelled divided bythetime required, asthe time taken getssmaller andsmaller, adinfinitum. This idea wasinvented by Newton andbyLeibnitz, independently, andisthebeginning ofanewbranch ofmathematics, called thedtflerential calculus. Calculus wasinvented inorder to describe motion, anditsfirstapplication wastotheproblem ofdefining what is meant bygoing “60miles anhour.” Letustrytodefine velocity alittle better. Suppose thatinashort time, e,thecarorother body goes ashort distance x;then thevelocity, v,isdefined as v=x/e, anapproximation thatbecomes better andbetter astheeistaken smaller and smaller. Ifamathematical expression isdesired, wecansaythatthevelocity equals thelimit astheeismade togosmaller andsmaller intheexpression x/e,or v=lim5- (33)e_.@e Wecannot dothesame thing with thelady inthecar,because thetable isin- complete. Weknow only where shewasatintervals ofoneminute; wecanget arough ideathatshewasgoing 5000 ft/minduring the7thminute, butwedonot know, atexactly themoment 7minutes, whether shehadbeen speeding upandthe speed was4900 ft/min atthebeginning ofthe6thminute, andisnow5100 ft/min, orsomething else, because wedonothave theexact details inbetween. Soonly ifthetable were completed with aninfinite number ofentries could wereally calculate thevelocity from such atable. Ontheother hand, when wehave acom- plete mathematical formula, asinthecase ofafalling body (Eq. 8.1), then itis possible tocalculate thevelocity, because wecancalculate theposition atanytime whatsoever. Letustake asanexample theproblem ofdetermining thevelocity ofthe falling ballattheparticular time 5seconds. Onewaytodothisistoseefrom 8-4 Table 8-2what itdidinthe5thsecond; itwent 400—256=144ft,soitisgoing 144ft/sec; however, thatiswrong, because thespeed ischanging; ontheaverage itis144ft/sec during thisinterval, buttheballisspeeding upandisreally going faster than 144ft/sec. Wewant tofindoutexactly howfast. Thetechnique in- volved inthisprocess isthefollowing: Weknow where theballwasat5sec. At5.1sec,thedistance thatithasgone alltogether is16(5.1)2 =416.16 ft(see Eq.8.1). At5secithadalready fallen 400ft;inthelasttenth ofasecond itfell 416.16 —400=16.16 ft.Since 16.16 ftin0.1secisthesame as161.6 ft/sec, thatisthespeed more orless,butitisnotexactly correct. Isthatthespeed at 5,orat5.1,orhalfway between at5.05sec,orwhen isthatthespeed‘? Never mind ——the problem wastofindthespeed at5seconds, andwedonothave exactly that; wehave todoabetter job. So,wetakeone-thousandth ofasecond more than 5sec,or5.001 sec,andcalculate thetotal fallas S=l6(5.0O1)2 =1e(25.010001) =400160016 rt. Inthelast0.001 sectheballfell0.160016 ft,andifwedivide thisnumber by0.001 secweobtain thespeed as160.016 ft/sec. That iscloser, very close, butitis stillnotexact. Itshould nowbeevident what wemust dotofindthespeed exactly. Toperform themathematics westate theproblem alittle more abstractly: to findthevelocity ataspecial time, to,which intheoriginal problem was5sec. Now thedistance atto,which wecallso,is16t%, or400ftinthiscase. Inorder tofindthevelocity, weask,“Atthetime to+(alittle bit), orto+e,where is thebody?” Thenewposition isl6(t0 -l—e)2=16t§+32t0e +1662. Soitis farther along than itwasbefore, because before itwasonly 16:3. This distance weshall callso+(alittle bitmore), orso—l—x(ifxistheextra bit). Now ifwe subtract thedistance attofrom thedistance atto+e,wegetx,theextra distance gone, asx=32tO-e+l6e2. Ourfirstapproximation tothevelocity is 7)=1:=321.,+16¢. (8.4) Thetruevelocity isthevalue ofthisratio, x/e,when ebecomes vanishingly small. Inother words, after forming theratio, wetake thelimit asegetssmaller and smaller, thatis,approaches 0.Theequation reduces to, 1)(at IIITIC to) =32tQ. Inourproblem, to=5sec,sothesolution isv=32X5=160ft/sec. Afew lines above, where wetook eas0.1and0.01secsuccessively, thevalue wegotfor 1)wasalittle more than this, butnow weseethattheactual velocity isprecisely 160ft/sec. 8-3Speed asaderivative Theprocedure wehavejustcarried outisperformed sooften inmathematics thatforconvenience special notations have been assigned toourquantities eandx. Inthisnotation, theeused above becomes Atandxbecomes As.This Atmeans “anextra bitoft,”andcarries animplication thatitcanbemade smaller. The prefix Aisnotamultiplier, anymore than sin0means s-i-n-0—it simply defines atime increment, andreminds usofitsspecial character. Ashasan analogous meaning forthedistance s.Since Aisnotafactor, itcannot becan- celled intheratio As/At togives/t,anymore than theratio sin6/sin 20canbe reduced to1/2bycancellation. Inthisnotation, velocity isequal tothelimit of As/At when Atgetssmaller, or .As=l—- 8.5v Al2l0 Al () Thisisreally thesame asourprevious expression (8.3)witheandx,butithasthe advantage ofshowing thatsomething ischanging, anditkeeps track ofwhat is changing. 8-5 Incidentally, toagood approximation wehave another law,which saysthat thechange indistance ofamoving point isthevelocity times thetime interval, orAs=vAt.This statement istrueonly ifthevelocity isnotchanging during thattime interval, andthiscondition istrueonly inthelimit asAtgoes to0. Physicists liketowrite itds=vdt,because byalttheymean Atincircumstances inwhich itisverysmall; withthisunderstanding, theexpression isvalid toaclose approximation. IfAtistoolong, thevelocity might change during theinterval, andtheapproximation would become lessaccurate. Foratime dt,approaching zero, ds=vdtprecisely. Inthisnotation wecanwrite (8.5) as 0-12:‘.§= Thequantity ds/dt which wefound above iscalled the“derivative ofswith respect tot”(thislanguage helps tokeep track ofwhat waschanged), andthecom- plicated process offinding itiscalled finding aderivative, ordifferentiating. Theds’sanddt’swhich appear separately arecalled dtflerentials. Tofamiliarize youwith thewords, wesaywefound thederivative ofthefunction 16t2, orthe derivative (with respect tot)of1622is32t. When wegetused tothewords, the ideas aremore easily understood. Forpractice, letusfindthederivative ofamore complicated function. Weshall consider theformula s=Ara+Bt+C,which might describe themotion ofapoint. Theletters A,B,andCrepresent constant numbers, asinthefamiliar general form ofaquadratic equation. Starting from theformula forthemotion, wewish tofindthevelocity atanytime. Tofind thevelocity inthemore elegant manner, wechange ttot+Atandnote that sisthenchanged tos—l—some As;then wefindtheAsinterms ofAt.That isto say, s+As=A(t+At)3—l—B(t+At)+ c =At3+Bt+c+3At2At +BAt+3At(At)2 +A(At)3, butsince s=At?’ +Bl+C, wefindthat As=3At2At+BA:+3At(At)2 +A(At)3. Butwedonotwant As-—we want Asdivided byAt.Wedivide thepreceding equa- tionbyAt,getting §=3,412+B+3At(At) +A(At)2. Table 8-3. AShort Table ofDerivatives s,u,v,warearbitrary functions oft;a,b,c,andnarearbitrary constants Function Derivative d _s=t" i=nt"1 £_ Qdt_cdt ds du dv dw-t=u+v+w+"' dt=Z+E+-t7+"' E s=c dt=0 _<=ic Q- 2% 11:12 ams_uvw"' dt_s<udt+vdt+wdt+ )S=CIl 8-6 AsAtgoes toward 0thelimit ofAs/At isds/dt andisequal to dS _ 2 Z1?-3At +B. Thisisthefundamental process ofcalculus, differentiating functions. Theprocess iseven more simple than itappears. Observe thatwhen these expansions con- tainanyterm withasquare oracube oranyhigher power ofAt,suchterms maybe dropped atonce, since theywillgoto0when thelimit istaken. After alittle prac- ticetheprocess getseasier because oneknows what toleave out. There aremany rules orformulas fordifferentiating various types offunctions. These canbe memorized. orcanbefound intables. Ashort listisfound inTable 8-3. 8-4Distance asanintegral Now wehave todiscuss theinverse problem. Suppose thatinstead ofatable of distances, wehave atable ofspeeds atdifferent times, starting from zero. Forthe falling ball, such speeds andtimes areshown inTable 8-4. Asimilar table could beconstructed forthevelocity ofthecar,byrecording thespeedometer reading every minute orhalf-minute. Ifweknow howfastthecarisgoing atanytime, can wedetermine howfaritgoes? This problem isjusttheinverse oftheonesolved above; wearegiven thevelocity andasked tofindthedistance. How canwefind thedistance ifweknow thespeed? Ifthespeed ofthecarisnotconstant, andthe ladygoes sixty miles anhour foramoment, then slows down, speeds up,andso on,howcanwedetermine howfarshehasgone? That iseasy. Weusethesame idea, andexpress thedistance interms ofinfinitesimals. Letussay,“Inthefirst second herspeed wassuch andsuch, andfrom theformula As=vAtwecan calculate how farthecarwent thefirstsecond atthatspeed.” Now inthenext second herspeed isnearly thesame, butslightly different; wecancalculate how farshewent inthenextsecond bytaking thenewspeed times thetime. Wepro- ceedsimilarly foreach second, totheendoftherun. Wenow have anumber oflittle distances, andthetotal distance willbethesumofallthese little pieces. That is,thedistance willbethesum ofthevelocities times thetimes, ors= XvAt,where theGreek letter Z(sigma) isused todenote addition. Tobemore precise, itisthesum ofthevelocity atacertain time, letussaythei-thtime, multiplied byAt. s=Zv(t,)At. (8.6) Theruleforthetimes isthatt,+1 =t,+At.However, thedistance weobtain bythismethod willnotbecorrect, because thevelocity changes during thetime interval At.Ifwetakethetimes short enough, thesumisprecise, sowetakethem smaller andsmaller until weobtain thedesired accuracy. Thetruesis s=lim Zv(t,) At. (8.7)At->0 1. Themathematicians have invented asymbol forthislimit, analogous tothesymbol forthedifferential. TheAturns intoadtoremind usthatthetime isassmall as itcanbe;thevelocity isthen called vatthetime t,andtheaddition iswritten asasumwith agreat “s,”j(from theLatin summa), which hasbecome distorted andisnow unfortunately justcalled anintegral sign. Thus wewrite S=f0(1)at. (8.8) Thisprocess ofadding allthese terms together iscalled integration, anditisthe opposite process todifferentiation. Thederivative ofthisintegral isv,soone operator (d)undoes theother (j). Onecangetformulas forintegrals bytaking theformulas forderivatives andrunning them backwards, because they arere- lated toeach other inversely. Thus onecanwork outhisown table ofintegrals bydifferentiating allsorts offunctions. Forevery formula with adifferential, wegetanintegral formula ifweturnitaround. 8-7Table 8-4 Velocity ofaFalling Ball t(sec) v(ft/sec) -l>o.>r~;>-O .-Ix)O0O32 64 96 Every function canbedifferentiated analytically, i.e.,theprocess canbecarried outalgebraically, andleads toadefinite function. Butitisnotpossible inasimple manner towrite ananalytical value foranyintegral atwill. Youcancalculate it, forinstance, bydoing theabove sum,andthendoing itagain withafinerinterval Atandagain with afiner interval until youhave itnearly right. Ingeneral, given some particular function, itisnotpossible tofind, analytically, what theintegral is.Onemayalways trytofindafunction which, when differentiated, gives some desired function; butonemaynotfindit,anditmaynotexist, inthesense ofbeing expressible interms offunctions thathave already been given names. 8-5Acceleration Thenext stepindeveloping theequations ofmotion istointroduce another idea which goes beyond theconcept ofvelocity tothat ofchange ofvelocity, andwenowask,“How does thevelocity change?” Inprevious chapters wehave discussed cases inwhich forces produce changes invelocity. You mayhave heard with great excitement about some carthatcangetfrom restto60miles anhour intenseconds flat. From such aperformance wecanseehow fastthespeed changes, butonly ontheaverage. What weshall nowdiscuss isthenextlevel of complexity, which ishow fastthevelocity ischanging. Inother words, byhow many feetpersecond does thevelocity change inasecond, thatis,howmany feet persecond, persecond? Wepreviously derived theformula forthevelocity of afalling body asv=32!,which ischarted inTable 8-4,andnow wewant to findouthow much thevelocity changes persecond ;thisquantity iscalled the acceleration. Acceleration isdefined asthetime rateofchange ofvelocity. From the preceding discussion weknow enough already towrite theacceleration asthe derivative dz)/dt, inthesame waythatthevelocity isthederivative ofthedistance. Ifwenowdifferentiate theformula v=321weobtain, forafalling body, a=5%=32. (8.9) [Todifferentiate theterm 321wecanutilize theresult obtained inaprevious problem, where wefound thatthederivative ofBtissimply B(aconstant). So byletting B=32,wehave atonce thatthederivative of32:is32.] This means thatthevelocity ofafalling body ischanging by32feetpersecond, persecond always. Wealsoseefrom Table 8-4thatthevelocity increases by32ft/secin each second. Thisisaverysimple case, foraccelerations areusually notconstant. Thereason theacceleration isconstant hereisthattheforce onthefalling body isconstant, andNewton’s lawsaysthattheacceleration isproportional totheforce. Asafurther example, letusfindtheacceleration intheproblem wehave already solved forthevelocity. Starting with s=At“+Bt+C weobtained, forv=ds/dt, v=3At2 +B. Since acceleration isthederivative ofthevelocity withrespect tothetime, weneed todifferentiate thelastexpression above. Recall therulethatthederivative ofthe twoterms ontheright equals thesumofthederivatives oftheindividual terms. Todifferentiate thefirstofthese terms, instead ofgoing through thefundamental process again wenote thatwehave already differentiated aquadratic term when wedifferentiated l6t2, andtheefiect wastodouble thenumerical coefiicient and change the12tot;letusassume thatthesame thing willhappen thistime, andyou cancheck theresult yourself. Thederivative of3Az2 willthen be6A1. Next we differentiate B,aconstant term; butbyarulestated previously, thederivative of Biszero; hence thisterm contributes nothing totheacceleration. Thefinal result, therefore, isa=dv/dt =6At. 8-8 Forreference, westate twovery useful formulas, which canbeobtained by integration. Ifabody starts from restandmoves with aconstant acceleration, g,itsvelocity vatanytime tisgiven by v=gt. Thedistance itcovers inthesame time is s=%gt2. Various mathematical notations areused inwriting derivatives. Since velocity isds/dt andacceleration isthetime derivative ofthevelocity, wecanalsowrite dd dz<1=7’ =Hg. (8.10) which arecommon ways ofwriting asecond derivative. Wehave another lawthatthevelocity isequal totheintegral oftheaccelera- tion. Thisisjusttheopposite ofa=dz»/dt; wehave already seenthatdistance is theintegral ofthevelocity, sodistance canbefound bytwice integrating theac- celeration. Intheforegoing discussion themotion wasinonly onedimension, andspace permits only abrief discussion ofmotion inthree dimensions. Consider aparticle Pwhich moves inthree dimensions inanymanner whatsoever. Atthebeginning ofthischapter, weopened ourdiscussion oftheone-dimensional caseofamoving carbyobserving thedistance ofthecarfrom itsstarting point atvarious times. Wethen discussed velocity interms ofchanges ofthese distances with time, and acceleration interms ofchanges invelocity. Wecantreat three-dimensional motion analogously. Itwillbesimpler toillustrate themotion onatwo-dimensional diagram, andthen extend theideas tothree dimensions. Weestablish apairof axesatright angles toeach other, anddetermine theposition oftheparticle atany moment bymeasuring howfaritisfrom each ofthetwoaxes. Thus each position isgiven interms ofanx-distance anday-distance, andthemotion canbedescribed byconstructing atable inwhich both these distances aregiven asfunctions oftime. (Extension ofthisprocess tothree dimensions requires only another axis, atright angles tothefirsttwo, andmeasuring athird distance, thez-distance. Thedis- tances arenow measured from coordinate planes instead oflines.) Having con- structed atable with x-andy-distances, how canwedetermine thevelocity? Wefirstfindthecomponents ofvelocity ineach direction. Thehorizontal partof thevelocity, orx-component, isthederivative ofthex-distance with respect to thetime, or 1»,=dx/dt. (8.11) Similarly, thevertical partofthevelocity, ory-component, is 1),,=dy/dz. (8.12) Inthethird dimension, v,=dz/dt. (8.13) Now, given thecomponents ofvelocity, howcanwefindthevelocity along the actual path ofmotion? Inthetwo-dimensional ease, consider twosuccessive positions oftheparticle, separated byashort distance Asandashort time in- terval t2—11=At.Inthetime Attheparticle moves horizontally adistance Ax~21,At,andvertically adistance Ay~vyAt. (The symbol “~” isread “isapproximately") Theactual distance moved isapproximately AS~\/(A102 +(AJ/)2. (3-14) asshown inFig.8-3. Theapproximate velocity during thisinterval canbeobtained bydividing byAtandbyletting Atgoto0,asatthebeginning ofthechapter. 8-9Y As#=_ /(Ax): +(Ayl2 AyflVyAt '"'1, 1, ' Axz VxAt X Fig. 8-3. Description ofthemotion of0body intwo dimensions and the computation ofitsvelocity. Y x \ Fig.8-4. Theparabola described by a‘l’O||ing body with aninitial horizontal velocity.Wethengetthevelocity as v=if};=\/(dx/a't)2 +(dy/dfl) =\/83+ 0% (8.15) Forthree dimensions theresult is 11=\/vi +113+vi. (8.16) Inthesame wayaswedefined velocities, wecandefine accelerations: wehave anx-component ofacceleration ax,which isthederivative of0,,thex-component ofthevelocity (that is,a,=d2x/dt2, thesecond derivative ofxwith respect to t),andsoon. Letusconsider oneniceexample ofcompound motion inaplane. Weshall takeamotion inwhich aballmoves horizontally with aconstant velocity u,and atthesame time goes vertically downward with aconstant acceleration —g; what isthemotion? Wecansaydx/dt =11,,=u.Since thevelocity 11,,isconstant, x=ut, (8.17) andsince thedownward acceleration —gisconstant, thedistance ytheobject fallscanbewritten as y=—%gt2. (8.18) What isthecurve ofitspath, i.e.,what istherelation between yandx?Wecan eliminate tfrom Eq.(8.18), since t=x/u. When wemake thissubstitution we findthat y=-2;-‘L;X2. (8.19) This relation between yandxmay beconsidered astheequation ofthepath of themoving ball. When thisequation isplotted weobtain acurve thatiscalled a parabola; anyfreely falling body thatisshot outinanydirection willtravel in aparabola, asshown inFig.8-4. 8—l0 9 Newton ’sLaws ofDynamics 9-1Momentum andforce Thediscovery ofthelaws ofdynamics, orthelaws ofmotion, wasadramatic moment inthehistory ofscience. Before Newton’s time, themotions ofthings liketheplanets were amystery, butafter Newton there wascomplete under- standing. Even theslight deviations from Kepler’s laws, duetotheperturbations oftheplanets, were computable. Themotions ofpendulums, oscillators with springs and weights inthem, and soon,could allbeanalyzed completely after Newton’s laws were enunciated. Soitiswith thischapter: before thischapter we could notcalculate how amass onaspring would move; much lesscould we calculate theperturbations ontheplanet Uranus duetoJupiter andSaturn. After thischapter wewillbeabletocompute notonlythemotion oftheoscillating mass, butalsotheperturbations ontheplanet Uranus produced byJupiter andSaturn! Galileo made agreat advance intheunderstanding ofmotion when he discovered theprinciple ofinertia: ifanobject isleftalone, isnotdisturbed, it continues tomove with aconstant velocity inastraight lineifitwasoriginally moving, oritcontinues tostand stillifitwasjuststanding still. Ofcourse this never appears tobethecaseinnature, forifweslide ablock across atable itstops, butthatisbecause itisnotlefttoitself—it isrubbing against thetable. Itrequired acertain imagination tofindtheright rule, andthatimagination wassupplied byGalileo. Ofcourse, thenextthing which isneeded isaruleforfinding howanobject changes itsspeed ifsomething isaffecting it.That isthecontribution ofNewton. Newton wrote down three laws: TheFirst Law wasamere restatement ofthe Galilean principle ofinertia justdescribed. TheSecond Law gave aspecific way ofdetermining how thevelocity changes under different influences called forces. TheThird Law describes theforces tosome extent, andweshall discuss thatat another time. Here weshall discuss only theSecond Law, which asserts thatthe motion ofanobject ischanged byforces inthisway: thetime-rate-of-change ofa quantity called momentum ISproportional totheforce. Weshall state thismathe- matically shortly, butletusfirstexplain theidea. Momentum isnotthesame asvelocity. Alotofwords areused inphysics, and'they allhave precise meanings inphysics, although they may nothave such precise meanings ineveryday language. Momentum isanexample, andwemust define itprecisely. Ifweexert acertain push with ourarms onanOb_]€Ct thatis light, itmoves easily; ifwepushJUSIashard onanother object thatismuch heavier intheusual sense, then itmoves much lessrapidly. Actually, wemust change the words from “light” and“heavy” tolessmassive andmore massive, because there isadifference tobeunderstood between theweight ofanobject anditsinertia. (How hard itistogetitgoing isonething, andhowmuch itweighs issomething else.) Weight andinertia areproportional, andontheearth’s surface areoften taken tobenumerically equal, which causes acertain confusion tothestudent. OnMars, weights would bedifferent buttheamount offorce needed toovercome inertia would bethesame. Weusetheterm mass asaquantitative measure ofinertia, andwemay measure mass, forexample, byswinging anobject inacircle atacertain speed and measuring howmuch force weneed tokeep itinthecircle. Inthiswaywefinda certain quantity ofmass forevery Ob_]€Ci. Now themomentum ofanobject isa product oftwoparts: itsmass anditsvelocity. Thus Newton’s Second Law may 9-19-1Momentum andforce 9-2Speed andvelocity 9-3Components ofvelocity, acceleration, andforce 9-4What istheforce? 9-5Meaning ofthedynamical equations 9-6Numerical solution ofthe equations 9-7Planetary motions X Fig. object.F5-"X_‘3}\U! \\.______‘:.lZ ____ //FAZ it" Y 9-1. Asmall displacement ofanbewritten mathematically thisway: F=%(mv). (9.1) Now there areseveral points tobeconsidered. Inwriting down anylawsuch as this, weusemany intuitive ideas, implications, andassumptions which areat firstcombined approximately intoour“law.” Later wemayhave tocome back andstudy ingreater detail exactly what each term means, butifwetrytodothis toosoon weshall getconfused. Thus atthebeginning wetakeseveral things for granted. First, thatthemass ofanobject isconstant; itisn’treally, butweshall start outwith theNewtonian approximation thatmass isconstant, thesame all thetime, andthat, further, when weputtwoobjects together, their masses add. These ideas were ofcourse implied byNewton when hewrote hisequation, for otherwise itismeaningless. Forexample, suppose themass varied inversely asthe velocity; thenthemomentum would never change inanycircumstance, sothelaw means nothing unless youknow how themass changes with velocity. Atfirst wesay,itdoesnotchange. Then there aresome implications concerning force. Asarough approximation wethink offorce asakind ofpush orpullthatwemake with ourmuscles, but wecandefine itmore accurately nowthatwehave thislawofmotion. Themost important thing torealize isthatthisrelationship involves notonly changes in themagnitude ofthemomentum orofthevelocity butalsointheir direction. Ifthemass isconstant, thenEq.(9.1) canalsobewritten as F=mg =ma. (9.2) Theacceleration aistherateofchange ofthevelocity, andNewton’s Second Law saysmore than thattheeffect ofagiven force varies inversely asthemass; itsaysalsothatthedirection ofthechange inthevelocity andthedirection ofthe force arethesame. Thus wemust understand thatachange inavelocity, oran acceleration, hasawider meaning than incommon language: Thevelocity ofa moving object canchange byitsspeeding up,slowing down (when itslows down, wesayitaccelerates with anegative acceleration), orchanging itsdirection of motion. Anacceleration atright angles tothevelocity wasdiscussed inChapter 7. There wesawthatanobject moving inacircle ofradius Rwith acertain speed v along thecircle fallsaway from astraightline path byadistance equal to%(v2/R)t2 iftisverysmall. Thus theformula foracceleration atright angles tothemotion is a=112/R, (9.3) andaforce atright angles tothevelocity willcause anobject tomove inacurved path whose radius ofcurvature canbefound bydividing theforce bythemass to gettheacceleration, andthenusing (9.3). 9-2Speed andvelocity Inorder tomake ourlanguage more precise, weshall make onefurther definition inouruseofthewords speed andvelocity. Ordinarily wethink ofspeed andvelocity asbeing thesame, andinordinary language theyarethesame. Butin physics wehave taken advantage ofthefactthatthere aretwowords andhave chosen tousethem todistinguish twoideas. Wecarefully distinguish velocity, which hasboth magnitude anddirection, from speed, which wechoose tomean themagnitude ofthevelocity, butwhich does notinclude thedirection. Wecan formulate thismore precisely bydescribing howthex-,y-,andz-coordinates of anobject change with time. Suppose, forexample, thatatacertain instant an object ismoving asshown inFig.9-l. Inagiven small interval oftime Atit willmove acertain distance Axinthex-direction, Ayinthey-direction, andAzin thez-direction. Thetotal effect ofthese three coordinate changes isadisplacement Asalong thediagonal ofaparallelepiped whose sides areAx,Ay,andAz.Interms 9-2 ofthevelocity, thedisplacement Axisthex-component ofthevelocity times At, andsimilarly forAyandA2: Ax=UIAr. Ay=vyAt, Az=v,At. (9.4) 9-3Components ofvelocity, acceleration, andforce InEq.(9.4) wehaveresolved thevelocity intocomponents bytelling howfastthe object ismoving inthex-direction, they-direction, andthez-direction. The velocity iscompletely specified, both astomagnitude anddirection, ifwegivethe numerical values ofitsthree rectangular components: ii,=dx/dt, vy=dy/dt, vz=dz/dt. (9.5) Ontheother hand, thespeed oftheobject is _ ._\/ds/dt -|v|- vi+vi+vi. (9.6) Next, suppose that, because oftheaction ofaforce, thevelocity changes to some other direction andadifferent magnitude, asshown inFig.9-2. Wecan analyze thisapparently complex situation rather simply ifweevaluate thechanges inthex-,y-,andz-components ofvelocity. Thechange inthecomponent ofthe velocity inthex-direction inatimeAtisAv,=axAt,where a,iswhat wecallthe x-component oftheacceleration. Similarly, weseethatAvy=ayAtandAv,= a,At.Inthese terms, weseethatNewton’s Second Law, insaying thattheforce isinthesame direction astheacceleration, isreally three laws, inthesense that thecomponent oftheforce inthex-,y-,or2-direction isequal tothemass times therateofchange ofthecorresponding component ofvelocity: F,=m(dv,,/dt) =m(d2x/dt2) =ma,,, F,,=m(dv,,/dt) =m(d2y/dt2) =may, (9.7) F2=m(dv,/dt) =m(d2z/dig) =maz. Just asthevelocity andacceleration have been resolved into components by projecting alinesegment representing thequantity anditsdirection onto three coordinate axes, so,inthesame way, aforce inagiven direction isrepresented bycertain components inthex-,y-,andz-directions: F,= F,= F,=Fcos (x,F), Fcos (y,F), (9.8) Fcos (z,F), where Fisthemagnitude oftheforce and(x,F)represents theangle between the x-axis andthedirection ofF,etc. Newton’s Second Lawisgiven incomplete form inEq.(9.7). Ifweknow the forces onanobject andresolve them intox-,y-,andz-components, then wecan findthemotion oftheobject from these equations. Letusconsider asimple example. Suppose there arenoforces inthey-andz-directions, theonly force being inthex-direction. sayvertically. Equation (9.7) tellsusthatthere would be changes inthevelocity inthevertical direction, butnochanges inthehorizontal direction. This wasdemonstrated with aspecial apparatus inChapter 7(see Fig.7-3). Afalling body moves horizontally without anychange inhorizontal motion, while itmoves vertically thesame wayasitwould move ifthehorizontal motion were zero. Inother words, motions inthex-,y-,andz-directions are independent iftheforces arenotconnected. 9-4What istheforce? Inorder touseNewton’s laws, wehave tohave some formula fortheforce; these lawssaypayattention totheforces. Ifanobject isaccelerating, some agency isatwork; findit.Ourprogram forthefuture ofdynamics must betofindthe 9-3X Fig.9-2. Achange invelocity which both themagnitude and direction change.l(;—\‘-—-T\\\1‘“T*“I1'-t-"I-'-'-ts_§y111:1II__~... l<;-1--F EQUILIBRIUMXPOSITION ma Fig.9-3. Amass onaspring.lawsfortheforce. Newton himself went ontogivesome examples. Inthecase ofgravity hegave aspecific formula fortheforce. Inthecaseofother forces he gave some partoftheinformation inhisThird Law, which wewillstudy inthe nextchapter, having todowith theequality ofaction andreaction. Extending ourprevious example, what aretheforces onobjects near the earth’s surface? Near theearth’s surface, theforce inthevertical direction due togravity isproportional tothemass oftheobject andisnearly independent of height forheights small compared withtheearth’s radius R:F=GmM/R2 =mg, where g=GM/R2iscalled theacceleration ofgravity. Thus thelawofgravity tellsusthatweight isproportional tomass; theforce isinthevertical direction andisthemass times g.Again wefindthatthemotion inthehorizontal direction isatconstant velocity. Theinteresting motion isinthevertical direction, and Newton’s Second Law tellsus mg=m(d2x/dtz). (9.9) Cancelling them’s,wefindthattheacceleration inthex-direction isconstant and equal tog.This isofcourse thewellknown lawoffreefallunder gravity, which leads totheequations vz=v0+gt, X=X0+Ugl +%gt2. Asanother example, letussuppose thatwehave been abletobuild agadget (Fig.9-3)which applies aforce proportional tothedistance anddirected oppositely —aspring. Ifweforget about gravity, which isofcourse balanced outbythe initial stretch ofthespring, andtalkonly about excess forces, weseethatifwe pullthemass down, thespring pulls up,while ifwepush itupthespring pulls down. This machine hasbeen designed carefully sothattheforce isgreater, the more wepullitup,inexact proportion tothedisplacement from thebalanced condition, andtheforce upward issimilarly proportional tohowfarwepulldown. Ifwewatch thedynamics ofthismachine, weseearather beautiful motion—up, down, up,down, ...Thequestion is,willNewton’s equations correctly describe thismotion? Letusseewhether wecanexactly calculate howitmoves with this periodic oscillation, byapplying Newton’s law(9.7). Inthepresent instance, theequation is —kx =m(dv,,/dt). (9.11) Here wehave asituation where thevelocity inthex-direction changes atarate proportional tox.Nothing willbegained byretaining numerous constants, so weshall imagine either thatthescale oftime haschanged orthatthere isan accident intheunits, sothatwehappen tohave k/m =1.Thus weshall tryto solve theequation dv,/dz =—x. (9.12) Toproceed, wemust know what v,is,butofcourse weknow thatthevelocity is therateofchange oftheposition. 9-5Meaning ofthedynamical equations Now letustrytoanalyze justwhat Eq.(9.12) means. Suppose thatata given time1theobject hasacertain velocity v,,andposition x.What isthevelocity andwhat istheposition ataslightly later time t+e?Ifwecananswer this question ourproblem issolved, forthenwecanstart withthegiven condition and compute howitchanges forthefirstinstant, thenextinstant, thenextinstant, and soon,andinthiswaywegradually evolve themotion. Tobespecific, letussuppose thatatthetime t=Owearegiven thatx=1andv,=0.Why does theobject move atall?Because there isaforce onitwhen itisatanyposition except x=0. Ifx>0,thatforce isupward. Therefore thevelocity which iszero starts to change, because ofthelawofmotion. Once itstarts tobuild upsome velocity theobject starts tomove up,andsoon.Now atanytime t,ifeisvery small, 9-4 wemayexpress theposition attime t+einterms oftheposition attime tand thevelocity attime ttoaverygood approximation as x(t+e)=x(t)+611,0). (9.13) Thesmaller thee,themore accurate thisexpression is,butitisstillusefully accurate even ifeisnotvanishingly small. Now what about thevelocity? Inorder toget thevelocity later, thevelocity atthetime t+e.weneed toknow howthevelocity changes, theacceleration. And howarewegoing tofindtheacceleration? That iswhere thelawofdynamics comes in.Thelawofdynamics tellsuswhat the acceleration is.Itsaystheacceleration is-x. v,,(t+e)=v,(t) +ea,,(t) (9.14) =v,(t) —ex(t). (9.15) Equation (9.14) ismerely kinematics; itsaysthatavelocity changes because of thepresence ofacceleration. ButEq.(9.15) isdynamics, because itrelates the acceleration totheforce; itsays thatatthisparticular time forthisparticular problem, youcanreplace theacceleration by—x(t). Therefore, ifweknow both thexandvatagiven time, weknow theacceleration, which tellsusthenew velocity, andweknow thenewposition—this ishow themachinery works. The velocity changes alittle bitbecause oftheforce, andtheposition changes alittle bitbecause ofthevelocity. 9-6 Numerical solution oftheequations Now letusreally solve theproblem. Suppose thatwetake e=0.100 sec. After wedoallthework ifwefindthatthisisnotsmall enough wemayhave to goback anddoitagain withe=0.010 sec.Starting with ourinitial value x(0) = 1.00, what isx(0.l)? Itistheoldposition x(0)plusthevelocity (which iszero) times 0.10 sec. Thus x(0.1) isstill1.00because ithasnotyetstarted tomove. Butthenewvelocity at0.10secwillbetheoldvelocity v(0)=0plusetimes the acceleration. Theacceleration is—x(0) =-1.00. Thus v(0.l) =0.00 —0.10 X1.00 =-0.10. Now at0.20sec x(0.2) =x(O.l) —l-ev(O.l) =1.00—0.10 X0.10 =0.99 and v(0.2) =v(0.l) +ea(0.l) =—0.10 —0.10 X1.00 =-0.20. And so,onandonandon,wecancalculate therestofthemotion, andthatis iustwhat weshall do.However, forpractical purposes there aresome little tricks bywhich wecanincrease theaccuracy. Ifwecontinued thiscalculation aswehave started it,wewould findthemotion only rather crudely because e=0.100 sec israther crude, andwewould have togotoavery small interval, saye=0.01. Then togothrough areasonable total time interval would takealotofcycles of computation. Soweshall organize thework inawaythatwillincrease thepre- cision ofourcalculations, using thesame coarse interval e=0.10sec. This can bedone ifwemake asubtle improvement inthetechnique oftheanalysis. Notice thatthenewposition istheoldposition plusthetime interval etimes thevelocity. Butthevelocity when? Thevelocity atthebeginning ofthetime interval isonevelocity andthevelocity attheendofthetime interval isanother velocity. Ourimprovement istousethevelocity halfway between. Ifweknow thespeed now, butthespeed ischanging, then wearenotgoing togettheright answer bygoing atthesame speed asnow. Weshould usesome speed between the“now” speed andthe“then” speed attheendoftheinterval. The same considerations also apply tothevelocity: tocompute thevelocity changes, we 9-5 Table 9-1 Solution ofdv,/dt =—x Interval: e=0.10sec I X U1 ax 0.0 1.000 0.1 0.995 0.2 0.980 0.3 0.955 0.4 0.921 0.5 0.877 7.? 0.825 0.7 0.764 0.8 0.696 0.9 0.621 1.0 0.540 11 0.453 1.2 0.362 1.3 0.267 1.4 0.169 1.5 0.070 1.6 -0.0300.000 -0.050 -0.150 -0.248 -0.343 -0.435 —-0.523 - -0.605 -0.682 -0.751 -0.814 --0.868 — -0.913 -0.949 -0.976 -0.993 --1.000--1.000 -0.995 -0.980 -0.955 -0.921 -0.877 -0.825 -0.764 -0.696 -0.621 -0.540 -0.453 -0.362 -0.267 -0.169 -0.070 -l-0.030 ii IO 05 OO5 IO Fig. 9-4. Graph ofthemotion ofa mass onaspring.L5 tlseclshould usetheacceleration midway between thetwotimes atwhich thevelocity istobefound. Thus theequations thatweshall actually usewillbesomething likethis:theposition laterisequal totheposition before plusetimes thevelocity atthetimeinthemiddle oftheinterval. Similarly, thevelocity atthishalfway point isthevelocity atatime ebefore (which isinthemiddle oftheprevious interval) plusetimes theacceleration atthetime i.That is,weusetheequations x(t+e)=x(t)+ev(t—l—e/2), v(t+e/2) =v(t-e/2) +ea(t), (9.16) a(t)=-x(t). There remains only oneslight problem: what isv(e/2)? Atthestart, wearegiven v(0), notv(—e/2). Togetourcalculation" started, weshall useaspecial equation, namely, v(e/2) =v(0) -l-(e/2)a(0). Now weareready tocarry through ourcalculation. Forconvenience, we mayarrange thework intheform ofatable, withcolumns forthetime, theposition, thevelocity, andtheacceleration, andthein-between lines forthevelocity, as shown inTable 9-1. Such atable is,ofcourse, justaconvenient wayofrepresenting thenumerical values obtained from thesetofequations (9.16), andinfactthe equations themselves need never bewritten. Wejustfillinthevarious spaces in thetable onebyone. This table now gives usavery good ideaofthemotion: itstarts from rest, firstpicks upalittle upward (negative) velocity anditloses some ofitsdistance. Theacceleration isthen alittle bitlessbutitisstillgaining speed. Butasitgoes onitgains speed more andmore slowly, until asitpasses x=0atabout t=1.50secwecanconfidently predict thatitwillkeep going, butnow itwillbeontheother side; theposition xwillbecome negative, theac- celeration therefore positive. Thus thespeed decreases. Itisinteresting tocompare these numbers with thefunction x=cost,which isdone inFig.9-4. Theagree- ment iswithin thethree significant figure accuracy ofourcalculation! Weshall seelater thatx=cost istheexact mathematical solution ofourequation of motion, butitisanimpressive illustration ofthepower ofnumerical analysis that such aneasycalculation should givesuch precise results. 9-7Planetary motions Theabove analysis isvery niceforthemotion ofanoscillating spring, but canweanalyze themotion ofaplanet around thesun? Letusseewhether we canarrive atanapproximation toanellipse fortheorbit. Weshall suppose that thesunisinfinitely heavy, inthesense thatweshall notinclude itsmotion. Suppose aplanet starts atacertain place andismoving with acertain velocity; itgoes around thesuninsome curve, andweshall trytoanalyze, byNewton’s laws of motion andhislawofgravitation, what thecurve is.How? Atagiven moment itisatsome position inspace. Iftheradial distance from thesuntothisposition iscalled r,thenweknow thatthere isaforce directed inward which, according to thelawofgravity, isequal toaconstant times theproduct ofthesun’s mass and theplanet's mass divided bythesquare ofthedistance. Toanalyze thisfurther wemust find outwhat acceleration willbeproduced bythisforce. Weshall need thecomponents oftheacceleration along twodirections, which wecallxandy. Thus ifwespecify theposition oftheplanet atagiven moment bygiving xandy (weshall suppose thatzisalways zero because there isnoforce inthez-direction and, ifthere isnoinitial velocity vz,there will benothing tomake 2other than zero), theforce isdirected along thelinejoining theplanet tothesun,asshown inFig.9-5. From thisfigure weseethat thehorizontal component oftheforce isrelated tothecomplete force inthesame manner asthehorizontal distance xistothe complete hypotenuse r,because thetwotriangles aresimilar. Also, ifx1Spositive, F,isnegative. That is,F1/IFI =-X/r, orF,=—|Flx/r =—GMmx/r3. Now weusethedynamical lawtofindthatthisforce component isequal tothemass of 9-6 theplanet times therateofchange ofitsvelocity inthex-direction. Thus wefind thefollowing laws: m(dv,/dt) =—GMmx/r3, m(dv,,/dr) =—GMmy/r3, (9_17) r=\/x2 +y2. This, then, isthesetofequations wemust solve. Again, inorder tosimplify the numerical work, weshall suppose thattheunitoftime, orthemass ofthesun,has been soadjusted (orluckiswithus)thatGME1.Forourspecific example we shall suppose thattheinitial position oftheplanet isatx=0.500 andy=0.000, andthatthevelocity isallinthey-direction atthestart, andisofmagnitude 1.6300. Now how dowemake thecalculation? Weagain make atable with columns forthetime, thex-position, thex-velocity 21,,andthex-acceleration a,; then, separated byadouble line,three columns forposition, velocity, andaccelera- tioninthey-direction. Inorder togettheaccelerations wearegoing toneed Eq.(9.17); ittellsusthattheacceleration inthex-direction is—x/r3, andthe acceleration inthey-direction is—y/rs, andthatristhesquare rootofx2+y2. Thus, given xandy,wemust doalittle calculating ontheside, taking thesquare rootofthesumofthesquares tofindrandthen, togetready tocalculate thetwo accelerations, itisuseful alsotoevaluate 1/r3. This work canbedone rather easily byusing atable ofsquares, cubes, andreciprocals: then weneed only multiply xby1/r3,which wedoonaslide rule. Ourcalculation thus proceeds bythefollowing steps, using time intervals e=0.100: Initial values att=0: x(0)=0.500 y(0)=0.000 6,10)=0.000 v,,(0)=+1630 From these wefind: r(0)=0.500 l/r3(0) =8.000 <1,=-4.000 11,,=0.000 Thus wemaycalculate thevelocities v,(0.05) andv,,(0.05):y F‘ PLANET (x,y) F Fy SUN ll Fig.9-5. Theforce ofgravity ona planet. v,,(0.05) =0.000-4.000><0.050=-0.200; 1.630+0.000><0.100=1.630. v,,(0.05) = Now ourmain calculations begin: x(0.l) = y(0.l) = ,-= l/r3 = a,(0.l) = a,,(0.l) = v,,(0.l5) = v,,(0.l5) = x(0.2) = y(0.2) =0.500 —0.20 X0.1 0.0+1.63 X0.1 \/0.4802 +0.1632 7.67 0.480 X7.67 -0.163 X7.67 -0.200 —3.68 X0.1 1.630 —1.26 X0.1 0.480 —0.568 X0.1 0.163 +1.50X0.1 etc.0.480 0.163 0.507 _3.68 1.-:0.-\' .../1-0.: |I|.5_\.0 °_‘ ‘. -1.256 . -0.568 -' 1.505 ""°".' '1-0—l.O -OI 0.423 0-313 Fig.9-6. Thecalculated motion ofa planet around thesun.I SUN Q5‘ Inthiswayweobtain thevalues given inTable 9-2,andin20steps orsowehave chased theplanet halfway around thesun! InFig.9-6areplotted thex-and y-coordinates given inTable 9-2. Thedotsrepresent thepositions atthesuccession oftimes atenth ofaunitapart; weseethatatthestart theplanet moves rapidly 9-7 Table 9-2 Solution ofdz),/dt =-x/r3, dv,/dt =-y/r3, r=\/x2 +y2. Interval: e=0.100 Orbit v,,=1.63 21,,=0 x=0.5 y=0 at i=0 X 7): az Y 91/ av 7' 1/r3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T 2 1.2 1.3 1.4 1.5 I __.i_ 1.7 1.8 1.9 2.0 _;1_ 2 2.2 2.30.500 0.480 0.423 0.337 0.232 0.115 -0.006 -0.127 -0.245 -0.357 -0.462 -0.559 -0.647 -0.726 -0.796 -0.857 -0.908 -0.950 -0.982 -1.005 -1.018 -1.022 -1.016-4.00 -0.200-3.68 -0.568-2.91 -0.859-1.96 -1.055-1.11 -1.166 -0.453 --1.211- i +0.020 -1.209 -1-0.344 -1.175 +0.562 -1.119 +0.705 -1.048 +0.796 --0.968- —i +0858-0.882 +0.90-0.792 +0.92-0.700 +0.93-0.607 +0.94--0.513- +0.95 -0.418 +0.95 -0.323 +0.95 -0.228 +0.95 -0.113 +0.96 --0.037- - +0.95 +0.058 +0.960.000 0.163 0.313 0.442 0.545 0.622 0.675 0.706 0.718 0.713 0.694 0.663 0.622 0.572 0.515 0.452 0.384 0.312 0.237 0.160 0.081 0.001 0.0791.630 1.505 1.290 1.033 0.771 -0.526- 0.306 0.115 -0.049 -0.190 --0.310- -0.412 -0.499 -0.570 -0.630 --0.680- -0.720 -0.751 -0.773 -0.778 --0.796~ -0.796 -0.7890.00 -1.25 -2.15 -2.57 -2.62 -2.45 -2.20 -1.91 -1.64 -1.41 -1.20 -1.02 -0.86 -0.72 -0.60 -0.50 -0.40 -0.31 -0.23 -0.15 -0.08 0.00 +0.070.500 0.507 0.526 0.556 0.592 0.633 E? 0.717 0.758 0.797 0.834 0.867 0.897 0.924 0.948 0.969 FE 1.000 1.010 1.018 1.021 1.022 1.0198.000 7.675 6.873 5.824 4.81 3.942 Z 2.712 2.296 1.975 1.723 1.535 1.385 1.267 1.173 1.099 E 1.000 0.970 0.948 0.939 0.936 0.945 Crossed x-axis at2.101 sec, period =4.20sec. 11,=0at2.086 sec. 6,,=0.796 Predicted time 1r(0.761)3/2 =1r(0.663) =2.082.Cross xat1.022, semimajor axis= =0.761. 9-8 andattheenditmoves slowly, andsotheshape ofthecurve isdetermined. Thus weseethatwereally doknow howtocalculate themotion ofplanets! Now letusseehowwecancalculate themotion ofNeptune, Jupiter, Uranus, oranyother planet. Ifwehave agreat many planets, andletthesunmove too, canwedothesame thing? Ofcourse wecan.Wecalculate theforce onaparticular planet, letussayplanet number i,which hasaposition x,,y,,z,(i=1mayrepre- sentthesun,i=2Mercury, i=3Venus, andsoon).Wemust know thepositions ofalltheplanets. Theforce acting ononeisduetoalltheother bodies which arelocated, letussay,atpositions x,,y,,z,.Therefore theequations are dun, N Gm,m,(x, —x,)m,i = ———-ii 1.1. ,3 2 Gm1m7(y. —J’) ,=dun, Jm,-= -—--i 9.18 d, I 6] <> N —-ml ZZ_Gm,m,(§, 2,)_ J=l 7'11 Further, wedefine r,,asthedistance between thetwoplanets 1'andj;thisisequal to '11=\/(X. —X.)2+0/.—M2+(Z1—202- (9-19) Also, Zmeans asum over allvalues ofj—-all other bodies——except, ofcourse, forj=i.Thus allwehave todoistomake more columns, lotsmore columns. Weneed ninecolumns forthemotions ofJupiter, nineforthemotions ofSaturn, andsoon.Then when wehave allinitial positions andvelocities wecancalculate alltheaccelerations from Eq.(9.18) byfirstcalculating allthedistances, using Eq.(9.19). How long willittake todoit?Ifyoudoitathome, itwilltakea verylong time! Butinmodern times wehave machines which doarithmetic very rapidly; avery good computing machine may take 1microsecond, that is,a millionth ofasecond, todoanaddition. Todoamultiplication takes longer, say10microseconds. Itmay bethatinonecycle ofcalculation, depending on theproblem, wemayhave 30multiplications, orsomething likethat, soonecycle willtake300microseconds. That means thatwecando3000 cycles ofcomputation persecond. Inorder togetanaccuracy, of,say,onepartinabillion, wewould need 4X105cycles tocorrespond toonerevolution ofaplanet around thesun. That corresponds toacomputation time of130seconds orabout twominutes. Thus ittake only twominutes tofollow Jupiter around thesun, with allthe perturbations ofalltheplanets correct toonepartinabillion, bythismethod! (Itturns outthattheerror varies about asthesquare oftheinterval 6.Ifwemake theinterval athousand times smaller, itisamillion times more accurate. So,let usmake theinterval 10,000 times smaller.) So,aswesaid, webegan thischapter notknowing howtocalculate even the motion ofamass onaspring. Now, armed withthetremendous power ofNewton’s laws, wecannotonlycalculate such simple motions butalso, given onlyamachine tohandle thearithmetic, even thetremendously complex motions oftheplanets, toashigh adegree ofprecision aswewish! 9-9 I0 Conservation ofMomentum 10-1 Newton’s Third Law Onthebasis ofNewton’s second lawofmotion, which gives therelation between theacceleration ofanybody andtheforce acting onit,anyproblem in mechanics canbesolved inprinciple. Forexample, todetermine themotion ofa fewparticles, onecanusethenumerical method developed inthepreceding chapter. Butthere aregood reasons tomake afurther study ofNewton’s laws. First, there arequite simple cases ofmotion which canbeanalyzed notonly bynumerical methods, butalsobydirect mathematical analysis. Forexample, although we know thattheacceleration ofafalling body is32ft/secz, andfrom thisfactcould calculate themotion bynumerical methods, itismuch easier andmore satisfactory toanalyze themotion andfindthegeneral solution, s=so+vol+1612. In thesame way, although wecanwork outthepositions ofaharmonic oscillator by numerical methods, itisalsopossible toshow analytically thatthegeneral solution isasimple cosine function oft,andsoitisunnecessary togotoallthatarithmetical trouble when there isasimple andmore accurate waytogettheresult. Inthe same manner, although themotion ofonebody around thesun,determined by gravitation, canbecalculated point bypoint bythenumerical methods ofChapter 9,which show thegeneral shape oftheorbit, itisnicealsotogettheexact shape, which analysis reveals asaperfect ellipse. Unfortunately, there arereally veryfewproblems which canbesolved exactly byanalysis. Inthecaseoftheharmonic oscillator, forexample, ifthespring force isnotproportional tothedisplacement, butissomething more complicated, one must fallback onthenumerical method. Orifthere aretwobodies going around thesun,sothatthetotal number ofbodies isthree, thenanalysis cannot produce a simple formula forthemotion, andinpractice theproblem must bedone numeri- cally. That isthefamous three-body problem, which solong challenged human powers ofanalysis; itisvery interesting how long ittook people toappreciate thefactthatperhaps thepowers ofmathematical analysis were limited andit might benecessary tousethenumerical methods. Today anenormous number of problems thatcannot bedone analytically aresolved bynumerical methods, and theoldthree-body problem, which wassupposed tobesodifiicult, issolved asa matter ofroutine inexactly thesame manner thatwasdescribed inthepreceding chapter, namely, bydoing enough arithmetic. However, there arealsosituations where both methods fail: thesimple problems wecandobyanalysis, andthe moderately difficult problems bynumerical, arithmetical methods, butthevery complicated problems wecannot dobyeither method. Acomplicated problem is, forexample, thecollision oftwoautomobiles, oreven themotion ofthemolecules ofagas. There arecountless particles inacubic millimeter ofgas,anditwould beridiculous totrytomake calculations with somany variables (about l0”— ahundred million billion). Anything likethemotion ofthemolecules oratoms of agasorablock oriron, orthemotion ofthestars inaglobular cluster, instead of justtwoorthree planets going around thesun——such problems wecannot do directly, sowehave toseekother means. Inthesituations inwhich wecannot follow details, weneed toknow some general properties, thatis,general theorems orprinciples which areconsequences ofNewton’s laws. One ofthese istheprinciple ofconservation ofenergy, which wasdiscussed inChapter 4.Another istheprinciple ofconservation ofmomentum, theSUlI)_]CC[ ofthischapter. Another reason forstudying mechanics further isthat there arecertain patterns ofmotion thatarerepeated inmany difierent circum- 10-110-1 Newton’s Third Law 10-2 Conservation ofmomentum 10-3 Momentum isconserved! 10-4 Momentum andenergy 10-5 Relativistic momentum stances, soitisgood tostudy these patterns inoneparticular circumstance. For example, weshall study collisions; different kinds ofcollisions have much in common. Intheflowoffiuids, itdoes notmake much difference what thefluid is, thelaws oftheflowaresimilar. Other problems thatweshall study arevibrations andoscillations and,inparticular, thepeculiar phenomena ofmechanical waves- sound, vibrations ofrods, andsoon. Inourdiscussion ofNewton’s lawsitwasexplained thatthese lawsareakind ofprogram thatsays“Pay attention totheforces,” andthatNewton toldusonly twothings about thenature offorces. Inthecaseofgravitation, hegave usthe complete lawoftheforce. Inthecase ofthevery complicated forces between atoms, hewasnotaware oftheright laws fortheforces; however, hediscovered onerule, onegeneral property offorces, which isexpressed inhisThird Law, and thatisthetotal knowledge thatNewton hadabout thenature offorces-—the law ofgravitation andthisprinciple, butnoother details. Thisprinciple isthataction equals reaction. What ismeant issomething ofthiskind: Suppose wehave twosmall bodies, sayparticles, andsuppose that thefirstoneexerts aforce onthesecond one, pushing itwith acertain force. Then, simultaneously, according toNewton’s Third Law, thesecond particle willpush onthefirstwith anequal force, inthe opposite direction; furthermore, these forces effectively actinthesame line. This isthehypothesis, orlaw, thatNewton proposed, anditseems tobequite accurate, though notexact (weshall discuss theerrors later). Forthemoment weshall takeittobetruethataction equals reaction. Ofcourse, ifthere isathird particle, notonthesame lineastheother two,thelawdoes notmean thatthetotal force onthefirstoneisequal tothetotalforce onthesecond, since thethird particle, forinstance, exerts itsownpush oneach oftheother two. Theresult isthatthe total effect onthefirsttwoisinsome other direction, andtheforces onthefirst twoparticles are,ingeneral, neither equal noropposite. However, theforces on each particle canberesolved intoparts, there being onecontribution orpartdue toeach other interacting particle. Then each pairofparticles hascorresponding components ofmutual interaction thatareequal inmagnitude andopposite in direction. 10-2 Conservation ofmomentum Now what aretheinteresting consequences oftheabove relationship? Sup- pose, forsimplicity, thatwehavejusttwointeracting particles, possibly ofdifferent mass, andnumbered 1and2.Theforces between them areequal andopposite; what aretheconsequences? According toNewton’s Second Law, force isthetime rateofchange ofthemomentum, soweconclude thattherateofchange ofmomen- tump1ofparticle 1isequal tominus therateofchange ofmomentum p2ofparticle 2,or dp1/dt=—dp2/dt. (10.1) Now iftherareofchange isalways equal andopposite, itfollows thatthetotal change inthemomentum ofparticle 1isequal andopposite tothetotal change in themomentum ofparticle 2;thismeans thatifweaddthemomentum ofparticle 1tothemomentum ofparticle 2,therateofchange ofthesumofthese, dueto themutual forces (called internal forces) between particles, iszero; thatis d(P1+P2)/dl =0- (10-2) There isassumed tobenoother force intheproblem. Iftherateofchange ofthis sumisalways zero, thatisjustanother wayofsaying thatthequantity (pl+p2) does notchange. (This quantity isalsowritten mlvl +77221)2,andiscalled the total momentum ofthetwoparticles.) Wehave now obtained theresult thatthe total momentum ofthetwoparticles does notchange because ofanymutual interactions between them. This statement expresses thelawofconservation of 10-2 momentum inthatparticular example. Weconclude thatifthere isanykind of force, nomatter how complicated, between twoparticles, andwemeasure or calculate mlvl +7712112, thatis,thesum ofthetwomomenta, both before and after theforces act,theresults should beequal, i.e.,thetotal momentum isa constant. Ifweextend theargument tothree ormore interacting particles inmore com- plicated circumstances, itisevident thatsofarasinternal forces areconcerned, the total momentum ofalltheparticles stays constant, since anincrease inmomentum ofone, duetoanother, isexactly compensated bythedecrease ofthesecond, duetothefirst. That is,alltheinternal forces willbalance out,andtherefore cannot change thetotal momentum oftheparticles. Then ifthere arenoforces from theoutside (external forces), there arenoforces thatcanchange thetotal momentum; hence thetotal momentum isaconstant. Itisworth describing what happens ifthere areforces thatdonotcome from themutual actions oftheparticles inquestion: suppose weisolate theinteracting particles. Ifthere areonlymutual forces, then, asbefore, thetotal momentum of theparticles does notchange, nomatter howcomplicated theforces. Ontheother hand, suppose there arealsoforces coming from theparticles outside theisolated group. Anyforce exerted byoutside bodies oninside bodies, wecallanexternal force. Weshall later demonstrate thatthesumofallexternal forces equals therate ofchange ofthetotal momentum ofalltheparticles inside, averyuseful theorem. Theconservation ofthetotal momentum ofanumber ofinteracting particles canbeexpressed as mlvl +"T2112 +m3113 + =aconstant, (10.3) ifthere arenonetexternal forces. Here themasses andcorresponding velocities oftheparticles arenumbered 1,2,3,4,...Thegeneral statement ofNewton’s Second Law foreach particle, f=;€(mv), (10.4) istruespecifically forthecomponents offorce andmomentum inanygiven direc- tion; thusthex-component oftheforce onaparticle isequal tothex-component oftherateofchange ofmomentum ofthatparticle, or /.=§',<m1».>. (10.5) andsimilarly forthey-andz-directions. Therefore Eq.(10.3) isreally three equations, oneforeach direction. Inaddition tothelawofconservation ofmomentum, there isanother inter- esting consequence ofNewton’s Second Law, tobeproved later, butmerely stated now. This principle isthatthelaws ofphysics willlook thesame whether weare standing stillormoving with auniform speed inastraight line. Forexample, a child bouncing aballinanairplane finds thattheballbounces thesame asthough hewere bouncing itontheground. Even though theairplane ismoving with a very high velocity, unless itchanges itsvelocity, thelaws look thesame tothe child astheydowhen theairplane isstanding still. This istheso-called relativity principle. Asweuseithereweshall callit“Galilean relativity” todistinguish it from themore careful analysis made byEinstein, which weshall study later. Wehave just derived thelawofconservation ofmomentum from Newton’s laws, andwecould goonfrom heretofindthespecial laws thatdescribe impacts andcollisions. Butforthesake ofvariety, andalso asanillustration ofakind of reasoning thatcanbeused inphysics inother circumstances where, forexample, onemight notknow Newton’s laws andmight take adifferent approach, weshall discuss thelaws ofimpacts andcollisions from acompletely different point of view. Weshall base ourdiscussion ontheprinciple ofGalilean relativity, stated above, andshall endupwith thelawofconservation ofmomentum. Weshall start byassuming thatnature would look thesame ifwerunalong atacertain speed andwatch itasitwould ifwewere standing still. Before dis- 10-3 cussing collisions inwhich twobodies collide andstick together, orcome together andbounce apart, weshall firstconsider twobodies thatareheld together bya spring orsomething else,andarethensuddenly released andpushed bythespring orperhaps byalittle explosion. Further, weshall consider motion inonly one direction. First, letussuppose thatthetwoobjects areexactly thesame, arenice symmetrical objects, andthen wehave alittle explosion between them. After the explosion, oneofthebodies willbemoving, letussaytoward theright, with a velocity 1/.Then itappears reasonable thattheother body ismoving toward the leftwithavelocity 0,because iftheobjects arealike there isnoreason forright or lefttobepreferred andsothebodies would dosomething thatissymmetrical. This isanillustration ofakind ofthinking thatisvery useful inmany problems but would notbebrought outifwejuststarted with theformulas. Thefirstresult from ourexperiment isthatequal objects willhave equal speed, butnow suppose that wehave two objects made ofdifferent materials, saycopper andaluminum, andwemake thetwomasses equal. Weshall now suppose thatifwedotheexperiment with twomasses thatareequal, even though theobjects arenotidentical, thevelocities willbeequal. Someone might object: “But youknow, youcould doitbackwards, youdidnothave tosuppose that. You could define equal masses tomean twomasses thatacquire equal velocities inthisexperiment.” Wefollow thatsuggestion andmake alittleexplosion between thecopper andaverylarge piece ofaluminum, soheavy thatthecopper fliesout andthealuminum hardly budges. That istoomuch aluminum, sowereduce the amount until there isjust avery tiny piece, then when wemake theexplosion the aluminum goesflying away, andthecopper hardly budges. That isnotenough alu- minum. Evidently there issome right amount inbetween; sowekeep adjusting theamount until thevelocities come outequal. Very wellthen—let usturn it around, andsaythatwhen thevelocities areequal, theniasses areequal. This appears tobejustadefinition, anditseems remarkable thatwecantransform physical laws into mere definitions. Nevertheless, there aresome physical laws involved, andifweaccept thisdefinition ofequal masses, weimmediately findone ofthelaws, asfollows. Suppose weknow from theforegoing experiment thattwopieces ofmatter, Aand B(ofcopper andaluminum), have equal masses, andwecompare athird body, sayapiece ofgold, with thecopper inthesame manner asabove, making surethatitsmass isequal tothemass ofthecopper. Ifwenowmake theexperiment between thealuminum andthegold, there isnothing inlogic thatsaysthese masses must beequal; however, theexperiment shows that they actually are. Sonow, by experiment, wehave found anewlaw. Astatement ofthislawmight be:Iftwo masses areeach equal toathird mass (asdetermined byequal velocities inthis experiment), then they areequal toeach other. (This statement does norfollow atallfrom asimilar statement used asapostulate regarding mathematical quanti- ties.) From thisexample wecanseehowquickly westart toinfer things ifweare careless. Itisnotjustadefinition tosaythemasses areequal when thevelocities areequal, because tosaythemasses areequal istoimply themathematical laws ofequality, which inturnmakes aprediction about anexperiment. Asasecond example, suppose that AandBarefound tobeequal bydoing theexperiment with onestrength ofexplosion, which gives acertain velocity; if wethen useastronger explosion, willitbetrue ornottrue that thevelocities now obtained areequal? Again, inlogic there isnothing thatcandecide thisquestion, butexperiment shows that itistrue. So,here isanother law, which might be stated: Iftwobodies have equal masses, asmeasured byequal velocities atone velocity, they willhave equal masses when measured atanother velocity. From these examples weseethatwhat appeared tobeonly adefinition really involved some laws ofphysics. Inthedevelopment thatfollows weshall assume itistruethatequal masses have equal andopposite velocities when anexplosion occurs between them. We shall make another assumption intheinverse case: Iftwoidentical objects, moving inopposite directions withequal velocities, collide andstick together bysome kind ofglue, then which way willthey bemoving after thecollision? This isagain a 10-4 symmetrical situation, with nopreference between right andleft,soweassume thatthey stand still. Weshall alsosuppose thatanytwoobjects ofequal mass, eveniftheobjects aremade ofdifferent materials, which collide andstick together, when moving with thesame velocity inopposite directions willcome torestafter thecollision. 10-3 Momentum isconserved! Wecanverify theabove assumptions experimentally: first, thatiftwostation- aryobjects ofequal mass areseparated byanexplosion theywillmove apart with thesame speed, andsecond, iftwoobjects ofequal mass, coming together withthe same speed, collide andstick together theywillstop. Thiswecandobymeans of amarvelous invention called anairtrough,* which getsridoffriction, thething which continually bothered Galileo (Fig. 10-1). Hecould notdoexperiments by sliding things because they donotslide freely, but,byadding amagic touch, we cantoday getridoffriction. Ourobjects willslide without difficulty, onandonat aconstant velocity, asadvertised byGalileo. Thisisdone bysupporting theobjects onair. Because airhasvery lowfriction, anobject glides along with practically constant velocity when there isnoapplied force. First, weusetwoglide blocks which have been made carefully tohave thesame weight, ormass (their weight wasmeasured really, butweknow thatthisweight isproportional tothemass), andweplace asmall explosive capinaclosed cylinder between thetwoblocks (Fig. 10-2). Weshall starttheblocks from restatthecenter point ofthetrack and force them apart byexploding thecapwithanelectric spark. What should happen? Ifthespeeds areequal when theyflyapart, theyshould arrive attheends ofthe trough atthesame time. Onreaching theends theywillboth bounce back with practically opposite velocity, andwillcome together andstopatthecenter where they started. Itisagood test; when itisactually done theresult isjustaswe have described (Fig. 10-3). Now thenextthing wewould liketofigure outiswhat happens inalesssimple situation. Suppose wehave twoequal masses, onemoving withvelocity vandthe other standing still, andtheycollide andstick; what isgoing tohappen? There isamass 2maltogether when wearefinished, drifting with anunknown velocity. What velocity? That istheproblem. Tofindtheanswer, wemake theassumption thatifweridealong inacar,physics willlook thesame asifwearestanding still. Westart withtheknowledge thattwoequal masses, moving inopposite directions with equal speeds v,willstop dead when they collide. Now suppose thatwhile thishappens, weareriding byinanautomobile, atavelocity —v.Then what does itlooklike? Since weareriding along withoneofthetwomasses which arecoming together, thatoneappears toustohave zerovelocity. Theother mass, however, going theother waywithvelocity v,willappear tobecoming toward usatavelocity 2v(Fig. 10-4). Finally. thecombined masses after collision willseem tobepassing bywith velocity 0.Wetherefore conclude thatanobject with velocity 2v,hitting anequal oneatrest,willendupwithvelocity 7),orwhat ismathematically exactly thesame, anobject with velocity vhitting andsticking tooneatrestwillproduce anobject moving with velocity v/2. Note thatifwemultiply themass andthe velocity beforehand andaddthem together, ma+0,wegetthesame answer as when wemultiply themass andthevelocity ofeverything afterwards, 2mtimes 0/2. Sothattellsuswhat happens when amass ofvelocity vhitsonestanding still. Inexactly thesame manner wecandeduce what happens when equal objects having anytwovelocities hiteach other. Suppose wehave twoequal bodies with velocities 01and02,respectively, which collide andstick together. What istheir velocity vafter thecollision? Again weridebyinanautomobile, sayatvelocity 112,sothatonebody appears to beatrest. Theother then appears tohave avelocity 01—02,andwehave the same case thatwehadbefore. When itisallfinished they willbemoving at %(v1 —02)withrespect tothecar. What thenistheactual speed ontheground? *H.V.Neher andR.B.Leighton, Amer. Jour. ofPhys. 3],255(1963). 10-5SMALL HOLES (JETS) COMPRESSED AIRSUPPLY Fig. 10-1. End view oflinear air trough. BUMPER SPRING TOY PISTOL CAP SPARK ELECTRODE w\\y/\\_\\\\\\‘ 7 //4 CYLINDER PISTON BUMPER SPRING Fig. 10-2. Sectional view ofgliders withexplosive interaction cylinder attach- ment. lE o Illdl 1- -V V -§ 13> 12:1 =1-:1» __1](bl <.i EP 4+vl l;-6 1310V‘O [P ‘l l:l l’ ‘Dill Fig. 10-3. Schematic view ofaction- reaction experiment with equal masses. VIEW FROM VIEW FROM CENTER OFMASS MOVlNG CAR (CAR VELOCITY I—V) v-> 4--v 2v-> 0 El ll]BEFORECOLLISION IE LE1 v=o v->[E AFTER COLLISION [E11] Fig. 10-4. Two views ofaninelastic collision between equal masses. VIEWFROM"LAB" VIEWFROM CAR vi—>v¢—> Vt-V2-P 0 E [E] serene cotustou [E] [E v—> |/2(v.- 1-»m AFTER COLLISION Fig. 10-5. Two views ofanother inelastic collision between equal masses. lA*ll‘ fl -v n»~—2~——-FrLi—1t_—1-1»-a <--v v'-> Diml [2m U< 5+30 Fig. lO-6. Anexperiment toverify that amass mwith velocity vstriking a mass mwith zero velocity gives 2mwith velocity v/2. VIEW FROM VIEW FROM CM SYSTEM CAR v -v/2 sv/2 o-O §— -O [El [1 serene cottislou IE ljjfl o v/2-> E03] AFTER COLLlSl0N [E Fig. 10-7. Two views ofaninelastic collision between mand2m. o o\I/0 o 0 lml lmlml lml lmlI_ -v __ lt —<3+30 30 BOE7 4-—V/2 V/2 -0 0 ILTIE Eli] [El 4-“V/2 V/3 -D GEE] lfilfilll Fig. 10-8. Action and reaction be- tween 2mand3m.Itis0=%(01 —02)+02or%(v1 +02)(Fig. 10-5). Again wenote that W101+m02 =2m(01 +02)/2. (10.6) Thus, using thisprinciple, wecananalyze anykind ofcollision inwhich two bodies ofequal mass hiteach other andstick. Infact,although wehave worked onlyinonedimension, wecanfindoutagreat dealabout much more complicated collisions byimagining thatweareriding byinacarinsome oblique direction. Theprinciple isthesame, butthedetails getsomewhat complicated. Inorder totestexperimentally whether anobject moving with velocity 0, colliding with anequal oneatrest,forms anobject moving with velocity 0/2,we may perform thefollowing experiment with ourair-trough apparatus. Weplace inthetrough three equally massive objects, twoofwhich areinitially joined to- gether with ourexplosive cylinder device, thethird being verynear tobutslightly separated from these andprovided with asticky bumper sothatitwillstick to another object which hitsit.Now, amoment after theexplosion, wehave two objects ofmass mmoving with equal andopposite velocities 0.Amoment after that, oneofthese collides with thethird object andmakes anobject ofmass 2m moving, sowebelieve, withvelocity 0/2. How dowetestwhether itisreally 0/2? Byarranging theinitial positions ofthemasses onthetrough sothat thedistances totheends arenotequal, butareintheratio 2:1. Thus ourfirstmass, which continues tomove with velocity 0,should cover twice asmuch distance inagiven timeasthetwowhich arestuck together (allowing forthesmall distance travelled bythesecond object before itcollided with thethird). Themass mandthemass 2mshould reach theends atthesame time, andwhen wetryit,wefindthatthey do(Fig. 10-6). Thenextproblem thatwewant towork outiswhat happens ifwehave two different masses. Letustake amass mandamass 2mandapply ourexplosive interaction. What willhappen then? If,asaresult oftheexplosion, mmoves with velocity 0,with what velocity does 2mmove? Theexperiment wehave justdone may berepeated with zero separation between thesecond andthird masses. and when wetryitwegetthesame result, namely, thereacting masses mand2m attain velocities —0and0/2. Thus thedirect reaction between mand2mgives thesame result asthesymmetrical reaction between mandm,followed byacollision between mandathird mass minwhich they stick together. Furthermore, wefind thatthemasses mand2mreturning from theends ofthetrough, withtheir veloci- ties(nearly) exactly reversed, stop dead ifthey stick together. Now thenextquestion wemayaskisthis. What willhappen ifamass mwith velocity 0,say,hitsandsticks toanother mass 2matrest? This isvery easy to answer using ourprinciple ofGalilean relativity. forwesimply watch thecollision which wehave justdescribed from acarmoving with velocity -0/2 (Fig. 10-7). From thecar,thevelocities are 0'1=0—0(car) =0+0/2=30/2 and 053=-0/2 —0(car) =-0/2 +0/2=0. After thecollision, themass 3mappears toustobemoving with velocity 0/2. Thus wehave theanswer, i.e.,theratio ofvelocities before andafter collision is 3tol:ifanobject ofmass mcollides withastationary object ofmass 2m,thenthe whole thing moves ofl,stuck together, with avelocity l/3asmuch. The general ruleagain isthatthesumoftheproducts ofthemasses andthevelocities stays the same: mii+0equals 3mtimes 0/3,sowearegradually building upthetheorem oftheconservation ofmomentum, piece bypiece. Now wehave oneagainst two. Using thesame arguments, wecanpredict the result ofoneagainst three. twoagainst three, etc. Thecase oftwoagainst three, starting from rest,isshown inFig.10-8. Inevery case wefind that themass ofthefirst object times itsvelocity, plus themass ofthesecond object times itsvelocity, isequal tothetotal mass ofthe final object times itsvelocity. These areallexamples, then, oftheconservation of 10-6 momentum. Starting from simple, symmetrical cases, wehave demonstrated the lawformore complex cases. Wecould, infact, doitforanyrational mass ratio, andsince every ratio isexceedingly close toarational ratio, wecanhandle every ratio asprecisely aswewish. 10-4 Momentum andenergy Alltheforegoing examples aresimple cases where thebodies collide andstick together, orwere initially stuck together andlater separated byanexplosion. However, there aresituations inwhich thebodies donotcohere, as,forexample, twobodies ofequal mass which collide with equal speeds andthen rebound. Forabrief moment theyareincontact andboth arecompressed. Attheinstant ofmaximum compression theyboth have zerovelocity andenergy isstored inthe elastic bodies, asinacompressed spring. This energy isderived from thekinetic energy thebodies hadbefore thecollision, which becomes zeroattheinstant their velocity iszero. Thelossofkinetic energy isonly momentary, however. The compressed condition isanalogous tothecapthatreleases energy inanexplosion. Thebodies areimmediately decompressed inakind ofexplosion, andflyapart again; butwealready know thatcase—the bodies flyapart with equal speeds. However, thisspeed ofrebound isless,ingeneral, than theinitial speed, because notalltheenergy isavailable fortheexplosion, depending onthematerial. Ifthe material isputty nokinetic energy isrecovered, butifitissomething more rigid, some kinetic energy isusually regained. Inthecollision therestofthekinetic energy istransformed intoheat andvibrational energy—the bodies arehotand vibrating. Thevibrational energy alsoissoon transformed intoheat. Itispossible tomake thecolliding bodies from highly elastic materials, such assteel, with carefully designed spring bumpers, sothatthecollision generates very little heat andvibration. Inthese circumstances thevelocities ofrebound arepractically equal totheinitial velocities; such acollision iscalled elastic. That thevelocities before andafter anelastic collision areequal isnotamatter ofconservation ofmomentum, butamatter ofconservation ofkinetic energy. That thespeeds ofthebodies rebounding after asymmetrical collision areequal toeachother, however. isamatter ofconservation ofmomentum. Wemight similarly analyze collisions between bodies ofdifferent masses, different initial velocities, andvarious degrees ofelasticity, anddetermine thefinal velocities andthelossofkinetic energy, butweshall notgointothedetails of theseprocesses. Elastic collisions areespecially interesting forsystems thathave nointernal “gears, wheels, orparts.” Then when there isacollision there isnowhere forthe energy tobeimpounded, because theobjects thatmove apart areinthesame condition aswhen theycollided. Therefore, between veryelementary objects, the collisions arealways elastic orvery nearly elastic. Forinstance, thecollisions between atoms ormolecules inagasaresaidtobeperfectly elastic. Although this isanexcellent approximation, even such collisions arenotperfectly elastic; other- wiseonecould notunderstand howenergy intheform oflight orheatradiation could come outofagas. Once inawhile, inagascollision, alow-energy infrared rayisemitted, butthisoccurrence isveryrareandtheenergy emitted isverysmall. So,formost purposes, collisions ofmolecules ingases areconsidered tobeper- fectly elastic. Asaninteresting example, letusconsider anelastic collision between two objects ofequal mass. 1fthey come together with thesame speed, they would come apart atthatsame speed, bysymmetry. Butnow look atthisinanother circumstance, inwhich oneofthem ismoving withvelocity 0andtheother oneis atrest. What happens? Wehave been through thisbefore. Wewatch thesym- metrical collision from acarmoving along with oneoftheobjects, andwefind thatifastationary body isstruck elastically byanother body ofexactly thesame mass, themoving body stops, andtheonethatwasstanding stillnowmoves away withthesame speed thattheother onehad; thebodies simply exchange velocities. Thisbehavior caneasily bedemonstrated withasuitable impact apparatus. More 10-7 generally, ifboth bodies aremoving, withdifferent velocities, theysimply exchange velocity atimpact. Another example ofanalmost elastic interaction ismagnetism. Ifwearrange apairofU-shaped magnets inourglide blocks, sothattheyrepel eachother, when onedrifts quietly uptotheother, itpushes itaway andstands perfectly still, andnow theother goes along, frictionlessly. Theprinciple ofconservation ofmomentum isveryuseful, because itenables ustosolve many problems without knowing thedetails. Wedidnotknow the details ofthegasmotions inthecapexplosion, yetwecould predict thevelocities with which thebodies came apart, forexample. Another interesting example is rocket propulsion. Arocket oflarge mass, M,ejects asmall piece, ofmass m,with aterrific velocity Vrelative totherocket. After thistherocket, ifitwere originally standing still,willbemoving with asmall velocity, 0.Using theprinciple ofcon- servation ofmomentum, wecancalculate thisvelocity tobe 0=%~V. Solong asmaterial isbeing ejected, therocket continues topick upspeed. Rocket propulsion isessentially thesame astherecoil ofagun: there isnoneed foranyairtopush against. 10-5 Relativistic momentum Inmodern times thelawofconservation ofmomentum hasundergone certain modifications. However, thelawisstilltruetoday, themodifications being mainly inthedefinitions ofthings. Inthetheory ofrelativity itturns outthatwedohave conservation ofmomentum; theparticles have mass andthemomentum isstill given bym0,themass times thevelocity, butthemass changes withthevelocity, hence themomentum alsochanges. Themass varies with velocity according to thelaw m=il—- (10.7)\/l —02/c2 where moisthemass ofthebody atrestandcisthespeed oflight. Itiseasyto seefrom theformula thatthere isnegligible difierence between mandmounless 0isvery large, andthat forordinary velocities theexpression formomentum reduces totheoldformula. Thecomponents ofmomentum forasingle particle arewritten as "IOU; P7101/‘y H101);I Z - Z , Z _-1' , P(/1-02/62 P"\/l-02/62 P\/1-02/82 () where 02=0f+0:+03.Ifthex-components aresummed over alltheinter- acting particles, both before andafter acollision, thesums areequal; that is, momentum isconserved inthex-direction. Thesame holds trueinanydirection. InChapter 4wesawthatthelawofconservation ofenergy isnotvalid unless werecognize thatenergy appears indifferent forms, electrical energy, mechanical energy, radiant energy, heatenergy, andsoon.Insome ofthese cases, heatenergy forexample, theenergy might besaidtobe“hidden.” Thisexample might suggest thequestion, “Are there alsohidden forms ofmomentum—perhaps heatmomen- tum?” Theanswer isthatitisveryhard tohide momentum forthefollowing reasons. Therandom motions oftheatoms ofabody furnish ameasure ofheatenergy, ifthesquares ofthevelocities aresummed. This sumwillbeapositive result, having nodirectional character. Theheatisthere, whether ornotthebody moves asawhole, andconservation ofenergy intheform ofheat isnotvery obvious. Ontheother hand, ifonesums thevelocities, which have direction, andfinds a result thatisnotzero, thatmeans thatthere isadrift oftheentire body insome particular direction, andsuch agross momentum isreadily observed. Thus there isnorandom internal lostmomentum, because thebody hasnetmomentum only 10-8 when itmoves asawhole. Therefore momentum, asamechanical quantity, is difficult tohide. Nevertheless, momentum canbehidden—in theelectromagnetic field, forexample. This caseisanother effect ofrelativity. One ofthepropositions ofNewton wasthatinteractions atadistance are instantaneous. Itturns outthat such isnotthecase; insituations involving electrical forces, forinstance, ifanelectrical charge atonelocation issuddenly moved, theeffects onanother charge, atanother place, donotappear instantane- ously—there isalittle delay. Inthose circumstances, even iftheforces areequal themomentum willnotcheck out; there willbeashort time during which there willbetrouble, because forawhile thefirstcharge willfeelacertain reaction force, say,andwillpickupsome momentum, butthesecond charge hasfeltnothing and hasnotyetchanged itsmomentum. Ittakes time fortheinfluence tocross the intervening distance, which itdoes at186,000 miles asecond. Inthattinytime themomentum oftheparticles isnotconserved. Ofcourse after thesecond charge hasfelttheeffect ofthefirstoneandallisquieted down, themomentum equation willcheck outallright, butduring thatsmall interval momentum isnotconserved. Werepresent thisbysaying thatduring thisinterval there isanother kind ofmo- mentum besides thatoftheparticle, m0,andthatismomentum intheelectro- magnetic field. Ifweaddthefield momentum tothemomentum oftheparticles, thenmomentum isconserved atanymoment allthetime. Thefactthattheelectro- magnetic field canpossess momentum andenergy makes thatfield veryreal, and so,forbetter understanding, theoriginal ideathatthere arejusttheforces between particles hastobemodified totheidea thataparticle makes afield, andafield actsonanother particle, andthefield itself hassuch familiar properties asenergy content andmomentum, justasparticles canhave. Totakeanother example: an electromagnetic field haswaves, which wecalllight; itturns outthatlight also carries momentum with it,sowhen light impinges onanobject itcarries ina certain amount ofmomentum persecond; thisisequivalent toaforce, because if theilluminated object ispicking upacertain amount ofmomentum persecond, itsmomentum ischanging andthesituation isexactly thesame asifthere were a force onit.Light canexert pressure bybombarding anobject; thispressure is verysmall, butwith sufficiently delicate apparatus itismeasurable. Now inquantum mechanics itturns outthatmomentum isadifferent thing- itisnolonger m0. Itishard todefine exactly what ismeant bythevelocity ofa particle, butmomentum stillexists. Inquantum mechanics thedifference isthat when theparticles arerepresented asparticles, themomentum isstillm0,butwhen theparticles arerepresented aswaves, themomentum ismeasured bythenumber ofwaves percentimeter: thegreater thisnumber ofwaves, thegreater themomen- tum. Inspite ofthedifferences, thelawofconservation ofmomentum holds also inquantum mechanics. Even though thelawf=maisfalse, andallthederiva- tions ofNewton were wrong fortheconservation ofmomentum, inquantum mechanics, nevertheless, intheend, thatparticular lawmaintains itself! 10-9 I1 Vectors 11-1 Symmetry inphysics Inthischapter weintroduce asubject thatistechnically known inphysics as symmetry inphysical law. Theword “symmetry” isused here with aspecial meaning, andtherefore needs tobedefined. When isathing symmetrical—how canwedefine it?When wehave apicture thatissymmetrical, onesideissomehow thesame astheother side. Professor Hermann Weyl hasgiven thisdefinition of symmetry: athing issymmetrical ifonecansubject ittoacertain operation and itappears exactly thesame after theoperation. Forinstance, ifwelook atavase that isleft-and-right symmetrical, then turn it180° around thevertical axis, it looks thesame. Weshall adopt thedefinition ofsymmetry inWeyl’s more general form, andinthatform weshall discuss symmetry ofphysical laws. Suppose webuild acomplex machine inacertain place, with alotofcompli- cated interactions, andballs bouncing around withforces between them, andsoon. Now suppose webuild exactly thesame kind ofequipment atsome other place, matching partbypart, with thesame dimensions andthesame orientation, every- thing thesame only displaced laterally bysome distance. Then, ifwestart the twomachines inthesame initial circumstances, inexact correspondence, weask: willonemachine behave exactly thesame astheother? Willitfollow allthemo- tions inexact parallelism? Ofcourse theanswer may wellbeno,because ifwe choose thewrong place forourmachine itmight beinside awallandinterferences from thewallwould make themachine notwork. Allofourideas inphysics require acertain amount ofcommon sense intheir application; theyarenotpurely mathematical orabstract ideas. Wehave tounder- stand what wemean when wesaythatthephenomena arethesame when wemove theapparatus toanew position. Wemean thatwemove everything thatwe believe isrelevant; ifthephenomenon isnotthesame, wesuggest thatsomething relevant hasnotbeen moved, andweproceed tolook forit.Ifwenever findit, thenweclaim thatthelaws ofphysics donothave thissymmetry. Ontheother hand, wemay findit—we expect tofindit——if thelaws ofphysics dohave this symmetry; looking around, wemaydiscover, forinstance, thatthewallispushing ontheapparatus. Thebasic question is,ifwedefine things wellenough, ifallthe essential forces areincluded inside theapparatus, ifalltherelevant parts aremoved from oneplace toanother, willthelaws bethesame? Will themachinery work thesame way? Itisclear thatwhat wewant todoistomove alltheequipment andessential influences, butnoteverything intheworld—planets, stars, andall—for ifwedo that, wehave thesame phenomenon again forthetrivial reason thatweareright back where westarted. No,wecannot move everything. Butitturns outin practice thatwith acertain amount ofintelligence about what tomove, thema- chinery willwork. Inother words, ifwedonotgoinside awall, ifweknow the origin oftheoutside forces, andarrange thatthose aremoved too,then thema- chinery willwork thesame inonelocation asinanother. 11-2 Translations Weshall limit ouranalysis tojustmechanics, forwhich wenowhave suiiicient knowledge. Inprevious chapters wehave seen thatthelaws ofmechanics canbe summarized byasetofthree equations foreach particle: m(d2x/dt2) =F,,m(d2y/a't2) =F,,,m(d2z/dt2) =F,.(11.1) 11-1ll-1 Symmetry inphysics 11-2 Translations 11-3 Rotations 11-4 Vectors ll-5 Vector algebra 11-6 Newton’s lawsinvector notation 11-7 Scalar product ofvectors y v' JOE MOE 0 £ l___i‘ x x' Fig. ll—l. Two parallel coordinate systems.Now thismeans thatthere exists awaytomeasure x,y,andzonthree perpendicu- laraxes, andtheforces along those directions, such thatthese lawsaretrue. These must bemeasured from some origin, butwhere doweputtheorigin? All thatNewton would tellusatfirstisthatthere issome place thatwecanmeasure from, perhaps thecenter oftheuniverse, such thatthese laws arecorrect. ButWe canshow immediately thatwecannever findthecenter, because ifweusesome other origin itwould make nodifference. Inother words, suppose thatthere are twopeople-—Joe, who hasanorigin inoneplace, andMoe, who hasaparallel system whose origin issomewhere else(Fig. 11-1). Now when Joemeasures the location ofthepoint inspace, hefinds itatx,y,andz(weshall usually leave zout because itistooconfusing todraw inapicture). Moe, ontheother hand, when measuring thesame point, willobtain adilierent x(inorder todistinguish it,we willcallitx’),andinprinciple adifferent y,although inourexample they are numerically equal. Sowehave x’=x—a, y’=y, z’=z. (11.2) Now inorder tocomplete ouranalysis wemust know what Moe would obtain for theforces. Theforce issupposed toactalong some line,andbytheforce inthe x-direction wemean thepart ofthetotal which is1nthex-direction, which is themagnitude oftheforce times thiscosine ofitsangle with thex-axis. Now we seethatMoe would useexactly thesame projection asJoewould use,sowehave asetofequations F,’=Fx, F1,’=Fy, F,’=F,. (11.3) These would betherelationships between quantities asseenbyJoeandMoe. Thequestion is,ifJoeknows Newton’s laws, andifMoe triestowrite down Newton’s laws, willtheyalsobecorrect forhim? Does itmake anydiflerence from which origin wemeasure thepoints? Inother words, assuming thatequations (11.1) aretrue, andtheEqs. (11.2) and(11.3) givetherelationship ofthemeasure- ments, isitorisitnottruethat (a)m(a'2x’/dtz) = (b)m(d2y’/dtz) =F,., (11.4) (C)m(d2z’/dtz) =F,/? Inorder totestthese equations weshall dilferentiate theformula forx’ twice. First ofall £__‘£( _)_@[email protected]_dtx "‘at at Now weshall assume thatMoe’s origin isfixed (not moving) relative toJoe’s; therefore aisaconstant andda/dt =0,sowefindthat dx’/dt =dx/dt’ andtherefore dzx’/dtz =d2x/dt2; therefore weknow thatEq.(ll.4a) becomes m(d2x/dtz) =Fat. (Wealsosuppose thatthemasses measured byJoeandMoe areequal.) Thus the acceleration times themass isthesame astheother fellow’s. Wehave alsofound theformula forFx»,for,substituting from Eq.(11.1), wefindthat F,»=F,. Therefore thelaws asseenbyMoe appear thesame; hecanwrite Newton’s laws too,with different coordinates, andtheywillstillberight. That means that 11-2 there isnounique waytodefine theorigin oftheworld, because thelaws will appear thesame, from whatever position theyareobserved. This isalsotrue: ifthere isapiece ofequipment inoneplace with acertain kind ofmachinery init,thesame equipment inanother place willbehave inthe same way. Why? Because onemachine, when analyzed byMoe, hasexactly the same equations astheother one,analyzed byJoe. Since theequations arethesame, thephenomena appear thesame. Sotheproof thatanapparatus inanewposition behaves thesame asitdidintheoldposition isthesame astheproof thatthe equations when displaced inspace reproduce themselves. Therefore wesaythat thelaws ofphysics aresymmetrical fortranslational displacements, symmetrical inthesense thatthelaws donotchange when wemake atranslation ofourco- ordinates. Ofcourse itisquite obvious intuitively thatthisistrue, butitisinter- esting andentertaining todiscuss themathematics ofit. 11-3 Rotations Theabove isthefirstofaseries ofevermore complicated propositions con- cerning thesymmetry ofaphysical law. Thenext proposition isthatitshould make nodifference inwhich direction wechoose theaxes. Inother words, ifwe build apiece ofequipment insome place andwatch itoperate, andnearby we build thesame kind ofapparatus butputituponanangle, willitoperate inthe same way? Obviously itwillnotifitisaGrandfather clock, forexample! Ifa pendulum clock stands upright, itworks fine,butifitistilted thependulum falls against thesideofthecase andnothing happens. Thetheorem isthen false in thecaseofthependulum clock, unless weinclude theearth, which ispulling on thependulum. Therefore wecanmake aprediction about pendulum clocks if webelieve inthesymmetry ofphysical lawforrotation: something elseisinvolved intheoperation ofapendulum clock besides themachinery oftheclock, something outside itthatweshould look for.Wemayalsopredict thatpendulum clocks will notwork thesame waywhen located indifferent places relative tothismysterious source ofasymmetry, perhaps theearth. Indeed, weknow thatapendulum clock upinanartificial satellite, forexample, would nottickeither, because there isno effective force, andonMars itwould goatadifferent rate. Pendulum clocks do involve something more than justthemachinery inside, they involve something ontheoutside. Once werecognize thisfactor, weseethatwemust turntheearth along withtheapparatus. Ofcourse wedonothave toworry about that, itiseasy todo;onesimply waits amoment ortwoandtheearth turns; thenthependulum clock ticks again inthenewposition thesame asitdidbefore. While weare rotating inspace ourangles arealways changing, absolutely; thischange does not seem tobother usvery much, forinthenewposition weseem tobeinthesame condition asintheold. This hasacertain tendency toconfuse one, because itis truethatinthenewturned position thelaws arethesame asintheunturned position, butitisnottruethatasweturnathing itfollows thesame lawsasitdoes when wearenotturning it.Ifweperform sufficiently delicate experiments, we cantellthattheearth isrotating, butnotthatithadrotated. Inother words, We cannot locate itsangular position, butwecantellthatitischanging. Now wemay discuss theeffects ofangular orientation upon physical laws. Letusfindoutwhether thesame game withJoeandMoe works again. This time, toavoid needless complication, weshall suppose thatJoeandMoe usethesame origin (wehave already shown thattheaxescanbemoved bytranslation toanother place). Assume that Moe’s axes have rotated relative toJoe’s byanangle 0. Thetwocoordinate systems areshown inFig. 11-2, which isrestricted totwo dimensions. Consider anypoint Phaving coordinates (x,y)inJoe’s system and (x',y’)inMoe’s system. Weshall begin, asintheprevious case, byexpressing thecoordinates x’andy’interms ofx,y,and0.Todoso,wefirstdrop perpendic- ulars from Ptoallfouraxesanddraw ABperpendicular toPQ. Inspection ofthe figure shows thatx’canbewritten asthesumoftwolengths along thex’-axis, andy’asthedifference oftwolengths along AB. Allthese lengths areexpressed 11-31' y (x,y) __.._ P(x',y') z/”%\ _’T\\ WM woe)0 \\] ,' xcoa0 0\ 1.106) 0 x Fig. ll-2. Two coordinate systems having different angular orientations. y Y F,__/_i ,/’ l\F; I\ x' \ F,- ii F, x Fig. ll-3. Components ofaforce in thetwosystems.interms ofx,y,and0inequations (11.5), towhich wehave added anequation forthethirddimension. x’=xcos6 +ysin6, y’=ycos6 —xsin 0, (11.5) z’=z. Thenextstepistoanalyze therelationship offorces asseenbythetwoobservers, following thesame general method asbefore. Letusassume thataforce F,which hasalready been analyzed ashaving components F,andF,(asseen byJoe), is acting onaparticle ofmass m,located atpoint PinFig.11-2. Forsimplicity, let usmove both setsofaxessothattheorigin isatP,asshown inFig.ll-3. Moe seesthecomponents ofFalong hisaxesasF,’andF,/.F,hascomponents along both thex’-andy’-axes, andFylikewise hascomponents along both these axes. Toexpress F11interms ofF,andF”,wesumthese components along thex’-axis, andinalikemanner wecanexpress F,‘interms ofF,andFy;Theresults are F,’=F,cos9+F,sin0, F,’=Fycos0—F,sin6, (11.6) F,»=F,. Itisinteresting tonote anaccident ofsorts, which isofextreme importance: the formulas (11.5) and(11.6), forcoordinates ofPandcomponents ofF,respectively, areofidentical form. Asbefore, Newton’s laws areassumed tobetrueinJoe’s system, andare expressed byequations (11.1). Thequestion, again, iswhether Moe canapply Newton’s 1aws—will theresults becorrect forhissystem ofrotated axes? Inother words, ifweassume thatEqs.(11.5) and(11.6) givetherelationship ofthemeasure- ments, isittrueornottruethat m(d2x’/dt2) =F,/, m(d2y’/dtz) =Fy», (11.7) m(d2z’/dt2) =FZ»? Totestthese equations, wecalculate theleftandright sides independently, and compare theresults. Tocalculate theleftsides, wemultiply equations (11.5) bym, anddifferentiate twice with respect totime, assuming theangle 0tobeconstant. This gives m(d2x’/dt2) =m(d2x/dt2) cos0+m(d2y/dtz) sin0, m(d2y’/dtz) =m(d2y/dtz) cos0—m(d2x/dtz) sin0, (11.8) m(d2z’/dt2) =m(d2z/dt2). Wecalculate theright sides ofequations (11.7) bysubstituting equations (11.1) intoequations (11.6). This gives F,’=m(d2x/dt2) cos0+m(d2y/dt2) sin0, Fy'=m(d2y/dt2) cos0—m(d2x/dt2) sin0, (11.9) F,’=m(d2z/dt2). Behold! Theright sides ofEqs.(11.8) and(11.9) areidentical, soweconclude thatifNewton’s laws arecorrect ononesetofaxes, they arealsovalid onany other setofaxes. Thisresult, which hasnowbeen established forboth translation androtation ofaxes, hascertain consequences: first,noonecanclaim hisparticular axes areunique, butofcourse theycanbemore convenient forcertain particular problems. Forexample, itishandy tohave gravity along oneaxis, butthisisnot physically necessary. Second, itmeans that anypiece ofequipment which is completely self-contained, with alltheforce-generating equipment completely in- sidetheapparatus, would work thesame when turned atanangle. 11-4 11-4 Vectors NotonlyNewton’s laws, butalsotheother lawsofphysics, sofarasweknow today, have thetwoproperties which wecallinvariance (orsymmetry) under translation ofaxesandrotation ofaxes. These properties aresoimportant thata mathematical technique hasbeen developed totakeadvantage ofthem inwriting andusing physical laws. The foregoing analysis involved considerable tedious mathematical work. Toreduce thedetails toaminimum intheanalysis ofsuch questions, averypower- fulmathematical machinery hasbeen devised. This system, called vector analysis, supplies thetitleofthischapter; strictly speaking, however, thisisachapter on thesymmetry ofphysical laws. Bythemethods ofthepreceding analysis wewere able todoeverything required forobtaining theresults thatwesought, butin practice weshould liketodothings more easily andrapidly, soweemploy the vector technique. Webegan bynoting some characteristics oftwokinds ofquantities thatare important inphysics. (Actually there aremore than two, butletusstart outwith two.) Oneofthem, likethenumber ofpotatoes inasack, wecallanordinary quantity, oranundirected quantity, orascalar. Temperature isanexample of such aquantity. Other quantities thatareimportant inphysics dohave direction, forinstance velocity: wehave tokeep track ofwhich wayabody isgoing, notjust itsspeed. Momentum andforce alsohave direction, asdoes displacement: when someone steps from oneplace toanother inspace, wecankeep track ofhow far hewent, butifwewish alsotoknow where hewent, wehave tospecify adirection. Allquantities thathave adirection, likeastepinspace, arecalled vectors. Avector isthree numbers. Inorder torepresent astepinspace, sayfrom the origin tosome particular point Pwhose location is(x,y,z),wereally need three numbers, butwearegoing toinvent asingle mathematical symbol, r,which is unlike anyother mathematical symbols wehave sofarused.* Itisnotasingle number, itrepresents three numbers: x,y,andz.Itmeans three numbers, but notreally onlythose three numbers, because ifwewere touseadifferent coordinate system, thethree numbers would bechanged tox’,y’,andz’.However, wewant tokeep ourmathematics simple andsowearegoing tousethesame mark torepre- sentthethree numbers (x,y,z)andthethree numbers (x’,y’,z’).That is,weuse thesame mark torepresent thefirstsetofthree numbers foronecoordinate system, butthesecond setofthree numbers ifweareusing theother coordinate system. This hastheadvantage thatwhen wechange thecoordinate system, wedonot have tochange theletters ofourequations. Ifwewrite anequation interms of x,y,z,andthen useanother system, wehave tochange tox’,y’,z’,butweshall justwrite 1-,withtheconvention thatitrepresents (x,y,z)ifweuseonesetofaxes, or(x’,y’,z’)ifweuseanother setofaxes, andsoon.Thethree numbers which describe thequantity inagiven coordinate system arecalled thecomponents ofthe vector inthedirection ofthecoordinate axesofthatsystem. That is,weusethe same symbol forthethree letters thatcorrespond tothesame object, asseenfrom different axes. Theveryfactthatwecansay“the same object” implies aphysical intuition about thereality ofastepinspace, thatisindependent ofthecomponents interms ofwhich wemeasure it.Sothesymbol rwillrepresent thesame thing nomatter howweturntheaxes. Now suppose there isanother directed physical quantity, anyother quantity, which also hasthree numbers associated with it,like force, and these three numbers change tothree other numbers byacertain mathematical rule, ifwe change theaxes. Itmust bethesame rulethatchanges (x,y,z)into(x’,y’,z’).In other words, anyphysical quantity associated withthree numbers which transform asdothecomponents ofastepinspace isavector. Anequation like F=r would thusbetrueinanycoordinate system ifitwere trueinone. This equation, *Intype, vectors arerepresented byboldface; inhandwritten form anarrow isusedzi. 11-5 ofcourse, stands forthethree equations Fxzxs Fu=y, FZ=zi or,alternatively, for F,’=x’, F,’=y’, F,’=z’. Thefactthataphysical relationship canbeexpressed asavector equation assures ustherelationship isunchanged byamere rotation ofthecoordinate system. That isthereason whyvectors aresouseful inphysics. Now letusexamine some oftheproperties ofvectors. Asexamples ofvectors wemaymention velocity, momentum, force, andacceleration. Formany purposes itisconvenient torepresent avector quantity byanarrow thatindicates thedirec- tioninwhich itisacting. Why canwerepresent force, say,byanarrow? Because ithasthesame mathematical transformation properties asa“step inspace.” We thusrepresent itinadiagram asifitwere astep, using ascale such thatoneunit offorce, oronenewton, corresponds toacertain convenient length. Once we have done this,allforces canberepresented aslengths, because anequation like F=kr, where kissome constant, isaperfectly legitimate equation. Thus wecanalways represent forces bylines, which isvery convenient, because once wehave drawn thelinewenolonger need theaxes. Ofcourse, wecanquickly calculate thethree components astheychange upon turning theaxes, because thatisjustageometric problem. 11-5 Vector algebra Now wemust describe thelaws, orrules, forcombining vectors invarious ways. Thefirstsuch combination istheaddition oftwovectors: suppose that aisavector which insome particular coordinate system hasthethree components (a,,av,az),andthatbisanother vector which hasthethree components (b,,bu,b,). Now letusinvent three newnumbers (a,+bx,a,,+b,,,a,+b,).Dothese form avector? “Well,” wemight say,“they arethree numbers, andevery three numbers form avector.” No,notevery three numbers form avector! Inorder forittobea vector, notonlymust there bethree numbers, butthese must beassociated with a coordinate system insuch awaythatifweturnthecoordinate system, thethree numbers “revolve” oneach other, get“mixed up”ineach other, bytheprecise lawswehave already described. Sothequestion is,ifwenowrotate thecoordinate system sothat(az,ay,a,)become (a,',av’,azl)and(b,,by,b,)become (b,',bu’,b3’), what do(a,+b,,,a,,+by,a,+b,)become? Dothey become (a,t+bx», a,’+bu’,azt+b3’)ornot? Theanswer is,ofcourse, yes,because theprototype transformations ofEq.(11.5) constitute what Wecallalinear transformation. Ifweapply those transformations toa,andb,togetax»+bx’,wefind that thetransformed a,+b,isindeed thesame asa,’+b,/. When aandbare “added together” inthissense, theywi1l,form avector which wemay callc.We would write thisas c=a+b. Now chastheinteresting property c=b+a, aswecanimmediately seefrom itscomponents. Thus also, a+(b+¢)= (=1+b)+c. Wecanaddvectors inanyorder. What isthegeometric significance ofa+b‘?Suppose that aandbwere represented bylines onapiece ofpaper, what would clook like? Thisisshown in 11-6 Fig.11-4. Weseethatwecanaddthecomponents ofbtothose ofamost con- veniently ifweplace therectangle representing thecomponents ofbnext tothat representing thecomponents ofainthemanner indicated. Since bjust“fits” intoitsrectangle, asdoes aintoitsrectangle, thisisthesame asputting the“tail” ofbonthe“head” ofa,thearrow from the“tail” ofatothe“head” ofbbeing thevector c.Ofcourse, ifweadded atobtheother wayaround, wewould putthe “tail” ofaonthe“head” ofb,andbythegeometrical properties ofparallelograms wewould getthesame result forc.Note thatvectors canbeadded inthisway without reference toanycoordinate axes. Suppose wemultiply avector byanumber a,what does thismean? We define ittomean anewvector whose components areaax,flay,andaaz. Weleave itasaproblem forthestudent toprove thatitisavector. Now letusconsider vector subtraction. Wemay define subtraction inthe same wayasaddition, butinstead ofadding, wesubtract thecomponents. Or wemight define subtraction bydefining anegative vector, —b=—lb,andthen wewould addthecomponents. Itcomes tothesame thing. Theresult isshown inFig.ll-5. This figure shows d=a—b=a+(—b); wealsonote thatthe difference a-bcanbefound very easily from aandbbyusing theequivalent relation a=b+d.Thus thedifference iseven easier tofindthan thesum: we justdraw thevector from btoa,togeta—b! Next wediscuss velocity. Why isvelocity avector? Ifposition isgiven bythe three coordinates (x,y,z),what isthevelocity? Thevelocity isgiven bydx/dt, dy/dt, anddz/dt Isthatavector, ornot? Wecanfindoutbydifferentiating the expressions inEq.(11.5) tofindoutwhether dx'/dt transforms intheright way. Weseethat thecomponents dx/dt anddy/dt dotransform according tothesame lawasxandy,andtherefore thetime derivative isavector. Sothevelocity isa vector. Wecanwrite thevelocity inaninteresting wayas v=dr/dt. What thevelocity is,andwhyitisavector. canalsobeunderstood more pictorially: How fardoes aparticle move inashort time At? Answer: Ar,soifaparticle is “here” atoneinstant and“there” atanother instant, then thevector difference ofthepositions Ar=r2-r1,which isinthedirection ofmotion shown inFig. ll-6, divided bythetime interval At=t2-11,isthe“average velocity” vector. Inother words, byvector velocity wemean thelimit, asAtgoes to0,ofthe difference between theradius vectors atthetime t+Atandthetime t,divided by At: v=limo(Ar/At) =dr/dt. (11.10)A-v Thus velocity isavector because itisthedifference oftwovectors. Itisalsothe right definition ofvelocity because itscomponents aredx/dt, dy/dt, anddz/dt. Infact,weseefrom thisargument thatifwedifferentiate anyvector with respect totime weproduce anew vector. Sowehave several ways ofproducing new vectors: (1)multiply byaconstant, (2)differentiate with respect totime, (3)add orsubtract twovectors. 11-6 Newton’s lawsinvector notation Inorder towrite Newton’s laws invector form, wehave togojustonestep further, anddefine theacceleration vector. This isthetime derivative ofthevelocity vector, anditiseasytodemonstrate thatitscomponents arethesecond derivatives ofx,y,andzwith respect tot: tl dd a2a=%= (EXQ =#, (11.11) db, d2x day d2y dv, d2z,=—=-, =-=—, ,=—=—- 11.12a dr dz? a” dz dt2 a dz dt2 ( ) 11-7i______Y F" 1 I ,1 Fig. 11-4. Theaddition ofvectors. a I -t 3=E—F Fig. ll-5. Thesubtraction ofvectors. I A°r=7,—?, 7, 1 l 7. o Fig. 11-6. The displacement ofa particle inashort time interval At= I2—ti. 1Fig. 11-7. Acurved trajectory.0 7' A17 Fig. 11-8. Diagram forcalculating theacceleration.With thisdefinition, then, Newton’s laws canbewritten inthisway: ma=F (11.13) or m(d2r/dtz) =F. (11.14) Now theproblem ofproving theinvariance ofNewton’s laws under rotation ofcoordinates isthis:prove thataisavector; thiswehavejustdone. Prove thatF isavector; wesuppose itis.Soifforce isavector, then, since weknow acceleration isavector, Eq.(11.13) willlook thesame inanycoordinate system. Writing itin aform which does notexplicitly contain x’s,y’s,andz’shastheadvantage that from nowonweneed notwrite three laws every timewewrite Newton’s equations orother laws ofphysics. Wewrite what looks likeonelaw,butreally, ofcourse, itisthethree laws foranyparticular setofaxes, because anyvector equation involves thestatement thateach ofthecomponents isequal. Thefactthattheacceleration istherateofchange ofthevector velocity helps ustocalculate theacceleration insome rather complicated circumstances. Suppose, forinstance, thataparticle ismoving onsome complicated curve (Fig. 11-7) and that, atagiven instant t,ithadacertain velocity v1,butthatwhen wegotoanother instant t2alittle later, ithasadifferent velocity v2.What istheacceleration? Answer: Acceleration isthedifference inthevelocity divided bythesmall time interval, soweneed thedifference ofthetwovelocities. How dowegetthediffer- ence ofthevelocities? Tosubtract twovectors, weputthevector across theends ofv2andv1;thatis,wedraw Aasthedifference ofthetwovectors, right? No! That onlyworks when thetailsofthevectors areinthesame place! Ithasnomean- ingifwemove thevector somewhere elseandthendraw alineacross, sowatch out! Wehave todraw anewdiagram tosubtract thevectors. InFig. ll-8, v1andv2 areboth drawn parallel andequal totheir counterparts inFig.11-7, andnow we candiscuss theacceleration. Ofcourse theacceleration issimply Av/At. Itis interesting tonote thatwecancompose thevelocity difference outoftwoparts; wecanthink ofacceleration ashaving twocomponents, Av“inthedirection tangent tothepath andAvLatright angles tothepath, asindicated inFig. ll-8. The acceleration tangent tothepath is,ofcourse, justthechange inthelength ofthe vector, i.e.,thechange inthespeed v: an=dv/dl. (11.15) Theother component ofacceleration, atright angles tothecurve, iseasy tocal- culate, using Figs. 11-7 andll-8. Intheshort time Atletthechange inangle between v1andV2bethesmall angle A6.Ifthemagnitude ofthevelocity iscalled 11,then ofcourse Av,=1»A0 andtheacceleration awillbe ai=11(A0/At). Now weneed toknow A0/At, which canbefound thisway: If,atthegiven moment, thecurve isapproximated asacircle ofacertain radius R,then inatime Atthe distance sis,ofcourse, vAt,where visthespeed. A0=v(At/R), or A0/At =v/R. Therefore, wefind a=02/R, (11.16) aswehave seenbefore. 11-7 Scalar product ofvectors Now letusexamine alittle further theproperties ofvectors. Itiseasytosee thatthelength ofastepinspace would bethesame inanycoordinate system. That is,ifaparticular steprisrepresented byx,y,z,inonecoordinate system, 11-8 andbyx’,y’,z’inanother coordinate system, surely thedistance r=Ir]would bethesame inboth. Now r=\/X2+1/2+2? rr=4/x/2 +_y/2 +Z/2. Sowhatwewishtoverify isthatthese twoquantities areequal. Itismuch more convenient nottobother totakethesquare root, soletustalkabout thesquare of thedistance; thatis,letusfindoutwhetherandalso x2+y2+22=x'2+y’2+2'2. (11.17) Ithadbetter be—and ifwesubstitute Eq.(11.5) wedoindeed findthatitis.So weseethatthereareother kinds ofequations which aretrueforanytwocoordinate systems. Something newisinvolved. Wecanproduce anewquantity, afunction of x,y,andz,called ascalar function, aquantity which hasnodirection butwhich is thesame inboth systems. Outofavector wecanmake ascalar. Wehave tofind ageneral ruleforthat. Itisclear what theruleisforthecasejustconsidered: addthesquares ofthecomponents. Letusnowdefine anewthing, which wecall a-a.Thisisnotavector, butascalar; itisanumber thatisthesame inallcoordi- natesystems, anditisdefined tobethesumofthesquares ofthethree components ofthevector: a-a=113+115+.12. (11.18) Now yousay,“But with what axes?” Itdoes notdepend ontheaxes, theanswer isthesame inevery setofaxes. Sowehave anewkindofquantity, anewinvariant orscalar produced byonevector “squared.” Ifwenow define thefollowing quantity foranytwovectors aandb: a~b=a,b,, +a,,b,, +a,b,, (11.19) wefindthatthisquantity, calculated intheprimed andunprimed systems, also stays thesame. Toprove itwenote thatitistrueofa-a,b-b,andc'c,where c=a+b.Therefore thesum ofthesquares (a,+b,,)2-1-(av+b,,)2+ (a,+b,)2willbeinvariant: (as: "1'b;1:)2 + (ay +bi/)2 + (az '1'bz)2 =(aa:’ '1‘b:c')2 -1-(fly —|-b,/)2 -1-(aZ' +l),r)2. (11.20) Ifboth sides ofthisequation areexpanded, there willbecross products ofjustthe typeappearing inEq.(11.19), aswellasthesums ofsquares ofthecomponents ofaandb.Theinvariance ofterms oftheform ofEq.(11.18) thenleaves thecross product terms (11.19) invariant also. Thequantity a-biscalled thescalar product oftwovectors, aandb,andit hasmany interesting anduseful properties. Forinstance, itiseasily proved that a~(b+c)=a-b+a-c. (11.21) Also, there isasimple geometrical waytocalculate a-b,without having tocal- culate thecomponents ofaandb:a-bistheproduct ofthelength ofaandthe length ofbtimes thecosine oftheangle between them. Why? Suppose thatwe choose aspecial coordinate system inwhich thex-axis liesalong a;inthose cir- cumstances, theonly component ofathatwillbethere isa,,which isofcourse thewhole length ofa.Thus Eq.(11.19) reduces toa-b=a,,b,, forthiscase, andthisisthelength ofatimes thecomponent ofbinthedirection ofa,thatis, bcos6: a-b =abcos9. Therefore, inthat special coordinate system, wehave proved that a-bisthe ll-9 Lngthofatimes thelength ofbtimes cos0.Butifitistrueinonecoordinate system, izistrueinall,because a-bisindependent ofthecoordinate system; thatisour argument. What good isthedotproduct? Arethere anycases inphysics where weneed it?Yes,weneed itallthetime. Forinstance, inChapter 4thekinetic energy was called %mv2, butiftheobject ismoving inspace itshould bethevelocity squared inthex-direction, they-direction, andthez-direction, andsotheformula for kinetic energy according tovector analysis is K.E.=%m(v-v) =%m(v:+11;+vi). (11.22) Energy does nothave direction. Momentum hasdirection; itisavector, anditis themass times thevelocity vector. Another example ofadotproduct isthework done byaforce when something ispushed from oneplace totheother. Wehave notyetdefined work, butitis equivalent totheenergy change, theweights lifted, when aforce Factsthrough adistance s: Work =F-s (11.23) Itissometimes veryconvenient totalkabout thecomponent ofavector ina certain direction (saythevertical direction because thatisthedirection ofgravity). Forsuch purposes, itisuseful toinvent what wecallaunitvector inthedirection that wewant tostudy. Byaunit vector wemean onewhose dotproduct with itself isequal tounity. Letuscallthisunitvector i;then i-i=1.Then, ifwewant thecomponent ofsome vector inthedirection ofi,weseethatthedotproduct a-iwillbeacos0,i.e.,thecomponent ofainthedirection ofi.This isanice waytogetthecomponent; infact,itpermits ustogetallthecomponents andto write arather amusing formula. Suppose thatinagiven system ofcoordinates, x,y,andz,weinvent three vectors: i,aunitvector inthedirection x;j,aunitvector inthedirection y;andk,aunitvector inthedirection z.Note firstthati-i=l. What isi'j?When twovectors areatright angles, their dotproduct iszero. Thus i'i=1 =0 j-k=0 k-k=l (11.24) ii~l xv1-C1-1-,._i Now with these definitions, anyvector whatsoever canbewritten thisway: a=a,,i+ayj-1-a,k. (11.25) Bythismeans wecangofrom thecomponents ofavector tothevector itself. This discussion ofvectors isbynomeans complete. However, rather than trytogomore deeply intotheSLlb_]CCi now, weshall firstlearn touseinphysical situations some oftheideas sofardiscussed. Then, when wehaveproperly mastered thisbasic material, weshall finditeasier topenetrate more deeply intothesubject without getting tooconfused. Weshall later findthatitisuseful todefine another kind ofproduct oftwovectors, called thevector product, andwritten asa><b. However, weshall undertake adiscussion ofsuch matters inalater chapter. 11-10 712 Characteristics ofForce 12-1 What isaforce? Although itisinteresting andworth while tostudy thephysical laws simply because theyhelpustounderstand andtousenature, oneought tostopevery once inawhile andthink, “What dothey really mean?” Themeaning ofanystatement isasubject thathasinterested andtroubled philosophers from timeimmemorial, andthemeaning ofphysical lawsiseven more interesting, because itisgenerally believed thatthese lawsrepresent some kind ofrealknowledge. Themeaning of knowledge isadeep problem inphilosophy, anditisalways important toask, “What does itmean?" Letusask,“What isthemeaning ofthephysical lawsofNewton, which we write asF=ma? What isthemeaning offorce, mass, andacceleration?” Well, wecanintuitively sense themeaning ofmass, andwecandefine acceleration ifwe know themeaning ofposition andtime. Weshall notdiscuss those meanings, butshall concentrate onthenewconcept offorce. Theanswer isequally simple: “Ifabody isaccelerating, thenthere isaforce onit.”That iswhat Newton’s laws say,sothemost precise andbeautiful definition offorce imaginable might simply betosaythatforce isthemass ofanobject times theacceleration. Suppose we have alawwhich saysthattheconservation ofmomentum isvalid ifthesum ofalltheexternal forces iszero; then thequestion arises, “What does itmean, thatthesumofalltheexternal forces iszero?” Apleasant waytodefine that statement would be:“When thetotalmomentum isaconstant, thenthesumofthe external forces iszero.” There must besomething wrong with that, because itis justnotsaying anything new. Ifwehave discovered afundamental law,which asserts thattheforce isequal tothemass times theacceleration, andthendefine the force tobethemass times theacceleration, wehave found outnothing. Wecould alsodefine force tomean thatamoving object withnoforce acting onitcontinues tomove with constant velocity inastraight line. Ifwethen observe anobject notmoving inastraight linewith aconstant velocity, wemight saythat there isaforce onit.Now suchthings certainly cannot bethecontent ofphysics, because theyaredefinitions going inacircle. TheNewtonian statement above, however, seems tobeamost precise definition offorce, andonethatappeals to themathematician; nevertheless, itiscompletely useless, because noprediction whatsoever canbemade from adefinition. Onemight sitinanarmchair allday longanddefine words atwill,buttofindoutwhat happens when twoballs push against each other, orwhen aweight ishung onaspring, isanother matter al- together, because theway thebodies behave issomething completely outside any choice ofdefinitions. Forexample, ifwewere tochoose tosaythatanobject lefttoitself keeps its position anddoes notmove, then when weseesomething drifting, wecould say thatmust beduetoa“gorce”——a gorce istherateofchange ofposition. Now we haveawonderful newlaw,everything stands stillexcept when agorce isacting. You see,thatwould beanalogous totheabove definition offorce, anditwould contain no information. Therealcontent ofNewton’s lawsisthis:thattheforce issupposed tohave some independent properties, inaddition tothelawF=ma;butthe specific independent properties thattheforce haswere notcompletely described byNewton orbyanybody else, andtherefore thephysical lawF=maisan incomplete law. Itimplies thatifwestudy themass times theacceleration and calltheproduct theforce, i.e.,ifwestudy thecharacteristics offorce asaprogram 12-112-1 What isaforce? 12-2 Friction 12-3 Molecular forces 12-4 Fundamental forces. Fields 12-5 Pseudo forces 12-6 Nuclear forces,,__.__i;-__f_._...1l .,., 111 Lf. 11‘..111: it 1111 l 1 11 i Q i l 1'1l 1 1 1t 1 _1_.__-...AT"~,‘11 1I t1 -=-$11.5-P‘:(<a-am.‘1.-1-.A~ ‘t 1f I r lI i E Qt‘li itt l 1-t _....___-_».;a._-‘_.__1- 1; 1ofinterest, thenweshall findthatforces have some simplicity; thelawisagood program foranalyzing nature, itisasuggestion thattheforces willbesimple. Now thefirstexample ofsuch forces wasthecomplete lawofgravitation, which wasgiven byNewton, andinstating thelawheanswered thequestion, “What istheforce?” Ifthere were nothing butgravitation, thenthecombination ofthislawandtheforce law(second lawofmotion) would beacomplete theory, butthere ismuch more than gravitation, andwewant touseNewton’s laws in many different situations. Therefore inorder toproceed wehave totellsomething about theproperties offorce. Forexample, indealing with force thetacitassumption isalways made that theforce isequal tozerounless some physical body ispresent, thatifwefinda force thatisnotequal tozerowealsofindsomething intheneighborhood that isasource oftheforce. Thisassumption isentirely dilierent from thecaseofthe “gorce” that weintroduced above. One ofthemost important characteristics of force isthatithasamaterial origin, andthisisnotjustadefinition. Newton alsogave oneruleabout theforce: thattheforces between interacting bodies areequal andopposite—action equals reaction; that rule, itturns out, is notexactly true. Infact, thelawF=maisnotexactly true; ifitwere adefinition weshould have tosaythatitisalways exactly true; butitisnot. Thestudent may object, “Idonotlikethisimprecision, Ishould liketohave everything defined exactly; infact, itsaysinsome books thatanyscience isanexact subject, inwhich everything isdefined.” Ifyouinsist upon aprecise definition of force, youwillnever getit‘First, because Newton's Second Lawisnotexact, and second, because inorder tounderstand physical laws youmust understand that theyareallsome kindofapproximation. Any simple ideaisapproximate; asanillustration, consider anobject, ... what isanobject? Philosophers arealways saying, “Well, justtakeachair for example.” Themoment theysaythat, youknow thattheydonotknow what theyaretalking about anymore. What isachair? Well, achair isacertain thing overthere ...certain?, howcertain? Theatoms areevaporating from itfrom time totime—-not many atoms, butafew—d1rt fallsonitandgetsdissolved inthepaint; sotodefine achair precisely, tosayexactly which atoms arechair, andwhich atoms areair,orwhich atoms aredirt,orwhich atoms arepaint thatbelongs to thechair isimpossible. Sothemass ofachair canbedefined onlyapproximately. Inthesame way, todefine themass ofasingle object isimpossible, because there arenotanysingle, left-alone objects intheworld—every object isamixture ofa lotofthings, sowecandealwithitonlyasaseries ofapproximations andidealiza- tions. Thetrick istheidealizations. Toanexcellent approximation ofperhaps one partin101°, thenumber ofatoms inthechair doesnotchange inaminute, andif wearenottooprecise wemayidealize thechair asadefinite thing; inthesame way weshall learn about thecharacteristics offorce, inanideal fashion, ifwearenot tooprecise. Onemaybedissatisfied with theapproximate view ofnature that physics tries toobtain (theattempt isalways toincrease theaccuracy ofthe approximation), andmay prefer amathematical definition; butmathematical definitions cannever work intherealworld. Amathematical definition willbe good formathematics, inwhich allthelogic canbefollowed outcompletely, but thephysical world iscomplex, aswehave indicated inanumber ofexamples, such asthose oftheocean waves andaglass ofwine. When wetrytoisolate pieces ofit, totalkabout onemass, thewine andtheglass, howcanweknow which iswhich, when onedissolves intheother? Theforces onasingle thing already involve approximation, andifwehave asystem ofdiscourse about therealworld, then that system, atleast forthepresent day, must involve approximations ofsome kind. This system isquite unlike thecase ofmathematics, inwhich everything can bedefined, andthen wedonotknow what wearetalking about. Infact, theglory ofmathematics isthat wedonothave tosaywhat wearetalking about. Theglory isthat thelaws, thearguments, andthelogic areindependent ofwhat “it”is.If wehave anyother setofobjects that obey thesame system ofaxioms asEuclid‘s l2-2 geometry, thenifwemake newdefinitions andfollow them outwithcorrect logic, alltheconsequences willbecorrect, anditmakes nodilference what thesubject was. Innature, however, when wedraw alineorestablish alinebyusing alight beam andatheodolite, aswedoinsurveying, arewemeasuring alineinthesense ofEuclid? No,wearemaking anapproximation; thecross hairhassome width, butageometrical linehasnowidth, andso,whether Euclidean geometry canbe used forsurveying ornotisaphysical question, notamathematical question. However, from anexperimental standpoint, notamathematical standpoint, we need toknow whether thelaws ofEuclid apply tothekind ofgeometry thatwe useinmeasuring land; sowemake ahypothesis thatitdoes, anditworks pretty well; butitisnotprecise, because oursurveying lines arenotreally geometrical lines. Whether ornotthose lines ofEuclid, which arereally abstract, apply tothe lines ofexperience isaquestion forexperience; itisnotaquestion thatcanbe answered bysheer reason. Inthesame way, wecannot justcallF=maadefinition, deduce everything purely mathematically, and make mechanics amathematical theory, when me- chanics isadescription ofnature. Byestablishing suitable postulates itisalways possible tomake asystem ofmathematics, justasEuclid did,butwecannot make amathematics oftheworld, because sooner orlater wehave tofindoutwhether theaxioms arevalid fortheobjects ofnature. Thus weimmediately getinvolved with these complicated and“dirty” objects ofnature, butwith approximations everincreasing inaccuracy. 12-2 Friction Theforegoing considerations show thatatrueunderstanding ofNewton’s lawsrequires adiscussion offorces, anditisthepurpose ofthischapter tointroduce such adiscussion, asakind ofcompletion ofNewton’s laws. Wehave already studied thedefinitions ofacceleration andrelated ideas, butnowwehave tostudy theproperties offorce, andthischapter, unlike theprevious chapters, willnotbe veryprecise, because forces arequite complicated. Tobegin with aparticular force, letusconsider thedrag onanairplane flying through theair.What isthelawforthatforce? (Surely there isalawfor every force, wemust have alaw!) Onecanhardly think thatthelawforthatforce willbesimple. Trytoimagine what makes adrag onanairplane flying through theair—the airrushing overthewings, theswirling intheback, thechanges going onaround thefuselage, andmany other complications, andyouseethatthere is notgoing tobeasimple law. Ontheother hand, itisaremarkable factthatthe drag force onanairplane isapproximately aconstant times thesquare ofthe velocity, orF~cu”. Now what isthestatus ofsuch alaw, isitanalogous toF=ma‘? Notatall, because inthefirstplace thislawisanempirical thing thatisobtained roughly by tests inawind tunnel. You say,“Well F=mamight beempirical too.” That is notthereason thatthere isadifference. Thedifference isnotthatitisempirical, butthat, asweunderstand nature, thislawistheresult ofanenormous complexity ofevents andisnot,fundamentally, asimple thing. Ifwecontinue tostudy itmore andmore, measuring more andmore accurately, thelawwillcontinue tobecome more complicated, notless. Inother words, aswestudy thislawofthedragonan airplane more andmore closely, wefindoutthatitis“falser” and“falser,” and themore deeply westudy it,andthemore accurately wemeasure, themore compli- cated thetruth becomes; sointhatsense weconsider itnottoresult from asimple, fundamental process, which agrees withouroriginal surmise. Forexample, ifthe velocity isextremely low,solowthatanordinary airplane isnotflying, aswhen theairplane isdragged slowly through theair,thenthelawchanges, andthedrag friction depends more nearly linearly onthevelocity. Totakeanother example, thefrictional dragonaballorabubble oranything thatismoving slowly through aviscous liquid likehoney, isproportional tothevelocity, butformotion sofast thatthefluid swirls around (honey does notbutwater andairdo)thenthedrag becomes more nearly proportional tothesquare ofthevelocity (F=cv2), and 12-3i1 :.:5?-"Q;-mm‘ 111 ii 1 iil1 i ‘Z1 21 *1l 11 p5»‘._<__.._.___ . ".I?_...._....--r‘'glf11 ii!. W01-:-iii... ——=_—_-_—____s___*111 jl i 1’'1l1 411 *11l. -»<s.r“-e-1l1 111 ,1H1 1t 1 1 ,lI 1 1z11 1-,1,. 1-litll; -.-1-‘__.-.111'11’ti.»1I11 I 11'|. 1l 111 _.__§.__ st 11£1 til1 —> DIRECTION OFMOTION >5, R N 11 Fig.l2-l. Therelation between fric- tional force and thenormal force for sliding contact.ifthevelocity continues toincrease, theneventhislawbegins tofail. People who say,“Well thecoefficient changes slightly,” aredodging theissue. Second, there areother great complications: canthisforce ontheairplane bedivided oranalyzed asaforce onthewings, aforce onthefront, andsoon?Indeed, thiscanbedone, ifweareconcerned about thetorques here andthere, butthen wehave toget special laws fortheforce onthewings, andsoon. Itisanamazing factthatthe force onawing depends upon theother wing: inother words, ifwetake theairplane apart andputjustonewing intheair,then theforce isnotthesame asiftherest oftheplane were there. Thereason, ofcourse, isthat some ofthewind thathits thefront goes around tothewings andchanges theforce onthewings. Itseems a miracle thatthere issuch asimple, rough, empirical lawthatcanbeused inthe design ofairplanes, butthislawisnotinthesame class asthebasic lawsofphysics, andfurther study ofitwillonlymake itmore andmore complicated. Astudy of howthecoeflicient cdepends ontheshape ofthefront oftheairplane is,toput itmildly, frustrating. There justisnosimple lawfordetermining thecoefiicient interms oftheshape oftheairplane. Incontrast, thelawofgravitation issimple, andfurther study onlyindicates itsgreater simplicity. Wehavejustdiscussed twocases offriction, resulting from fastmovement in airandslow movement inhoney. There isanother kind offriction, called dry friction orsliding friction, which occurs when onesolid body slides onanother. Inthiscaseaforce isneeded tomaintain motion. Thisiscalled africtional force, anditsorigin, also, isaverycomplicated matter. Both surfaces ofcontact are irregular, onanatomic level. There aremany points ofcontact where theatoms seem tocling together, andthen, asthesliding body ispulled along, theatoms snap apart andvibration ensues; something likethathastohappen. Formerly themechanism ofthisfriction wasthought tobeverysimple, thatthesurfaces were merely fullofirregularities andthefriction originated inlifting theslider overthebumps; butthiscannot be,forthere isnolossofenergy inthatprocess, whereas power isinfactconsumed. Themechanism ofpower lossisthatasthe slider snaps over thebumps, thebumps deform andthen generate waves and atomic motions and,after awhile, heat, inthetwobodies. Now itisveryremark- ablethatagain, empirically, thisfriction canbedescribed approximately bya simple law. Thislawisthattheforce needed toovercome friction andtodragone object over another depends upon thenormal force (i.e., perpendicular tothe surface) between thetwosurfaces thatareincontact. Actually, toafairly good approximation, thefrictional force isproportional tothisnormal force, andhas amore orlessconstant coelficient; thatis, F=;.1N, (12.1) where 1.1iscalled thecoefiicient offriction (Fig. 12-1). Although thiscoefficient is notexactly constant, theformula isagood empirical ruleforjudging approxi- mately theamount offorce thatwillbeneeded incertain practical orengineering circumstances. Ifthenormal force orthespeed ofmotion getstoobig,thelawfails because oftheexcessive heat generated. Itisimportant torealize thateach ofthese empirical laws hasitslimitations, beyond which itdoes notreally work. That theformula F=,aNisapproximately correct canbedemonstrated by asimple experiment. Wesetupaplane, inclined atasmall angle 0,andplace a block ofweight Wontheplane. Wethen tilttheplane atasteeper angle, until theblock justbegins toslide from itsown weight. Thecomponent oftheweight downward along theplane isWsin6,andthismust equal thefrictional force F when theblock issliding uniformly. Thecomponent oftheweight normal tothe plane isWcos 0,andthisisthenormal force N.With these values, theformula becomes Wsin 0=11Wcos0,from which weget1.1=sin6/cos 0=tan0.If thislawwereexactly true,anobject would starttoslideatsome definite inclination. Ifthesame block isloaded byputting extra weight onit,then, although Wis increased, alltheforces intheformula areincreased inthesame proportion, and Wcancels out. Ifjastays constant, theloaded block willslide again atthesame slope. When theangle 0isdetermined bytrialwiththeoriginal weight, itisfound 12-4 thatwiththegreater weight theblock willslideatabout thesame angle. Thiswill betrueeven when oneweight ismany times asgreat astheother, andsowecon- clude thatthecoeflicient offriction isindependent oftheweight. Inperforming thisexperiment itisnoticeable that when theplane istilted atabout thecorrect angle 0,theblock does notslide steadily butinahalting fashion. Atoneplace itmay stop, atanother itmay move with acceleration. This behavior indicates thatthecoefficient offriction isonly roughly aconstant, andvaries from place toplace along theplane. Thesame erratic behavior isobserved whether the block isloaded ornot. Such variations arecaused bydifferent degrees ofsmooth- nessorhardness oftheplane, andperhaps dirt,oxides, orother foreign matter. Thetables thatlistpurported values of;.1for“steel onsteel,” “copper oncopper,” andthelike,areallfalse, because theyignore thefactors mentioned above, which really determine /.1.Thefriction isnever dueto“copper oncopper," etc.,butto theimpurities clinging tothecopper. Inexperiments ofthetypedescribed above, thefriction isnearly independent ofthevelocity. Many people believe thatthefriction tobeovercome toget something started (static friction) exceeds theforce required tokeep itsliding (sliding friction), butwithdrymetals itisveryhard toshow anydifference. The opinion probably arises from experiences where small bitsofoilorlubricant are present, orwhere blocks, forexample, aresupported bysprings orother flexible supports sothattheyappear tobind. Itisquite difficult todoaccurate quantitative experiments infriction, andthe lawsoffriction arestillnotanalyzed verywell, inspiteoftheenormous engineering value ofanaccurate analysis. Although thelawF=j.iNisfairly accurate once the surfaces arestandardized, thereason forthisform ofthelawisnotreally under- understood. Toshow thatthecoeflicient itisnearly independent ofvelocity requires some delicate experimentation, because theapparent friction ismuch reduced ifthelower surface vibrates veryfast. When theexperiment isdone at veryhighspeed, caremust betaken thattheobjects donotvibrate relative toone another, since apparent decreases ofthefriction athigh speed areoften dueto vibrations. Atanyrate, thisfriction lawisanother ofthose semiempirical laws thatarenotthoroughly understood. andinview ofallthework thathasbeen done itissurprising thatmore understanding ofthisphenomenon hasnotcome about. Atthepresent time, infact,itisimpossible eventoestimate thecoeflicient offriction between twosubstances. Itwaspointed outabove thatattempts tomeasure itbysliding puresubstances such ascopper oncopper willlead tospurious results, because thesurfaces in contact arenotpure copper, butaremixtures ofoxides and other impurities. Ifwetrytogetabsolutely pure copper, ifweclean andpolish thesurfaces, outgas thematerials inavacuum, andtake every conceivable precaution, westilldonot getii.Forifwetilttheapparatus even toavertical position, theslider willnot fallofl—the twopieces ofcopper stick together! Thecoefficient ii,which isordi- narily lessthan unity forreasonably hard surfaces, becomes several times unity! Thereason forthisunexpected behavior isthatwhen theatoms incontact areall ofthesame kind, there isnowayfortheatoms to“know” thattheyareindifferent pieces ofcopper. When there areother atoms, intheoxides andgreases and more complicated thin surface layers ofcontaminants inbetween, theatoms “know” when theyarenotonthesame part. When weconsider thatitisforces between atoms thathold thecopper together asasolid, itshould become clear thatitisimpossible togettheright coefficient offriction forpure metals. Thesame phenomenon canbeobserved inasimple home-made experiment withaflatglass plate andaglass tumbler. Ifthetumbler isplaced ontheplate and pulled along withaloopofstring, itslides fairly wellandonecanfeelthecoefiicient offriction; itisalittleirregular, butitisacoefficient. Ifwenowwettheglass plate andthebottom ofthetumbler andpullagain, wefindthatitbinds, andifwelook closely weshall findscratches, because thewater isabletoliftthegrease andthe other contaminants ofi"thesurface, andthenwereally haveaglass-to-glass contact; thiscontact issogood thatitholds tight andresists separation somuch thatthe glass istornapart; thatis,itmakes scratches. 12-5 F” 1 1 I 1 ‘L 1 L1‘ >11'1 llll l l 1 § 1‘ 1: 1 1 l 1 LF REPULSION O >v F=k/r7 ATTRACTION Fig. l2—2. The force between two atoms ascifunction oftheir distance of separation.12-3 Molecular forces Weshall nextdiscuss thecharacteristics ofmolecular forces. These areforces between theatoms, andaretheultimate origin offriction. Molecular forces have never been satisfactorily explained onabasis ofclassical physics; ittakes quantum mechanics tounderstand them fully. Empirically, however, theforce between atoms isillustrated schematically inFig. 12-2, where theforce Fbetween two atoms isplotted asafunction ofthedistance rbetween them. There aredifferent cases: inthewater molecule, forexample, thenegative charges sitmore onthe oxygen, andthemean positions ofthenegative charges andofthepositive charges arenotatthesame point; consequently, another molecule nearby feelsarelatively large force, which iscalled adipole-dipole force. However, formany systems thecharges areverymuch better balanced, inparticular foroxygen gas,which is perfectly symmetrical. Inthiscase, although theminus charges andtheplus charges aredispersed over themolecule, thedistribution issuch that thecenter oftheminus charges and thecenter oftheplus charges coincide. Amolecule where thecenters donotcoincide iscalled apolar molecule, andcharge times the separation between centers iscalled thedipole moment. Anonpolar molecule is oneinwhich thecenters ofthecharges coincide. Forallnonpolar molecules, in which alltheelectrical forces areneutralized, itnevertheless turns outthat the force atvery large distances isanattraction andvaries inversely astheseventh power ofthedistance, orF=k/r7, where kisaconstant that depends onthe molecules. Why thisisweshall learn only when welearn quantum mechanics. When there aredipoles theforces aregreater. When atoms ormolecules gettoo close theyrepel with averylarge repulsion; thatiswhat keeps usfrom falling through thefloor! These molecular forces canbedemonstrated inafairly direct way: oneof these isthefriction experiment withasliding glass tumbler; another istotaketwo verycarefully ground andlapped surfaces which areveryaccurately flat,sothat thesurfaces canbebrought veryclose together. Anexample ofsuch surfaces is theJohansson blocks thatareused inmachine shops asstandards formaking accurate length measurements. Ifonesuchblock isslidoveranother verycarefully andtheupper oneislifted, theother onewilladhere andalsobelifted bythemolec- ularforces, exemplifying thedirect attraction between theatoms ononeblock fortheatoms ontheother block. Nevertheless these molecular forces ofattraction arestillnotfundamental inthesense thatgravitation isfundamental; theyareduetothevastly complex interactions ofalltheelectrons andnuclei inonemolecule withalltheelectrons and nuclei inanother. Any simple-looking formula wegetrepresents asum- mation ofcomplications, sowestillhave notgotthefundamental phenomena. Since themolecular forces attract atlarge distances andrepel atshort dis- tances, asshown inFig.l2-2, wecanmake upsolids inwhich alltheatoms are heldtogether bytheirattractions andheldapart bytherepulsion thatsetsinwhen theyaretooclose together. Atacertain distance d(where thegraph inFig.12-2 crosses theaxis) theforces arezero, which means thattheyareallbalanced, sothat themolecules staythatdistance apart from oneanother. Ifthemolecules are pushed closer together thanthedistance dtheyallshow arepulsion, represented by theportion ofthegraph above ther-axis. Topush themolecules only slightly closer together requires agreat force, because themolecular repulsion rapidly becomes verygreat atdistances lessthan d.Ifthemolecules arepulled slightly apart there isaslight attraction, which increases astheseparation increases. If they arepulled sufficiently hard, they will separate permanently—-the bond is broken. Ifthemolecules arepushed only avery small distance closer, orpulled only averysmall distance farther than d,thecorresponding distance along thecurve of Fig. 12-2 isalso very small, andcanthen beapproximated byastraight line. Therefore, inmany circumstances, ifthedisplacement isnottoogreat theforce is proportional tothedisplacement. This principle isknown asHooke’s law, orthe lawofelasticity, which saysthattheforce inabody which triestorestore thebody 12-6 toitsoriginal condition when itisdistorted isproportional tothedistortion. This law,ofcourse, holds trueonlyifthedistortion isrelatively small; when itgetstoo large thebody willbetorn apart orcrushed, depending onthekind ofdistortion. Theamount offorce forwhich Hooke’s lawisvalid depends upon thematerial; for instance, fordough orputty theforce isverysmall, butforsteelitisrelatively large. H0oke’s lawcanbenicely demonstrated withalongcoilspring, made ofsteeland suspended vertically. Asuitable weight hung onthelower endofthespring produces atinytwist throughout thelength ofthewire, which results inasmall vertical deflection ineach turnandadds uptoalarge displacement ifthere are many turns. Ifthetotal elongation produced, say,bya100-gram weight, ismeas- ured, itisfound thatadditional weights of100grams willeach produce anaddi- tional elongation thatisverynearly equal tothestretch thatwasmeasured forthe first100grams. Thisconstant ratio offorce todisplacement begins tochange when thespring isoverloaded, i.e.,Hooke’s lawnolonger holds. 12-4 Fundamental forces. Fields Weshall nowdiscuss theonlyremaining forces thatarefundamental. We callthem fundamental inthesense that their laws arefundamentally simple. We shall firstdiscuss electrical force. Ob_]€CllS carry electrical charges which consist simply ofelectrons orprotons. Ifanytwobodies areelectrically charged, there isanelectrical force between them, andifthemagnitudes ofthecharges are qlandq2,respectively, theforce varies inversely asthesquare ofthedistance between thecharges, orF=(const) qlqg/r2. Forunlike charges, thislawislike thelawofgravitation, butforlikecharges theforce isrepulsive andthesign (direction) isreversed. Thecharges qlandq2canbeintrinsically either positive ornegative, andinanyspecific application oftheformula thedirection ofthe force willcome outright iftheq’saregiven theproper plusorminus sign; theforce isdirected along thelinebetween thetwocharges. Theconstant intheformula depends, ofcourse, upon what units areused fortheforce, thecharge, andthe distance. Incurrent practice thecharge ismeasured incoulombs, thedistance in meters, andtheforce innewtons. Then, inorder togettheforce tocome outprop- erlyinnewtons, theconstant (which forhistorical reasons iswritten l/41re0) takes thenumerical value en=8.854 XlO_12coul2/newton-m2 or l/41re(, =8.99 Xl09n -m2/coul2. Thus theforce lawforstatic charges is F=q1qgl'/4Tl'€0!'3. Innature, themost important charge ofallisthecharge onasingle electron, which is1.60 XlO_19 coulomb. Inworking with electrical forces between fundamental particles rather than with large charges, many people prefer thecombination (qe1)2/41re0, inwhich qoiisdefined asthecharge onanelectron. This combination occurs frequently, andtosimplify calculations ithasbeen defined bythesymbol e2;itsnumerical value inthemkssystem ofunits turns outtobe(1.52 XlO_1“)2. The advantage ofusing theconstant inthisform isthat theforce between two electrons innewtons canthen bewritten simply ase2/r2, with rinmeters, without alltheindividual constants. Electrical forces aremuch more complicated than this simple formula indicates, since theformula gives theforce between two objects only when theobjects arestanding still. Weshall consider themore general case shortly. Intheanalysis offorces ofthemore fundamental kinds (not such forces as friction, buttheelectrical force orthegravitational force), aninteresting andvery important concept hasbeen developed. Since atfirst sight theforces arevery much more complicated than isindicated bytheinverse-square laws andthese laws hold trueonly when theinteracting bodies arestanding still, animproved I2-7at i. Y ,i ‘i A 1 I 4 v. I‘ ii~:‘ il , l 1 I -1l 0 l i 4 l I method isneeded todealwiththeverycomplex forces thatensue when thebodies start tomove inacomplicated way. Experience hasshown thatanapproach known astheconcept ofa“field” isofgreat utility fortheanalysis offorces ofthis type. Toillustrate theideafor,say,electrical force, suppose wehave twoelectrical charges, qlandqg,located atpoints PandRrespectively. Then theforce between thecharges isgiven by F=qlqgr/r3. (12.3) Toanalyze thisforce bymeans ofthefield concept, wesaythatthecharge qlatP produces a“condition” atR,such thatwhen thecharge q2isplaced atRit“feels” theforce. This isoneway, strange perhaps, ofdescribing it;wesaythattheforce Fonq2atRcanbewritten intwoparts. Itisq2multiplied byaquantity Ethat would bethere whether q2were there ornot(provided wekeep alltheother charges intheir right places). Eisthe“condition” produced byql,wesay,andFisthe response ofq2toE.Eiscalled anelectric field, anditisavector. Theformula for theelectric field Ethatisproduced atRbyacharge qlatPisthecharge qltimes theconstant 1/41re0 divided byr2(risthedistance fromP toR),anditisacting in thedirection oftheradius vector (theradius vector rdivided byitsownlength). The expression forEisthus E=qlr/4-rre0r3. (12.4) Wethenwrite F=q,E, (12.5) which expresses theforce, thefield, andthecharge inthefield. What isthepoint ofallthis? Thepoint istodivide theanalysis into twoparts. One part says that something produces afield. The other part says that something isacted onby thefield. Byallowing ustolook atthetwoparts independently, thisseparation oftheanalysis simplifies thecalculation ofaproblem inmany situations. If many charges arepresent, wefirstwork outthetotal electric field produced atR byallthecharges, andthen, knowing thecharge thatisplaced atR,wefind the force onit. Inthecaseofgravitation, wecandoexactly thesame thing. Inthiscase, where theforce F=—Gm1m2r/r3, wecanmake ananalogous analysis, asfollows: theforce onabody inagravitational fieldisthemass ofthatbody times thefield C.The force onm2isthemass m2times thefield Cproduced byml; that is, F=m2C. Then thefieldCproduced byabody ofmass mlisC=—Gm1r/r3 anditisdirected radially, asintheelectrical case. Inspite ofhowitmight atfirstseem, thisseparation ofonepartfrom another isnotatriviality. Itwould betrivial, justanother wayofwriting thesame thing, ifthelaws offorce were simple, butthelaws offorce aresocomplicated thatit turns outthatthefields have areality thatisalmost independent oftheobjects which create them. Onecandosomething likeshake acharge andproduce an effect, afield, atadistance; ifonethenstops moving thecharge, thefieldkeeps track ofallthepast, because theinteraction between twoparticles isnotin- stantaneous. Itisdesirable tohave some waytoremember what happened previ- ously. Iftheforce upon some charge depends upon where another charge was yesterday, which itdoes, thenweneed machinery tokeep track ofwhat went on yesterday, andthatisthecharacter ofafield. Sowhen theforces getmore compli- cated, thefieldbecomes more andmore real,andthistechnique becomes lessand lessofanartificial separation. Inanalyzing forces bytheuseoffields, weneed twokinds oflawspertaining tofields. Thefirstistheresponse toafield, andthatgives theequations ofmotion. Forexample, thelawofresponse ofamass toagravitational fieldisthattheforce isequal tothemass times thegravitational field; or,ifthere isalsoacharge onthebody, theresponse ofthecharge totheelectric fieldequals thecharge times theelectric field. Thesecond partoftheanalysis ofnature inthese situations isto formulate thelawswhich determine thestrength ofthefieldandhowitisproduced. These laws aresometimes called thefield equations. Weshall learn more about them induetime, butshall write down afewthings about them now. l2—8 First, themost remarkable factofall,which istrueexactly andwhich can beeasily understood, isthatthetotalelectric fieldproduced byanumber ofsources isthevector sumoftheelectric fields produced bythefirstsource, thesecond source, andsoon.Inother words, ifwehave numerous charges making afield, andifall byitself oneofthem would make thefieldEl,another would make thefieldE2, andsoon,thenwemerely addthevectors togetthetotal field. This principle canbeexpressed as or,inview ofthedefinition given above, E—Z‘”’- (12.7)T —is,47T€()l', Canthesame methods beapplied togravitation? Theforce between two masses mlandm2wasexpressed byNewton asF=Gmlmgr/r3. Butaccording tothefield concept, wemay saythat mlcreates afield Cinallthesurrounding space, such that theforce onm2isgiven by F=m2C. (12.8) Bycomplete analogy withtheelectrical case, c=—Gm,r,/r? (12.9) andthegravitational fieldproduced byseveral masses is C=Cl+C2+C3+"' (12.10) InChapter 7,inworking outacase ofplanetary motion, weused thisprinciple in essence. Wesimply added alltheforce vectors togettheresultant force ona planet. Ifwedivide outthemass oftheplanet inquestion, wegetEq.(12.10). Equations (12.6) and(12.10) express what isknown astheprinciple ofsuper- position offields. Thisprinciple states thatthetotal fieldduetoallthesources is thesumofthefields duetoeach source. Sofarasweknow today, forelectricity thisisanabsolutely guaranteed law,which istrueeven when theforce lawis complicated because ofthemotions ofthecharges. There areapparent violations, butmore careful analysis hasalways shown these tobeduetotheoverlooking of certain moving charges. However, although theprinciple ofsuperposition applies exactly forelectrical forces, itisnotexact forgravity ifthefieldistoostrong, and Newton’s equation (12.10) isonly approximate, according toEinstein’s gravita- tional theory. ‘ Closely related toelectrical force isanother kind, called magnetic force, and thistooisanalyzed interms ofafield. Some ofthequalitative relations between electrical andmagnetic forces canbeillustrated byanexperiment withanelectron- raytube(Fig. 12-3). Atoneendofsuchatubeisasource thatemits astream of electrons. Within thetube arearrangements foraccelerating theelectrons toa highspeed andsending some ofthem inanarrow beam toafluorescent screen at theother endofthetube. Aspotoflight glows inthecenter ofthescreen where theelectrons strike, andthisenables ustotrace theelectron path. Onthewayto thescreen theelectron beam passes through anarrow space between apairof parallel metal plates, which arearranged, say,horizontally. Avoltage canbe applied across theplates, sothateither plate canbemade negative atwill. When suchavoltage ispresent, there isanelectric fieldbetween theplates. Thefirst part oftheexperiment istoapply anegative voltage tothelower plate, which means thatextra electrons have beenplaced onthelower plate. Since likecharges repel, thelight spotonthescreen instantly shifts upward. (Wecould alsosaythisinanother way—that theelectrons “felt” thefield, andresponded by deflecting upward.) Wenext reverse thevoltage, making theupper plate negative. Thelight spot onthescreen nowjumps below thecenter, showing thattheelectrons inthebeam were repelled bythose intheplate above them. (Orwecould sayagain 12-9OM____4‘| +V II IIlIIl4=1 L/,-:—lJ\(‘TK-3v1'\\\-’-*5\‘——’l—lELECTRON GUN HOT FlLAMENT— etzcmou souncs \ Rises", -— an Fig. 12-3. Anelectron-beam tube thattheelectrons had“responded” tothefield, which isnowinthereverse direc- tion.) Thesecond part oftheexperiment istodisconnect thevoltage from the plates andtesttheefl"ect ofamagnetic fieldontheelectron beam. Thisisdone by means ofahorseshoe magnet, whose poles arefarenough apart tomore orless straddle thetube. Suppose weholdthemagnet below thetubeinthesame orienta- tionastheletter U,withitspoles upandpartofthetubeinbetween. Wenote thatthelight spotisdeflected, say,upward, asthemagnet approaches thetube from below. Soitappears thatthemagnet repels theelectron beam. However, it isnotthatsimple, forifweinvert themagnet without reversing thepoles side-for- side, andnowapproach thetubefrom above, thespotstillmoves upward, sothe electron beam isnotrepelled; instead, itappears tobeattracted thistime. Now westart again, restoring themagnet toitsoriginal Uorientation andholding it below thetube, asbefore. Yes, thespotisstilldeflected upward; butnowturn themagnet 180degrees around avertical axis,sothatitisstillintheUposition butthepoles arereversed side-for-side. Behold, thespotnowjumps downward, andstays down, evenifweinvert themagnet andapproach from above, asbefore. Tounderstand thispeculiar behavior, wehave tohave anewcombination offorces. Weexplain itthus: Across themagnet from onepoletotheother there isamagnetic field. This field hasadirection which isalways away from one particular pole(which wecould mark) andtoward theother. Inverting themagnet didnotchange thedirection ofthefield, butreversing thepoles side-for-side did reverse itsdirection. Forexample, iftheelectron velocity were horizontal inthe x-direction andthemagnetic fieldwere alsohorizontal butinthey-direction, the magnetic force onthemoving electrons would beinthez-direction, i.e.,upordown, depending onwhether thefieldwasinthepositive ornegative y-direction. Although weshallnotatthepresent timegivethecorrect lawofforce between charges moving inanarbitrary manner, onerelative totheother, because itistoo complicated, weshall give oneaspect ofit:thecomplete lawoftheforces ifthe fields areknown. The force onacharged object depends upon itsmotion; if, when theobject isstanding stillatagiven place, there issome force, thisistaken tobeproportional tothecharge, thecoefficient being what wecalltheelectric field. When theobject moves theforce may bediflerent, and thecorrection, thenew “piece” offorce, turns outtobedependent exactly linearly onthevelocity, butat right angles tovand toanother vector quantity which wecall themagnetic induction B.Ifthecomponents oftheelectric field Eandthemagnetic induction Bare,respectively, (E,,,Ey,E2)and(BI,By,B2),andifthevelocity vhasthe components (11,,v,,,0,),then thetotal electric andmagnetic force onamoving charge qhasthecomponents Fa: : +vyBz —vzBy): Fl,=q(E,, +v,B, ——1123,), (12.11) F,=q(E, +v,B,, —v,,B,,). If,forinstance, theonly component ofthemagnetic field were Byandtheonly component ofthevelocity were 21,,thentheonlyterm leftinthemagnetic force would beaforce inthez-direction, atright angles toboth Bandv. 12-5 Pseudo forces Thenextkind offorce weshall discuss might becalled apseudo force. In Chapter llwediscussed therelationship between twopeople, JoeandMoe, who usediflerent coordinate systems. Letussuppose thatthepositions ofaparticle asmeasured byJoearexandbyMoe arex’;thenthelawsareasfollows: x=x’+s, y=y’, Z=Z’, where sisthedisplacement ofMoe’s system relative toJoe’s. Ifwesuppose that 12-10 thelaws ofmotion arecorrect forJoe,how dothey look forMoe? Wefindfirst, that dx/df=ax’/at+ds/dt. Previously, weconsidered thecase where swasconstant, and wefound that s made nodiflerence inthelaws ofmotion, since ds/dt =0;ultimately, therefore, thelaws ofphysics were thesame inboth systems. Butanother case wecantake is thats=ut,where uisauniform velocity inastraight line. Then sisnotconstant, andds/dt isnotzero, butisu,aconstant. However, theacceleration d2x/1112 isstillthesame asd2x'/dl2, because du/dz =O.This proves thelawthatweused inChapter 10,namely, thatifwemove inastraight linewith uniform velocity the laws ofphysics willlook thesame tousaswhen wearestanding still. That is theGalilean transformation. Butwewish todiscuss theinteresting casewhere s isstillmore complicated, says=at2/2. Then ds/dt =atandd2s/dt2 =a,a uniform acceleration; orinastillmore complicated case, theacceleration might beafunction oftime. Thismeans thatalthough thelawsofforce from thepoint ofview ofJoewould look like dgx"'1HF ==F1, thelawsofforce aslooked upon byMoe would appear as d2x’m-at? =F,—ma. That is,since Moe’s coordinate system isaccelerating with respect toJoe’s, the extra term macomes in,andMoe willhave tocorrect hisforces bythat amount inorder togetNewton’s lawstowork. Inother words, hereisanapparent, mysteri- ousnewforce ofunknown origin which arises, ofcourse, because Moe hasthe wrong coordinate system. This isanexample ofapseudo force; other examples occur incoordinate systems thatarerotating. Another example ofpseudo force iswhat isoften called “centrifugal force.” Anobserver inarotating coordinate system, e.g., inarotating box, willfind mysterious forces, notaccounted forbyanyknown origin offorce, throwing things outward toward thewalls. These forces areduemerely tothefactthatthe observer does nothave Newton’s coordinate system, which isthesimplest coordi- natesystem. Pseudo force canbeillustrated byaninteresting experiment inwhich wepush ajarofwater along atable, with acceleration. Gravity, ofcourse, actsdownward onthewater, butbecause ofthehorizontal acceleration there isalsoapseudo force acting horizontally andinadirection opposite totheacceleration. Theresultant ofgravity and pseudo force makes anangle with thevertical, and during the acceleration thesurface ofthewater willbeperpendicular totheresultant force, i.e.,inclined atanangle with thetable, with thewater standing higher inthe rearward sideofthejar. When thepush onthejarstops andthejardecelerates because offriction, thepseudo force isreversed, andthewater stands higher in theforward sideofthejar(Fig. 12-4). Oneveryimportant feature ofpseudo forces isthattheyarealways propor- tional tothemasses; thesame istrue ofgravity. Thepossibility exists, therefore, thatgravity itself isapseudo force. Isitnotpossible thatperhaps gravitation is duesimply tothefactthatwedonothave theright coordinate system? After all, wecanalways getaforce proportional tothemass ifweimagine thatabody is accelerating. Forinstance, aman shutupinaboxthatisstanding stillonthe earth finds himself heldtothefloor oftheboxwithacertain force thatispropor- tional tohismass. Butifthere were noearth atallandtheboxwere standing still, theman inside would float inspace. Ontheother hand, ifthere were no earth atalland something were pulling theboxalong with anacceleration g, then theman inthebox, analyzing physics, would find apseudo force which would pullhimtothefloor, justasgravity does. 12-llL>i___.><i 3 E Fig.l2-4. Illustration of0pseudo force Einstein putforward thefamous hypothesis thataccelerations giveanimita- tion ofgravitation, that theforces ofacceleration (the pseudo forces) cannot be distinguished from those ofgravity; itisnotpossible totellhow much ofagiven force isgravity andhowmuch ispseudo force. Itmight seem allright toconsider gravity tobeapseudo force. tosaythatwe areallhelddown because weareaccelerating upward, buthowabout thepeople inMadagascar, ontheother sideoftheearth—are theyaccelerating too? Einstein found thatgravity could beconsidered apseudo force onlyatonepoint atatime, andwasledbyhisconsiderations tosuggest thatthegeometry oftheworld ismore complicated than ordinary Euclidean geometry. Thepresent discussion isonly qualitative, anddoesnotpretend toconvey anything more thanthegeneral idea. Togivearough ideaofhowgravitation could betheresult ofpseudo forces, we present anillustration which ispurely geometrical anddoesnotrepresent thereal situation. Suppose thatwealllived intwodimensions, andknew nothing ofa third. Wethink weareonaplane, butsuppose wearereally onthesurface ofa sphere. And suppose thatweshoot anobject along theground, with noforces onit.Where willitgo?Itwillappear togoinastraight line,butithastoremain onthesurface ofasphere, where theshortest distance between twopoints isalong agreat circle; soitgoesalong agreat circle. Ifweshoot another object similarly, butinanother direction, itgoesalong another great circle. Because wethink we areonaplane, weexpect thatthese twobodies willcontinue todiverge linearly withtime, butcareful observation willshow thatiftheygofarenough theymove closer together again, asthough theywere attracting each other. Buttheyarenot attracting each other—there isjustsomething “weird” about thisgeometry. This particular illustration does notdescribe correctly theway inwhich Euclid’s geometry is“weird,” butitillustrates thatifwedistort thegeometry sufiiciently itispossible thatallgravitation isrelated insome waytopseudo forces; thatisthe general ideaoftheEinsteinian theory ofgravitation. 12-6 Nuclear forces Weconclude thischapter with abrief discussion oftheonly other known forces, which arecalled nuclear forces. These forces arewithin thenuclei ofatoms, and although they aremuch discussed, noonehasever calculated theforce between twonuclei, andindeed atpresent there isnoknown lawfornuclear forces. These forces have avery tinyrange which isjustabout thesame asthesizeofthenucleus, perhaps 10”” centimeter. With particles sosmall andatsuch atiny distance, only thequantum-mechanical laws arevalid, nottheNewtonian laws. Innuclear analysis wenolonger think interms offorces, andinfactwecanreplace theforce concept withaconcept oftheenergy ofinteraction oftwoparticles, asubject that willbediscussed later. Anyformula thatcanbewritten fornuclear forces isa rather crude approximation which omits many complications; onemight be somewhat asfollows: forces within anucleus donotvaryinversely asthesquare ofthedistance, butdieoffexponentially overacertain distance r,asexpressed by F=(1/r2) exp(—r/r0), where thedistance r0isoftheorder of10*‘ llcentimeter. Inother words, theforces disappear assoon astheparticles areanygreat distance apart, although theyareverystrong within the10-13 centimeter range. Sofar astheyareunderstood today, thelawsofnuclear force areverycomplex; wedo notunderstand them inanysimple way, andthewhole problem ofanalyzing the fundamental machinery behind nuclear forces isunsolved. Attempts atasolution haveledtothediscovery ofnumerous strange particles, the1r-mesons, forexample, buttheorigin ofthese forces remains obscure. 12-12i .4\- l I3 Work and Potential Energy (A) 13-1 Energy ofafalling body InChapter 4wediscussed theconservation ofenergy. Inthatdiscussion, we didnotuseNewton’s laws, butitis,ofcourse, ofgreat interest toseehowitcomes about thatenergy isinfactconserved inaccordance with these laws. Forclarity weshall start with thesimplest possible example, andthen develop harder and harder examples. Thesimplest example oftheconservation ofenergy isavertically falling object, onethat moves only inavertical direction. Anobject which changes itsheight under theinfluence ofgravity alone hasakinetic energy T(orK.E.) duetoits motion during thefall,andapotential energy mgh, abbreviated UorP.E., whose sumisconstant: %mv2 -1-mgh =const,KE i>E or T+ U=const. (13.1) Now wewould liketoshow thatthisstatement istrue. What dowemean, show itistrue? From Newton’s Second Law wecaneasily tellhow theobject moves, anditiseasytofindouthowthevelocity varies withtime, namely, thatitincreases proportionally with thetime, andthattheheight varies asthesquare ofthetime. Soifwemeasure theheight from azero point where theobject isstationary, itisnomiracle thattheheight turns outtobeequal tothesquare ofthevelocity times anumber ofconstants. However, letuslook atitalittle more closely. Letusfindoutdirectly from Newton’s Second Law how thekinetic energy should change, bytaking thederivative ofthekinetic energy with respect totime andthen using Newton's laws. When wedifferentiate %mv2 with respect totime, weobtain dT_ d_2__ dv_ dz)71,?-2,-t(§,mv)-§m2v;fi-mv-if (13.2) since misassumed constant. Butfrom Newton’s Second Law, m(dv/dt) =F, sothat dT/dt=Fv. (13.3) Ingeneral, itwillcome outtobeF-v,butinourone-dimensional caseletusleave itastheforce times thevelocity. Now inoursimple example theforce isconstant, equal to—mg, avertical force (theminus signmeans thatitactsdownward), andthevelocity, ofcourse, is therateofchange ofthevertical position, orheight h,with time. Thus therate ofchange ofthekinetic energy is—mg(dh/dt), which quantity, miracle ofmiracles, istherateofchange ofsomething else! Itisthetime rateofchange ofmgh! Therefore, astimegoes on,thechanges inkinetic energy andinthequantity mgh areequal andopposite, sothatthesum ofthetwoquantities remains constant. Q.E.D. Wehave shown, from Newton’s second lawofmotion, that energy iscon- served forconstant forces when weaddthepotential energy mgh tothekinetic energy %mv2. Now letuslookintothisfurther andseewhether itcanbegeneralized, andthusadvance ourunderstanding. Does itwork onlyforafreely falling body, orisitmore general? Weexpect from ourdiscussion oftheconservation ofenergy 13-113-1 Energy ofafalling body 13-2 Work done bygravity 13-3 Summation ofenergy 13-4 Gravitational fieldoflarge objects h dh/d1 X Fig. l3~l. Anobject moving onu frictionless curve under theinfluence of gravity.thatitwould work foranobject moving from onepoint toanother insome kind offrictionless curve, under theinfluence ofgravity (Fig. 13—l). Iftheobject reaches acertain height hfrom theoriginal height H,thenthesame formula should again beright, even though thevelocity isnow insome direction other than the vertical. Wewould liketounderstand whythelawisstillcorrect. Letusfollow thesame analysis, finding thetime rateofchange ofthekinetic energy. This willagain bemv(dv/dt), butm(dv/dt) istherateofchange ofthemagnitude ofthe momentum, i.e.,theforce inthedirection ofmotion—the tangential force Fl.Thus dT d5,? :WIU =FIU. Now thespeed istherateofchange ofdistance along thecurve, ds/dt, andthe tangential force F,isnotmgbutisweaker bytheratio ofthedistance dsalong the path tothevertical distance dh.Inother words, . dhF,=—mgsin0 =—mgE» sothat p£’£__,,, %>€~S)__.,,, %H11“ gatat'gut’ since theds’scancel. Thus weget—mg(dh/dt), which isequal totherateofchange ofmgh, asbefore. Inorder tounderstand exactly how theconservation ofenergy works in general inmechanics, weshall nowdiscuss anumber ofconcepts which willhelp ustoanalyze it. First, wediscuss therateofchange ofkinetic energy ingeneral inthree dimensions. Thekinetic energy inthree dimensions is T=%m(vf —l—of-l—112). When wedifferentiate thiswith respect totime, wegetthree terrifying terms: dT it, d 4,7,;=m(v,,-5-1+U,2”?”+U; (13.4) Butm(dv,,/dt) istheforce F,acting ontheobject inthex-direction. Thus theright sideofEq.(13.4) isF,v,, +F,/1,, —l—F,v_,. Werecall ourvector analysis andrecog- nizethisasF-v;therefore tn"/at=F-v. (13.5) This result canbederived more quickly asfollows: ifaandbaretwovectors, both ofwhich maydepend upon thetime, thederivative ofa~bis,ingeneral, d(a-b)/dt =a~db/dt +(da/dt) -b. (13.6) Wethen usethisintheform a=b=v: €(%""@ =51%.' d ds =m-!-v=F-v=F---- (13.7)vv) dt dt dt dt Because theconcepts ofkinetic energy, andenergy ingeneral, aresoimpor- tant, various names have been given totheimportant terms inequations such as these. %mv2 is,asweknow, called kinetic energy. F-viscalled power: theforce acting onanobject times thevelocity oftheobject (vector “dot” product) isthe power being delivered totheobject bythatforce. Wethus have amarvelous theorem: therate ofchange ofkinetic energy ofanobject tsequal tothepower expended bytheforces acting onit. However, tostudy theconservation ofenergy, wewant toanalyze thisstill more closely. Letusevaluate thechange inkinetic energy inaveryshort timedt. Ifwemultiply both sides ofEq.(13.7) bydt,wefindthatthedifferential change in 13-2 thekinetic energy istheforce “dot” thediflerential distance moved: dT=F-ds. (13.8) Ifwenowintegrate, weget AT=/2F-ds. (13.9)1 What does thismean? Itmeans thatifanobject ismoving inanywayunder the influence ofaforce, moving insome kind ofcurved path, then thechange inK.E. when itgoes from onepoint toanother along thecurve isequal totheintegral of thecomponent oftheforce along thecurve times thedifferential displacement ds, theintegral being carried outfrom onepoint totheother. This integral alsohasa name; itiscalled thework done bytheforce ontheobject. Weseeimmediately thatpower equals work done persecond. Wealsoseethatitisonly acomponent offorce inthedirection ofmotion thatcontributes tothework done. Inoursimple example theforces were only vertical, andhadonly asingle component, sayF,, equal to—mg. Nomatter how theobject moves inthose circumstances, falling inaparabola forexample, F'ds,which canbewritten asF,dx+Fydy+F,dz, hasnothing leftofitbutF,dz=—mg dz,because theother components offorce arezero. Therefore, inoursimple case, fr-ds=f2-mgdz =——mg(z2 -2,), (13.10)1 zl soagain wefindthatitisonly thevertical height from which theobject falls that counts toward thepotential energy. Aword about units. Since forces aremeasured innewtons, andwemultiply byadistance inorder toobtain work, work ismeasured innewton -meters (n-m), butpeople donotliketosaynewton-meters, they prefer tosayjoules (j). A newton-meter iscalled ajoule; work ismeasured injoules. Power, then, isjoules persecond, andthatisalsocalled awatt (w). Ifwemultiply watts bytime, the result isthework done. Thework done bytheelectrical company inourhouses, technically, isequal tothewatts times thetime. That iswhere wegetthings like kilowatt hours, 1000 watts times'3600 seconds, or3.6X106joules. Now wetakeanother example ofthelawofconservation ofenergy. Consider anobject which initially haskinetic energy andismoving very fast, andwhich slides against thefloor with friction. Itstops. Atthestart thekinetic energy isnot zero, butattheenditiszero; there iswork done bytheforces, because whenever there isfriction there isalways acomponent offorce inadirection opposite tothat ofthemotion, andsoenergy issteadily lost. Butnowletustakeamass ontheend ofapivot swinging inavertical plane inagravitational fieldwithnofriction. What happens hereisdiflerent, because when themass isgoing uptheforce isdownward, andwhen itiscoming down, theforce isalsodownward. Thus F-dshasone signgoing upandanother signcoming down. Ateach corresponding point ofthe downward andupward paths thevalues ofF-dsareexactly equal insizebutof opposite sign, sothenetresult oftheintegral willbezero forthiscase. Thus the kinetic energy withwhich themass comes back tothebottom isthesame asithad when itleft,thatistheprinciple oftheconservation ofenergy. (Note thatwhen there arefriction forces theconservation ofenergy seems atfirstsight tobeinvalid. Wehave tofindanother form ofenergy. Itturns out,infact,thatheatisgenerated inanobject when itrubs another with friction, butatthemoment wesupposedly donotknow that.) 13-2 Work done bygravity Thenext problem tobediscussed ismuch more difficult than theabove; ithastodowith thecasewhen theforces arenotconstant, orsimply vertical, as they were inthecases wehave worked out. Wewant toconsider aplanet, for example, moving around thesun,orasatellite inthespace around theearth. 13-3 M mpa__.4__iin . 2 1 Fig. 13-2. Asmall mass mfalls under theinfluence ofgravity toward alarge mass M. ix U 2 s I 3'4 4 Fig. 13-3. Aclosed path inagravi- tational field.Weshall firstconsider themotion ofanobject which starts atsome point 1 andfalls, say,directly toward thesunortoward theearth (Fig. 13-2). Will there bealawofconservation ofenergy inthese circumstances? Theonly difference is thatinthiscase, theforce ischanging aswegoalong, itisnotjustaconstant. As weknow, theforce isGM/r2times themass m,where misthemass thatmoves. Now certainly when abody fallstoward theearth, thekinetic energy increases as thedistance fallen increases, justasitdoes when wedonotworry about the variation offorce withheight. Thequestion iswhether itispossible tofindanother formula forpotential energy different from mgh, adifferent function ofdistance away from theearth, sothatconservation ofenergy willstillbetrue. This one-dimensional caseiseasy totreat because weknow thatthechange inthekinetic energy isequal totheintegral, from oneendofthemotion totheother, of—GMm/r2 times thedisplacement dr: 2 T2-Tl=- GMm (13.11)1 F2 There arenocosines needed forthiscasebecause theforce andthedisplacement areinthesame direction. Itiseasy tointegrate dr/r2; theresult is-1/r, so Eq.(13.11) becomes T2—Tl=+GMm — (13.12) Thus wehave adifferent formula forpotential energy. Equation (13.12) tellsus thatthequantity (§mv2 —GMm/r) calculated atpoint 1,atpoint 2,oratany other place, hasaconstant value. Wenow have theformula forthepotential energy inagravitational field for vertical motion. Now wehave aninteresting problem. Canwemake perpetual motion inagravitational field? Thegravitational field varies; indifferent places itisindiflerent directions andhasdifferent strengths. Could wedosomething likethis,using afixed, frictionless track: start atsome point andliftanobject out tosome other point, then move itaround anarctoathird point, then lower ita certain distance, then move itinatacertain slope andpullitoutsome other way, sothatwhen webring itback tothestarting point, acertain amount ofwork has been done bythegravitational force, andthekinetic energy oftheobject isin- creased? Canwedesign thecurve sothatitcomes back moving alittle bitfaster than itdidbefore, sothatitgoes around andaround andaround, andgives us perpetual motion? Since perpetual motion isimpossible, weought tofindoutthat thisisalsoimpossible. Weought todiscover thefollowing proposition: since there isnofriction theobject should come back with neither higher norlower velocity- itshould beable tokeep going around andaround anyclosed path. Stated in another way, thetotal work done ingoing around acomplete cycle should bezero forgravity forces, because ifitisnotzero wecangetenergy outbygoing around. (Ifthework turns outtobelessthan zero, sothatwegetlessspeed when wego around oneway, then wemerely goaround theother way, because theforces, of course, depend only upon theposition, notupon thedirection; ifonewayisplus, theother waywould beminus, sounless itiszero wewillgetperpetual motion bygoing around either way.) Isthework really zero? Letustrytodemonstrate thatitis.First weshall explain more orlesswhy itiszero, andthen weshall examine italittle better mathematically. Suppose thatweuseasimple path such asthatshown inFig. 13-3, inwhich asmall mass iscarried from point 1topoint 2,andthen ismade togoaround acircle to3,back to4,then to5,6,7,and8,andfinally back to1. Allofthelines areeither purely radial orcircular, with Masthecenter. How much work isdone incarrying maround thispath? Between points 1and2,itis GMm times thedifference of1/rbetween these twopoints: 2 2 W12=f F-ds=l -cMmi§= -c;Mm(l-l)-1 1 F F2 '1 13-4 From 2to3theforce isexactly atright angles tothecurve, sothat W23 E0. Thework from 3to4is 4 W34=/F'dS=-cMm(l -l)-a Y4 '3 Inthesame fashion, wefind that W45 =0,W56 =—GMm(l/r6 —l/r5), W67 = O, W78 = 1 1/1'7), and W81 = O. Thus W=GM,,,(1_1,.1_1+1_1+1_1).7'1 7'2 7'3 V4 1'5 7'6 T7 7'8 Butwenote thatr2=r3,r4=r5,rl,=r7,andrs=rl.Therefore W=0. Ofcourse wemaywonder whether thisistootrivial acurve. What ifweuse arealcurve? Letustryitonarealcurve. First ofall,wemight liketoassert that arealcurve could always beimitated sufficiently well byaseries ofsawtooth jiggles likethose ofFig.13-4, andthattherefore, etc.,Q.E.D., butwithout alittle analysis, itisnotobvious atfirstthatthework done going around even asmall triangle iszero. Letusmagnify oneofthetriangles, asshown inFig.13-4. Isthe work done ingoing from atobandbtoconatriangle thesame asthework done ingoing directly from atoc?Suppose thattheforce isacting inacertain direction; letustakethetriangle such thatthesidebcisinthisdirection, justasanexample. Wealsosuppose thatthetriangle issosmall thattheforce isessentially constant over theentire triangle. What isthework done ingoing from atoc?Itis WM =[CF-ds =Fscos0,Ll since theforce isconstant. Now letuscalculate thework done ingoing around theother twosides ofthetriangle. Onthevertical sideabtheforce isperpendicular tods,sothathere thework iszero. Onthehorizontal sidebc, W,,,=/IF-ds =Fx. Thus weseethat thework done ingoing along thesides ofasmall triangle is thesame asthatdone going onaslant, because scos0isequal tox.Wehave proved previously that theanswer iszero foranypath composed ofaseries of notches likethose ofFig.13-3, andalsothatwedothesame work ifwecutacross thecorners instead ofgoing along thenotches (solong asthenotches arefine enough, andwecanalways make them very fine); therefore, thework done in going around anypath inagravitational field iszero. This isavery remarkable result. Ittellsussomething wedidnotpreviously know about planetary motion. Ittellsusthatwhen aplanet moves around the sun(without anyother objects around, noother forces) itmoves insuch amanner thatthesquare ofthespeed atanypoint minus some constants divided bythe radius atthatpoint isalways thesame atevery point ontheorbit. Forexample, thecloser theplanet istothesun, thefaster itisgoing, butbyhow much? By thefollowing amount: ifinstead ofletting theplanet goaround thesun,wewere tochange thedirection (butnotthemagnitude) ofitsvelocity andmake itmove radially, andthen weletitfallfrom some special radius totheradius ofinterest, thenewspeed would bethesame asthespeed ithadintheactual orbit, because thisisjustanother example ofacomplicated path. Solong aswecome back tothe same distance, thekinetic energy willbethesame. So,whether themotion isthe real, undisturbed one, orischanged indirection bychannels, byfrictionless con- straints, thekinetic energy withwhich theplanet arrives atapoint willbethesame. Thus, when wemake anumerical analysis ofthemotion oftheplanet inits orbit, aswedidearlier, wecancheck whether ornotwearemaking appreciable errors bycalculating thisconstant quantity, theenergy, atevery step, anditshould notchange. Fortheorbit ofTable 9-2theenergy does change,* itchanges by *Theenergy is%(v§-1-vi)-1/r intheunits ofTable 9-2. 13-5MO O ‘Zfl’C X D Fig. 13-4. A"smooth" closed path, showing amagnified segment ofitap- proximated byaseries ofradial and circumferential steps, and anenlarged view ofonestep. some 1.5percent from thebeginning totheend. Why? Either because forthe numerical method weusefinite intervals, orelsebecause wemade aslight mistake somewhere inarithmetic. Letusconsider theenergy inanother case: theproblem ofamass onaspring. When wedisplace themass from itsbalanced position, therestoring force is proportional tothedisplacement. Inthose circumstances, canwework outa lawforconservation ofenergy? Yes, because thework done bysuch aforce is W=/xFdx =fl—kxdx =—%kx2. (13.13)O O Therefore, foramass onaspring wehave thatthekinetic energy oftheoscillating mass plus%kx2 isaconstant. Letusseehowthisworks. Wepullthemass down; itisstanding stillandsoitsspeed iszero. Butxisnotzero, xisatitsmaximum, sothere issome energy, thepotential energy, ofcourse. Now werelease themass andthings begin tohappen (thedetails nottobediscussed), butatanyinstant the kinetic pluspotential energy must beaconstant. Forexample, after themass is onitswaypasttheoriginal equilibrium point, theposition xequals zero, butthat iswhen ithasitsbiggest v2,andasitgetsmore x2itgetsless02,andsoon.So thebalance ofx2andv2ismaintained asthemass goes upanddown. Thus we have another rulenow, thatthepotential energy foraspring is%kx2, iftheforce is—kx. 13-3 Summation ofenergy Now wegoontothemore general consideration ofwhat happens when there arelarge numbers ofobjects. Suppose wehave thecomplicated problem ofmany objects, which welabel i=1,2,3,...,allexerting graviational pulls oneach other. What happens then? Weshall prove thatifweaddthekinetic energies of alltheparticles, andaddtothisthesum, over allpairs ofparticles, oftheir mutual gravitational potential energy, —GMm/r,,, thetotal isaconstant: 2%m,vf + Z — =const. (13.14)r1, (pairs u) ‘J How doweprove it?Wedifferentiate each sidewith respect totime andgetzero. When wedifferentiate %m,z/3, wefindderivatives ofthevelocity thataretheforces, justasinEq.(13.5). Wereplace these forces bythelawofforce thatweknow from Newton’s lawofgravity andthenwenotice thatwhat isleftisthesame asthetime derivative of 2_Gm,m,_ pairs 7'” Thetime derivative ofthekinetic energy is d d,Ei %m1v? =$ mtvz ‘“J7 =ZF,-v, (13.15)1 7 Thetime derivative ofthepotential energy is d _Gm,m, _ Gm,m, dtgs rt] _in 7?; dl But 7'11=\/(xi _xJ)2 +(yr_y;|)2 +(Z1"ZJ)2> 13-6 sothat d’~_L _ E1_’2_£iZ‘__'E"m”P“* x9(a ma +20».—y.) - V, V]=|--.___i1] rm V1 V] =r1]"—'+ r]t'__’1'” 1'77, since r,,=—r,,, while r,,=r,,.Thus d Gm,m, _ Gm,m, r,, Gm,m,r,, 1316 -2--"_-Z T'"+?"""1' (-)dtpairs r” pairs 1J Now wemust note carefully what Z{Z} and Zmean. InEq.(13.15),1 pairs Z{Z} means thatitakes onallvalues i41,2,3,...inturn, andforeach value 1 1 ofi,theindex jtakes onallvalues except i.Thus ifi=3,jtakes onthevalues 1,2,4,... InEq.(13.16), ontheother hand, Zmeans thatgiven values ofiandjoccurpairs only once. Thus theparticle pair 1and3contributes only oneterm tothesum. Tokeep track ofthis,wemight agree toletirange over allvalues 1,2,3,...,and foreach iletjrange only over values greater than i.Thus ifi=3,jcould only have values 4,5,6,...Butwenotice thatforeach i,jvalue there aretwocontribu- tions tothesum, oneinvolving v,,andtheother v,,andthatthese terms havethe same appearance asthose ofEq.(13.14), where allvalues ofiandj(except i=j) areincluded inthesum. Therefore, bymatching theterms onebyone,wesee thatEqs. (13.16) and(13.15) areprecisely thesame, butofopposite sign, sothat thetime derivative ofthekinetic plus potential energy isindeed zero. Thus we seethat, formany objects, thekinetic energy isthesumofthecontributions from each individual object, andthatthepotential energy isalso simple, itbeing also justasumofcontributions, theenergies between allthepairs. Wecanunderstand whyitshould betheenergy ofevery pairthisway: Suppose thatwewant tofind thetotal amount ofwork thatmust bedone tobring theobjects tocertain distances from each other. We\may dothisinseveral steps, bringing them infrom infinity where there isnoforce, onebyone. First webring innumber one,which requires nowork, since noother objects areyetpresent toexert force onit.Next webring innumber two, which does take some work, namely W12 =—Gm1m2/r12. Now, andthisisanimportantpoint, suppose webring inthenextobject toposition three. Atanymoment theforce onnumber 3canbewritten asthesumoftwo forces—the force exerted bynumber 1andthatexerted bynumber 2.Therefore thework done isthesumoftheworks done byeach, because ifF3canberesolved intothesumoftwoforces, F3=F13+F23, then thework is /F,-ds =[F13-ds+ [F23-as =W13+ W23. That is,thework done isthesumofthework done against thefirstforce andthe second force, asifeach acted independently. Proceeding inthisway, weseethat thetotal work required toassemble thegiven configuration ofobjects isprecisely thevalue given inEq.(13.14) asthepotential energy. Itisbecause gravity obeys theprinciple ofsuperposition offorces thatwecanwrite thepotential energy as asumover each pairofparticles. 13-7 +|dp|<—P ——>1O dm -6 'lac. P X Fig. 13-5. Thegravitational force F onamoss point produced byaninfinite plane sheet ofmatter.13-4 Gravitational fieldoflarge objects Now weshall calculate thefields which aremetinafewphysical circumstances involving distributions ofmass. Wehave notsofarconsidered distributions of mass, only particles, soitisinteresting tocalculate theforces when they are produced bymore thanjustoneparticle. First weshall findthegravitational force onamass thatisproduced byaplane sheet ofmaterial, infinite inextent. The force onaunitmass atagiven point P,produced bythissheet ofmaterial (Fig. 13-5), willofcourse bedirected toward thesheet. Letthedistance ofthepoint from thesheet bea,andlettheamount ofmass perunitareaofthishuge sheet beit. Weshall suppose ,utobeconstant; itisauniform sheet ofmaterial. Now, what small fielddCisproduced bythemass dmlying between pandp+dpfrom the point 0ofthesheet nearest point P?Answer: dC=G(dmr/r3). Butthisfield isdirected along r,andweknow thatonlythex-component ofitwillremain when weaddallthelittle vector dC’s toproduce C.Thex-component ofdCis dCx= G.‘.iL'fi= GFLE.r3 r3 Now allmasses dmwhich areatthesame distance rfrom Pwillyield thesame dC,,sowemayatonce write fordmthetotal mass intheringbetween pand p+dp,namely dm=u.2trp dp(21rp dpisthearea ofaringofradius pand width dp,ifdp<<p).Thus dC,,=G;.l.21rp 4;?- Then, since r2=p2+a2,pdp=rdr. Therefore, W d 11c,=2110,” ;§=27I'Gp.6l(5-3)=210“. (13.17) Thus theforce isindependent ofdistance a!Why? Have wemade amistake? Onemight think thatthefarther away wego,theweaker theforce would be.But no!Ifweareclose, most ofthematter ispulling atanunfavorable angle; ifweare faraway, more ofthematter issituated more favorably toexert apulltoward theplane. Atanydistance, thematter which ismost effective liesinacertain cone. When wearefarther away theforce issmaller bytheinverse square, butinthe same cone, inthesame angle, there ismuch more matter, larger byjustthesquare ofthedistance! This analysis canbemade rigorous byjustnoticing that the differential contribution inanygiven cone isinfactindependent ofthedistance, because ofthereciprocal variation ofthestrength oftheforce from agiven mass, andtheamount ofmass included inthecone, withchanging distance. Theforce isnotreally constant ofcourse, because when wegoontheother sideofthesheet itisreversed insign. Wehavealso,ineffect, solved anelectrical problem: ifwehaveanelectrically charged plate, with anamount trofcharge perunitarea, then theelectric field at apoint outside thesheet isequal to0/2e0, andisintheoutward direction ifthe sheet ispositively charged, andinward ifthesheet isnegatively charged. Toprove this,wemerely notethatG,gravity, plays thesame roleas1/41re0 forelectricity. Now suppose thatwehave twoplates, withapositive charge +o'onone andanegative charge -0onanother atadistance Dfrom thefirst. What isthe field? Outside thetwoplates itiszero. Why? Because oneattracts andtheother repels, theforce being independent ofdistance, sothatthetwobalance out! Also, theforce between thetwoplates isclearly twice asgreat asthatfrom oneplate, namely E=0'/so, andisdirected from thepositive plate tothenegative one. Now wecome toamost interesting andimportant problem, whose solution wehave been assuming allthetime, namely, thattheforce produced bytheearth atapoint onthesurface oroutside itisthesame asifallthemass oftheearth were located atitscenter. Thevalidity ofthisassumption isnotobvious, because when weareclose, some ofthemass isveryclose tous,andsome isfarther away, 13-8 andsoon.When weaddtheeffects alltogether, itseems amiracle thatthenet force isexactly thesame aswewould getifweputallthemass inthemiddle! Wenowdemonstrate thecorrectness ofthismiracle. Inorder todoso,how- ever, weshall consider athinuniform hollow shell instead ofthewhole earth. Letthetotal mass oftheshell bem,andletuscalculate thepotential energy ofa particle ofmass m’adistance Raway from thesphere (Fig. 13-6) andshow thatthe potential energy isthesame asitwould beifthemass mwere apoint atthecenter. (The potential energy iseasier towork with than isthefield because wedonot havetoworry about angles, wemerely addthepotential energies ofallthepieces ofmass.) Ifwecallxthedistance ofacertain plane section from thecenter, then allthemass thatisinaslicedxisatthesame distance rfrom P,andthepotential energy duetothisringis—Gm’ dm/r. How much mass isinthesmall slicedx? Anamount 2 d 2 ddm=21ry#dS =—————1:;f'0x =_-iwyi; xa=21ra#dx, where it=m/41ra2 isthesurface density ofmass onthespherical shell. (Itisa general rulethattheareaofazoneofasphere isproportional toitsaxial width.) Therefore thepotential energy duetodmis dW=_W =__¢mm/~*l><.I‘ I‘ Butweseethat r2=y2+(R—-x)2=y2+x2+R2—2Rx =a2+R2-2Rx. Thus 2rdr =—2Rdx or dx_dr 7'7? Therefore, 4W=_@llii’ ,R andso R+a __Gm'21rap./W- ——————R R_a dr :_Gm’21ra;.t 2a=_Gm'(41l'd2p.) R R Gm’m_—T (13.18) Thus, forathinspherical shell, thepotential energy ofamass m’,external tothe shell, isthesame asthough themass oftheshellwereconcentrated atitscenter. Theearth canbeimagined asaseries ofspherical shells, each oneofwhich con- tributes anenergy which depends only onitsmass andthedistance from the center; adding them alltogether wegetthetotal mass, andtherefore theearth acts asthough allthematerial were atthecenter! Butnotice what happens ifourpoint isontheinside oftheshell. Making thesame calculation, butwithPontheinside, westillgetthedifference ofthetwo r’s,butnow intheform a+R—(a —R)=2R,ortwice thedistance from the center. Inother words, Wcomes outtobeW=—Gm’m/a, which isindependent ofRandindependent ofposition, i.e.,thesame energy nomatter where weare inside. Therefore noforce; nowork isdone when wemove about inside. Ifthe potential energy isthesame nomatter where anobject isplaced inside thesphere, there canbenoforce onit.Sothere isnoforce inside, there isonlyaforce outside, andtheforce outside isthesame asthough themass wereallatthecenter. 13-9ds fit .I? Fig. 13-6. Athin spherical shell of mass orcharge. 14 Work and Potential Energy (conclusion) 14-1 Work Inthepreceding chapter wehave presented agreat many new ideas and results thatplay acentral roleinphysics. These ideas aresoimportant thatit seems worth while todevote awhole chapter toacloser examination ofthem. Inthepresent chapter weshall notrepeat the“proofs” orthespecific tricks by which theresults were obtained, butshall concentrate instead upon adiscussion oftheideas themselves. Inlearning anysubject ofatechnical nature where mathematics plays arole, oneisconfronted with thetaskofunderstanding andstoring away inthememory ahuge body offacts andideas, held together bycertain relationships which can be“proved” or“shown” toexist between them. Itiseasy toconfuse theproof itself with therelationship which itestablishes. Clearly, theimportant thing to learn andtoremember istherelationship, nottheproof. Inanyparticular cir- cumstance wecaneit_her say“itcanbeshown that” such andsuch istrue, orwe canshow it.Inalmost allcases, theparticular proof thatisused isconcocted, firstofall,insuchform thatitcanbewritten quickly andeasily onthechalkboard oronpaper, andsothatitwillbeassmooth-looking aspossible. Consequently, theproof may look deceptively simple, when infact, theauthor might have worked forhours trying different ways ofcalculating thesame thing until hehas found theneatest way, soastobeabletoshow thatitcanbeshown intheshortest amount oftime! Thething toberemembered, when seeing aproof, isnottheproof itself, butrather thatitcanbeshown thatsuch andsuch istrue. Ofcourse, ifthe proof involves some mathematical procedures or“tricks” thatonehasnotseen before, attention should begiven nottothetrick exactly, buttothemathematical ideainvolved. Itiscertain thatinallthedemonstrations thataremade inacourse such as this,notonehasbeen remembered from thetime when theauthor studied fresh- man physics. Quite thecontrary: hemerely remembers that such andsuch is true, andtoexplain howitcanbeshown heinvents ademonstration atthemoment itisneeded. Anyone whohasreally learned asubject should beabletofollow a similar procedure, butitisnouseremembering theproofs. That iswhy, inthis chapter, weshall avoid theproofs ofthevarious statements made previously, and merely summarize theresults. 1, Thefirstideathathastobedigested iswork done byaforce. Thephysical word “work” isnottheword intheordinary sense of“Workers oftheworld unite!,” butisadifferent idea. Physical work isexpressed asjF-ds,called “the lineintegral ofFdotds,”which means thatiftheforce, forinstance, isinone direction andtheobject onwhich theforce isworking isdisplaced inacertain direction, then only thecomponent offorce inthedirection ofthedisplacement does anywork. If,forinstance, theforce were constant andthedisplacement were afinite distance As,thenthework done inmoving theconstant force through that distance isonlythecomponent offorce along Astimes As.Theruleis“force times distance,” butwereally mean only thecomponent offorce inthedirection ofthe displacement times Asor,equivalently, thecomponent ofdisplacement inthe direction offorce times F.Itisevident thatnowork whatsoever isdone bya force which isatright angles tothedisplacement. Now ifthevector displacement Asisresolved into components, inother words, iftheactual displacement isAsandwewant toconsider iteffectively asa component ofdisplacement Axinthex-direction, Ayinthey-direction, andAz 14-114-1 Work 14-2 Constrained motion 14-3 Conservative forces 14-4 Nonconservative forces 14-5 Potentials andfieldsf 1 I 1 1 1 1 »1 1 . 51. inthez-direction, then thework done incarrying anobject from oneplace to another canbecalculated inthree parts, bycalculating thework done along x, along y,andalong z.Thework done ingoing along xinvolves onlythatcomponent offorce, namely Fx,andsoon,sothework isF,Ax+F,Ay+F,Az.When theforce isnotconstant, andwehave acomplicated curved motion, thenwemust resolve thepath intoalotoflittle As’s, addthework done incarrying theobject along each As,andtakethelimit asAsgoes tozero. This isthemeaning ofthe “line integral.” Everything wehave justsaidiscontained intheformula W=IF-ds.It isallvery welltosaythatitisamarvelous formula, butitisanother thing to understand what itmeans, orwhat some oftheconsequences are. Theword “work” inphysics hasameaning sodifferent from thatoftheword asitisused inordinary circumstances thatitmust beobserved carefully thatthere aresome peculiar circumstances inwhich itappears nottobethesame. For\ex- ample, according tothephysical definition ofwork, ifoneholds ahundred-pound weight offtheground forawhile, heisdoing nowork. Nevertheless, everyone knows thathebegins tosweat, shake, andbreathe harder, asifhewere running upaflight ofstairs. Yetrunning upstairs isconsidered asdoing work (inrunning downstairs, onegetswork outoftheworld, according tophysics), butinsimply holding anobject inafixed position, nowork isdone. Clearly, thephysical defini- tionofwork differs from thephysiological definition, forreasons weshall briefly explore. Itisafactthatwhen oneholds aweight hehastodo“physiological” work. Why should hesweat? Why should heneed toconsume food tohold theweight up’? Why isthemachinery inside himoperating atfullthrottle, justtohold the weight up? Actually, theweight could beheldupwithnoeffort byjustplacing it onatable; then thetable, quietly andcalmly, without anysupply ofenergy, is abletomaintain thesame weight atthesame height! Thephysiological situation issomething likethefollowing. There aretwokinds ofmuscles inthehuman body andinother animals: onekind, called striated orskeletal muscle, isthetype ofmuscle wehave inourarms, forexample, which isunder voluntary control; theother kind, called smooth muscle, islikethemuscle intheintestines or,inthe clam, thegreater adductor muscle thatcloses theshell. The smooth muscles work very slowly, butthey canhold a“set”; thatistosay,iftheclam tries to close itsshell inacertain position, itwillhold thatposition, even ifthere isavery great force trying tochange it.Itwillhold aposition under load forhours and hours without getting tired because itisverymuch likeatable holding upaweight, it“sets” intoacertain position, andthemolecules justlock there temporarily withnowork being done, noeffort being generated bytheclam. Thefactthatwe have togenerate effort tohold upaweight issimply duetothedesign ofstriated muscle. What happens isthatwhen anerve impulse reaches amuscle fiber, the fiber gives alittle twitch andthen relaxes, sothatwhen wehold something up, enormous volleys ofnerve impulses arecoming intothemuscle, large numbers oftwitches aremaintaining theweight, while theother fibers relax. Wecansee this, ofcourse: when wehold aheavy weight andgettired, webegin toshake. Thereason isthatthevolleys arecoming irregularly, andthemuscle istired and notreacting fastenough. Why suchaninefficient scheme? Wedonotknow exactly why, butevolution hasnotbeen able todevelop fast smooth muscle. Smooth muscle would bemuch more effective forholding upweights because youcould juststand there anditwould lock in;there would benowork involved andno energy would berequired. However, ithasthedisadvantage thatitisveryslow- operating. Returning now tophysics, wemay askwhywewant tocalculate thework done. Theanswer isthatitisinteresting anduseful todoso,since thework done onaparticle bytheresultant ofalltheforces acting onitisexactly equal tothe change inkinetic energy ofthatparticle. That is,ifanobject isbeing pushed, it picks upspeed, and 2_Z .. A(v)-mF As 14-2 1 _.._.....r_.,_ 11 l l fE . E14-2 Constrained motion Another interesting feature offorces andwork isthis:suppose thatwehave asloping oracurved track, andaparticle thatmust move along thetrack, but without friction. Orwemayhave apendulum withastring andaweight; thestring constrains theweight tomove inacircle about thepivot point. Thepivot point maybechanged byhaving thestring hitapeg,sothatthepath oftheweight is along twocircles ofdifferent radii. These areexamples ofwhat wecallfixed, frictionless constraints. Inmotion withafixed frictionless constraint, nowork isdone bytheconstraint because theforces ofconstraint arealways atright angles tothemotion. Bythe “forces ofconstraint” wemean those forces which areapplied totheobject directly bytheconstraint itself—the contact force withthetrack, orthetension inthestring. Theforces involved inthemotion ofaparticle onaslope moving under the influence ofgravity arequite complicated, since there isaconstraint force, a gravitational force, andsoon.However, ifwebaseourcalculation ofthemotion onconservation ofenergy andthegravitational force alone, wegettheright result. This seems rather strange, because itisnotstrictly theright waytodoit—we should usetheresultant force. Nevertheless, thework done bythegravitational force alone willturnouttobethechange inthekinetic energy, because thework done bytheconstraint partoftheforce iszero (Fig. 14-1). Theimportant feature here isthatifaforce canbeanalyzed asthesumof twoormore “pieces” then thework done bytheresultant force ingoing along a certain curve isthesumoftheworks done bythevarious “component” forces intowhich theforce isanalyzed. Thus ifweanalyze theforce asbeing thevector sumofseveral effects, gravitational plusconstraint forces, etc.,orthex-component ofallforces andthey-component ofallforces, oranyother waythatwewish tosplititup,thenthework done bythenetforce isequal tothesumoftheworks done byalltheparts intowhich wehave divided theforce inmaking theanalysis. 14-3 Conservative forces Innature there arecertain forces, thatofgravity, forexample, which have avery remarkable property which wecall“conservative” (nopolitical ideas involved, itisagain oneofthose “crazy words”). Ifwecalculate howmuch work isdone byaforce inmoving anobject from onepoint toanother along some curved path, ingeneral thework depends upon thecurve, butinspecial cases it doesnot. Ifitdoes notdepend upon thecurve, wesaythattheforce isaconserva- tiveforce. Inother words, iftheintegral oftheforce times thedistance ingoing from position 1toposition 2inFig.14-2 iscalculated along curve Aandthen along B,wegetthesame number ofjoules, andifthisistrueforthispairofpoints onevery curve, andifthesame proposition works nomatter which pair ofpoints weuse,then. wesaytheforce isconservative. Insuch circumstances, thework integral going from 1to2canbeevaluated inasimple manner, andwecangive aformula fortheresult. Ordinarily itisnotthiseasy, because wealsohave to specify thecurve\, butwhen wehave acasewhere thework does notdepend on thecurve, then, ‘fcourse, thework depends only upon thepositions of1and2. Todemonstjate thisidea, consider thefollowing. Wetake a“standard” point P,atanaribitrary location (Fig. 14-2). Then, thework line-integral from 1to2,which wewant tocalculate, canbeevaluated asthework done ingoing from 1toPplusthework done ingoing from Pto2,because theforces arecon- servative andthework does notdepend upon thecurve. Now, thework done in going from position Ptoaparticular position inspace isafunction ofthatposition inspace. Ofcourse itreally depends onPalso, butwehold thearbitrary point P fixed permanently fortheanalysis. Ifthatisdone, then thework done ingoing from point Ptopoint 2issome function ofthefinalposition of2.Itdepends upon where 2is;ifwegotosome other point wegetadifferent answer. Weshall callthisfunction ofposition —U(x,y,z),andwhen wewish torefer tosome particular point 2whose coordinates are(x2,yz,Z2),weshall write U(2), 14-3DIRECTION OFMOTION FORCE OF CONSTRAINT FQRCE 0|: GRAVITY Fig. 14-1. Forces acting onc1sliding body (nofriction). .2Fig. 14-2. Possible paths between twopoints inafield offorce. asanabbreviation forU(x2, yz,Z2). Thework done ingoing from point 1to point Pcanbewritten alsobygoing theother wayalong theintegral, reversing alltheds’s. That is,thework done ingoing from 1toPisminus thework done ingoing from thepoint Pto1: LPF-ds =/P113-(-as) =—L:F'ds. Thus thework done ingoing from Pto1is—U(l), andfrom Pto2thework is —U(2). Therefore theintegral from 1to2isequal to—U(2) plus[—U(l) back- wards], or+U(l) —U(2): 0(1)=—L:F-ds, U(2)=—LfF-ds, [FF-ds =0(1)-U(2). (14.1) Thequantity U(l) —U(2) iscalled thechange inthepotential energy, andwe callUthepotential energy. Weshall saythatwhen theobject islocated atposition 2,ithaspotential energy U(2) andatposition 1ithaspotential energy U(l). Ifitislocated atposition P,ithaszeropotential energy. Ifwehadused anyother point, sayQ,instead ofP,itwould turnout(and weshall leave ittoyoutodemon- strate) thatthepotential energy ischanged onlybytheaddition ofaconstant. Since theconservation ofenergy depends only upon changes, itdoes notmatter ifwe addaconstant tothepotential energy. Thus thepoint Pisarbitrary. Now, wehave thefollowing twopropositions: (1)thatthework done bya force isequal tothechange inkinetic energy oftheparticle, but(2)mathematically, foraconservative force, thework done isminus thechange inafunction Uwhich wecallthepotential energy. Asaconsequence ofthese two, wearrive atthe proposition thatifonlyconservative forces act,thekinetic energy Tplusthepotential energy Uremains constant: T+U=constant. (14.2) Letusnowdiscuss theformulas forthepotential energy foranumber ofcases. Ifwehave agravitational field thatisuniform, ifwearenotgoing toheights comparable with theradius oftheearth, then theforce isaconstant vertical force andthework done issimply theforce times thevertical distance. Thus U(2) =mgz, (14.3) andthepoint Pwhich corresponds tozero potential energy happens tobeany point intheplane z=0.Wecould alsohave saidthatthepotential energy is mg(z —6)ifwehadwanted to—all theresults would, ofcourse, bethesame in ouranalysis except that thevalue ofthepotential energy atz=Owould be —mg6. Itmakes nodifference, because onlydzflerences inpotential energy count. Theenergy needed tocompress alinear spring adistance xfrom anequilibrium point is U(x) =%kx2, (14.4) andthezeroofpotential energy isatthepoint x=0,theequilibrium position of thespring. Again wecould addanyconstant wewish. Thepotential energy ofgravitation forpoint masses Mandm,adistance r apart, is U(r) =—GMm/r. (14.5) Theconstant hasbeen chosen here sothatthepotential iszero atinfinity. Of course thesame formula applies toelectrical charges, because itisthesame law: U(r) =qlqz/41re0r. (14.6) Now letusactually useoneofthese formulas, toseewhether weunderstand what itmeans. Question: How fastdowehave toshoot arocket away from the 14-4 I earth inorder forittoleave? Solution: Thekinetic pluspotential energy must be aconstant; when itleaves, itwillbemillions ofmiles away, andifitisjust barely abletoleave, wemaysuppose thatitismoving with zerospeed outthere, justbarely going. Letabetheradius oftheearth, andMitsmass. Thekinetic pluspotential energy isthen initially given by%mo2 —GmM/a. Attheendof themotion thetwoenergies must beequal. Thekinetic energy istaken tobe zeroattheendofthemotion, because itissupposed tobejustbarely drifting away atessentially zero speed, andthepotential energy isGmM divided byinfinity, which iszero. Soeverything iszeroononesideandthattellsusthatthesquare of thevelocity must be2GM/a. ButGM/a2 iswhat wecalltheacceleration of gravity, g.Thus 212=2ga. Atwhat speed must asatellite travel inorder tokeep going around theearth? Weworked thisoutlong agoandfound that 02=GM/a. Therefore togo away from theearth, weneed \/2times thevelocity weneed tojustgoaround theearth nearitssurface. Weneed, inother words, twice asmuch energy (because energy goesasthesquare ofthevelocity) toleave theearth aswedotogoaround it. Therefore thefirstthing thatwasdone historically with satellites wastogetone togoaround theearth, which requires aspeed offivemiles persecond. Thenext thing wastosend asatellite away from theearth permanently; thisrequired twice theenergy, orabout seven miles persecond. Now, continuing ourdiscussion ofthecharacteristics ofpotential energy, let usconsider theinteraction oftwomolecules, ortwoatoms, twooxygen atoms for instance. When theyareveryfarapart, theforce isoneofattraction, which varies astheinverse seventh power ofthedistance, andwhen theyareveryclose theforce isaverylarge repulsion. Ifweintegrate theinverse seventh power tofindthework done, wefindthatthepotential energy U,which isafunction oftheradial distance between thetwooxygen atoms, varies astheinverse sixth power ofthedistance forlarge distances. Ifwesketch thecurve ofthepotential energy U(r) asinFig. 14-3, wethus start outatlarge rwith aninverse sixth power, butifwecome insufficiently near wereach apoint dwhere there isaminimum ofpotential energy. Theminimum of potential energy atr=dmeans this: ifwestart atdandmove asmall distance, averysmall distance, thework done, which isthechange inpotential energy when wemove thisdistance, isnearly zero, because there isverylittle change inpotential energy atthebottom ofthecurve. Thus there isnoforce atthispoint, andsoitis theequilibrium point. Another waytoseethatitistheequilibrium point isthat ittakes work tomove away from dineither direction. When thetwooxygen atoms have settled down, sothatnomore energy canbeliberated from theforce between them, theyareinthelowest energy state, andtheywillbeatthisseparation d.This isthewayanoxygen molecule looks when itiscold. When weheatitup, theatoms shake andmove farther apart, andwecaninfactbreak them apart, but todosotakes acertain amount ofwork orenergy, which isthepotential energy difference between r=dandr=oo.When wetrytopush theatoms veryclose together theenergy goes upveryrapidly, because theyrepel each other. Thereason webring thisoutisthattheideaofforce isnotparticularly suitable forquantum mechanics; there theidea ofenergy ismost natural. Wefindthat although forces andvelocities “dissolve” anddisappear when weconsider the more advanced forces between nuclear matter andbetween molecules andsoon, theenergy concept remains. Therefore wefind curves ofpotential energy in quantum mechanics books, butvery rarely doweeverseeacurve fortheforce between twomolecules, because bythattime people who aredoing analyses are thinking interms ofenergy rather than offorce. Next wenote thatifseveral conservative forces areacting onanobject atthe same time, then thepotential energy oftheobject isthesum ofthepotential energies from each oftheseparate forces. This isthesame proposition thatwe mentioned before, because iftheforce canberepresented asavector sumofforces, then thework done bythetotal force isthesumoftheworks done bythepartial 14-5U(I'l F D \Ufr)~ |/r‘ (|Fr>>u) d—>l Fig. 14-3. Thepotential energy be- tween two atoms asafunction ofthe distance between them. F l1.at 1I4 i 1 l lforces, anditcantherefore beanalyzed aschanges inthepotential energies ofeach ofthem separately. Thus thetotal potential energy isthesumofallthelittlepieces. Wecould generalize thistothecaseofasystem ofmany objects interacting with oneanother, likeJupiter, Saturn, Uranus, etc.,oroxygen, nitrogen, carbon, etc.,which areacting withrespect tooneanother inpairs duetoforces allofwhich areconservative. Inthese circumstances thekinetic energy intheentire system issimply thesumofthekinetic energies ofalloftheparticular atoms orplanets or whatever, andthepotential energy ofthesystem isthesum, over thepairs of particles, ofthepotential energy ofmutual interaction ofasingle pair, asthough theothers were notthere. ("Phis isreally nottrueformolecular forces, andthe formula issomewhat more complicated; itcertainly istrueforNewtonian gravita- tion, anditistrueasanapproximation formolecular forces. Formolecular forces there isapotential energy, butitissometimes amore complicated function ofthe positions oftheatoms than simply asum ofterms from pairs.) Inthespecial caseofgravity, therefore, thepotential energy isthesum, overallthepairs iandj, of—Gm,m,/r,,, aswasindicated inEq.(13.14). Equation (13.14) expressed mathematically thefollowing proposition: thatthetotal kinetic energy plus the total potential energy does notchange with time. Asthevarious planets wheel about, andturnandtwist andsoon,ifwecalculate thetotal kinetic energy and thetotal potential energy wefindthatthetotal remains constant. 14-4 Nonconservative forces Wehave spent aconsiderable timediscussing conservative forces; what about nonconservative forces? Weshall takeadeeper view ofthisthanisusual, andstate thatthere arenononconservative forces! Asamatter offact,allthefundamental forces innature appear tobeconservative. This isnotaconsequence ofNewton's laws. Infact,sofarasNewton himself knew, theforces could benonconservative, asfriction apparently is.When wesayfriction apparently is,wearetaking a modern view, inwhich ithasbeen discovered thatallthedeep forces, theforces between theparticles atthemost fundamental level, areconservative. If,forexample, weanalyze asystem likethatgreat globular starcluster that wesawapicture of,with thethousands ofstars allinteracting, then theformula forthetotal potential energy issimply oneterm plusanother term, etc.,summed over allpairs ofstars, andthekinetic energy isthesumofthekinetic energies of alltheindividual stars. Buttheglobular cluster asawhole isdrifting inspace too, and,ifwewere farenough away from itanddidnotseethedetails, could bethought ofasasingle object. Then ifforces were applied toit,some ofthose forces might endupdriving itforward asawhole, andwewould seethecenter ofthewhole thing moving. Ontheother hand, some oftheforces canbe,sotospeak, “wasted” inincreasing thekinetic orpotential energy ofthe“particles” inside. Letus suppose, forinstance, thattheaction ofthese forces expands thewhole cluster andmakes theparticles move faster. The total energy ofthewhole thing is really conserved, butseenfrom theoutside with ourcrude eyeswhich cannot see theconfusion ofmotions inside, andjustthinking ofthekinetic energy ofthe motion ofthewhole object asthough itwere asingle particle, itwould appear that energy isnotconserved, butthisisduetoalackofappreciation ofwhat itisthat wesee. And that, itturns out,isthecase: thetotal energy oftheworld, kinetic pluspotential, isaconstant when welook closely enough. When westudy matter inthefinest detail attheatomic level, itisnotalways easy toseparate thetotal energy ofathing intotwoparts, kinetic energy and potential energy, andsuch separation isnotalways necessary. Itisalmost always possible todoit,soletussaythatitisalways possible, andthatthepotential- plus-kinetic energy oftheworld isconstant. Thus thetotal potential-plus-kinetic energy inside thewhole world isconstant, andifthe“world” isapiece ofisolated material, theenergy isconstant ifthere arenoexternal forces. Butaswehave seen, some ofthekinetic andpotential energy ofathing may beinternal, for instance theinternal molecular motions, inthesense thatwedonotnotice it.We know thatinaglass ofwater everything isjiggling around, alltheparts aremoving 14»-6 allthetime, sothere isacertain kinetic energy inside, which weordinarily maynot payanyattention to.Wedonotnotice themotion oftheatoms, which produces heat, andsowedonotcallitkinetic energy, butheatisprimarily kinetic energy. Internal potential energy mayalsobeintheform, forinstance, ofchemical energy: when weburn gasoline energy isliberated because thepotential energies ofthe atoms inthenewatomic arrangement arelower than intheoldarrangement. It isnotstrictly possible totreat heatasbeing pure kinetic energy, foralittle ofthe potential getsin,andviceversa forchemical energy, soweputthetwotogether andsaythatthetotal kinetic andpotential energy inside anobject ispartly heat, partly chemical energy, andsoon.Anyway, allthese different forms ofinternal energy aresometimes considered as“lost” energy inthesense described above; thiswillbemade clearer when westudy thermodynamics. Asanother example, when friction ispresent itisnottruethatkinetic energy islost,even though asliding object stops andthekinetic energy seems tobelost. Thekinetic energy isnotlostbecause, ofcourse, theatoms inside arejiggling with agreater amount ofkinetic energy than before, andalthough wecannot seethat, wecanmeasure itbydetermining thetemperature. Ofcourse ifwedisregard the heatenergy, then theconservation ofenergy theorem willappear tobefalse. Another situation inwhich energy conservation appears tobefalse iswhen westudy only partofasystem. Naturally, theconservation ofenergy theorem willappear nottobetrueifsomething isinteracting with something elseonthe outside andweneglect totakethatinteraction intoaccount. Inclassical physics potential energy involved only gravitation andelectricity, butnow wehave nuclear energy andother energies also. Light, forexample, would involve anewform ofenergy intheclassical theory, butwecanalso, ifwe want to,imagine thattheenergy oflight isthekinetic energy ofaphoton, andthen ourformula (14.2) would stillberight. 14-5 Potentials andfields Weshall nowdiscuss afewoftheideas associated with potential energy and with theideaofafield. Suppose wehave twolarge objects AandBandathird very small onewhich isattracted gravitationally bythetwo, with some resultant force F.Wehave already noted inChapter 12thatthegravitational force ona particle canbewritten asitsmass, m,times another vector, C,which isdependent only upon theposition oftheparticle: F=mC. Wecananalyze gravitation, then, byimagining thatthere isacertain vector Cat every position inspace which “acts” upon amass which wemayplace there, but which isthere itself whether weactually supply amass foritto“act” onornot. Chasthree components, andeach ofthose components isafunction of(x,y,z), afunction ofposition inspace. Such athing wecallafield, andwesaythatthe objects AandBgenerate thefield, i.e.,they“make” thevector C.When anobject isputinafield, theforce onitisequal toitsmass times thevalue ofthefieldvector atthepoint where theobject isput. Wecanalsodothesame withthepotential energy. Since thepotential energy, theintegral of(force) -(ds)canbewritten asmtimes theintegral ofthe(field) -(ds), amere change ofscale, weseethatthepotential energy U(x, y,z)ofanobject located atapoint (x,y,2,)inspace canbewritten asmtimes another function which wemaycallthepotential \I/.Theintegral fC-ds=—\I/,justasfF~ds=—-U; there isonlyascale factor between thetwo: U=—/F-ds=—m[C'ds=m\I/. (14.7) Byhaving thisfunction \I/(x,y,z)atevery point inspace, wecanimmediately calculate thepotential energy ofanobject atany point inspace, namely, U(x,y,z)=m\I/(x, y,z)—rather atrivial business, itseems. Butitisnotreally trivial, because itissometimes much nicer todescribe thefieldbygiving thevalue 14-7 F l 1l I lt6u—-1 >r \-¢(r) =—Gm/r Mr)=CONSTANT =—Gm/0 Fig. 14-4. Potential duetoaspher- icalshell ofradius a.of\I/everywhere inspace instead ofhaving togiveC.Instead ofhaving towrite three complicated components ofavector function, wecangiveinstead thescalar function \I/.Furthermore, itismuch easier tocalculate \I/thananygiven component ofCwhen thefieldisproduced byanumber ofmasses, forsince thepotential isa scalar wemerely add, without worrying about direction. Also, thefield Ccanbe recovered easily from \I/,asweshall shortly see. Suppose wehave point masses m1,m2, ...atthepoints 1,2,...andwewish toknow thepotential \I/atsome arbitrary point p.Thisissimply thesumofthepotentials atPduetotheindividual masses taken onebyone: G.».\I/(p)=2—-I-:—:—, l=l,2,... (14.8) Inthelastchapter weused thisformula, thatthepotential isthesumofthe potentials from allthedifferent objects, tocalculate thepotential duetoaspherical shell ofmatter byadding thecontributions tothepotential atapoint from all parts oftheshell. Theresult ofthiscalculation isshown graphically inFig.14-4. Itisnegative, having thevalue zero atr=ooandvarying as1/rdown tothe radius a,andthen isconstant inside theshell. Outside theshell thepotential is —Gm/r, where misthemass oftheshell, which isexactly thesame asitwould have been ifallthemass were located atthecenter. Butitisnoteverywhere exactly thesame, forinside theshell thepotential turns outtobe—Gm/a, andisa constant! When thepotential isconstant, there isnofield, orwhen thepotential energy isconstant there isnoforce, because ifwemove anobject from oneplace toanother anywhere inside thesphere thework done bytheforce isexactly zero. Why? Because thework done inmoving theobject from oneplace totheother isequal tominus thechange inthepotential energy (or,thecorresponding field integral isthechange ofthepotential). Butthepotential energy isthesame at anytwopoints inside, sothere iszero change inpotentiai energy, andtherefore nowork isdone ingoing between anytwopoints inside theshell. Theonly way thework canbezeroforalldirections ofdisplacement isthatthere isnoforce atall. This gives usaclueastohowwecanobtain theforce orthefield, given the potential energy. Letussuppose thatthepotential energy ofanobject isknown attheposition (x,y,z)andwewant toknow what theforce ontheobject is.It willnotdotoknow thepotential atonlythisonepoint, asweshall see;itrequires knowledge ofthepotential atneighboring points aswell. Why? How canwe calculate thex-component oftheforce? (Ifwecandothis,ofcourse, wecanalso findthey-andz-components, andwewillthen know thewhole force.) Now, if wewere tomove theobject asmall distance Ax,thework done bytheforce onthe object would bethex-component oftheforce times Ax,ifAxissufficiently small, andthisshould equal thechange inpotential energy ingoing from onepoint to theother: AW=—AU =F,Ax. (14.9) Wehave merely used theformula fF-ds=—AU, butforavery short path. Now wedivide byAxandsofindthattheforce is F,,=—AU/Ax. (14.10) Ofcourse thisisnotexact. What wereally want isthelimit of(14.10) asAx getssmaller andsmaller, because itisonlyexactly right inthelimit ofinfinitesimal Ax.This werecognize asthederivative ofUwith respect tox,andwewould be inclined, therefore, towrite —dU/dx. ButUdepends onx,y,andz,andthe mathematicians have invented adifferent symbol toremind ustobevery careful when wearedifferentiating such afunction, soastoremember thatwearecon- sidering that only xvaries, andyandzdonotvary. Instead ofadthey simply make a“backwards 6,”or6.(A6should have been used inthebeginning of calculus because wealways want tocancel thatd,butwenever want tocancel a6!) Sotheywrite 8U/6x, andfurthermore, inmoments ofduress, ifthey want tobe verycareful, they putalinebeside itwith alittle yzatthebottom (6U/6x|,,), 14-8 which means “Take thederivative ofUwithrespect tox,keeping yandzconstant.” Most often weleave outtheremark about what iskept constant because itis usually evident from thecontext, soweusually donotusethelinewith theyand z.However, always usea6instead ofadasawarning thatitisaderivative with some other variables keptconstant. Thisiscalled apartial derivative ;itisaderiva- tiveinwhich wevary only x. Therefore, wefindthattheforce inthex-direction isminus thepartial deriva- tiveofUwith respect tox: F,=—6U/6x. (14.11) Inasimilar way, theforce inthey-direction canbefound bydiflerentiating U with respect toy,keeping xandzconstant, andthethird component, ofcourse, isthederivative with respect toz,keeping yandxconstant: F,=—6U/6y, F,=—6U/62. (14.12) Thisisthewaytogetfrom thepotential energy totheforce. Wegetthefield from thepotential inexactly thesame way: C,=-—d\I//6x, C,=—t'i‘I//dy, C,=—6\I//dz. (14.13) Incidentally, weshall mention here another notation, which weshall not actually useforquite awhile: Since Cisavector andhasx-,y-,andz-components, thesymbolized 6/6x, 6/6y, and6/oz which produce thex-,y-,andz-compo- nents aresomething likevectors. Themathematicians have invented aglorious newsymbol, V,called “grad” or“gradient” which isnotaquantity butanoperator which makes avector from ascalar. Ithasthefollowing “components”: The x-component ofthis“grad” is6/6x, they-component is6/6y, andthez-component is6/oz, andthen wehave thefunofwriting ourformulas thisway: F=-vt/, c=—V\I/. (14.14) Using Vgives usaquick wayoftesting whether wehave arealvector equation or not,butactually Eq.(14.14) means precisely thesame asEqs. (14.11) and(14.12); itisjustanother wayofwriting them, andsince wedonotwant towrite three equations every time, wejustwrite VUinstead. Onemore example offields andpotentials hastodowith theelectrical case. Inthecase ofelectricity theforce onastationary object isthecharge times the electric field: F=qE. (Ingeneral, ofcourse, thex-component offorce inan electrical problem hasalsoapartwhich depends onthemagnetic field. Itiseasy toshow from Eq.(12.10) thattheforce onaparticle duetomagnetic fields is always atright angles toitsvelocity, andalsoatright angles tothefield. Since theforce duetomagnetism onamoving charge isatright angles tothevelocity, nowork isdone bythemagnetism onthemoving charge because themotion isat right angles totheforce. Therefore, incalculating theorems ofkinetic energy in electric andmagnetic fields wecandisregard thecontribution from themagnetic field, since itdoes notchange thekinetic energy.) Wesuppose thatthere isonly anelectric field. Then wecancalculate theenergy, orwork done, inthesame way asforgravity, andcalculate aquantity ¢which isminus theintegral ofE-ds, from thearbitrary fixed point tothepoint where wemake thecalculation, andthen thepotential energy inanelectric field isjustcharge times thisquantity ¢: ¢(r)=[Eds U=q¢. Letustake, asanexample, thecaseoftwoparallel metal plates, each with a surface charge ofin‘perunitarea. This iscalled aparallel-plate capacitor. We found previously thatthere iszero force outside theplates andthatthere isa constant electric field between them, directed from +to—andofmagnitude o"/so (Fig. 14-5). Wewould liketoknow how much work would bedone in 14-9 yr- |+++++++t ,_______1 Fig. 14-5. Field between parallel plates. l carrying acharge from oneplate totheother. Thework would bethe(force) -(ds) integral, which canbewritten ascharge times thepotential value atplate 1minus thatatplate 2: 2 W=] F-ds=qo.-4.). 1 Wecanactually work outtheintegral because theforce isconstant, andifwecall theseparation oftheplates d,then theintegral iseasy: 2 2 l/>F.ds=£[ 1 601 E0 Thedifference inpotential, A¢>=ad/co, iscalled thevoltage diflerence, and¢ ismeasured involts. When wesayapairofplates ischarged toacertain voltage, what wemean isthatthedifference inelectrical potential ofthetwoplates isso- and-so many volts. Foracapacitor made oftwoparallel plates carrying asurface charge :l:tT,thevoltage, ordifference inpotential, ofthepairofplates isad/co. 14-10 I5 The Special Theory ofRelativity 15-1 Theprinciple ofrelativity Forover200years theequations ofmotion enunciated byNewton were be- lieved todescribe nature correctly, andthefirsttime thatanerror inthese laws wasdiscovered, thewaytocorrect itwasalsodiscovered. Both theerror andits correction were discovered byEinstein in1905. Newton’s Second Law, which wehave expressed bytheequation F=d(mv)/dt, wasstated withthetacitassumption thatmisaconstant, butwenowknow that thisisnottrue, andthatthemass ofabody increases with velocity. InEinstein’s corrected formula mhasthevalue =__'_”<>__,”’,/_{_,2/,2 (15.1) where the“rest mass” morepresents themass ofabody thatisnotmoving and cisthespeed oflight, which isabout 3X105km-sec“ orabout 186,000 mi'sec_1. Forthose whowant tolearn justenough about itsotheycansolve problems, thatisallthere istothetheory ofrelativity—it justchanges Newton’s laws by introducing acorrection factor tothemass. From theformula itself itiseasyto seethatthismass increase isverysmall inordinary circumstances. Ifthevelocity iseven asgreat asthatofasatellite, which goes around theearth at5mi/sec, then v/c=5/186,000: putting thisvalue intotheformula shows thatthecor- rection tothemass isonlyonepartintwotothree billion, which isnearly impossible toobserve. Actually, thecorrectness oftheformula hasbeen amply confirmed by theobservation ofmany kinds ofparticles, moving atspeeds ranging uptopracti- cally thespeed oflight. However, because theeffect isordinarily sosmall, it seems remarkable thatitwasdiscovered theoretically before itwasdiscovered experimentally. Empirically, atasufficiently highvelocity, theeffect isverylarge, butitwasnotdiscovered thatway. Therefore itisinteresting toseehowalaw thatinvolved sodelicate amodification (atthetimewhen itwasfirstdiscovered) wasbrought tolight byacombination ofexperiments andphysical reasoning. Contributions tothediscovery were made byanumber ofpeople, thefinal result ofwhose work wasEinstein’s discovery. There arereally twoEinstein theories ofrelativity. Thischapter isconcerned with theSpecial Theory ofRelativity, which dates from 1905. In1915 Einstein published anadditional theory, called theGeneral Theory ofRelativity. This latter theory deals withtheextension oftheSpecial Theory tothecaseofthelaw ofgravitation; weshall notdiscuss theGeneral Theory here. Theprinciple ofrelativity wasfirststated byNewton, inoneofhiscorollaries tothelawsofmotion: “The motions ofbodies included inagiven space arethe same among themselves, whether thatspace isatrestormoves uniformly forward inastraight line.” Thismeans, forexample, thatifaspace shipisdrifting along atauniform speed, allexperiments performed inthespace shipandallthephenom- enainthespace shipwillappear thesame asiftheshipwere notmoving, pro- vided, ofcourse, thatonedoesnotlookoutside. That isthemeaning oftheprinci- pleofrelativity. Thisisasimple enough idea, andtheonlyquestion iswhether it istruethatinallexperiments performed inside amoving system thelawsofphysics 15-115-1 Theprinciple ofrelativity 15-2 TheLorentz transformation 15-3 TheMichelson-Morley experiment 15-4 Transformation oftime 15-5 TheLorentz contraction 15-6 Siinultaneity 15-7 Four-vectors 15-8 Relativistic dynamics 15-9 Equivalence ofmass andenergyif1 1 1 111’ 111 11 l 4;-.5- ‘1 ‘ it<1 1. —,.:—_-._?-r-v—_—~———---_-;<‘1.-:1__L._""_‘l__“"'_~h-—-—TT;_‘-4.;>:=:_._-_ 1 .,1 I l 11 _vs»... —.:'~r-in-.=-..A;),_-. ll ' l it 1 .1 1 V1 11 +111:1| l 1 l 1 1 1 11 = 1 |J | 1 ll .1 1 E 1lY Y’ JOE MOE 11_. _P(X.3;2’) " (X.Ml)ut——> r >,X X Fig.15-1. Two coordinate systems inuniform relative motion along their x-axes.willappear thesame astheywould ifthesystem were standing still. Letusfirst investigate whether Newton’s laws appear thesame inthemoving system. Suppose thatMoe ismoving inthex-direction withauniform velocity u,and hemeasures theposition ofacertain point, shown inFig.15-1. Hedesignates the “x-distance” ofthepoint inhiscoordinate system asx’.Joeisatrest,andmeasures theposition ofthesame point, designating itsx-coordinate inhissystem asx. Therelationship ofthecoordinates inthetwosystems isclear from thediagram. After timezMoe’s origin hasmoved adistance ut,andifthetwosystems originally coincided, x’=x-ut, , Z yy’ (15.2)z’=z, t'=l. Ifwesubstitute thistransformation ofcoordinates into Newton’s laws wefind thatthese laws transform tothesame laws intheprimed system; thatis,thelaws ofNewton areofthesame form inamoving system asinastationary system, and therefore itisimpossible totell,bymaking mechanical experiments, whether the system ismoving ornot. Theprinciple ofrelativity hasbeen usedinmechanics foralongtime. Itwas employed byvarious people, inparticular Huygens, toobtain therules forthe collision ofbilliard balls, inmuch thesame wayasweused itinChapter 10to discuss theconservation ofmomentum. Inthepastcentury interest initwas heightened astheresult ofinvestigations intothephenomena ofelectricity, mag- netism, andlight. Along series ofcareful studies ofthese phenomena bymany people culminated inMaxwell’s equations oftheelectromagnetic field, which describe electricity, magnetism, andlight inoneuniform system. However, the Maxwell equations didnotseem toobey theprinciple ofrelativity. That is,ifwe transform Maxwell’s equations bythesubstitution ofequations 15.2, theirform does notremain thesame; therefore, inamoving space shiptheelectrical and optical phenomena should bedifferent from those inastationary ship. Thus onecould usethese optical phenomena todetermine thespeed oftheship; in particular, onecould determine theabsolute speed oftheshipbymaking suitable optical orelectrical measurements. Oneoftheconsequences ofMaxwell’s equa- tions isthatifthere isadisturbance inthefieldsuchthatlight isgenerated, these electromagnetic waves gooutinalldirections equally andatthesame speed c,or 186,000 mi/sec. Another consequence oftheequations isthat ifthesource ofthedisturbance ismoving, thelight emitted goes through space atthesame speed c.This isanalogous tothecaseofsound, thespeed ofsound waves being likewise independent ofthemotion ofthesource. Thisindependence ofthemotion ofthesource, inthecaseoflight, brings up aninteresting problem: Suppose weareriding inacarthatisgoing ataspeed u,andlight from the rearisgoing past thecarwith speed c.Differentiating thefirstequation in(15.2) gives dx’/dt =dx/dt —u, which means thataccording totheGalilean transformation theapparent speed of thepassing light, aswemeasure itinthecar,should notbecbutshould bec—u. Forinstance, ifthecarisgoing 100,000 mi/sec, andthelight isgoing 186,000 mi/sec, then apparently thelight going past thecarshould go86,000 mi/sec. Inanycase, bymeasuring thespeed ofthelightgoing pastthecar(iftheGalilean transformation iscorrect forlight), onecould determine thespeed ofthecar. A number ofexperiments based onthisgeneral ideawere performed todetermine thevelocity oftheearth, butthey allfailed—they gave novelocity atall. We shall discuss oneofthese experiments indetail, toshow exactly what wasdone andwhat wasthematter; something wasthematter, ofcourse, something was wrong withtheequations ofphysics. What could itbe? 15-2 15-2 TheLorentz transformation When thefailure oftheequations ofphysics intheabove casecame tolight, thefirstthought that occurred wasthat thetrouble must lieinthenew Maxwell equations ofelectrodynamics, which were only20years oldatthetime. Itseemed almost obvious thatthese equations must bewrong, sothething todowasto change them insuch awaythatunder theGalilean transformation theprinciple ofrelativity would besatisfied. When thiswastried, thenewterms thathadto beputintotheequations ledtopredictions ofnewelectrical phenomena thatdid notexist atallwhen tested experimentally, sothisattempt hadtobeabandoned. Then itgradually became apparent thatMaxwell’s laws ofelectrodynamics were correct, andthetrouble must besought elsewhere. Inthemeantime, H.A.Lorentz noticed aremarkable andcurious thing when hemade thefollowing substitutions intheMaxwell equations: x/=M, \/1—u?/c2 y’=y, Z1ZZ, (15.3) ,/=ll/Z,\/1—u2/c2 namely, Maxwell’s equations remain inthesame form when thistransformation isapplied tothem! Equations (15.3) areknown asaLorentz transformation. Einstein, following asuggestion originally made byPoincare, then proposed that allthephysical laws should beofsuch akind that they remain unchanged under a Lorentz transformation. Inother words, weshould change, notthelaws ofelectro- dynamics, but thelaws ofmechanics. How shall wechange Newton’s laws sothat they Wlll remain unchanged bytheLorentz transformation? Ifthis goal isset,wethen have torewrite Newton’s equations insuch away that the conditions wehave imposed aresatisfied. Asitturned out,theonly requirement is that themass minNewton’s equations must bereplaced bytheform shown in Eq.(15.1). When thischange ismade, Newton’s lawsandthelawsofelectrody- namics willharmonize. Then ifweusetheLorentz transformation incomparing Moe’s measurements withJoe’s, weshall never beabletodetect whether either is moving, because theform ofalltheequations willbethesame inboth coordinate systems! Itisinteresting todiscuss what itmeans thatwereplace theoldtransformation between thecoordinates andtime with anew one, because theoldone(Galilean) seems tobeself-evident, andthenew one(Lorentz) looks peculiar. Wewish to know whether itislogically andexperimentally possible that thenew, andnotthe old,transformation canbecorrect. Tofind thatout,itisnotenough tostudy the laws ofmechanics but,asEinstein did,wetoomust analyze ourideas ofspace andtimeinorder tounderstand thistransformation. Weshall have todiscuss these ideas andtheir implications formechanics atsome length, sowesayin advance thattheeffort willbejustified, since theresults agree withexperiment. 15-3 TheMichelson-Morley experiment Asmentioned above, attempts were made todetermine theabsolute velocity oftheearth through thehypothetical “ether” that wassupposed topervade all space. The most famous ofthese experiments isoneperformed byMichelson andMorley in1887. Itwas18years later before thenegative results oftheexperi- ment were finally explained, byEinstein. TheMichelson-Morley experiment wasperformed withanapparatus likethat shown schematically inFig. 15-2. This apparatus isessentially comprised ofa light source A,apartially silvered glass plate B,andtwomirrors CandE,all mounted onarigid base. Themirrors areplaced atequal distances Lfrom B. Theplate Bsplits anoncoming beam oflight, andthetworesulting beams con- 15-3If \\\\,:»m /11I1I11. / \1 iL I \/ \/L. \/ \ ’ \ \A ,\/a Source *1*0—-—---A ‘!@§s-ar4'‘ /~_§-,_\),.<.-\‘\\V 1 M Waves \ l|'\phase \L, Waves out 31'l/ olphase t\,t 0-v\ q-\/~—~.,,\_\Ht>)1 Fig. 15-2. Schematic diogrom ofthe Michelson-Morley experiment.':i1I_'flll rl‘ >. ll 1 1Y” I111 ll‘ 11 l :_<5;-J.x__....¢.;_-:-:_.=._.,;.t¢m..=:,.'_r.}:“% :1 1 1) .t .1 1t11 I 1 9 F 1 -4 1tinue inmutually perpendicular directions tothemirrors, where theyarereflected back toB.Onarriving back atB,thetwobeams arerecombined astwosuperposed beams, DandF.Ifthetimetaken forthelight togofrom BtoEandback isthe same asthetimefrom BtoCandback, theemerging beams DandFwillbein phase andwillreinforce each other, butifthetwotimes differ slightly, thebeams willbeslightly outofphase andinterference willresult. Iftheapparatus is“at rest” intheether, thetimes should beprecisely equal, butifitismoving toward theright withavelocity u,there should beadifference inthetimes. Letusseewhy. First, letuscalculate thetimerequired forthelight togofrom BtoEand back. Letussaythatthetimeforlight togofrom plate Btomirror Eist1,and thetimeforthereturn ist2.Now, while thelightisonitswayfrom Btothemirror, theapparatus moves adistance utl,sothelight must traverse adistance L-1-ut,, atthespeed c.Wecanalsoexpress thisdistance asctl,sowehave ctl=L+ut,, or 11=L/(c —u). (This result isalsoobvious from thepoint ofviewthatthevelocity oflightrelative totheapparatus isc—u,sothetimeisthelength Ldivided byc—u.)Inalike manner, thetime t2canbecalculated. During thistime theplate Badvances a distance utg,sothereturn distance ofthelight isL—U12.Then wehave ct2=L—I112, or t2=L/(c -1-u). Then thetotal time is t1—l~t2=2Lc/(c2 —uz). Forconvenience inlater comparison oftimes wewrite thisas 2L/ctl-l-I2= (15-4) Oursecond calculation willbeofthetimet3forthelighttogofrom Btothe mirror C.Asbefore, during timet3themirror Cmoves totheright adistance ut3 totheposition C’;inthesame time, thelight travels adistance ct3along the hypotenuse ofatriangle, which isBC’. Forthisright triangle wehave (vi-.02 =L2+(uta)2Or L2=c2z§—u’r§=(c2—u2)t§, from which weget t3=L/\/c2 —u2. Forthereturn tripfrom C’thedistance isthesame, ascanbeseen from the symmetry ofthefigure; therefore thereturn timeisalsothesame, andthetotal timeis2t3.With alittle rearrangement oftheform wecanwrite 2L 2L/c2=la =-i———- 15.5 ta \/c2 —u2 \/1—uz/c2 ( ) Wearenowabletocompare thetimes taken bythetwobeams oflight. In expressions (15.4) and(15.5) thenumerators areidentical, andrepresent thetime thatwould betaken iftheapparatus were atrest. Inthedenominators, theterm u2/c2 willbesmall, unless uiscomparable insizetoc.Thedenominators represent themodifications inthetimes caused bythemotion oftheapparatus. Andbehold, these modifications arenotthesame—the timetogotoCandback isalittle less thanthetimetoEandback, even though themirrors areequidistant from B,and allwehave todoistomeasure thatdifference withprecision. Here aminor technical point arises—suppose thetwolengths Larenot exactly equal? Infact,wesurely cannot make them exactly equal. Inthatcase wesimply turntheapparatus 90degrees, sothatBCisinthelineofmotion and BEisperpendicular tothemotion. Anysmalldifference inlength thenbecomes 15-4 unimportant, andwhat welook forisashift intheinterference fringes when we rotate theapparatus. Incarrying outtheexperiment, Michelson andMorley oriented theapparatus sothatthelineBEwasnearly parallel totheearth’s motion initsorbit (atcertain times ofthedayandnight). This orbital speed isabout 18miles persecond, and any“ether drift” should beatleast thatmuch atsome time ofthedayornight and atsome time during theyear. Theapparatus wasamply sensitive toobserve such aneffect, butnotime difference wasfound—the velocity oftheearth through the ether could notbedetected. Theresult oftheexperiment wasnull. Theresult oftheMichelson-Morley experiment wasverypuzzling andmost disturbing. Thefirstfruitful idea forfinding awayoutoftheimpasse came from Lorentz. Hesuggested that material bodies contract when they aremoving, and that thisforeshortening isonly inthedirection ofthemotion, andalso, that if thelength isLOwhen abody isatrest, then when itmoves with speed uparallel toitslength, thenewlength, which wecallL||(L-parallel), isgiven by L||=L0\/1—u2/c2. (15.6) When thismodification isapplied totheMichelson-Morley interferometer appara- tusthedistance from BtoCdoes notchange, butthedistance from BtoEis shortened toL\/l —u2/c2. Therefore Eq.(15.5) isnotchanged, buttheLof Eq.(15.4) must bechanged inaccordance with Eq.(15.6). When thisisdone we obtain __. 2 2 Comparing thisresult with Eq.(15.5), weseethat t1-1-t2=2t3. Soiftheap- paratus shrinks inthemanner justdescribed, wehave awayofunderstanding why theMichelson-Morley experiment gives noeffect atall.Although thecontraction hypothesis successfully accounted forthenegative result oftheexperiment, itwas open totheobjection that itwasinvented fortheexpress purpose ofexplaining away thedifficulty, andwastooartificial. However, inmany other experiments todiscover anether wind, similar difficulties arose, until itappeared thatnature wasina“conspiracy” tothwart man byintroducing some newphenomenon to undo every phenomenon thathethought would permit ameasurement ofu. Itwasultimately recognized, asPoincaré pointed out,thatacomplete conspiracy isitsehf alawofnature! Poincare then proposed thatthere issuch alawofnature, thatitisnotpossible todiscover anether wind byanyexperiment; that is,there isnowaytodetermine anabsolute velocity. 15-4 Transformation oftime Inchecking outwhether thecontraction ideaisinharmony withthefacts in other experiments, itturns outthat everything iscorrect provided that thetimes arealsomodified, inthemanner expressed inthefourth equation oftheset(15.3). That isbecause thetime t3,calculated forthetripfrom BtoCandback, isnotthe same when calculated byaman performing theexperiment inamoving space shipaswhen calculated‘by astationary observer whoiswatching thespace ship. Totheman intheship thetime issimply 2L/c, buttotheother observer itis (2L/c)/\/l —M2/C2 (Eq. 15.5). Inother words, when theoutsider seestheman inthespace shiplighting acigar, alltheactions appear tobeslower thannormal, while totheman inside, everything moves atanormal rate. Sonotonly must the lengths shorten, butalso thetime-measuring instruments (“clocks”) must appar- ently slow down. That is,when theclock inthespace shiprecords lsecond elapsed, asseenbytheman intheship, itshows l/\/1 —uz/c2 second tothe manoutside. Thisslowing oftheclocks inamoving system isaverypeculiar phenomenon, andisworth anexplanation. Inorder tounderstand this,wehave towatch the machinery oftheclock andseewhat happens when itismoving. Since thatis 15-5 > i 1 i i I. 1li4Mirror \ $4.=1 ii(I Photocull \ (0)Pulse r______ reflected __ t - r"‘1 tutkw Ssystnm .cf gs‘ti, A D isira jgpp"J‘t‘7"‘»-~-‘la lJPurse ‘U-> —___ pm” omitted (b) received iyjnCnC u (C) Fig.15-3. (ci)A"Iight clock" atrest intheS’system. lb)Thesome clock, moving through theSsystem. (c)Illustra- tionofthediogonol path token bythe light becim incimoving “light c|ock."- -r V e rather difficult, weshall takeaverysimple kind ofclock. Theonewechoose is rather asillykind ofclock, butitwillwork inprinciple: itisarod(meter stick) withamirror ateach end,andwhen westart alight signal between themirrors, thelight keeps going upanddown, making aclick every timeitcomes down, like astandard ticking clock. Webuild twosuchclocks, withexactly thesame lengths, andsynchronize them bystarting them together; thentheyagree always thereafter, because theyarethesame inlength, andlight always travels with speed c.We giveoneofthese clocks tothemantotakealong inhisspace ship, andhemounts therodperpendicular tothedirection ofmotion oftheship; then thelength of therodwillnotchange. How doweknow thatperpendicular lengths donot change? Themencanagree tomake marks oneach other’s y-meter stick asthey pass each other. Bysymmetry, thetwomarks must come atthesame y-and y’-coordinates, since otherwise, when they gettogether tocompare results, one mark willbeabove orbelow theother, andsowecould tellwho wasreally moving. Now letusseewhat happens tothemoving clock. Before theman took it aboard, heagreed that itwasanice, standard clock, andwhen hegoes along in thespace ship hewillnotseeanything peculiar. Ifhedid,hewould know hewas moving—if anything atallchanged because ofthemotion, hecould tellhewas moving. Buttheprinciple ofrelativity saysthisisimpossible inauniformly moving system, sonothing haschanged. Ontheother hand, when theexternal observer looks attheclock going by,heseesthatthelight, ingoing from mirror tomirror, is“really” taking azigzag path, since therodismoving sidewise allthewhile. Wehave already analyzed suchazigzag motion inconnection withtheMichelson- Morley experiment. Ifinagiven timetherodmoves forward adistance propor- tional touinFig.15-3, thedistance thelight travels inthesame timeispropor- tional toc,andthevertical distance istherefore proportional to\/c2 —uz. That is,ittakes alonger timeforlight togofrom endtoendinthemoving clock than inthestationary clock. Therefore theapparent timebetween clicks is longer forthemoving clock, inthesame proportion asshown inthehypotenuse ofthetriangle (that isthesource ofthesquare rootexpressions inourequations). From thefigure itisalsoapparent thatthegreater uis,themore slowly themoving clock appears torun. Notonlydoesthisparticular kindofclock runmore slowly, butifthetheory ofrelativity iscorrect, anyother clock, operating onanyprinciple whatsoever, would alsoappear torunslower, andinthesame proportion—we cansaythiswithout further analysis. Why isthisso‘? Toanswer theabove question, suppose wehadtwoother clocks made exactly alike withwheels andgears, orperhaps based onradioactive decay, orsomething else. Then weadjust these clocks sotheyboth runinprecise synchronism with ourfirstclocks. When light goesupandback inthefirstclocks andannounces itsarrival with aclick, thenewmodels alsocomplete some sortofcycle, which theysimultaneously announce bysome doubly coincident flash, orbong, orother signal. Oneofthese clocks istaken intothespace ship, along withthefirstkind. Perhaps thisclock willnotrunslower, butwillcontinue tokeep thesame timeas itsstationary counterpart, andthusdisagree withtheother moving clock. Ahno, ifthatshould happen, theman intheshipcould usethismismatch between his twoclocks todetermine thespeed ofhisship, which wehave been supposing isimpossible. Weneed notknow anything about themachinery ofthenewclock that might cause theeffect—we simply know that whatever thereason, itwill appear torunslow, justlikethefirstone. Now ifallmoving clocks runslower, ifnoway ofmeasuring time gives any- thing butaslower rate, weshall just have tosay, inacertain sense, that time itself appears tobeslower inaspace ship. Allthephenomena there—the man’s pulse rate, histhought processes, thetimehetakes tolight acigar, how long it takes togrow upandgetold--all these things must beslowed down inthesame proportion, because hecannot tellheismoving. Thebiologists andmedical men sometimes sayitisnotquite certain thatthetimeittakes foracancer todevelop willbelonger inaspace ship, butfrom theviewpoint ofamodern physicist itis nearly certain; otherwise onecould usetherateofcancer development todetermine thespeed oftheship! 15-6 Averyinteresting example oftheslowing oftime with motion isfurnished bymu-mesons (muons), which areparticles that disintegrate spontaneously after anaverage lifetime of2.2XlO_‘* sec. They come totheearth incosmic rays, and can also beproduced artificially inthelaboratory. Some ofthem disintegrate inmidair, buttheremainder disintegrate only after they encounter apiece ofmaterial andstop. Itisclear thatinitsshort lifetime amuon cannot travel, even atthespeed oflight, much more than 600meters. Butalthough the muons arecreated atthetopoftheatmosphere, some 10kilometers up,yettheyare actually found inalaboratory down here, incosmic rays. How canthatbe? Theanswer isthatdifferent muons move atvarious speeds, some ofwhich are veryclose tothespeed oflight. While from theirownpoint ofviewtheyliveonly about 2iisec, from ourpoint ofviewtheyliveconsiderably longer—enough longer thatthey may reach theearth. Thefactor bywhich thetime isincreased hasalready beengiven as1/\/1—112/c2.Theaverage lifehasbeen measured quite accurately formuons ofdifferent velocities, andthevalues agree closely with theformula. Wedonotknow whythemeson disintegrates orwhat itsmachinery is,but wedoknow itsbehavior satisfies theprinciple ofrelativity. That istheutility of theprinciple ofrelativity——it permits ustomake predictions, even about things that otherwise wedonotknow much about. Forexample, before wehave any ideaatallabout what makes themeson disintegrate, wecanstillpredict thatwhen itismoving atnine-tenths of f liglg, theapparent length oftimethat itlasts is(2.2X10f_1[\/1 —92/102 sec;andourprediction works—that is thegoodllfiigabout it. 7 ‘ - 15-5 TheLorentz contraction Now letusreturn totheLorentz transformation (15.3) andtrytogetabetter understanding oftherelationship between the(x,y,2,t)and the(x’,y’,z', I’) coordinate systems, which weshall calltheSand S’systems, orJoeandMoe systems, respectively. Wehave already noted thatthefirstequation isbased on theLorentz suggestion ofcontraction along thex-direction ;howcanweprove that acontraction takes place? IntheMichelson-Morley experiment, wenowappre- ciatethatthetransverse armBCcannot change length, bytheprinciple ofrelativity; yetthenullresult oftheexperiment demands thatthetimes must beequal. So,in order fortheexperiment togiveanullresult, thelongitudinal armBEmust appear shorter, bythesquare root\/1—u2/c2. What doesthiscontraction mean, interms ofmeasurements made byJoeandMoe? Suppose thatMoe, moving withtheS’system inthex-direction, ismeasuring thex’-coordinate ofsome point withameter stick. Helaysthestick down x’times, sohethinks thedistance is x’meters. From theviewpoint ofJoeintheSsystem, however, Moe isusing a foreshortened ruler, sothe“real” distance measured isx’\/1 —uz/c2 meters. Then iftheS’system hastravelled adistance utaway from theSsystem, theS observer would saythatthesame point, measured inhiscoordinates, isata distance x=x’\/1 —u2/c2 +ut,or X—lll \/1-U2/C2, which isthefirstequation oftheLorentz transformation.x’= 15-6 Simultaneity Inananalogous way, because ofthedifference intimescales, thedenominator expression isintroduced intothefourth equation oftheLorentz transformation. Themost interesting term inthatequation istheux/c2 inthenumerator, because thatisquite newandunexpected. Now what does thatmean? Ifwelook atthe situation carefully weseethatevents thatoccur attwoseparated places atthesame time, asseenbyMoe inS’,donothappen atthesame timeasviewed byJoeinS. Ifoneevent occurs atpoint x1attime toandtheother event atx2andto(thesame 15-7 l l1 4l 4 V | I . Il 1 v l l l1tl ‘. l 1 1 r r l 1 v l .l ‘1time), wefindthatthetwocorresponding times ifandtédiffer byanamount u(x—X2)/<12t’—t’1=iii- 2 \/l—u2/c2 This circumstance iscalled “failure ofsimultaneity atadistance,” andtomake theidea alittle clearer letusconsider thefollowing experiment. Suppose thataman moving inaspace ship (system S’)hasplaced aclock at each endoftheship andisinterested inmaking sure that theltwoclocks arein synchronism. How can theclocks besynchronized? There aremany ways. One way, involving very little calculation, would befirst tolocate exactly the midpoint between theclocks. Then from thisstation wesend outalight signal which willgoboth ways atthesame speed andwillarrive atboth clocks, clearly, atthesame time. This simultaneous arrival ofthesignals canbeused tosyn- chronize theclocks. Letusthensuppose thatthemaninS’synchronizes hisclocks bythisparticular method. Letusseewhether anobserver insystem Swould agree thatthetwoclocks aresynchronous. ThemaninS’hasaright tobelieve theyare, because hedoesnotknow thatheismoving. ButthemaninSreasons thatsince theshipismoving forward, theclock inthefront endwasrunning away from the light signal, hence thelighthadtogomore thanhalfway inorder tocatch up;the rearclock, however, wasadvancing tomeet thelight signal, sothisdistance was shorter. Therefore thesignal reached therearclock first, although themaninS’ thought thatthesignals arrived simultaneously. Wethus seethatwhen aman ina space shipthinks thetimes attwolocations aresimultaneous, equal values oft’ inhiscoordinate system must correspond todiflerenz values oftintheother coordinate system! 15-7 Four-vectors Letusseewhat elsewecandiscover intheLorentz transformation. Itis interesting tonote thatthetransformation between thex’sandt’sisanalogous in form tothetransformation ofthex’sandy’sthatwestudied inChapter llfora rotation ofcoordinates. Wethen had x/=xcos 0+ysin 0, (158) y’=ycos0 -xs1n0, inwhich thenew x’mixes theoldxandy,andthenewy’also mixes theoldx andy;similarly, intheLorentz transformation wefindanewx’which isamixture ofxandt,andanewt’which isamixture oftandx.SotheLorentz transforma- tionisanalogous toarotation, only itisa“rotation” inspace andtime, which appears tobeastrange concept. Acheck oftheanalogy torotation canbemade bycalculating thequantity x12 +y/2 +Z12 _C21/2 :x2 __|_y2 +Z2 ___C212’ Inthisequation thefirstthree terms oneach siderepresent, inthree-dimensional geometry, thesquare ofthedistance between apoint andtheorigin (surface ofa sphere) which remains unchanged (invariant) regardless ofrotation oftheco- ordinate axes. Similarly, Eq.(15.9) shows that there isacertain combination which includes time, that isinvariant toaLorentz transformation. Thus, the analogy toarotation iscomplete, andisofsuchakindthatvectors, i.e.,quantities involving “components” which transform thesame wayasthecoordinates and time, arealsouseful inconnection withrelativity. Thus wecontemplate anextension oftheideaofvectors, which wehave so farconsidered tohave only space components, toinclude atime component. That is,weexpect thatthere willbevectors with four components, three ofwhich arelikethecomponents ofanordinary vector, andwith these willbeassociated afourth component, which 1Stheanalog ofthetimepart. 15-8 This concept willbeanalyzed further inthenextchapters, where weshall findthatiftheideas ofthepreceding paragraph areapplied tomomentum, the transformation gives three space parts thatarelikeordinary momentum com- ponents, andafourth component, thetimepart, which istheenergy. 15-8 Relativistic dynamics Wearenow ready toinvestigate, more generally, what form thelaws of mechanics takeunder theLorentz transformation. [Wehave thusfarexplained howlength andtime change, butnothowwegetthemodified formula form (Eq. 15.1). Weshall dothisinthenextchapter.] Toseetheconsequences of Einstein’s modification ofmforNewtonian mechanics, westartwiththeNewtonian lawthatforce istherateofchange ofmomentum, or F=d(mv)/dz. Momentum isstillgiven bymv,butwhen weusethenewmthisbecomes p=mv= (15.10) This isEinstein’s modification ofNewton’s laws. Under thismodification, if action andreaction arestillequal (which they may notbeindetail, butareinthe long run), there willbeconservation ofmomentum inthesame wayasbefore, butthequantity thatisbeing conserved isnottheoldmvwithitsconstant mass, butinstead isthequantity shown in(15.10), which hasthemodified mass. When thischange ismade intheformula formomentum, conservation ofmomentum stillworks. Now letusseehowmomentum varies withspeed. InNewtonian mechanics itisproportional tothespeed and,according to(15.10), overaconsiderable range ofspeed, butsmall compared withc,itisnearly thesame inrelativistic mechanics, because thesquare-root expression differs only slightly from 1.Butwhen vis almost equal toc,thesquare-root expression approaches zero, andthemomentum therefore goestoward infinity. What happens ifaconstant force actsonabody foralongtime? InNewtonian mechanics thebody keeps picking upspeed until itgoesfaster thanlight. Butthis isimpossible inrelativistic mechanics. Inrelativity, thebody keeps picking up, notspeed, butmomentum, which cancontinually increase because themass is increasing. After awhile there ispractically noacceleration inthesense ofachange ofvelocity, butthemomentum continues toincrease. Ofcourse, whenever aforce produces verylittle change inthevelocity ofabody, wesaythatthebody hasa great deal ofinertia, andthatisexactly what ourformula forrelativistic mass says(seeEq.l5.l0)—it saysthattheinertia isverygreat when visnearly asgreat asc.Asanexample ofthiseffect, todeflect thehigh-speed electrons inthesyn- chrotron thatisused hereatCaltech, weneed amagnetic fieldthatis2000 times stronger than would beexpected onthebasis ofNewton’s laws. Inother words, themass oftheelectrons inthesynchrotron is2000 times asgreat astheirnormal mass, andisasgreat asthatofaproton! That mshould be2000 times momeans thatI—v2/c2 must bel/4,000,000, andthatmeans thatv2/c2 differs from 1 byonepartin4,000,000, orthatvdiffers from cbyonepartin8,000,000, sothe electrons aregetting pretty close tothespeed oflight. Iftheelectrons andlight were both tostart from thesynchrotron (estimated as700feetaway) andrush outtoBridge Lab, which would arrive first? Thelight, ofcourse, because light always travels faster.* How much earlier? That istoohard totel1—instead, we tellbywhat distance thelightisahead: itisabout l/1000 ofaninch, or%;thethick- nessofapiece ofpaper! When theelectrons aregoing thatfasttheir masses are enormous, buttheir speed cannot exceed thespeed oflight. *Theelectrons would actually wintheraceversus visible light because oftheindex of refraction ofair.Agamma raywould make outbetter. 15-9 1 1 5. M1‘ 1 I 1 l1 1 . r p 1 l 1l1 l 1 l Jr Now letuslook atsome further consequences ofrelativistic change ofmass. Consider themotion ofthemolecules inasmall tank ofgas. When thegasis heated, thespeed ofthemolecules isincreased, andtherefore themass isalso increased andthegasisheavier. Anapproximate formula toexpress theincrease ofmass, forthecase when thevelocity issmall, canbefound byexpanding mo/\/1 —v2/c2 =m0(1 —122/c2)_1/2 inapower series, using thebinomial theorem. Weget mu(l -112/02)"/2 =mo(1 +%v2/62 +%v“/C‘ +--')- Weseeclearly from theformula thattheseries converges rapidly when vissmall, andtheterms after thefirsttwoorthree arenegligible. Sowecanwrite l m2mo—l—%mOv2(c—2) (15.11) inwhich thesecond term ontheright expresses theincrease ofmass duetomo- lecular velocity. When thetemperature increases thev2increases proportionately, sowecansaythattheincrease inmass isproportional totheincrease intempera- ture. Butsince %m0v2 isthekinetic energy intheold-fashioned Newtonian sense, wecanalsosaythattheincrease inmass ofallthisbody ofgasisequal tothe increase inkinetic energy divided byc2,orAm=A(K.E.)/c2. 15-9 Equivalence ofmass andenergy Theabove observation ledEinstein tothesuggestion thatthemass ofabody canbeexpressed more simply than bytheformula (15.1), ifwesaythatthemass isequal tothetotal energy content divided by02.IfEq.(15.11) ismultiplied by c2theresult is mcz=m0c2 +émovz + (15.12) Here, theterm ontheleftexpresses thetotal energy ofabody, andwerecognize thelastterm astheordinary kinetic energy. Einstein interpreted thelarge constant term, m0c2, tobepartofthetotal energy ofthebody, anintrinsic energy known asthe“rest energy.” Letusfollow outtheconsequences ofassuming, with Einstein, that the energy ofabody always equals mc2. Asaninteresting result, weshall findthe formula (15.1) forthevariation ofmass withspeed, which wehave merely assumed uptonow. Westart with thebody atrest,when itsenergy ismocz. Then we apply aforce tothebody, which starts itmoving andgives itkinetic energy; therefore, since theenergy hasincreased, themass hasincreased—this isimplicit intheoriginal assumption. Solongastheforce continues, theenergy andthemass bothcontinue toincrease. Wehavealready seen(Chapter 13)thattherateofchange ofenergy with time equals theforce times thevelocity, or dE—= F-. 15.13dz V ( ) Wealsohave (Chapter 9,Eq.9.1)thatF=d(mv)/dt. When these relations are puttogether withthedefinition ofE,Eq.(15.13) becomes d(mc2) _d(mv)-71- _v-7-- (15.14) Wewish tosolve thisequation form.Todothiswefirstusethemathematical trick ofmultiplying both sides by2m,which changes theequation to ¢’(2m) 9%=Zmvi(;"T”- (15.15) Weneed togetridofthederivatives, which canbeaccomplished byintegrating 15—10 both sides. Thequantity (Zm) dm/dt canberecognized asthetime derivative of m2,and(2mv) -d(mv)/dt isthetime derivative of(mv)2. So,Eq.(15.15) isthe same as 2 22 C251%’-l =1’%)- (15.16) Ifthederivatives oftwoquantities areequal, thequantities themselves differ at most byaconstant, sayC.Thispermits ustowrite m2c2 =mzvz -1-C. (15.17) Weneed todefine theconstant Cmore explicitly. Since Eq.(15.17) must betrue forallvelocities, wecanchoose aspecial casewhere v=0,andsaythatinthis casethemass ismo.Substituting these values intoEq.(15.17) gives mficz =0+C. Wecannowusethisvalue ofCinEq.(15.17), which becomes m2c2:m2v2+mgcz. (15.18) Dividing by02andrearranging terms gives m"’<1—12/C’)=mt,from which weget m=mo/\/l —-v2/02. (15.19) Thisistheformula (15.1), andisexactly what isnecessary fortheagreement be- tween mass andenergy inEq.(15.12). Ordinarily these energy changes represent extremely slight changes inmass, because most ofthetimewecannot generate much energy from agiven amount ofmaterial; butinanatomic bomb ofexplosive energy equivalent to20kilotons ofTNT, forexample, itcanbeshown thatthedirtafter theexplosion islighter by 1gram than theinitial mass ofthereacting material, because oftheenergy thatwas released, i.e.,thereleased energy hadamass of1gram, according totherelationship AE=A(mc 2).This theory ofequivalence ofmass andenergy hasbeen beautifully verified byexperiments inwhich matter isannihilated—converted totally toenergy: Anelectron andapositron come together atrest, each with arestmass mo. When theycome together theydisintegrate andtwogamma raysemerge, each with the measured energy ofmocz. This experiment furnishes adirect determination of theenergy associated withtheexistence oftherestmass ofaparticle. 15-11l I6 Relativistic Energy and Momentum 16-1 Relativity andthephilosophers Inthischapter weshall continue todiscuss theprinciple ofrelativity of Einstein andPoincaré, asitaffects ourideas ofphysics andother branches of human thought. Poincaré made thefollowing statement oftheprinciple ofrelativity: “Accord- ingtotheprinciple ofrelativity, thelaws ofphysical phenomena must bethesame forafixed observer asforanobserver who hasauniform motion oftranslation relative tohim, sothatwehave not,norcanwepossibly have, anymeans of discerning whether ornotwearecarried along insuch amotion.” When thisideadescended upon theworld, itcaused agreat stiramong philos- ophers, particularly the“cocktail-party philosophers,” who say, “Oh, itisvery simple: Einstein’s theory saysallisrelative!” Infact, asurprisingly large number ofphilosophers, notonlythose found atcocktail parties (butrather thanembarrass them, weshall justcallthem “cocktail-party philosophers”), willsay,“That all isrelative isaconsequence ofEinstein, andithasprofound influences onour ideas.” Inaddition, theysay“lthasbeen demonstrated inphysics thatphenomena depend upon your frame ofreference.” Wehearthatagreat deal. butitisdifficult tofindoutwhat itmeans. Probably theframes ofreference thatwere originally referred towere thecoordinate systems which weuseintheanalysis ofthetheory ofrelativity. Sothefactthat“things depend upon your frame ofreference” is supposed tohave hadaprofound effect onmodern thought. One might well wonder why, because, after all,thatthings depend upon one’s point ofview isso simple anideathatitcertainly cannot have been necessary togotoallthetrouble ofthephysical relativity theory inorder todiscover it.That what oneseesdepends upon hisframe ofreference iscertainly known toanybody who walks around, because heseesanapproaching pedestrian firstfrom thefront andthen from the back; there isnothing deeper inmost ofthephilosophy which issaidtohave come from thetheory ofrelativity than theremark that“Aperson looks different from thefront than from theback.” Theoldstory about theelephant thatseveral blind mendescribe indifferent ways isanother example, perhaps, ofthetheory ofrela- tivity from thephilosopher’s point ofview. Butcertainly there must bedeeper things inthetheory ofrelativity thanjust thissimple remark that“Aperson looks different from thefront than from the back." Ofcourse relativity isdeeper than this, because wecanmake definite predictions withit.Itcertainly would berather remarkable ifwecould predict the behavior ofnature from such asimple observation alone. There isanother school ofphilosophers who feelvery uncomfortable about thetheory ofrelativity, which asserts that wecannot determine ourabsolute velocity without looking atsomething outside, andwho would say,“Itisobvious thatonecannot measure hisvelocity without looking outside. Itisself-evident that itismeaningless totalkabout thevelocity ofathing without looking outside; the physicists arerather stupid forhaving thought otherwise, butithasjustdawned onthem thatthisisthecase. Ifonly wephilosophers hadrealized what theprob- lems were thatthephysicists had, wecould have decided immediately bybrain- work thatitisimpossible totellhow fastoneismoving without looking outside, andwecould have made anenormous contribution tophysics.” These philosophers arealways withus,struggling intheperiphery totrytotellussomething, butthey never really understand thesubtleties anddepths oftheproblem. 16-116-1 Relativity andthephilosophers 16-2 Thetwinparadox 16-3 Transformation ofvelocities 16-4 Relativistic mass 16-5 Relativistic energy Ourinability todetect absolute motion isaresult ofexperiment andnota result ofplain thought, aswecaneasily illustrate. Inthefirstplace, Newton believed thatitwastruethatonecould nottellhowfastheisgoing ifheismoving with uniform velocity inastraight line. Infact, Newton firststated theprinciple ofrelativity, andonequotation made inthelastchapter wasastatement ofNew- ton’s. Why thendidthephilosophers notmake allthisfussabout “allisrelative,” orwhatever, inNewton’s time? Because itwasnotuntil Maxwell’s theory of electrodynamics wasdeveloped thatthere were physical laws thatsuggested that onecould measure hisvelocity without looking outside; soon itwasfound experi- mentally thatonecould not. Now, isitabsolutely, definitely, philosophically necessary that oneshould notbeable totellhow fastheismoving without looking outside? Oneofthe consequences ofrelativity was thedevelopment ofaphilosophy which said, “You canonlydefine what youcanmeasure! Since itisself-evident thatonecan- notmeasure avelocity without seeing what heismeasuring itrelative to,therefore itisclear thatthere isnomeaning toabsolute velocity. Thephysicists should have realized thattheycantalkonlyabout what theycanmeasure." Butthatisthewhole problem: whether ornotonecandefine absolute velocity isthesame astheproblem ofwhether ornotonecandetect inanexperiment, without looking outside, whether heismoving. Inother words, whether ornotathing ismeasurable isnotsomething tobedecided apriori bythought alone, butsomething thatcanbedecided only byexperiment. Given thefactthatthevelocity oflight is186,000 mi/sec, onewill findfewphilosophers whowillcalmly state thatitisself-evident thatiflight goes 186,000 mi/sec inside acar,andthecarisgoing 100,000 mi/sec, thatthelight also goes 186,000 mi/sec past anobserver ontheground. That isashocking facttothem; thevery ones whoclaim itisobvious find, when yougivethem a specific fact, thatitisnotobvious. Finally, there iseven aphilosophy which says thatonecannot detect any motion except bylooking outside. Itissimply nottrueinphysics. True, onecan- notperceive auniform motion inastraight line,butifthewhole room were ro- tating wewould certainly know it,foreverybody would bethrown tothewall-—- there would beallkinds of“centrifugal” effects. That theearth isturning onits axiscanbedetermined without looking atthestars, bymeans oftheso-called Foucault pendulum, forexample. Therefore itisnottruethat“allisrelative”; itisonlyuniform velocity thatcannot bedetected without looking outside. Uniform rotation about afixed axiscanbe.When thisistoldtoaphilosopher, heisvery upset thathedidnotreally understand it,because tohimitseems impossible that oneshould beabletodetermine rotation about anaxiswithout looking outside. Ifthephilosopher isgood enough, after some time hemay come back andsay, “Iunderstand. Wereally donothave such athing asabsolute rotation; weare really rotating relative tothestars, yousee.And sosome influence exerted bythe stars ontheobject must cause thecentrifugal force.” Now, forallweknow, thatistrue; wehave noway, atthepresent time, of telling whether there would have been centrifugal force ifthere were nostars and nebulae around. Wehave notbeen abletodotheexperiment ofremoving allthe nebulae andthen measuring ourrotation, sowesimply donotknow. Wemust admit thatthephilosopher mayberight. Hecomes back, therefore, indelight and says, “Itisabsolutely necessary thattheworld ultimately turnouttobethisway: absolute rotation means nothing; itisonly relative tothenebulae.” Then wesay tohim, “Now, myfriend, isitorisitnotobvious thatuniform velocity inastraight line,relative tothenebulae should produce noeffects inside acar?” Now thatthe motion isnolonger absolute, butisamotion relative tothenebulae, itbecomes a mysterious question, andaquestion thatcanbeanswered only byexperiment. What, then, arethephilosophic influences ofthetheory ofrelativity? Ifwe limit ourselves toinfluences inthesense ofwhat kindofnewideas andsuggestions aremade tothephysicist bytheprinciple ofrelativity, wecould describe some of them asfollows. Thefirstdiscovery is,essentially, thateven those ideas which have been heldforaverylong time andwhich have been veryaccurately verified might bewrong. Itwasashocking discovery, ofcourse, thatNewton’s laws are 16-2 wrong, after alltheyears inwhich they seemed tobeaccurate. Ofcourse itis clear, notthattheexperiments were wrong, butthattheywere done over only a limited range ofvelocities, sosmall thattherelativistic eflects would nothave been evident. Butnevertheless, wenowhave amuch more humble point ofview ofourphysical 1aws—-everything canbewrong! Secondly, ifwehave asetof“strange” ideas, such asthattime goes slower when onemoves, andsoforth, whether welikethem ordonotlikethem isan irrelevant question. Theonlyrelevant question iswhether theideas areconsistent with what isfound experimentally. Inother words, the“strange ideas” need only agree with experiment, andtheonly reason thatwehave todiscuss thebe- havior ofclocks andsoforth istodemonstrate thatalthough thenotion ofthe timedilation isstrange, itisconsistent with thewaywemeasure time. Finally, there isathird suggestion which isalittle more technical butwhich hasturned outtobeofenormous utility inourstudy ofother physical laws, and thatistolook atthesymmetry ofthelaws or,more specifically, tolook forthe ways inwhich thelawscanbetransformed andleave their form thesame. When wediscussed thetheory ofvectors, wenoted thatthefundamental laws ofmotion arenotchanged when werotate thecoordinate system, andnowwelearn thatthey arenotchanged when wechange thespace andtimevariables inaparticular way, given bytheLorentz transformation. Sothisidea ofstudying thepatterns or operations under which thefundamental laws arenotchanged hasproved tobea veryuseful one. 16-2 Thetwinparadox Tocontinue ourdiscussion oftheLorentz transformation andrelativistic effects, weconsider afamous so-called “paradox” ofPeter andPaul, who are supposed tobetwins, born atthesame time. When theyareoldenough todrive a space ship, Paul fliesaway atveryhigh speed. Because Peter, whoisleftonthe ground, seesPaul going sofast,allofPaul’s clocks appear togoslower, hisheart beats goslower, histhoughts goslower, everything goes slower, from Peter’s point ofview. Ofcourse, Paul notices nothing unusual, butifhetravels around andabout forawhile andthencomes back, hewillbeyounger than Peter, theman ontheground! That isactually right; itisoneoftheconsequences ofthetheory ofrelativity which hasbeen clearly demonstrated. Justasthemu-mesons last longer when theyaremoving, soalsowillPaullastlonger when heismoving. This iscalled a“paradox” onlybythepeople whobelieve thattheprinciple ofrelativity means thatallmotion isrelative; theysay,“Heh, heh,heh,from thepoint ofview ofPaul, can’t wesaythatPeter wasmoving andshould therefore appear toage more slowly? Bysymmetry, theonly possible result isthatboth should bethe same agewhen theymeet.” Butinorder forthem tocome back together andmake thecomparison, Paulmust either stopattheendofthetripandmake acomparison ofclocks or,more simply, hehastocome back, andtheonewho comes back must betheman who wasmoving, andheknows this, because hehadtoturn around. When heturned around, allkinds ofunusual things happened inhis space ship—-the rockets went off,things jammed upagainst onewall, andsoon— while Peter feltnothing. Sothewaytostate theruleistosaythatthemanwhohasfelttheaccelerations, whohasseenthings fallagainst thewalls, andsoon,istheonewhowould bethe younger; thatisthedifference between them inan“absolute” sense, anditis certainly correct. When wediscussed thefactthatmoving mu-mesons livelonger, weused asanexample their straight-line motion intheatmosphere. Butwecan alsomake mu-mesons inalaboratory andcause them togoinacurve with a magnet, andeven under thisaccelerated motion, theylastexactly asmuch longer astheydowhen theyaremoving inastraight line. Although noonehasarranged anexperiment explicitly sothatwecangetridoftheparadox, onecould compare amu-meson which isleftstanding with onethathadgone around acomplete circle, anditwould surely befound thattheonethatwent around thecircle lasted longer. Although wehave notactually carried outanexperiment using acomplete 16-3 circle, itisreally notnecessary, ofcourse, because everything fitstogether allright. Thismaynotsatisfy those whoinsist thatevery single factbedemonstrated directly, butweconfidently predict theresult oftheexperiment inwhich Paul goes ina complete circle. 16-3 Transformation ofvelocities Themain diflerence between therelativity ofEinstein andtherelativity of Newton isthatthelaws oftransformation connecting thecoordinates andtimes between relatively moving systems aredifferent. Thecorrect transformation law, thatofLorentz, is \/l —u?/c2 y,=ya Z’=z, (16.1) I,=t—ux/c2 _ \/1—u2/c2 These equations correspond totherelatively simple case inwhich therelative motion ofthetwoobservers isalong their common x-axes. Ofcourse other direc- tions ofmotion arepossible, butthemost general Lorentz transformation is rather complicated, withallfourquantities mixed uptogether. Weshall continue tousethissimpler form, since itcontains alltheessential features ofrelativity. Letusnow discuss more oftheconsequences ofthistransformation. First, itisinteresting tosolve these equations inreverse. That is,here isasetoflinear equations, fourequations with fourunknowns, andtheycanbesolved inreverse, forx,y,z,tinterms ofx’,y’,z’,t’.Theresult isvery interesting, since ittellsus howasystem ofcoordinates “atrest” looks from thepoint ofview ofonethatis “moving.” Ofcourse, since themotions arerelative andofuniform velocity, the manwhois“moving” cansay,ifhewishes, thatitisreally theother fellow whois moving andhehimself who isatrest. And since heismoving intheopposite direction, heshould getthesame transformation, butwith theopposite signof velocity. That isprecisely what wefindbymanipulation, sothatisconsistent. Ifitdidnotcome outthatway, wewould have realcause toworry!, x—utX s x’ ut'x=__L_,\/1—u2/c2 y=y’, Z=Z’, t_ t’+ux’/02. _\/l —u2/c2 Next wediscuss theinteresting problem oftheaddition ofvelocities inrela- tivity. Werecall thatoneoftheoriginal puzzles wasthatlight travels at186,000 mi/secinallsystems, even when theyareinrelative motion. Thisisaspecial case ofthemore general problem exemplified bythefollowing. Suppose thatanobject inside aspace shipisgoing at100,000 mi/sec andthespace shipitself isgoing at 100,000 mi/sec; howfastistheobject inside thespace shipmoving from thepoint ofview ofanobserver outside? Wemight want tosay200,000 mi/sec, which is faster than thespeed oflight. This isveryunnerving, because itisnotsupposed tobegoing faster than thespeed oflight! Thegeneral problem isasfollows. Letussuppose thattheobject inside theship, from thepoint ofview ofthe maninside, ismoving with velocity v,andthatthespace shipitself hasavelocity uwith respect totheground. Wewant toknow with what velocity 1),,thisobject ismoving from thepoint ofview ofthemanontheground. Thisis,ofcourse, still butaspecial caseinwhich themotion isinthex-direction. There willalso bea 16-4 transformation forvelocities inthey-direction, orforanyangle; these canbe worked outasneeded. Inside thespace shipthevelocity isv,/,which means that thedisplacement xisequal tothevelocity times thetime: x’=vat’. (16.3) Now wehave only tocalculate what theposition andtime arefrom thepoint of view oftheoutside observer foranobject which hastherelation (16.2) between x’andt’.Sowesimply substitute (16.3) into(16.2), andobtain aft’ + ll x= (16.4) Butherewefindxexpressed interms oft’.Inorder togetthevelocity asseen by theman ontheoutside, wemust divide hisdistance byhistime, notbytheother man’s time! Sowemust alsocalculate thetimeasseenfrom theoutside, which is 1= (16.5) Now wemust findtheratio ofxtot,which is 11,== (16.6) thesquare roots having cancelled. This isthelawthatweseek: theresultant ve- locity, the“summing” oftwovelocities, isnotjustthealgebraic sumoftwoveloc- ities (weknow that itcannot beorwegetintrouble), butis“corrected” by 1+uv/02. Now letusseewhat happens. Suppose thatyouaremoving inside thespace shipathalfthespeed oflight, andthatthespace shipitself isgoing athalfthespeed oflight. Thus uisécand11isea,butinthedenominator uvisone-fourth, sothat — 4C1): Z-5-»NF >—-ts+1-1->Pr-Mr-(‘I So,inrelativity, “half” and“half” does notmake “one,” itmakes only “4/5.” Ofcourse lowvelocities canbeadded quite easily inthefamiliar way, because so long asthevelocities aresmall compared with thespeed oflight wecanforget about the(1+uv/c2) factor; butthings arequite different andquite interesting athigh velocity. Letustakealimiting case. Justforfun,suppose thatinside thespace shipthe man wasobserving light itsef Inother words, v=c,andyetthespace shipis moving. How willitlook totheman ontheground? Theanswer willbe _ u+c _u+c_ U~1+uc/¢‘2_cu+C_c' Therefore, ifsomething ismoving atthespeed oflight inside theship, itwillappear tobemoving atthespeed oflight from thepoint ofview oftheman ontheground too! This isgood, foritis,infact, what theEinstein theory ofrelativity was designed todointhefirstplace-so ithadbetter work! Ofcourse, there arecases inwhich themotion isnotinthedirection ofthe uniform translation. Forexample, there may beanobject inside theshipwhich isjustmoving “upward” with thevelocity 1),,’withrespect totheship, andtheship ismoving “horizontally.” Now, wesimply gothrough thesame thing, only using y’sinstead ofx’s,with theresult y= y, Z U1/1,, sothatif11,1=0, 6,,=§=6,,\/1-u2/c2. (16.7) 16-5 \LlGHT \ PARTICLE IIC\\\ //) ‘X I*~ 1->/I / Fig 16-1. Trajectories described by olight rciy and pcirticle inside cimoving clock. Z 29 2 2 2, a/2 e/2 \a 0/2 aX/2' 161 1 1lb) Fig. 16-2. Two views ofonelastic collision between equal objects moving atthesome speed inopposite directions.Thus asidewise velocity isnolonger vy»,but0,/\/l -u2/c2. Wefound thisresult bysubstituting andcombining thetransformation equations, butwecanalsosee theresult directly from theprinciple ofrelativity forthefollowing reason (itis always good tolook again toseewhether wecanseethereason). Wehave already (Fig. 15-3) discussed how apossible clock might work when itismoving; the light appears totravel atanangle atthespeed cinthefixed system, while itsimply goes vertically with thesame speed inthemoving system. Wefound thatthe vertical component ofthevelocity inthefixed system islessthanthatoflight bythe factor \/1—u?/c2 (seeEq.15-3). Butnow suppose that weletamaterial particle goback andforth inthissame “clock,” butatsome integral fraction l/nofthespeed oflight (Fig. 16-1). Then when theparticle hasgone back and forth once, thelight willhave gone exactly ntimes. That is,each “click” ofthe “particle” clock willcoincide with each nth“click” ofthelight clock. Thisfact must stillbetruewhen thewhole system ISmoving, because thephysical phenomenon ofcoincidence willbeacoincidence inanyframe. Therefore, since thespeed cyis lessthan thespeed oflight, thespeed vyoftheparticle must beslower than the corresponding speed bythesame square-root ratio! That iswhythesquare root appears inanyvertical velocity. 16-4 Relativistic mass Welearned inthelastchapter that themass ofanobject increases with velocity, butnodemonstration ofthiswasgiven, inthesense thatwemade no arguments analogous tothose about thewayclocks have tobehave. However, wecanshow that,asaconsequence ofrelativity plusafewother reasonable assump- tions, themass must varyinthisway. (Wehave tosay“afewother assumptions” because wecannot prove anything unless wehave some laws which weassume to betrue, ifweexpect tomake meaningful deductions.) Toavoid theneed tostudy thetransformation lawsofforce, weshall analyze acollision, where weneed know nothing about thelaws offorce, except thatweshall assume theconservation of momentum andenergy. Also, weshall assume thatthemomentum ofaparticle which ismoving isavector andisalways directed inthedirection ofthevelocity. However, weshall notassume thatthemomentum isaconstant times thevelocity, asNewton did,butonly thatitissome function ofvelocity. Wethus write the momentum vector asacertain coefficient times thevector velocity: p=m,.v. (16.8) Weputasubscript vonthecoefficient toremind usthatitisafunction ofvelocity, andweshall agree tocallthiscoefficient m,the“mass.” Ofcourse, when the velocity issmall, itisthesame mass thatwewould measure intheslow-moving experiments thatweareused to.Now weshall trytodemonstrate thattheformula form,,must bemo/\/1 —112/c2, byarguing from theprinciple ofrelativity that thelaws ofphysics must bethesame inevery coordinate system. Suppose thatwehave twoparticles, liketwoprotons, that areabsolutely equal, andtheyaremoving toward each other withexactly equal velocities. Their total momentum iszero. Now what canhappen? After thecollision, their direc- tions ofmotion must beexactly opposite toeach other, because ifthey arenot exactly opposite, there willbeanonzero total vector momentum, andmomentum would nothave been conserved. Also theymust have thesame speeds, since they areexactly similar objects; infact,theymust have thesame speed theystarted with, since wesuppose thattheenergy isconserved inthese collisions. Sothediagram of anelastic collision, areversible collision, willlook likeFig.16-2(a): allthearrows arethesame length, allthespeeds areequal. Weshall suppose thatsuchcollisions canalways bearranged, thatanyangle 19canoccur, andthatanyspeed could be used insuch acollision. Next, wenotice thatthissame collision canbeviewed differently byturning theaxes, andjust forconvenience weshall turn theaxes, sothat thehorizontal splits itevenly, asinFig. l6—2(b). Itisthesame collision redrawn, only with theaxesturned. 16-6 Now hereistherealtrick: letuslook atthiscollision from thepoint ofview ofsomeone riding along inacarthatismoving withaspeed equal tothehorizontal component ofthevelocity ofoneparticle. Then how does thecollision look? Itlooks asthough particle 1isjustgoing straight up,because ithaslostitshori- zontal component, anditcomes straight down again, alsobecause itdoes nothave thatcomponent. That is,thecollision appears asshown inFig.16-3(a).Particle 2, however, wasgoing theother way, andasweridepastitappears toflybyatsome terrific speed andatasmaller angle, butwecanappreciate thattheangles before andafter thecollision arethesame. Letusdenote byuthehorizontal component ofthevelocity ofparticle 2,andbywthevertical velocity ofparticle 1. Now thequestion is,what isthevertical velocity utanoi?Ifweknew that, we could getthecorrect expression forthemomentum, using thelawofconservation ofmomentum inthevertical direction. Clearly, thehorizontal component ofthe momentum isconserved: itisthesame before andafter thecollision forboth particles, andiszero forparticle 1.Soweneed usetheconservation lawonly for theupward velocity utan(1.Butwecangettheupward velocity, simply bylooking atthesame collision going theother way! Ifwelook atthecollision ofFig. 16-3(a)from acartotheleftmoving withspeed u,weseethesame collision, except “turned over,” asshown inFig.16-3(b). Now particle 2istheonethatgoesup anddown with speed w,andparticle 1haspicked upthehorizontal speed u.Of course, nowweknow what thevelocity utanozis:itisw\/1 —u?/c2 (seeEq.16.7). Weknow thatthechange inthevertical momentum ofthevertically moving par- ticleis Ap=2m,,,w (2,because itmoves upandback down). Theobliquely moving particle hasa certain velocity vwhose components wehave found tobeuandw\/1—u2/c2, andwhose mass ism,.Thechange invertical momentum ofthisparticle istherefore Ap’=2m,,w\/l -uz/c2 because, inaccordance with ourassumed law(16.8), themomentum component isalways themass corresponding tothemagnitude of thevelocity times thecomponent ofthevelocity inthedirection ofinterest. Thus inorder forthetotal momentum tobezerothevertical momenta must cancel and theratio ofthemass moving with speed vandthemass moving with speed w must therefore be %=\/1-U2/C2. (16.9)Inn Letustakethelimiting casethatwisinfinitesimal. Ifwisverytinyindeed, it isclear that vanduarepractically equal. Inthiscase, mu,—>moandm,,-—>mu. Thegrand result is m,= (16.10) Itisaninteresting exercise nowtocheck whether ornotEq.(16.9) isindeed truefor arbitrary values ofw,assuming thatEq.(16.10) istheright formula forthemass. Note thatthevelocity vneeded inEq.(16.9) canbecalculated from theright-angle triangle: 112=112+w2(l —u2/c2). Itwillbefound tocheck outautomatically, although weused itonlyinthelimit ofsmall w. Now, letusaccept that momentum isconserved andthat themass depends upon thevelocity according to(16.10) andgoontofindwhat elsewecanconclude. Letusconsider what iscommonly called aninelastic collision. For simplicity, weshall suppose thattwoobjects ofthesame kind, moving oppositely withequal speeds w,hiteachother andstick together, tobecome some new, stationary object, asshown inFig.16-4(a). Themass mofeach corresponds tow,which, aswe know, ismo/\/1 —w2/c2. Ifweassume theconservation ofmomentum and theprinciple ofrelativity, wecandemonstrate aninteresting factabout themass ofthenewobject which hasbeen formed. Weimagine aninfinitesimal velocity 16-7Z1 T1,. ZZ 2 2 v v W ° ° X’ u u x"u u , W Iy y I W 1°) '' lb) Fig. 16-3. Two more views ofthe collision, from moving cars. _. _. 37+" 5’* -—->-'"’ -<—-—3""" azroas lip %m, m, 0 AFTER (O) M M (b) Fig. 16-4. Two views ofoninelastic collision between equally massive objects. uatright angles tow(wecandothesame withfinite values ofu,butitiseasier to understand with aninfinitesimal velocity), then look atthissame collision aswe ridebyinanelevator atthevelocity —u. What weseeisshown inFig.l6—4(b). Thecomposite object hasanunknown mass M.Now object 1moves with an upward component ofvelocity uandahorizontal component which ispractically equal tow,andsoalsodoes object 2.After thecollision wehave themass M moving upward withvelocity u,considered verysmall compared withthespeed of light, andalsosmall compared with w.Momentum must beconserved, soletus estimate themomentum intheupward direction before andafter thecollision. Before thecollision wehave p~2m,,,u, andafter thecollision themomentum is evidently p’=M,,u, butMuisessentially thesame asM0because uissosmall. These momenta must beequal because oftheconservation ofmomentum, andthere- fore M0=2m,,,. (16.11) Themass oftheobject which isformed when twoequal objects collide must betwice themass oftheobjects which come together. You might say, “Yes, ofcourse, thatistheconservation ofmass.” Butnot“Yes, ofcourse,” soeasily, because these masses have been enhanced over themasses thattheywould beiftheywere standing still,yettheystillcontribute, tothetotal M,notthemass theyhave when standing still, butmore. Astonishing asthatmay seem, inorder fortheconser- vation ofmomentum towork when twoobjects come together, themass that they form must begreater than therestmasses oftheobjects, even though the objects areatrestafter thecollision! 16-5 Relativistic energy Inthelastchapter wedemonstrated thatasaresult ofthedependence ofthe mass onvelocity andNewton’s laws, thechanges inthekinetic energy ofanobject resulting from thetotal work done bytheforces onitalways comes outtobe AT=(mu-m0)c2=_’;2:__ -mocz. (16.12)\/1—u?/c2 Weeven went further, andguessed thatthetotal energy isthetotal mass times 02. Now wecontinue thisdiscussion. Suppose thatourtwoequally massive objects thatcollide canstillbe“seen” inside M.Forinstance, aproton andaneutron are“stuck together,” butarestill moving about inside ofM.Then, although wemight atfirstexpect themass M tobe2m0, wehave found thatitisnotZmo, but2m,,,. Since 2m,,,iswhat isputin, but2m0aretherestmasses ofthethings inside, theexcess mass ofthecomposite object isequal tothekinetic energy brought in.This means, ofcourse, that energy hasinertia. Inthelastchapter wediscussed theheating ofagas,andshowed thatbecause thegasmolecules aremoving andmoving things areheavier, when weputenergy intothegasitsmolecules move faster andsothegasgetsheavier. Butinfacttheargument iscompletely general, andourdiscussion oftheinelastic collision shows thatthemass isthere whether ornotitiskinetic energy. Inother words, iftwoparticles come together andproduce potential oranyother form of energy; ifthepieces areslowed down byclimbing hills, doing work against internal forces, orwhatever; then itisstilltruethatthemass isthetotal energy thathas been putin.Soweseethattheconservation ofmass which wehave deduced above isequivalent totheconservation ofenergy, andtherefore there isnoplace inthe theory ofrelativity forstrictly inelastic collisions, asthere wasinNewtonian mechanics. According toNewtonian mechanics itisallright fortwothings to collide andsoform anobject ofmass 2m0which isinnowaydistinct from theone thatwould result from putting them together slowly. Ofcourse weknow from the lawofconservation ofenergy thatthere ismore kinetic energy inside, butthat does notaffect themass, according toNewton’s laws. Butnowweseethatthisis impossible; because ofthekinetic energy involved inthecollision, theresulting 16-8 object willbeheavier; therefore, itwillbeadzflerent object. When weput theobjects together gently they make something whose mass is2m0; when weputthem together forcefully, they make something whose mass isgreater. When themass isdifferent, wecantellthatitisdifferent. So,necessarily, the conservation ofenergy must goalong with theconservation ofmomentum inthe theory ofrelativity. This hasinteresting consequences. Forexample, suppose thatwehave an object whose mass Mismeasured, andsuppose something happens sothatitflies intotwoequal pieces moving with speed w,sothatthey each have amass mw. Now suppose that these pieces encounter enough material toslow them up until they stop; then they willhave mass mo. How much energy will they have given tothematerial when they have stopped? Each willgiveanamount (m,,,—mO)c2, bythetheorem thatweproved before. This much energy isleft inthematerial insome form, asheat, potential energy, orwhatever. Now 2m,,, = M,sotheliberated energy isE=(M—2m0)C2. This equation wasused to estimate how much energy would beliberated under fission intheatomic bomb, forexample. (Although thefragments arenotexactly equal, theyarenearly equal.) Themass oftheuranium atom wasknown—it hadbeen measured ahead oftime—- andtheatoms intowhich itsplit, iodine, xenon, andsoon,allwere ofknown mass. Bymasses, wedonotmean themasses while theatoms aremoving, wemean the masses when theatoms areatrest. Inother words, both Mandmoareknown. Sobysubtracting thetwonumbers onecancalculate howmuch energy willbe released ifMcanbemade tosplit in“half.” Forthisreason poor oldEinstein wascalled the“father” oftheatomic bomb inallthenewspapers. Ofcourse, allthatmeant wasthathecould tellusahead oftime howmuch energy would be released ifwetold himwhat process would occur. Theenergy thatshould be liberated when anatom ofuranium undergoes fission wasestimated about six months before thefirstdirect test,andassoon astheenergy wasinfactliberated, someone measured itdirectly (andifEinstein’s formula hadnotworked, theywould have measured itanyway), andthemoment theymeasured ittheynolonger needed theformula. Ofcourse, weshould notbelittle Einstein, butrather should criticize thenewspapers andmany popular descriptions ofwhat causes what inthehistory ofphysics andtechnology. Theproblem ofhow togetthething tooccur inan effective andrapid manner isacompletely different matter. Theresult isjustassignificant inchemistry. Forinstance, ifwewere toweigh thecarbon dioxide molecule andcompare itsmass with thatofthecarbon and theoxygen, wecould findouthowmuch energy would beliberated when carbon andoxygen form carbon dioxide. Theonly trouble hereisthatthedifferences in masses aresosmall thatitistechnically verydiflicult todo. Now letusturntothequestion ofwhether weshould addm0c2 tothekinetic energy andsayfrom now onthatthetotal energy ofanobject ismc2. First, if wecanstillseethecomponent pieces ofrestmass moinside M,then wecould saythatsome ofthemass Mofthecompound object isthemechanical restmass oftheparts, part ofitiskinetic energy oftheparts, andpart ofitispotential energy oftheparts. Butwehave discovered, innature, particles ofvarious kinds which undergo reactions justliketheonewehave treated above, inwhich with all thestudy intheworld, wecannot seetheparts inside. Forinstance, when aK-meson disintegrates intotwopions itdoes soaccording tothelaw(16.11), buttheidea thataKismade outof21r’sisauseless idea, because italsodisintegrates into31r’s! Therefore wehave anewidea: wedonothave toknow what things aremade ofinside; wecannot andneed notidentify, inside aparticle, which oftheenergy isrestenergy oftheparts intowhich itisgoing todisintegrate. Itisnotconvenient andoften notpossible toseparate thetotal mc2energy ofanobject intorestenergy oftheinside pieces, kinetic energy ofthepieces, andpotential energy ofthepieces; instead, wesimply speak ofthetotal energy oftheparticle. We“shift theorigin” ofenergy byadding aconstant mocz toeverything, andsaythatthetotal energy ofa particle isthemass inmotion times c2,andwhen theobject isstanding still, the energy isthemass atresttimes c2. 16-9 Finally, wefindthatthevelocity v,momentum P,andtotal energy Earere- lated inarather simple way. That themass inmotion atspeed visthemass mo atrestdivided by\/I—v2/c2, surprisingly enough, israrely used. Instead, the following relations areeasily proved, andturnouttobeveryuseful: d E2-—P202 =m§c4 (l6.l3) an Pc=Ev/c. (16.14) 16-l0 I7 Spuce- Time 17-1 Thegeometry ofspace-time Thetheory ofrelativity shows usthattherelationships ofpositions andtimes asmeasured inonecoordinate system andanother arenotwhat wewould have expected onthebasis ofourintuitive ideas. Itisveryimportant thatwethoroughly understand therelations ofspace andtimeimplied bytheLorentz transformation, andtherefore weshall consider thismatter more deeply inthischapter. TheLorentz transformation between thepositions andtimes (x,y,z,t)as measured byanobserver “standing still,” andthecorresponding coordinates and time (x’,y’,z’,t’)measured inside a“moving” space ship, moving with velocity uare x—utx’=-\/l—u2/c2, y’=11, z’=z,(17.1) t—-ux/c2t’= n \/1— u2/c2 Letuscompare these equations with Eq.(11.5), which alsorelates measurements intwosystems, oneofwhich inthisinstance isrotated relative totheother: x’=xcosli +ysin6, y’=ycos 0—xsin6, (17.2) z’=z. Inthisparticular case, Moe andJoearemeasuring with axes having anangle 6 between thex’-andx-axes. Ineach case, wenote thatthe“primed” quantities are “mixtures” ofthe“unprimed” ones: thenewx’isamixture ofxandy,andthe newy’isalsoamixture ofxandy. Ananalogy isuseful: When welook atanobject, there isanobvious thing we might callthe“apparent width,” andanother wemight callthe“depth.” Butthe twoideas, width anddepth, arenotfundamental properties oftheobject, because ifwestepaside andlook atthesame thing from adifferent angle, wegetadifferent width andadifferent depth, andwemaydevelop some formulas forcomputing the newones from theoldones andtheangles involved. Equations (17.2) arethese formulas. Onemight saythatagiven depth isakind of“mixture” ofalldepth andallwidth. Ifitwere impossible ever tomove, andwealways sawagiven object from thesame position, then thiswhole business would beirrelevant—we would always seethe“true” width andthe“true” depth, andtheywould appear to have quite different qualities, because oneappears asasubtended optical angle andtheother involves some focusing oftheeyesoreven intuition; theywould seem tobeverydifferent things andwould never getmixed up.Itisbecause wecanwalk around thatwerealize thatdepth andwidth are,somehow orother, justtwodiffer- entaspects ofthesame thing. CanwenotlookattheLorentz transformations inthesame way?Here alsowe have amixture—of positions andthetime. Adifference between aspace measure- ment andatimemeasurement produces anewspace measurement. Inother words, inthespace measurements ofoneman there ismixed inalittle bitofthetime, as seenbytheother. Ouranalogy permits ustogenerate thisidea: The“reality” of 17-117-1 Thegeometry ofspace-time 17-2 Space-time intervals 17-3 Past, present, andfuture 17-4 More about four-vectors 17-5 Four-vector algebra CtSLOW (0) (bl 1 (Cl FAST x,, x Fig. l7—l. Three particle paths in space-time: la)aparticle atrest at x=x0;(blaparticle which starts at x=xoand moves with constant speed; (c)aparticle which starts athigh speed butslows down. ¢1' ct ct / ct’ / XI X’ ct’ xi >" >x(0)NOTconnect (blCORRECT Fig. l7-2. Two views ofadisinte- grating particle.anobject thatwearelooking atissomehow greater (speaking crudely andintui- tively) than its“width” andits“depth” because theydepend upon howwelook atit;when wemove toanewposition, ourbrain immediately recalculates the width andthedepth. Butourbrain does notimmediately recalculate coordinates andtimewhen wemove athigh speed, because wehave hadnoeffective experience ofgoing nearly asfastaslight toappreciate thefactthattime andspace arealso ofthesame nature. Itisasthough wewere always stuck intheposition ofhaving tolook atjustthewidth ofsomething, notbeing abletomove ourheads appreci- ablyonewayortheother; ifwecould, weunderstand now, wewould seesome of theother man’s time—we would see“behind,” sotospeak, alittle bit. Thus weshall trytothink ofobjects inanewkind ofworld, ofspace andtime mixed together, inthesame sense thattheobjects inourordinary space-world arereal, andcanbelooked atfrom different directions. Weshall then consider thatobjects occupying space andlasting foracertain length oftime occupy akind ofa“blob” inanewkind ofworld, andthatwelook atthis“blob” from different points ofview when wearemoving atdifferent velocities. This newworld, this geometrical entity inwhich the“blobs” exist byoccupying position andtaking upa certain amount oftime, iscalled space-time. Agiven point (x,y,z,t)inspace-time iscalled anevent. Imagine, forexample, thatWeplotthex-positions horizontally, yandzintwoother directions, both mutually at“right angles” andat“right angles” tothepaper (l),andtime, vertically. Now, how does amoving particle, say,look onsuch adiagram? Iftheparticle isstanding still, then ithasacertain x,andastime goes on,ithasthesame x,thesame x,thesame x;soits“path” is alinethatruns parallel tothet-axis (Fig. 17-1 a).Ontheother hand, ifitdrifts outward, then asthetime goes onxincreases (Fig. 17-1 b).Soaparticle, forex- ample, which starts todriftoutandthen slows upshould have amotion something likethatshown inFig. 17-1(c).Apatricle, inother words, which ispermanent anddoes notdisintegrate isrepresented byalineinspace-time. Aparticle which disintegrates would berepresented byaforked line, because itwould turn into twoother things which would start from thatpoint. What about light? Light travels atthespeed c,andthatwould berepresented byalinehaving acertain fixed slope (Fig. 17-1 d). Now according toournewidea, ifagiven event occurs toaparticle, sayifit suddenly disintegrates atacertain space-time point intotwonewones which follow some newtracks, andthisinteresting event occurred atacertain value ofxanda certain value oft,thenwewould expect that, ifthismakes anysense, wejusthave totakeanewpairofaxesandturnthem, andthatwillgiveusthenewtandthe newxinournewsystem, asshown inFig.17-2(a). Butthisiswrong, because Eq.(17.1) isnotexactly thesame mathematical transformation asEq.(17.2). Note, forexample, thedifference insignbetween thetwo, andthefactthatoneis written interms ofcos0andsin0,while theother iswritten with algebraic quanti- ties. (Ofcourse, itisnotimpossible thatthealgebraic quantities could bewritten as cosine andsine, butactually they cannot.) Butstill,thetwoexpressions arevery similar. Asweshall see,itisnotreally possible tothink ofspace-time asareal, ordinary geometry because ofthatdifference insign. Infact, although weshall notemphasize thispoint, itturns outthatamanwhoismoving hastouseasetof axeswhich areinclined equally tothelight ray,using aspecial kind ofprojection parallel tothex’-andt’-axes, forhisx’andt’,asshown inFig.l7—2(b). Weshall notdealwith thegeometry, since itdoes nothelpmuch; itiseasier towork with theequations. 17-2 Space-time intervals Although thegeometry ofspace-time isnotEuclidean intheordinary sense, there isageometry which isvery similar, butpeculiar incertain respects. Ifthis ideaofgeometry isright, there ought tobesome functions ofcoordinates andtime which areindependent ofthecoordinate system. Forexample, under ordinary rotations, ifwetaketwopoints, oneattheorigin, forsimplicity, andtheother one somewhere else, both systems would have thesame origin, andthedistance from 17-2 heretotheother point isthesame inboth. That isoneproperty thatisinde- pendent oftheparticular way ofmeasuring it.Thesquare ofthedistance is x2-1-y2-1-22.Now what about space-time? Itisnothard t_odemonstrate that wehave here, also, something which stays thesame, namely, thecombination c2t2 —x2—y2—z2isthesame before andafter thetransformation: c2tr2 _x/2 _y!2 _212 =c2t2 _X2 _y2 _Z2. Thisquantity istherefore something which, likethedistance, is“real” insome sense; itiscalled theinterval between thetwospace-time points, oneofwhich is, inthiscase, attheorigin. (Actually, ofcourse, itistheinterval squared, justas x2+yz+22isthedistance squared.) Wegiveitadifferent name because itis inadifferent geometry, buttheinteresting thing isonlythatsome signsarereversed andthere isacinit. Letusgetridofthec;thatisanabsurdity ifwearegoing tohave awonderful space withx’sandy’sthatcanbeinterchanged. Oneoftheconfusions thatcould becaused bysomeone withnoexperience would betomeasure widths,.say, bythe angle subtended attheeye,andmeasure depth inadifferent way, likethestrain on themuscles needed tofocus them, sothatthedepths would bemeasured infeet andthewidths inmeters. Then onewould getanenormously complicated messof equations inmaking transformations such as(17.2), andwould notbeabletosee theclarity andsimplicity ofthething foraverysimple technical reason, thatthe same thing isbeing measured intwodifferent units. NowinEqs.(17.1) and(17.3) nature istelling usthattime andspace areequivalent; time becomes space; they should bemeasured inthesame units. What distance isa“second”? Itiseasyto figure outfrom (17.3) what itis.Itis3X108meters, thedistance thatlight would goinonesecond. Inother words, ifweweretomeasure alldistances andtimes inthesame units, seconds, then ourunitofdistance would be3X108meters, andtheequations would besimpler. Oranother waythatwecould make theunits equal istomeasure timeinmeters. What isameter oftime? Ameter oftime isthetime ittakes forlight togoonemeter, andistherefore 1/3X10's sec,or 3.3billionths ofasecond! Wewould like,inother words, toputallourequations inasystem ofunits inwhich c=1.Iftime andspace aremeasured inthesame units, assuggested, then theequations areobviously much simplified. They are x—utx’=i——,\/l—u2 y’=y,Z,=Z, (17.4) t,= t—ux_ \/1—u2 t/2 __x/2 ___y;2 __Z/2 =t2_x2 _y2 _Z2. Ifweareeverunsure or“frightened” thatafterwehavethissystem withc=1 weshall never beabletogetourequations right again, theanswer isquite the opposite. Itismuch easier toremember them without thec’sinthem, anditis always easytoputthec’sback, bylooking after thedimensions. Forinstance, in \/1—u2,weknow thatwecannot subtract avelocity squared, which hasunits, fromthepurenumber 1,soweknow thatwemustdivide u2byc2inorder tomake thatunitless, andthatisthewayitgoes. Thedifference between space-time andordinary space, andthecharacter of aninterval asrelated tothedistance, isvery interesting. According toformula (17.5), ifweconsider apoint which inagiven coordinate system hadzerotime, and only space, then theinterval squared would benegative andwewould have an imaginary interval, thesquare rootofanegative number. Intervals canbeeither realorimaginary inthetheory. Thesquare ofaninterval maybeeither positive ornegative, unlike distance, which hasapositive square. When aninterval is imaginary, wesaythatthetwopoints have aspace-like interval between them 17-3 1FUTURE® LIGHT —CONE O / X / R0 / /6p® ucm-cons PAST Fig. 17-3. The space-time region surrounding apoint attheorigin.(instead ofimaginary), because theinterval ismore likespace than liketime. Ontheother hand, iftwoobjects areatthesame place inagiven coordinate system, butdiffer onlyintime, thenthesquare ofthetimeispositive andthedistances are zeroandtheinterval squared ispositive; thisiscalled atime-like interval. Inour diagram ofspace-time, therefore, wewould have arepresentation something like this: at45°there aretwolines (actually, infourdimensions these willbe“cones,” called light cones) andpoints onthese lines areallatzerointerval from theorigin. Where light goes from agiven point isalways separated from itbyazerointerval, asweseefrom Eq.(17.5). Incidentally, wehave justproved thatiflight travels with speed cinonesystem, ittravels with speed cinanother, foriftheinterval is thesame inboth systems, i.e.,zeroinoneandzerointheother, then tostate that thepropagation speed oflight isinvariant isthesame assaying thattheinterval 1Szero. 17-3 Past, present, andfuture Thespace-time region surrounding agiven space-time point canbeseparated intothree regions, asshown inFig.17-3. Inoneregion wehave space-like inter- vals, andintworegions, time-like intervals. Physically, these three regions into which space-time around agiven point isdivided have aninteresting physical relationship tothatpoint: aphysical pbject orasignal cangetfrom apoint in region 2totheevent 0bymoving along ataspeed lessthan thespeed oflight. Therefore events inthisregion canaffect thepoint O,canhave aninfluence onit from thepast. Infact, ofcourse, anobject atPonthenegative t-axis isprecisely inthe“past” withrespect to0;itisthesame space-point as0,onlyearlier. What happened there then, affects Onow. (Unfortunately, thatisthewaylifeis.)An- other object atQcangetto0bymoving withacertain speed lessthan c,soifthis object were inaspace shipandmoving, itwould be,again, thepastofthesame space-point. That is,inanother coordinate system, theaxisoftime might go through both 0andQ.Soallpoints ofregion 2areinthe“past” of0,andany- thing thathappens inthisregion canaffect O.Therefore region 2issometimes called theaffective past, oraffecting past; itisthelocus ofallevents which can affect point 0inanyway. Region 3,ontheother hand, isaregion which wecanaffect from 0,wecan “hit” things byshooting “bullets” outatspeeds lessthan c.Sothisistheworld whose future canbeaffected byus,andwemaycallthattheafiective future. Now theinteresting thing about alltherestofspace-time, i.e.,region 1,isthatwecan neither affect itnowfrom 0,norcanitaffect usnowat0,because nothing cango faster than thespeed oflight. Ofcourse, what happens atRcanaffect uslater; thatis,ifthesunisexploding “right now,” ittakes eight minutes before weknow about it,anditcannot possibly affect usbefore then. What wemean by“right now” isamysterious thing which wecannot define andwecannot affect, butitcanaffect uslater, orwecould have affected itifwe haddone something farenough inthepast. When welook atthestarAlpha Centauri, weseeitasitwasfouryears ago;wemight wonder what itislike“now.” “Now” means atthesame time from ourspecial coordinate system. Wecanonly seeAlpha Centauri bythelight thathascome from ourpast, uptofouryears ago, butwedonotknow what itisdoing “now”; itwilltakefouryears before what it isdoing “now” canaffect us.Alpha Centauri “now” isanideaorconcept ofour mind; itisnotsomething thatisreally definable physically atthemoment, because wehave towaittoobserve it;wecannot even define itright “now.” Furthermore, the“now” depends onthecoordinate system. If,forexample, Alpha Centauri were moving, anobserver there would notagree with usbecause hewould put hisaxes atanangle, andhis“now” would beadiflerent time. Wehave already talked about thefactthatsimultaneity isnotaunique thing. There arefortune tellers, orpeople whotellustheycanknow thefuture, and there aremany wonderful stories about theman who suddenly discovers thathe hasknowledge about theaffective future. Well, there arelotsofparadoxes pro- duced bythatbecause ifweknow something isgoing tohappen, thenwecanmake 17-4 surewewillavoid itfbydoing theright thing attheright time, andsoon.But actually there isnofortune teller whocaneven tellusthepresent! There isnoone who cantelluswhat isreally happening right now, atanyreasonable distance, because thatisunobseryable. Wemight askourselves thisquestion, which we leave tothestudent totrytoanswer: Would anyparadox beproduced ifitwere suddenly tobecome possible toknow things thatareinthespace-like intervals of region 1? 17-4 More about four-vectors Letusnow return toourconsideration oftheanalogy oftheLorentz trans- formation androtations ofthespace axes. Wehave learned theutility ofcollecting together other quantities which have thesame transformation properties asthe coordinates, toform what wecallvectors, directed lines. Inthecaseofordinary rotations, there aremany quantities thattransform thesame wayasx,y,andz under rotation: forexample, thevelocity hasthree components, anx,y,and z-component; when seenina‘different coordinate system, none ofthecomponents isthesame, instead theyarealltransformed tonewvalues. But,somehow orother, thevelocity “itself” hasagreater reality than doanyofitsparticular components, andwerepresent itbyadirected line. Wetherefore ask:lsitorisitnottruethatthere arequantities which transform, orwhich arerelated, inamoving system andinanonmoving system, inthesame wayasx,y,z,andt?From ourexperience with vectors, weknow thatthree of thequantities, likex,y,2,would constitute thethree components ofanordinary space-vector, butthefourth quantity would look likeanordinary scalar under space rotation, because itdoes notchange solong aswedonotgointoamoving coordinate system. lsitpossible, then, toassociate with some ofourknown “three-vectors” afourth object, thatwecould callthe“time component,” insuch a manner thatthefour objects together would “rotate” thesame wayasposition andtimeinspace-time? Weshall nowshow thatthere is,indeed, atleast onesuch thing (there aremany ofthem, infact): thethree components ofmomentum, andthe energy asthetime component, transform together tomake what wecalla“four- vector.” Indemonstrating this, since itisquite inconvenient tohave towrite c’s everywhere, weshall usethesame trick concerning units oftheenergy, themass, andthemomentum, thatweused inEq.(17.4). Energy andmass, forexample; differ onlybyafactor c2which ismerely aquestion ofunits, sowecansayenergy isthemass. Instead ofhaving towrite thecf,weputE=m,andthen, ofcourse, ifthere were anytrouble wewould putintheright amounts ofcsothattheunits would straighten outinthelastequation, butnotintheintermediate ones. Thus ourequations forenergy andmomentum are = = / __2 E m m0,\/l v, (17.6) p=mv=mov/\/1—— v2. Also inthese units, wehave E2-p2=mg. (17.7) Forexample, ifwemeasure energy inelectron volts, what does amass of1electron voltmean? Itmeans themass whose restenergy is1electron volt, thatis,mocz isoneelectron volt. Forexample, therestmass ofanelectron is0.511 X106ev. Now what would themomentum andenergy look likeinanewcoordinate system? Tofindout,weshall have totransform Eq.(17.6), which wecando because weknow howthevelocity transforms. Suppose that, aswemeasure it,an object hasavelocity v,butwelook upon thesame object from thepoint ofview ofaspace shipwhich itself ismoving withavelocity u,andinthatsystem weusea prime todesignate thecorresponding thing. Inorder tosimplify things atfirst, weshalltakethecasethatthevelocity visinthedirection ofu.(Later, wecandothe more general case.) What isv’,thevelocity asseenfrom thespace ship? Itisthe 17-5 composite velocity, the“difference” between vandu.Bythelawwhich weworked outbefore,1)'—ll 1)’—G’ Now letuscalculate thenewenergy E’,theenergy asthefellow inthespace ship would seeit.Hewould usethesame restmass, ofcourse, buthewould usev’for thevelocity. What Weh3)VC todoissquare v’,subtract itfrom one,takethesquare root, andtakethereciprocal: ,,12=£ir"’,1—2uv+u2v2 I_v,2=l—2uv+u2v2—v2+2uv—u2 1—2uv+u2v2 _l—v2—u2—l—u2v2 _ l— 2uv+u2v2 =<1-v”><1-u’>_(1—uv)2 Therefore 1 1—uv= - 17.9\/1-1/2 \/l—v2\/1-u2 () Theenergy E’isthensimply motimes theabove expression. Butwewant to express theenergy interms oftheunprimed energy andmomentum, andwenote that E,= mo—mouv =(mo/V1 —122)—(mov/\/1 —v2)u, \/M5 \/M112 \/Ti OI‘ E—up,E’=-i- 17.10\/ii? ‘) which werecognize asbeing exactly ofthesame form as t,=t—ux_ \/l—u2 Next wemust findthenewmomentum p;.This isjusttheenergy Etimes v’,and isalsosimply expressed interms ofEandp: p,=E,v,= m0(l —uv) _v—u= mov——mou _ ’ \/l—v2\/l—u2 (1—uv) \/l—v2\/l—u2 Thus I #1_ Ep,=5/-1-%-"75. (17.11) which werecognize asbeing ofprecisely thesame form as x—ut x/l—u2' Thus thetransformations forthenewenergy andmomentum interms of theoldenergy andmomentum areexactly thesame asthetransformations for t’interms oftandx,andx’interms ofxand1:allwehavetodois,every timewe seetin(17.4) substitute E,andevery time weseexsubstitute p,,,andthen the equations (17.4) willbecome thesame asEqs. (17.10) and(17.11). This would imply, ifeverything works right, anadditional rulethatp,’,=p,,andthatpi=p,. Toprove thiswould require ourgoing back andstudying thecase ofmotion up anddown. Actually, wedidstudy thecase ofmotion upanddown inthelast 17-6x’= chapter. Weanalyzed acomplicated collision andwenoticed that, infact, the transverse momentum isnotchanged when viewed from amoving system; sowe1 I . have already verified thatpy=pgandp;=pz.Thecomplete transformation, then, is P;=———P’—”E»\/1—u2 Pf=pt.pg:P2 (17.12) E,:E—up,,_ \/l—u2 Inthese transformations, therefore, wehave discovered fourquantities which transform likex,y,z,andt,andwhich wecallthefour-vector momentum. Since themomentum isafour-vector, itcanberepresented onaspace-time diagram ofa moving particle asan“arrow” tangent tothepath, asshown inFig.17-4. This arrow hasatimecomponent equal totheenergy, anditsspace components repre- sentitsthree-vector momentum; thisarrow ismore “real” thaneither theenergy or themomentum, because those justdepend onhowwelook atthediagram. 17-5 Four-vector algebra Thenotation forfour-vectors isdifferent than itisforthree-vectors. Inthe caseofthree-vectors, ifwewere totalkabout theordinary three-vector momentum wewould write itp.Ifwewanted tobemore specific, wecould sayithasthree components which are,fortheaxesinquestion, p,,,pg’andp,,orwecould simply refer toageneral component asp,-,andsaythaticould either bex,y,orz,andthat these arethethree components; thatis,imagine thatiisanyoneofthree directions, x,y,orz.Thenotation thatweuseforfour-vectors isanalogous tothis: wewrite p,,forthefour-vector, and/.1stands forthefour possible directions t,x,y,orz. Wecould, ofcourse, useanynotation wewant; donotlaugh atnotations; invent them, theyarepowerful. Infact,mathematics is,toalarge extent, invention ofbetter notations. Thewhole ideaofafour-vector, infact,isanimprovement in notation sothatthetransformations canberemembered easily. A,,,then, isa general four-vector, butforthespecial caseofmomentum, thep,isidentified as theenergy, p,isthemomentum inthex-direction, p,,isthatinthey-direction, and p,isthatinthez-direction. Toaddfour-vectors, weaddthecorresponding com- ponents. Ifthere isanequation among four-vectors, then theequation istrue for eachcomponent. Forinstance, ifthelawofconservation ofthree-vector momentum istobetrueinparticle collisions, i.e.,ifthesumofthemomenta foralarge number ofinteracting orcolliding particles istobeaconstant, thatmust mean thatthe sums ofallmomenta inthex-direction, inthey-direction, andinthez-direction, for alltheparticles, must each beconstant. This lawalone would beimpossible in relativity because itisincomplete; itisliketalking about onlytwoofthecomponents ofathree-vector. Itisincomplete because ifwerotate theaxes, wemixthevarious components, sowemust include allthree components inourlaw. Thus, inrela- tivity, wemust complete thelawofconservation ofmomentum byextending itto include thetime component. This isabsolutely necessary togowith theother three, orthere cannot berelativistic invariance. Theconservation ofenergy isthe fourth equation which goes with theconservation ofmomentum tomake avalid four-vector relationship inthegeometry ofspace andtime. Thus thelawofcon- servation ofenergy andmomentum infour-dimensional notation is Z1n= Emparticles particles (11 In Ollll or,inaslightly different notation Z1”"=Pit" (17.14)7 1 17-71 P»- X Fig. 17-4. Thefour-vector momentum ofaparticle. where i=1,2,. ..refers totheparticles going intothecollision, j=1,2,... refers totheparticles coming outofthecollision, and;1=x,y,z,ort.You say, “Inwhich axes?” Itmakes nodifference. Thelawistrueforeach component, using anyaxes. Invector analysis wediscussed oneother thing, thedotproduct oftwovectors. Letusnow consider thecorresponding thing inspace-time. Inordinary rotation wediscovered there wasanunchanged quantity x2+y2+22.Infour dimen- sions, wefindthatthecorresponding quantity ist2—x2—y2—22(Eq. 17.3). How canwewrite that? Onewaywould betowrite some kindoffour-dimensional thing with asquare dotbetween, likeA,,<>B,.;oneofthenotations which is actually used is 224,4, =A?-A3-.4;-AZ. (17.15)I4 Theprime onZmeans thatthefirstterm, the“time” term, ispositive, butthe other three terms have minus signs. This quantity, then, willbethesame inany coordinate system, andwemaycallitthesquare ofthelength ofthefour-vector. Forinstance, what isthesquare ofthelength ofthefour-vector momentum ofa single particle? This willbeequal topf—pf—pi—pfor,inother words, E2—p2,because weknow thatp, isE.What isE2—p2? Itmust besomething which isthesame inevery coordinate system. Inparticular, itmust bethesame foracoordinate system which ismoving right along with theparticle, inwhich theparticle isstanding still. Iftheparticle isstanding still, itwould have no momentum. Sointhatcoordinate system, itispurely itsenergy, which isthesame asitsrestmass. Thus E2—p2=mg.Soweseethatthesquare ofthelength of thisvector, thefour-vector momentum, isequal tomg. From thesquare ofavector, wecangoontoinvent the“dot product,” orthe product which isascalar: ifa,isonefour-vector andb,isanother four-vector, then thescalar product is Z’a,b,, =a,b,-11,11,-a,b,-a,,b,. (17.16) Itisthesame inallcoordinate systems. Finally, weshall mention certain things whose restmass moiszero. Aphoton oflight, forexample. Aphoton islikeaparticle, inthatitcarries anenergy and amomentum. Theenergy ofaphoton isacertain constant, called Planck’s con- stant, times thefrequency ofthephoton: E=hv.Such aphoton alsocarries a momentum, andthemomentum ofaphoton (orofanyother particle, infact) ish divided bythewavelength: p=h/>1. But,foraphoton, there isadefinite relation- shipbetween thefrequency andthewavelength: 1/=c/)\. (The number ofwaves persecond, times thewavelength ofeach, isthedistance thatthelight goes in onesecond, which, ofcourse, isc.)Thus weseeimmediately thattheenergy ofa photon must bethemomentum times c,orifc=1,theenergy andmomentum areequal. That istosay,therestmass iszero. Letuslook atthatagain; thatis quite curious. Ifitisaparticle ofzero restmass, what happens when itstops? Itnever stops! Italways goes atthespeed c.Theusual formula forenergy is mo/\/l —v2.Now canwesaythatmo=0andv=1,sotheenergy is0?We cannot saythatitiszero; thephoton really can(anddoes) have energy even though ithasnorestmass, butthisitpossesses byperpetually going atthespeed oflight! Wealsoknow thatthemomentum ofanyparticle isequal toitstotal energy times itsvelocity: ifc=1,p=vEor,inordinary units, p=11E/c2. Forany particle moving atthespeed oflight,p =Eifc =1.Theformulas fortheenergy ofaphoton asseenfrom amoving system are,ofcourse, given byEq.(17.12), but forthemomentum wemust substitute theenergy times c(ortimes 1inthiscase). Thedifferent energies after transformation means thatthere aredifferent frequen- cies. This iscalled theDoppler effect, andonecancalculate iteasily from Eq. (17.12), using alsoE=pandE=hv. AsMinkowski said, “Space ofitself, andtime ofitself willsink intomere shadows, andonly akind ofunion between them shall survive.” 17-8 18 Rotation inTwo Dimensions 18-1 Thecenter ofmass Intheprevious chapters wehave been studying themechanics ofpoints, or small particles whose internal structuredoes notconcern us.Forthenext few chapters weshall study theapplication ofNewton’s laws tomore complicated things. When theworld becomes more complicated, italsobecomes more inter- esting, andweshall findthatthephenomena associated with themechanics ofa more complex object than justapoint arereally quite striking. Ofcourse these phenomena involve nothing butcombinations ofNewton’s laws, butitissome- times hard tobelieve thatonly F=maisatwork. Themore complicated objects wedealwithcanbeofseveral kinds: water flowing, galaxies whirling, andsoon.Thesimplest “complicated” object toanalyze, atthestart, iswhat wecallarigid body, asolid object thatisturning asitmoves about. However, even such asimple object mayhave amost complex motion, and weshall therefore firstconsider thesimplest aspects ofsuch motion, inwhich an extended body rotates about afixed axis. Agiven point onsuch abody thenmoves inaplane perpendicular tothisaxis. Such rotation ofabody about afixed axisis called plane rotation orrotation intwodimensions. Weshall later generalize the results tothree dimensions, butindoing soweshall findthat, unlike thecase of ordinary particle mechanics, rotations aresubtle andhard tounderstand unless wefirstgetasolid grounding intwodimensions. Thefirstinteresting theorem concerning themotion ofcomplicated objects canbeobserved atwork ifwethrow anobject made ofalotofblocks andspokes, heldtogether bystrings, intotheair. Ofcourse weknow itgoes inaparabola, because westudied thatforaparticle. Butnow ourobject isnotaparticle; it wobbles anditjiggles, andsoon.Itdoes goinaparabola though; onecansee that. What goesinaparabola? Certainly notthepoint onthecorner oftheblock, because thatisjiggling about; neither isittheendofthewooden stick, orthemiddle ofthewooden stick, orthemiddle oftheblock. Butsomething goesinaparabola, there isaneffective “center” which moves inaparabola. Soourfirsttheorem about complicated objects istodemonstrate thatthere isamean position which is mathematically definable, butnotnecessarily apoint ofthematerial itself, which goesinaparabola. That iscalled thetheorem ofthecenter ofthemass, andthe proof ofitisasfollows. Wemayconsider anyobject asbeing made oflotsoflittle particles, theatoms, withvarious forces among them. Letirepresent anindex which defines oneofthe particles. (There aremillions ofthem, soigoes to1023, orsomething.) Then the force ontheithparticle is,ofcourse, themass times theacceleration ofthat particle: F,=m,-(d2r,-/dt2). (18.1) Inthenext fewchapters ourmoving objects willbeones inwhich allthe parts aremoving atspeeds very much slower than thespeed oflight, andweshall usethenonrelativistic approximation forallquantities. Inthese circumstances themass isconstant, sothat F,=d2(m,-r,-)/dt2 (18.2) Ifwenow addtheforce onalltheparticles, thatis,ifwetake thesumofallthe F,-’sforallthedifferent indexes, wegetthetotal force, F.Ontheother sideofthe 18-118-1 Thecenter ofmass 18-2 Rotation ofarigid body 18-3 Angular momentum 18-4 Conservation ofangular momentum equation, wegetthesame thing asthough weadded before thedifferentiation: 2 - . - zFi=F= . (183) 1 dt Therefore thetotal force isthesecond derivative ofthemasses times their positions, added together. Now thetotal force onalltheparticles isthesame astheexternal force. Why‘? Although there areallkinds offorces ontheparticles because ofthestrings, the wigglings, thepullings andpushings, andtheatomic forces, andwhoknows what, andwehave toaddallthese together, wearerescued byNewton’s Third Law. Between anytwoparticles theaction andreaction areequal, sothatwhen weadd alltheequations together, ifanytwoparticles have forces between them itcancels outinthesum; therefore thenetresult isonly those forces which arise from other particles which arenotincluded inwhatever object wedecide tosumover. Soif Eq.(18.3) isthesum over acertain number oftheparticles, which together are called “the object,” then theexternal force onthetotal object isequal tothesum ofalltheforces onallitsconstituent particles. Now itwould beniceifwecould write Eq.(18.3) asthetotal mass times some acceleration. Wecan. LetussayMisthesumofallthemasses, i.e.,thetotal mass. Then ifwedefine acertain vector Rtobe R=Zm,-i',~/M, (18.4) Eq.(18.3)willbesimply ‘ F=d2(MR)/dt2 =M(d2R/dt2), (18.5) since Misaconstant. Thus wefindthattheexternal force isthetotal mass times theacceleration ofanimaginary point whose location isR.This point iscalled thecenter ofmass ofthebody. Itisapoint somewhere inthe“middle” ofthe object, akind ofaverage rinwhich thedifferent r,~’shave weights orimportances proportional tothemasses. Weshall discuss thisimportant theorem inmore detail inalater chapter, and weshall therefore limit ourremarks totwopoints: First, iftheexternal forces are zero, iftheobject were floating inempty space, itmight whirl, andjiggle, andtwist, anddoallkinds ofthings. Butthecenter ofmass, thisartificially invented, cal- culated position, somewhere inthemiddle, willmove with aconstant velocity. Inparticular, ifitisinitially atrest, itwillstayatrest. Soifwehave some kind ofabox, perhaps aspace ship, with people init,andwecalculate thelocation of thecenter ofmass andfinditisstanding still,then thecenter ofmass willcontinue tostand stillifnoexternal forces areacting onthebox. Ofcourse, thespace ship may move alittle inspace, butthatisbecause thepeople arewalking back and forth inside; when onewalks toward thefront, theshipgoestoward theback soas tokeep theaverage position ofallthemasses inexactly thesame place. Isrocket propulsion therefore absolutely impossible because onecannot move thecenter ofmass? No; butofcourse wefindthattopropel aninteresting part oftherocket, anuninteresting partmust bethrown away. Inother words, ifwe start witharocket atzerovelocity andwespitsome gasouttheback end,thenthis little blob ofgasgoes onewayastherocket shipgoes theother, butthecenter of mass isstillexactly where itwasbefore. Sowesimply move thepartthatweare interested inagainst thepartwearenotinterested in. The second point concerning thecenter ofmass, which isthereason we introduced itintoourdiscussion atthistime, isthatitmay betreated separately from the“internal” motions ofanobject, andmay therefore beignored inour discussion ofrotation. 18-2 Rotation ofarigid body Now letusdiscuss rotations. Ofcourse anordinary object does notsimply rotate, itwobbles, shakes, and bends, sotosimplify matters weshall discuss the motion ofanonexistent ideal object which wecallarigid body. This means an 18-2 object inwhich theforces between theatoms aresostrong, andofsuch character, thatthelittle forces thatareneeded tomove itdonotbend it.Itsshape stays essentially thesame asitmoves about. Ifwewishtostudy themotion ofsucha body, andagree toignore themotion ofitscenter ofmass, there isonly onething leftforittodo,andthatistoturn. Wehave todescribe that. How? Suppose there issome lineinthebody which stays put(perhaps itincludes thecenter of mass andperhaps not), andthebody isrotating about thisparticular lineasan axis. How dowedefine therotation? That iseasyenough, forifwemark apoint somewhere ontheobject, anywhere except ontheaxis, wecanalways tellexactly where theobject is,ifweonly know where thispoint hasgone to.Theonly thing needed todescribe theposition ofthatpoint isanangle. Sorotation consists ofa study ofthevariations oftheangle with time. Inorder tostudy rotation, weobserve theangle through which abody has turned. Ofcourse, wearenotreferring toanyparticular angle inside theobject itself; itisnotthatwedraw some angle ontheobject. Wearetalking about the angular change oftheposition ofthewhole thing, from onetime toanother. First, letusstudy thekinematics ofrotations. Theangle willchange with time, andjustaswetalked about position andvelocity inonedimension, wemay talk about angular position andangular velocity inplane rotation. Infact, there isa veryinteresting relationship between rotation intwodimensions andone-dimen- sional displacement, inwhich almost every quantity hasitsanalog. First, wehave theangle 0which defines how farthebody hasgone around; thisreplaces the distance y,which defines howfarithasgone along. Inthesame manner, wehave a velocity ofturning, at=d0/dt, which tellsushow much theangle changes ina second, justasv=ds/dt describes how fastathing moves, orhow faritmoves inasecond. Iftheangle ismeasured inradians, then theangular velocity wwill besoandsomany radians persecond. Thegreater theangular velocity, thefaster theobject isturning, thefaster theangle changes. Wecangoon: wecandiffer- entiate theangular velocity with respect totime, andwecancallor=dw/dt = d20/dt2 theangular acceleration. That would betheanalog oftheordinary accel- eration. Now ofcourse weshall have torelate thedynamics ofrotation tothelaws of dynamics oftheparticles ofwhich theobject ismade, sowemust findouthow a particular particle moves when theangular velocity issuch andsuch. Todothis, letustakeacertain particle which islocated atadistance rfrom theaxisandsay itisinacertain location P(x,y)atagiven instant, intheusual manner (Fig. 18-1). Ifatamoment Atlater theangle ofthewhole object hasturned through A0,then thisparticle iscarried withit.Itisatthesame radius away from Oasitwasbefore, butiscarried toQ.Thefirstthing wewould liketoknow ishowmuch thedistance xchanges andhowmuch thedistance ychanges. IfOPiscalled r,thenthelength PQisrA0,because ofthewayangles aredefined. Thechange inx,then, issimply theprojection ofrA0inthex-direction: Ax=—PQ sin6=—rA0- (y/r) =—yA0. (18.6) Similarly, Ay=+xA0. (18.7) Iftheobject isturning with agiven angular velocity w,wefind, bydividing both sides of(18.6) and(18.7) byAt,thatthevelocity oftheparticle is 11,,=—wy and 11,,=+wx. (18.8) Ofcourse ifwewant tofindthemagnitude ofthevelocity, wejustwrite U=_ 1/U12: + vi = -y/w2y2 + w2X2 = an/X2 + y2 = wf_ Itshould notbemysterious that thevalue ofthemagnitude ofthisvelocity is wr;infact,itshould beself-evident, because thedistance thatitmoves isrA19and thedistance itmoves persecond isrA0/At, orrw. 18-30 x,yl A y Fig. 18-1. Kinematics oftwo-dimen sional rotation.Qk X<< '44‘1-<@ X Letusnow move ontoconsider thedynamicsof rotation. Here anew concept, force, must beintroduced. Letusinquire whether wecaninvent something which weshallcallthetorque (L.torquere, totwist) which bears thesame relation- ship torotation asforce does tolinear movement. Aforce isthething thatis needed tomake linear motion, andthething that makes something rotate isa “rotary force” ora“twisting force,” i.e.,atorque. Qualitatively, atorque isa “twist”; what isatorque quantitatively? Weshall gettothetheory oftorques quantitatively bystudying thework done inttirrifng anobject, foronevery nice wayofdefining aforce istosayhow much work itdoes when itactsthrough a given displacement. Wearegoing totrytomaintain theanalogy between linear andangular quantities byequating thework thatwedowhen weturnsomething a little bitwhen there areforces acting onit,tothetorque times theangle itturns through. Inother words, thedefinition ofthetorque isgoing tobesoarranged thatthetheorem ofwork hasanabsolute analog: force times distance iswork, and torque times angle isgoing tobework. That tellsuswhat torque is.Consider, forinstance, arigid body ofsome kind with various forces acting onit,andan axisabout which thebody rotates. Letusatfirstconcentrate ononeforce and suppose thatthisforce isapplied atacertain point (x,y).How much work would bedone ifwewere toturn theobject through avery small angle? That iseasy. Thework done is AW=FmAx+F”Ay. (18.10) Weneed only tosubstitute Eqs. (18.6) and(18.7) forAxandAytoobtain AW=(xF,, —yF,,) A0. (18.11) That is,theamount ofwork thatwehave done is,infact,equal totheangle through which wehave turned theobject, multiplied byastrange-looking combination of theforce andthedistance. This “strange combination” iswhat wecallthetorque. So,defining thechange inwork asthetorque times theangle, wenow have the formula fortorque interms oftheforces. (Obviously, torque isnotacompletely new idea independent ofNewtonian mechanics—torque must have adefinite definition interms oftheforce.) When there areseveral forces acting, thework thatisdone is,ofcourse, the sumoftheworks done byalltheforces, sothatAWwillbeawhole lotofterms, alladded together, foralltheforces, each ofwhich isproportional, however, toA0. Wecantake theA0outside andtherefore cansaythatthechange inthework is equal tothesumofallthetorques duetoallthedifferent forces thatareacting, times A0.This sum wemight callthetotal torque, -r.Thus torques addbythe ordinary laws ofalgebra, butweshall later seethatthisisonly because weare working inaplane. Itislikeone-dimensional kinematics, where theforces simply addalgebraically, butonly because they areallinthesame direction. Itismore complicated inthree dimensions. Thus, fortwo-dimensional rotation, 7',=x,-F,,,~ —y,-F,,,- (18.12) and 1'=21,-. (18.13) Itmust beemphasized thatthetorque isabout agiven axis. Ifadifferent,axis is chosen, sothatallthex,-andy,~arechanged, thevalue ofthetorque is(usually) changed too. Now wepause briefly tonote that ourforegoing introduction oftorque, through theideaofwork, gives usamost important result foranobject inequilib- rium: ifalltheforces onanobject areinbalance both fortranslation androtation, notonly isthenetforce zero, butthetotal ofallthetorques isalsozero, because ifanobject isinequilibrium, nowork isdone bytheforces forasmall displacement. Therefore, since AW=1-A0=0,thesum ofallthetorques must bezero. So there aretwoconditions forequilibrium: thatthesumoftheforces iszero, and thatthesumofthetorques iszero. Prove thatitsuffices tobesurethatthesum oftorques about anyoneaxis(intwodimensions) iszero. 18-4 Now letusconsider asingle force, andtrytofigure out,geometrically, what thisstrange thing xF,,—yF,amounts to.InFig.18-2 weseeaforce Facting at apoint 1-.When theobject hasrotated through asmall angle A0,thework done, ofcourse, isthecomponent offorce inthedirection ofthedisplacement times the displacement. Inother words, itisonly thetangential component oftheforce thatcounts, andthismust bemultiplied bythedistance rA0. Therefore wesee that_the torque isalsoequal tothetangential component offorce (perpendicular totheradius) times theradius. That makes sense interms ofourordinary idea ofthetorque, because iftheforce were completely radial, itwould notputany “twist” onthebody; itisevident thatthetwisting effect should involve only the partoftheforce which isnotpulling outfrom thecenter, andthatmeans the tangential component. Furthermore, itisclear thatagiven force ismore effective onalong armthan near theaxis. Infact, ifwetakethecasewhere wepush right ontheaxis, wearenottwisting atall!Soitmakes sense thattheamount oftwist, ortorque, isproportional both totheradial distance andtothetangential com- ponent oftheforce. There isstillathird formula forthetorque which isveryinteresting. Wehave justseenthatthetorque istheforce times theradius times thesineoftheangle oz, inFig.18-2. Butifweextend thelineofaction oftheforce anddraw thelineOS, theperpendicular distance tothelineofaction oftheforce (thelever armofthe force) wenotice thatthislever armisshorter than rinjustthesame proportion asthetangential partoftheforce islessthan thetotal force. Therefore theformula forthetorque canalsobewritten asthemagnitude oftheforce times thelength ofthelever arm. Thetorque isalsooften called themoment oftheforce. Theorigin ofthis term isobscure, butitmayberelated tothefactthat“moment” isderived from theLatin movimentum, andthatthecapability ofaforce tomove anobject (using theforce onalever orcrowbar) increases with thelength ofthelever arm. In mathematics ‘m0ment” means weighted byhowfaraway itisfrom anaxis. 18-3 Angular momentum Although wehave sofarconsidered only thespecial caseofarigid body, the properties oftorques andtheir mathematical relationships areinteresting also evenwhen anobject isnotrigid. Infact,wecanprove averyremarkable theorem: justasexternal force istherateofchange ofaquantity p,which wecallthetotal momentum ofacollection ofparticles, sotheexternal torque istherateofchange ofaquantity Lwhich wecalltheangular momentum ofthegroup ofparticles. Toprove this, weshall suppose thatthere isasystem ofparticles onwhich there aresome forces acting andfindoutwhat happens tothesystem asaresult of thetorques duetothese forces. First, ofcourse, weshould consider justone particle. InFig.18-3 isoneparticle ofmass m,andanaxis0;theparticle isnot necessarily rotating inacircle about O,itmaybemoving inanellipse, likeaplanet going around thesun,orinsome other curve. Itismoving somehow, andthere areforces onit,anditaccelerates according totheusual formula thatthex-com- ponent offorce isthemass times thex-component ofacceleration, etc.Butletus seewhat thetorque does. Thetorque equals xF,,—yF,,, andtheforce inthe x-ory-direction isthemass times theacceleration inthex-ory-direction: T=xF,,—yF,, =xm(d2y/dt2) —ym(d2x/dt2). (18.14) Now, although thisdoes notappear tobethederivative ofanysimple quantity, it isinfactthederivative ofthequantity xm(dy/dt) —ym(dx/dt): d dy dx _ d2y dx dy a(ail7Y” -""’(tel+la)'"la)d2x dy dx ,d2y d2x —ym 221? —df EZxm Tfl —ym fl? i(18.15) 5 18-50 Ftr,, * Fr rA9 ° I P Fs Fig. 18-2. Thetorque produced by aforce. R 0..V m\ P\\f O Fig. 18-3. Aparticle moves about anaxis O. Soitistruethatthetorque istherateofchange ofsomething withtime! Sowe payattention tothe“something,” wegive itaname: wecallitL,theangular momentum: L=xm(dy/dt) —ym(dx/dt) =xpy-—yp,. (18.16) Although ourpresent discussion isnonrelativistic, thesecond form forL given above isrelativistically correct. Sowehave found thatthere isalsoarota- tional analog forthemomentum, andthatthisanalog, theangular momentum, is given byanexpression interms ofthecomponents oflinear momentum thatis justliketheformula fortorque interms oftheforce components! Thus, ifwewant toknow theangular momentum ofaparticle about anaxis, wetake only the component ofthemomentum thatistangential, andmultiply itbytheradius. In other words, what counts forangular momentum isnothow fastitisgoing away from theorigin, buthow much itisgoing around theorigin. Only thetangential partofthemomentum counts forangular momentum. Furthermore, thefarther outthelineofthemomentum extends, thegreater theangular momentum. And also, because thegeometrical facts arethesame whether thequantity islabeled porF,itistruethatthere isalever arm(notthesame asthelever armoftheforce ontheparticle!) which isobtained byextending thelineofthemomentum and finding theperpendicular distance totheaxis. Thus theangular momentum isthe magnitude ofthemomentum times themomentum lever arm. Sowehave three formulas forangular momentum, justaswehave three formulas forthetorque: L=X111—J/in =rptang =p-lever arm. (18.17) Like torque, angular momentum depends upon theposition oftheaxis about which itistobecalculated. Before proceeding toatreatment ofmore than oneparticle, letusapply the above results toaplanet going around thesun. Inwhich direction istheforce? Theforce istoward thesun. What, then, isthetorque ontheobject? Ofcourse, thisdepends upon where wetake theaxis, butwegetavery simple result ifwe takeitatthesunitself, forthetorque istheforce times thelever arm, orthecom- ponent offorce perpendicular tor,times r.Butthere isnotangential force, so there isnotorque about anaxisatthesun! Therefore, theangular momentum of theplanet going around thesunmust remain constant. Letusseewhat thatmeans. Thetangential component ofvelocity, times themass, times theradius, willbe constant, because thatistheangular momentum, andtherateofchange ofthe angular momentum isthetorque, and, inthisproblem, thetorque iszero. Of course since themass isalsoaconstant, thismeans thatthetangential velocity times theradius isaconstant. Butthisissomething wealready knew forthemotion ofaplanet. Suppose weconsider asmall amount oftime At.How farwillthe planet move when itmoves from PtoQ(Fig. 18-3)? How much areawillitsweep through? Disregarding thevery tinyarea QQ’P compared with themuch larger area OPQ, itissimply halfthebasePQtimes theheight, OR. Inother words, the areathatisswept through inunittime willbeequal tothevelocity times thelever armofthevelocity (times one-half). Thus therateofchange ofareaisproportional totheangular momentum, which isconstant. SoKepler’s lawabout equal areas inequal times isaword description ofthestatement ofthelawofconservation of angular momentum, when there isnotorque produced bytheforce. 18-4 Conservation ofangular momentum Now weshall goontoconsider what happens when there isalarge number ofparticles, when anobject ismade ofmany pieces withmany forces acting between them andonthem from theoutside. Ofcourse, wealready know that, about any given fixed axis, thetorque ontheithparticle (which istheforce ontheithparticle 18-6 times thelever armofthatforce) isequal totherateofchange oftheangular momentum ofthatparticle, andthattheangular momentum oftheithparticle isitsmomentum times itsmomentum lever arm. Now suppose weaddthetorques 1-,~foralltheparticles andcallitthetotal torque -r.Then thiswillbetherateof change ofthesumoftheangular momenta ofalltheparticles L,-,andthatdefines anewquantity which wecallthetotal angular momentum L.Just asthetotal momentum ofanobject isthesumofthemomenta ofalltheparts, sotheangular momentum isthesumoftheangular momenta ofalltheparts. Then therateof change ofthetotal Listhetotal torque: ¢=g..=z%=£i§. (18.18) Now itmight seem thatthetotal torque isacomplicated thing. There areall those internal forces andalltheoutside forces tobeconsidered. But, ifwetake Newton’s lawofaction andreaction tosay,notsimply thattheaction andreaction areequal, butalso that they aredirected exactly oppositely along thesame line (Newton may ormaynotactually have saidthis,buthetacitly assumed it),then thetwotorques onthereacting objects, duetotheir mutual interaction, willbe equal andopposite because thelever arms foranyaxisareequal. Therefore the internal torques balance outpairbypair, andsowehave theremarkable theorem thattherateofchange ofthetotal angular momentum about anyaxisisequal to theexternal torque about thataxis! T=Z-.~=T,“=at/at. (18.19) Thus wehaveaverypowerful theorem concerning themotion oflarge collections ofparticles, which permits ustostudy theover-all motion without having tolook atthedetailed machinery inside. This theorem istrueforanycollection ofobjects, whether they form arigid body ornot. Oneextremely important caseoftheabove theorem isthelawofconservation ofangular momentum: ifnoexternal torques actupon asystem ofparticles, the angular momentum remains constant. Aspecial caseofgreat importance isthatofarigidbody, thatis,anobject ofa definite shape thatisjustturning around. Consider anobject thatisfixed inits geometrical dimensions, andwhich isrotating about afixed axis. Various parts of theobject bear thesame relationship tooneanother atalltimes. Now letustry tofindthetotal angular momentum ofthisobject. Ifthemass ofoneofitsparticles ism,-,anditsposition orlocation isat(x,-,y,~),then theproblem istofindthe angular momentum ofthatparticle, because thetotal angular momentum isthe sumoftheangular momenta ofallsuch particles inthebody. Foranobject going around inacircle, theangular momentum, ofcourse, isthemass times thevelocity times thedistance from theaxis, andthevelocity isequal totheangular velocity times thedistance from theaxis: Li =m,-v,-r,- =m,-rfw, or,summing over alltheparticles i,weget L=Iw, (18.21) where 1=Zm,-rt. (18.22) This istheanalog ofthelawthat themomentum ismass times velocity. Velocity isreplaced byangular velocity, andweseethatthemass isreplaced by anewthing which wecallthemoment ofinertia I,which isanalogous tothemass. Equations (18.21) and(18.22) saythatabody hasinertia forturning which depends, notjustonthemasses, butonhowfaraway theyarefrom theaxis. So,ifwehave twoobjects ofthesame mass, when weputthemasses farther away from theaxis, theinertia forturning willbehigher. This iseasily demonstrated bytheapparatus 18-7 K m\-’\ \_J 1 "1 Fig. 18-4. The "inertia forturning" depends upon thelever armofthemasses.shown inFig.18-4, where aweight Miskept from falling very fastbecause ithas toturn thelarge weighted rod. Atfirst, themasses mareclose totheaxis, and Mspeeds upatacertain rate. Butwhen wechange themoment ofinertia by putting thetwomasses mmuch farther away from theaxis, then weseethatM accelerates much lessrapidly than itdidbefore, because thebody hasmuch more inertia against turning. Themoment ofinertia istheinertia against turning, and isthesum ofthecontributions ofallthemasses, times their distances squared, from theaxis. There isoneimportant difference between mass andmoment ofinertia which isvery dramatic. Themass ofanobject never changes, butitsmoment ofinertia canbechanged. Ifwestand onafrictionless rotatable stand with ourarms out- stretched, andhold some weights inourhands aswerotate slowly, wemaychange ourmoment ofinertia bydrawing ourarms in,butourmass does notchange. When wedothis, allkinds ofwonderful things happen, because ofthelawofthe conservation ofangular momentum: Iftheexternal torque iszero, then theangular momentum, themoment ofinertia times omega, remains constant. Initially, we were rotating with alarge moment ofinertia I1atalowangular velocity (.01,and theangular momentum was110.11. Then wechanged ourmoment ofinertia by pulling ourarms in,saytoasmaller value I2.Then theproduct Iw,which hasto stay thesame because thetotal angular momentum hastostay thesame, was 120.12. SoI1w1=12:112. That is,ifwereduce themoment ofinertia, wehave to increase theangular velocity. 18-8 I9 Center ofMass; Moment ofInertia 19-1 Properties ofthecenter ofmass Intheprevious chapter wefound thatifagreat many forces areacting ona complicated mass ofparticles, whether theparticles comprise arigid oranonrigid body, oracloud ofstars, oranything else, andwefindthesumofalltheforces (that is,ofcourse, theexternal forces, because theinternal forces balance out), then ifweconsider thebody asawhole, andsayithasatotal mass M,there isacertain point “inside” thebody, called thecenter ofmass, such that thenetresulting external force produces anacceleration ofthispoint, justasthough thewhole mass were concentrated there. Letusnow discuss thecenter ofmass inalittle more detail. Thelocation ofthecenter ofmass (abbreviated CM) isgiven bytheequation RCM= (19.1)m. This is,ofcourse, avector equation which isreally three equations, oneforeach of thethree directions. Weshall consider only thex-direction, because ifwecan understand that one, wecanunderstand theother two. What does XCM = Em,-x,~/Em, mean? Suppose foramoment thattheobject isdivided intolittle pieces, allofwhich havethesame massm;thenthetotalmassissimply thenumber Nofpieces times themass ofonepiece, sayonegram, oranyunit. Then this equation simply says thatweaddallthex’s,andthen divide bythenumber of things that wehave added: XCM =mZx,-/mN =Xx,/N. Inother words, XCMistheaverage ofallthex’s,ifthemasses areequal. Butsuppose oneofthem were twice asheavy astheothers. Then inthesum, thatxwould come intwice. This iseasy tounderstand, forwecanthink ofthisdouble mass asbeing split intotwoequal ones, justliketheothers; then intaking theaverage, ofcourse, we have tocount thatxtwice because there aretwomasses there. Thus Xisthe average position, inthex-direction, ofallthemasses, every mass being counted a number oftimes proportional tothemass, asthough itwere divided into “little grams.” From thisitiseasy toprove that Xmust besomewhere between the largest andthesmallest x,and, therefore liesinside theenvelope including the entire body. Itdoes nothave tobeinthematerial ofthebody, forthebody could beacircle, likeahoop, andthecenter ofmass isinthecenter ofthehoop, notin thehoop itself. Ofcourse, ifanobject issymmetrical insome way, forinstance, arectangle, sothatithasaplane ofsymmetry, thecenter ofmass liessomewhere ontheplane ofsymmetry. Inthecase ofarectangle there aretwoplanes, andthat locates it uniquely. Butifitisjustanysymmetrical object, then thecenter ofgravity lies somewhere ontheaxisofsymmetry, because inthose circumstances there areas many positive asnegative x’s. Another interesting proposition isthefollowing very curious one. Suppose thatweimagine anobject tobemade oftwopieces, AandB(Fig. 19-1). Then thecenter ofmass ofthewhole object canbecalculated asfollows. First, findthe center ofmass ofpiece A,andthen ofpiece B.Also, findthetotal mass ofeach piece, MAandMB.Then consider anewproblem, inwhich apoint mass MAis atthecenter ofmass ofobject A,andanother point mass MBisatthecenter of mass ofobject B.Thecenter ofmass ofthese twopoint masses isthen thecenter ofmass ofthewhole object. Inother words, ifthecenters ofmass ofvarious parts 19-119-1 Properties ofthecenter ofmass 19-2 Locating thecenter ofmass 19-3 Finding themoment ofinertia 19-4 Rotational kinetic energy \\ CM‘I Fig. 19-1. The CM of0compound body liesonthelinejoining theCM's of thetwocomposite ports. ofanobject have been worked out,wedonothave tostart allover again tofind thecenter ofmass ofthewhole object; wejusthave toputthepieces together, treating each oneasapoint mass situated atthecenter ofmass ofthat piece. Letusseewhythatis.Suppose thatwewanted tocalculate thecenter ofmass of acomplete object, some ofwhose particles areconsidered tobemembers of object Aandsome members ofobject B.Thetotal sum Em,-x, canthen besplit into twopieces—the sum ZAm,-x,- fortheAobject only, andthesum Zgm,-xi forobject Bonly. Now ifwewere computing thecenter ofmass ofobject Aalone, wewould have exactly thefirstofthese sums, andweknow thatthisbyitself is MAXA,thetotal mass ofalltheparticles inAtimes theposition ofthecenter of mass ofA,because thatisthetheorem ofthecenter ofmass, applied toobject A. Inthesame manner, justbylooking atobject B,wegetMBXB,andofcourse, adding thetwoyields MX: MXCM =2mtxt +Zmixt A B =MAXA + MBXB. Now since Misevidently thesumofMAandMB,weseethatEq.(19.2) canbe interpreted asaspecial example ofthecenter ofmass formula fortwopoint objects, oneofmass MAlocated atXAandtheother ofmass MBlocated atXB. Thetheorem concerning themotion ofthecenter ofmass isvery interesting, andhasplayed animportant part inthedevelopment ofourunderstanding of physics. Suppose weassume thatNewton’s lawisright forthesmall component parts ofamuch larger object. Then thistheorem shows thatNewton’s lawisalso correct forthelarger object, even ifwedonotstudy thedetails oftheobject, but only thetotal force acting onitanditsmass. Inother words, Newton’s lawhas thepeculiar property thatifitisright onacertain small scale, then itwillberight onalarger scale. Ifwedonotconsider abaseball asatremendously complex thing, made ofmyriads ofinteracting particles, butstudy only themotion ofthe center ofmass andtheexternal forces ontheball, wefind F=ma,where Fisthe external force onthebaseball, misitsmass, andaistheacceleration ofitscenter ofmass. SoF=maisalawwhich reproduces itself onalarger scale. (There ought tobeagood word, outoftheGreek, perhaps, todescribe alawwhich reproduces thesame lawonalarger scale.) Ofcourse, onemight suspect thatthefirstlaws thatwould bediscovered by human beings would bethose thatwould reproduce themselves onalarger scale. Why? Because theactual scale ofthefundamental gears andwheels oftheuniverse areofatomic dimensions, which aresomuch finer than ourobservations thatwe arenowhere near thatscale inourordinary observations. Sothefirstthings thatwe would discover must betrueforobjects ofnospecial sizerelative toanatomic scale. Ifthelaws forsmall particles didnotreproduce themselves onalarger scale, wewould notdiscover those laws very easily. What about thereverse problem? Must thelaws onasmall scale bethesame asthose onalarger scale? Ofcourse itisnotnecessarily soinnature, thatatanatomic level thelawshave tobethesame asonalarge scale. Suppose thatthetruelaws ofmotion ofatoms were given by some strange equation which does nothave theproperty thatwhen wegotoa larger scale wereproduce thesame law, butinstead hastheproperty thatifwe gotoalarger scale, wecanapproximate itbyacertain expression such that, ifwe extend thatexpression upandup,itkeeps reproducing itself onalarger andlarger scale. That ispossible, andinfactthatisthewayitworks. Newton’s laws arethe “tail end” oftheatomic laws, extrapolated toaverylarge size. Theactual laws of motion ofparticles onafinescale areverypeculiar, butifwetakelarge numbers of them andcompound them, they approximate, butonly approximate, Newton’s laws. Newton’s laws then permit ustogoontoahigher andhigher scale, and itstillseems tobethesame law. Infact, itbecomes more andmore accurate as thescale getslarger andlarger. This self-reproducing factor ofNewton’s laws is thus really notafundamental feature ofnature, butisanimportant historical feature. Wewould never discover thefundamental laws oftheatomic particles at firstobservation because thefirstobservations aremuch toocrude. Infact,itturns 19-2 outthatthefundamental atomic laws, which wecallquantum mechanics, arequite different from Newton’s laws, andaredifficult tounderstand because allourdirect experiences arewith large-scale objects andthesmall-scale atoms behave like nothing weseeonalarge scale. Sowecannot say,“An atom isjustlikeaplanet going around thesun,” oranything likethat. Itislikenothing wearefamiliar with because there isnothing likeit.Asweapply quantum mechanics tolarger andlarger things, thelaws about thebehavior ofmany atoms together donotreproduce themselves, butproduce newlaws, which areNewton’s laws, which then continue toreproduce themselves from, say, micro-microgram size, which stillisbillions andbillions ofatoms, onuptothesizeoftheearth, andabove. Letusnow return tothecenter ofmass. Thecenter ofmass issometimes called thecenter ofgravity, forthereason that, inmany cases, gravity may be considered uniform. Letussuppose thatwehave small enough dimensions that thegravitational force isnotonly proportional tothemass, butiseverywhere parallel tosome fixed line. Then consider anobject inwhich there aregravitational forces oneach ofitsconstituent masses. Letm,-bethemass ofonepart. Then the gravitational force onthatpartism,-times g.Now thequestion is,where canwe apply asingle force tobalance thegravitational force onthewhole thing, sothatthe entire object, ifitisarigid body, willnotturn? Theanswer isthatthisforce must gothrough thecenter ofmass, andweshow thisinthefollowing way. Inorder that thebody willnotturn, thetorque produced byalltheforces must adduptozero, because ifthere isatorque, there isachange ofangular momentum, andthus a rotation. Sowemust calculate thetotal ofallthetorques onalltheparticles, and seehow much torque there isabout anygiven axis; itshould bezero ifthisaxisis atthecenter ofmass. Now, measuring xhorizontally andyvertically, weknow thatthetorques aretheforces inthey-direction, times thelever armx(that isto say,theforce times thelever armaround which wewant tomeasure thetorque). Now thetotal torque isthesum T=2mtgxi =82 mtxt, (19-3) soifthetotal torque istobezero, thesumZm,~x,- must bezero. ButZm,~x,- =MX, thetotal mass times thedistance ofthecenter ofmass from theaxis. Thus the x-distance ofthecenter ofmass from theaxisiszero. Ofcourse, wehave checked theresult only forthex-distance, butifweuse thetruecenter ofmass theobject willbalance inanyposition, because ifweturned it90degrees, wewould have y’sinstead ofx’s. Inother words, when anobject issupported atitscenter ofmass, there isnotorque onitbecause ofaparallel gravitational field. Incase theobject issolarge that thenonparallelism ofthe gravitational forces issignificant, then thecenter where onemust apply thebalanc- ingforce isnotsimple todescribe, anditdeparts slightly from thecenter ofmass. That iswhy onemust distinguish between thecenter ofmass andthecenter of gravity. Thefactthatanobject supported exactly atthecenter ofmass willbalance inallpositions hasanother interesting consequence. If,instead ofgravitation, wehave apseudoforce duetoacceleration, wemay useexactly thesame mathe- matical procedure tofindtheposition tosupport itsothatthere arenotorques produced bytheinertial force ofacceleration. Suppose thattheobject isheld in some manner inside abox, andthat thebox, andeverything contained init,is accelerating. Weknow that, from thepoint ofview ofsomeone atrestrelative to thisaccelerating box, there willbeaneffective force duetoinertia. That is,to make theobject goalong with thebox, wehave topush onittoaccelerate it,and thisforce is“balanced” bythe“force ofinertia,” which isapseudoforce equal to themass times theacceleration ofthebox. Totheman inthebox, thisisthesame situation asiftheobject were inauniform gravitational field whose “g”value is equal totheacceleration a.Thus theinertial force duetoaccelerating anobject has notorque about thecenter ofmass. This facthasavery interesting consequence. Inaninertial frame thatisnot accelerating, thetorque isalways equal totherateofchange oftheangular mo- mentum. However, about anaxisthrough thecenter ofmass ofanobject which 19-3 / // H .\:7> </" \>~/I \\ //\\\_______\<// Fig. 19-2. Aright triangle and CI right circular cone generated byro- tating thetriangle.isaccelerating, itisstilltruethatthetorque isequal totherateofchange ofthe angular momentum. Even ifthecenter ofmass isaccelerating, wemaystillchoose onespecial axis, namely, onepassing through thecenter ofmass, such thatitwill stillbetruethatthetorque isequal totherateofchange ofangular momentum around thataxis. Thus thetheorem thattorque equals therateofchange ofangular momentum istrueintwogeneral cases: (1)afixed axisininertial space, (2)anaxis through thecenter ofmass, even though theobject may beaccelerating. 19-2 Locating thecenter ofmass Themathematical techniques forthecalculation ofcenters ofmass areinthe province ofamathematics course, andsuch problems provide good exercise in integral calculus. After onehaslearned calculus, however, andwants toknow how tolocate centers ofmass, itisnicetoknow certain tricks which canbeused todoso.Onesuch trick makes useofwhat iscalled thetheorem ofPappus. It works likethis: ifwetakeanyclosed areainaplane andgenerate asolid bymoving itthrough space such thateach point isalways moved perpendicular totheplane ofthearea, theresulting solid hasatotal volume equal tothearea ofthecross section times thedistance thatthecenter ofmass moved! Certainly thisistrueif wemove thearea inastraight lineperpendicular toitself, butifwemove itina circle orinsome other curve, then itgenerates arather peculiar volume. Fora curved path, theoutside goes around farther, andtheinside goes around less,and these effects balance out. Soifwewant tolocate thecenter ofmass ofaplane sheet ofuniform density, wecanremember thatthevolume generated byspinning itabout anaxisisthedistance thatthecenter ofmass goes around, times thearea ofthesheet. Forexample, ifwewish tofindthecenter ofmass ofaright triangle ofbase Dandheight H(Fig. 19-2), wemight solve theproblem inthefollowing way. Imagine anaxisalong H,androtate thetriangle about thataxisthrough afull 360degrees. This generates acone. Thedistance that thex-coordinate ofthe center ofmass hasmoved is21rx. Theareawhich isbeing moved istheareaof thetriangle, %HD. Sothex-distance ofthecenter ofmass times theareaofthe triangle isthevolume swept out,which isofcourse 1rD2H /3.Thus (21rx)(=}HD) = l/31rD2H, orx=D/3.Inasimilar manner, byrotating about theother axis, orby symmetry, wefindy=H/3.Infact, thecenter ofmass ofanyuniform triangular area iswhere thethree medians, thelines from thevertices through thecenters of theopposite sides, allmeet. That point isl/3ofthewayalong each median. Clue: Slice thetriangle upintoalotoflittle pieces, each parallel toabase. Note thatthe median linebisects every piece, andtherefore thecenter ofmass must lieonthisline. Now letustryamore complicated figure. Suppose thatitisdesired tofind theposition ofthecenter ofmass ofauniform semicircular disc—a discsliced in half. Where isthecenter ofmass? Forafulldisc, itisatthecenter, ofcourse, but ahalf-disc ismore difficult. Letrbetheradius andxbethedistance ofthecenter ofmass from thestraight edge ofthedisc. Spin itaround thisedge asaxisto generate asphere. Then thecenter ofmass hasgone around 21rx, thearea is 1rr2/2 (because itisonlyhalfacircle). Thevolume generated is,ofcourse, 41rr3/3, from which wefindthat (21rx)(%1rr2) =41rr3/3, or x=4r/31r. There isanother theorem ofPappus which isaspecial caseoftheabove one, andtherefore equally true. Suppose that, instead ofthesolid semicircular disc, wehave asemicircular piece ofwire with uniform mass density along thewire, andwewant tofinditscenter ofmass. Inthiscasethere isnomass intheinterior, only onthewire. Then itturns outthattheareawhich isswept byaplane curved line, when itmoves asbefore, isthedistance thatthecenter ofmass moves times thelength oftheline. (The linecanbethought ofasavery narrow area, andthe previous theorem canbeapplied toit.) 19-4 19-3 Finding themoment ofinertia Now letusdiscuss theproblem offinding themoments ofinertia ofvarious objects. Theformula forthemoment ofinertia about thez-axis ofanobject is 1=Zmix?+y?) OI‘ 1=f(x2+y2)a’m =f(x2+y2)pdv. (19.4) That is,wemust sumthemasses, each onemultiplied bythesquare ofitsdistance (x?+yf)from theaxis. Note thatitisnotthethree-dimensional distance, only thetwo-dimensional distance squared, even forathree-dimensional object. For themost part, weshall restrict ourselves totwo-dimensional objects, butthe formula forrotation about thez-axis isjustthesame inthree dimensions. Asasimple example, consider arodrotating about aperpendicular axis through oneend(Fig. 19-3). Now wemust sumallthemasses times thex-distances squared (they’sbeing allzero inthiscase). What wemean by“the sum,” of course, istheintegral ofx2times thelittle elements ofmass. Ifwedivide therod intosmall elements oflength dx,thecorresponding elements ofmass arepropor- tional todx,andifdxwere thelength ofthewhole rodthemass would beM. Therefore dm=Mdx/L L L Md M ML21=f0x2%=f/0x2dx=7-- (19.5) Thedimensions ofmoment ofinertia arealways mass times length squared, so allwereally hadtowork outwasthefactor 1/3. Now what isIiftherotation axisisatthecenter oftherod? Wecould just dotheintegral overagain, letting xrange from —%L to+§L. Butletusnotice a fewthings about themoment ofinertia. Wecanimagine therodastworods, each ofmass M/2 andlength L/2; themoments ofinertia ofthetwosmall rods areequal, andareboth given bytheformula (19.5). Therefore themoment of inertia isandso 1=LL”/2§(L/2)2 = (19.6) Thus itismuch easier toturnarodabout itscenter, than toswing itaround anend. Ofcourse, wecould goontocompute themoments ofinertia ofvarious other bodies ofinterest. However, while such computations provide acertain amount ofimportant exercise inthecalculus, they arenotbasically ofinterest tousas such. There is,however, aninteresting theorem which isvery useful. Suppose wehave anobject, andwewant tofinditsmoment ofinertia around some axis. That means wewant theinertia needed tocarry itbyrotation about that axis. Now ifwesupport theobject onpivots atthecenter ofmass, sothattheobject does notturn asitrotates about theaxis(because there isnotorque onitfrom inertial effects, andtherefore itwillnotturn when westart moving it),then the forces needed toswing itaround arethesame asthough allthemass were concen- trated atthecenter ofmass, andthemoment ofinertia would besimply I1= MR%;M, where RCM isthedistance from theaxis tothecenter ofmass. Butof course thatisnottheright formula forthemoment ofinertia ofanobject which isreally being rotated asitrevolves, because notonly isthecenter ofitmoving in acircle, which would contribute anamount I1tothemoment ofinertia, butalso wemust turn itabout itscenter ofmass. Soitisnotunreasonable thatwemust addtoI1themoment ofinertia I,about thecenter ofmass. Soitisagood guess thatthetotal moment ofinertia about anyaxiswillbe 1=1,,+MR%M. (19.7) 19-S'-W1 x——>l dx Fig. 19-3. Astraight rod oflength Lrotating about anaxisthrough oneend. This theorem iscalled theparallel-axis theorem, andmay beeasily proved. Themoment ofinertia about anyaxisisthemass times thesum ofthex,~’sand they,-’s,each squared: I=Z(xf+yf)m,-. Weshall concentrate onthex’s,but ofcourse they’swork thesame way. Now xisthedistance ofaparticular point mass from theorigin, butletusconsider howitwould look ifwemeasured x’from theCM, instead ofxfrom theorigin. Togetready forthisanalysis, wewrite Xi=X1"+Xen- Then wejustsquare thistofind X?=X92-1-2XCMx1' -1-X?:M- So,when thisismultiplied bym,~andsummed over alli,what happens? Taking theconstants outside thesummation sign, weget L.=Zmix’.-2 +2XCM 2mix?+XénZmt- Thethird sumiseasy; itisjustMX31“. Inthesecond sumthere aretwopieces, one ofthem isZm,-xi-, which isthetotal mass times thex’-coordinate ofthecenter of mass. Butthiscontributes nothing, because x’ismeasured from thecenter ofmass, andinthese axes theaverage position ofalltheparticles, weighted bythemasses, iszero. Thefirstsum, ofcourse, isthexpart ofIA.Thus wearrive atEq.(19.7), justasweguessed. Letuscheck (19.7) foroneexample. Letusjustseewhether itworks forthe rod. Foranaxisthrough oneend, themoment ofinertia should bemL2/3, for wecalculated that. Thecenter ofmass ofarod,ofcourse, isinthecenter ofthe rod, atadistance L/2. Therefore weshould find that ML2/3 =ML2/l2 + M(L/2)2. Since one-quarter plus one-twelfth isone-third, wehave made no fundamental error. Incidentally, wedidnotreally need touseanintegral tofindthemoment of inertia (19.5). Ifwesimply assume thatitisML2 times ‘Y,anunknown coefficient, andthenusetheargument about thetwohalves togetitfor(19.6), thenfrom our argument about transferring theaxeswecould prove that‘Y=it+>1,so‘Y must be1/3. There isalways another waytodoit! Inapplying theparallel-axis theorem, itisofcourse important toremember thattheaxisforI,must beparallel totheaxisabout which themoment ofinertia iswanted. One further property ofthemoment ofinertia isworth mentioning because itisoften helpful infinding themoment ofinertia ofcertain kinds ofobjects. This property isthatifonehasaplane figure andasetofcoordinate axes with origin intheplane andz-axis perpendicular totheplane, then themoment of inertia ofthisfigure about thez-axis isequal tothesumofthemoments ofinertia about thex-andy-axes. This iseasily proved bynoting that 1.=Zm.-of+2%)=Zmy? (since z,-=0).Similarly, n=Zm@+m=Zm% It=Zmt-(X? +yi)=2mi-Xi+Zmty? =I,+1,. Asanexample, themoment ofinertia ofauniform rectangular plate ofmass M,width w,andlength L,about anaxisperpendicular totheplate andthrough its center issimplybut I=M(w2 +L2)/l2, because itsmoment ofinertia about anaxisinitsplane andparallel toitslength isMw2/12, i.e.,justasforarodoflength w,andthemoment ofinertia about the other axisinitsplane isML2/12, justasforarodoflength L. 19-6 Tosummarize, themoment ofinertia ofanobject about agiven axis, which weshall callthez-axis, hasthefollowing properties: (1)Themoment ofinertia is I,= m,(x? +ya-)=f(x2 —l—y2)dm. (2)Iftheobject ismade ofanumber ofparts, each ofwhose moment ofinertia isknown, thetotal moment ofinertia isthesumofthemoments ofinertia ofthepieces. (3)The moment ofinertia about anygiven axisisequal tothemoment of inertia about aparallel axisthrough theCMplus thetotal mass times the square ofthedistance from theaxistotheCM. (4)Iftheobject isaplane figure, themoment ofinertia about anaxisperpendicu- lartotheplane isequal tothesumofthemoments ofinertia about anytwo mutually perpendicular axes lying intheplane and intersecting atthe perpendicular axis. Themoments ofinertia ofanumber ofelementary shapes having uniform mass densities aregiven inTable 19-1, andthemoments ofinertia ofsome other objects, which may bededuced from Table 19-1, using theabove properties, aregiven in Table 19-2. Table 19-1 Objéct z-axis It Thin rod, length L Thin concentric circular ring, radii r1andF2 Sphere, radius r_Lrodatcenter J.ringatcenter through centerML2/12 M(ri +Pi)/2 2Mr2/5 Table 19-2 Object z-axis It Rect. sheet, sides a,b Rect. sheet, sides a,b Thin annular ring, radii r1,rg Rect. parallelepiped, sides a,b,c Rt.circ.cyl.,radius r,length L Rt.circ. cyl., radius r,length L[Ibatcenter J_sheet at center anydiameter ||c,through center ||L,through center _l_L,through centerMa2/12 M(a2 -l-b2)/12 Mtri+rt)/4 M(a2+112)/12 Mrz/2 M(r2/4 +L2/12) 19-4 Rotational kinetic energy Now letusgoontodiscuss dynamics further. Intheanalogy between linear motion andangular motion thatwediscussed inChapter 18,weused thework theorem, butwedidnottalkabout kinetic energy. What isthekinetic energy ofa rigid body, rotating about acertain axiswith anangular velocity w?Wecanim- mediately guess thecorrect answer byusing ouranalogies. Themoment ofinertia corresponds tothemass, angular velocity corresponds tovelocity, andsothe kinetic energy ought tobe%Iw2, andindeed itis,aswillnow bedemonstrated. Suppose theobject isrotating about some axissothateach point hasavelocity whose magnitude iswr,-,where r,-istheradius from theparticular point totheaxis. 19-7 Then ifm,-isthemass ofthatpoint, thetotal kinetic energy ofthewhole thing is justthesumofthekinetic energies ofallofthelittle pieces: T=gzmin?=gZm,~(r,-w)2. Now 0:2isaconstant, thesame forallpoints. Thus T=@122 m,-r?=%Iw2. (19.8) AttheendofChapter 18wepointed outthat there aresome interesting phenomena associated with anobject which isnotrigid, butwhich changes from onerigid condition with adefinite moment ofinertia, toanother rigid condition. Namely, inourexample oftheturntable, wehadacertain moment ofinertia I1 with ourarms stretched out,andacertain angular velocity wl.When wepulled ourarms in,wehadadifferent moment ofinertia, I2,andadifferent angular veloc- ity,(1)2,butagain wewere “rigid.” Theangular momentum remained constant, since there wasnotorque about thevertical axisoftheturntable. This means that Ilwl =120:2. Now what about theenergy? That isaninteresting question. With ourarms pulled in,weturn faster, butourmoment ofinertia isless,andit looks asthough theenergies might beequal. Buttheyarenot,because what does balance isIw,notI092. Soifwecompare thekinetic energy before andafter, the kinetic energy before is51lwf=%Lw1, where L=Ila», =12¢»-2 istheangular momentum. Afterward, bythesame argument, wehave T=%Lw2, andsince L02>(01thekinetic energy ofrotation isgreater than itwasbefore. Sowehada certain energy when ourarms were out,andwhen wepulled them in,wewere turn- ingfaster andhadmore kinetic energy. What happened tothetheorem ofthe conservation ofenergy? Somebody must have done some work. Wedidwork! When didwedoanywork? When wemove aweight horizontally, wedonotdo anywork. Ifwehold athing outandpullitin,wedonotdoanywork. Butthat iswhen wearenotrotating! When wearerotating, there iscentrifugal force on theweights. They aretrying toflyout,sowhen wearegoing around wehave to pulltheweights inagainst thecentrifugal force. So,thework wedoagainst the centrifugal force ought toagree with thedifference inrotational energy, andof course itdoes. That iswhere theextra kinetic energy comes from. There isstillanother interesting feature which wecantreat onlydescriptively, asamatter ofgeneral interest. This feature isalittle more advanced, butisworth pointing outbecause itisquite curious andproduces many interesting effects. Consider thatturntable experiment again. Consider thebody andthearms separately, from thepoint ofview oftheman who isrotating. After theweights arepulled in,thewhole object isspinning faster, butobserve, thecentral part of thebody isnotchanged, yet1tisturning faster after theevent than before. So,if wewere todraw acircle around theinner body, andconsider onlyobjects inside the circle, their angular momentum would change; they aregoing faster. Therefore there must beatorque exerted onthebody while wepullinourarms. Notorque canbeexerted bythecentrifugal force, because that isradial. Sothat means thatamong theforces thataredeveloped inarotating system, centrifugal force is nottheentire story, there isanother force. This other force iscalled Coriolis force, andithasthevery strange property thatwhen wemove something inarotating system, itseems tobepushed sidewise. Like thecentrifugal force, itisanapparent force. Butifweliveinasystem thatisrotating, andmove something radially, we findthatwemust alsopush itsidewise tomove itradially. Thissidewise push which wehave toexert iswhat turned ourbody around. Now letusdevelop aformula toshow how thisCoriolis force really works. Suppose Moe issitting onacarousel thatappears tohimtobestationary. Butfrom thepoint ofview ofJoe,who isstanding ontheground andwhoknows theright laws ofmechanics, thecarousel isgoing around. Suppose thatwehave drawn a radial lineonthecarousel, andthatMoe ismoving some mass radially along this line. Wewould liketodemonstrate thatasidewise force isrequired todothat. Wecandothisbypaying attention totheangular momentum ofthemass. Itis 19-8 always going around with thesame angular velocity w,sothattheangular mo- mentum is L=mvtangr =mwr-r =mwr2. Sowhen themass isclose tothecenter, ithasrelatively little angular momentum, butifwemove ittoanewposition farther out,ifweincrease r,mhasmore angular momentum, soatorque must beexerted inorder tomove italong theradius. (Towalk along theradius inacarousel, onehastoleanover andpush sidewise. Tryitsometime.) Thetorque thatisrequired istherateofchange ofLwith time asmmoves along theradius. Ifmmoves only along theradius, omega stays con- stant, sothatthetorque is 2 'r=F,r= %=-g'5:r)= Zmwrg, where F,istheCoriolis force. What wereally want toknow iswhat sidewiseforce hastobeexerted byMoe inorder tomove moutatspeed v,=dr/dt. This is F,=1'/r=2mwv,. Now thatwehave aformula fortheCoriolis force, letuslook atthesituation alittle more carefully, toseewhether wecanunderstand theorigin ofthisforce from amore elementary point ofview. Wenote thattheCoriolis force isthesame atevery radius, andisevidently present even attheorigin! Butitisespecially easy tounderstand itattheorigin, justbylooking atwhat happens from thein- ertial system ofJoe, who isstanding ontheground. Figure 19-4 shows three successive views ofmjustasitpasses theorigin att=0.Because oftherotation ofthecarousel, weseethatmdoes notmove inastraight line,butinacurved path tangent toadiameter ofthecarousel where r=0.Inorder formtogoinacurve, there must beaforce toaccelerate itinabsolute space. This istheCoriolis force. This isnottheonly case inwhich theCoriolis force occurs. Wecanalso show thatifanobject ismoving with constant speed around thecircumference of acircle, there isalsoaCoriolis force. Why? Moe seesavelocity vMaround the circle. Ontheother hand, Joeseesmgoing around thecircle with thevelocity v;=vM+wr,because misalsocarried bythecarousel. Therefore weknow what theforce really is,namely, thetotal centripetal force duetothevelocity 0],or mi/2}/r; thatistheactual force. Now from Moe’s point ofview, thiscentripetal force hasthree pieces. Wemaywrite italloutasfollows: 2 2mv mvF,= ——r-J—= ——rfi—2mvMw—mw2r. Now, F,istheforce thatMoe would see.Letustrytounderstand it.Would Moe appreciate thefirstterm? “Yes,” hewould say,“even ifIwere notturning, there would beacentripetal force ifIwere torunaround acircle with velocity vM.” This issimply thecentripetal force thatMoe would expect, having/nothing todo with rotation. Inaddition, Moe isquite aware thatthere isanother centripetal force thatwould acteven onobjects which arestanding stillonhiscarousel. This isthethird term. Butthere isanother term inaddition tothese, namely thesecond term, which isagain Zmwv. TheCoriolis force F,wastangential when thevelocity wasradial, andnow itisradial when thevelocity istangential. Infact, oneex- pression hasaminus signrelative totheother. Theforce isalways inthesame direction, relative tothevelocity, nomatter inwhich direction thevelocity is. Theforce isatright angles tothevelocity, andofmagnitude Zmwv. 19-9| 3 I 3 2 2 2 3 I Fig. 19-4. Three successive views of apoint moving radially onarotating turntable. 20 Rotation inspace 20-1 Torques inthree dimensions Inthischapter weshall discuss oneofthemost remarkable andamusing consequences ofmechanics, thebehavior ofarotating wheel. Inorder todothis Wemust first extend themathematical formulation ofrotational motion, the principles ofangular momentum, torque, andsoon,tothree-dimensional space. Weshall notusethese equations inalltheir generality andstudy alltheir conse- quences, because thiswould take many years, andwemust soon turn toother subjects. Inanintroductory course wecanpresent onlythefundamental laws and apply them toavery fewsituations ofspecial interest. First, wenotice thatifwehave arotation inthree dimensions, whether ofa rigid body oranyother system, what wededuced fortwodimensions isstillright. That is,itisstilltrue thatxF,,—yF,,isthetorque “inthexy-plane,” orthe torque “around thez-axis.” Italsoturns outthatthistorque isstillequal totherate ofchange ofxpy—,yp,,,forifwegoback over thederivation ofEq.(18.15) from Newton’s laws weseethatwedidnothave toassume thatthemotion wasina plane; when wedifferentiate xpy-yp,,,wegetxF,,—yF,,, sothistheorem is stillright. Thequantity xpy—yp,,,then, wecalltheangular momentum belonging tothexy-plane, ortheangular momentum about thez-axis. This being true, we canuseanyother pairofaxes andgetanother equation. Forinstance, wecan usetheyz-plane, anditisclear from symmetry thatifwejustsubstitute yforx andzfory,wewould findyF,—zF,,forthetorque andyp,—zp,would bethe angular momentum associated withtheyz-plane. Ofcourse wecould have another plane, thezx-plane, andforthiswewould findzF,,-xF,=d/dt (zp,—xp,). That these three equations canbededuced forthemotion ofasingle particle isquite clear. Furthermore, ifweadded such things asxp,,—yp,together for many particles andcalled itthetotal angular momentum, wewould have three kinds forthethree planes xy,yz,andzx,andifwedidthesame with theforces, wewould talkabout thetorque intheplanes xy,yz,andzxalso. Thus wewould have laws thattheexternal torque associated with anyplane isequal totherate ofchange oftheangular momentum associated with thatplane. This isjusta generalization ofwhat wewrote intwodimensions. Butnow onemay say,“Ah, butthere aremore planes; after all,canwenot takesome other plane atsome angle, andcalculate thetorque onthatplane from theforces? Since wewould have towrite another setofequations forevery such plane, wewould have alotofequations!” Interestingly enough, itturns outthat ifwewere towork outthecombination x’F,,» -y’F,,' foranother plane, measuring thex’,F,/,etc.,inthatplane, theresult canbewritten assome combination ofthe three expressions forthexy-,yz-andzx-planes. There isnothing new. Inother words, ifweknow what thethree torques inthexy-,yz-,andzx-planes are,then thetorque inanyother plane, andcorrespondingly theangular momentum also, canbewritten assome combination ofthese: sixpercent ofoneandninety-two percent ofanother, andsoon.This property weshall nowanalyze. Suppose thatinthexyz-axes, Joehasworked outallhistorques andhisangu- larmomenta inhisplanes. ButMoe hasaxesx’,y’,z’insome other direction. To make italittleeasier, weshall suppose thatonlythex-andy-axes have been turned. Moe’s x’andy’arenew, buthisz’happens tobethesame. That is,hehasnew planes, letussay,foryzandzx.Hetherefore hasnewtorques andangular momenta which hewould work out. Forexample, historque inthex’y’-plane would be equal tox’F,,/ —y'F,,/ andsoforth. What wemust nowdoistofindtherelation- shipbetween thenewtorques andtheoldtorques, sowewillbeabletomake a 20-120-1 Torques inthree dimensions 20-2 Therotation equations using cross products 20-3 Thegyroscope 20-4 Angular momentum ofasolid body connection from onesetofaxestotheother. Someone maysay,“That looks just likewhat wedidwith vectors.” And indeed, thatisexactly what weareintending todo.Then hemay say,“Well, isn’t torque justavector?” ltdoes turn outto beavector, butwedonotknow thatright away without making ananalysis. So inthefollowing steps weshall make theanalysis. Weshall notdiscuss every slep indetail, since weonly want toillustrate how itworks. Thetorques calculated byJoeare A 1,,=xF,,—yF,, Tyz=yF, —zF,,, (20.1) 1,,=zF,,—xF,. Wedigress atthispoint tonote thatinsuch cases asthisonemaygetthewrong sign forsome quantity ifthecoordinates arenothandled intheright way. Why notwrite 11,,=zF,,—yF,? Theproblem arises from thefactthatacoordinate system may be either “right-handed" or“left-handed.” Having chosen (arbitrarily) asign for,say 1,1,,then thecorrect expressions fortheother twoquantities may always befound by interchanging theletters xyzineither order x or x .L\. /_\. Moe nowcalculates thetorques inhissystem: 1./1,’=x’F;.,'-J/Fez 1,41,’ =y’F,' —z'F,/, (20.2) 1,1,,» =z’F,,/ —x’F,t. Now wesuppose thatonecoordinate system isrotated byafixed angle 0,such thatthez-andz’-axes arethesame. (This angle 0hasnothing todowith rotating objects orwhat isgoing oninside thecoordinate system. Itismerely therelation- ship between theaxes used byoneman andtheaxes used bytheother, andis supposedly constant.) Thus thecoordinates ofthetwosystems arerelated by x’=xcos6 +ysin 0, y’=ycos 0—xsin0, (20.3) z’=2. Likewise, because force isavector ittransforms intothenewsystem inthesame wayasdox,y,andz,since athing isavector ifandonlyifthevarious components transform inthesame wayasx,y,andz: F,,/=F,cos0+F,sin0, Fl,’=F,cos19—F,sin0, (20.4) F,»=F2. Now wecanfind outhow thetorque transforms bymerely substituting for x’,y’,andz’theexpressions (20.3), andforF11,Fyl,F,’those given by(20.4), all into(20.2). So,wehave arather longstring ofterms for1,,»,,/ and(rather surpris- ingly atfirst) itturns outthatitcomes right down toxF_,,—yF,,,which werecog- nizetobethetorque inthexy-plane: 1,1,,’ =(xcos0+ysin6)(F,, cos0—F,sin19) —(ycos19—xsin9)(F,, cos0+Fitsin0) =xF,,(cos2 0+sin26)—yF,(sin2 6—l—cos” 6) +xF,(—sin 6cos 0+sin0cos 6) +yFy(sin 0cos0—sin0cos B) =xFy —yF,, =1,3,. (20.5) 20-2 That result isclear, forifweonly turn ouraxes intheplane, thetwist around z inthatplane isnodifferent than itwasbefore, because itisthesame plane! What willbemore interesting istheexpression for1,»,», because thatisanewplane. Wenowdoexactly thesame thing withthey’z’-plane, anditcomes outasfollows: 1,1,» =(ycos0—xsin0)F, —z(F, cos6-F,sin0) =(yF, -zF,)cos0+(zF, —xF,,)sin0 =1,,cos0+1,,sin0. (20.6) Finally, wedoitforz’x': 1,/,1 =z(F,,cos0+F,sin0) -(xcos0—l—ysin19)F, =(zF, —xF,)cos9—(yF, —zF,)sin0 =12,,cos0—1,,sin0. (20.7) Wewanted togetaruleforfinding torques innewaxesinterms oftorques inoldaxes, andnow wehave therule. How canweever remember thatrule? Ifwelook carefully at(20.5), (20.6), and(20.7), weseethatthere isaclose relation- shipbetween these equations andtheequations forx,y,andz.If,somehow, we could call1,,thez-component ofsomething, letuscallitthez-component of1, thenitwould beallright; wewould understand (20.5) asavector transformation, since thez-component would beunchanged, asitshould be. Likewise, ifwe associate with theyz-plane thex-component ofournewly invected vector, and withthezx-plane, they-component, then these transformation expressions would read T2’ :T29 1,,»=1,cos0+1,sin0, (20.8) 1,1=1,cos0-1,sin6, which isjusttheruleforvectors! Therefore wehave proved thatwemayidentify thecombination ofxF,—yF, with what weordinarily callthez-component ofacertain artificially invented vector. Although atorque isatwist onaplane, andithasnoapriori vector char- acter, mathematically itdoes behave likeavector. This vector isatright angles to theplane ofthetwist, anditslength isproportional tothestrength ofthetwist. The three components ofsuch aquantity willtransform likearealvector. Sowerepresent torques byvectors; with each plane onwhich thetorque is supposed tobeacting, weassociate alineatright angles, byarule. But“atright angles” leaves thesignunspecified. Togetthesignright, wemust adopt arule which willtellusthatifthetorque were inacertain sense onthexy-plane, then theaxisthatwewant toassociate with itisinthe“up” z-direction. That is,some- body hastodefine “right” and“left” forus.Supposing thatthecoordinate system isx,y,zinaright-hand system, then therulewillbethefollowing: ifwethink of thetwist asifwewere turning ascrew having aright-hand thread, thenthedirection ofthevector thatwewillassociate with thattwist isinthedirection thatthescrew would advance. Why istorque avector? Itisamiracle ofgood luck thatwecanassociate a single axiswith aplane, andtherefore thatwecanassociate avector with the torque; itisaspecial property ofthree-dimensional space. Intwodimensions, the torque isanordinary scalar, andthere need benodirection associated with it. Inthree dimensions, itisavector. Ifwehadfourdimensions, wewould beingreat difficulty, because (ifwehadtime, forexample, asthefourth dimension) wewould notonlyhave planes likexy,yz,andzx,wewould alsohave tx-,ty-,andtz-planes. There would besixofthem, andonecannot represent sixquantities asonevector infourdimensions. Wewillbeliving inthree dimensions foralong time, soitiswelltonotice thattheforegoing mathematical treatment didnotdepend upon thefactthatx 20-3 wasposition andFwasforce; itonly depended onthetransformation laws for vectors. Therefore if,instead ofx,weused thex-component ofsome other vector, itisnotgoing tomake anydifference. Inother words, ifwewere tocalculate a,b, —a,b,, where aandbarevectors, andcallitthez-component ofsome new quantity c,then these newquantities form avector c.Weneed amathematical notation fortherelationship ofthenewvector, with itsthree components, tothe vectors aandb.Thenotation thathasbeen devised forthisisc=aXb.We have then, inaddition totheordinary scalar product inthetheory ofvector analysis, anewkind ofproduct, called thevector product. Thus, ifc=aXb, thisisthesame aswriting c,=a,b, —a,b,, c,=azb, —a,,b,, (20.9) c,=azb, -a,b,. Ifwereverse theorder ofaandb,calling a,bandb,a,wewould have thesign ofcreversed, because c,would beb,a, —b,a,. Therefore thecross product is unlike ordinary multiplication, where ab=ba;forthecross product, bXa= —aXb.From this, wecanprove atonce thatifa=b,thecross product is zero. Thus, aXa=0. Thecross product isveryimportant forrepresenting thefeatures ofrotation, anditisimportant thatweunderstand thegeometrical relationship ofthethree vectors a,b,andc.Ofcourse therelationship incomponents isgiven inEq.(20.9) andfrom thatonecandetermine what therelationship isingeometry. Theanswer is,first, thatthevector cisperpendicular toboth aandb.(Try tocalculate c-a, andseeifitdoes notreduce tozero.) Second, themagnitude ofcturns outtobe themagnitude ofatimes themagnitude ofbtimes thesineoftheangle between thetwo. Inwhich direction does cpoint? Imagine thatweturnaintobthrough anangle lessthan 180°; ascrew with aright-hand thread turning inthiswaywill advance inthedirection ofc.Thefactthatwesayaright-hand screw instead ofa left-hand screw isaconvention, andisaperpetual reminder thatifaandbare “honest” vectors intheordinary sense, thenewkind of“vector” which wehave created byaXbisartificial, orslightly different initscharacter from aandb, because itwasmade upwith aspecial rule. Ifaandbarecalled ordinary vectors, wehave aspecial name forthem, wecallthem polar vectors. Examples ofsuch vectors arethecoordinate r,force F,momentum p,velocity v,electric fieldE,etc.; these areordinary polar vectors. Vectors which involve justonecross product in their definition arecalled axial vectors orpseudovectors. Examples ofpseudovectors are,ofcourse, torque 1andtheangular momentum L.Italsoturns outthatthe angular velocity atisapseudovector, asisthemagnetic fieldB. Inorder tocomplete themathematical properties ofvectors, weshould know alltherules fortheir multiplication, using dotandcross products. Inourapplica- tions atthemoment, wewillneed verylittle ofthis,butforthesakeofcompleteness weshall write down alloftherules forvector multiplication sothatwecanuse theresults later. These are (a) aX(b+c)= (b) (9) (<1) (e) (f)(aa)Xb a-(bXc) aX(bXc) aXa a-(aXb)aXb+aXc, a(a Xb), (aXb)-c, (20.10) b(a-c)—c(a-b), 0, 0. 20-2 Therotation equations using cross products Now letusaskwhether anyequations inphysics canbewritten using the cross product. Theanswer, ofcourse, isthatagreat many equations canbeso written. Forinstance, weseeimmediately thatthetorque isequal totheposition 20-4 vector cross theforce: 1=rXF. (20.11) This isavector summary ofthethree equations 1,=yF,-zF,, etc. Bythe same token, theangular momentum vector, ifthere isonly oneparticle present, isthedistance from theorigin multiplied bythevector momentum: L=r><p. (20.12) Forthree-dimensional space rotation, thedynamical lawanalogous tothelaw F=dp/dt ofNewton, isthatthetorque vector istherateofchange with time of theangular momentum vector: 1=dL/dt. (20.13) Ifwesum(20.13) over many particles, theexternal torque onasystem istherate ofchange ofthetotal angular momentum: Text =dLto(-,/dt. Another theorem: Ifthetotal external torque iszero, then thetotal vector angular momentum ofthesystem isaconstant. This iscalled thelawofconserva- tionofangular momentum. Ifthere isnotorque onagiven system, itsangular momentum cannot change. What about angular velocity? Isitavector? Wehave already discussed turning asolid object about afixed axis, butforamoment suppose thatweare turning itsimultaneously about twoaxes. Itmight beturning about anaxisinside abox, while theboxisturning about some other axis. Thenetresult ofsuch combined motions isthat theobject simply turns about some new axis! The wonderful thing about thisnewaxisisthatitcanbefigured outthisway. Ifthe rateofturning inthexy-plane iswritten asavector inthez-direction whose length isequal totherateofrotation intheplane, andifanother vector isdrawn inthe y-direction, say,which istherateofrotation inthezx-plane, then ifweaddthese together asavector, themagnitude oftheresult tellsushow fasttheobject is turning, andthedirection tellsusinwhat plane, bytheruleoftheparallelogram. That istosay,simply, angular velocity isavector, where wedraw themagnitudes oftherotations inthethree planes asprojections atright angles tothose planes.* Asasimple application oftheuseoftheangular velocity vector, wemayevalu- atethepower being expended bythetorque acting onarigid body. Thepower, of course, istherateofchange ofwork with time; inthree dimensions, thepower turns outtobeP=1-co. Alltheformulas thatwewrote forplane rotation canbegeneralized tothree dimensions. Forexample, ifarigid body isturning about acertain axiswith angular velocity w,wemight ask, “What isthevelocity ofapoint atacertain radial position r?”Weshall leave itasaproblem forthestudent toshow thatthe velocity ofaparticle inarigid body isgiven byv=wXr,where wistheangular velocity andristheposition. Also, asanother example ofcross products, wehada formula forCoriolis force, which canalso bewritten using cross products: F,=2mv Xw.That is,ifaparticle ismoving with velocity vinacoordinate system which is,infact, rotating with angular velocity w,andwewant tothink in terms oftherotating coordinate system, then wehave toaddthepseudoforce F,. 20-3 Thegyroscope Letusnow return tothelawofconservation ofangular momentum. This lawmaybedemonstrated with arapidly spinning wheel, orgyroscope, asfollows (seeFig.20-1). Ifwesitonaswivel chair andhold thespinning wheel with its axishorizontal, thewheel hasanangular momentum about thehorizontal axis. *That thisistruecanbederived bycompounding thedisplacements oftheparticles ofthebody during aninfinitesimal timeAt.Itisnotself-evident, andislefttothose who areinterested totrytofigure itout. 20-5ut, t§?".i/ 9 .1 rj. 4-it‘ _ *1, , ,BEFORE AFTER Fig. 20-1. Before: axis ishorizontal; moment about vertical axis =0.After: axis isvertical; momentum about vertical axis isstillzero; man and chair spin in direction opposite tospinofthewheel. 2 -I.W1 § .- A-1: F F at,-X Lo Y Q?Fig.20-2. Agyroscope. \d> Fl ‘F Fig.20-3. Arapidly spinning top. Note that thedirection ofthetorque vector isthedirection oftheprecession.bi /LATER /// ’/ /K? NOW \\\ \ \ v’ \EARLIER\/\/\/ I~\/A /.‘*\1*’3Ti"7V1 Fig. 20-4. Themotion ofparticles in thespinning wheel ofFig.20-2, whose axis isturning, isincurved lines.Angular momentum around avertical axiscannot change because ofthe(friction- less)pivot ofthechair, soifweturntheaxisofthewheel intothevertical, thenthe wheel would have angular momentum about thevertical axis, because itisnow spinning about thisaxis. Butthesystem (wheel, ourself, andchair) cannot have a vertical component, soweandthechair have toturn inthedirection opposite tothespinofthewheel, tobalance it. First letusanalyze inmore detail thething wehave justdescribed. What is surprising, andwhat wemust understand, istheorigin oftheforces which turn usandthechair around asweturntheaxisofthegyroscope toward thevertical. Figure 20-2 shows thewheel spinning rapidly about they-axis. Therefore its angular velocity isabout thataxisand, itturns out,itsangular momentum islike- wise inthatdirection. Now suppose thatwewish torotate thewheel about the x-axis atasmall angular velocity S2;what forces arerequired? After ashort time At,theaxishasturned toanewposition, tilted atanangle A0with thehorizontal. Since themajor partoftheangular momentum isduetothespinontheaxis(very little iscontributed bytheslow turning), weseethattheangular momentum vector haschanged. What isthechange inangular momentum? Theangular momentum does notchange inmagnitude, butitdoes change indirection byanamount A0. Themagnitude ofthevector AListhus AL=L0A0,sothatthetorque, which is thetime rateofchange oftheangular momentum, is1=AL/At =L0A0/At = L09. Taking thedirections ofthevarious quantities intoaccount, weseethat ¢=oxm. ems Thus, if£2andL0areboth horizontal, asshown inthefigure, 1-isvertical. To produce such atorque, horizontal forces Fand—Fmust beapplied attheends of theaxle. How arethese forces applied? Byourhands, aswetrytorotate the axisofthewheel intothevertical direction. ButNewton’s Third Law demands thatequal andopposite forces (and equal andopposite torques) actonus.This causes ustorotate intheopposite sense about thevertical axisz. This result canbegeneralized forarapidly spinning top. Inthefamiliar case ofaspinning top,gravity acting onitscenter ofmass furnishes atorque about the point ofcontact with thefloor (seeFig.20-3). This torque isinthehorizontal direction, andcauses thetoptoprecess withitsaxismoving inacircular cone about thevertical. IfQisthe(vertical) angular velocity ofprecession, weagain findthat Thus, when weapply atorque toarapidly spinning top, thedirection ofthe precessional motion isinthedirection ofthetorque, oratright angles tothe forces producing thetorque. Wemay now claim tounderstand theprecession ofgyroscopes, andindeed wedo,mathematically. However, thisisamathematical thing which, inasense, appears asa“miracle.” Itwillturn out,aswegotomore andmore advanced physics, that many simple things canbededuced mathematically more rapidly than they canbereally understood inafundamental orsimple sense. This isa strange characteristic, andaswegetintomore andmore advanced work there are circumstances inwhich mathematics willproduce results which noonehasreally been abletounderstand inanydirect fashion. Anexample istheDirac equation, which appears inavery simple andbeautiful form, butwhose consequences are hard tounderstand. Inourparticular case, theprecession ofatoplooks likesome kind ofamiracle involving right angles andcircles, andtwists andright-hand screws. What weshould trytodoistounderstand itinamore physical way. How canweexplain thetorque interms oftherealforces andtheaccelerations? Wenote thatwhen thewheel isprecessing, theparticles thataregoing around the wheel arenotreally moving inaplane because thewheel isprecessing (seeFig. 20-4). Asweexplained previously (Fig. 19-4), theparticles which arecrossing through theprecession axisaremoving incurved paths, andthisrequires application ofalateral force. This issupplied byourpushing ontheaxle, which then com- 20-6 municates theforce totherimthrough thespokes. “Wait,” someone says, “what about theparticles thataregoing back ontheother side?” Itdoes nottakelong todecide thatthere must beaforce intheopposite direction onthatside. Thenet force thatwehave toapply istherefore zero. Theforces balance out,butoneof them must beapplied atonesideofthewheel, andtheother must beapplied atthe other sideofthewheel. Wecould apply these forces directly, butbecause thewheel issolid weareallowed todoitbypushing ontheaxle, since forces canbecarried upthrough thespokes. What wehave sofarproved isthatifthewheel isprecessing, itcanbalance thetorque duetogravity orsome other applied torque. Butallwehave shown is thatthisisasolution ofanequation. That is,ifthetorque isgiven, andifwe get thespinning started right, then thewheel willprecess smoothly anduniformly. Butwehave notproved (and itisnottrue) thatauniform precession isthemost general motion aspinning body canundergo astheresult ofagiven torque. The general motion involves also a“wobbling” about themean precession. This “wobbling” iscalled nutation. Some people liketosaythatwhen oneexerts atorque onagyroscope, itturns anditprecesses, andthatthetorque produces theprecession. Itisverystrange that when onesuddenly letsgoofagyroscope, itdoesnotfallunder theaction ofgravity, butmoves sidewise instead! Why isitthatthedownward force ofthegravity, which weknow andfeel, makes itgosidewise? Alltheformulas intheworld like(20.15) arenotgoing totellus,because (20.15) isaspecial equation, valid only after the gyroscope isprecessing nicely. What really happens, indetail, isthefollowing. Ifwewere tohold theaxisabsolutely fixed, sothatitcannot precess inanymanner (but thetopisspinning) then there isnotorque acting, noteven atorque from gravity, because itisbalanced byourfingers. Butifwesuddenly letgo,thenthere willinstantaneously beatorque from gravity. Anyone inhisright mind would think that thetopwould fall, andthat iswhat itstarts todo,ascanbeseen ifthe topisnotspinning toofast. Thegyro actually does fall,aswewould expect. Butassoon asitfalls, itis then turning, andifthisturning were tocontinue, atorque would berequired. Intheabsence ofatorque inthisdirection, thegyro begins to“fall” inthedirection opposite thatofthemissing force. This gives thegyro acomponent ofmotion around thevertical axis,asitwould have insteady precession. Buttheactual motion “overshoots” thesteady precessional velocity, andtheaxisactually rises again to thelevel from which itstarted. Thepath followed bytheendoftheaxleisacycloid (thepath followed byapebble thatisstuck inthetread. ofanautomobile tire). Ordinarily, thismotion istooquick fortheeyetofollow, anditdamps outquickly because ofthefriction inthegimbal bearings, leaving only thesteady preces- sional drift (Fig. 20-5). Theslower thewheel spins, themore obvious thenu- tation is. When themotion settles down, theaxisofthegyro isalittle bitlower than it wasatthestart. Why? (These arethemore complicated details, butwebring them inbecause wedonotwant thereader togettheideathatthegyroscope isanabso- lutemiracle. ltisawonderful thing, butitisnotamiracle.) Ifwewere holding theaxisabsolutely horizontally, andsuddenly letgo,then thesimple precession equation would tellusthatitprecesses, thatitgoes around inahorizontal plane. Butthatisimpossible! Although weneglected itbefore, itistruethatthewheel has some moment ofinertia about theprecession axis, andifitismoving about that axis, even slowly, ithasaweak angular momentum about theaxis. Where didit come from? Ifthepivots areperfect, there isnotorque about thevertical axis. How then does itgettoprecess ifthere isnochange intheangular momentum? Theanswer isthatthecycloidal motion oftheendoftheaxisdamps down tothe average, steady motion ofthecenter oftheequivalent rolling circle. That is,itset- tlesdown alittle bitlow. Because itislow,thespinangular momentum now has asmall vertical component, which isexactly what isneeded fortheprecession. So youseeithastogodown alittle, inorder togoaround. Ithastoyield alittle bit tothegravity; byturning itsaxisdown alittle bit,itmaintains therotation about thevertical axis. That, then, isthewayagyroscope works. [email protected]* Fig. 20-5. Actual motion oftipof axis ofgyroscope under gravity just after releasing axispreviously held fixed. L,II,w,“H \‘ E Ll /- L,-I,w cu "\ ~\\\\\\\\\\\.~'8 Fig.20-6. Theangular momentum of arotating body isnotnecessarily parallel totheangular velocity. Z \//ml~\_\ \ >.F____A %-1I ‘wI.I Q%'\\\“E_.___K_7|__..E:\J!_____§./W5 I //fix X Fig.20-7. Theangular velocity and angular momentum ofarigid body (A>B>C).20-4 Angular momentum ofasolid body Before weleave thesubject ofrotations inthree dimensions, weshall discuss, atleast qualitatively, afeweffects thatoccur inthree-dimensional rotations thatare notself-evident. Themain effect isthat, ingeneral, theangular momentum ofa rigid body isnotnecessarily inthesame direction astheangular velocity. Consider awheel thatisfastened onto ashaft inalopsided fashion, butwiththeaxisthrough thecenter ofgravity, tobesure(Fig. 20-6). When wespinthewheel around the axis, anybody knows that there willbeshaking atthebearings because ofthe lopsided waywehave itmounted. Qualitatively, weknow thatintherotating system there iscentrifugal force acting onthewheel, trying tothrow itsmass as faraspossible from theaxis. Thistends tolineuptheplane ofthewheel sothatit isperpendicular totheaxis. Toresist thistendency, atorque isexerted bythe bearings. Ifthere isatorque exerted bythebearings, there must bearateofchange ofangular momentum. How canthere bearateofchange ofangular momentum when wearesimply turning thewheel about theaxis? Suppose webreak the angular velocity 0.»intocomponents 401and(.02perpendicular andparallel tothe plane ofthewheel. What istheangular momentum? Themoments ofinertia about these twoaxes aredifferent, sotheangular momentum components, which (inthese particular, special axesonly) areequal tothemoments ofinertia times the corresponding angular velocity components, areinadififerent ratio than arethe angular velocity components. Therefore theangular momentum vector isina direction inspace notalong theaxis. When weturntheobject, wehave toturnthe angular momentum vector inspace, sowemust exert torques ontheshaft. Although itismuch toocomplicated toprove here, there isaveryimportant andinteresting property ofthemoment ofinertia which iseasytodescribe andto use,andwhich isthebasis ofourabove analysis. This property isthefollowing: Any rigid body, even anirregular onelikeapotato, possesses three mutually perpendicular axesthrough theCM, such thatthemoment ofinertia about oneof these axeshasthegreatest possible value foranyaxisthrough theCM, themoment ofinertia about another oftheaxes hastheminimum possible value, andthe moment ofinertia about thethird axisisintermediate between these two(orequal tooneofthem). These axesarecalled theprincipal axesofthebody, andtheyhave theimportant property thatifthebody isrotating about oneofthem, itsangular momentum isinthesame direction astheangular velocity. Forabody having axes ofsymmetry, theprincipal axesarealong thesymmetry axes. Ifwetake thex-,y-,andz-axes along theprincipal axes, andcallthecorre- sponding principal moments ofinertia A,B,andC,wemay easily evaluate the angular momentum andthekinetic energy ofrotation ofthebody foranyangular velocity w.Ifweresolve wintocomponents cox,(Dy,andav,along thex-,y-,z-axes, anduseunitvectors i,j,k,alsoalong x,y,z,wemaywrite theangular momentum as L=Awxi+Bwyj+Cwzk. (20.16) Thekinetic energy ofrotation is KE=%(/mi+Ba;+cwi) (20.17) =-%L'w. 20-8 21 The Harmonic llscillutor 21-1 Linear differential equations Inthestudy ofphysics, usually thecourse isdivided intoaseries ofsubjects, such asmechanics, electricity, optics, etc., andonestudies onesubject after the other. Forexample, thiscourse hassofardealt mostly with mechanics. Buta strange thing occurs again andagain: theequations which appear indifferent fields ofphysics, andeven inother sciences, areoften almost exactly thesame, so thatmany phenomena have analogs inthese different fields. Totake thesimplest example, thepropagation ofsound waves isinmany ways analogous tothepropaga- tionoflight waves. Ifwestudy acoustics ingreat detail wediscover thatmuch of thework isthesame asitwould beifwewere studying optics ingreat detail. So thestudy ofaphenomenon inonefield maypermit anextension ofourknowledge inanother field. Itisbesttorealize from thefirstthatsuch extensions arepossible, forotherwise onemight notunderstand thereason forspending agreat deal of time andenergy onwhat appears tobeonly asmall partofmechanics. Theharmonic oscillator, which weareabout tostudy, hasclose analogs in many other fields; although westart with amechanical example ofaweight ona spring, orapendulum with asmall swing, orcertain other mechanical devices, we arereally studying acertain differential equation. This equation appears again andagain inphysics andinother sciences, andinfactitisapart ofsomany phenomena thatitsclose study iswellworth ourwhile. Some ofthephenomena involving thisequation aretheoscillations ofamass onaspring; theoscillations ofcharge flowing back andforth inanelectrical circuit; thevibrations ofatuning fork which isgenerating sound waves; theanalogous vibrations oftheelectrons inanatom, which generate light waves; theequations fortheoperation ofa servosystem, such asathermostat trying toadjust atemperature; complicated interactions inchemical reactions; thegrowth ofacolony ofbacteria ininteraction with thefood supply andthepoisons thebacteria produce; foxes eating rabbits eating grass, andsoon;allthese phenomena follow equations which arevery similar tooneanother, andthisisthereason whylwe study themechanical oscillator insuch detail. Theequations arecalled linear dzflerential equations withconstant coefiicients. Alinear differential equation withconstant coefficients isadifferential equation consisting ofasumofseveral terms, each term being aderivative ofthe dependent variable with respect totheindependent variable, andmultiplied by some constant. Thus ana'"x/dt” +a,,_,d""‘x/d1"—‘ + +<1,dx/dt+aox=fa) (21.1) iscalled alinear difi'erential equation oforder nwith constant coefficients (each a,-isconstant). Perhaps thesimplest mechanical system whose motion follows alinear differ- ential equation with constant coefficients isamass onaspring: firstthespring stretches tobalance thegravity; once itisbalanced, wethen discuss thevertical displacement ofthemass from itsequilibrium position (Fig. 21-1). Weshall call thisupward displacement x,andweshall alsosuppose thatthespring isperfectly linear, inwhich case theforce pulling back when thespring isstretched ispre- cisely proportional totheamount ofstretch. That is,theforce is—kx (with a 21-121-1 Linear differential equations 21-2 Theharmonic oscillator 21-3 Harmonic motion andcircular motion 21-4 Initial conditions 21-5 Forced oscillations O 0 O —r_- I L__ Fig. 2'l—l. Ama4 L__| ssonaspring a 21-2 Theharmonic oscillator simple example ofaharmonic oscillator minus signtoremind usthatitpulls back). Thus themass times theacceleration must equal —~kx: mdzx/dtz =—kx. (21.2) Forsimplicity, suppose ithappens (orwechange ourunitoftime measurement) thattheratio k/m =l.Weshall firststudy theequation d2x/dt2 =——x. (21.3) Later weshall come back toEq.(21.2) with thekandmexplicitly present. Wehave already analyzed Eq.(21.3) indetail numerically; when wefirst introduced thesubject ofmechanics wesolved thisequation (seeEq.9.12) tofind themotion. Bynumerical integration wefound acurve (Fig. 9-4) which showed thatifmwasinitially displaced, butatrest, itwould come down andgothrough zero; wedidnotthen follow itanyfarther, butofcourse weknow thatitjust keeps going upanddown——it oscillates. When wecalculated themotion numer~ ically, wefound thatitwent through theequilibrium point att=1.570. The length ofthewhole cycle isfour times thislong, orto=6.28 “sec.” This was found numerically, before weknew much calculus. Weassume thatinthemeantime theMathematics Department hasbrought forth afunction which, when differ- entiated twice, isequal toitself with aminus sign. (There are,ofcourse, ways of getting atthisfunction inadirect fashion, butthey aremore complicated than already knowing what theanswer is.)Thefunction isx=cost.Ifwedifferentiate thiswefinddx/dt =—sint anddzx/dtz =—cost =—x. The function x= coststarts, att=0,with x=1,andnoinitial velocity; thatwasthesituation with which westarted when wedidournumerical work. Now thatweknow that x=cost,wecancalculate aprecise value forthetime atwhich itshould pass x=0.Theanswer ist=1r/2, or1.57108. Wewere wrong inthelastfigure because oftheerrors ofnumerical analysis, butitwasveryclose! Now togofurther with theoriginal problem, werestore thetime units to realseconds. What isthesolution then? First ofall,wemight think thatwecan gettheconstants kandminbymultiplying costbysomething. Soletustrythe equation x=Acost;then wefind dx/dt =—Asint,and dzx/dt2 =—A cost=—x. Thus wediscover toourhorror thatwedidnotsucceed insolving Eq.(21.2), butwegotEq.(21.3) again! That factillustrates oneofthemost important properties oflinear differential equations: ifwemultiply asolution of theequation byanyconstant, itisagain asolution. Themathematical reason for thisisclear. Ifxisasolution, andwemultiply both sides oftheequation, saybyA, weseethatallderivatives arealsomultiplied byA,andtherefore Axisjustasgood asolution oftheoriginal equation asxwas. Thephysics ofitisthefollowing. Ifwehave aweight onaspring, andpullitdown twice asfar,theforce istwice asmuch, theresulting acceleration istwice asgreat, thevelocity itacquires ina given time istwice asgreat, thedistance covered inagiven time istwice asgreat; butithastocover twice asgreat adistance inorder togetback totheorigin because itispulled down twice asfar.Soittakes thesame timetogetback tothe origin, irrespective oftheinitial displacement. Inother words, with alinear equa- tion, themotion hasthesame timepattern, nomatter how “strong” itis. That wasthewrong thing todo—it only taught usthatwecanmultiply the solution byanything, anditsatisfies thesame equation, butnotadifferent equation. After alittle cutandtrytogettoanequation with adifferent constant multiplying x,wefindthatwemust alter thescale oftime. Inother words, Eq.(21.2) hasa solution oftheform x=coswot. (21.4) (Itisimportant torealize thatinthepresent case, weisnotanangular velocity ofa spinning body, butwerunoutofletters ifwearenotallowed tousethesame letter formore than onething.) Thereason weputasubscript “O”oncoisthatweare going tohave more omegas before long; letusremember thatworefers tothe natural motion ofthisoscillator. Now wetryEq.(21.4) andthistime wearemore successful, because dx/dt =—w0 sinwotandd2x/dt2 =—w§ coswot=—w§x. 21-2 Soatlastwehave solved theequation thatwereally wanted tosolve. Theequation d2x/dt2 =—w§x isthesame asEq.(21.2) ifwg=k/m. Thenext thing wemust investigate isthephysical significance ofwo. We know thatthecosine function repeats itself when theangle itrefers tois21r.So x=coswotwillrepeat itsmotion, itwillgothrough acomplete cycle, when the “angle” changes by21r.Thequantity wotisoften called thephase ofthemotion. Inorder tochange wotby21r,thetime must change byanamount to,called the period ofonecomplete oscillation; ofcourse tomust besuch thatwoto =21r. That is,woto must account foronecycle oftheangle, andthen everything will repeat itself—if weincrease tbyto,weadd21rtothephase. Thus to=21r/wo =21r\/ m/k. (21.5) Thus ifwehadaheavier mass, itwould takelonger tooscillate back andforth on aspring. That isbecause ithasmore inertia, andso,while theforces arethesame, ittakes longer togetthemass moving. Or,ifthespring isstronger, itwillmove more quickly, andthatisright: theperiod islessifthespring isstronger. Note thattheperiod ofoscillation ofamass onaspring does notdepend in anywayonhowithasbeen started, how fardown wepullit.Theperiod isdeter- mined, buttheamplitude oftheoscillation isnotdetermined bytheequation of motion (21.2). Theamplitude isdetermined, infact, byhow weletgoofit,by what wecalltheinitial conditions orstarting conditions. Actually, wehave notquite found themost general possible solution ofEq. (21.2). There areother solutions. Itshould beclear why: because allofthecases covered byx=acoswot‘start with aninitial displacement andnoinitial velocity. Butitispossible, forinstance, forthemass tostart atx=0,andwemay then giveitanimpulsive kick, sothatithassome speed att=0.Such amotion isnot represented byacosine—it isrepresented byasine. Toputitanother way, if x=coswotisasolution, then isitnotobvious thatifwewere tohappen towalk intotheroom atsome time (which wewould call“t=0”)andsawthemass as itwaspassing x=0,itwould keep ongoing justthesame? Therefore, x=cos wotcannot bethemost general solution; itmust bepossible toshift thebeginning oftime, sotospeak. Asanexample, wecould write thesolution thisway: x= acoswo(t —t1),where t1issome constant. This alsocorresponds toshifting the origin oftimetosome newinstant. Furthermore, wemayexpand cos(wot +A)=coswotcosA —sinwotsinA, andwrite x=Acoswot-1-Bsin wot, where A=acosAandB=—asinA.Any oneofthese forms isapossible waytowrite thecomplete, general solution of(21.2): thatis,every solution ofthe differential equation dzx/dtz =—w§x thatexists intheworld canbewritten as (a) x=acos wo(t —t1), or (b) x=acos (wot +A), (21.6) or (c) x=Acoswot+Bsinwot. Some ofthequantities in(21.6) have names: woiscalled theangular frequency; itisthenumber ofradians bywhich thephase changes inasecond. That isdeter- mined bythedifferential equation. Theother constants arenotdetermined bythe equation, butbyhow themotion isstarted. Ofthese constants, ameasures the maximum displacement attained bythemass, andiscalled theamplitude ofoscilla- tion. Theconstant Aissometimes called thephase oftheoscillation, butthatisa confusion, because other people callwot+Athephase, andsaythephase changes with time. Wemight saythatAisaphase shift from some defined zero. Letusput itdifferently. Different A’scorrespond tomotions indifferent phases. That is true, butwhether wewant tocallAthephase, ornot,isanother question. 21-3 .s ‘V’ X Fig. 21-2. Aparticle moving ina circular path atconstant speed. J > 2A ' Light / /. ___i9_ of flan.ii?_)_ Projector Shadowsit-A. ) Screen Fig. 21-3. Demonstration of the equivalence between simple harmonic motion anduniform circular motion.21-3 Harmonic motion andcircular motion Thefactthatcosines areinvolved inthesolution ofEq.(21.2) suggests that there might besome relationship tocircles. This isartificial, ofcourse, because there isnocircle actually involved inthelinear motion—it justgoes upanddown. Wemay point outthatwehave, infact, already solved thatdifferential equation when wewere studying themechanics ofcircular motion. Ifaparticle moves ina circle with aconstant speed v,theradius vector from thecenter ofthecircle tothe particle turns through anangle whose sizeisproportional tothetime. Ifwecall thisangle 0=vt/R (Fig. 21-2) thend0/dt =wo=v/R. Weknow thatthere is anacceleration a=112/R =w§Rtoward thecenter. Now wealsoknow thatthe position x,atagiven moment, istheradius ofthecircle times cos0,andthatyis theradius times sin0: x=Rcos0, y=Rsin0. Now what about theacceleration? What isthex-component ofacceleration, dzx/dtg? Wehave already worked thatoutgeometrically; itisthemagnitude of theacceleration times thecosine oftheprojection angle, with aminus signbecause itistoward thecenter. ax=—acos 0=—w2R cos0=—w2x. (21.7) Inother words, when aparticle ismoving inacircle, thehorizontal component of itsmotion hasanacceleration which isproportional tothehorizontal displacement from thecenter. Ofcourse wealso have thesolution formotion inacircle: x=Rcoswot. Equation (21.7) does notdepend upon theradius ofthecircle, soforacircle ofanyradius, onefinds thesame equation foragiven wo.Thus, forseveral reasons, weexpect thatthedisplacement ofamass onaspring willturn outtobeproportional tocoswot,andwill,infact, beexactly thesame motion as wewould seeifwelooked atthex-component oftheposition ofanobject rotating inacircle with angular velocity wo.Asacheck onthis,onecandevise anexperi- ment toshow thattheup-and-down motion ofamass onaspring isthesame as thatofapoint going around inacircle. InFig.21-3 anarclight projected ona screen casts shadows ofacrank pinonashaft andofavertically oscillating mass, sidebyside. Ifweletgoofthemass attheright time from theright place, andif theshaft speed iscarefully adjusted sothatthefrequencies match, each should follow theother exactly. Onecanalsocheck thenumerical solution weobtained earlier with thecosine function, andseewhether thatagrees very well. Here wemaypoint outthatbecause uniform motion inacircle issoclosely related mathematically tooscillatory up-and-down motion, wecan analyze oscillatory motion inasimpler wayifweimagine ittobeaprojection ofsomething going inacircle. Inother words, although thedistance ymeans nothing inthe oscillator problem, wemay stillartificially supplement Eq.(21.2) with another equation using y,andputthetwotogether. Ifwedothis,wewillbeabletoanalyze ourone-dimensional oscillator with circular motions, which isaloteasier than having tosolve adifferential equation. Thetrick indoing thisistousecomplex numbers, aprocedure weshall introduce inthenextchapter. 21-4 Initial conditions Now letusconsider what determines theconstants AandB,oraandA.Of course these aredetermined byhow westart themotion. Ifwestart themotion withjustasmall displacement, thatisonetype ofoscillation; ifwestart with an initial displacement andthen push upwhen weletgo,wegetstilladifferent motion. Theconstants AandB,oraandA,oranyother wayofputting it,are determined, ofcourse, bythewaythemotion started, notbyanyother features of thesituation. These arecalled theinitial conditions. Wewould liketoconnect the initial conditions with theconstants. Although thiscanbedone using anyone oftheforms (21.6), itturns outtobeeasiest ifweuseEq.(2l.6c). Suppose thatat t=0wehave started with aninitial displacement xoandacertain velocity vo. 21-4 This isthemost general waywecanstart themotion. (We cannot specify the acceleration with which itstarted, true, because thatisdetermined bythespring, once wespecify xo.) Now letuscalculate AandB.Westart with theequation forx, x=Acoswot+Bsin wot. Since weshall later need thevelocity also, wedifferentiate xandobtain v=—woA sinwot+woBcoswot. These expressions arevalid forallt,butwehave special knowledge about xand vatt=0.Soifweputt=0intothese equations, ontheleftwegetxoandvo, because thatiswhat xandvareatt=0;also, weknow thatthecosine ofzerois unity, andthesineofzero iszero. Therefore weget xo=A-l+B-0=A and vo= —woA-0-1-woB-1= woB. Soforthisparticular casewefindthat A=X0, B=U0/(.00. From these values ofAandB,wecangetaandAifwewish. That istheendofoursolution, butthere isonephysically interesting thing tocheck, andthatistheconservation ofenergy. Since there arenofrictional losses, energy ought tobeconserved. Letususetheformula x=acos (wot +A); then U=—-woa sin(wot +A). Now letusfindoutwhat thekinetic energy Tis,andwhat thepotential energy Uis.Thepotential energy atanymoment is%kx2,where xisthedisplacement and kistheconstant ofthespring. Ifwesubstitute forx,using ourexpression above, weget U=%kx2 =%ka2 cosz (wot +A). Ofcourse thepotential energy isnotconstant; thepotential never becomes negative, naturally—there isalways some energy inthespring, buttheamount ofenergy fluctuates with x.Thekinetic energy, ontheother hand, is%mv2, andbysub- stituting forvweget 2 22*2T==§mv =%mwoa sin(wot +A). Now thekinetic energy iszerowhen xisatthemaximum, because thenthere isno velocity; ontheother hand, itismaximal when xispassing through zero, because then itismoving fastest. This variation ofthekinetic energy isjusttheopposite ofthatofthepotential energy. Butthetotal energy ought tobeaconstant. If wenotethatk=mwfi,weseethat T+U=Q-mwgaz [cos2 (wot +A)+sinz(wot +A)]=~§mw§a2. Theenergy isdependent onthesquare oftheamplitude; ifwehave twice the amplitude, wegetanoscillation which hasfour times theenergy. Theaverage potential energy ishalfthemaximum and,therefore, halfthetotal, andtheaverage kinetic energy islikewise halfthetotal energy. 21-5 Forced oscillations Next weshall discuss theforced harmonic oscillator, i.e.,oneinwhich there is anexternal driving force acting. Theequation then isthefollowing: mdzx/dt2 =—kx +F(t). (21.8) 21-5 Wewould liketofindoutwhat happens inthese circumstances. Theexternal driv- ingforce canhave various kinds offunctional dependence onthetime; thefirst onethatweshall analyze isverysimple—we shall suppose thattheforce isoscillat- ing: F(t) =Focoswt. (21.9) Notice, however, thatthiswisnotnecessarily wo:wehave wunder ourcontrol; theforcing may bedone atdifferent frequencies. Sowetrytosolve Eq.(21.8) with thespecial force (21.9). What isthesolution of(21.8)? Onespecial solution, (weshall discuss themore general cases later) is x=Ccos wt, (21.10) where theconstant istobedetermined. Inother words, wemight suppose thatif wekept pushing back andforth, themass would follow back andforth instep with theforce. Wecantryitanyway. Soweput(21.10) into (21.9), andget —mw2C coswt=—mw§C coswt+Focoswt. (21.11) Wehave alsoputink=mwg, sothatwewillunderstand theequation better at theend. Now because thecosine appears everywhere, wecandivide itout,and thatshows that(21.10) is,infact, asolution, provided wepick Cjustright. The answer isthatCmust be c=F0/m(a?, -w2). (21.12) That is,moscillates atthesame frequency astheforce, butwith anamplitude which depends onthefrequency oftheforce, andalsoupon thefrequency ofthe natural motion oftheoscillator. Itmeans, first, thatifwisvery small compared with wo,then thedisplacement andtheforce areinthesame direction. Onthe other hand, ifweshake itback andforth very fast,then (21.12) tellsusthatCis negative ifwisabove thenatural frequency wooftheharmonic oscillator. (We willcallwothenatural frequency oftheharmonic oscillator, andwtheapplied frequency.) Atvery high frequency thedenominator maybecome very large, and there isthen notmuch amplitude. Ofcourse thesolution wehave found isthesolution only ifthings arestarted justright, forotherwise there isapartwhich usually diesoutafter awhile. This other part iscalled thetransient response toF(t), while (21.10) and(21.12) are called thesteady-state response. According toourformula (21.12), averyremarkable thing should alsooccur: ifwisalmost exactly thesame aswo,then Cshould approach infinity. Soifwe adjust thefrequency oftheforce tobe“intime” with thenatural frequency, then weshould getanenormous displacement. This iswellknown toanybody whohas pushed achild onaswing. Itdoes notwork verywelltoclose oureyesandpush at acertain speed atrandom. Ifwehappen togettheright timing, then theswing goesveryhigh, butifwehave thewrong timing, thensometimes wemaybepushing when weshould bepulling, andsoon,anditdoes notwork. Ifwemake wexactly equal towo,wefindthatitshould oscillate ataninfinite amplitude, which is,ofcourse, impossible. Thereason itdoes notisthatsomething goes wrong with theequation, there aresome other frictional terms, andother forces, which arenotin(21.8) butwhich occur intherealworld. Sotheamplitude does notreach infinity forsome reason; itmaybethatthespring breaks! 21-6 22 Algebra 22-1 Addition andmultiplication Inourstudy ofoscillating systems weshall have occasion touseoneofthe most remarkable, almost astounding, formulas inallofmathematics. From the physicist’s point ofview wecould bring forth thisformula intwominutes orso, andbedone with it.Butscience isasmuch forintellectual enjoyment asforprac- tical utility, soinstead ofjustspending afewminutes onthisamazing jewel, we shall surround thejewel byitsproper setting inthegrand design ofthatbranch of mathematics which iscalled elementary algebra. Now youmay ask,“What ismathematics doing inaphysics lecture?” We have several possible excuses: first, ofcourse, mathematics isanimportant tool, butthatwould onlyexcuse usforgiving theformula intwominutes. Ontheother hand, intheoretical physics wediscover thatallourlaws canbewritten inmathe- matical form-, andthat thishasacertain simplicity andbeauty about it.So, ultimately, inorder tounderstand nature itmay benecessary tohave adeeper understanding ofmathematical relationships. Buttherealreason isthatthesubject isenjoyable, andalthough wehumans cutnature upindifferent ways, andwehave different courses indifferent departments, such compartmentalization isreally artificial, andweshould take ourintellectual pleasures where wefindthem. Another reason forlooking more carefully atalgebra now, even though most ofusstudied algebra inhigh school, isthatthatwasthefirsttimewestudied it;all theequations were unfamiliar, anditwashard work, justasphysics isnow. Every sooften itisagreat pleasure tolook back toseewhat territory hasbeen covered, andwhat thegreat maporplan ofthewhole thing is.Perhaps some daysomebody intheMathematics Department willpresent alecture onmechanics insuch away astoshow what itwaswewere trying tolearn inthephysics course! Thesubject ofalgebra willnotbedeveloped from thepoint ofview ofamathe- matician, exactly, because themathematicians aremainly interested inhowvarious mathematical facts aredemonstrated, andhow many assumptions areabsolutely required, andwhat isnotrequired. They arenotsointerested intheresult ofwhat theyprove. Forexample, wemayfindthePythagorean theorem quite interesting, thatthesumofthesquares ofthesides ofaright triangle isequal tothesquare of thehypotenuse; thatisaninteresting fact, acuriously simple thing, which maybe appreciated without discussing thequestion ofhowtoprove it,orwhat axioms are required. So,inthesame spirit, weshall describe qualitatively, ifwemay putit thatway, thesystem ofelementary algebra. Wesayelementary algebra because there isabranch ofmathematics called modern algebra inwhich some oftherules such asab=ba,areabandoned, anditisstillcalled algebra, butweshall not discuss that. Todiscuss thissubject westart inthemiddle. Wesuppose thatwealready know what integers are,what zero is,andwhat itmeans toincrease anumber by oneunit. You maysay,“That isnotinthemiddle!” Butitisthemiddle from a mathematical standpoint, because wecould goeven further back anddescribe the theory ofsetsinorder toderive some ofthese properties ofintegers. Butweare notgoing inthatdirection, thedirection ofmathematical philosophy andmathe- matical logic, butrather intheother direction, from theassumption thatweknow what integers areandweknow how tocount. Ifwestart with acertain number a,aninteger, andwecount successively one unitbtimes, thenumber wearrive atwecalla+b,andthatdefines addition of integers. 22-122-1 Addition andmultiplication 22-2 Theinverse operations 22-3 Abstraction andgeneralization 22-4 Approximating irrational numbers 22-5 Complex numbers 22-6 Imaginary exponents Once wehave defined addition, then wecanconsider this: ifwestart with nothing andaddatoit,btimes insuccession, wecalltheresult multiplication of integers; wecallitbtimes a. Now wecanalsohave asuccession ofmultiplications: ifwestartwith1and multiply bya,btimes insuccession, wecallthatraising toapower: ab. Now asaconsequence ofthese definitions itcanbeeasily shown thatallof thefollowing relationships aretrue: (a) a+b=b+a (b) a+(b+c)=(a+b)+c (c) ab=ba (d) a(b+c)=ab+ac (e) (ab)c =a(bc) (f) (ab)° =a‘b” (22.1) (g) abac =a<1>+o (h) (ab)c =a(bc) (i) a+0=a (j) a-1=a (k) a1=a These results arewellknown andweshall notbelabor thepoint, wemerely list them. Ofcourse, 1and0have special properties; forexample, a+0isa,atimes 1=a,andatothefirstpower isa. Inthisdiscussion wemust alsoassume afewother properties likecontinuity andordering, which areveryhard todefine; wewilllettherigorous theory doit. Furthermore, itisdefinitely truethatwehave written down toomany “rules”; some ofthem may bededucible from theothers, butweshall notworry about suchmatters. 22-2 Theinverse operations Inaddition tothedirect operations ofaddition, multiplication, andraising toapower, wehave also theinverse operations, which aredefined asfollows. Letusassume thataandcaregiven, andthatwewish tofindwhat values ofb satisfy such equations asa+b=c,ab=c,b“=c.Ifa+b=c,bisdefined asc—a,which iscalled subtraction. Theoperation called division isalsoclear: ifab=c,then b=c/adefines division—a solution oftheequation ab=c “backwards.” Now ifwehave apower b“=candweaskourselves, “What is b?,” itiscalled theathrootofc:b= Forinstance, ifweaskourselves the following question, “What integer, raised tothethird power, equals 8?,” then the answer iscalled thecuberootof8;itis2.Because b“andabarenotequal, there aretwoinverse problems associated with powers, andtheother inverse problem would be,“Towhat power must weraise 2toget8?” This iscalled taking the logarithm. Ifab=c,wewrite b=logac. Thefactthatithasacumbersome nota- tionrelative totheothers does notmean thatitisanylesselementary, atleast applied tointegers, than theother processes. Although logarithms come latein analgebra class, inpractice they are,ofcourse, justassimple asroots; they are justadifferent kind ofsolution ofanalgebraic equation. Thedirect andinverse operations aresummarized asfollows: (a) addition (a') subtraction a+b=c b=c—a (b) multiplication (b’) division ab=c b=c/a (22.2) (c)power (c’) root b“=c b=\“/Z (d)power (d’) logarithm a°=c b=logac Now here istheidea. These relationships, orrules, arecorrect forintegers, since they follow from thedefinitions ofaddition, multiplication, andraising toa power. Wearegoing todiscuss whether ornotwecanbroaden theclass ofobjects 22-2 which a,b,andcrepresent sothattheywillobey these same rules, although theproc- esses fora+b,andsoon,willnotbedefinable interms ofthedirect action of adding 1,forinstance, orsuccessive multiplications byintegers. 22-3 Abstraction andgeneralization When wetrytosolve simple algebraic equations using allthese definitions, wesoon discover some insoluble problems, suchasthefollowing. Suppose that wetrytosolve theequation b=3—5.That means, according toourdefinition ofsubtraction, thatwemust findanumber which, when added to5,gives 3.And ofcourse there isnosuch number, because weconsider only positive integers; thisisaninsoluble problem. However, theplan, thegreat idea, isthis: abstraction andgeneralization. From thewhole structure ofalgebra, rules plus integers, we abstract theoriginal definitions ofaddition andmultiplication, butweleave the rules (22.1) and(22.2), andassume these tobetrueingeneral onawider class of numbers, even though they areoriginally derived onasmaller class. Thus, rather thanusing integers symbolically todefine therules, weusetherules asthedefinition ofthesymbols, which then represent amore general kind ofnumber. Asanex- ample, byworking with therules alone wecanshow that3—-5=0—2.In factwecanshow thatonecanmake allsubtractions, provided wedefine awhole setofnewnumbers: O—1,0-2,0—3,0—4,andsoon,called thenegative integers. Then wemay usealltheother rules, likea(b+c)=ab+acandso forth, tofindwhat therules areformultiplying negative numbers, andwewill discover, infact, thatalloftherules canbemaintained with negative aswellas positive integers. Sowehave increased therange ofobjects over which therules work, butthe meaning ofthesymbols isdifferent. Onecannot say,forinstance, that-2times 5really means toadd5together successively —-2times. That means nothing. Butnevertheless everything willwork outallright according totherules. Aninteresting problem comes upintaking powers. Suppose thatwewish to discover what a(3_5’ means. Weknow onlythat3—5isasolution oftheproblem, (3—5)+5=3.Knowing that, weknow thata‘3_5’a5 =a3. Therefore a‘3_5) =a3/a5, bythedefinition ofdivision. With alittle more work, thiscan bereduced to1/a2. Sowefindthatthenegative powers arethereciprocals ofthe positive powers, butl/a2 isameaningless symbol, because ifaisapositive or negative integer, thesquare ofitisgreater than l,andwedonotyetknow what wemean by1divided byanumber greater than 1! Onward! Thegreat plan istocontinue theprocess ofgeneralization; whenever wefindanother problem thatwecannot solve weextend ourrealm ofnumbers. Consider division: wecannot findanumber which isaninteger, even anegative integer, which isequal totheresult ofdividing 3by5.Butifwesuppose thatall fractional numbers alsosatisfy therules, then wecantalkabout multiplying and adding fractions, andeverything works aswellasitdidbefore. Take another example ofpowers: what isa3/5?Weknow onlythat(3/5)5= 3,since thatwasthedefinition of3/5. Soweknow alsothat(am“)5=am5)“) = a3,because thisisoneoftherules. Then bythedefinition ofroots wefindthata<a/st =xi/as Inthisway, then, wecandefine what wemean byputting fractions inthe various symbols, byusing therules themselves tohelpusdetermine thedefinition— itisnotarbitrary. Itisaremarkable factthatalltherules stillwork forpositive andnegative integers, aswellasforfractions! Wegoonintheprocess ofgeneralization. Arethere anyother equations we cannot solve? Yes, there are. Forexample, itisimpossible tosolve thisequation: b=21/2=\/2. Itisimpossible tofindanumber which isrational (afraction) whose square isequal to2.Itisvery easy forusinmodern days toanswer this question. Weknow thedecimal system, andsowehave nodifficulty inappreciating themeaning ofanunending decimal asatypeofapproximation tothesquare root of2.Historically, thisidea presented great difficulty totheGreeks. Toreally 22-3 define precisely what ismeant here requires thatweaddsome substance ofcon- tinuity andordering, anditis,infact, quite themost difficult stepintheprocesses ofgeneralization justatthispoint. Itwasmade, formally andrigorously, by Dedekind. However, without worrying about themathematical rigor ofthething, itisquite easy tounderstand thatwhat wemean isthatwearegoing tofinda whole sequence ofapproximate fractions, perfect fractions (because anydecimal, when stopped somewhere, isofcourse rational), which justkeeps ongoing, getting closer andcloser tothedesired result. That isgood enough forwhat wewish to discuss, anditpermits ustoinvolve ourselves inirrational numbers, andtocal- culate things likethesquare rootof2toanyaccuracy thatwedesire, with enough work. 22-4 Approximating irrational numbers Thenextproblem comes with what happens with theirrational powers. Sup- pose thatwewant todefine, forinstance, 10‘/Y. Inprinciple, theanswer issimple enough. Ifweapproximate thesquare root of2toacertain number ofdecimal places, then thepower isrational, andwecantake theapproximate root, using theabove method, andgetanapproximation to10*/2. Then wemay runitupa fewmore decimal places (itisagain rational), take theappropriate root, thistime amuch higher root because there isamuch bigger denominator inthefraction, andgetabetter approximation. Ofcourse wearegoing togetsome enormously high roots involved here, andthework isquite diflicult. How canwecope with thisproblem? Inthecomputations ofsquare roots, cube roots, andother small roots, there isanarithmetical process available bywhich wecangetonedecimal place after another. Buttheamount oflabor needed tocalculate irrational powers andthe logarithms thatgowiththem (theinverse problem) issogreat thatthere isnosimple arithmetical process wecanuse. Therefore tables have been built upwhich permit ustocalculate these powers, andthese arecalled thetables oflogarithms, orthe tables ofpowers, depending onwhich waythetable issetup.Itismerely aquestion ofsaving time; ifwemust raise some number toanirrational power, wecanlook ituprather than having tocompute it.Ofcourse, such acomputationis justa technical problem, butitisaninteresting one, ahdofgreat historical value. In thefirstplace, notonlydowehave theproblem ofsolving x=10”, butwealso have theproblem ofsolving 10”‘=2,orx=loglo 2.This isnotaproblem where wehave todefine anewkind ofnumber fortheresult, itismerely acomputational problem. Theanswer issimply anirrational nuhiber, anunending decimal, nota newkind ofanumber. Letusnow discuss theproblem ofcalculating solutions ofsuch equations. Thegeneral ideaisreally very simple. Ifwecould calculate 10‘,and104/1°,and 101/10°, and104/1°00 andsoon,andmultiply them alltogether, wewould get l01"““"' or10*/7, andthatisthegeneral ideaonwhich things work. Butinstead ofcalculating 10”1°andsoon,weshall calculate 101/2, 10”4,andsoon.Before westart, weshould explain whywemake somuch work with 10,instead ofsome other number. Ofcourse, werealize thatlogarithm tables areofgreat practical utility, quite aside from themathematical problem oftaking roots, since with anybase atall, logo(ac)=logoa+logoc. (22.3) Weareallfamiliar with thefactthatonecanusethisfactinapractical wayto multiply numbers ifwehave atable oflogarithms. Theonlyquestion is,withwhat base bshall wecompute? Itmakes nodifference what baseisused; wecanusethe same principle allthetime, andifweareusing logarithms toanyparticular base, wecanfindlogarithms toanyother basemerely byachange inscale, amultiplying factor. Ifwemultiply Eq.(22.3) by61,itisjustastrue, andifwehadatable of logswithabaseb,andsomebody elsemultiplied allofourtable by61,there would benoessential difference. Suppose thatweknow thelogarithms ofallthenumbers tothebase b.Inother words, wecansolve theequation b“=cforanycbecause 22-4 wehave tatable. Theproblem istofindthelogarithm ofthesame number cto some other base, letussaythebase x.Wewould liketosolve x“'=c.Itiseasy todo,because wecanalways write x=bi,which defines t,knowing xandb.Asa matter offact, t=logox.Then ifweputthatinandsolve fora’,weseethat (b‘)“' =b'“'=c.Inother words, ta’isthelogarithm ofcinbase b.Thus a’=a/t. Thus logstobase xarejust1/t,which isaconstant, times thelogsto thebase, b.Therefore anylogtable isequivalent toanyother logtable ifwe multiply byaconstant, andtheconstant is1/log), x.This permits ustochoose a particular base, andforconvenience wetakethebase 10.(The question mayarise astowhether there isanynatural base, anybase inwhich things aresomehow simpler, andweshall trytofindananswer tothatlater. Atthemoment weshall justusethebase 10.) Table 22-1 Successive Square Roots ofTen Power s 1024 s 10’ (10‘ —1)/s 1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024 A/10241024 512 256 128 64 32 16 8 4 --to A10.00000 3.16228 1.77828 1.33352 1.15478 1.074607 1.036633 1.018152 1.0090350 1.0045073 1.0022511 1+.0022486A<--9.00 4.32 3.113 2.668 2.476 2.3874 2.3445 2.3234211 2.3130104 2.3077 53 2.3051 26 l 26 2.3025 (A—>0) Now letusseehowtocalculate logarithms. Webegin bycomputing successive square roots of10,bycutandtry. Theresults areshown inTable 22-1. The powers of10aregiven inthefirstcolumn, andtheresult, 10*,isgiven inthethird column. Thus 10‘=10.Theone-half power of10wecaneasily work out,because thatisthesquare rootof10,andthere isaknown, simple process fortaking square roots ofanynumber.* Using thisprocess, wefind thefirst square root tobe 3.16228. What good isthat? Italready tellsussomething, ittellsushow totake l0°"5, sowenow know atleast onelogarithm, ifwehappen toneed thelogarithm of3.16228, weknow theanswer isclose to0.50000. Butwemust doalittle bit better than that; weclearly need more information. Sowetakethesquare root again, andfind 10‘/4, which is1.77828. Now wehave thelogarithm ofmore numbers than wehadbefore, 1.250 isthelogarithm of17.78 and, incidentally, ifithappens that somebody asks for10°“, wecangetit,because that is 10‘°'5+°"25’; itistherefore theproduct ofthesecond andthird numbers. Ifwe cangetenough numbers incolumn stobeable tomake upalmost anynumber, then bymultiplying’the proper things incolumn 3,wecanget10toanypower; thatistheplan. Soweevaluate tensuccessive square roots of10,andthatisthe main work which isinvolved inthecalculations. Why don’t wekeep ongoing formore andmore accuracy? Because webegin tonotice something. When weraise 10toaverysmall power, weget1plusasmall amount. Thereason forthisisclear, because wearegoing tohave totake the *There isadefinite arithmetic procedure, buttheeasiest way tofindthesquare root ofanynumber Nistochoose some afairly close, findN/a, average a’=%[a+(N/a)], andusethisaverage a’forthenext choice fora.Theconvergence isvery rapid—the number ofsignificant figures doubles each time. 22-5 l000th power of101/‘°°° togetback to10,sowehadbetter notstart with too biganumber; ithastobeclose to1.What wenotice isthatthesmall numbers thatareadded to1begin tolook asthough wearemerely dividing by2each time; wesee1815becomes 903,then450,225;soitisclear that,toanexcellent approxi- mation, ifwetake another root, weshall get1.00112 something, andrather than actually takeallthesquare roots, weguess attheultimate limit. When wetakea small fraction Aof1024 asAapproaches zero, what willtheanswer be?Ofcourse itwillbesome number close to0.002251 1A.Notexactly 0.002251 1A,however—we cangetabetter value bythefollowing trick: wesubtract the1,andthen divide by thepower s.This ought tocorrect alltheexcesses tothesame value. Weseethat theyareveryclosely equal. Atthetopofthetable theyarenotequal, butasthey come down, they getcloser andcloser toaconstant value. What isthevalue? Again welook toseehowtheseries isgoing, howithaschanged withs.Itchanged by211,by104,by53,by26.These changes areobviously halfofeach other, very closely, aswegodown. Therefore, ifwekeptgoing, thechanges would be13,7,3, 2and1,more orless,oratotal of26.Thus wehave only 26more togo,andso wefindthatthetruenumber is2.3025. (Actually, weshall later seethattheexact number should be2.3026, buttokeep itrealistic, weshall notalter anything inthe arithmetic.) From thistable wecannowcalculate anypower of10,bycompound- ingthepower outof1024ths. Letusnow actually calculate alogarithm, because theprocess weshall useis where logarithm tables actually come from. Theprocedure isshown inTable 22-2, andthenumerical values areshown inTable 22-1 (columns 2and3). Table 22-2 Calculation ofalogarithm: logl02 2+1.77828 =1.124682 1.124682 +1.074607 =1.046598, etc. 2=(1.77828)(1.074607)(1.036633)(l.090350)(l.000573) 1 308.254=10_256 321640.254 =10?i1024( ++++ i [1024i =10°-3°1°3 33=0.2542249 10g1() 2=0.30103 Suppose wewant thelogarithm of2.That is,wewant toknow towhat power wemust raise 10toget2.Canweraise 10tothel/2power? No;thatistoobig. Inother words, wecanseethattheanswer isgoing tobebigger than 1/4,andless than 1/2. Letustake thefactor 101/4 out; wedivide 2by1.778. ..,andget 1.124 ...,andsoon,andnow weknow thatwehave taken away 0.250000 from thelogarithm. Thenumber 1.124 ...isnowthenumber whose logarithm weneed. When wearefinished weshall addback the1/4,or256/1024. Now welook inthe table forthenextnumber justbelow 1.124 ...,andthatis1.074607. Wetherefore divide by1.074607 andget1.046598. From thatwediscover that2canbemade upofaproduct ofnumbers thataneinTable 22-1, asfollows: 2=(1.77828)(1 .O74607)(l .036633)(l .0090350)(l .000573). There wasonefactor (1.0_00573) leftover, naturally, which isbeyond therange ofourtable. Togetthelogarithm ofthisfactor, weuseourresult that10“1°“ ==-. 1+2.3025 A/1024. WefindA=0.254. Therefore ouranswer is10tothefollow- ingpower: (256 +32+16+4+0.254)/1024. Adding those together, we get308.254/1024. Dividing, weget0.30103, soweknow thatthelog102=0.30103, which happens toberight to5figures! This ishow logarithms were originally computed byMr.Briggs ofHalifax, in1620. Hesaid, “Icomputed successively 54square roots of10." Weknow he 22-6 really computed only thefirst27,because therestofthem canbeobtained bythis trick with A.Hiswork involved calculating thesquare root of10twenty-seven times, which isnotmuch more than thetentimes wedid; however, itwasmore work because hecalculated tosixteen decimal places, andthenreduced hisanswer tofourteen when hepublished it,sothatthere were norounding errors. Hemade tables oflogarithms tofourteen decimal places bythismethod, which isquite tedious. Butalllogarithm tables forthree hundred years were borrowed from Mr. Briggs’ tables byreducing thenumber ofdecimal places. Only inmodern times, with theWPA andcomputing machines, have new tables been independently computed. There aremuch more efficient methods ofcomputing logarithms today, using certain series expansions. Intheabove process, wediscovered something rather interesting, andthatis thatforvery small powers ewecancalculate 10‘easily; wehave discovered that 10‘=1+2.3025e, bysheer numerical analysis. Ofcourse thisalsomeans that 10”/2'30” =1+nifnisvery small. Now logarithms toanyother base are merely multiples oflogarithms tothebase 10.Thebase 10wasused only because wehave 10fingers, andthearithmetic ofitiseasy, butifweaskforamathematically natural base, onethathasnothing todowith thenumber offingers onhuman beings, wemight trytochange ourscale oflogarithms insome convenient and natural manner, andthemethod which people have chosen istoredefine thelog- arithms bymultiplying allthelogarithms tothebase 10by2.3025 ...This then corresponds tousing some other base, andthisiscalled thenatural base, orbase e. Notethatlog,(l +n)zn,ore" z1+nasn——>0. Itiseasy enough tofindoutwhat eis:e=101/2'30“ or10°'434294'"', an irrational power. Ourtable ofthesuccessive square roots of10canbeused to compute, notjustlogarithms, butalso10toanypower, soletususeittocalculate thisnatural base e.Forconvenience wetransform 0.434294 ...into444.73/1024. Now, 444.73 is256+128+32+16+2+0.73. Therefore e,since itisan exponent ofasum, willbeaproduct ofthenumbers (1.77828)(l.33352)(1.074607)(l.036633)(l.0l8l52)(l.O09035)(1.00l643) =2.7184. (The only problem isthelastone,which is0.73, andwhich isnotinthetable, but weknow thatifAissmall enough, theanswer is1+2.3025 A.)When wemultiply allthese together, weget2.7184 (itshould be2.7183, butitisgood enough). The useofsuch tables, then, isthewayinwhich irrational powers andthelogarithms ofirrational numbers areallcalculated. That takes careoftheirrationals. 22-5 Complex numbers Now itturns outthatafter allthatwork westillcannot solve every equation! Forinstance, what isthesquare rootof-1? Suppose wehave tofindx2=-1. Thesquare ofnorational, ofnoirrational, ofnothing thatwehave discovered so far,isequal to—1.Soweagain have togeneralize ournumbers toastillwider class. Letussuppose thataspecific solution ofx2=~liscalled something, weshall calliti;ihastheproperty, bydefinition, thatitssquare is—l. That is about allwearegoing tosayabout it;ofcourse, there ismore than onerootofthe equation x2=—l.Someone could write i,butanother could say,“No, Iprefer —i.Myiisminus your i.”Itisjustasgood asolution, andsince theonlydefinition thatihasisthati2=—1,itmust betruethatanyequation wecanwrite isequally true ifthesign ofiischanged everywhere. This iscalled taking thecomplex conjugate. Now wearegoing tomake upnumbers byadding successive i’s,and multiplying i’sbynumbers, andadding other numbers, andsoon,according to allofourrules. Inthiswaywefindthatnumbers cananbewritten intheform p+iq,where pandqarewhat wecallrealnumbers, i.e.,thenumbers wehave been defining upuntil now. Thenumber iiscalled theunitimaginary number. Anyrealmultiple ofiiscalled pure imaginary. Themost general number, a,isof theform p+iqandiscalled acomplex number. Things donotgetanyworse if, forinstance, wemultiply twosuch numbers, letussay(r+is)(p +iq).Then, 22-7 using therules, weget (r+is)(1>+iq)=rp+r(iq)+(is)P+(is)(iq) =#11+i(rq)+i(s11)+(ii)(sq) =(rp—sq)+i(rq+SP), (22-4) since ii=i2=-1. Therefore allthenumbers that now belong intherules (22.1) have thismathematical form. Now yousay,“This cangoonforever! Wehave defined powers ofimaginaries andalltherest,andwhen weareallfinished, somebody elsewillcome along with another equation which cannot besolved, likex6-1-3x2=-2. Then wehave to generalize allover again!” Butitturns outthatwiththisonemore invention, just thesquare root of-1,every algebraic equation canbesolved! This isafantastic fact, which wemust leave totheMathematics Department toprove. Theproofs arevery beautiful andvery interesting, butcertainly notself-evident. Infact, the most obvious supposition isthatwearegoing tohave toinvent again andagain andagain. Butthegreatest miracle ofallisthatwedonot.Thisisthelastinvention. After thisinvention ofcomplex numbers, wefindthattherules stillwork with complex numbers, andwearefinished inventing newthings. Wecanfindthecom- plex power ofanycomplex number, wecansolve anyequation that iswritten algebraically, interms ofafinite number ofthose symbols. Wedonotfindany newnumbers. Thesquare root ofi,forinstance, hasadefinite result, itisnot something new; andiiissomething. Wewilldiscuss thatnow. Wehave already discussed multiplication, andaddition isalsoeasy; ifweadd twocomplex numbers, (p+iq)+(r+is),theanswer is(p+r)+i(q+s). Now wecanaddandmultiply complex numbers. Buttherealproblem, ofcourse, istocompute complex powers ofcomplex numbers. Itturns outthattheproblem isactually nomore difficult than computing complex powers ofrealnumbers. So letusconcentrate now ontheproblem ofcalculating 10toacomplex power, not justanirrational power, but10"+i”. Ofcourse, wemust atalltimes useour rules (22.1) and(22.2). Thus 10<'+":’ =10'10:'". (22.5) But10'wealready know how tocompute, andwecanalways multiply anything byanything else; therefore theproblem istocompute only 10“. Letuscallitsome complex number, x+iy.Problem: given s,findx,findy.Now if 10"“=X+iy, then thecomplex conjugate ofthisequation must alsobetrue, sothat 104“ =x—iy. (Thus weseethatwecandeduce anumber ofthings without actually computing anything, byusing ourrules.) Wededuce another interesting thing bymultiplying these together: 10:'*'10-“ =10°=1=(x+iy)(x-iy)=X2+y2. (22.6) Thus ifwefindx,wehave yalso. Now theproblem ishowtocompute 10toanimaginary power. What guide isthere? Wemay work over ourrules until wecangonofurther, buthere isa reasonable guide: ifwecancompute itforanyparticular s,wecangetitforallthe rest. Ifweknow 10"”foranyonesandthen wewant itfortwice thats,wecan square thenumber, andsoon.Buthowcanwefind10"’foreven onespecial value ofs‘?Todosoweshall make oneadditional assumption, which isnotquite inthe category ofalltheother rules, butwhich leads toreasonable results andpermits ustomake progress: when thepower issmall, weshall suppose thatthe“law” 10‘=1+2.3025e isright, asegetsverysmall, notonlyforreale,butforcomplex easwell. Therefore, webegin with thesupposition thatthislawistrueingeneral, andthattellsusthat10“=l-1-2.3025 -is,fors->0.Soweassume thatifsis very small, sayonepartin1024, wehave arather good approximation to10“. 22-8 Now wemake atable bywhich wecancompute alltheimaginary powers of 10,thatis,compute xandy.Itisdone asfollows. Thefirstpower westart withis the1/1024 power, which wepresume isvery nearly 1-1-2.3025i/1024. Thus we start with 10”‘°“ =1.00000 +0.0022480", (22.7) andifwekeep multiplying thenumber byitself, wecangettoahigher imaginary power. Infact,wemayjustreverse theprocedure weused inmaking ourlogarithm table, andcalculate thesquare, 4thpower, 8thpower, etc., of(22.7), andthus build upthevalues shown inTable 22-3. Wenotice aninteresting thing, thatthe xnumbers arepositive atfirst, butthen swing negative. Weshall look intothata little bitmore inamoment. Butfirstwemaybecurious tofindforwhat number stherealpartof10”iszero. They-value would bei,andsowewould have 10“= i,oris=loglo i.Asanexample ofhow tousethistable, justaswecalculated logm 2before, letusnow useTable 22-3 tofindloglo i. Which ofthenumbers inTable 22-3 dowehave tomultiply together togeta pure imaginary result? After alittle trial anderror, wediscover thattoreduce xthemost, itisbesttomultiply “5l2” by“l28.” This gives 0.13056 +0.99l44i. Then wediscover thatweshould multiply thisbyanumber whose imaginary part isabout equal tothesizeoftherealpartwearetrying toremove. Thus wechoose “64” whose y-value is0.14349, since thatisclosest to0.13056. This then gives —0.0l350 -1-0.999931’. Now wehave overshot, andmust divide by0.99996 + 0.00900i. How dowedothat? Bychanging thesignofiandmultiplying by 0.99996 —0.00900i (which works ifx2+y2=1).Continuing inthisway, we findthattheentire power towhich 10must beraised togiveiisi(512 +128-1- 64—4—2+0.20)/1024, or698.20i/1024. Ifweraise 10tothatpower, we cangeti.Therefore logloi =O.68226i. 22-6 Imaginary exponents Tofurther investigate thesubject oftaking complex imaginary powers, let uslook atthepowers of10taking successive powers, notdoubling thepower each time, inorder tofollow Table 22-3 further andtoseewhat happens tothose minus signs. This isshown inTable 22-4, inwhich wetake 10"/8, andjustkeep multiply- ingit.Weseethatxdecreases, passes through zero, swings almost to—l(ifwe could getinbetween p=10andp=llitwould obviously swing to-1), and swings back. They-value isgoing back andforth too. InFig.22-1 thedots represent thenumbers thatappear inTable 22-4, and thelines arejustdrawn tohelpyouvisually. Soweseethatthenumbers xandy oscillate; 10“repeats itself, itisaperiodic thing, andassuch, itiseasy enough to explain, because ifacertain power isi,then thefourth power ofthatwould be1'2 squared. Itwould be+1again, andtherefore, since l0°'68i isequal toi,bytaking thefourth power wediscover that10”“ isequal to+1. Therefore, ifwewanted l03"°°‘, forinstance, wecould write itas10272" times 10°23‘. Inother words, it hasaperiod, itrepeats. Ofcourse, werecognize what thecurves look like! They look likethesineandcosine, andweshall callthem, forawhile, thealgebraic sine andalgebraic cosine. However, instead ofusing thebase 10,weshall putthem into ournatural base, which only changes thehorizontal scale; sowedenote 2.3025s byt,andwrite 101’=eh‘,where tisarealnumber. Now e“=x+iy, andweshall write thisasthealgebraic cosine oftplusitimes thealgebraic sineof t.Thus Q”=Qst+isi_n1. (22.8) What aretheproperties ofc_ostandQ_nt‘?First, weknow, forinstance, thatx2+ yzmust be1;wehave proved thatbefore, anditisjustastrue forbase easforbase 10.Therefore @2t +Q1121 =1.Wealsoknow that, forsmall t,eh‘=1-1-it, andtherefore gstisnearly 1,andsi_ntisnearly t,andsoitgoes, thatallofthe various properties ofthese remarkable functions, which come from taking imaginary powers, arethesame asthesineandcosine oftrigonometry. 22-9Table 22-3 Successive Squares of 10’!/1°24 =l+0.0022486i Power is1024s loia i/1024 i/512 1'/256 i/128 i/64 i/32 i/16 i/8 i/4 i/2 i/lO0-(#5-I’-‘ 16 32 64 128 256 512 10241.00000 +0.00225i" 1.00000 -1-0.00450i 0.99996 -1-0.00900i 0.99984 +0.0l800i 0.99936 +0.03599i 0.99742 +0.07193i 0.98967 +0.l4349i 0.95885 +0.28402i 0.83872 -1-0.54467i 0.40679 +0.9l365i 0.66928 -1-0.74332i *Should be0.0022486i Table 22-4 Successive Powers of101/8 p=power -8i 10"!’/8 ®\lO'\LI|->0-)l\)>-‘Q 9 10 11 12 14 16 18 20 22 24+ + -1- +1.00000 0.95882 0.83867 0.64944 0.40672 0.13050 0.15647 0.43055 0.66917 0.85268 0.96596 0.99969 0.95104 0.62928 0.10447 0.45454 0.86648 0.99884 0.808900.00000i 0.28402i 0.54465i 0.76042i 0.91356i 0.991461 0.98770i 0.90260i 0.74315i 0.52249i 0.25880i 0.02620i 0.30905i 0.777l7i 0.994531’ 0.89098i 0.49967i -1-0.05287i -1-0.588361’+ + + + + + + + + + + 0.!IO vln=x+iy 2.5 3.0Y ,>__o.s |s_ 2 OD -| x Figure. .| S 22-l Y f yI X |g22-2. x%—iy =rem.Istheperiod thesame? Letusfindout. etowhat power isequal toi?What isthelogarithm ofitothebase e?Weworked itoutbefore, inthebase 10itwas 0.68226i, butwhen wechange ourlogarithmic scale toe,wehave tomultiply by 2.3025, andifwedothatitcomes out1.5709. SothisW111becalled “algebraic 1r/2.” But, wesee,itdifiers from theregular 1r/2byonly oneplace inthelast point, andthat, ofcourse, istheresult oferrors inourarithmetic! Sowehave created twonewfunctions inapurely algebraic manner, thecosine andthesine, which belong toalgebra, andonly toalgebra. Wewake upattheendtodiscover thevery functions thatarenatural togeometry. Sothere isaconnection, ulti- mately, between algebra andgeometry. Wesummarize with this,themost remarkable formula inmathematics: ei”=cos0+isin0. (22.9) This isourjewel. Wemayrelate thegeometry tothealgebra byrepresenting complex numbers inaplane; thehorizontal position ofapoint isx,thevertical position ofapoint isy(Fig. 22-2). Werepresent every complex number, x+iy.Then iftheradial distance tothispoint iscalled randtheangle iscalled 0,thealgebraic lawisthat x-1-iyiswritten intheform re”,where thegeometrical relationships between x,y,r,and0areasshown. This, then, istheunification ofalgebra andgeometry. When webegan thischapter, armed only with thebasic notions ofintegers andcounting, wehadlittle ideaofthepower oftheprocesses ofabstraction and generalization. Using thesetofalgebraic “laws,” orproperties ofnumbers, Eq. (22.1), andthedefinitions ofinverse operations (22.2), wehave been able here, ourselves, tomanufacture notonly numbers butuseful things liketables oflog- arithms, powers, andtrigonometric functions (forthese arewhat theimaginary powers ofrealnumbers are), allmerely byextracting tensuccessive square roots often! 22-10 23 Resonance 23-1 Complex numbers andharmonic motion Inthepresent chapter weshall continue ourdiscussion oftheharmonic oscillator and, inparticular, theforced harmonic oscillator, using anewtechnique intheanalysis. Inthepreceding chapter weintroduced theideaofcomplex num- bers, which have realandimaginary parts andwhich canberepresented ona diagram inwhich theordinate represents theimaginary part andtheabscissa represents therealpart. Ifaisacomplex number, wemaywrite itasa=a,+ia,-, where thesubscript rmeans therealpart ofa,andthesubscript imeans the imaginary partofa.Referring toFig.23-1, weseethatwemayalsowrite acom- plex number a=x-1-iyintheform x+iy=re“, where r2=x2+y2= (x+iy)(x —iy)=aa*. (The complex conjugate ofa,written a*,isobtained byreversing thesignofiina.)Soweshall represent acomplex number ineither oftwoforms, arealplus animaginary part, oramagnitude randaphase angle 0,so-called. Given rand0,xandyareclearly rcos0andrsin0and, inreverse, given acomplex number x+iy,r=\/x2 +y2andtan0=y/x, theratio of theimaginary totherealpart. Wearegoing toapply complex numbers toouranalysis ofphysical phenomena bythefollowing trick. Wehave examples ofthings thatoscillate; theoscillation may have adriving force which isacertain constant times coswt.Now such a force, F=F0coswt,canbewritten astherealpart ofacomplex number F= FGel“because em‘=coswt+isinwt.Thereason wedothisisthatitiseasier towork with anexponential function than with acosine. Sothewhole trick istorepresent ouroscillatory functions astherealparts ofcertain complex func- tions. Thecomplex number Fthatwehave sodefined isnotarealphysical force, because noforce inphysics isreally complex; actual forces have noimaginary part, only arealpart. Weshall, however, speak ofthe“force” FDel“, butofcourse theactual force istherealpart ofthatexpression. Letustake another example. Suppose wewant torepresent aforce which is acosine wave thatisoutofphase with adelayed phase A.This, ofcourse, would betherealpartofFOe““"““), butexponentials being what they are,wemaywrite e“""'*“ =ei“‘e_“. Thus weseethatthealgebra ofexponentials ismuch easier than thatofsines andcosines; thisisthereason wechoose tousecomplex numbers. Weshall often write F=Foe-“e‘"‘ =F@"'"‘- (23.1) Wewrite alittle caret (~)over theFtoremind ourselves thatthisquantity isa complex number: here thenumber is F=F0e_iA. Now letussolve anequation, using complex numbers, toseewhether wecan work outaproblem forsome realcase. Forexample, letustrytosolve d2x kx F F21? +‘E =E =-rfi) COS (Of, where Fistheforce which drives theoscillator andxisthedisplacement. Now, absurd though itmay seem, letussuppose thatxandFareactually complex numbers, foramathematical purpose only. That istosay,xhasarealpartand animaginary part times i,andFhasarealpart andanimaginary part times i. 23-123-1 Complex numbers and harmonic motion 23-2 Theforced oscillator with damping 23-3 Electrical resonance 23-4 Resonance innature IMAGINARY AXIS O r ll 1‘REALAXIS Fig. 23-l. Acomplex number may berepresented byGpoint inthe“complex plane." Now ifwehadasolution of(23.2) with complex numbers, andsubstituted the complex numbers intheequation, wewould get d2(xr +ix.-)+ k(xt+ix.-)=F.+iFt alt? m m OI‘ (12.76, kx, .d2x,- kX,' F, dt2+m+l(dt2 +m)=m+ m. Now, since iftwocomplex numbers areequal, their realparts must beequal and their complex parts must beequal, wededuce thattherealpart ofxsatisfies the equation with therealpart oftheforce. Wemust emphasize, however, thatthis separation intoarealpartand animaginary part isnotvalid ingeneral, butis valid only forequations which arelinear, thatis,forequations inwhich xappears inevery term only inthefirstpower orthezeroth power. Forinstance, ifthere were intheequation aterm )\x2, then when wesubstitute x,+ix,-,wewould get >\(x, -1-ix,-)2, butwhen separated intorealandimaginary parts thiswould yield )\(x§ —xf)astherealpartand2i)\x,x,~ astheimaginary part. Soweseethatthe realpartoftheequation would notinvolve justAxf,butalso —>\xf. Inthiscase wegetadifferent equation than theonewewanted tosolve, withx,-,thecompletely artificial thing weintroduced inouranalysis, mixed in. Letusnowtryournewmethod fortheproblem oftheforced oscillator, that wealready know howtosolve. Wewant tosolve Eq.(23.2) asbefore, butwesay thatwearegoing totrytosolve d2x kx Fem an+F=T’ <23-3) where Few isacomplex number. Ofcourse xwillalsobecomplex, butremember therule: taketherealparttofindoutwhat isreally going on.Sowetrytosolve (23.3) fortheforced solution; weshall discuss other solutions later. Theforced solution hasthesame frequency astheapplied force, andhassome amplitude of oscillation andsome phase, andsoitcanberepresented also bysome complex number itwhose magnitude represents theswing ofxandwhose phase represents thetime delay inthesame wayasfortheforce. Now awonderful feature ofan exponential function isthat d(xe'_‘°‘)/dt =t<.»2e:'~'. When wedifferentiate an exponential function, webring down theexponent asasimple multiplier. The second derivative does thesame thing, itbrings down another iw,andsoitisvery simple towrite immediately, byinspection, what theequation isfor>2;every time weseeadifferentiation, wesimply multiply byiw.(Differentiation isnowaseasyas multiplication! This idea ofusing exponentials inlinear differential equations is almost asgreat astheinvention oflogarithms, inwhich multiplication isreplaced byaddition. 1-Iere differentiation isreplaced bymultiplication.) Thus ourequation becomes (i<.>)2x +(kx/m) =F/m. (23.4) (We have cancelled thecommon factor e“"‘.) Seehow simple itis!Differential equations areimmediately converted, bysight, into mere algebraic equations; wevirtually have thesolution bysight, that _ F/m *-mi’ since (iw)2 =—w2. This may beslightly simplified bysubstituting k/m =wfi, which gives x=F/m(a3 -(.02). (23.5) This, ofcourse, isthesolution wehadbefore; forsince m(w§ —wz)isarealnum- ber,thephase angles ofFandof3arethesame (orperhaps 180°apart, ifw2>wfi), asadvertised previously, Themagnitude of>2,which measures howfaritoscillates, isrelated tothesizeoftheFbythefactor 1/m(w§ —<02),andthisfactor becomes 23-2 enormous when wisnearly equal towo.Sowegetavery strong response when weapply theright frequency w(ifwehold apendulum ontheendofastring and shake itatjust theright frequency, wecanmake itswing very high). 23-2 Theforced oscillator withdamping That, then, ishow weanalyze oscillatory motion with themore elegant mathematical technique. Buttheelegance ofthetechnique isnotatallexhibited insuch aproblem thatcanbesolved easily byother methods. Itisonly exhibited when oneapplies ittomore difficult problems. Letustherefore solve another, more difficult problem, which furthermore adds arelatively realistic feature tothe previous one. Equation (23.5) tellsusthatifthefrequency wwere exactly equal to wo,wewould have aninfinite response. Actually, ofcourse, nosuch infinite response occurs because some other things, likefriction, which wehave sofar ignored, limits theresponse. Letustherefore addtoEq.(23.2) afriction term. Ordinarily such aproblem isvery diflicult because ofthecharacter and complexity ofthefrictional term. There are,however, many circumstances in which thefrictional force isproportional tothespeed with which theobject moves. Anexample ofsuch friction isthefriction forslow motion ofanobject inoilora thick liquid. There isnoforce when itisjuststanding still,butthefaster itmoves thefaster theoilhastogopast theobject, andthegreater istheresistance. So weshall assume thatthere is,inaddition totheterms in(23.2), another term, a resistance force proportional tothevelocity: F,»=—cdx/dt. Itwillbeconvenient, inourmathematical analysis, towrite theconstant casmtimes 'Ytosimplify the equation alittle. This isjustthesame trick weusewith kwhen wereplace itby mwg, justtosimplify thealgebra. Thus ourequation willbe m(d2x/dtz) +c(dx/dt) -1-kx=F (23.6) or,writing c=mvandk=mwganddividing outthemass m, (d2x/dt2) +v(dx/dt) +wfix=F/m. (23.6a) Now wehave theequation inthemost convenient form tosolve. If7isvery small, thatrepresents very little friction; ifVisvery large, there isatremendous amount offriction. How dowesolve thisnewlinear differential equation? Suppose thatthedriving force isequal toF0cos(wt-1-A);wecould putthisinto(23.6a) andtrytosolve it,butweshall instead solve itbyournewmethod. Thus we write FastherealpartofFe!“ andxastherealpartof52¢“, andsubstitute these intoEq.(23.6a). Itisnoteven necessary todotheactual substituting, forwecan seebyinspection thattheequation would become [(i@)%e +v(r@)5e +ttfix].-fw‘ =(F/m)ei"'. (23.7) [Asamatter offact,ifwetried tosolve Eq.(23.6a) byouroldstraightforward way, wewould really appreciate themagic ofthe“complex” method.] Ifwedivide by em‘onboth sides. then wecanobtain theresponse >2tothegiven force F;itis 2=F/m(a5 -C02+rm). (23.8) Thus again 52isgiven byFtimes acertain factor. There isnotechnical name forthisfactor, noparticular letter forit,butwemaycallitRfordiscussion pur- poses: l R=arm.m(w0 —w+I'Yw) and 52=FR. (23.9) (Although theletters Yandwoareinvery common use,thisRhasnoparticular name.) This factor Rcaneither bewritten asp+iq,orasacertain magnitude ptimes e".Ifitiswritten asacertain magnitude times e",letusseewhat itmeans. 23-3 P2 _>r I f | l 8’. 0 I (0 Fig. 23-2. PlotofpfVersus w. O 8 -90‘ 1 1 1 1 1 1 1 W. Q‘, -I80 — Fig. 23-3. Plotof0versus w.Now F=FGe“, andtheactual force Fistherealpart ofF0e‘Ael°", that is, F0cos(wt-1-A).Next, Eq.(23.9) tells usthat >2isequal toFR. So,writing R=pe”asanother name forR,weget x=RF=pe“F0eM =pF0em+A). Finally, going even further back, weseethatthephysical x,which istherealpart ofthecomplex 52,isequal totherealpartofpF(,e‘(’+“)e“". ButpandF0arereal, andtherealpartofe‘“'+A+”” issimply cos(wt-1-A+6).Thus x=pF0cos(wt+A+0). (23.10) This tellsusthattheamplitude oftheresponse isthemagnitude oftheforce F multiplied byacertain magnification factor, p;thisgives usthe“amount” of oscillation. Italsotellsus,however, thatxisnotoscillating inphase with the force, which hasthephase A,butisshifted byanextra amount 0.Therefore p and0represent thesizeoftheresponse andthephase shift oftheresponse. Now letuswork outwhat pis.Ifwehave acomplex number, thesquare of themagnitude isequal tothenumber times itscomplex conjugate; thus 2 l P= . .m2(w§ —w2+1Vw)(w§ —w2—z'Yw) I (23.11) =m21(~»’-4%?+v’421' Inaddition, thephase angle 0iseasy tofind, forifwewrite 1/R=1/pe”=(1/we-"” =m(<»%—4*+in»).weseethat tan0=-—“/w/(wg —w2). (23.12) Itisminus because tan(-0) =—tan 0.Anegative value for0results forallw, andthiscorresponds tothedisplacement xlagging theforce F. Figure 23-2 shows howp2varies asafunction offrequency (p2isphysically more interesting than p,because p2isproportional tothesquare oftheamplitude, ormore orlesstotheenergy thatisdeveloped intheoscillator bytheforce). We seethatif7isverysmall, then 1/(wfi—w2)2 isthemost important term, andthe response triestogouptoward infinity when wequals wo.Now the“infinity” isnot actually infinite because ifw=wo,then 1/V2802 isstillthere. Thephase shift varies asshown inFig.23-3. Incertain circumstances wegetaslightly different formula than (23.8), also called a“resonance” formula, andonemight think thatitrepresents adifferent phenomenon, butitdoes not. Thereason isthatif'Yisverysmall themost interest- ingpartofthecurve isnearw=wo,andwemayreplace (23.8) byanapproximate formula which isvery accurate if7issmall andwisnear wo. Since 853—m2= (wo—w)(w(, +w),ifwisnear wothisisnearly thesame as2w0(w0 —w)and 'Ywisnearly thesame as'Yw0. Using these in(23.8), weseethatw§—wz+ivw= 2w0(w0 -—w-1-i7/2), sothat >2zF/2mw0(w0 -8.+iv/2)11v<<8.,and1.5==850.(23.13) Itiseasy tofindthecorresponding formula forp2.Itis p2z1/411121.13 [(w0 —w)2+'Y2/4]. Weshall leave ittothestudent toshow thefollowing: ifwecallthemaximum height ofthecurve ofp2vs.woneunit, andweaskforthewidth Awofthecurve, atonehalfthemaximum height, thefullwidth athalfthemaximum height of thecurve isAw=‘Y,supposing that 7issmall. Theresonance issharper and sharper asthefrictional effects aremade smaller andsmaller. 23-4 Asanother measure ofthewidth, some people useaquantity Qwhich is defined asQ=wo/'Y. Thenarrower theresonance, thehigher theQ:Q=1000 means aresonance whose width isonly l000th ofthefrequency scale. TheQof theresonance curve shown inFig.23-2 is5. Theimportance oftheresonance phenomenon isthatitoccurs inmany other circumstances, andsotherestofthischapter willdescribe some ofthese other circumstances. 23-3 Electrical resonance Thesimplest andbroadest technical applications ofresonance areinelectricity. Intheelectrical world there areanumber ofobjects which canbeconnected to make electric circuits. These passive circuit elements, asthey areoften called, are ofthree main types, although each onehasalittle bitoftheother twomixed in. Before describing them ingreater detail, letusnote thatthewhole idea ofour mechanical oscillator being amass ontheendofaspring isonlyanapproximation. Allthemass isnotactually atthe“mass”; some ofthemass isintheinertia ofthe spring. Similarly, allofthespring isnotatthe“spring”; themass itself hasalittle elasticity, andalthough itmay appear so,itisnotabsolutely rigid, andasitgoes upanddown, itflexes ever soslightly under theaction ofthespring pulling it. Thesame thing istrueinelectricity. There isanapproximation inwhich wecan lump things into“circuit elements” which areassumed tohave pure, ideal char- acteristics. Itisnottheproper time todiscuss thatapproximation here, weshall simply assume thatitistrueinthecircumstances. Thethree main kinds ofcircuit elements arethefollowing. Thefirstiscalled acapacitor (Fig. 23-4); anexample istwoplane metallic plates spaced averysmall distance apart byaninsulating material. When theplates arecharged there isa certain voltage difference, thatis,acertain difference inpotential, between them. Thesame difference ofpotential appears between theterminals AandB,because ifthere were anydifference along theconnecting wire, electricity would flowright away. Sothere isacertain voltage difierence Vbetween theplates ifthere isa certain electric charge +qand -qonthem, respectively. Between theplates there willbeacertain electric field; wehave even found aformula forit(Chapters 13and14): V=ad/so =qd/e0A, (23.14) where disthespacing andAisthearea oftheplates. Note thatthepotential difference isalinear function ofthecharge. Ifwedonothave parallel plates, but insulated electrodes which areofanyother shape, thedifference inpotential is stillprecisely proportional tothecharge, buttheconstant ofproportionality may notbesoeasy tocompute. However, allweneed toknow isthatthepotential difference across acapacitor isproportional tothecharge: V=q/C;thepropor- tionality constant is1/C,where Cisthecapacitance oftheobject. The second kind ofcircuit element iscalled aresistor; itoffers resistance tothefiow ofelectrical current. ltturns outthat metallic wires and many other substances resist theflow ofelectricity inthismanner: ifthere isavoltage difference across apiece ofsome substance, there exists anelectric current I= dq/dt thatisproportional totheelectric voltage difference: V=RI=Rdq/dt. (23.15) Theproportionality coefficient iscalled theresistance R.This relationship may already befamiliar toyou; itisOhm’s law. Ifwethink ofthecharge qonacapacitor asbeing analogous tothedisplace- ment xofamechanical system, weseethatthecurrent, I=dq/dt, isanalogous tovelocity, 1/Cisanalogous toaspring constant k,andRisanalogous tothe resistive coefficient V.Now itisvery interesting thatthere exists another circuit element which istheanalog ofmass! This isacoilwhich builds upamagnetic field within itself when there isacurrent init.Achanging magnetic fielddevelops inthe coilavoltage thatisproportional toall/dt (this ishow atransformer works, in 23-5A C E CAPACITOR RESISTOR INDUCTOR Fig. 23-4. The three passive circuit elements. Fig. 23-5. Anoscillatory electrical circuit with resistance, inductance, and capacitance.fact). Themagnetic field isproportional toacurrent, andtheinduced voltage (so-called) insuch acoilisproportional totherateofchange ofthecurrent: V=Ldl/dt =Ldzq/dt2. (23.16) Thecoefficient Listheself-inductance, andisanalogous tothemass inamechanical oscillating circuit. Suppose wemake acircuit inwhich wehave connected thethree circuit elements inseries (Fig. 23-5); then thevoltage across thewhole thing from 1to2 isthework done incarrying acharge through, anditconsists ofthesumofseveral pieces: across theinductor, V1,=Ld2q/dt2; across theresistance, VR=Rdq/dt; across thecapacitor, V,=q/C.Thesumofthese isequal totheapplied voltage, V: Ldzq/dtz +Rdq/dt +q/C=V(t). (23.17) Now weseethatthisequation isexactly thesame asthemechanical equation (23.6), andofcourse itcanbesolved inexactly thesame manner. Wesuppose that V(t) isoscillatory: wearedriving thecircuit with agenerator with apure sinewave oscillation. Then wecanwrite ourV(t)asacomplex I7with theunderstanding thatitmust beultimately multiplied byem,andtherealpart taken inorder to findthetrue V.Likewise, thecharge qcanthus beanalyzed, andthen inexactly thesame manner asinEq.(23.8) wewrite thecorresponding equation: thesecond derivative of1;is(iw)2q; thefirstderivative is(iw)q. Thus Eq.(23.17) translates to [L(iw)2 +R(iw)+2;=V OI‘ V11=mm? L(iw)2 +R(iw)+6 which wecanwrite intheform q=V/L(o§ -82+mi), (23.18) where wf,=1/LC andV=R/L. Itisexactly thesame denominator aswehadin themechanical case, with exactly thesame resonance properties! Thecorrespond- ence between theelectrical andmechanical cases isoutlined inTable 23-1. Table 23-1 General Mechanical Electrical characteristic property property indep. variable time (t) time (t) dep.variable position (x) charge (q) inertia mass (m) inductance (L) resistance drag coeff. (c='Ym) resistance (R=7L) stiffness stiffness (k) (capacitance)_1 (1/C) resonant frequency wfi=k/m 8,3=1/LC period to=21r\/m?/It to=21r\/l._C‘ figure ofmerit Q=w()/‘Y Q=w()L/R Wemust mention asmall technical point. Intheelectrical literature, adifferent notation isused. (From onefieldtoanother, thesubject isnotreally anydifferent, butthewayofwriting thenotations isoften different.) First, jiscommonly used instead ofiinelectrical engineering, todenote \/Tl. (After all,imust betheeur- rent!) Also, theengineers would rather have arelationship between Vandithan between Vand.7,justbecause they aremore used toitthatway. Thus, since l=dq/dt =iwq,wecanjustsubstitute F/iwfor2;andget V=(z<.5L+R+1/iwC)l= Zi. (23.19) 23-6 Another wayistorewrite Eq.(23.17), sothatitlooks more familiar; oneoften sees itwritten thisway: Ldl/dt +RI+(1/C)/‘rd: =v(t). (23.20) Atanyrate, wefindtherelation (23.19) between voltage Vandcurrent fwhich is justthesame as(23.18) except divided byiw,andthatproduces Eq.(23.19). The quantity R-1-iwL-1-1/iwC isacomplex number, andisused somuch inelectrical engineering thatithasaname: itiscalled thecomplex impedance, Z.Thus wecan write V=Zl. Thereason thattheengineers liketodothisisthatthey learned something when theywere young: V=RIforresistances, when they only knew about resistances andDC. Now they have become more educated andhave AC circuits, sotheywant theequation tolook thesame. Thus theywrite V=Zl,the only difference being thattheresistance isreplaced byamore complicated thing, acomplex quantity. Sothey insist thattheycannot usewhat everyone elseinthe world usesforimaginary numbers, theyhave touseajforthat; itisamiracle that theydidnotinsist alsothattheletter ZbeanR!(Then theygetintotrouble when theytalkabout current densities, forwhich theyalsousej.Thedifficulties ofscience aretoalarge extent thedifficulties ofnotations, theunits, andalltheother arti- ficialities which areinvented byman, notbynature.) 23-4 Resonance innature Although wehave discussed theelectrical case indetail, wecould alsobring upcaseafter caseinmany fields, andshow exactly howtheresonance equation is thesame. There aremany circumstances innature inwhich something is“oscilla- ting” andinwhich theresonance phenomenon occurs. Wesaidthatinanearlier chapter; letusnow demonstrate it.Ifwewalk around ourstudy, pulling books offtheshelves andsimply looking through them tofindanexample ofacurve thatcorresponds toFig.23-2 andcomes from thesame equation, what dowefind? Justtodemonstrate thewide range obtained bytaking thesmallest possible sample, ittakes only fiveorsixbooks toproduce quite aseries ofphenomena which show resonances. Thefirsttwoarefrom mechanics, thefirstonalarge scale: theatmosphere ofthewhole earth. Iftheatmosphere, which wesuppose surrounds theearth evenly onallsides, ispulled toonesidebythemoon or,rather, squashed prolate intoadouble tide, andifwecould thenletitgo,itwould gosloshing upanddown; itisanoscillator. This oscillator isdriven bythemoon, which iseffectively re- volving about theearth; anyonecomponent oftheforce, sayinthex-direction, has acosine component, andsotheresponse oftheearth’s atmosphere tothetidal pull ofthemoon isthatofanoscillator. Theexpected response oftheatmosphere is shown inFig.23-6, curve b(curve aisanother theoretical curve under discussion inthebook from which thisistaken outofcontext). Now onemight think thatwe only have onepoint onthisresonance curve, since weonlyhave theonefrequency, corresponding totherotation oftheearth under themoon, which occurs ata period of12.42 hours-12 hours fortheearth (thetideisadouble bump), plusa little more because themoon isgoing around. Butfrom thesizeoftheatmospheric tides, andfrom thephase, theamount ofdelay, wecangetboth pand6.From those wecangetwoandV,andthusdraw theentire curve! This isanexample of very poor science. From twonumbers weobtain twonumbers, andfrom those twonumbers wedraw abeautiful curve, which ofcourse goes through thevery point thatdetermined thecurve! Itisofnouseunless wecanmeasure something else,andinthecaseofgeophysics thatisoften verydifficult. Butinthisparticular case there isanother thing which wecanshow theoretically must have thesame timing asthenatural frequency wo:thatis,ifsomeone disturbed theatmosphere, itwould oscillate with thefrequency wo.Now there wassuch asharp disturbance in1883; theKrakatoa volcano exploded andhalftheisland blew off,anditmade such aterrific explosion intheatmosphere that theperiod ofoscillation ofthe atmosphere could bemeasured. Itcame outto10%hours. Thewoobtained from 23-710Cycles perday --3I a 11 1 /’b 5 D _:r:::_'%8 121142 10820 Fig. 23-6. Response ofthe atmos- phere toexternal excitation. aisthere- quired response iftheatmospheric S2-tide isofgravitational origin; peak amplifi- cation islOO=l. bisderived from observed magnification cind phase ofM;-tide. [Munk and MacDonald, “Rotation of theEarth," Cambridge University Press (195011 Iransmssaon(5)88ioo so zo 0 140455055606570 Wavelength inmicrons (104cm) Fig. 23-7. Transmission ofinfrared radiation through athin(0.17 /.1)sodium chloride film. [After R.B.Barnes, Z. Physik 75,723 119321. Kitlel, Introduc- tiontoSolid State Physics, Wiley, 1956.] 20 M%N§'$INIKGYK35ISite MQLiii’:biI‘CAVFTYAMIML‘23Zis5»z‘-V5!4;G. OIRSTEDS 0.2 0 _._A '.I3| ‘F. 53¢ =".. aw ;'-'.. ;1'. Y STATIC MOGNCYIC WILD INOIRITIDS Fig. 23-8. Magnetic energy loss in paramagnetic organic compound asfunc- tion ofapplied magnetic field intensity. [Holden etciI.,Phys. Rev.75,1614 11949)]Fig.23-6 comes out10hours and20minutes, sothere wehave atleast onecheck onthereality ofourunderstanding oftheatmospheric tides. Next wegotothesmall scale ofmechanical oscillation. Thistimewetake asodium chloride crystal, which hassodium ions andchlorine ions next toeach other, aswedescribed inanearly chapter. These ions areelectrically charged, alternately plusandminus. Now there isaninteresting oscillation possible. Sup- pose thatwecould drive allthepluscharges totheright andallthenegative charges totheleft,andletgo;theywould then oscillate back andforth, thesodium lattice against thechlorine lattice. How canweeverdrive such athing? That iseasy, for ifweapply anelectric field onthecrystal, itwillpush thepluscharge onewayand theminus charge theother way! So,byhaving anexternal electric field wecan perhaps getthecrystal tooscillate. Thefrequency oftheelectric fieldneeded isso high, however, thatitcorresponds toinfrared radiation! Sowetrytofindareso- nance curve bymeasuring theabsorption ofinfrared light bysodium chloride. Such acurve isshown inFig.23-7. Theabscissa isnotfrequency, butisgiven in terms ofwavelength, butthatisjustatechnical matter, ofcourse, since forawave there isadefinite relation between frequency andwavelength; soitisreally a frequency scale, andacertain frequency corresponds totheresonant frequency. Butwhat about thewidth? What determines thewidth? There aremany cases inwhich thewidth thatisseen onthecurve isnotreally thenatural width 7thatonewould have theoretically. There aretworeasons why there canbea wider curve than thetheoretical curve. Iftheobjects donotallhave thesame frequency, asmight happen ifthecrystal were strained incertain regions, sothat inthose regions theoscillation frequency were slightly difierent than inother regions, then what wehave ismany resonance curves ontopofeach other; sowe apparently getawider curve. Theother kind ofwidth issimply this: perhaps we cannot measure thefrequency precisely enough—if weopen theslitofthespectrom- eterfairly wide, soalthough wethought wehadonly onefrequency, weactually hadacertain range Aw,then wemaynothave theresolving power needed toseea narrow curve. Ofi"hand, wecannot saywhether thewidth inFig.23-7 isnatural, orwhether itisduetoinhomogeneities inthecrystal orthefinite width oftheslit ofthespectrometer. Now weturntoamore esoteric example, andthatistheswinging ofamagnet. Ifwehave amagnet, with north andsouth poles, inaconstant magnetic field, the Nendofthemagnet willbepulled onewayandtheSendtheother way, andthere willingeneral beatorque onit,soitwillvibrate about itsequilibrium position, likeacompass needle. However, themagnets wearetalking about areatoms. These atoms have anangular momentum, thetorque does notproduce asimple motion inthedirection ofthefield, butinstead, ofcourse, aprecession. Now, looked at from theside, anyonecomponent is“swinging,” andwecandisturb ordrive that swinging andmeasure anabsorption. Thecurve inFig.23-8 represents atypical such resonance curve. What hasbeen done here isslightly different technically. Thefrequency ofthelateral field thatisused todrive thisswinging isalways kept thesame, while wewould have expected thattheinvestigators would varythatand plotthecurve. They could have done itthatway, buttechnically itwaseasier for them toleave thefrequency wfixed, andchange thestrength oftheconstant magnetic field, which corresponds tochanging woinourformula. They have plotted theresonance curve against wq.Anyway, thisisatypical resonance with a certain woandY. Now wegostillfurther. Ournextexample hastodowith atomic nuclei. The motions ofprotons andneutrons innuclei areoscillatory incertain ways, andwe candemonstrate thisbythefollowing experiment. Webombard alithium atom with protons, andwediscover thatacertain reaction, producing V-rays, actually hasavery sharp maximum typical ofresonance. Wenote inFig.23-9, however, onedifference from other cases: thehorizontal scale isnotafrequency, itisan energy! Thereason isthatinquantum mechanics what wethink ofclassically as theenergy willturn outtobereally related toafrequency ofawave amplitude. When weanalyze something which insimple large-scale physics hastodowith a frequency, wefindthatwhen wedoquantum-mechanical experiments with atomic 23-8 Y50IO ‘ I-RA OF Fig. 23-9. The intensity ofgamma- YELDA radiation from lithium ascifunction ofthe energy ofthebombarding protons. The dashed curve isatheoretical one cal- culated for protons with anangular1 II 1I t\I e - I \ i\\ . ° 2 I \\ . \ _'.¢' n ‘~~ 1:2j- momentum Z: O.[Bonner and Evans, Phys. Rev. 73,666 (1948)]300 O00 matter, wegetthecorresponding curve asafunction ofenergy. Infact, thiscurve isademonstration ofthisrelationship, inasense. Itshows that frequency and energy have some deep interrelationship, which ofcourse they do. Now weturntoanother example which alsoinvolves anuclear energy level, but nowamuch, much narrower one. ThewoinFig.23-10 corresponds toanenergy of 100,000 electron volts, while thewidth 7isapproximately IOT5 electron volt; inother words, thishasaQof101°! When thiscurve wasmeasured itwasthe largest Qofanyoscillator thathadeverbeen measured. Itwasmeasured byDr. Moessbauer, anditwasthebasis ofhisNobel prize. Thehorizontal scale here is velocity, because thetechnique forobtaining theslightly difierent frequencies was tousetheDoppler effect, bymoving thesource relative totheabsorber. Onecan seehowdelicate theexperiment iswhen werealize thatthespeed involved isafew centimeters persecond! Ontheactual scale ofthefigure, zero frequency would correspond toapoint about 101° cmtotheleft—slightly offthepaper! Finally, ifwelook inanissue ofthePhysical Review, saythatofJanuary 1, 1962, willwefindaresonance curve? Every issue hasaresonance curve, andFig. 23-11 istheresonance curve forthisone. This resonance curve turns outbebe very interesting. Itistheresonance found inacertain reaction among strange particles, areaction inwhich aK’andaproton interact. Theresonance isde- tected byseeing howmany ofsome kinds ofparticles come out,anddepending on what andhow many come out,onegetsdifferent curves, butofthesame shape andwiththepeak atthesame energy. Wethusdetermine thatthere isaresonance atacertain energy fortheK‘meson. That presumably means thatthere issome kind ofastate, orcondition, corresponding tothisresonance, which canbeattained byputting together aK‘andaproton. Thisisanewparticle, orresonance. Today wedonotknow whether tocallabump likethisa“particle” orsimply aresonance. When there isavery sharp resonance, itcorresponds toavery definite energy, justasthough there were aparticle ofthatenergy present innature. When the resonance getswider, thenwedonotknow whether tosaythere isaparticle which does notlastvery long, orsimply aresonance inthereaction probability. Inthe second chapter, thispoint ismade about theparticles, butwhen thesecond chapter waswritten thisresonance wasnotknown, soourchart should now have still another particle init! 23~9I00 O PROTON ENERGY INKEV AI;z-lo" o 2-|o'° 4-* A5 O-1.<2.+.*@"~'~= »~ "0.4% '0.8% -L27, Fig. 23-10. [Courtesy ofDr. R. Mossbauer] 3 1’ Is ‘ I I 0 _ I.w ll slb A. 5 9‘ 0 P.llnvlsl Fig. 23-11. Momentum dependence ofthecross section forthereactions la) K‘-l—p—>A-1-1r++1r“and (b) K‘-1-p-—>K°+n.The lower curves in(a)and (b)represent thepresumed nonresonant backgrounds, while theupper curves contain inaddition thesuperposed resonance. [Ferro-Luzzi etal.,Phys. Rev., Lett. 8,28(1962)] 24 Transients 24-1 Theenergy ofanoscillator Although thischapter isentitled “transients,” certain parts ofitare,inaway, partofthelastchapter onforced oscillation. Oneofthefeatures ofaforced oscilla- tionwhich wehave notyetdiscussed istheenergy intheoscillation. Letusnow consider thatenergy. Inamechanical oscillator, howmuch kinetic energy isthere? Itisproportional tothesquare ofthevelocity. Now wecome toanimportant point. Consider an arbitrary quantity A,which maybethevelocity orsomething elsethatwewant to discuss. When wewrite A=Ael”‘,acomplex number, thetrueandhonest A, inthephysical world, isonly therealpart; therefore if,forsome reason, wewant tousethesquare ofA,itisnotright tosquare thecomplex number andthen take therealpart, because therealpartofthesquare ofacomplex number isnotjust thesquare oftherealpart, butalsoinvolves theimaginary part. Sowhen wewish tofindtheenergy wehave togetaway from thecomplex notation forawhile to seewhat theinner workings are. Now thetruephysical Aistherealpart ofA0e““‘+“>, that is,A=A0cos (wt+A),where A,thecomplex number, iswritten asAge“. Now thesquare of thisrealphysical quantity isA2=Agcos2 (wt+A).Thesquare ofthequantity, then, goes upanddown from amaximum tozero, likethesquare ofthecosine. Thesquare ofthecosine hasamaximum of1andaminimum of0,anditsaverage value is1/2. Inmany circumstances wearenotinterested intheenergy atanyspecific moment during theoscillation; foralarge number ofapplications wemerely want theaverage ofA2,themean ofthesquare ofAoveraperiod oftimelarge compared with theperiod ofoscillation. Inthose circumstances, theaverage ofthecosine squared may beused, sowehave thefollowing theorem: ifAisrepresented bya complex number, then themean ofA2isequal to%A?,. Now A2,isthesquare of themagnitude ofthecomplex A.(This canbewritten inmany ways—some people liketowrite |A|2;others write, AA*, Atimes itscomplex conjugate.) Weshall use thistheorem several times. Now letusconsider theenergy inaforced oscillator. Theequation forthe forced oscillator is md2x/dt2 +7mdx/dt+mafix=F(t). (24.1) Inourproblem, ofcourse, F(t)isacosine function oft.Now letusanalyze the situation: howmuch work isdone bytheoutside force F?Thework done bythe force persecond, i.e.,thepower, istheforce times thevelocity. (We know that thedifierential work inatime dtisFdx,andthepower isFdx/dt.) Thus dx dx dzx dx dx2 Butthefirsttwoterms ontheright canalso bewritten asd/dt[%m(dx/dz)2 + %mw§x2], asisimmediately verified bydifferentiating. That istosay,theterm in brackets isapure derivative oftwoterms thatareeasy tounderstand—one isthe kinetic energy ofmotion, andtheother isthepotential energy ofthespring. Let uscallthisquantity thestored energy, thatis,theenergy stored intheoscillation. Suppose thatwewant theaverage power over many cycles when theoscillator is being forced andhasbeen running foralong time. Inthelong run, thestored 24-124-1 Theenergy ofanoscillator 24-2 Damped oscillations 24-3 Electrical transients energy doesnotchange—its derivative gives zeroaverage effect. Inother words, ifweaverage thepower inthelong run, alltheenergy ultimately ends upinthe resistive term’Ym(dx/dt)2. There issome energy stored intheoscillation, butthat doesnotchange withtime, ifweaverage overmany cycles. Therefore themean power (P)is (P)=('Ym(dx/dt)2). (24.3) Using ourmethod ofwriting complex numbers, andourtheorem that(A2)= §A§, wemay findthismean power. Thus ifx=few‘, then dx/dt =iwJ?e““‘. Therefore, inthese circumstances, theaverage power could bewritten as (P)=%'Ymw2x%. (24.4) Inthenotation forelectrical circuits, dx/dt isreplaced bythecurrent I(Iis dq/dt, where qcorresponds tox),andm7corresponds totheresistance R.Thus therateoftheenergy loss—the power used upbytheforcing function—is the resistance inthecircuit times theaverage square ofthecurrent: (P)=R(I2)=R-s15. (24.5) This energy, ofcourse, goes intoheating theresistor; itissometimes called the heating lossortheJoule heating. Another interesting feature todiscuss ishow much energy isstored. That is notthesame asthepower, because although power wasatfirstused tostore up some energy, after thatthesystem keeps onabsorbing power, insofar asthere are anyheating (resistive) losses. Atanymoment there isacertain amount ofstored energy, sowewould liketocalculate themean stored energy (E)also. Wehave already calculated what theaverage of(dx/dt)2 is,sowefind <E>=%m<<dx/do”> +%m<»%<x”> (2,6) =%m(w2 +w§)%X§- Now, when anoscillator isvery efficient, andifwisnear wo,sothat[flislarge, thestored energy isvery high—we cangetalarge stored energy from arelatively small force. Theforce does agreat dealofwork ingetting theoscillation going, butthentokeep itsteady, allithastodoistofight thefriction. Theoscillator can have agreat dealofenergy ifthefriction isverylow,andeven though itisoscillating strongly, notmuch energy isbeing lost. Theefficiency ofanoscillator canbe measured byhowmuch energy isstored, compared with howmuch work theforce does peroscillation. Howdoesthestored energy compare withtheamount ofwork thatisdone in onecycle? Thisiscalled theQofthesystem, andQisdefined as21rtimes the mean stored energy, divided bythework done percycle. (Ifwewere tosaythe work done perradian instead ofpercycle, thenthe21rdisappears.) %"1(w2 +w§)'(X2) wz+<03Q=2” 'Ymw2(x2) -21r/w = 2'Yw l (241) Qisnotaveryuseful number unless itisverylarge. When itisrelatively large, it gives ameasure ofhowgood theoscillator is.People have tried todefine Qinthe simplest andmost useful way; various definitions differ abitfrom oneanother, butifQisverylarge, alldefinitions areinagreement. Themost generally accepted definition isEq.(24.7), which depends onw.Foragood oscillator, close toreso- nance, wecansimplify (24.7) alittle bysetting w=wo,andwethen have Q= coo/’Y, which isthedefinition ofQthatweusedbefore. What isQforanelectrical circuit? Tofindout,wemerely have totranslate Lform,Rformv,and1/Cformwfi(seeTable 23-1). TheQatresonance is Loo/R, where wistheresonance frequency. Ifweconsider acircuit withahighQ, thatmeans thattheamount ofenergy stored intheoscillation isverylarge compared with theamount ofwork done percycle bythemachinery thatdrives theoscilla- tions. 24-2 24-2 Damped oscillations Wenowturntoourmaintopic ofdiscussion: transients. Byatransient ismeant asolution ofthedifferential equation when there isnoforce present, butwhen the system isnotsimply atrest. (Ofcourse, ifitisstanding stillattheorigin with no force acting, thatisaniceproblem—it stays there!) Suppose theoscillation starts another way: sayitwasdriven byaforce forawhile, andthenweturnofftheforce. What happens then? Letusfirstgetarough ideaofwhat willhappen foravery high Qsystem. Solong asaforce isacting, thestored energy stays thesame, and there isacertain amount ofwork done tomaintain it.Now suppose weturn ofi" theforce, andnomore work isbeing done; then thelosses which areeating upthe energy ofthesupply arenolonger eating upitsenergy—there isnomore driver. Thelosses willhave toconsume, sotospeak, theenergy thatisstored. Letus suppose thatQ/21r =1000. Then thework done percycle is1/1000 ofthestored energy. Isitnotreasonable, since itisoscillating withnodriving force, thatinone cycle thesystem willstillloseathousandth ofitsenergy E,which ordinarily would have been supplied from theoutside, andthatitwillcontinue oscillating, always losing 1/1000 ofitsenergy percycle? So,asaguess, forarelatively high Q system, wewould suppose that thefollowing equation might beroughly right (wewilllater doitexactly, anditwillturnoutthatitwasright!): dE/dt=—wE/Q. (24.8) This isrough because itistrueonly forlarge Q.Ineach radian thesystem loses a fraction 1/Qofthestored energy E.Thus inagiven amount oftimedttheenergy willchange byanamount wdt/Q,since thenumber ofradians associated with the time dtiswdt.What isthefrequency? Letussuppose thatthesystem moves so nicely, withhardly anyforce, thatifweletgoitwilloscillate atessentially thesame frequency allbyitself. SoWewillguess thatwistheresonant frequency (.00.Then wededuce from Eq.(24.8) thatthestored energy willvary as E=E0e_"’<>”Q =E0e'". (24.9) Thiswould bethemeasure oftheenergy atanymoment. What would theformula be,roughly, fortheamplitude oftheoscillation asafunction ofthetime? The same? No! Theamount ofenergy inaspring, say,goes asthesquare ofthedis- placement; thekinetic energy goes asthesquare ofthevelocity; sothetotal energy goes asthesquare ofthedisplacement. Thus thedisplacement, theamplitude of oscillation, willdecrease halfasfastbecause ofthesquare. Inother words, we guess thatthesolution forthedamped transient motion willbeanoscillation of frequency close totheresonance frequency wo,inwhich theamplitude ofthesine- wave motion willdiminish ase""” 2: x=A0e_”/2 coswot. (24.10) This equation andFig.24-l giveusanideaofwhat weshould expect; nowletus trytoanalyze themotion precisely bysolving thedifferential equation ofthe motion itself. So,starting with Eq.(24.1), with nooutside force, howdowesolve it?Being physicists, wedonothave toworry about themethod asmuch aswedoabout what thesolution is.Armed with ourprevious experience, letustryasasolution an exponential curve, x=Ae'2"‘. (Why dowetrythis? Itistheeasiest thing to differentiate!) Weputthisinto (24.1) (with F(t) =0),using therulethateach timewedifferentiate xwith respect totime, wemultiply byia.Soitisreally quite simple tosubstitute. Thus ourequation looks likethis: (-a2+i'Yd+w%)Ae‘“‘ =0. (24.11) Thenetresult must bezero foralltimes, which isimpossible unless (a)A=0, which isnosolution atall—it stands still, or(b) -<12+iofl+60%=0. (24.12) 24-3\ X\\ Fi tion.\\\/e /_e-rt/2 _7l/2 coswot\\\\ ,—""' /’// // / / 24-1. Adamped cosine oscilla Ifwecansolve thisandfindana,thenwewillhave asolution inwhich Aneed notbezero! 0.=iv/2i\/4,3-)2/4. (24.13) Forawhile weshall assume that7isfairly small compared with wo,sothat wfi—'Y2/4 isdefinitely positive, andthere isnothing thematter with taking the square root. Theonly bothersome thing isthatwegettwosolutions! Thus <11=i'Y/2+x/a3-12/4=iv/2+40-, (24.14) 0.2=iv/2-\/<43-)2/4=iv/2-41.. (24.15) Letusconsider thefirstone,supposing thatwehadnotnoticed thatthesquare root hastwopossible values. Then weknow that asolution forxisx1=Ae‘2‘1‘, where Aisanyconstant whatever. Now, insubstituting a1,because itisgoing to come somany times andittakes solongtowrite, weshall call\/wfi -Y2/4=w.,- Thus i011=-7/2 +iw,,andwegetx=Ae‘_2'/2""““')‘, orwhat isthesame, because ofthewonderful properties ofanexponential,and x1=Ae_2”/2e"“’“/'. (24.16) First, werecognize thisasanoscillation, anoscillation atafrequency w.,which is notexactly thefrequency wo,butisrather close to(.00ifitisagood system. Second, theamplitude oftheoscillation isdecreasing exponentially! Ifwetake, forinstance, therealpartof(24.16), weget x1=Ae_"‘/2cos w.,t. (24.17) This isvery much likeourguessed-at solution (24.10), except thatthefrequency really is(11,.This istheonly error, soitisthesame thing—we have theright idea. Buteverything isnotallright! What isnotallright isthatthere isanother solution. Theother solution is0:2,andweseethatthedifference isonly thatthesign ofco,isreversed: x2=Be_"'2e_"“"". (24.18) What does thismean? Weshall soon prove thatifx1andx2areeach apossible solution ofEq.(24.1) with F=0,then x1+x2isalsoasolution ofthesame equation! Sothegeneral solution xisofthemathematical form x=e_"”2(Ae’2"“/‘ +Be_’2""). (24.19) Now wemaywonder whywebother togivethisother solution, since wewere so happy with thefirstoneallbyitself. What istheextra onefor,because ofcourse weknow weshould only taketherealpart? Weknow thatwemust takethereal part, buthowdidthemathematics know thatweonlywanted therealpart? When wehadanonzero driving force F(t),weputinanartificial force togowith it,and theimaginary partoftheequation, sotospeak, wasdriven inadefinite way. But when weputF(t)E0,ourconvention thatxshould beonly therealpart of whatever wewrite down ispurely ourown, andthemathematical equations do notknow ityet. Thephysical world hasarealsolution, buttheanswer thatwe were sohappy with before isnotreal, itiscomplex. Theequation does notknow thatwearearbitrarily going totake therealpart, soithastopresent us,soto speak, with acomplex conjugate typeofsolution, sothatbyputting them together wecanmake atruly realsolution; thatiswhat 012isdoing forus.Inorder forx tobereal, Be*’2’*‘ willhave tobethecomplex conjugate ofAe“‘Y‘, sothat the imaginary parts disappear. Soitturns outthatBisthecomplex conjugate ofA, andourrealsolution is x=er"/2(Ae*~~' +A*e-W). (24.20) Soourrealsolution isanoscillation with aphase shift andadamping—just as advertised. 24-4 24-3 Electrical transients Now letusseeiftheabove really works. Weconstruct theelectrical circuit shown inFig.24-2, inwhich weapply toanoscilloscope thevoltage across the inductance Lafter wesuddenly turnonavoltage byclosing theswitch S.Itisan oscillatory circuit, anditgenerates atransient ofsome kind. Itcorresponds toa circumstance inwhich wesuddenly apply aforce andthesystem starts tooscillate. Itistheelectrical analog ofadamped mechanical oscillator, andwewatch the oscillation onanoscilloscope, where weshould seethecurves thatwewere trying toanalyze. (The horizontal motion oftheoscilloscope isdriven atauniform speed, while thevertical motion isthevoltage across theinductor. Therestofthe circuit isonly atechnical detail. Wewould liketorepeat theexperiment many, many times, since thepersistence ofvision isnotgood enough toseeonly one trace onthescreen. Sowedotheexperiment again andagain byclosing the switch 60times asecond; each time weclose theswitch, wealsostart theoscillo- scope horizontal sweep, anditdraws thecurve over andover.) InFigs. 24-3 to 24-6 weseeexamples ofdamped oscillations, actually photographed onanoscillo- scope screen. Figure 24-3 shows adamped oscillation inacircuit which hasa high Q,asmall Y.Itdoes notdieoutvery fast; itoscillates many times onthe waydown. Butletusseewhat happens aswedecrease Q,sothattheoscillation diesout more rapidly. Wecandecrease Qbyincreasing theresistance Rinthecircuit. When weincrease theresistance inthecircuit, itdiesoutfaster (Fig. 24-4). Then ifweincrease theresistance inthecircuit stillmore, itdiesoutfaster still(Fig. 24-5). Butwhen weputinmore than acertain amount, wecannot seeanyoscilla- tionatall!Thequestion is,isthisbecause oureyesarenotgood enough? Ifwe increase theresistance stillmore, wegetacurve likethatofFig.24-6, which does notappear tohave anyoscillations, except perhaps one. Now, howcanweexplain thatbymathematics? The resistance is,ofcourse, proportional tothe'Yterm inthemechanical device. Specifically, 1isR/L. Now ifweincrease the"Yinthesolutions (24.14) and(24.15) thatwewere sohappy with before, chaos setsinwhen 'Y/2exceeds wo;wemust write itadifferent way, as iv/2+ix/‘Y2/4 -44% and iv/2-i\/'Y2/4 -413. Those arenow thetwosolutions and, following thesame lineofmathematical reasoning aspreviously, weagain findtwosolutions: e“"1‘ ande“'2‘. Ifwenow substitute fora1,weget X=Ae_('y/2+‘/'12/4—wg)t’ aniceexponential decay with nooscillations. Likewise, theother solution is xIBe_(.,/2_w/-12/4_»§)i_ Note thatthesquare root cannot exceed 7/2, because even ifwe=0,oneterm justequals theother. Butwfiistaken away from V2/4, sothesquare root isless than 7/2, andtheterm inparentheses is,therefore, always apositive number. Thank goodness! Why? Because ifitwere negative, wewould finderaised toa positive factor times t,which would mean itwasexploding! Inputting more and more resistance into thecircuit, weknow itisnotgoing toexplode—quite the contrary. Sonow wehave twosolutions, each onebyitself adying exponential, butonehaving amuch faster “dying rate” than theother. Thegeneral solution is ofcourse acombination ofthetwo; thecoefficients inthecombination depending upon how themotion starts—what theinitial conditions oftheproblem are. In theparticular waythiscircuit happens tobestarting, theAisnegative andtheB ispositive, sowegetthedifference oftwoexponential curves. Now letusdiscuss howwecanfindthetwocoefficients AandB(orAandA*), ifweknow howthemotion wasstarted. 24-5I’ L R-£'—'!*‘is=2 _@__ Fig.24-2. Anelectrical circuit for demonstrating transients. Figure 24-3 Figure 24-4 Figure 24-5 Figure 24-6 Suppose thatatt=0weknow thatx=xo,anddx/dt =v0.Ifweput t=0,x=xo,anddx/dt =v0intotheexpressions x=e—‘Yt/2(Ae1§w'yl_‘_ A*e—1k»7t), dx/dt=e_"/2[(—“//2 +iw,)Ae“’*‘ +(-'1/2 -i<»,)A*e-W], wefind, since e°=em=l, XQ=A+A*=2AR, 110=(-—'Y/2)(A +A*) +iw,(A —A*) =—7x0/2 +iw.,(2iA1), whereA =AR+iA1,andA*=AR—iA1. Thus wefind AR =X0/2 and A1=(2)0+'Yx0/2)/2w.,. (24.21) This completely determines Aand/1*,andtherefore thecomplete curve ofthe transient solution, interms ofhowitbegins. Incidentally, wecanwrite thesolution another wayifwenote that e”+e_" =2cos 0and e“—e““ =2isin0. Wemaythen write thecomplete solution as x=e'"'2 [xocosw-,t+———iU°+0;/xo/2 sinom], (24.22)‘Y where w.,=+\/wg —"12/4. This isthemathematical expression fortheway anoscillation diesout. Weshall notmake direct useofit,butthere areanumber ofpoints weshould liketoemphasize thataretrueinmore general cases. First ofallthebehavior ofsuch asystem with noexternal force isexpressed byasum, orsuperposition, ofpure exponentials intime (which wewrote asei°“). This isagood solution totryinsuch circumstances. Thevalues ofozmaybecom- plex ingeneral, theimaginary parts representing damping. Finally theintimate mathematical relation ofthesinusoidal and exponential function discussed in Chapter 22often appears physically asachange from oscillatory toexponential behavior when some physical parameter (inthiscase resistance, 'Y)exceeds some critical value. 24-6 25 Linear Systems and Review 25-1 Linear differential equations Inthischapter weshall discuss certain aspects ofoscillating systems thatare found somewhat more generally than justintheparticular systems wehave been discussing. Forourparticular system, thedifferential equation that wehave been solving is dz 4mfi+’YmBit‘+mwgx=F(t). (25.1) Now thisparticular combination of“operations” onthevariable xhastheinterest- ingproperty thatifwesubstitute (x+y)forx,then wegetthesumofthesame operations onxandy;or,ifwemultiply xbya,then wegetjustatimes thesame combination. This iseasy toprove. Justasa“shorthand” notation, because we gettired ofwriting down allthose letters in(25.1), weshall usethesymbol L.(x) instead. When weseethis,itmeans theleft-hand sideof(25.1), with xsubstituted in.With thissystem ofwriting, §(x+y)would mean thefollowing: 2 L(x+y)=m%;“—”) +rm"(_"§jl) +mwtrx+y).(25.2) (Weunderline theLsoastoremind ourselves thatitisnotanordinary function.) Wesometimes callthisanoperator notation, butitmakes nodifi‘erence what we callit,itisjust“shorthand.” Ourfirststatement wasthat L(x+J/)=L(x)+LO’), (25-3) which ofcourse follows from thefactthata(x+y)=ax+ay,d(x+y)/dt = dx/dt +dy/dt, etc. Oursecond statement was, forconstant a, L(ax) =a_L(x). (25.4) [Actually, (25.3) and(25.4) areveryclosely related, because ifweputx+xinto (25.3), thisisthesame assetting a=2in(25.4), andsoon.] Inmore complicated problems, there may bemore derivatives, andmore terms inL;thequestion ofinterest iswhether thetwoequations (25.3) and(25.4) aremaintained ornot. Ifthey are,wecallsuch aproblem alinear problem. In thischapter weshall discuss some oftheproperties thatexist because thesystem islinear, toappreciate thegenerality ofsome oftheresults thatwehave obtained inourspecial analysis ofaspecial equation. Now letusstudy some oftheproperties oflinear differential equations, having illustrated them already with thespecific equation (25.1) thatwehave stud- iedsoclosely. Thefirstproperty ofinterest isthis: suppose thatwehave tosolve thedifferential equation foratransient, thefreeoscillation with nodriving force. That is,wewant tosolve §(x) =0. (25.5) Suppose that, bysome hook orcrook, wehave found aparticular solution, which weshall callx1.That is,wehave anx1forwhich L(x1) =0.Now wenotice that ax,isalsoasolution tothesame equation; wecanmultiply thisspecial solution byanyconstant whatever, andgetanewsolution. Inother words, ifwehada 25-125-1 Linear differential equations 25-2 Superposition ofsolutions 25-3 Oscillations inlinear systems 25-4 Analogs inphysics 25-5 Series andparallel impedances motion ofacertain “size,” then amotion twice as“big” isagain asolution. Proof.'L(ax1) =aL(x1) =a-0=O. Next, suppose that,byhook orbycrook, wehavenotonlyfound onesolution x1,butalsoanother solution, x2.(Remember thatwhen wesubstituted x=e"“‘ forfinding thetransients, wefound twovalues foroz,thatis,twosolutions, x1 andx2.) Now letusshow that thecombination (x1+x2)isalsoasolution. Inother words, ifweputx=x1+x2,xisagain asolution oftheequation. Why? Because, if§(x1) =0and §(x2) =0,then §(x1 +x2)=§(x1) + I;(x2) =0+0=0.Soifwehave found anumber ofsolutions forthemotion ofalinear system wecanaddthem together. Combining these twoideas, wesee,ofcourse, thatwecanalsoaddsixofone andtwooftheother: ifx1isasolution, soisaxl. Therefore anysumofthese two solutions, such as(ax, +BX2), isalso asolution. Ifwehappen tobeable to findthree solutions, then wefindthatanycombination ofthethree solutions is again asolution, andsoon.Itturns outthatthenumber ofwhat wecallinde- pendent soluti0ns* thatwehave obtained forouroscillator problem isonly two. Thenumber ofindependent solutions thatonefinds inthegeneral case depends upon what iscalled thenumber ofdegrees offreedom. Weshall notdiscuss thisin detail now, butifwehave asecond-order dilierential equation, there areonly two independent solutions, andwehave found both ofthem; sowehave themost general solution. Now letusgoontoanother proposition, which applies tothesituation in which thesystem issubjected toanoutside force. Suppose theequation is L(X)=F(1), (255) andsuppose thatwehave found aspecial solution ofit.Letussaythat Joe’s solution isX],andthat §(xJ) =F(t). Suppose wewant tofind yetanother solution; suppose weaddtoJoe’s solution oneofthose thatwasasolution ofthe freeequation (25.5), sayx1.Then weseeby(25.3) that _L(XJ+X1)=L(XJ) +L(X1) =F(l)+0=F(l)- (25-7) Therefore, tothe“forced” solution wecanaddany“free” solution, andwestill have asolution. Thefreesolution iscalled atransient solution. When wehave noforce acting, andsuddenly turn oneon,wedonotim- mediately getthesteady solution thatwesolved forwith thesinewave solution, butforawhile there isatransient which sooner orlater diesout,ifwewait long enough. The“forced” solution does notdieout,since itkeeps onbeing driven by theforce. Ultimately, forlong periods oftime, thesolution isunique, butinitially themotions aredifferent fordifierent circumstances, depending onhowthesystem wasstarted. 25-2 Superposition ofsolutions Now wecome toanother interesting proposition. Suppose thatwehave a certain particular driving force Fa(letussayanoscillatory onewith acertain w=00,1,butourconclusions willbetrueforanyfunctional form ofFa)andwe have solved fortheforced motion (with orwithout thetransients; itmakes no difference). Now suppose some other force isacting, letussayFb,andwesolve thesame problem, butforthisdifferent force. Then suppose someone comes along andsays, “Ihave anewproblem foryoutosolve; Ihave theforce Fa+F;,.” Can wedoit‘?Ofcourse wecandoit,because thesolution isthesum ofthetwo solutions xaandx1,fortheforces taken separately—a most remarkable circum- stance indeed. Ifweuse(25.3), weseethat §(xa +xb)=§(x,,) +§(xb) =F,,(t) +F;,(t). (25.8) *Solutions which cannot beexpressed aslinear combinations ofeach other arecalled independent. 25-2 This isanexample ofwhat iscalled theprinciple ofsuperposition forlinear systems, anditisveryimportant. Itmeans thefollowing: ifwehave acomplicated force which canbebroken upinanyconvenient manner intoasum ofseparate pieces, each ofwhich isinsome waysimple, inthesense thatforeach special piece intowhich wehave divided theforce wecansolve theequation, then theanswer is available forthewhole force, because wemaysimply addthepieces ofthesolution back together, inthesame manner asthetotalforce iscompounded outofpieces (Fig. 25-1). Letusgiveanother example oftheprinciple ofsuperposition. InChapter 12 wesaidthatitwasoneofthegreat facts ofthelaws ofelectricity thatifwehave a certain distribution ofcharges qaandcalculate theelectric fieldEaarising from these charges atacertain place P,andif,ontheother hand, wehave another setofcharges qbandwecalculate thefieldE1,duetothese atthecorresponding place, thenifboth charge distributions arepresent atthesame time, thefield EatPisthesum of E,duetoonesetplusEbduetotheother. Inother words, ifweknow thefield duetoacertain charge, then thefield duetomany charges ismerely thevector sumofthefields ofthese charges taken individually. This isexactly analogous to theabove proposition thatifweknow theresult oftwogiven forces taken atone time, then iftheforce isconsidered asasumofthem, theresponse isasumofthe corresponding individual responses. Thereason whythisistrueinelectricity isthatthegreat laws ofelectricity, Maxwell’s equations, which determine theelectric field, turn outtobedifferential equations which arelinear, i.e.,which have theproperty (25.3). What corresponds totheforce isthecharge generating theelectric field, andtheequation which deter- mines theelectric field interms ofthecharge islinear. Asanother interesting example ofthisproposition, letusaskhowitispossible to“tune in"toaparticular radio station atthesame time asalltheradio stations arebroadcasting. The radio station transmits, fundamentally, anoscillating electric field ofvery high frequency which actsonourradio antenna. Itistrue thattheamplitude oftheoscillation ofthefield ischanged, modulated, tocarry thesignal ofthevoice, butthatisvery slow, andwearenotgoing toworry about it.When onehears “This station isbroadcasting atafrequency of780kilocycles,” thisindicates that780,000 oscillations persecond isthefrequency oftheelectric field ofthestation antenna, andthisdrives theelectrons upanddown atthat frequency inourantenna. Now atthesame time wemay have another radio station inthesame town radiating atadifferent frequency, say550kilocycles per second; then theelectrons inourantenna arealsobeing driven bythatfrequency. Now thequestion is,howisitthatwecanseparate thesignals coming intotheone radio at780kilocycles from those coming inat550kilocycles? Wecertainly do nothear both stations atthesame time. Bytheprinciple ofsuperposition, theresponse oftheelectric circuit inthe radio, thefirstpartofwhich isalinear circuit, totheforces thatareacting dueto theelectric fieldFa+Fb,isx,,+xb.Ittherefore looks asthough wewillnever disen- tangle them. Infact, theveryproposition ofsuperposition seems toinsist thatwe cannot avoid having both ofthem inoursystem. Butremember, foraresonant circuit, theresponse curve, theamount ofxperunitF,asafunction ofthefre- quency, looks likeFig.25-3. Ifitwere avery high Qcircuit, theresponse would show averysharp maximum. Now suppose thatthetwostations arecomparable instrength, thatis,thetwoforces areofthesame magnitude. Theresponse that wegetisthesumofx,,andxb.But, inFig.25-3, xaistremendous, while xbis small. So,inspite ofthefactthatthetwosignals areequal instrength, when they gothrough thesharp resonant circuit oftheradio tuned forma,thefrequency of thetransmission ofonestation, then theresponse tothisstation ismuch greater than totheother. Therefore thecomplete response, with both signals acting, is almost allmade upofwa,andwehave selected thestation wewant. Now what about thetuning? How dowetune it?Wechange wobychanging theLortheCofthecircuit, because thefrequency ofthecircuit hastodowith the combination ofLandC.Inparticular, most radios arebuilt sothat onecan change thecapacitance. When weretune theradio, wecanmake anewsetting of 25-3F.+Pb '1 Pb X.**1. ‘I ‘n Fig. 25-1. Anexample oftheprinci- pleofsuperposition forlinear systems. Fig. 25-2. The principle ofsuper position inelectrostatics. lX| 4—X¢ x rtl >(Uh we (U0 (II,/D \U\..Jm Fig. 25-3. Asharply tuned reso- FIGFICG curve. F X Fig.25-4. Acomplicated force may betreated asosuccession ofsharp impulses.thedial,sothatthenatural frequency ofthecircuit isshifted, say,towc.Inthose circumstances wehearneither onestation northeother; wegetsilence, provided there isnoother station atfrequency we.Ifwekeeponchanging thecapacitance untiltheresonance curve isatwb,thenofcourse weheartheother station. That ishowradio tuning works; itisagain theprinciple ofsuperposition, combined with aresonant response)“ Toconclude thisdiscussion, letusdescribe qualitatively what happens ifwe proceed further inanalyzing alinear problem with agiven force, when theforce is quite complicated. Outofthemany possible procedures, there aretwoespecially useful general ways thatwecansolve theproblem. Oneisthis: suppose thatwe cansolve itforspecial known forces, such assinewaves ofdifferent frequencies. Weknow itischild’s play tosolve itforsinewaves. Sowehave theso-called “child’s play” cases. Now thequestion iswhether ourvery complicated force canberepresented asthesumoftwoormore “child’s play” forces. InFig.25-l wealready hadafairly complicated curve, andofcourse wecanmake itmore complicated stillifweaddinmore sinewaves. Soitiscertainly possible toobtain very complicated curves. And, infact, thereverse isalsotrue: practically every curve canbeobtained byadding together infinite numbers ofsinewaves ofdifferent wavelengths (orfrequencies) foreach oneofwhich weknow theanswer. Wejust have toknow howmuch ofeach sinewave toputintomake thegiven F,andthen ouranswer, x,isthecorresponding sumoftheFsinewaves, each multiplied byits effective ratio ofxtoF.This method ofsolution iscalled themethod ofFourier transforms orFourier analysis. Wearenotgoing toactually carry outsuch an analysis justnow; weonly wish todescribe theideainvolved. Another wayinwhich ourcomplicated problem canbesolved isthefollowing very interesting one. Suppose that, bysome tremendous mental effort, itwere possible tosolve ourproblem foraspecial force, namely animpulse. Theforce is quickly turned onandthen off;itisallover. Actually weneed only solve foran impulse ofsome unitstrength, anyother strength canbegotten bymultiplication byanappropriate factor. Weknow thattheresponse xforanimpulse isadamped oscillation. Now what canwesayabout some other force, forinstance aforce likethatofFig.25-4? Such aforce canbelikened toasuccession ofblows with ahammer. First there isnoforce, andallofasudden there isasteady force——impulse, impulse, impulse, impulse, ...andthenitstops. Inother words, weimagine thecontinuous force tobeaseries ofimpulses, veryclose together. Now, weknow theresult for animpulse, sotheresult forawhole series ofimpulses willbeawhole series of damped oscillations: itwillbethecurve forthefirstimpulse, andthen (slightly later) weaddtothatthecurve forthesecond impulse, andthecurve forthethird impulse, andsoon.Thus wecanrepresent, mathematically, thecomplete solution forarbitrary functions ifweknow theanswer foranimpulse. Wegettheanswer foranyother force simply byintegrating. This method iscalled theGreen’s function method. AGreen’s function isaresponse toanimpulse, andthemethod ofanalyzing anyforce byputting together theresponse ofimpulses iscalled the Green’s function method. Thephysical principles involved inboth ofthese schemes aresosimple, involv- ingjustthelinear equation, thatthey canbereadily understood, butthemathe- matical problems thatareinvolved, thecomplicated integrations andsoon,area little tooadvanced forustoattack right now. You willmost likely return tothis some daywhen youhave hadmore practice inmathematics. Buttheideaisvery simple indeed. Finally, wemake some remarks onwhylinear systems aresoimportant. The answer issimple: because wecansolve them! Somost ofthetimewesolve linear *Inmodern superheterodyne receivers theactual operation ismore complex. The amplifiers arealltuned toafixed frequency (called IFfrequency) andanoscillator of variable tunable frequency iscombined with theinput signal inanonlinear circuit to produce anewfrequency (thedifference ofsignal andoscillator frequency) equal tothe IFfrequency, which isthenamplified. Thiswillbediscussed inChapter 50. 25-4 problems. Second (and most important), itturns outthatthefundamental lawsof physics areoften linear. The Maxwell equations forthelaws ofelectricity are linear, forexample. Thegreat laws ofquantum mechanics turn out,sofaraswe know, tobelinear equations. That iswhywespend somuch time onlinear equa- tions: because ifweunderstand linear equations, weareready, inprinciple, to understand alotofthings. Wemention another situation where linear equations arefound. When displacements aresmall, many functions canbeapproximated linearly. Forexample, ifwehave asimple pendulum, thecorrect equation foritsmotion is a’20/dtz =—(g/L) sin0. (25.9) This equation canbesolved byelliptic functions, buttheeasiest waytosolve itis numerically, aswasshown inChapter 9onNewton’s Laws ofMotion. Anon- linear equation cannot besolved, ordinarily, anyother waybutnumerically. Now forsmall 0,sin6ispractically equal to0,andwehave alinear equation. Itturns out thatthere aremany circumstances where small effects arelinear: fortheexample heretheswing ofapendulum through small arcs. Asanother example, ifwepull alittle bitonaspring, theforce isproportional totheextension. Ifwepullhard, webreak thespring, andtheforce isacompletely different function ofthedistance! Linear equations areimportant. Infactthey aresoimportant thatperhaps fifty percent ofthetime wearesolving linear equations inphysics andinengineering. 25-3 Oscillations inlinear systems Letusnow review thethings wehave been talking about inthepast few chapters. Itisvery easy forthephysics ofoscillators tobecome obscured bythe mathematics. Thephysics isactually verysimple, andifwemayforget themathe- matics foramoment weshall seethatwecanunderstand almost everything that happens inanoscillating system. First, ifwehave only thespring andtheweight, itiseasy tounderstand whythesystem oscillates-—it isaconsequence ofinertia. Wepullthemass down andtheforce pulls itback up;asitpasses zero, which is theplace itlikes tobe,itcannot justsuddenly stop; because ofitsmomentum it keeps ongoing andswings totheother side, andback andforth. So,ifthere were nofriction, wewould surely expect anoscillatory motion, andindeed wegetone. Butifthere iseven alittle bitoffriction, then onthereturn cycle, theswing will notbequite ashigh asitwasthefirsttime. Now what happens, cycle bycycle? That depends onthekind andamount offriction. Suppose thatwecould concoct akind offriction force thatalways remains inthesame proportion totheother forces, ofinertia andinthespring, as theamplitude ofoscillation varies. Inother words, forsmaller oscillations the friction should beweaker than forbigoscillations. Ordinary friction does nothave thisproperty, soaspecial kind offriction must becarefully invented forthevery purpose ofcreating afriction thatisdirectly proportional tothevelocity—so that forbigoscillations itisstronger andforsmall oscillations itisweaker. Ifwehappen tohave thatkind offriction, then attheendofeach successive cycle thesystem is inthesame condition asitwasatthestart, except alittle bitsmaller. Alltheforces aresmaller inthesame proportion: thespring force isreduced, theinertial effects arelower because theaccelerations arenow weaker, andthefriction islesstoo, byourcareful design. When weactually have thatkind offriction, wefindthat each oscillation isexactly thesame asthefirstone, except reduced inamplitude. Ifthefirstcycle dropped theamplitude, say,to90percent ofwhat itwasatthe start, thenext willdrop itto90percent of90percent, andsoon:thesizes ofthe oscillations arereduced bythesame fraction ofthemselves inevery cycle. An exponential function isacurve which doesjustthat. Itchanges bythesame factor ineach equal interval oftime. That istosay,iftheamplitude ofonecycle, relative tothepreceding one, iscalled a,then theamplitude ofthenext isa2,andofthe next, a3.Sotheamplitude issome constant raised toapower equal tothenumber ofcycles traversed: A=Aoa". (25.10) 25-5 X 1.‘ I,\ /"\, )uo “’ Fig.25-5. Resonance curves with various amounts offriction present.Butofcourse n~t,soitisperfectly clear thatthegeneral solution willbesome kind ofanoscillation, sineorcosine wt,times anamplitude which goes asb‘more orless. Butbcanbewritten ase‘°,ifbispositive andlessthan l.Sothisiswhy thesolution looks likee"“coswt.Itisvery simple. What happens ifthefriction isnotsoartificial; forexample, ordinary rubbing onatable, sothatthefriction force isacertain constant amount, andisindependent ofthesizeoftheoscillation thatreverses itsdirection each half-cycle? Then the equation isnolonger linear, itbecomes hard tosolve, andmust besolved bythe numerical method given inChapter 2,orbyconsidering each half-cycle separately. Thenumerical method isthemost powerful method ofall,andcansolve any equation. Itisonly when wehave asimple problem thatwecanusemathematical analysis. Mathematical analysis isnotthegrand thing itissaidtobe;itsolves onlythe simplest possible equations. Assoon astheequations getalittle more complicated, justashade—they cannot besolved analytically. Butthenumerical method, which wasadvertised atthebeginning ofthecourse, cantake care ofanyequation of physical interest. Next, what about theresonance curve? Why isthere aresonance‘? First, imagine foramoment thatthere isnofriction, andwehave something which could oscillate byitself. Ifwetapped thependulum justright each time itwent by,of course wecould make itgolikemad. Butifweclose oureyesanddonotwatch it,andtapatarbitrary equal intervals, what isgoing tohappen? Sometimes we willfindourselves tapping when itisgoing thewrong way. When wehappen to have thetiming justright, ofcourse, each tapisgiven atjusttheright time, and soitgoes higher andhigher andhigher. Sowithout friction wegetacurve which looks likethesolid curve inFig.25-5 fordifferent frequencies. Qualitatively, we understand theresonance curve; inorder togettheexact shape ofthecurve itis probably justaswelltodothemathematics. Thecurve goes toward infinity as w—>wo,where L00isthenatural frequency oftheoscillator. Now suppose there isalittle bitoffriction; then when thedisplacement of theoscillator issmall, thefriction does notaffect itmuch; theresonance curve is thesame, except when wearenear resonance. Instead ofbecoming infinite near resonance, thecurve isonlygoing togetsohigh thatthework done byourtapping each time isenough tocompensate forthelossofenergy byfriction during the cycle. Sothetopofthecurve isrounded off—it does notgotoinfinity. Ifthere is more friction, thetopofthecurve isrounded offstillmore. Now someone might say,“Ithought thewidths ofthecurves depended onthefriction.” That isbecause thecurve isusually plotted sothatthetopofthecurve iscalled oneunit. However, themathematical expression iseven simpler tounderstand ifwejustplotallthe curves onthesame scale; then allthathappens isthatthefriction cutsdown the top! Ifthere islessfriction, wecangofarther upintothatlittle pinnacle before thefriction cutsitoff,soitlooks relatively narrow. That is,thehigher thepeak ofthecurve, thenarrower thewidth athalfthemaximum height. Finally, wetake thecasewhere there isanenormous amount offriction. It turns outthatifthere istoomuch friction, thesystem does notoscillate atall. Theenergy inthespring isbarely abletomove itagainst thefrictional force, and soitslowly oozes down totheequilibrium point. 25-4 Analogs inphysics Thenext aspect ofthisreview istonote thatmasses andsprings arenotthe only linear systems; there areothers. Inparticular, there areelectrical systems called linear circuits, inwhich wefindacomplete analog tomechanical systems. Wedidnotlearn exactly whyeach oftheobjects inanelectrical circuit works in thewayitdoes—that isnottobeunderstood atthepresent moment; wemayassert itasanexperimentally verifiable factthatthey behave asstated. Forexample, letustakethesimplest possible circumstance. Wehave apiece ofwire, which isjustaresistance, andwehave applied toitadifference inpotential, V.Now theVmeans this: ifwecarry acharge qthrough thewirefrom oneterminal 25-6 toanother terminal, thework done isqV. Thehigher thevoltage difference, the more work wasdone when thecharge, aswesay,“falls” from thehigh potential endoftheterminal tothelowpotential end. Socharges release energy ingoing from oneendtotheother. Now thecharges donotsimply fiyfrom oneendstraight totheother end; theatoms inthewireoffer some resistance tothecurrent, andthis resistance obeys thefollowing lawforalmost allordinary substances: ifthere isa current I,thatis,soandsomany charges persecond tumbling down, thenumber persecond thatcomes tumbling through thewire isproportional tohowhard we push them—-in other words, proportional tohow much voltage there is: V=IR=R(dq/dt). (25.11) Thecoefficient Riscalled theresistance, andtheequation iscalled Ohm’s Law. Theunitofresistance istheohm; itisequal toonevoltperampere. Inmechanical situations, togetsuch africtional force inproportion tothevelocity isdifficult; in anelectrical system itisvery easy, andthislawisextremely accurate formost metals. Weareoften interested inhowmuch work isdone persecond, thepower loss, ortheenergy liberated bythecharges asthey tumble down thewire. When we carry acharge qthrough avoltage V,thework isqV,sothework done persecond would beV(dq/dt), which isthesame asV1,oralsoIR~I=12R. This iscalled theheating loss—this ishowmuch heat isgenerated intheresistance persecond, bytheconservation ofenergy. Itisthisheatthatmakes anordinary incandescent light bulb work. Ofcourse, there areother interesting properties ofmechanical systems, such asthemass (inertia), anditturns outthatthere isanelectrical analog toinertia also. Itispossible tomake something called aninductor, having aproperty called inductance, such that acurrent, once started through theinductance, does not want tostop. Itrequires avoltage inorder tochange thecurrent! Ifthecurrent is constant, there isnovoltage across aninductance. DCcircuits donotknow any- thing about inductance; itisonly when wechange thecurrent thattheeffects of inductance show up.Theequation is V=L(dI/dt) =L(d2q/dt2), (25.12) andtheunitofinductance, called thehenry, issuch thatonevoltapplied toan inductance ofonehenry produces achange ofoneampere persecond inthecurrent. Equation (25.12) istheanalog ofNewton’s lawforelectricity, ifwewish: Vcorre- sponds toF,Lcorresponds tom,andIcorresponds tovelocity! Allofthecon- sequent equations forthetwokinds ofsystems willhave thesame derivations because, inalltheequations, wecanchange anyletter toitscorresponding analog letter andwegetthesame equation; everything wededuce willhave acorrespond- ence inthetwosystems. Now what electrical thing corresponds tothemechanical spring, inwhich there wasaforce proportional tothestretch? Ifwestart with F=kxandreplace F—>Vandx-+q,wegetV=aq.Itturns outthatthere issuch athing, infact itistheonly oneofthethree circuit elements wecanreally understand, because wedidstudy apairofparallel plates, andwefound thatifthere were acharge of certain equal, opposite amounts oneach plate, theelectric fieldbetween them would beproportional tothesizeofthecharge. Sothework done inmoving aunitcharge across thegapfrom oneplate totheother isprecisely proportional tothecharge. This work isthedefinition ofthevoltage difference, anditisthelineintegral ofthe electric field from oneplate toanother. Itturns out,forhistorical reasons, that theconstant ofproportionality isnotcalled C,butl/C.Itcould have been called C,butitwasnot. Sowehave V=q/C. (25.13) Theunitofcapacitance, C,isthefarad; acharge ofonecoulomb oneach plate of aone-farad capacitor yields avoltage difference ofonevolt. 25-7 There areouranalogies, andtheequation corresponding totheoscillating circuit becomes thefollowing, bydirect substitution ofLform,qforx,etc: m(d2x/dtz) +1/m(dx/dt) +kx=F, (25.14) L(d2q/dt2) +R(dq/dt) +q/C=V. (25.15) Now everything welearned about (25.14) canbetransformed toapply to(25.15). Every consequence isthesame; somuch thesame thatthere isabrilliant thing we cando. Suppose wehave amechanical system which isquite complicated, notjust onemass onaspring, butseveral masses onseveral springs, allhooked together. What dowedo? Solve it?Perhaps; butlook, wecanmake anelectrical circuit which willhave thesame equations asthething wearetrying toanalyze! For instance, ifwewanted toanalyze amass onaspring, why canwenotbuild an electrical circuit inwhich weuseaninductance proportional tothemass, are- sistance proportional tothecorresponding m7,1/Cproportional tok,allinthe same ratio? Then, ofcourse, thiselectrical circuit willbetheexact analog ofour mechanical one,inthesense thatwhatever qdoes, inresponse toV(Valsoismade tocorrespond totheforces thatareacting), sothexwould doinresponse tothe force! Soifwehave acomplicated thing with awhole lotofinterconnecting ele- ments, wecaninterconnect awhole lotofresistances, inductances, andcapacitances, toimitate themechanically complicated system. What istheadvantage tothat? One problem isjustashard (oraseasy) astheother, because they areexactly equivalent. Theadvantage isnotthatitisanyeasier tosolve themathematical equations after wediscover thatwehave anelectrical circuit (although thatisthe method used byelectrical engineersl), butinstead, therealreason forlooking at theanalog isthatitiseasier tomake theelectrical circuit, andtochange something inthesystem. Suppose wehave designed anautomobile, andwant toknow how much itis going toshake when itgoes over acertain kind ofbumpy road. Webuild an electrical circuit with inductances torepresent theinertia ofthewheels, spring constants ascapacitances torepresent thesprings ofthewheels, andresistors to represent theshock absorbers, andso‘nfortheother parts oftheautomobile. Then weneed abumpy road. Allright, weapply avoltage from agenerator, which represents such andsuch akind ofbump, andthen look athow theleftwheel jiggles bymeasuring thecharge onsome capacitor. Having measured it(itiseasy todo),wefindthatitisbumping toomuch. Doweneed more shock absorber, orlessshock absorber? With acomplicated thing likeanautomobile, dowe actually change theshock absorber, andsolve itallover again? Nol, wesimply turn adial; dialnumber tenisshock absorber number three, soweputinmore shock absorber. Thebumps areworse—all right, wetryless. Thebumps arestill worse; wechange thestiffness ofthespring (dial 17),andweadjust allthese things electrically, with merely theturn ofaknob. This iscalled ananalog computer. Itisadevice which imitates theproblem thatwewant tosolve bymaking another problem, which hasthesame equation, butinanother circumstance ofnature, andwhich iseasier tobuild, tomeasure, toadjust, andtodestroy! 25-5 Series andparallel impedances Finally, there isanimportant item which isnotquite inthenature ofreview. This hastodowith anelectrical circuit inwhich there ismore than onecircuit element. Forexample, when wehave aninductor, aresistor, andacapacitor connected asinFig.24-2, wenote thatallthecharge went through every oneof thethree, sothatthecurrent insuch asingly connected thing isthesame atall points along thewire. Since thecurrent isthesame ineach one,thevoltage across RisIR,thevoltage across LisL(dI/dt), andsoon.So,thetotal voltage drop isthe sum ofthese, andthisleads toEq.(25.15). Using complex numbers, wefound that wecould solve theequation forthesteady-state motion inresponse toa 25-8 A AA A sinusoidal force. Wethusfound thatV=Z1.Now Ziscalled theimpedance of thisparticular circuit. Ittellsusthatifweapply asinusoidal voltage, V,wegeta current I. Now suppose wehave amore complicated circuit which hastwopieces, which bythemselves have certain impedances, Z1andZ2,andweputthem inseries (Fig. 25-6a) andapply avoltage. What happens? Itisnow alittle more compli- cated, butifIisthecurrent through Z1,thevoltage difference across Z1isV1= IZ1; similarly, thevoltage across Z2isV2=IZ2. Thesame currentgoes through both. Therefore thetotal voltage isthesumofthevoltages across thetwosections andisequal toV=V1—l-V2=(Z1+Z2)I. Thismeans thatthevoltage on thecomplete circuit canbewritten V=IZ,where theZ,ofthecombined system inseries isthesumofthetwoZ’softheseparate pieces: Z.=2,+Z2. (25.16) This isnottheonly waythings maybeconnected. Wemayalsoconnect them inanother way, called aparallel connection (Fig. 25-6b). Now weseethatagiven voltage across theterminals, iftheconnecting wires areperfect conductors, is effectively applied toboth oftheimpedances, andwillcause currents ineach independently. Therefore thecurrent through Z1isequal toI1=V/Z1. The current inZ2isI2=V/Z2. Itisthesame voltage. Now thetotal current which issupplied totheterminals isthesumofthecurrents inthetwosections: I= V/Z1 —l—V/Z2. This canbewritten as v=-.-.—’-—.—=r2,..(1/Z1)+ (1/Z2) 1/2,,=1/2,+1/22. (25.17)Thus More complicated circuits cansometimes besimplified bytaking pieces of them, working outthesuccession ofimpedances ofthepieces, andcombining the circuit together stepbystep, using theabove rules. Ifwehave anykind ofcircuit with many impedances connected inallkinds ofways, andifweinclude thevolt- ages intheform oflittle generators having noimpedance (when wepass charge through it,thegenerator adds avoltage V),then thefollowing principles apply: (1)Atanyjunction, thesum ofthecurrents intoajunction iszero. That is,all thecurrent which comes inmust come back out. (2)Ifwecarry acharge around anyloop, andback towhere itstarted, thenetwork done iszero. These rules are called Kirchhofl"s laws forelectrical circuits. Their systematic application to complicated circuits often simplifies theanalysis ofsuch circuits. Wemention them here inconjunction with Eqs. (25.16) and(25.17), incaseyouhave already come across such circuits thatyouneed toanalyze inlaboratory work. They will bediscussed again inmore detail next year. 25-9NIH(e)Serfel (ti)Parallel Fig.25-6. Two impedances, con nected inseries andinparallel. 26 Opties: The Principle ofLeast Time 26-1 Light This isthefirstofanumber ofchapters onthesubject ofelectromagnetic radiation. Light, with which wesee,isonly onesmall partofavastspectrum of thesame kind ofthing, thevarious parts ofthisspectrum being distinguished by different yalues ofacertain quantity which varies. This variable quantity could becalled the“wavelength.” Asitvaries inthevisible spectrum, thelight apparently changes color from redtoviolet. Ifweexplore thespectrum systematically, from longwavelengths toward shorter ones, wewould begin withwhat areusually called radiowaves. Radiowaves aretechnically available inawide range ofwavelengths, some even longer than those used inregular broadcasts; regular broadcasts have wavelengths corresponding toabout 500meters. Then there aretheso-called “short waves,” i.e.,radar waves, millimeter waves, andsoon.There arenoactual boundaries between onerange ofwavelengths andanother, because nature did notpresent uswith sharp edges. Thenumber associated with agiven name for thewaves areonly approximate and, ofcourse, soarethenames wegivetothe different ranges. Then, along waydown through themillimeter waves, wecome towhat we calltheinfrared, andthence tothevisible spectrum. Then going intheother direction, wegetintoaregion which iscalled theultraviolet. Where theultraviolet stops, thex-rays begin, butwecannot define precisely where thisis;itisroughly at l0_8 m,orl0_2 ].L.These are“soft” x-rays; then there areordinary x-rays andvery hard x-rays; then‘Y-rays, andsoon,forsmaller andsmaller values ofthisdimension called thewavelength. Within thisvast range ofwavelengths, there arethree ormore regions of approximation which areespecially interesting. Inoneofthese, acondition exists inwhich thewavelengths involved arevery small compared with thedimensions oftheequipment available fortheir study; furthermore, thephoton energies, using thequantum theory, aresmall compared with theenergy sensitivity oftheequip- ment. Under these conditions wecanmake arough firstapproximation bya method called geometrical optics. If,ontheother hand, thewavelengths arecom- parable tothedimensions oftheequipment, which isdifficult toarrange with visible light buteasier with radiowaves, andifthephoton energies arestillnegligi- blysmall, then averyuseful approximation canbemade bystudying thebehavior ofthewaves, stilldisregarding thequantum mechanics. This method isbased on theclassical theory ofelectromagnetic radiation, which willbediscussed inalater chapter. Next, ifwegotovery short wavelengths, where wecandisregard the wave character butthephotons have avery large energy compared with the sensitivity ofourequipment, things getsimple again. This isthesimple photon picture, which wewilldescribe only very roughly. Thecomplete picture, which unifies thewhole thing intoonemodel, willnotbeavailable tousforalong time. Inthischapter ourdiscussion islimited tothegeometrical optics region, in which weforget about thewavelength andthephoton character ofthelight, which willallbeexplained induetime. Wedonoteven bother tosaywhat thelight is, butjustfindouthowitbehaves onalarge scale compared with thedimensions of interest. Allthismust besaidinorder toemphasize thefactthatwhat wearegoing totalkabout isonly avery crude approximation; thisisoneofthechapters that weshall have to“unlearn” again. Butweshall very quickly unlearn it,because weshall almost immediately goontoamore accurate method. 26-126-1 Light 26-2 Reflection andrefraction 26-3 Fermat’s principle ofleast time 26-4 Applications ofFermat’s principle 26-5 Amore precise statement of Fermat’s principle 26-6 How itworks Fig.26-1. Theangle ofincidence is9|id, equal totheangle ofreflection. Fig. 26-2. Alight ray isrefracted when itpasses from one medium into another.l91 10II Table 26-1 Angle inair Angle inwater 10° 20° 30° 400 50° 60° 70° 80°80 15-1/2° 22-1/2° 29° 35° 40-1/2° 45-1/2° 50°Although geometrical optics isjust anapproximation, itisofvery great importance technically andofgreat interest historically. Weshall present this subject more historically than some oftheothers inorder togivesome ideaofthe development ofaphysical theory orphysical idea. First, light is,ofcourse, familiar toeverybody, andhasbeen familiar since time immemorial. Now oneproblem is,bywhat process doweseelight? There have been many theories, butitfinally settled down toone, which isthatthere is something which enters theeye—-which bounces offobjects intotheeye. Wehave heard thatidea solong thatweaccept it,anditisalmost impossible forusto realize thatveryintelligent menhave proposed contrary theories—that something comes outoftheeyeandfeels fortheobject, forexample. Some other important observations arethat, aslight goes from oneplace toanother, itgoes instraight lines, ifthere isnothing intheway, andthattheraysdonotseem tointerfere with oneanother. That is,light iscrisscrossing inalldirections intheroom, butthe light thatispassing across ourlineofvision does notaffect thelight thatcomes tousfrom some object. This wasonce amost powerful argument against the corpuscular theory; itwasused byHuygens. Iflight were likealotofarrows shooting along, how could other arrows gothrough them soeasily? Such philo- sophical arguments arenotofmuch weight. Onecould always saythatlight is made upofarrows which gothrough each other! 26-2 Reflection andrefraction Thediscussion above gives enough ofthebasic ideaofgeometrical optics- nowwehave togoalittle further intothequantitative features. Thus farwehave light going onlyinstraight lines between twopoints; nowletusstudy thebehavior oflight when ithitsvarious materials. Thesimplest object isamirror, andthe lawforamirror isthatwhen thelight hitsthemirror, itdoes notcontinue ina straight line, butbounces offthemirror intoanewstraight line, which changes when wechange theinclination ofthemirror. Thequestion fortheancients was, what istherelation between thetwoangles involved? This isaverysimple relation, discovered long ago. Thelight striking amirror travels insuch awaythatthetwo angles, between each beam andthemirror, areequal. Forsome reason itis customary tomeasure theangles from thenormal tothemirror surface. Thus the so-called lawofreflection is 0,=0,. (26.1) That isasimple enough proposition, butamore difficult problem isencoun- tered when light goes from onemedium intoanother, forexample from airinto water; herealso, weseethatitdoes notgoinastraight line. Inthewater theray isataninclination toitspath intheair;ifwechange theangle 0,~sothatitcomes down more nearly vertically, then theangle of“breakage” isnotasgreat. But ifwetiltthebeam oflight atquite anangle, then thedeviation angle isverylarge. Thequestion is,what istherelation ofoneangle totheother? This alsopuzzled theancients foralong time, andheretheynever found theanswer! Itis,however, oneofthefewplaces inallofGreek physics thatonemay findanyexperimental results listed. Claudius Ptolemy made alistoftheangle inwater foreach ofa number ofdifferent angles inair.Table 26-1 shows theangles intheair,indegrees, andthecorresponding angle asmeasured inthewater. (Ordinarily itissaidthat Greek scientists never didanyexperiments. Butitwould beimpossible toobtain thistable ofvalues without knowing theright law, except byexperiment. It should benoted, however, thatthese donotrepresent independent careful measure- ments foreach angle butonly some numbers interpolated from afewmeasure- ments, forthey allfitperfectly onaparabola.) This, then, isoneoftheimportant steps inthedevelopment ofphysical law: firstweobserve aneffect, then wemeasure itandlistitinatable; then wetryto find therulebywhich onething canbeconnected with another. The above numerical table wasmade in140A.D., butitwasnotuntil 1621 that someone finally found theruleconnecting thetwoangles! Therule, found byWillebrord 26-2 Snell, aDutch mathematician, isasfollows: if0,istheangle inairand0,isthe angle inthewater, then itturns outthatthesineofB,isequal tosome constant multiple ofthesineof0,: sin19,-=nsin0,. (26.2) Forwater thenumber nisapproximately 1.33. Equation (26.2) iscalled Snell’s law; itpermits ustopredict how thelight isgoing tobend when itgoes from air intowater. Table 26-2 shows theangles inairandinwater according toSnell’s law. Note theremarkable agreement with Ptolemy’s list. 26-3 Fermat’s principle ofleast time Now inthefurther development ofscience, wewant more thanjustaformula. First wehave anobservation, then wehave numbers thatwemeasure, thenwe have alawwhich summarizes allthenumbers. Buttherealglory ofscience isthat wecanfindawayofthinking such thatthelawisevident. Thefirstwayofthinking thatmade thelawabout thebehavior oflight evident wasdiscovered byFermat inabout 1650, anditiscalled theprinciple ofleast time, orFermat’s principle. Hisideaisthis: thatoutofallpossible paths thatitmight take togetfrom onepoint toanother, light takes thepath which requires the shortest time. Letusfirstshow thatthisistrueforthecase ofthemirror, thatthissimple principle contains both thelawofstraight-line propagation andthelawforthe mirror. So,wearegrowing inourunderstanding! Letustrytofindthesolution tothefollowing problem. InFig.26-3 areshown twopoints, AandB,anda plane mirror, MM’. What isthewaytogetfrom AtoBintheshortest time? Theanswer istogostraight from AtoB!Butifweaddtheextra rulethatthelight hastostrike themirror andcome back intheshortest time, theanswer isnotso easy. Onewaywould betogoasquickly aspossible tothemirror andthen goto B,onthepath ADB. Ofcourse, wethen have along path DB. Ifwemove over a little totheright, toE,weslightly increase thefirstdistance, butwegreatly decrease thesecond one,andsothetotal path length, andtherefore thetravel time, isless. How canwefindthepoint Cforwhich thetime istheshortest? Wecanfindit verynicely byageometrical trick. Weconstruct ontheother sideofMM’ anartificial point B’,which isthe same distance below theplane MM’ asthepoint Bisabove theplane. Then we draw thelineEB’. Now because BFM isaright angle andBF=FB’, EBis equal toEB’. Therefore thesumofthetwodistances, AE+EB,which ispropor- tional tothetime itwilltake ifthelight travels with constant velocity, isalsothe sumofthetwolengths AE+EB’. Therefore theproblem becomes, when isthe sumofthese twolengths theleast? Theanswer iseasy: when thelinegoes through point Casastraight linefrom AtoB’!Inother words, wehave tofindthepoint where wegotoward theartificial point, andthatwillbethecorrect one. Now if ACB’ isastraight line,then angle BCF isequal toangle B’CF andthence toangle ACM. Thus thestatement thattheangle ofincidence equals theangle ofreflection isequivalent tothestatement thatthelight goes tothemirror insuch awaythat itcomes back tothepoint B’intheleast possible time. Originally, thestatement wasmade byHero ofAlexandria thatthelight travels insuch awaythatitgoes tothemirror andtotheother point intheshortest possible distance, soitisnota modern theory. Itwasthisthatinspired Fermat tosuggest tohimself thatperhaps refraction operated onasimilar basis. Butforrefraction, light obviously does not usethepath ofshortest distance, soFermat tried theideathatittakes theshortest time. Before wegoontoanalyze refraction, weshould make onemore remark about themirror. Ifwehave asource oflight atthepoint Banditsends light to- ward themirror, thenweseethatthelight which goes toAfrom thepoint Bcomes toAinexactly thesame manner asitwould have come toAifthere were anobject atB’,andnomirror. Now ofcourse theeyedetects only thelight which enters it physically, soifwehave anobject atBandamirror which makes thelight come 26-3Angle inair Angle inwater 10° 20° 30°40° 50° 600 70° 80° IEv’1\it\\Table 26-2 7-1/2° 15° 22° 29° 35° 40-1/2° 45° 48° __-_p—__--" 4’ //’ UM Fig. 26-3. lllustrotion oftheprinciple ofleast time\ //I’//'\ \ / §\ A N \ \\ \ AIR \\ Ec 'GLASS \x ’i<\F\ \ \ \ \\ N‘ B Fig.26-4. Illustration of Fermat's principle forrefraction. Fig. 26-5. The minimum time corre- sponds topoint C,but nearby points correspond tonearly thesame time.intotheeyeinexactly thesame manner asitwould have come intotheeyeifthe object were atB’,then theeye-brain system interprets that, assuming itdoes not know toomuch, asbeing anobject atB’.Sotheillusion thatthere isanobject behind themirror ismerely duetothefactthat thelight which isentering the eyeisentering inexactly thesame manner, physically, asitwould have entered hadthere been anobject back there (except forthedirtonthemirror, andour knowledge oftheexistence ofthemirror, andsoon,which iscorrected inthebrain). Now letusdemonstrate thattheprinciple ofleast time willgiveSnell’s law ofrefraction. Wemust, however, make anassumption about thespeed oflight inwater. Weshall assume thatthespeed oflight inwater islower than thespeed oflight inairbyacertain factor, n. InFig.26-4, ourproblem isagain togofrom AtoBintheshortest time. Toillustrate thatthebestthing todoisnotjusttogoinastraight line, letus imagine thatabeautiful girlhasfallen outofaboat, andsheisscreaming forhelp inthewater atpoint B.Thelinemarked Xistheshoreline. Weareatpoint Aonland, andweseetheaccident, andwecanrunandcanalsoswim. Butwecanrunfaster than wecanswim. What dowedo? Dowegoinastraight line? (Yes, nodoubt!) However, byusing alittle more intelligence wewould realize thatitwould beadvan- tageous totravel alittle greater distance onland inorder todecrease thedistance inthewater, because wegosomuch slower inthewater. (Following thislineof reasoning out,wewould saytheright thing todoistocompute very carefully what should bedone!) Atanyrate, letustrytoshow thatthefinal solution tothe problem isthepath ACB, andthatthispath takes theshortest time ofallpossible ones. Ifitistheshortest path, thatmeans thatifwetakeanyother, itwillbelonger. So,ifwewere toplotthetimeittakes against theposition ofpoint X,wewould get acurve something likethatshown inFig.26-5, where point Ccorresponds tothe shortest ofallpossible times. This means thatifwemove thepoint Xtopoints near C,inthefirstapproximation there isessentially nochange intime because the slope iszero atthebottom ofthecurve. Soourwayoffinding thelawwillbeto consider that wemove theplace byavery small amount, andtodemand that there beessentially nochange intime. (Ofcourse there isaninfinitesimal change ofasecond order; weought tohave apositive increase fordisplacements ineither direction from C.)Soweconsider anearby point Xandwecalculate howlongit would take togofrom AtoBbythetwopaths, andcompare thenewpath with theoldpath. Itisveryeasytodo.Wewant thedifference, ofcourse, tobenearly zero ifthedistance XCisshort. First, look atthepath onland. Ifwedraw a perpendicular XE,weseethatthispath isshortened bytheamount EC. Letus saywegainbynothaving togothatextra distance. Ontheother hand, inthewater, bydrawing acorresponding perpendicular, CF,wefindthatwehave togothe extra distance XF,andthatiswhat welose. Or,intime, wegainthetime itwould have taken togothedistance EC,butwelosethetime itwould have taken togo thedistance XF.Those times must beequal since, inthefirstapproximation, there istobenochange intime. Butsupposing thatinthewater thespeed isl/ntimes asfastasinair,then wemust have EC=n-XF. (26.3) Therefore weseethatwhen wehave theright point, XCsinEXC=n-XCsinXCF or,cancelling thecommon hypotenuse length XCandnoting that EXC =ECN =0, and XCF =BCN’ =6,, wehave sin0,=nsin6,. (26.4) Soweseethattogetfrom onepoint toanother intheleast time when theratio ofspeeds isn,thelight should enter atsuch anangle thattheratio ofthesines of theangles 19,-and0,istheratio ofthespeeds inthetwomedia. 26-4 26-4 Applications ofFermat’s principle Now letusconsider some oftheinteresting consequences oftheprinciple of least time. First istheprinciple ofreciprocity. Iftogofrom AtoBwehave found thepath oftheleast time, then togointheopposite direction (assuming thatlight goes atthesame speed inanydirection), theshortest time willbethesame path, andtherefore, iflight canbesentoneway, itcanbesenttheother way. Anexample ofinterest isaglass block with plane parallel faces, setatanangle toalight beam. Light, ingoing through theblock from apoint Atoapoint B (Fig. 26-6) does notgothrough inastraight line,butinstead itdecreases thetime intheblock bymaking theangle intheblock lessinclined, although itloses alittle bitintheair. Thebeam issimply displaced parallel toitself because theangles inandoutarethesame. Athird interesting phenomenon isthefactthatwhen weseethesunsetting, itisalready below thehorizon! Itdoes notlookasthough itisbelow thehorizon, butitis(Fig. 26-7). Theearth’s atmosphere isthinatthetopanddense atthe bottom. Light travels more slowly inairthan itdoes inavacuum, andsothelight ofthesuncangettopoint Sbeyond thehorizon more quickly if,instead ofjust going inastraight line,itavoids thedense regions where itgoes slowly bygetting through them atasteeper tilt. When itappears togobelow thehorizon, itis actually already wellbelow thehorizon. Another example ofthisphenomenon is themirage thatoneoften seeswhile driving onhotroads. Onesees“water” onthe road, butwhen hegetsthere, itisasdryasthedesert! Thephenomenon isthe following. What wearereally seeing istheskylight “reflected” ontheroad: light from thesky,heading fortheroad, canendupintheeye,asshown inFig. 26-8. Why? Theairisvery hotjustabove theroad butitiscooler uphigher. Hotter airismore expanded than cooler airandisthinner, andthisdecreases the speed oflight less. That istosay,light goes faster inthehotregion than inthe coolregion. Therefore, instead ofthelight deciding tocome inthestraightforward way, italsohasaleast-time path bywhich itgoes intotheregion where itgoes faster forawhile, inorder tosave time. So,itcangoinacurve. Asanother important example oftheprinciple ofleast time, suppose thatwe would liketoarrange asituation where wehave allthelight thatcomes outofone point, P,collected back together atanother point, P’(Fig. 26-9). That means, ofcourse, thatthelight cangoinastraight linefrom PtoP’.That isallright. Buthowcanwearrange thatnotonly does itgostraight, butalsosothatthelight starting outfrom Ptoward Qalsoends upatP’? Wewant tobring allthelight back towhat wecallafocus. How? Ifthelight always takes thepath ofleast time, then certainly itshould notwant togoover allthese other paths. Theonly way thatthelight canbeperfectly satisfied totake several adjacent paths istomake those times exactly equal! Otherwise, itwould select theoneofleast time. There- foretheproblem ofmaking afocusing system ismerely toarrange adevice sothat ittakes thesame time forthelight togoonallthedifferent paths! This iseasy todo.Suppose thatwehadapiece ofglass inwhich light goes slower than itdoes intheair(Fig. 26-10). Now consider araywhich goes inair inthepath PQP’. That isalonger path than from Pdirectly toP’andnodoubt takes alonger time. Butifwewere toinsert apiece ofglass ofjusttheright thick- ness (weshall later figure outhow thick) itmight exactly compensate theexcess time thatitwould takethelight togoatanangle! Inthose circumstances wecan arrange thatthetime thelight takes togostraight through isthesame asthetime ittakes togointhepath PQP’. Likewise, ifwetakearayPRR’P’ which ispartly inclined, itisnotquite aslong asPQP’, andwedonothave tocompensate as much asforthestraight one,butwedohave tocompensate somewhat. Weend upwith apiece ofglass thatlooks likeFig.26-10. With thisshape, allthelight which comes from PwillgotoP’.This, ofcourse, iswellknown tous,andwecall such adevice aconverging lens. Inthenext chapter weshall actually calculate what shape thelenshastohave tomake aperfect focus. Take another example: suppose wewish toarrange some mirrors sothat the light from Palways goes toP’(Fig. 26-l1).Onanypath, itgoes tosome mirror 26-5A B' _ ———— --0 B Fig.26-6. Abeam oflight isoffset as itpasses through Cltransparent block. roAPPARENT sunI/ AmosPueRE/_/ “Gm"T" / *_ TOTRUE > \ sun E4 \ Fig. 26-7. Near thehorizon, theap- parent sunishigher than thetrue sunby about l/2degree. 1-torROAD onSAND Fig. 26-8. Amirage. r— __ 1, / zz r \ P P' PTICAL SYSTE l’_;—_-IIIlllzl____..__ Fig.26-9. Anoptical “black box." P P‘ R- O Fig. 26-lO. Afocusing optical sys- tem. Fig.26-1 l.Anellipsoidal mirror. “A a c x K At :_i_‘l__°-2'41'i§l_l_lct.0..>:r— W_r- ><__ Fig. 26-12. Aparaboloidal mirror.andcomes back, andalltimes must beequal. Here thelight always travels inair, sothetime andthedistance areproportional. Therefore thestatement thatall thetimes arethesame isthesame asthestatement thatthetotal distance isthe same. Thus thesumofthetwodistances r1andr2must beaconstant. Anellipse isthatcurve which hastheproperty thatthesumofthedistances from twopoints isaconstant forevery point ontheellipse; thuswecanbesurethatthelight from onefocus willcome totheother. Thesame principle works forgathering thelight ofastar. Thegreat 200-inch Palomar telescope isbuilt onthefollowing principle. Imagine astarbillions of miles away; wewould liketocause allthelight thatcomes intocome toafocus. Ofcourse wecannot draw theraysthatgoallthewayuptothestar, butwestill want tocheck whether thetimes areequal. Ofcourse weknow thatwhen thevari- ousrayshave arrived atsome plane KK’, perpendicular totherays, allthetimes inthisplane areequal (Fig. 26-12). Theraysmust then come down tothemirror andproceed toward P’inequal times. That is,wemust findacurve which hasthe property thatthesumofthedistances XX’ +X’P’isaconstant, nomatter where Xischosen. Aneasywaytofinditistoextend thelength ofthelineXX’down to aplane LL’. Now ifwearrange ourcurve sothatA’A” =A’P’, B'B” =B’P’, C’C” =C'P’, andsoon,wewillhave ourcurve, because then ofcourse, AA’+ A’P’ =AA’ +A’A” willbeconstant. Thus ourcurve isthelocus ofallpoints equidistant from alineandapoint. Such acurve iscalled aparabola; themirror ismade intheshape ofaparabola. Theabove examples illustrate theprinciple upon which such optical devices canbedesigned. Theexact curves canbecalculated using theprinciple that, to focus perfectly, thetravel times must beexactly equal foralllight rays, aswellas being lessthan foranyother nearby path. Weshall discuss these focusing optical devices further inthenext chapter; letusnowdiscuss thefurther development ofthetheory. When anewtheoretical principle isdeveloped, such astheprinciple ofleast time, ourfirstinclination might betosay,“Well, thatisvery pretty; itisdelightful; butthequestion is,does it help atallinunderstanding thephysics?” Someone may say,“Yes, look athow many things wecannowunderstand!” Another says, “Very well, butIcanunder- stand mirrors, too. Ineed acurve such thatevery tangent plane makes equal angles with thetworays. Icanfigure outalens, too,because every raythatcomes toit isbent through anangle given bySnell’s law.” Evidently thestatement ofleast time andthestatement thatangles areequal onreflection, andthatthesines of theangles areproportional onrefraction, arethesame. Soisitmerely aphilo- sophical question, oroneofbeauty? There canbearguments onboth sides. However, theimportance ofapowerful principle isthatitpredicts newthings. Itiseasytoshow thatthere areanumber ofnewthings predicted byFermat’s principle. First, suppose thatthere arethree media, glass, water, andair,andwe perform arefraction experiment andmeasure theindex nforonemedium against another. Letuscalln12theindex ofair(1)against water (2);n13theindex ofair (1)against glass (3). Ifwemeasured water against glass, weshould findanother index, which weshall calln23.Butthere isnoapriori reason whythere should be anyconnection between n12,n13,andn23. Ontheother hand, according tothe ideaofleast time, there isadefinite relationship. Theindex n12istheratio oftwo things, thespeed inairtothespeed inwater; n13istheratio ofthespeed inairto thespeed inglass; n23istheratio ofthespeed inwater tothespeed inglass. Therefore wecancel outtheair,andget 1/2 111/T13 "13n=—=i = - (26.523 Us 111/l/2 "12 ) Inother words, wepredict thattheindex foranewpairofmaterials canbeob- tained from theindexes oftheindividual materials, both against airoragainst vacuum. Soifwemeasure thespeed oflight inallmaterials, andfrom thisgeta single number foreach material, namely itsindex relative tovacuum, called n,- 26-6 (n1isthespeed inairrelative tothespeed invacuum, etc.), then ourformula is easy. Theindex foranytwomaterials iandj is *'Z E i Q 0ni, vi ni (26.6) Using only Snell’s law, there isnobasis foraprediction ofthiskind.* Butof course thisprediction works. Therelation (26.5) wasknown veryearly, andwasa very strong argument fortheprinciple ofleast time. Another argument fortheprinciple ofleast time, another prediction, isthat ifwemeasure thespeed oflight inwater, itwillbelower than inair. This isa prediction ofacompletely different type. Itisabrilliant prediction, because all wehave sofarmeasured areangles; herewehave atheoretical prediction which is quite different from theobservations from which Fermat deduced theideaofleast time. ltturns out,infact, thatthespeed inwater isslower than thespeed inair, byjusttheproportion thatisneeded togettheright index! 26-5 Amore precise statement ofFermat’s principle Actually, wemust make thestatement oftheprinciple ofleast time alittle more accurately. Itwasnotstated correctly above. Itisincorrectly called the principle ofleast time andwehave gone along with theincorrect description for convenience, butwemust nowseewhat thecorrect statement is.Suppose wehad amirror asinFig.26-3. What makes thelight think ithastogotothemirror? Thepath ofleast time isclearly AB. Sosome people might say,“Sometimes itisa maximum time.” Itisnotamaximum time, because certainly acurved path would take astilllonger time! Thecorrect statement isthefollowing: araygoing ina certain particular path hastheproperty that ifwemake asmall change (say a onepercent shift) intherayinanymanner whatever, sayinthelocation atwhich itcomes tothemirror, ortheshape ofthecurve, oranything, there willbenofirst- order change inthetime; there willbeonly asecond-order change inthetime. Inother words, theprinciple isthatlight takes apath such thatthere aremany other paths nearby which take almost exactly thesame time. Thefollowing isanother difficulty with theprinciple ofleast time, andone which people who donotlikethiskind ofatheory could never stomach. With Snell’s theory wecan“understand” light. Light goes along, itseesasurface, it bends because itdoes something atthesurface. Theideaofcausality, thatitgoes from onepoint toanother, andanother, andsoon,iseasy tounderstand. But theprinciple ofleast time isacompletely different philosophical principle about thewaynature works. Instead ofsaying itisacausal thing, thatwhen wedoone thing, something elsehappens, andsoon,itsaysthis: wesetupthesituation, and light decides which istheshortest time, ortheextreme one,andchooses thatpath. Butwhatdoes itdo,howdoes itfindout? Does itsmell thenearby paths, andcheck them against each other? Theanswer is,yes,itdoes, inaway. That isthefeature which is,ofcourse, notknown ingeometrical optics, andwhich isinvolved in theidea ofwavelength; thewavelength tellsusapproximately how faraway the light must “smell” thepath inorder tocheck it.Itishard todemonstrate thisfact onalarge scale with light, because thewavelengths aresoterribly short. Butwith radiowaves, say3-cm waves, thedistances overwhich theradiowaves arechecking arelarger. lfwehave asource ofradiowaves, adetector, andaslit,asinFig.26-13, theraysofcourse gofrom StoDbecause itisastraight line, andifweclose down theslititisallright-they stillgo.Butnow ifwemove thedetector aside toD’, thewaves willnotgothrough thewide slitfrom StoD’,because theycheck several paths nearby, andsay,“No, myfriend, those allcorrespond todifferent times.” Ontheother hand, ifweprevent theradiation from checking thepaths byclosing theslitdown toaverynarrow crack, then there isbutonepath available, andthe *Although itcanbededuced iftheadditional assumption ismade thatadding alayer ofonesubstance tothesurface ofanother does notchange theeventual angle ofrefraction inthelatter material. 26-7-q>- -<i>jj ~'" A Fig. 26-13. The passage ofradio- waves through anarrow slit. E c 0 Fig. 26-14. Thesummation ofproba- bility omplitudes formany neighboring paths.radiation takes it!With anarrow slit,more radiation reaches D’than reaches it with awide slit! Onecandothesame thing withlight, butitishardtodemonstrate onalarge scale. Theeffect canbeseenunder thefollowing simple conditions. Find asmall, bright light, sayanunfrosted bulb inastreet light faraway orthereflection ofthe suninacurved automobile bumper. Then puttwofingers infront ofoneeye,so astolook through thecrack, andsqueeze thelight tozero very gently. You will seethattheimage ofthelight, which wasalittle dotbefore, becomes quite elon- gated, andeven stretches intoalong line. Thereason isthatthefingers arevery close together, andthelight which issupposed tocome inastraight lineisspread outatanangle, sothatwhen itcomes intotheeyeitcomes infrom several direc- tions. Also youwillnotice, ifyouarevery careful, sidemaxima, alotoffringes along theedges too, Furthermore, thewhole thing iscolored. Allofthiswillbe explained induetime, butforthepresent itisademonstration thatlight does not always goinstraight lines, anditisonethatisvery easily performed. 26-6 How itworks Finally, wegiveavery crude view ofwhat actually happens, how thewhole thing really works, from what wenowbelieve isthecorrect, quantum-dynamically accurate viewpoint, butofcourse only qualitatively described. Infollowing the light from AtoBinFig.26-3, wefindthatthelight does notseem tobeinthe form ofwaves atall.Instead theraysseem tobemade upofphotons, andthey actually produce clicks inaphoton counter, ifweareusing one. Thebrightness of thelight isproportional totheaverage number ofphotons thatcome inpersecond, andwhat wecalculate isthechance thataphoton getsfrom AtoB,saybyhitting themirror. Thelawforthatchance isthefollowing very strange one. Take any path andfindthetimeforthatpath; then make acomplex number, ordraw alittle complex vector, pe”, whose angle 0isproportional tothetime. Thenumber of turns persecond isthefrequency ofthelight. Now takeanother path; ithas,for instance, adifferent time, sothevector foritisturned through adifferent angle- theangle being always proportional tothetime. Take alltheavailable paths and addonalittle vector foreach one; then theanswer isthatthechance ofarrival ofthephoton isproportional tothesquare ofthelength ofthefinal vector, from thebeginning totheend! Now letusshow howthisimplies theprinciple ofleast time foramirror. We consider allrays, allpossible paths ADB, AEB, ACB, etc.,inFig.26-3. Thepath ADB makes acertain small contribution, butthenext path, AEB, takes aquite different time, soitsangle 0isquite different. Letussaythatpoint Ccorresponds to minimum time, where ifwechange thepaths thetimes donotchange. Soforawhile thetimes dochange, andthentheybegin tochange lessandlessaswegetnearpoint C(Fig. 26-14). Sothearrows which wehave toaddarecoming almost exactly at thesame angle forawhile near C,andthen gradually thetime begins toincrease again, andthephases goaround theother way, andsoon.Eventually, wehave quite atight knot. Thetotal probability isthedistance from oneendtotheother, squared. Almost allofthataccumulated probability occurs intheregion where all thearrows areinthesame direction (orinthesame phase). Allthecontributions from thepaths which have verydifferent times aswechange thepath, cancel them- selves outbypointing indiflerent directions. That iswhy, ifwehide theextreme parts ofthemirror, itstillreflects almost exactly thesame, because allwedidwas totake outapiece ofthediagram inside thespiral ends, andthatmakes only a very small change inthelight. Sothisistherelationship between theultimate picture ofphotons with aprobability ofarrival depending onanaccumulation of arrows, andtheprinciple ofleast time. 26-8 27 Geometrical Optics 27-1 Introduction Inthischapter weshall discuss some elementary applications oftheideas of theprevious chapter toanumber ofpractical devices, using theapproximation called geometrical optics. This isamost useful approximation inthepractical design ofmany optical systems andinstruments. Geometrical optics iseither verysimple orelseitisverycomplicated. Bythatwemean thatwecaneither study itonly superficially, sothatwecandesign instruments roughly, using rules that aresosimple thatwehardly need dealwith them here atall,since theyarepracti- cally ofhigh school level, orelse, ifwewant toknow about thesmall errors in lenses andsimilar details, thesubject getssocomplicated thatitistooadvanced todiscuss here! Ifonehasanactual, detailed problem inlensdesign, including analysis ofaberrations, then heisadvised toread about thesubject orelsesimply totrace theraysthrough thevarious surfaces (which iswhat thebook tellshowto do),using thelawofrefraction from onesidetotheother, andtofindoutwhere theycome outandseeiftheyform asatisfactory image. People have saidthatthis istootedious, buttoday, with computing machines, itistheright waytodoit. Onecansetuptheproblem andmake thecalculation foronerayafter another very easily. Sothesubject isreally ultimately quite simple, andinvolves nonew principles. Furthermore, itturns outthattherules ofeither elementary oradvanced optics areseldom characteristic ofother fields, sothatthere isnospecial reason tofollow thesubject very far,with oneimportant exception. Themost advanced andabstract theory ofgeometrical optics wasworked outbyHamilton, anditturns outthatthishasvery important applications in mechanics. Itisactually even more important inmechanics than itisinoptics, andsoweleave Hamilton’s theory forthesubject ofadvanced analytical mechanics, which isstudied inthesenior year oringraduate school. So,appreciating that geometrical optics contributes verylittle, except foritsownsake, wenow goonto discuss theelementary properties ofsimple optical systems onthebasis ofthe principles outlined inthelastchapter. Inorder togoon,wemust have onegeometrical formula, which isthefollow- ing:ifwehave atriangle withasmall altitude handalong based,thenthediagonal s(wearegoing toneed ittofindthedifference intimebetween twodifferent routes) islonger than thebase (Fig. 27-1). How much longer? Thedifference A=s—d canbefound inanumber ofways. Onewayisthis. Weseethats2—d2=h2, or(s—d)(s-1-d)=h2.Buts—d=A,ands -1-d~2s.Thus A~hz/2s. (27.1) This isallthegeometry weneed todiscuss theformation ofimages bycurved surfaces! 27-2 Thefocal length ofaspherical surface Thefirstandsimplest situation todiscuss isasingle refracting surface, sep- arating twomedia with different indices ofrefraction (Fig. 27-2). Weleave the caseofarbitrary indices ofrefraction tothestudent, because ideas arealways the most important thing, notthespecific situation, andtheproblem iseasy enough todoinanycase. Soweshall suppose that, ontheleft,thespeed is1andonthe right itis1/n,where nistheindex ofrefraction. Thelight travels more slowly in theglass byafactor n. 27-127-1 Introduction 27-2 Thefocal length ofaspherical surface 27-3 Thefocal length ofalens 27-4 Magnification 27-5 Compound lenses 27-6 Aberrations 27-7 Resolving power A \S h d Figure 27-1 /Po vo c o’ AIR GLASS Fig. 27-2. Focusing byasingle re- fracting surface. Now suppose thatwehave apoint at0,atadistance sfrom thefront surface oftheglass, andanother point 0’atadistance s’inside theglass, andwedesire to arrange thecurved surface insuchamanner thatevery rayfrom 0which hitsthe surface, atanypoint P,willbebent soastoproceed toward thepoint 0’.Forthat tobetrue, wehave toshape thesurface insuch awaythatthetime ittakes for thelight togofrom 0toP,thatis,thedistance OPdivided bythespeed oflight (thespeed hereisunity), plusn~O’P,which isthetime ittakes togofrom Pto0’, isequal toaconstant independent ofthepoint P.This condition supplies uswith anequation fordetermining thesurface. Theanswer isthatthesurface isavery complicated fourth-degree curve, andthestudent mayentertain himself bytrying tocalculate itbyanalytic geometry. Itissimpler totryaspecial casethatcorre- sponds tos—>oo,because then thecurve isasecond-degree curve andismore recognizable. Itisinteresting tocompare thiscurve with theparabolic curve we found forafocusing mirror when thelight iscoming from infinity. Sotheproper surface cannot easily bemade-to focus thelight from one point toanother requires arather complicated surface. Itturns outinpractice that wedonottrytomake such complicated surfaces ordinarily, butinstead wemake acompromise. Instead oftrying togetalltheraystocome toafocus, we arrange itsothatonly theraysfairly close totheaxisO0’come toafocus. The farther ones may deviate iftheywant to,unfortunately, because theideal surface iscomplicated, andweuseinstead aspherical surface with theright curvature at theaxis. Itissomuch easier tofabricate asphere than other surfaces thatitis profitable forustofindoutwhat happens torays striking aspherical surface, supposing that only therays near theaxis aregoing tobefocused perfectly. Those rayswhich arenear theaxisaresometimes called paraxial rays, andwhat weareanalyzing aretheconditions forthefocusing ofparaxial rays. Weshall discuss later theerrors thatareintroduced bythefactthatallraysarenotalways close totheaxis. Thus, supposing Pisclose totheaxis, wedrop aperpendicular PQsuch that theheight PQish.Foramoment, weimagine thatthesurface isaplane passing through P.Inthatcase, thetimeneeded togofrom OtoPwould exceed thetime from 0toQ,andalso, thetime from Pto0’would exceed thetime from QtoO’. Butthatiswhy theglass must becurved, because thetotal excess time must be compensated bythedelay inpassing from VtoQ!Now theexcess time along route OPishz/2s, andtheexcess time ontheother route isnh2/2s’. This excess time, which must bematched bythedelay ingoing along VQ,differs from what itwould have been inavacuum, because there isamedium present. Inother words, thetime togofrom VtoQisnotasifitwere straight intheair,butitis slower bythefactor n,sothattheexcess delay inthisdistance isthen (n—1)VQ. And now, how large isVQ? Ifthepoint Cisthecenter ofthesphere andif itsradius isR,weseebythesame formula that thedistance VQisequal to h2/2R. Therefore wediscover thatthelawthat connects thedistances sands’, andthatgives ustheradius ofcurvature Rofthesurface thatweneed, is (112/2s) +(nh2/2s’) =(n-1)h2/2R (21.2) OI‘ (l/S)+(n/s’)=(n-1)/R. (27.3) Ifwehave aposition 0andanother position 0’,andwant tofocus light from 0 toO’,then wecancalculate therequired radius ofcurvature Rofthesurface by thisformula. Now itturns out,interestingly, thatthesame lens, with thesame curvature R,willfocus forother distances, namely, foranypairofdistances such thatthe sumofthetworeciprocals, onemultiplied byn,isaconstant. Thus agiven lens will(solong aswelimit ourselves toparaxial rays) focus notonly from 0to0’, butbetween aninfinite number ofother pairs ofpoints, solong asthose pairs of points beartherelationship that1/s+n/s’isaconstant, characteristic ofthelens. Inparticular, aninteresting caseisthatinwhich s—>oo.Wecanseefrom the formula thatasonesincreases, theother decreases. Inother words, ifpoint O 27-2 goesout,point 0’comes in,andviceversa. Aspoint 0goestoward infinity, point 0’keeps moving inuntil itreaches acertain distance, called thefocal length f’, inside thematerial. Ifparallel rayscome in,they willmeet theaxisatadistance f’.Likewise, wecould imagine ittheother way. (Remember thereciprocity rule: iflight willgofrom 0to0’,ofcourse itwillalsogofrom 0’to0.)Therefore, if wehadalight source inside theglass, wemight want toknow where thefocus is. Inparticular, ifthelight intheglass were atinfinity (same problem) where would itcome toafocus outside? This distance iscalled f.Ofcourse, wecanalsoput ittheother way. Ifwehadalight source atfandthelight went through thesurface, then itwould gooutasaparallel beam. Wecaneasily findoutwhat fandf’are: n/f’=(H-1)/R orf’=Rn/(n -1), (27.4) 1/;=(I1-1)/R orf=R/(n-1). (27.5) Weseeaninteresting thing: ifwedivide each focal length bythecorresponding index ofrefraction wegetthesame result! This theorem, infact, isgeneral. Itis true ofanysystem oflenses, nomatter how complicated, soitisinteresting to remember. Wedidnotprove herethatitisgeneral-we merely noted itforasingle surface, butithappens tobetrueingeneral thatthetwofocal lengths ofasystem arerelated inthisway. Sometimes Eq.(27.3) iswritten intheform 1/s+n/s’ =l/f. (27.6) This ismore useful than (27.3) because wecanmeasure fmore easily than wecan measure thecurvature andindex ofrefraction ofthelens: ifwearenotinterested indesigning alensorinknowing howitgotthatway, butsimply liftitoffashelf, theinteresting quantity isf,notthenandthe1andtheR! Now aninteresting situation occurs ifsbecomes lessthanf.What happens then? Ifs<f,then (1/s) >(l/f), andtherefore s’isnegative; ourequation saysthatthelight willfocus only with anegative value ofs’,whatever thatmeans! Itdoes mean something veryinteresting andverydefinite. Itisstillauseful formula, inother words, even when thenumbers arenegative. What itmeans isshown in Fig.27-3. Ifwedraw therayswhich arediverging from 0,theywillbebent, itis true, atthesurface, andtheywillnotcome toafocus, because Oissoclose inthat they are“beyond parallel." However, they diverge asifthey hadcome from a point 0’outside theglass. This isanapparent image, sometimes called avirtual image. Theimage 0’inFig.27-2 iscalled arealimage. Ifthelight really comes to apoint, itisarealimage. Butifthelight appears tobecoming from apoint, a fictitious point different from theoriginal point, itisavirtual image. Sowhen s’comes outnegative, itmeans that0’isontheother sideofthesurface, andevery- thing isallright. ~ Now consider theinteresting case where Risequal toinfinity; then wehave (1/s) +(n/s’) =0.Inother words, s’=—ns, which means thatifwelook from adense medium intoararemedium andseeapoint intheraremedium, itappears tobedeeper byafactor n.Likewise, wecanusethesame equation backwards, sothatifwelook intoaplane surface atanobject thatisatacertain distance inside thedense medium, itwillappear asthough thelight iscoming from notas farback (Fig. 27-4). When welook atthebottom ofaswimming pool from above, itdoes notlook asdeep asitreally is,byafactor 3/4,which isthereciprocal ofthe index ofrefraction ofwater. Wecould goon,ofcourse, todiscuss thespherical mirror. Butifoneappreci- atestheideas involved, heshould beable towork itoutforhimself. Therefore weleave ittothestudent towork outtheformula forthespherical mirror, but wemention thatitiswell toadopt certain conventions concerning thedistances involved: (1)Theobject distance sispositive ifthepoint Oistotheleftofthesurface. (2)Theimage distance s’ispositive ifthepoint O’istotheright ofthesurface. (3)Theradius ofcurvature ofthesurface ispositive ifthecenter istotheright ofthesurface. 27-31_¢_’_-._ »-:--- O’ I::5-FIJI :9I-I_ -.\. Fig. 27-3. Avirtual image. 1 / AIR GLASS Fig.27-4. Aplane surface re-images thelight from O’toO. {L 2 Fig.27-5. Image formation bya two-surface lens. P h \ T .0 Q 0 HIH2 HI Fig.27-6. Athinlenswith twoposi- tiveradii.InFig.27-2, forexample, s,s’,andRareallpositive; inFig.27-3, sandRare positive, buts’isnegative. Ifwehadused aconcave surface, ourformula (27.3) would stillgivethecorrect result ifwemerely make Ranegative quantity. Inworking outthecorresponding formula foramirror, using theabove conventions, youwillfindthatifyouputn=—1throughout theformula (27.3) (asthough thematerial behind themirror hadanindex -1), theright formula for amirror results! Although thederivation offormula (27.3) issimple andelegant, using least time, onecanofcourse work outthesame formula using Snell’s law,remembering thattheangles aresosmall thatthesines ofangles canbereplaced bytheangles themselves. 27-3 Thefocal length ofalens Now wegoontoconsider another situation, avery practical one. Most of thelenses that weusehave twosurfaces, notjust one. How does thisaffect matters? Suppose that wehave twosurfaces ofdifferent curvature, with glass filling thespace between them (Fig. 27-5). Wewant tostudy theproblem of focusing from apoint Otoanalternate point 0’.How canwedothat? Theanswer isthis: First, useformula (27.3) forthefirstsurface, forgetting about thesecond surface. This willtellusthatthelight which wasdiverging from Owillappear tobeconverging ordiverging, depending onthesign, from some other point, sayO’.Now weconsider anewproblem. Wehave adifferent surface, between glass andair,inwhich raysareconverging toward acertain point O’.Where will they actually converge? Weusethesame formula again! Wefindthattheycon- verge at0". Thus, ifnecessary, wecangothrough 75surfaces byjustusing the same formula insuccession, from onetothenext! There aresome rather high-class formulas thatwould save usconsiderable energy inthefewtimes inourlives thatwemight have tochase thelight through fivesurfaces, butitiseasier justtochase itthrough fivesurfaces when theproblem arises than itistomemorize alotofformulas, because itmaybewewillnever have tochase itthrough anysurfaces atall! Inanycase, theprinciple isthatwhen wegothrough onesurface wefinda newposition, anewfocal point, andthen take thatpoint asthestarting point for thenextsurface, andsoon.Inorder toactually dothis,since onthesecond surface wearegoing from nto1rather than from 1ton,andsince inmany systems there ismore than onekind ofglass, sothatthere areindices n1,n2,...,wereally need ageneralization offormula (27.3) foracasewhere there aretwodifferent indices, n1andn2,rather than only n.Then itisnotdiflicult toprove thatthegeneral form of(27.3) is (H1/S) +('12/S’) =(I12—P11)/R (27-7) Particularly simple isthespecial caseinwhich thetwosurfaces areveryclose together—so close thatwemayignore small errors duetothethickness. Ifwedraw thelensasshown inFig.27-6, wemay askthisquestion: How must thelensbe built soastofocus light from OtoO’? Suppose thelight comes exactly to theedge ofthelens, atpoint P.Then theexcess time ingoing from 0to0'is (nlhz/2s) +(n1h2/2s’), ignoring foramoment thepresence ofthethickness T ofglass ofindex n2.Now, tomake thetime forthedirect path equal tothatfor thepath OP0’, wehave touseapiece ofglass whose thickness Tatthecenter issuch thatthedelay introduced ingoing through thisthickness isenough tocompensate fortheexcess time above. Therefore thethickness ofthelensatthecenter must begiven bytherelationship (n1h2/2s) -l—(n1h2/2s’) =(n2—n1)T. (27.8) Wecanalsoexpress Tinterms oftheradii R1andR2ofthetwosurfaces. Paying attention toourconvention (3),wethusfind, forR1<R2(aconvex lens), T=(h2/2R1) —(hz/2R2). (27.9) 27-4 Therefore, wefinally get ('11/S) +(H1/S’) =(I12—n1)(1/R1 -1/R2) (27-10) Now wenote again thatifoneofthepoints isatinfinity, theother willbeata point which wewillcallthefocal length f.Thefocal length fisgiven by 1/f=('1-1)(1/R1 —1/R2). (27-11) wheren =n2/n1. Now, ifwetake theopposite case, where sgoes toinfinity, weseethats’isat thefocal length j’.This time thefocal lengths areequal. (This isanother special case ofthegeneral rulethattheratio ofthetwofocal lengths istheratio ofthe indices ofrefraction inthetwomedia inwhich therays focus. Inthisparticular optical system, theinitial andfinal indices arethesame, sothetwofocal lengths areequal.) Forgetting foramoment about theactual formula forthefocal length, ifwe bought alensthatsomebody designed with certain radii ofcurvature andacertain index, wecould measure thefocal length, say,byseeing where apoint atinfinity focuses. Once wehadthefocal length, itwould bebetter towrite ourequation in terms ofthefocal length directly, andtheformula then is (l/s) -1-(1/s’) =l/f. (27.12) Now letusseehowtheformula works andwhat itimplies indifferent circum- stances. First, itimplies thatifsors’isinfinite theother oneisf.That means that parallel light focuses atadistance f,andthisineffect defines f.Another interesting thing itsays isthatboth points move inthesame direction. Ifonemoves tothe right, theother does also. Another thing itsaysisthatsands’areequal iftheyare both equal to2f.Inother words, ifwewant asymmetrical situation, wefindthat theywillboth focus atadistance 2fl 27-4 Magnification Sofarwehave discussed thefocusing action only forpoints ontheaxis. Now letusdiscuss alsotheimaging ofobjects notexactly ontheaxis, butalittle bitoff, sothatwecanunderstand theproperties ofmagnification. When wesetupalens soastofocus light from asmall filament onto a“point” onascreen, wenotice thatonthescreen wegeta“picture” ofthesame filament, except ofalarger or smaller sizethan thetruefilament. This must mean thatthelight comes toafocus from eachpoint ofthefilament. Inorder tounderstand thisalittle better, letus analyze thethinlenssystem shown schematically inFig.27-7. Weknow thefollow- ingfacts: (1)Any raythatcomes inparallel ononesideproceeds toward acertain par- ticular point called thefocus ontheother side, atadistance ffrom thelens. (2)Anyraythatarrives atthelensfrom thefocus ononesidecomes outparallel totheaxisontheother side. This isallweneed toestablish formula (27.12) bygeometry, asfollows: Suppose wehave anobject atsome distance xfrom thefocus; lettheheight oftheobject bey.Then weknow thatoneoftherays, namely PQ,willbebent soastopass through thefocus Rontheother side. Now ifthelenswillfocus point Patall,we canfindoutwhere ifwefindoutwhere justoneother raygoes, because thenew focus willbewhere thetwointersect again. Weneed only useouringenuity to findtheexact direction ofoneother ray. Butweremember thataparallel raygoes through thefocus andviceversa: araywhich goes through thefocus willcome out parallel! Sowedraw rayPTthrough U.(Itistruethattheactual rayswhich are doing thefocusing may bemuch more limited than thetwowehave drawn, but they areharder tofigure, sowemake believe thatwecanmake thisray.) Since it would come outparallel, wedraw TSparallel toXW.Theintersection Sisthe point weneed. This willdetermine thecorrect place andthecorrect height. Let 27-5it . Fig. 27-7. Thegeometry ofimaging byathinlens.<XC><~< Y __—T—— I xX _ —i‘_ y, f f 1 2 Fig. 27-8. Illustration oftheprincipal planes ofanoptical system.uscalltheheight y’andthedistance from thefocus, x’.Now wemay derive a lensformula. Using thesimilar triangles PVUandTXU,wefind I J’ J’-=-' (27.13 rx ) Similarly, from triangles SWRandQXR, weget I J’ J’—=-- 2.14 X,f (7) Solving each fory'/y,wefindthat I I ’i=3%= (27.15) Equation (27.15) isthefamous lensformula; initiseverything weneed toknow about lenses: Ittellsusthemagnification, y’/y,interms ofthedistances andthe focal lengths. Italsoconnects thetwodistances xandx’withf1 xx’=f2, (27.16) which isamuch neater form towork with than Eq.(27.12). Weleave ittothe student todemonstrate thatifwecalls=x+fands’=x’+f,Eq.(27.12) isthesame asEq.(27.16). 27-5 Compound lenses Without actually deriving it,weshall briefly describe thegeneral result when wehave anumber oflenses. Ifwehave asystem ofseveral lenses, how canwe possibly analyze it?That iseasy. Westart with some object andcalculate where itsimage isforthefirstlens, using formula (27.16)or(27.12) oranyother equivalent formula, orbydrawing diagrams. Sowefindanimage. Then wetreat thisimage asthesource forthenext lens, andusethesecond lenswith whatever itsfocal length istoagain findanimage. Wesimply chase thething through thesuccession oflenses. That isallthere istoit.Itinvolves nothing newinprinciple, soweshall notgointoit.However, there isavery interesting netresult oftheeffects ofany sequence oflenses onlight thatstarts andends upinthesame medium, sayair. Any optical instrument—a telescope oramicroscope with anynumber oflenses and mirrors—has thefollowing property: There exist two planes, called the principal planes ofthesystem (these planes areoften fairly close tothefirstsurface ofthefirstlensandthelastsurface ofthelastlens), which have thefollowing prop- erties: (l)Iflight comes intothesystem parallel from thefirstside, itcomes out atacertain focus, atadistance from thesecond principal plane equal tothefocal length, justasthough thesystem were athinlenssituated atthisplane. (2)If parallel light comes intheother way, itcomes toafocus atthesame distance f from thefirst principal plane, again asifathinlenswhere situated there. (See Fig.27-8.) Ofcourse, ifwemeasure thedistances xandx’,andyandy’asbefore, the formula (27.16) thatwehave written forthethinlensisabsolutely general, pro- vided thatwemeasure thefocal length from theprincipal planes andnotfrom the center ofthelens. Itsohappens thatforathinlenstheprincipal planes arecoin- cident. Itisjustasthough wecould takeathinlens, sliceitdown themiddle, and separate it,andnotnotice thatitwasseparated. Every raythatcomes inpops out immediately ontheother sideofthesecond plane from thesame point asitwent intothefirstplane! Theprincipal planes andthefocal length maybefound either byexperiment orbycalculation, andthen thewhole setofproperties oftheoptical system aredescribed. Itisvery interesting thattheresult isnotcomplicated when weareallfinished with such abig,complicated optical system. 27-6 27-6 Aberrations Before wegettooexcited about howmarvelous lenses are,wemust hasten toadd that there arealso serious limitations, because ofthefact that wehave limited ourselves, strictly speaking, toparaxial rays, theraysnear theaxis. Areal lenshaving afinite sizewill, ingeneral, exhibit aberrations. Forexample, aray thatisontheaxis, ofcourse, goes through thefocus; araythatisveryclose tothe axiswillstillcome tothefocus verywell. Butaswegofarther out,theraybegins todeviate from thefocus, perhaps byfalling short, andaraystriking near the topedge comes down andmisses thefocus byquite awide margin. So,instead of getting apoint image, wegetasmear. This effect iscalled spherical aberration, because itisaproperty ofthespherical surfaces weuseinplace oftheright shape. Thiscould beremedied, foranyspecific object distance, byre-forming theshape of thelenssurface, orperhaps byusing several lenses arranged sothattheaberrations oftheindividual lenses tend tocancel each other. Lenses have another fault: light ofdifferent colors hasdifferent speeds, or different indices ofrefraction, intheglass, andtherefore thefocal length ofa given lensisdifferent fordifferent colors. Soifweimage awhite spot, theimage willhave colors, because when wefocus forthered,theblue isoutoffocus, or viceversa. This property iscalled chromatic aberration. There arestillother faults. Iftheobject isofftheaxis, then thefocus really isn‘t perfect anymore, when itgetsfarenough offtheaxis. Theeasiest wayto verify thisistofocus alensandthentiltitsothattheraysarecoming inatalarge angle from theaxis. Then theimage thatisformed willusually bequite crude, andthere may benoplace where itfocuses well. There arethus several kinds of errors inlenses thattheoptical designer tries toremedy byusing many lenses to compensate each other’s errors. How careful dowehave tobetoeliminate aberrations? Isitpossible tomake anabsolutely perfect optical system? Suppose wehadbuilt anoptical system that issupposed tobring light exactly toapoint. Now, arguing from thepoint ofview ofleast time, canwefindacondition onhow perfect thesystem hastobe? The system willhave some kind ofanentrance opening forthelight. Ifwetake the farthest rayfrom theaxisthatcancome tothefocus (ifthesystem isperfect, of course), thetimes forallrays areexactly equal. Butnothing isperfect, sothe question is,how wrong canthetime beforthisrayandnotbeworth correcting anyfurther? That depends onhow perfect wewant tomake theimage. But suppose wewant tomake theimage asperfect asitpossibly canbemade. Then, ofcourse, ourimpression isthatwehave toarrange thatevery raytakes asnearly thesame time aspossible. Butitturns outthat thisisnottrue, thatbeyond a certain point wearetrying todosomething thatistoofine, because thetheory of geometrical optics does notwork! Remember that theprinciple ofleast time isnotanaccurate formulation, unlike theprinciple ofconservation ofenergy ortheprinciple ofconservation of momentum. Theprinciple ofleast time isonly anapproximation, anditisinter- esting toknow how much error canbeallowed andstillnotmake anyapparent difference. Theanswer isthatifwehave arranged thatbetween themaximal ray— theworst ray,theraythatisfarthest out—and thecentral ray,thedifference in time islessthan about theperiod thatcorresponds tooneoscillation ofthelight, then there isnouseimproving itanyfurther. Light isanoscillatory thing with a definite frequency thatisrelated tothewavelength, andifwehave arranged that thetime difference fordifferent rays islessthan about aperiod, there isnouse going anyfurther. 27-7 Resolving power Another interesting question—a very important technical question with all optical instruments—is how much resolving power theyhave. Ifwebuild amicro- scope, wewant toseetheobjects thatwearelooking at.That means, forinstance, thatifwearelooking atabacterium with aspot oneach end,wewant toseethat 27-7 op,‘I 4eP1,S R Fig. 27-9. Theresolving power ofon optical system.there aretwodots when wemagnify them. Onemight think thatallwehave to doistogetenough magnification—we canalways addanother lens, andwecan always magnify again andagain, andwith thecleverness ofdesigners, allthe spherical aberrations andchromatic aberrations canbecancelled out,andthere isnoreason whywecannot keep onmagnifying theimage. Sothelimitations ofa microscope arenotthatitisimpossible tobuild alensthatmagnifies more than 2000 diameters. Wecanbuild asystem oflenses thatmagnifies 10,000 diameters, butwestillcould notseetwopoints thataretooclose together because ofthe limitations ofgeometrical optics, because ofthefactthatleast time isnotprecise. Todiscover therulethatdetermines how farapart twopoints have tobeso thatattheimage they appear asseparate points canbestated inavery beautiful wayassociated with thetime ittakes fordifferent rays. Suppose thatwedisregard theaberrations now, andimagine thatforaparticular point P(Fig. 27-9) allthe raysfrom object toimage Ttakeexactly thesame time. (Itisnottrue, because it isnotaperfect system, butthatisanother problem.) Now take another nearby point, P’,andaskwhether itsimage willbedistinct from T.Inother words, whether wecanmake outthedifference between them. Ofcourse, according togeometrical optics, there should betwopoint images, butwhat weseemay berather smeared andwemaynotbeabletomake outthatthere aretwopoints. Thecondition that thesecond point isfocused inadistinctly different place from thefirstoneisthat thetwotimes fortheextreme raysP’ST andP’RT oneach sideofthebigopening ofthelenses togofrom oneendtotheother, must notbeequal from thetwo possible object points toagiven image point. Why? Because, ifthetimes were equal, ofcourse both wouldfocus atthesame point. Sothetimes arenotgoing to beequal. Butbyhowmuch dotheyhave todiffer sothatwecansaythatboth do notcome toacommon focus, sothatwecandistinguish thetwoimage points‘? Thegeneral rulefortheresolution ofanyoptical instrument isthis: twodifi"erent point sources canberesolved only ifonesource isfocused atsuch apoint that thetimes forthemaximal rays from theother source toreach that point, as compared with itsown trueimage point, differ bymore than oneperiod. Itis necessary thatthedifference intime between thetoprayandthebottom rayto thewrong focus shall exceed acertain amount, namely, approximately theperiod ofoscillation ofthelight: I2— T1>1/V, where 1/isthefrequency ofthelight (number ofoscillations persecond; alsospeed divided bywavelength). Ifthedistance ofseparation ofthetwopoints iscalled D,andiftheopening angle ofthelensiscalled 0,then onecandemonstrate that (27.17) isexactly equivalent tothestatement thatDmust exceed A/nsin9,where nistheindex ofrefraction atPandAisthewavelength. Thesmallest things that Wecanseearetherefore approximately thewavelength oflight. Acorresponding formula exists fortelescopes, which tellsusthesmallest difference inangle between twostars thatcanjustbedistinguished.* *Theangle isabout A/D, where Disthelensdiameter. Canyouseewhy? 27-8 28 Electromagnetic Radiation 28-1 Electromagnetism Themost dramatic moments inthedevelopment ofphysics arethose inwhich great syntheses take place, where phenomena which previously hadappeared to bedifferent aresuddenly discovered tobebutdifferent aspects ofthesame thing. Thehistory ofphysics isthehistory ofsuch syntheses, andthebasis ofthesuccess ofphysical science ismainly thatweareabletosynthesize. Perhaps themost dramatic moment inthedevelopment ofphysics during the l9thcentury occurred toJ.C.Maxwell onedayinthe1860’s, when hecombined thelaws ofelectricity andmagnetism with thelaws ofthebehavior oflight. Asa result, theproperties oflight were partly unravelled——that oldandsubtle stuff thatissoimportant andmysterious thatitwasfeltnecessary toarrange aspecial creation foritwhen writing Genesis. Maxwell could say, when hewas finished with hisdiscovery, “Let there beelectricity andmagnetism, andthere islight!” For this culminating moment there was along preparation inthegradual discovery andunfolding ofthelaws ofelectricity andmagnetism. This story we shall reserve fordetailed study nextyear. However, thestory is,briefly, asfollows. Thegradually discovered properties ofelectricity andmagnetism, ofelectric forces ofattraction andrepulsion, andofmagnetic forces, showed thatalthough these forces were rather complex, theyallfelloffinversely asthesquare ofthedistance. Weknow, forexample, thatthesimple coulomb lawforstationary charges isthat theelectric force field varies inversely asthesquare ofthedistance. Asaconse- quence, forsulhciently great distances there isverylittle influence ofonesystem ofcharges onanother. Maxwell noted thattheequations orthelaws thathadbeen discovered uptothistime were mutually inconsistent when hetried toputthem alltogether, andinorder forthewhole system tobeconsistent, hehadtoaddan- other term tohisequations. With thisnewterm there came anamazing prediction, which wasthatapartoftheelectric andmagnetic fields would falloffmuch more slowly with thedistance than theinverse square, namely, inversely asthefirst power ofthedistance! And soherealized thatelectric currents inoneplace can affect other charges faraway, andhepredicted thebasic effects withwhich weare familiar today——radio transmission, radar, andsoon. ltseems amiracle thatsomeone talking inEurope can, with mere electrical influences, beheard thousands ofmiles away inLosAngeles. How isitpossible? ltisbecause thefields donotvary astheinverse square, butonly inversely asthe firstpower ofthedistance. Finally, then, even light itself wasrecognized tobe electric andmagnetic influences extending over vast distances, generated byan almost incredibly rapid oscillation oftheelectrons intheatoms. Allthese phe- nomena wesummarize bytheword radiation or,more specifically, electromagnetic radiation, there being one ortwo other kinds ofradiation also. Almost always, radiation means electromagnetic radiation. And thus istheuniverse knittogether. Theatomic motions ofadistant star stillhave sulficient influence atthisgreat distance tosettheelectrons inoureye inmotion, andsoweknow about thestars. Ifthislawdidnotexist, wewould allbeliterally inthedark about theexterior world! And theelectric surgings ina galaxy fivebillion light years away—which isthefarthest object wehave found sofar——can stillinfluence inasignificant anddetectable waythecurrents inthe great “dish” infront ofaradio telescope. And soitisthatweseethestars andthe galaxies. 28-128-1 Electromagnetism 28-2 Radiation 28-3 Thedipole radiator 28-4 Interference This remarkable phenomenon iswhat weshall discuss inthepresent chapter. Atthebeginning ofthiscourse inphysics weoutlined abroad picture oftheworld, butwearenowbetter prepared tounderstand some aspects ofit,andsoweshall now goover some parts ofitagain ingreater detail. Webegin bydescribing the position ofphysics attheendofthe19thcentury. Allthatwasthen known about thefundamental laws canbesummarized asfollows. First, there were laws offorces: oneforce wasthelawofgravitation, which wehave written down several times; theforce onanobject ofmass m,dueto another ofmass M,isgiven by F=GmMe,/r2, (28.1) where e,isaunitvector directed from mtoM,andristhedistance between them. Next, thelaws ofelectricity andmagnetism, asknown attheendofthe19th century, arethese: theelectrical forces acting onacharge qcanbedescribed by twofields, called EandB,andthevelocity vofthecharge q,bytheequation F=q(E+VXB). (28.2) Tocomplete thislaw,wehave tosaywhat theformulas forEandBareinagiven circumstance: ifanumber ofcharges arepresent, EandtheBareeach thesumof contributions, onefrom each individual charge. SoifwecanfindtheEandB produced byasingle charge, weneed onlytoaddalltheeffects from allthecharges intheuniverse togetthetotal EandB!This istheprinciple ofsuperposition. What istheformula fortheelectric andmagnetic field produced byonein- dividual charge? Itturns outthatthisisverycomplicated, andittakes agreat deal ofstudy andsophistication toappreciate it.Butthatisnotthepoint. Wewrite down thelawnowonlytoimpress thereader with thebeauty ofnature, sotospeak, i.e.,thatitispossible tosummarize allthefundamental knowledge ononepage, withnotations thatheisnowfamiliar with. This lawforthefields ofanindividual charge iscomplete andaccurate, sofarasweknow (except forquantum mechanics) butitlooks rather complicated. Weshall notstudy allthepieces now; weonly write itdown togiveanimpression, toshow thatitcanbewritten, andsothatwe canseeahead oftime roughly what itlooks like. Asamatter offact, themost useful waytowrite thecorrect laws ofelectricity andmagnetism isnotthewaywe shall now write them, butinvolves what arecalled field equations, which weshall learn about nextyear. Butthemathematical notations forthese aredifferent and new, andsowewrite thelawinaninconvenient form forcalculation, butinnota- tions thatwenow know. Theelectric field, E,isgiven by — e,’ r’de,’ ldz E=F?.,lrT2+za(;'-2>+am°"l' <2“) What dothevarious terms tellus‘?Take thefirstterm, E=—qe,//41re0r’2. That, ofcourse, isCoulomb’s law,which wealready know: qisthecharge thatispro- ducing thefield; e,»istheunitvector inthedirection from thepoint Pwhere Eis measured, risthedistance from Ptoq.But, Coulomb’s lawiswrong. Thedis- coveries ofthe19th century showed thatinfluences cannot travel faster than a certain fundamental speed c,which wenowcallthespeed oflight. Itisnotcorrect thatthefirstterm isCoulomb’s law, notonly because itisnotpossible toknow where thecharge isnowandatwhat distance itisnow, butalsobecause theonly thing thatcanaffect thefieldatagiven place andtimeisthebehavior ofthecharges inthepast. How farinthepast? Thetime delay, orretarded time, so-called, is thetime ittakes, atspeed c,togetfrom thecharge tothefieldpoint P.Thedelay isr’/c. Sotoallow forthistime delay, weputalittle prime onr,meaning how far away itwaswhen theinformation now arriving atPleftq.Justforamoment suppose thatthecharge carried alight, andthatthelight could only come toP atthespeed c.Then when welook atq,wewould notseewhere itisnow, ofcourse, butwhere itwasatsome earlier time. What appears inourformula istheapparent 28-2 direction 0,!—the direction itused tobe——the so-called retarded direction—and attheretarded distance r’.That would beeasyenough tounderstand, too,butit isalsowrong. Thewhole thing ismuch more complicated. There areseveral more terms. Thenextterm isasthough nature were trying toallow forthefactthattheeffect isretarded, ifwemight putitverycrudely. It suggests thatweshould calculate thedelayed coulomb field andaddacorrection toit,which isitsrateofchange times thetime delay thatweuse. Nature seems to beattempting toguess what thefield atthepresent time isgoing tobe,bytaking therateofchange andmultiplying bythetime thatisdelayed. Butwearenot yetthrough. There isathird term—the second derivative, with respect tot,ofthe unitvector inthedirection ofthecharge. Now theformula isfinished, andthatis allthere istotheelectric field from anarbitrarily moving charge. Themagnetic field isgiven by B=—e.' XE/c. (28.4) Wehave written these down only forthepurpose ofshowing thebeauty ofnature or,inaway, thepower ofmathematics. Wedonotpretend tounderstand whyit ispossible towrite somuch insuch asmall space, but(28.3) and(28.4) contain themachinery bywhich electric generators work, how light operates, allthephe- nomena ofelectricity andmagnetism. Ofcourse, tocomplete thestory wealso need toknow something about thebehavior ofthematerials involved——the prop- erties ofmatter—which arenotdescribed properly by(28.3). Tofinish with ourdescription oftheworld ofthe19th century wemust mention oneother great synthesis which occurred inthatcentury, onewith which Maxwell hadagreat dealtodoalso, andthatwasthesynthesis ofthephenomena ofheatandmechanics. Weshall study thatsubject soon. What hadtobeadded inthe20th century waisflthat thedynamical laws of Newton were found tobeallwrong, andquantum mechanics hadtobeintroduced tocorrect them. Newton’s laws areapproximately valid when thescale ofthings is sufficiently large. These quantum-mechanical laws, combined with thelaws of electricity, have only recently been combined toform asetoflaws called quantum electrodynamics. Inaddition, there were discovered anumber ofnewphenomena, ofwhich thefirst wasradioactivity, discovered byBecquerel in1898——he just sneaked itinunder the19th century.» This phenomenon ofradioactivity was followed uptoproduce ourknowledge ofnuclei andnewkinds offorces thatare notgravitational andnotelectrical, butnewparticles with different interactions, asubject which hasstillnotbeen unravelled. Forthose purists who know more (theprofessors who happen tobereading this), weshould addthatwhen wesaythat(28.3) isacomplete expression ofthe knowledge ofelectrodynamics, wearenotbeing entirely accurate. There wasa problem thatwasnotquite solved attheendofthe19thcentury. When wetryto calculate thefield from allthecharges including thecharge itself thatwewant the field toacton,wegetintotrouble trying tofindthedistance, forexample, ofa charge from itself, anddividing something bythatdistance, which iszero. The problem ofhow tohandle thepart ofthisfield which isgenerated bythevery charge onwhich wewant thefield toactisnotyetsolved today. Soweleave it there; wedonothave acomplete solution tothatpuzzle yet,andsoweshall avoid thepuzzle foraslong aswecan. 28-2 Radiation That, then, isasummary oftheworld picture. Now letususeittodiscuss thephenomena called radiation. Todiscuss these phenomena, wemust select from Eq.(28.3) only thatpiece which varies inversely asthedistance andnotas thesquare ofthedistance. Itturns outthatwhen wefinally dofindthatpiece, it issosimple initsform thatitislegitimate tostudy optics andelectrodynamics inan elementary waybytaking itas“thelaw" oftheelectric fieldproduced byamoving charge faraway. Weshall take ittemporarily asagiven lawwhich wewilllearn about indetail nextyear. 28-3 Oftheterms appearing in(28.3), thefirstoneevidently goes inversely asthe square ofthedistance, andthesecond isonly acorrection fordelay, soitiseasy toshow thatbothofthem varyinversely asthesquare ofthedistance. Allofthe effects weareinterested income from thethird term, which isnotverycomplicated, after all.What thisterm saysis:look atthecharge andnote thedirection ofthe unitvector (wecanproject theendofitonto thesurface ofaunitsphere). Asthe charge moves around, theunit vector wiggles, andtheacceleration ofthatunit vector iswhat wearelooking for. That isall.Thus _—q d2e,1 E"new71;? <28-5) isastatement ofthelaws ofradiation, because thatistheonly important term when wegetfarenough away thatthefields arevarying inversely asthedistance. (The parts thatgoasthesquare have fallen offsomuch thatwearenotinterested inthem.) Now wecangoalittle bitfurther instudying (28.5) toseewhat itmeans. Suppose acharge ismoving inanymanner whatsoever, andweareobserving it from adistance. Weimagine foramoment thatinasense itis“1itup”(although itislight thatwearetrying toexplain); weimagine itasalittle white dot. Then wewould seethiswhite dotrunning around. Butwedon’t seeexactly how itis running around right now, because ofthedelay thatwehave been talking about. What counts ishowitwasmoving earlier. Theunitvector e,’ispointed toward the apparent position ofthecharge. Ofcourse, theendofe,’goes onaslight curve, sothatitsacceleration hastwocomponents. Oneisthetransverse piece, because theendofitgoesupanddown, andtheother isaradial piece because itstays ona sphere. Itiseasy todemonstrate thatthelatter ismuch smaller andvaries asthe inverse square ofrwhen risvery great. This iseasy tosee,forwhen weimagine thatwemove agiven source farther andfarther away, then thewigglings ofe,’ look smaller andsmaller, inversely asthedistance, buttheradial component ofacceleration isvarying much more rapidly than inversely asthedistance. So forpractical purposes allwehave todoisproject themotion onaplane atunit distance. Therefore wefindthefollowing rule: Imagine thatwelook atthemoving charge andthateverything weseeisdelayed—like apainter trying topaint ascene onascreen ataunitdistance. Arealpainter, ofcourse, does nottakeintoaccount thefactthatlight isgoing atacertain speed, butpaints theworld asheseesit. Wewant toseewhat hispicture would look like. Soweseeadot,representing the charge, moving about inthepicture. Theacceleration ofthatdotisproportional totheelectric field. That isall—all weneed. Thus Eq.(28.5) isthecomplete andcorrect formula forradiation; even relativity effects areallcontained init.However, weoften want toapply ittoa stillsimpler circumstance inwhich thecharges aremoving only asmall distance atarelatively slow rate. Since they aremoving slowly, they donotmove anap- preciable distance from where they start, sothat thedelay time ispractically constant. Then thelawisstillsimpler, because thedelay time isfixed. Thus we imagine thatthecharge isexecuting avery tinymotion ataneffectively constant distance. Thedelay atthedistance risr/c. Then ourrulebecomes thefollowing: Ifthecharged object ismoving inaverysmall motion anditislaterally displaced bythedistance x(t), then theangle thattheunitvector e,’isdisplaced isx/r,and since rispractically constant, thex-component ofd2e,t/dt2 issimply theaccelera- tionofxitself atanearlier time, andsofinally wegetthelawwewant, which is E(t)-if a(1- (22.6)Z 41reoc2r ’ Only thecomponent ofaxperpendicular tothelineofsight isimportant. Let usseewhythatis.Evidently, ifthecharge ismoving inandoutstraight atus,the unitvector inthatdirection does notwiggle atall,andithasnoacceleration. So itisonly thesidewise motion which isimportant, only theacceleration thatwe seeprojected onthescreen. 28-4 28-3 Thedipole radiator Asourfundamental “law” ofelectromagnetic radiation, wearegoing to assume that(28.6) istrue, i.e.,thattheelectric field produced byanaccelerating charge which ismoving nonrelativistically atavery large distance rapproaches thatform. Theelectric fieldvaries inversely asrandisproportional totheaccelera- tionofthecharge, projected onto the“plane ofsight,” andthisacceleration isnot today’s acceleration, buttheacceleration thatithadatanearlier time, theamount ofdelay being atime, r/c. Intheremainder ofthischapter weshall discuss this lawsothatwecanunderstand itbetter physically, because wearegoing touse ittounderstand allofthephenomena oflight andradio propagation, such as reflection, refraction, interference, diffraction, andscattering. Itisthecentral law,andisallweneed. AlltherestofEq.(28.3) waswritten down only tosetthe stage, sothatwecould appreciate where (28.6) fitsandhow itcomes about. Weshall discuss (28.3) further nextyear. Inthemeantime, weshall accept it astrue, butnotjust onatheoretical basis. Wemaydevise anumber ofexperiments which illustrate thecharacter ofthelaw. Inorder todoso,weneed anaccelerating charge. Itshould beasingle charge, butifwecanmake agreat many charges move together, allthesame way, weknow thatthefield willbethesum ofthe eflects ofeach oftheindividual charges; wejust addthem together. Asanexample, consider twopieces ofwire connected toagenerator, asshown inFig. 28-1. Theideaisthatthegenerator makes apotential difference, orafield, which pulls electrons away from piece Aandpushes them into Batonemoment, andthen, an infinitesimal time later, itreverses theeffect andpulls theelectrons outofBand pumps them back into A!Sointhese twowires charges, letussay,areaccelerating upward inwireAandupward inwireBforonemoment, andamoment later they areaccelerating downward inwire Aanddownward inwire B.The factthat we need twowires andagenerator ismerely thatthisisawayofdoing it.Thenet result isthatwemerely have acharge accelerating upanddown asthough AandB were onesingle wire. Awire thatisvery short compared with thedistance light travels inoneoscillation period iscalled anelectric dipole oscillator. Thus we have thecircumstance thatweneed toapply ourlaw,which tellsusthatthischarge makes anelectric field, andsoweneed aninstrument todetect anelectric field. andtheinstrument weuseisthesame thing—a pairofwires likeAandBlIfan electric field isapplied tosuch adevice, itwillproduce aforce which willpull theelectrons uponboth wires ordown onboth wires. This signal isdetected bymeans ofarectifier mounted between AandB,andatiny, finewirecarries the information into anamplifier, where itisamplified sowecanhear theaudio- frequency tonewithwhich theradiofrequency ismodulated. When thisprobe feels anelectric field, there willbealoud noise coming outoftheloudspeaker, andwhen there isnoelectric field driving it,there willbenonoise. Because theroom inwhich thewaves wearemeasuring hasother objects init, ourelectric fieldwillshake electrons inthese other objects; theelectric fieldmakes these other charges goupanddown, andingoing upanddown, these alsoproduce aneffect onourprobe. Thus forasuccessful experiment wemust hold things fairly close together, sothat theinfluences from thewalls andfrom ourselves——the reflected waves——are relatively small. Sothephenomena willnotturnouttoappear tobeprecisely andperfectly inaccord withEq.(28.6), butwillbeclose enough that weshall beabletoappreciate thelaw. Now weturn thegenerator onand hear theaudio signal. Wefind astrong fieldwhen thedetector Disparallel tothegenerator Gatpoint 1(Fig. 28-2). We find thesame amount offield also atanyother azimuth angle about theaxis of G,because ithasnodirectional effects. Ontheother hand, when thedetector isat 3thefield iszero. That isallright, because ourformula saidthatthefield should betheacceleration ofthecharge projected perpendicular tothelineofsight. Therefore when welook down onG,thecharge ismoving toward andaway from D,andthere isnoeffect. Sothatchecks thefirstrule, thatthere isnoeffect when thecharge ismoving directly toward us.Secondly, theformula saysthattheelectric field should beperpendicular torandintheplane ofGandr;soifweputD atlbutrotate it90°,weshould getnosignal. And thisisjustwhat wefind, the 28-5-1’ Ti GEN.li.............. --T Fig. 28-1. Ahigh-frequency signal generator drives charges upand down ontwowires. -3.-/ \ / \2l“tie \ / Fig.28-2. Theinstantaneous electric field onasphere centered atalocalized, linearly oscillating charge. A5| DI $2 D TOPD3 VIEW Fig.28-3. Illustration ofinterference ofsources. ZR I 5| 52 Fig. 28-4. Illustration ofthevector character ofthecombination ofsources.electric fieldisindeed vertical, andnothorizontal. When wemove Dtosome inter- mediate angle, weseethatthestrongest signal occurs when itisoriented asshown, because although Gisvertical, itdoesnotproduce afieldthatissimply parallel toitself—it istheprojection oftheacceleration perpendicular tothelineofsight thatcounts. Thesignal isweaker at2than itisatl,because oftheprojection effect. 28-4 Interference Next, wemaytestwhat happens when wehave twosources sidebysideafew centimeters apart (Fig. 28-3). Thelawisthatthetwosources should addtheir effects atpoint 1when both ofthesources areconnected tothesame generator andareboth moving upanddown thesame way, sothatthetotal electric field is thesumofthetwoandistwice asstrong asitwasbefore. Now comes aninteresting possibility. Suppose wemake thecharges inS1 andS2both accelerate upanddown, butdelay thetiming ofS2sothatthey are 180°outofphase. Then thefield produced byS1willbeinonedirection andthe fieldproduced byS2willbeintheopposite direction atanyinstant, andtherefore weshould getnoeffect atpoint 1.Thephase ofoscillation isneatly adjustable by means ofapipewhich iscarrying thesignal toS2.Bychanging thelength ofthis pipewechange thetime ittakes thesignal toarrive atS2andthuswechange the phase ofthatoscillation. Byadjusting thislength, wecanindeed findaplace where there isnomore signal left,inspite ofthefactthatboth S1andS2aremoving! Thefactthatthey areboth moving canbechecked, because ifwecutoneout,we canseethemotion oftheother. Sothetwoofthem together canproduce zeroif everything isadjusted correctly. Now, itisvery interesting toshow thattheaddition ofthetwofields isin factavector addition. Wehave justchecked itforupanddown motion, butlet uscheck twononparallel directions. First, werestore S1andS2tothesame phase; that is,they areagain moving together. Butnow weturn S,through 90°,as shown inFig.28-4. Now weshould have atpoint lthesumoftwoefl'ects, oneof which isvertical andtheother horizontal. Theelectric field isthevector sumof these twoin-phase signals—they areboth strong atthesame time andgothrough zero together; thetotal field should beasignal Rat45°. Ifweturn Dtogetthe maximum noise, itshould beatabout 45°,andnotvertical. And ifweturn itat right angles tothatdirection, weshould getzero, which iseasytomeasure. Indeed, weobserve justsuch behavior! Now, how about theretardation? How canwedemonstrate thatthesignal isretarded? Wecould, with agreat dealofequipment, measure thetime atwhich itarrives, butthere isanother, very simple way. Referring again toFig.28-3, suppose thatS1andS2areinphase. They areboth shaking together, andthey produce equal electric fields atpoint 1.Butsuppose wegotoacertain place 2 which iscloser toS2andfarther from S1.Then, inaccordance with theprinciple thattheacceleration should beretarded byanamount equal tor/c,iftheretarda- tions arenotequal, thesignals arenolonger inphase. Thus itshould bepossible tofindaposition atwhich thedistances ofDfrom S1andS2differ bysome amount A,insuch amanner thatthere isnonetsignal. That is,thedistance Aistobe thedistance light goes inone-half anoscillation ofthegenerator. Wemay go stillfurther, andfindapoint where thedifference isgreater byawhole cycle; thatistosay,thesignal from thefirstantenna reaches point 3with adelay intime thatisgreater than thatofthesecond antenna byjustthelength oftime ittakes fortheelectric current tooscillate once, andtherefore thetwoelectric fields pro- duced at3areinphase again. Atpoint 3thesignal isstrong again. This completes ourdiscussion oftheexperimental verification ofsome ofthe important features ofEq.(28.6). Ofcourse wehave notreally checked the1/r variation oftheelectric field strength, orthefactthat there isalso amagnetic field thatgoes along with theelectric field. Todosowould require rather sophis- ticated techniques andwould hardly addtoourunderstanding atthispoint. In anycase, wehave checked those features thatareofthegreatest importance for ourlater applications, andweshall come back tostudy some oftheother properties ofelectromagnetic waves next year. 28-6 29 Interference 29-1 Electromagnetic waves Inthischapter weshall discuss thesubject ofthepreceding chapter more mathematically. Wehave qualitatively demonstrated that there aremaxima and minima intheradiation fieldfrom twosources, andourproblem nowistodescribe thefield inmathematical detail, notjust qualitatively. Wehave already physically analyzed themeaning offormula (28.6) quite satisfactorily, butthere areafewpoints tobemade about itmathematically. In thefirstplace, ifacharge isaccelerating upanddown along aline,inamotion of verysmall amplitude, thefield atsome angle 0from theaxisofthemotion isina direction atright angles tothelineofsight andintheplane containing both the acceleration andthelineofsight (Fig. 29-1). lfthedistance iscalled r,then at time rtheelectric field hasthemagnitude E(;) =: , (29_1) where a(t-r/c) istheacceleration atthetime (t-r/c), called theretarded acceleration. Now itwould beinteresting todraw apicture ofthefield under different conditions. Thething thatisinteresting, ofcourse, isthefactor a(t-r/c), andto understand itwecantakethesimplest case, 0=90°,andplotthefieldgraphically. What wehadbeen thinking ofbefore isthatwestand inoneposition andaskhow thefieldthere changes withtime. Butinstead ofthat, wearenowgoing toseewhat thefield looks likeatdifferent positions inspace atagiven instant. Sowhat we want isa“snapshot” picture which tellsuswhat thefield isindifl'erent places. Ofcourse itdepends upon theacceleration ofthecharge. Suppose thatthecharge atfirsthadsome particular motion: itwasinitially standing still, anditsuddenly accelerated insome manner, asshown inFig. 29-2, andthen stopped. Then, alittle bitlater, wemeasure thefield atadifferent place. Then wemayassert that thefield willappear asshown inFig.29-3. Ateach point thefield isdetermined bytheacceleration ofthecharge atanearlier time, theamount earlier being the delay r/c.Thefieldatfarther andfarther points isdetermined bytheacceleration at earlier andearlier times. Sothecurve inFig.29-3 isreally, inasense, a“reversed” plotoftheacceleration asafunction oftime; thedistance isrelated totime bya constant scale factor c,which weoften takeasunity. Thisiseasily seenbyconsider- ingthemathematical behavior ofa(t—r/c). Evidently, ifweaddalittle time At,wegetthesame value fora(t—r/c)aswewould have ifwehadsubtracted a little distance: Ar=—cAt. Stated another way: ifweaddalittle timeAt,wecanrestore a(t-r/c)toits former value byadding alittle distance Ar=cAt. That is,astime goes onthe fieldmoves asawave outward from thesource. That isthereason whywesometimes saylight ispropagated aswaves. ltisequivalent tosaying thatthefieldisdelayed, ortosaying thattheelectric field ismoving outward astime goes on. Aninteresting special caseisthatwhere thecharge qismoving upanddown inanoscillatory manner. Thecase which westudied experimentally inthelast chapter wasoneinwhich thedisplacement xatanytime twasequal toacertain constant xn.themagnitude oftheoscillation, times coswt.Then theacceleration is a=—w2x0 coswt=ancoswt, (29.2) 29—ll29-1 Electromagnetic waves 29-2 Energy ofradiation 29-3 Sinusoidal waves 29-4 Two dipole radiators 29-5 Themathematics ofinterference E7 atI l Fig. 29—l. The electric field Edue toapositive charge whose retarded acceleration isa’. weFig. 29-2. The acceleration ofa certain charge asafunction oftime. Fig. 29-3. The electric fleld asa function ofposition atalater time. (The l/rvariation isignored.)> I’ “' 5 rl Fig.29-4. Theenergy flowing within thecone OABCD isindependent ofthe distance ratwhich itismeasured.where aoisthemaximum acceleration, —w2x0. Putting thisformula into(29.1), wefind _ .a0cosw(t-—r/c)E— qS111 0Wé Now, ignoring theangle 6andtheconstant factors, letusseewhat thatlooks like asafunction ofposition orasafunction oftime. 29-2 Energy ofradiation First ofall,atanyparticular moment orinanyparticular place, thestrength ofthefield varies inversely asthedistance r,aswementioned previously. Now wemust point outthattheenergy content ofawave, ortheenergy effects thatsuch anelectric field canhave, areproportional tothesquare ofthefield, because if, forinstance, wehave some kind ofacharge oranoscillator intheelectric field, then ifweletthefield actontheoscillator, itmakes itmove. Ifthisisalinear oscillator, theacceleration, velocity, anddisplacement produced bytheelectric field acting onthecharge areallproportional tothefield. Sothekinetic energy which isdeveloped inthecharge isproportional tothesquare ofthefield. S0we shall take itthattheenergy thatafield candeliver toasystem isproportional somehow tothesquare ofthefield. This means thattheenergy thatthesource candeliver decreases asweget farther away; infact, itvaries inversely asthesquare ofthedistance. Butthathas averysimple interpretation: ifwewanted topickupalltheenergy wecould from thewave inacertain cone atadistance r1(Fig. 29-4), andwedothesame atan- other distance r2,wefindthat theamount ofenergy perunit area atanyone place goes inversely asthesquare ofr,butthearea ofthesurface intercepted by thecone goesdirectly asthesquare ofr.Sotheenergy thatwecantakeoutofthe wave within agiven conical angle isthesame, nomatter how faraway weare! Inparticular, thetotal energy thatwecould takeoutofthewhole wave byputting absorbing oscillators allaround isacertain fixed amount. Sothefactthatthe amplitude ofEvaries as1/risthesame assaying thatthere isanenergy fiux which isnever lost,anenergy which goes onandon,spreading over agreater and greater effective area. Thus weseethatafter acharge hasoscillated, ithaslost some energy which itcannever recover; theenergy keeps going farther andfarther away without diminution. S0ifwearefarenough away thatourbasic approxima- tionisgood enough, thecharge cannot recover theenergy which hasbeen, aswe say,radiated away. Ofcourse theenergy stillexists somewhere, andisavailable tobepicked upbyother systems. Weshall study thisenergy “loss” further in Chapter 32. Letusnowconsider more carefully how thewave (29.3) varies asafunction oftime atagiven place, andasafunction ofposition atagiven time. Again we ignore thel/rvariation andtheconstants. 29-3 Sinusoidal waves First letusfixtheposition r,andwatch thefield asafunction oftime. Itis oscillatory attheangular frequency cu.Theangular frequency wcanbedefined astherateofchange ofphase with time (radians persecond). Wehave already studied such athing, soitshould bequite familiar tousbynow. Theperiod is thetime needed foroneoscillation, onecomplete cycle, andwehave worked that outtoo;itis21r/w, because wtimes theperiod isonecycle ofthecosine. Now weintroduce anewquantity which isused agreat dealinphysics. This hastodowith theopposite situation, inwhich wefixtandlook atthewave asa function ofdistance r.Ofcourse wenotice that, asafunction ofr,thewave (29.3) isalsooscillatory. That is,aside from l/r,which weareignoring, weseethatE oscillates aswechange theposition. So,inanalogy withw,wecandefine aquantity called thewave number, symbolized ask.This isdefined astherateofchange of phase with distance (radians permeter). That is,aswemove inspace atafixed time, thephase changes. 29—2 There isanother quantity thatcorresponds totheperiod, andwemight call ittheperiod inspace, butitisusually called thewavelength, symbolized X.The wavelength isthedistance occupied byonecomplete cycle. Itiseasytosee,then, thatthewavelength is21r/k, because ktimes thewavelength would bethenumber ofradians thatthewhole thing changes, being theproduct oftherateofchange oftheradians permeter, times thenumber ofmeters, andwemust make a21r change foronecycle. Sok>\=21risexactly analogous towto=21r. Now inourparticular wave there isadefinite relationship between thefre- quency andthewavelength, buttheabove definitions ofkandcoareactually quite general. That is,thewavelength andthefrequency may notberelated inthe same way inother physical circumstances. However, inourcircumstance the rateofchange ofphase with distance iseasily determined, because ifwecall ¢=w(t—r/c) thephase, anddifferentiate (partially) with respect todistance r,therateofchange, 0¢>/0r, is <’¢_-92. -5;‘e/<_C (29.4) There aremany ways torepresent thesame thing, such as >\=ct‘, (29.5) Au=c (29.7) w:ck (29.6) wk=21rc/ (29.8) Why isthewavelength equal toctimes theperiod? That’s veryeasy, ofcourse, because ifwesitstillandwait foroneperiod toelapse, thewaves, travelling atthe speed c,willmove adistance ct“,andwillofcourse have moved over justone wavelength. lnaphysical situation other than thatoflight, kisnotnecessarily related to winthissimple way. Ifwecallthedistance along anaxisx,then theformula for acosine wave moving inadirection xwith awave number kandanangular fre- quency wwillbewritten ingeneral ascos(wt—kx). Now thatwehave introduced theideaofwavelength, wemay saysomething more about thecircumstances inwhich (29.1) isalegitimate formula. Werecall thatthefieldismade upofseveral pieces, oneofwhich varies inversely asr,another partwhich varies inversely asr2,andothers which vary even faster. Itwould be worth while toknow inwhat circumstances the1/rpart ofthefield isthemost important part, andtheother parts arerelatively small. Naturally, theanswer is “ifwego‘farenough’ away,” because terms which vary inversely asthesquare ultimately become negligible compared withthe1/rterm. How faris“farenough”? Theanswer is,qualitatively, thattheother terms areoforder A/rsmaller than the l/rterm. Thus, solong aswearebeyond afewwavelengths, (29.1) isanexcellent approximation tothefield. Sometimes theregion beyond afewwavelengths is called the“wave zone.” 29-4 Twodipole radiators Next letusdiscuss themathematics involved incombining theeffects oftwo oscillators tofindthenetfield atagiven point. This isveryeasy inthefewcases that weconsidered intheprevious chapter. Weshall first describe theeffects qualitatively, andthen more quantitatively. Letustakethesimple case, where the oscillators aresituated with their centers inthesame horizontal plane asthede- tector, andthelineofvibration isvertical. Figure 29—5(a) represents thetopview oftwosuch oscillators, andinthis particular example they arehalfawavelength apart inaN—S direction, andare oscillating together inthesame phase, which wecallzero phase. Now wewould liketoknow theintensity oftheradiation invarious directions. Bytheintensity wemean theamount ofenergy thatthefield carries pastuspersecond, which is proportional tothesquare ofthefield, averaged intime. Sothething tolook at, when wewant toknow how bright thelight is,isthesquare oftheelectric field, nottheelectric fielditself. (The electric fieldtellsthestrength oftheforce feltbya 29-3O 4 2 ix.2 2 _l_/4 x/2—4 o —o 2 \2 2 \2 0 4 a=O a=1r mi lb) Fig. 29-5. The intensities invarious directions from two dipole oscillators one-half wavelength apart. Left: in phase (a=O). Right: one-half period outofphase la=1r). 4 2 )./4i2 O G-1r/2 Fig. 29-6. Apair ofdipole antennas giving maximum power inonedirection.stationary charge, buttheamount ofenergy thatisgoing past, inwatts persquare meter, isproportional tothesquare oftheelectric field. Weshall derive theconstant ofproportionality inthenextchapter.) Ifwelookatthearray from theWside, both oscillators contribute equally andinphase, sotheelectric field istwice as strong asitwould befrom asingle oscillator. Therefore theintensity isfour times asstrong asitwould beifthere were only oneoscillator. (The numbers inFig. 29-5 represent howstrong theintensity would beinthiscase, compared with what itwould beifthere were only asingle oscillator ofunitstrength.) Now, ineither theNorSdirection along thelineoftheoscillators, since theyarehalfawavelength apart, theefiect ofoneoscillator turns outtobeoutofphase byexactly halfan oscillation from theother, andtherefore thefields addtozero. Atacertain par- ticular intermediate angle (infact, at30°)theintensity is2,anditfallsoff,4,2,0, andsoforth. Wehave tolearn howtofindthese numbers atother angles. Itisa question ofadding twooscillations with different phases. Letusquickly look atsome other cases ofinterest. Suppose theoscillators are again one-half awavelength apart, butthephase aofoneissethalfaperiod behind theother initsoscillation (Fig. 29—5b). IntheWdirection theintensity isnow zero, because oneoscillator is“pushing” when theother oneis“pulling.” Butin theNdirection thesignal from thenear onecomes atacertain time, andthatof theother comes halfaperiod later. Butthelatter wasoriginally halfaperiod behind intiming, andtherefore itisnowexactly intimewith thefirstone,andso theintensity inthisdirection is4units. Theintensity inthedirection at30°is still2,aswecanprove later. Now wecome toaninteresting casewhich shows upapossibly useful feature. Letusremark thatoneofthereasons thatphase relations ofoscillators arein- teresting isforbeaming radio transmitters. Forinstance, ifwebuild anantenna system andwant tosend aradio signal, say,toHawaii, wesettheantennas upas inFig.29—5(a) andwebroadcast with ourtwoantennas inphase, because Hawaii istothewest ofus.Then wedecide thattomornow wearegoing tobroadcast toward Alberta, Canada. Since thatisnorth, notwest, allwehave todoisto reverse thephase ofoneofourantennas, andwecanbroadcast tothenorth. Sowecanbuild antenna systems with various arrangements. Ours isoneofthe simplest possible ones; wecanmake them much more complicated, andbychang- ingthephases inthevarious antennas wecansend thebeams invarious directions andsend most ofthepower inthedirection inwhich wewish totransmit, without evermoving theantenna! Inboth ofthepreceding cases, however, while weare broadcasting toward Alberta wearewasting alotofpower onEaster Island, and itwould beinteresting toaskwhether itispossible tosend itinonly onedirection. Atfirstsight wemight think thatwith apairofantennas ofthisnature theresult isalways going tobesymmetrical. Soletusconsider acase thatcomes outun- symmetrical, toshow thepossible variety. Iftheantennas areseparated byone-quarter wavelength, andiftheNone isone-fourth period behind theSoneintime, then what happens (Fig. 29-6)? IntheWdirection weget2,aswewillseelater. IntheSdirection wegetzero, because thesignal from Scomes atacertain time; thatfrom Ncomes 90°later in time, butitisalready 90°behind initsbuilt-in phase, therefore itarrives, altogether, 180° outofphase, andthere isnoefi'ect. Ontheother hand, intheNdirection, theNsignal arrives earlier than theSsignal by90°intime, because itisaquarter wavelength closer. Butitsphase issetsothatitisoscillating 90°behind intime, which justcompensates thedelay difference, andtherefore thetwosignals appear together inphase, making thefield strength twice aslarge, andtheenergy four times asgreat. Thus, byusing some cleverness inspacing andphasing ourantennas, wecan send thepower allinonedirection. Butstillitisdistributed over agreat range of angles. Canwearrange itsothatitisfocused stillmore sharply inaparticular direction? Letusconsider thecase ofHawaii again, where wearesending the beam eastandwest butitisspread overquite anangle, because even at30°weare stillgetting halftheintensity—we arewasting thepower. Canwedobetter than that? Letustake asituation inwhich theseparation istenwavelengths (Fig. 29-4 29-7), which ismore nearly comparable tothesituation inwhich weexperimented intheprevious chapter, with separations ofseveral wavelengths rather than a small fraction ofawavelength. Here thepicture isquite different. Iftheoscillators aretenwavelengths apart (wetakethein-phase casetomake iteasy), weseethatintheE—W direction, they areinphase, andwegetastrong intensity, fourtimes what wewould getifoneofthem were there alone. Onthe other hand, atavery small angle away, thearrival times differ by180° andthe intensity iszero. Tobeprecise, ifwedraw alinefrom each oscillator toadistant point andthedifference Ainthetwodistances is>\/2,halfanoscillation, then they willbeoutofphase. Sothisfirstnulloccurs when thathappens. (The figure is notdrawn toscale; itisonly arough sketch.) This means thatwedoindeed have avery sharp beam inthedirection wewant, because ifwejustmove over alittle bitwelose allourintensity. Unfortunately forpractical purposes, ifwewere thinking ofmaking aradio broadcasting array andwedoubled thedistance A,then wewould beawhole cycle outofphase, which isthesame asbeing exactly in phase again! Thus wegetmany successive maxima andminima, justaswefound with the2%)spacing inChapter 28. Now howcanwearrange togetridofallthese extra maxima, or“lobes,” as theyarecalled? Wecould getridofthe unwanted lobes inarather interesting way. Suppose thatwewere toplace another setofantennas between thetwothatwe already have. That is,theoutside ones arestilll0>\apart, butbetween them, say every 2)\,wehave putanother antenna, andwedrive them allinphase. There are nowsixantennas, andifwe looked attheintensity intheE—W direction, itwould, ofcourse, bemuch higher with sixantennas than with one. Thefield would be sixtimes andtheintensity thirty-six times asgreat (thesquare ofthefield). We get36units ofintensity inthatdirection. Now ifwelook atneighboring points, wefindazero asbefore, roughly, butifwegofarther, towhere weused togeta big“bump,” wegetamuch smaller “bump” now. Letustrytoseewhy. Thereason isthat although wemight expect togetabigbump when the distance Aisexactly equal tothewavelength, itistruethatdipoles 1and6arethen inphase andarecooperating intrying togetsome strength inthatdirection. But numbers 3and4areroughly éawavelength outofphase with 1and6,andalthough land6push together, 3and4push together too,butinopposite phase. Therefore there isvery little intensity inthisdirection—but there issomething; itdoes not balance exactly. This kind ofthing keeps onhappening; wegetverylittle bumps, andwehave thestrong beam inthedirection where wewant it.Butinthisparticu- larexample, something elsewillhappen: namely, since thedistance between suc- cessive dipoles is2)\,itispossible tofindanangle where thedistance 5between successive dipoles isexactly onewavelength, sothattheeffects from allofthem are inphase again. Each oneisdelayed relative tothenext oneby360°, sothey all come back inphase, andwehave another strong beam inthatdirection! Itis easytoavoid thisinpractice because itispossible toputthedipoles closer thanone wavelength apart. ifweputinmore antennas, closer than onewavelength apart, then thiscannot happen. Butthefactthatthiscanhappen atcertain angles, ifthe spacing isbigger than onewavelength, isaveryinteresting anduseful phenomenon inother applications—-not radio broadcasting, butindiffraction gratings. 29-5 Themathematics ofinterference Now wehave finished ouranalysis ofthephenomena ofdipole radiators qualitatively, andwemust learn how toanalyze them quantitatively. Tofindthe effect oftwosources atsome particular angle inthemost general case, where the twooscillators have some intrinsic relative phase ozfrom oneanother andthe strengths A1andA2arenotequal, wefindthat wehave toaddtwocosines having thesame frequency, butwith different phases. Itisvery easy tofindthisphase difference; itismade upofadelay duetothedifference indistance, andthe intrinsic, built-in phase oftheoscillation. Mathematically, wehave tofindthe sum Roftwowaves: R=A1cos(wt+¢1)+Agcos (wt—l-¢2)- l-low dowe doit? 29-54*A i0). I r mint Fig.29-7. The intensity pattern for twodipoles separated bylO)\. =58 AuraI> _T_ ‘lat¥\\ 0' 30‘ Fig. 29-8. Asix-dipole antenna ar- rayandpart ofitsintensity pattern. I K.‘= Z, W fa 4’, I Fig.29-9. Ageometrical method for combining twocosine waves. Theentire diagram isthought ofasrotating counter- clockwise with angular frequency w.Itisreally very easy, andwepresume thatwealready know how todoit. However, weshall outline theprocedure insome detail. First, wecan, ifweare clever with mathematics andknow enough about cosines andsines, simply work itout. Theeasiest such caseistheonewhere A1andA2areequal, letussaythey areboth equal toA.Inthose circumstances, forexample (wecould callthisthe trigonometric method ofsolving theproblem), wehave R=A[cos (wt+¢1)+cos(wt+¢2)]. (29.9) Once, inourtrigonometry class, wemayhave learned therulethat cosA+cosB=2cos%(A +B)cos%(A—B). (29.10) Ifweknow that, then wecanimmediately write Ras R=2Acos2(¢1 —¢2)cos(wt +%¢>1+%¢2). (29.11) Sowefindthatwehave anoscillatory wave withanewphase andanewamplitude. Ingeneral, theresult willbeanoscillatory wave with anewamplitude AR,which wemaycalltheresultant amplitude, oscillating atthesame frequency butwith a phase difference ¢R,called theresultant phase. Inview ofthis,ourparticular case hasthefollowing result: thattheresultant amplitude is AR=2A-cos %(¢1 —q52), (29.12) andtheresultant phase istheaverage ofthetwophases, andwehave completely solved ourproblem. Now suppose thatwecannot remember thatthesumoftwocosines istwice thecosine ofhalfthesumtimes thecosine ofhalfthedifference. Then wemayuse another method ofanalysis which ismore geometrical. Any cosine function of wtcanbeconsidered asthehorizontal projection ofarotating vector. Suppose there were avector A1oflength A1rotating with time, sothatitsangle with the horizontal axisiswt+451-(Weshall leave outthewtinaminute, andseethatit makes nodifference.) Suppose thatwetakeasnapshot atthetime t=0,although, infact, thepicture isrotating with angular velocity w(Fig. 29-9). Theprojection ofA1along thehorizontal axisisprecisely A1cos(wt+451). Now att=0the second wave could berepresented byanother vector, A2,oflength A2andatan angle 4:2,andalsorotating. They areboth rotating with thesame angular velocity w,andtherefore therelative positions ofthetwoarefixed. Thesystem goesaround likearigid body. Thehorizontal projection ofA2isA2cos(wt+¢2). Butwe know from thetheory ofvectors thatifweaddthetwovectors intheordinary way, bytheparallelogram rule, anddraw theresultant vector AR,thex-component oftheresultant isthesum ofthex-components oftheother twovectors. That solves ourproblem. Itiseasy tocheck thatthisgives thecorrect result forthe special case wetreated above, where A1=A2=A.Inthiscase, weseefrom Fig.29-9 thatARliesmidway between A1andA2andmakes anangle %(¢2 —4:1) with each. Therefore weseethatAR=2Acos%(4>2 —¢1),asbefore. Also, as weseefrom thetriangle, thephase ofAR,asitgoes around, istheaverage angle ofA1andA2when thetwoamplitudes areequal. Clearly, wecanalsosolve for thecasewhere theamplitudes arenotequal, justaseasily. Wecancallthatthe geometrical wayofsolving theproblem. There isstillanother wayofsolving theproblem, andthatistheanalytical way. That is,instead ofhaving actually todraw apicture likeFig.29-9, wecan write something down which saysthesame thing asthepicture: instead ofdrawing thevectors, wewrite acomplex number torepresent each ofthevectors. Thereal parts ofthecomplex numbers aretheactual physical quantities. Soinourpar- ticular case thewaves could bewritten inthisway: A1e““‘+‘”1’ [therealpart of thisisA1cos(wt+¢1)]andA2e““‘+¢2’. Now wecanaddthetwo: R:Alei(wl+¢1) +A2ei(wl+¢2):(A1ei¢1 +A2ei¢2)eiu2 OI‘ R=A1ei°’1 +A2ei“’2 =ARWR. (29.14) 29-6 Thissolves theproblem thatwewanted tosolve, because itrepresents theresult as acomplex number ofmagnitude ARandphase ¢R. Toseehowthismethod works, letusfindtheamplitude ARwhich isthe “length” ofR.Togetthe“length” ofacomplex quantity, wealways multiply thequantity byitscomplex conjugate, which gives thelength squared. The complex conjugate isthesame expression, butwith thesign ofthei'sreversed. Thus wehave /1?.=(A1e“‘ +A2e""”)(A1e“l"‘ +A2e_“’2). (29.15) Inmultiplying thisout,wegetA?+AZ(here thee'scancel), andforthecross terms wehave _ _A1A2(et(¢1—¢2) +ez(¢2—¢1))' Now _ , _ _ cl’+e_” =cost) +is1n6 +cos0 —isin0. That istosay,e”+e_“ =2cos0.Ourfinal result istherefore Air=Ai"l"Ag"l"2141/12C05(¢2 —¢1)- (29-16) Aswesee,thisagrees with thelength ofARinFig.29-9, using therules of trigonometry. Thus thesumofthetwoeffects hastheintensity A?wewould getwith oneof them alone, plus theintensity A2wewould getwith theother onealone, plus a correction. Thiscorrection wecalltheinterference eflect. Itisreally onlythediffer- ence between what wegetsimply byadding theintensities, andwhat actually happens. Wecallitinterference whether itispositive ornegative. (Interference in ordinary language usually suggests opposition orhindrance, butinphysics we often donotuselanguage thewayitwasoriginally designed!) Iftheinterference term ispositive, wecallthatcaseconstructive interference, horrible though itmay sound toanybody other than aphysicist! Theopposite case iscalled destructive interference. Now letusseehow toapply ourgeneral formula (29.16) forthecaseoftwo oscillators tothespecial situations which wehave discussed qualitatively. To apply thisgeneral formula, itisonly necessary tofind what phase difference, ¢1—452,exists between thesignals arriving atagiven point. (Itdepends only onthephase difference, ofcourse, andnotonthephase itself.) Soletusconsider thecasewhere thetwooscillators, ofequal amplitude, areseparated bysome dis- tance dandhave anintrinsic relative phase Ot.(When oneisatphase zero, the phase oftheother isa.)Then weaskwhat theintensity willbeinsome azimuth direction 0from theE—W line. [Note thatthisisnotthesame 0asappears in (29.1). Wearetorn between using anunconventional symbol likeH1orthecon- ventional symbol 0(Fig. 29-10).] Thephase relationship isfound bynoting that thedifference indistance from Ptothetwooscillators isdsin0,sothatthephase difference contribution from thisisthenumber ofwavelengths indsin0,multiplied by21r.(Those whoaremore sophisticated might want tomultiply thewave number k,which istherateofchange ofphase with distance, bydsin0;itisexactly the same.) Thephase difference duetothedistance difference isthus21rdsin0/)1,but, duetothetiming oftheoscillators, there isanadditional phase oz.Sothephase difference atarrival would be ¢2—¢1=or-1-21rdsin 6/)\. (29.17) This takes careofallthecases. Thus allwehave todoissubstitute thisexpression into(29.16) forthecaseA1=A2,andwecancalculate allthevarious results for twoantennas ofequal intensity. Now letusseewhat happens inourvarious cases. Thereason weknow, for example, thattheintensity is2at30°inFig.29-5 isthefollowing: thetwooscilla- torsare2).apart, soat30°,dsin0=>1/4.Thus ¢2—¢>1=21r>\/4>\ =1r/2,and sotheinterference term iszero. (Weareadding twovectors at90°.) Theresult is thehypotenuse ofa45°right-angle triangle, which is\/2times theunitamplitude; squaring it,wegettwice theintensity ofoneoscillator alone. Alltheother cases canbeworked outinthissame way. 29-7T0Paint P Aei(wt+ a) #°*!/Aeiot/l ’<d sin9 Fig. 29-10. Two oscillators ofequal amplitude, with aphase difference a between them. 30 Diffraction 30-1 Theresultant amplitude duetonequal oscillators This chapter isadirect continuation oftheprevious one,although thename hasbeen changed from Interference toDiflraction. Noonehasever been ableto define thedifference between interference anddiffraction satisfactorily. Itisjusta question ofusage, andthere isnospecific, important physical difference between them. Thebestwecando,roughly speaking, istosaythatwhen there areonlyafew sources, saytwo, interfering, then theresult isusually called interference, butif there isalarge number ofthem, itseems thattheword diffraction ismore often used. So,weshall notworry about whether itisinterference ordiffraction, but continue directly from where weleftoffinthemiddle ofthesubject inthelast chapter. Thus weshall now discuss thesituation where there arenequally spaced os- cillators, allofequal amplitude butdifferent from oneanother inphase, either because they aredriven differently inphase, orbecause wearelooking atthem atanangle such thatthere isadifference intime delay. Foronereason oranother, wehave toaddsomething likethis: R=A[cos wt+cos(wt +¢)+cos(wt +2¢)+ +cos(wt +(n—l)¢)], (30.1) where ¢isthephase difference between oneoscillator andthenextone,asseenin aparticular direction. Specifically, ¢=oz+21rdsin0/)1. Now wemust addall theterms together. Weshall dothisgeometrically. Thefirstoneisoflength A, andithaszero phase. Thenext isalsooflength Aandithasaphase equal to¢. Thenext oneisagain oflength Aandithasaphase equal to2¢,andsoon.Sowe areevidently going around anequiangular polygon with nsides (Fig. 30-1). Now thevertices, ofcourse, alllieonacircle, andwecanfindthenetamplitude most easily ifwefindtheradius ofthatcircle. Suppose thatQisthecenter ofthe circle. Then weknow thattheangle OQS isjustaphase angle ¢.(This isbecause theradius QSbears thesame geometrical relation toA2asQ0bears toA1,so they form anangle ¢between them.) Therefore theradius rmust besuch that A=2rsin¢>/2, which fixes r.Butthelarge angle OQT isequal ton¢,andwe thus findthatAR=2rsinn¢/2. Combining these tworesults toeliminate r,we get AR=A-iS;;:1’;‘f;/22- (30.2) Theresultant intensity isthus I=I0sinzn¢/2_. 30.3SIH2¢/2 ( ) Now letusanalyze thisexpression andstudy some ofitsconsequences. In thefirstplace, wecancheck itforn=1.Itchecks: I=I11.Next, wecheck it forn=2:writing sin¢ =2sin ¢/2cos¢/2, wefind that AR=2Acos¢/2, which agrees with (29.12). Now theideathatledustoconsider theaddition ofseveral sources wasthat wemight getamuch stronger intensity inonedirection than inanother; thatthe nearby maxima which would have been present ifthere were only twosources willhave gone down instrength. Inorder toseethiseffect, weplotthecurve that comes from (30.3), taking ntobeenormously large andplotting theregion near 30-130-1 Theresultant amplitude dueto nequal oscillators 30-2 Thediffraction grating 30-3 Resolving power ofagrating 30-4 Theparabolic antenna 30-5 Colored films; crystals 30-6 Diffraction byopaque screens 30-7 Thefieldofaplane of oscillating charges T Y K. F»O .7i= AK’ O A-I 3 1 Fig. 30-l .The resultant amplitude ofn=6equally spaced sources withnet successive phase differences ¢. Ln2I° '..‘/x IQ \~,/T ‘\\\11 //—<_,_, ‘\|_.’,_~\‘\_1_a——‘._ O Fig. 30-2. Theintensity asafunction ofphase angle foralarge number of oscillators ofequal strength. 6 8-A/n =dsin0 ‘ A l-S n |= L 1 Fig. 30-3. Alinear array ofnequal oscillators, driven with phases as=sot.I 2 3 4n¢/2” 5qs=0.Inthefirstplace, if¢isexactly 0,wehave 0/0,butif¢isinfinitesimal, the ratio ofthetwosines squared issimply n2,since thesineandtheangle areapproxi- mately equal. Thus theintensity ofthemaximum ofthecurve isequal ton2times theintensity ofoneoscillator. That iseasy tosee,because ifthey areallinphase, then thelittle vectors have norelative angle andallnofthem addupsotheampli- tude isntimes, andtheintensity n2times, stronger. Asthephase ¢increases, theratio ofthetwosines begins tofalloff,andthe firsttime itreaches zero iswhen n¢/2 =1r,because sin1r=0.Inother words, ¢=21r/n corresponds tothefirstminimum inthecurve (Fig. 30-2). Interms ofwhat ishappening with thearrows inFig.30-1, thefirstminimum occurs when allthearrows come back tothestarting point; thatmeans thatthetotal accumu- lated angle inallthearrows, thetotal phase difference between thefirstandlast oscillator, must be21rtocomplete thecircle. Now wegotothenext maximum, andwewant toseethatitisreally much smaller than thefirstone,aswehadhoped. Weshall notgoprecisely tothemaxi- mum position, because both thenumerator andthedenominator of(30.3) are variant, butsin¢/2varies quite slowly compared with sinn¢/2 when nislarge, sowhen sinn¢/2 =1wearevery close tothemaximum. Thenext maximum of singn¢/2 comes atn¢/2 =31r/2, or¢>=31r/n. This corresponds tothearrows having traversed thecircle oneandahalftimes. Onputting ¢>=31r/n intothe formula tofindthesizeofthemaximum, wefindthatsin231r/2 =1inthenu- merator (because thatiswhy wepicked thisangle), andinthedenominator we have sin231r/2n. Now ifnissufficiently large, then thisangle isvery small and thesineisequal totheangle; soforallpractical purposes, wecanputsin31r/2n = 31r/2n. Thus wefindthattheintensity atthismaximum ls1=I11(4n2/91r2). But n21O wasthemaximum intensity, andsowehave 4/91r2 times themaximum in- tensity, which isabout 0.047, lessthan 5percent, ofthemaximum intensity! Of course there aredecreasing intensities farther out. Sowehave averysharp central maximum with very weak subsidiary maxima onthesides. Itispossible toprove thattheareaofthewhole curve, including allthelittle bumps, isequal to21rnI0, ortwice theareaofthedotted rectangle inFig.30-2. Now letusconsider further how wemay apply Eq.(30.3) indifferent cir- cumstances, andtrytounderstand what ishappening. Letusconsider oursources tobeallonaline, asdrawn inFig.30-3. There arenofthem, allspaced bya distance d,andweshall suppose thattheintrinsic relative phase, onetothenext, isoz.Then ifweareobserving inagiven direction 6from thenormal, there isan additional phase 21rdsin6/>\because ofthetime delay between each successive two, which wetalked about before. Thus ¢=oz+2-rrdsin 6/A (304) =or-1-kdsin 6. First, weshall take thecase oz=0.That is,alloscillators areinphase, and wewant toknow what theintensity isasafunction oftheangle 6.Inorder to findout,wemerely have toput4>=kdsin6into formula (30.3) andseewhat happens. Inthefirstplace, there isamaximum when ¢=0.That means that when alltheoscillators areinphase there isastrong intensity inthedirection 6=0.Ontheother hand, aninteresting question is,where isthefirstminimum? That occurs when ¢=21r/n. Inother words, when 21rdsin6/A=21r/n, we getthefirstminumum ofthecurve. Ifwegetridofthe21r’s sowecanlook atita little better, itsaysthat ndsin6=)1. (30.5) Now letusunderstand physically whywegetaminimum atthatposition. ndis thetotal length Lofthearray. Referring toFig.30-3, weseethat ndsin6= Lsin6=A.What (30.5) saysisthatwhen Aisequal toonewavelength, wegeta minimum. Now whydowegetaminimum when A=)1?Because thecontribu- tions ofthevarious oscillators arethen uniformly distributed inphase from 0°to 30-2 360°. Thearrows (Fig. 30-1) aregoing around awhole circle—we areadding equal vectors inalldirections, andsuch asumiszero. Sowhen wehave anangle such thatA=)1,wegetaminimum. That isthefirstminimum. There isanother important feature about formula (30.3), which isthatifthe angle ¢isincreased byanymultiple of21r,itmakes nodifference totheformula. Sowewillgetother strong maxima at¢=21r,41r,61r,andsoforth. Near each of these great maxima thepattern ofFig.30-2 isrepeated. Wemay askourselves, what isthegeometrical circumstance that leads tothese other great maxima? Thecondition isthat¢=21rm, where misanyinteger. That is,21rdsin6/)1= 21rm. Dividing by21r,weseethat dsin 6=ma. (30.6) This looks liketheother formula, (30.5). No, that formula wasndsin6=)\. Thedifference isthathere wehave tolook attheindividual sources, andwhen we saydsin6=m)\,thatmeans thatwehave anangle 6such that6=m)\. Inother words, each source isnow contributing acertain amount, andsuccessive ones are outofphase byawhole multiple of360°, andtherefore arecontributing inphase, because outofphase by360° isthesame asbeing inphase. Sotheyallcontribute inphase andproduce justasgood amaximum astheoneform=0thatwedis- cussed before. Thesubsidiary bumps, thewhole shape ofthepattern, isjustlike theonenear ¢=0,with exactly thesame minima oneach side, etc.Thus such an array willsend beams invarious directions—each beam having astrong central maximum andacertain number ofweak “side lobes.” Thevarious strong beams arereferred toasthezero-order beam, thefirst-order beam, etc.,according tothe value ofm.miscalled theorder ofthebeam. Wecallattention tothefactthatifdislessthan A,Eq.(30.6) canhave no solution except m=0,sothatifthespacing istoosmall there isonly onepossible beam, thezero-order onecentered at6=0.(Ofcourse, there isalsoabeam in theopposite direction.) Inorder togetsubsidiary great maxima, wemust have thespacing dofthearray greater thanonewavelength. 30-2 Thediffraction grating Intechnical work with antennas andwires itispossible toarrange thatall thephases ofthelittle oscillators, orantennas, areequal. Thequestion iswhether andhowwecandoasimilar thing withlight. Wecannot atthepresent timeliterally make little optical-frequency radio stations andhook them upwith infinitesimal wires anddrive them allwith agiven phase. Butthere isaveryeasywaytodowhat amounts tothesame thing. Suppose thatwehadalotofparallel wires, equally spaced ataspacing d, andaradiofrequency source very faraway, practically atinfinity, which isgenerat- inganelectric field which arrives ateach oneofthewires atthesame phase (itis sofaraway thatthetime delay isthesame forallofthewires). (One canwork out cases with curved arrays, butletustake aplane one.) Then theexternal electric field willdrive theelectrons upanddown ineach wire. That is,thefield which is coming from theoriginal source willshake theelectrons upanddown, andin moving, these represent newgenerators. This phenomenon iscalled scattering: alight wave from some source caninduce amotion oftheelectrons inapiece of material, and these motions generate their own waves. Therefore allthat is necessary istosetupalotofwires, equally spaced, drive them witharadiofrequency source faraway, andwehave thesituation thatwewant, without awhole lotof special wiring. Iftheincidence islnormal, thephases willbeequal, andwewill getexactly thecircumstance wehave been discussing. Therefore, ifthewire spacing isgreater than thewavelength, wewillgetastrong intensity ofscattering inthenormal direction, andincertain other directions given by(30.6). Thiscanalsobedone withlight! Instead ofwires, weuseafiatpiece ofglass andmake notches initsuch thateach ofthenotches scatters alittle differently thantherestoftheglass. Ifwethen shine light ontheglass, each oneofthenotches 30-3 <1sinBane dsgn8;, mu 9,21 ‘_ r—~—+ Fig. 30-4. The path difference for rays scattered from adiacent rulings ofa grating isdsin021,1—dsin61,.willrepresent asource, andifwespace thelinesveryfinely, butnotcloser thana wavelength (which istechnically almost impossible anyway), then wewould expect amiraculous phenomenon: thelight notonly willpass straight through, butthere willalsobeastrong beam atafinite angle, depending onthespacing ofthenotches! Such objects have actually been made‘ andareincommon use—they arecalled dzflraction gratings. Inoneofitsforms, adiffraction grating consists ofnothing butaplane glass sheet, transparent andcolorless, with scratches onit.There areoften several hundred scratches tothemillimeter, verycarefully arranged soastobeequally spaced. Theeffect ofsuch agrating canbeseen byarranging aprojector soasto throw anarrow, vertical lineoflight (theimage ofaslit)onto ascreen. When we putthegrating intothebeam, with itsscratches vertical, weseethatthelineisstill there but,inaddition, oneach sidewehave another strong patch oflight which is colored. This, ofcourse, istheslitimage spread outover awide angular range, because theangle 6in(30.6) depends upon >1,andlights ofdifferent colors, aswe know, correspond todifferent frequencies, andtherefore different wavelengths. Thelongest visible wavelength isred,andsince dsin6=)1,thatrequires alarger 6.And wedo,infact,findthatredisatagreater angle outfrom thecentral image! There should alsobeabeam ontheother side, andindeed weseeoneonthescreen. Then, there might beanother solution of(30.6) when m=2.Wedoseethatthere issomething vaguely there—very weak—and there areeven other beams beyond. Wehave justargued thatallthese beams ought tobeofthesame strength, butweseethattheyactually arenotand,infact,noteven thefirstones ontheright andleftareequal! Thereason isthatthegrating hasbeen carefully built todojust this. How? Ifthegrating consists ofveryfinenotches, infinitesimally wide, spaced evenly, then alltheintensities would indeed beequal. But, asamatter offact, although wehave taken thesimplest case, wecould alsohave considered anarray ofpairs ofantennas, inwhich each member ofthepairhasacertain strength and some relative phase. Inthiscase, itispossible togetintensities which aredifferent inthedifferent orders. Agrating isoften made withlittle“sawtooth” cutsinstead oflittle symmetrical notches. Bycarefully arranging the“sawteeth,” more light may besent into oneparticular order ofspectrum than into theothers. Ina practical grating, wewould liketohave asmuch light aspossible inoneofthe orders. This mayseem acomplicated point tobring in,butitisaveryclever thing todo,because itmakes thegrating more useful. Sofar,wehave taken thecasewhere allthephases ofthesources areequal. Butwealso have aformula for¢when thephases differ from onetothenext byanangle oi.That requires wiring upourantennas with aslight phase shift between each one. Canwedothatwith light? Yes, wecandoitvery easily, for suppose thatthere were asource oflight atinfinity, atanangle such thatthelight iscoming inatanangle 61,1,andletussaythatwewish todiscuss thescattered beam, which isleaving atanangle 601,1. The601,1isthesame 6aswehave hadbefore, butthe61,,ismerely ameans forarranging that thephase ofeach source is different: thelight coming from thedistant driving source firsthitsonescratch, then thenext, then thenext, andsoon,with aphase shift from onetotheother, which, aswesee,isOZ=-dsin61,,/)1. Therefore wehave theformula foragrating inwhich light both comes inandgoes outatanangle: ¢=21rdsin601,1/A —21rdsin61“/)\. (30.7) Letustrytofindoutwhere wegetstrong intensity inthese circumstances. The condition forstrong intensities is,ofcourse, that¢should beamultiple of21r. There areseveral interesting points tobenoted. Onecase ofrather great interest isthatwhich corresponds tom=0,where dislessthan A;infact, thisistheonly solution. Inthiscaseweseethatsin601,1= sin61“,which means that thelight comes outinthesame direction asthelight which wasexciting thegrating. Wemight think thatthelight “goes right through.” No,itisdiflferent light thatwearetalking about. Thelight thatgoes right through isfrom theoriginal source; what wearetalking about isthenewlight which is 30-4 generated byscattering. Itturns outthatthescattered light isgoing inthesame direction astheoriginal light, infactitcaninterfere with it—a feature which we willstudy later. There isanother solution forthissame case. Foragiven 6111,60111may bethe supplement of611,.Sonotonly dowegetabeam inthesame direction asthein- coming beam butalsooneinanother direction, which, ifweconsider itcarefully, issuch thattheangle ofincidence isequal totheangle ofscattering. This wecallthe reflected beam. Sowebegin tounderstand thebasic machinery ofreflection: thelight that comes ingenerates motions oftheatoms inthereflector, andthereflector then regenerates anewwave, andoneofthesolutions forthedirection ofscattering, the onlysolution ifthespacing ofthescatterers issmall compared with onewavelength, isthattheangle atwhich thelight comes outisequal totheangle atwhich itcomes in! Next, wediscuss thespecial case when d—+0.That is,wehave justasolid piece ofmaterial, sotospeak, butoffinite length. Inaddition, wewant thephase shiftfrom onescatterer tothenexttogotozero. Inother words, weputmore and more antennas between theother ones, sothateach ofthephase differences is getting smaller, butthenumber ofantennas isincreasing insuch awaythatthe total phase difference, between oneendofthelineandtheother, isconstant. Letusseewhat happens to(30.3) ifwekeep thedifference inphase n¢from one endtotheother constant (sayn¢=<I>),letting thenumber gotoinfinity andthe phase shift ¢ofeach onegotozero. Butnow ¢issosmall thatsin¢=¢,andif wealsorecognize n2I(1 asI,,,,themaximum intensity atthecenter ofthebeam, wefind I=4I,,,sin224>/<I>2. (30.8) This limiting caseiswhat isshown inFig.30-2. Insuch circumstances wefindthesame general kind ofapicture asforfinite spacing withd>7\;allthesidelobes arepractically thesame asbefore, butthere arenohigher-order maxima. Ifthescatterers areallinphase, wegetamaximum inthedirection 60111,=0,andaminimum when thedistance Aisequal toX,just asforfinite dandn.Sowecaneven analyze acontinuous distribution of scatterers oroscillators, byusing integrals instead ofsumming. Asanexample, suppose there were along lineofoscillators, with thecharge oscillating along thedirection oftheline(Fig. 30-5). From such anarray the greatest intensity isperpendicular totheline. There isalittle bitofintensity up anddown from theequatorial plane, butitisvery slight. With thisresult, wecan handle amore complicated situation. Suppose wehave asetofsuch lines, each producing abeam only inaplane perpendicular totheline. Tofindtheintensity invarious directions from aseries oflong wires, instead ofinfinitesimal wires, is thesame problem asitwasforinfinitesimal wires, solong asweareinthecentral plane perpendicular tothewires; wejustaddthecontribution from each ofthe long wires. That iswhy, although weactually analyzed only tinyantennas, we might aswellhave used agrating with long, narrow slots. Each ofthelong slots produces aneffect only initsown direction, notupanddown, butthey areall setnext toeach other horizontally, sothey produce interference thatway. Thus wecanbuild upmore complicated situations byhaving various distribu- tions ofscatterers inlines, planes, orinspace. Thefirstthing wedidwastocon- sider scatterers inaline, andwehave justextended theanalysis tostrips; wecan work itoutbyjustdoing thenecessary summations, adding thecontributions from theindividual scatterers. Theprinciple isalways thesame. 30-3 Resolving power ofagrating Wearenow inaposition tounderstand anumber ofinteresting phenomena. Forexample, consider theuseofagrating forseparating wavelengths. Wenoticed thatthewhole spectrum wasspread outonthescreen, soagrating canbeused as aninstrument forseparating light into itsdifferent wavelengths. One ofthe 30-5___>. Fir-A»_.,_._._.1- _ a=O Fig. 30-5. Theintensity pattern ofa continuous lineofoscillators hasasingle strong maximum and many weak "side lobes." /\ /\It'\ , \ =-1-1")-<2-»—_' ~_/ _/\’:>c:-___-.= Fig. 30-6. Illustration oftheRayleigh criterion. Themaximum ofone pattern falls onthefirstminimum oftheother.interesting questions is:supposing thatthere were twosources ofslightly different frequency, orslightly different wavelength, howclose together inwavelength could they besuch thatthegrating would beunable totellthatthere were really two different wavelengths there? Theredandtheblue were clearly separated. But when onewave isredandtheother isslightly redder, veryclose, howclose canthey be? This iscalled theresolving power ofthegrating, andonewayofanalyzing theproblem isasfollows. Suppose thatforlight ofacertain color wehappen to have themaximum ofthediffracted beam occurring atacertain angle. Ifwevary thewavelength thephase 21rd sin6/)1isdifferent, soofcourse themaximum occurs atadifferent angle. That iswhy theredandblue arespread out. How different inangle must itbeinorder forustobeabletoseeit?Ifthetwomaxima areexactly ontopofeach other, ofcourse wecannot seethem. Ifthemaximum ofoneisfarenough away from theother, then wecanseethatthere isadouble bump inthedistribution oflight. Inorder tobeabletojustmake outthedouble bump, thefollowing simple criterion, called Rayleigh’s criterion, isusually used (Fig. 30-6). Itisthatthefirstminimum from onebump should sitatthemaximum oftheother. Now itisveryeasytocalculate, when oneminimum sitsontheother maximum, how much thedifference inwavelength is.Thebestway todoitis geometrically. Inorder tohave amaximum forwavelength )1’,thedistance A(Fig. 30-3) must ben>\’,andifwearelooking atthemth-order beam, itismn>\’. Inother words, 21rdsin6/).’ =21rm, sondsin6,which isA,isNtimes n,ormn>\’. Fortheother beam, ofwavelength )1,wewant tohave aminimum atthisangle. That is,wewant Atobeexactly onewavelength Amore than mn)\. That is,A=mn>\ -1->1=mn)\’. Thus ifN=A+Alt,wefind A>\/A =1/mn. (30.9) Theratio >1/AX iscalled theresolving power ofagrating; weseethatitisequal to thetotal number oflines inthegrating, times theorder. Itisnothard toprove thatthisformula isequivalent totheformula thattheerror infrequency isequal tothereciprocal timedifference between extreme paths thatareallowed tointerfere:* Av=1/T. Infact, thatisthebestway toremember it,because thegeneral formula works notonly forgratings, butforanyother instrument whatsoever, while thespecial formula (30.9) depends onthefactthatweareusing agrating. 30-4 Theparabolic antenna Now letusconsider another problem inresolving power. This hastodowith theantenna ofaradio telescope, used fordetermining theposition ofradio sources inthesky,i.e.,how large they areinangle. Ofcourse ifweuseanyoldantenna andfind signals, wewould notknow from what direction they came. Weare very interested toknow whether thesource isinoneplace oranother. Oneway wecanfindoutistolayoutawhole series ofequally spaced dipole wires onthe Australian landscape. Then wetake allthewires from these antennas andfeed them intothesame receiver, insuch awaythatallthedelays inthefeedlines are equal. Thus thereceiver receives signals from allofthedipoles inphase. That is, itadds allthewaves from every oneofthedipoles inthesame phase. Now what happens? Ifthesource isdirectly above thearray, atinfinity ornearly so,then itsradiowaves willexcite alltheantennas inthesame phase, sothey allfeedthe receiver together. Now suppose that theradio source isataslight angle 6from thevertical. Then thevarious antennas arereceiving signals alittle outofphase. Thereceiver adds allthese out-of-phase signals together, andsowegetnothing, iftheangle *InourcaseT=A/c=mn)\/c, where cisthespeed oflight. Thefrequency 1/=c/X, soAv=cA>\/X2. 30-6 6istoobig.How bigmaytheangle be?Answer: wegetzeroiftheangle A/L =6 (Fig. 30-3) corresponds toa360° phase shift, thatis,ifAisthewavelength >1. This isbecause thevector contributions form together acomplete polygon with zeroresultant. Thesmallest angle thatcanberesolved byanantenna array of length Lis6=)1/L. Notice thatthereceiving pattern ofanantenna such asthis isexactly thesame astheintensity distribution wewould getifweturned the receiver around andmade itinto atransmitter. This isanexample ofwhat is called areciprocity principle. Itturns out, infact, tobegenerally true forany arrangement ofantennas, angles, andsoon,thatifwefirstwork outwhat the relative intensities would beinvarious directions ifthereceiver were atransmitter instead, then therelative directional sensitivity ofareceiver with thesame external wiring, thesame array ofantennas, isthesame astherelative intensity ofemission would beifitwere atransmitter. Some radio antennas aremade inadifferent way. Instead ofhaving awhole lotofdipoles inalong line, with alotoffeedwires, wemay arrange them notina linebutinacurve, andputthereceiver atacertain point where itcandetect the scattered waves. This curve iscleverly designed sothat iftheradiowaves are coming down from above, andthewires scatter, making anew wave, thewires aresoarranged thatthescattered waves reach thereceiver allatthesame time (Fig. 26-12). Inother words, thecurve isaparabola, andwhen thesource isex- actly onitsaxis, wegetavery strong intensity atthefocus. Inthiscaseweunder- stand veryclearly what theresolving power ofsuch aninstrument is.Thearranging oftheantennas onaparabolic curve isnotanessential point. Itisonly acon- venient way togetallthesignals tothesame point with norelative delay and without feed wires. The angle such aninstrument canresolve isstill6=>1/L, where Listheseparation ofthefirstandlastantennas. Itdoes notdepend onthe spacing oftheantennas andthey may bevery close together orinfactbeallone piece ofmetal. Now wearedescribing atelescope mirror, ofcourse. Wehave found theresolving power ofatelescope! (Sometimes theresolving power iswritten 6=l.22)\/L, where Listhediameter ofthetelescope. Thereason thatitisnot exactly >1/Listhis:when weworked outthat6=>1/L,weassumed thatallthe lines ofdipoles were equal instrength, butwhen wehave acircular telescope, which isthewayweusually arrange atelescope, notasmuch signal comes from theoutside edges, because itisnotlikeasquare, where wegetthesame intensity allalong aside. Wegetsomewhat lessbecause weareusing only partofthetele- scope there; thus wecanappreciate thattheeffective diameter isalittle shorter thanthetruediameter, andthatiswhat the1.22factor tellsus.Inanycase, itseems alittle pedantic toputsuch precision intotheresolving power formula.*) 30-5 Colored films; crystals Theabove, then, aresome oftheeffects ofinterference obtained byadding thevarious waves. Butthere areanumber ofother examples, andeven though we donotunderstand thefundamental mechanism yet,wewillsome day,andwecan understand even now how theinterference occurs. Forexample, when alight wave hitsasurface ofamaterial with anindex n,letussayatnormal incidence, some ofthelight isreflected. Thereason forthereflection wearenotinaposition tounderstand right now; weshall discuss itlater. Butsuppose weknow thatsome ofthelight isreflected both onentering andleaving arefracting medium. Then, ifwelook atthereflection ofalight source inathinfilm, weseethesumoftwo waves; ifthethicknesses aresmall enough, these twowaves willproduce aninter- ference, either constructive ordestructive, depending onthesigns ofthephases. Itmight be,forinstance, thatforredlight, wegetanenhanced reflection, butfor *Thisisbecause Rayleigh’s criterion isarough ideainthefirstplace, Ittellsyouwhere itbegins togetveryhard totellwhether theimage wasmade byoneorbytwostars. Actually, ifsufficiently careful measurements oftheexact intensity distribution over the diffracted image spotcanbemade, thefactthattwosources make thespotcanbeproved evenif6islessthan>1/L. 30-7 i, C. , Ri. EF 1 D i A i A i ii P 1. el ' iv 0Z, A Opaque Screen Oblect Fig. 30-7. Adistant light source casts ashadow ofanopaque object ona screen.blue light, which hasadifferent wavelength, perhaps wegetadestructively inter- fering reflection, sothatweseeabright redreflection. Ifwechange thethickness, i.e.,ifwelook atanother place where thefilmisthicker, itmaybereversed, thered interfering andtheblue not,soitisbright blue, orgreen, oryellow, orwhatnot. Soweseecolors when welook atthinfilms andthecolors change ifwelook at different angles, because wecanappreciate thatthetimings aredifferent atdifferent angles. Thus wesuddenly appreciate another hundred thousand situations involv- ingthecolors thatweseeonoilfims, soap bubbles, etc.atdifferent angles. Butthe principle isallthesame: weareonly adding waves atdifferent phases. Asanother important application ofdiffraction, wemaymention thefollowing. Weused agrating andwesawthediffracted image onthescreen. Ifwehadused monochromatic light, itwould have been atacertain specific place. Then there were various higher-order images also. From thepositions oftheimages, wecould tellhow farapart thelines onthegrating were, ifweknew thewavelength ofthe light. From thedifference inintensity ofthevarious images, wecould findoutthe shape ofthegrating scratches, whether thegrating wasmade ofwires, sawtooth notches, orwhatever, without being abletoseethem. This principle isused todis- cover thepositions oftheatoms inacrystal. Theonlycomplication isthatacrystal isthree-dimensional; itisarepeating three-dimensional array ofatoms. Wecannot useordinary light, because wemust usesomething whose wavelength islessthan thespace between theatoms orwegetnoeffect; sowemust useradiation of very short wavelength, i.e.,x-rays. So,byshining x-rays into acrystal andby noticing how intense isthereflection inthevarious orders, wecandetermine the arrangement oftheatoms inside without everbeing abletoseethem with theeye! Itisinthiswaythatweknow thearrangement oftheatoms invarious substances, which permitted ustodraw those pictures inthefirstchapter, showing thearrange- ment ofatoms insalt,andsoon.Weshall later come back tothissubject anddis- cuss itinmore detail, andtherefore wesaynomore about thismost remarkable ideaatpresent. 30-6 Diffraction byopaque screens Now wecome toaveryinteresting situation. Suppose thatwehave anopaque sheet with holes init,andalight ononesideofit.Wewant toknow what the intensity isontheother side. What most people sayisthatthelight shines through theholes, andproduces aneffect ontheother side. Itwillturn outthatonegets theright answer, toanexcellent approximation, ifheassumes thatthere aresources distributed with uniform density across theopen holes, andthat thephases of these sources arethesame asthey would have been iftheopaque material were absent. Ofcourse, actually there arenosources attheholes, infactthatistheonly place thatthere arecertainly nosources. Nevertheless, wegetthecorrect diffraction patterns byconsidering theholes tobetheonly places thatthere aresources; that isarather peculiar fact. Weshall explain later whythisistrue, butfornowletus justsuppose thatitis. Inthetheory ofdiffraction there isanother kind ofdiffraction thatweshould briefly discuss. Itisusually notdiscussed inanelementary course asearly asthis, only because themathematical formulas involved inadding these little vectors are alittle elaborate. Otherwise itisexactly thesame aswehave been doing allalong. Alltheinterference phenomena arethesame; there isnothing very much more advanced involved, only thecircumstances aremore complicated anditisharder toaddthevectors together, thatisall. Suppose thatwehave light coming infrom infinity, casting ashadow ofan object. Figure 30-7 shows ascreen onwhich theshadow ofanobject ABismade byalight source very faraway compared with onewavelength. Now wewould expect thatoutside theshadow, theintensity isallbright, andinside it,itisalldark. Asamatter offact,ifweplottheintensity asafunction ofposition neartheshadow edge, theintensity rises andthen overshoots, andwobbles, andoscillates about inavery peculiar manner near thisedge (Fig. 30-8). Wenow shall discuss the reason forthis. Ifweusethetheorem thatwehave notyetproved, then wecan 30-8 replace theactual problem byasetofeffective sources uniformly distributed over theopen space beyond theobject. Weimagine alarge number ofveryclosely spaced antennas, andwewant the intensity atsome point P.That looks justlikewhat wehave been doing. Not quite; because ourscreen isnotatinfinity. Wedonotwant theintensity atinfinity, butatafinite point. Tocalculate theintensity atsome particular place, wehave to addthecontributions from alltheantennas. First there isanantenna atD, exactly opposite P;ifwegoupalittle bitinangle, letussayaheight h,then there isanincrease indelay (there isalsoachange inamplitude because ofthechange in distance, butthisisavery small effect ifweareatallfaraway, andismuch less important than thedifference inthephases). Now thepath difference EP—DP ish2/2s, sothatthephase difference isproportional tothesquare ofhow farwe gofrom D,while inourprevious work swasinfinite, andthephase difference was linearly proportional toh.When thephases arelinearly proportional, each vector adds ataconstant angle tothenext vector. What wenow need isacurve which ismade byadding alotofinfinitesimal vectors with therequirement thattheangle theymake shall increase, notlinearly, butasthesquare ofthelength ofthecurve. Toconstruct thatcurve involves slightly advanced mathematics, butwecanalways construct itbyactually drawing thearrows andmeasuring theangles. Inanycase, wegetthemarvelous curve (called Cornu’s spiral) shown inFig.30-8. Now how doweusethiscurve? Ifwewant theintensity, letussay,atpoint P,weaddalotofcontributions of different phases from point Donuptoinfinity, andfrom Ddown onlytopoint BP. Sowestart atBpinFig.30-8, anddraw aseries ofarrows ofever-increasing angle. Therefore thetotal contribution above point BPallgoes along thespiraling curve. Ifwewere tostop integrating atsome place, then thetotal amplitude would bea vector from Btothatpoint; inthisparticular problem wearegoing toinfinity, sothetotal answer isthevector Bpw. Now theposition onthecurve which corresponds topoint Bpontheobject depends upon where point Pislocated, since point D,theinflection point, always corresponds totheposition ofpoint P. Thus, depending upon where Pislocated above B,thebeginning point willfall atvarious positions onthelower leftpart ofthecurve, andtheresultant vector Bpmwillhave many maxima andminima (Fig. 30-9).'5‘? . @ Fig. 30-8. The addition ofampli- tudes formany in-phase oscillators whose phase delays vary asthesquare ofthe distance from point Dofthe previous figure. .2.10 1.0R Fig. 30-9. The intensity near the P edge ofashadow. The geometrical Q25 ______ ___ shadow edge isatxo. Q 1, I Ontheother hand, ifweareatQ,ontheother sideofP,then weareusing only oneendofthespiral curve, andnottheother end. Inother words, wedonot even start atD,butatBQ,soonthissidewegetanintensity which continuously fallsoffasQgoesfarther intotheshadow. Onepoint thatwecanimmediately calculate with ease, toshow thatwereally understand it,istheintensity exactly opposite theedge. Theintensity here is1/4 thatoftheincident light. Reason: Exactly attheedge (sotheendpoint Bofthe arrow isatDinFig.30-8) wehave halfthecurve thatwewould have hadifwe were farintothebright region. Ifourpoint Risfarintothelight wegofrom one endofthecurve totheother, thatis,onefullunitvector; butifweareattheedge oftheshadow, wehave only halftheamplitude—l/4 theintensity. Inthischapter wehave been finding theintensity produced invarious direc- tions from various distributions ofsources. Asafinal example weshall derive a formula which weshall need forthenext chapter onthetheory oftheindex of refraction. Uptothispoint relative intensities have been sufficient forourpurpose, butthistime weshall findthecomplete formula forthefieldinthefollowing situa- tion. 30-9 Oscil latinq charge gm . Sheet ofoscillating charges Fig. 30-IO. Radiation field of sheet ofoscillating charges.30-7 Thefieldofaplane ofoscillating charges Suppose that wehave aplane fullofsources, alloscillating together, with theirmotion intheplane andallhaving thesame amplitude andphase. What is thefield atafinite, butvery large, distance away from theplane? (We cannot getvery close, ofcourse, because wedonothave theright formulas forthefield close tothesources.) Ifwelettheplane ofthecharges betheXY-plane, then we want tofindthefieldatthepoint PfaroutontheZ-axis (Fig. 30-10). Wesuppose thatthere are1;charges perunitareaoftheplane, andthateach oneofthem hasa charge q.Allofthecharges move with simple harmonic motion, with thesame direction, amplitude, andphase. Weletthemotion ofeach charge, withrespect toitsownaverage position, be-x0 coswt.Or,using thecomplex notation andre- membering that therealpart represents theactual motion, themotion canbe described byxoei“. Now wefindthefield atthepoint Pfrom allofthecharges byfinding the field there from each charge q,andthen adding thecontributions from allthe charges. Weknow thattheradiation field isproportional totheacceleration of thecharge, which is—w2x0e“" (and isthesame forevery charge). Theelectric field thatwewant atthepoint Pduetoacharge atthepoint Qisproportional to theacceleration ofthecharge q,butwehave toremember thatthefieldatthepoint Pattheinstant tisgiven bytheacceleration ofthecharge attheearlier time t’=t—r/c,where r/cisthetime ittakes thewaves totravel thedistance r from QtoP.Therefore thefield atPisproportional to —w2x0ei‘°('_'/°)- (30.10) Using thisvalue fortheacceleration asseenfrom Pinourformula fortheelectric field atlarge distances from aradiating charge, weget ‘Electric field atP_ qw2x0e"“"’_'/"l (from charge atQ)_WM W (approX')' 60'“) Now this,formula isnotquite right, because weshould have used notthe acceleration ofthecharge butitscomponent perpendicular tothelineQP.Weshall suppose, however, thatthepoint Pissofaraway, compared with thedistance of thepoint Qfrom theaxis(thedistance pinFig.30-9), forthose changes thatwe need totake intoaccount, thatwecanleave outthecosine factor (which would benearly equal to1anyway). Togetthetotal fieldatP,wenowaddtheeffects ofallthecharges intheplane. Weshould, ofcourse, make avector sum. Butsince thedirection oftheelectric field isnearly thesame forallthecharges, wemay, inkeeping with theapproxima- tionwehave already made, justaddthemagnitudes ofthefields. Toourapproxi- mation thefield atPdepends only onthedistance r,soallcharges atthesame r produce equal fields. Soweadd, first, thefields ofthose charges inaringofwidth dpandradius p.Then, bytaking theintegral overallp,wewillobtain thetotal field. Thenumber ofcharges intheringistheproduct ofthesurface area ofthe ring, 21rpdp,and11,thenumber ofcharges perunitarea. Wehave, then, _ q w2x0e'iw(t—r/c) - Wewish toevaluate thisintegral from p=0top=oo.Thevariable t,of course, istobeheld fixed while wedotheintegral, sotheonly varying quantities arepandr.Leaving outalltheconstant factors, including thefactor e“"‘,forthe moment, theintegral wewish is /P=°° —iwr/c P_057 pdp. (30.13) Todothisintegral weneed tousetherelation between randp: r2=p2+22. (30.14) 30-10 Since zisindependent ofp,when wetake thedifferential ofthisequation, weget 2rdr=2pdp, which islucky, since inourintegral wecanreplace pdpbyrdrandtherwillcancel theoneinthedenominator. Theintegral wewant isthen thesimpler one T=W /6-""'/"clr. (30.15) Tointegrate anexponential isvery easy. Wedivide bythecoefficient ofrinthe exponent andevaluate theexponential atthelimits. Butthelimits ofrarenotthe same asthelimits ofp.When p=0,wehave r=z,sothelimits ofrarezto infinity. Wegetfortheintegral _i -—ino _ —(iw/c)2iw[e e ], (30.16) where wehave written oofor(r/c) oo,since they both just mean avery large number! Now e”i°° isamysterious quantity. Itsrealpart, forexample, iscos(—oo), which, mathematically speaking, iscompletely indefinite (although wewould expect ittobesomewhere—or everywhere (?)—between +1and—ll).Butina physical situation, itcanmean something quite reasonable, andusually canjust betaken tobezero. Toseethatthisissoinourcase, wegoback toconsider again theoriginal integral (30.15). Wecanunderstand (30.15) asasumofmany small complex numbers, each of magnitude Ar,andwith theangle 0=-wr/c inthecomplex plane. Wecantry toevaluate thesumbyagraphical method. InFig.30-11 wehave drawn thefirst fivepieces ofthesum. Each segment ofthecurve hasthelength Arandisplaced attheangle A0=—wAr/c withrespect tothepreceding piece. Thesumfor these firstfivepieces isrepresented bythearrow from thestarting point tothe endofthefifthsegment. Aswecontinue toaddpieces weshall trace outapolygon until wegetback tothestarting point (approximately) andthen start around once more. Adding more pieces, wejustgoround andround, staying close toacircle whose radius iseasily shown tobec/w. Wecanseenowwhytheintegral does not giveadefinite answer! Butnow wehave togoback tothephysics ofthesituation. Inanyreal situation theplane ofcharges cannot beinfinite inextent, butmust sometime stop. Ifitstopped suddenly, andwasexactly circular inshape, ourintegral would have some value onthecircle inFig.30-ll. If,however, weletthenumber ofcharges intheplane gradually taper offatsome large distance from thecenter (orelsestop suddenly butinanirregular shape soforlarger ptheentire ringofwidth dpno longer contributes), then thecoefficient 1;intheexact integral would decrease toward zero. Since weareadding smaller pieces butstillturning through thesame angle, thegraph ofourintegral would then become acurve which isaspiral. The spiral would eventually endupatthecenter ofouroriginal circle, asdrawn in Fig.30-12. Thephysically correct integral isthecomplex number Ainthefigure represented bytheinterval from thestarting point tothecenter ofthecircle, which isjustequal to i -iwz/ciw6 , (30.17) asyoucanwork outforyourself. This isthesame result wewould getfrom Eq. (30.16) ifwesete“i°° =0. (There isalsoanother reason why thecontribution totheintegral tapers off forlarge values ofr,andthatisthefactor wehave omitted fortheprojection of theacceleration ontheplane perpendicular tothelinePQ.) Weare,ofcourse, interested only inphysical situations, sowewilltake e'l°° equal tozero. Returning toouroriginal formula (30.12) forthefield andputting 30-11‘Imaginary Alil 0--flgl Ar A9B—90- RealAlla,-\._. , // -9/ ‘L/ A8 l\‘A8 CO \\\ / \\ // \--1 Fig. 30-ll. Graphical solution of co I e-1/Ar/ed,‘ Z Imaginary Alli _ Start; rI2 Rgal Axis <4/ Fig. 30-12. Graphical solution of w . I We—u-Jr/ed,-' 2 back allofthefactors thatgowiththeintegral, wehavetheresult TotalfieldatP=-2%iwx.,e""<‘-='°> (30.18) (remembering that1/i=—i). Itisinteresting tonotethat(iwxGem‘)isjustequal tothevelocity ofthecharges, sothatwecanalsowrite theequation forthefield as Total fieldatP=—2%[velocity ofcharges],,, ,_,,,, (30.19) which isalittle strange, because theretardation isjustbythedistance z,which is theshortest distance from Ptotheplane ofcharges. Butthatisthewayitcomes out—fortunately arather simple formula. (Wemayadd,bytheway, thatalthough ourderivation isvalid only fordistances farfrom theplane ofoscillatory charges, itturns outthattheformula (30.18) or(30.19) iscorrect atanydistance z,even forz<X.) 30-12 31 The Origin ofthe Refractive Index 31-1 Theindex ofrefraction Wehave saidbefore thatlight goes slower inwater than inair,andslower, slightly, inairthan invacuum. This effect isdescribed bytheindex ofrefraction n.Now wewould liketounderstand howsuch aslower velocity could come about. Inparticular, weshould trytoseewhat therelation istosome physical assumptions, orstatements, wemade earlier, which were thefollowing: (a)That thetotal electric field inanyphysical circumstance canalways be represented bythesumofthefields from allthecharges intheuniverse. (b)That thefieldfrom asingle charge isgiven byitsacceleration evaluated with aretardation atthespeed c,always (fortheradiation field). But, forapiece ofglass, youmight think: “Oh, no,youshould modify all this. You should sayitisretarded atthespeed c/n.” That, however, isnotright, andwehave tounderstand whyitisnot. Itisapproximately truethatlight oranyelectrical wave doesappear totravel atthespeed c/nthrough amaterial whose index ofrefraction isn,butthefields are stillproduced bythemotions ofallthecharges—including thecharges moving in thematerial—and with these basic contributions ofthefield travelling atthe ultimate velocity c.Our problem istounderstand how theapparently slower velocity comes about. Weshall trytounderstand theeffect inavery simple case. Asource which weshall call“the external source” isplaced alarge distance away from athin plate oftransparent material, sayglass. Weinquire about thefield atalarge distance ontheopposite side oftheplate. The situation isillustrated bythe diagram ofFig. 31-1, where SandPareimagined tobevery faraway from the plate. According totheprinciples wehave stated earlier, anelectric fieldanywhere thatisfarfrom allmoving charges isthe(vector) sumofthefields produced bythe external source (atS)andthefields produced byeach ofthecharges intheplate ofglass, every onewithitsproper retardation atthevelocity c.Remember thatthe contribution ofeach charge isnotchanged bythepresence oftheother charges. These areourbasic principles. Thefield atPcanbewritten thus: E: Z Eeach charge I.1) allchargcs 0!‘ E=Es+ Z Eeach charge’ allother charges where E,isthefield duetothesource alone andwould beprecisely thefield at Pifthere were nomaterial present. Weexpect thefield atPtobedifferent from E,ifthere areanyother moving charges. Why should there becharges moving intheglass? Weknow thatallmaterial consists ofatoms which contain electrons. When theelectric fieldofthesource acts onthese atoms itdrives theelectrons upanddown, because itexerts aforce onthe electrons. And moving electrons generate afie1d—they constitute newradiators. These newradiators arerelated tothesource S,because they aredriven bythe field ofthesource. Thetotal field isnotjustthefield ofthesource S,butitis modified bytheadditional contribution from theother moving charges. This means thatthefield isnotthesame astheonewhich wasthere before theglass wasthere, butismodified, anditturns outthatitismodified insuch awaythat 31-131-1 Theindex ofrefraction 31-2 Thefieldduetothematerial 31-3 Dispersion 31-4 Absorption 31-5 Theenergy carried byan electric wave 31-6 Diffraction oflightbyascreen Arriving Wave "Transmitted"Wave _. P :54 6 m Q‘ "1"":1''22..ICC Source of minl|QCiFIC wove "Reflected"Wave lass plate Fig. 3l—l .Electric waves passing through alayer oftransparent material. \\\ -/jg//,v\0/\\ \\\2-\\\\>’\\\\\\\ VACUUM/ ’/ cuss/ / 0/,\\ \\\\\ 4;'/ms,¢,,/;¢'x@,'\ Fig. 31-2. Relation between refrac- tionandvelocity change.thefieldinside theglassappears tobemoving atadifferent speed. Thatistheidea which wewould liketowork outquantitatively. Now thisis,intheexact case, pretty complicated, because although wehave saidthatalltheother moving charges aredriven bythesource field, thatisnot quite true. Ifwethink ofaparticular charge, itfeels notonly thesource, butlike anything elseintheworld, itfeels allofthecharges thataremoving. Itfeels, in particular, thecharges thataremoving somewhere elseintheglass. Sothetotal field which isacting onaparticular charge isacombination ofthefields from the other charges, whose motions depend onwhat thisparticular charge isdoing! You canseethatitwould takeacomplicated setofequations togetthecomplete and exact formula. Itissocomplicated thatwepostpone thisproblem until nextyear. Instead weshall work outavery simple case inorder tounderstand allthe physical principles veryclearly. Wetakeacircumstance inwhich theeffects from theother atoms isverysmall relative totheeffects from thesource. Inother words, wetakeamaterial inwhich thetotal fieldisnotmodified verymuch bythemotion oftheother charges. That corresponds toamaterial inwhich theindex ofrefraction isvery close to1,which willhappen, forexample, ifthedensity oftheatoms is verylow. Ourcalculation willbevalid foranycaseinwhich theindex isforany reason very close to1.Inthiswayweshall avoid thecomplications ofthemost general, complete solution. Incidentally, youshould notice thatthere isanother effect caused bythemotion ofthecharges intheplate. These charges willalsoradiate waves back toward the source S.This backward-going field isthelight weseereflected from thesurfaces oftransparent materials. Itdoes notcome from justthesurface. Thebackward radiation comes from everywhere intheinterior, butitturns outthatthetotal effect isequivalent toareflection from thesurfaces. These reflection efl"ects arebeyond ourapproximation atthemoment because weshall belimited toacalculation fora material with anindex soclose to1thatverylittle light isreflected. Before weproceed with ourstudy ofhowtheindex ofrefraction comes about, weshould understand thatallthatisrequired tounderstand refraction istounder- stand why theapparent wave velocity isdifferent indifferent materials. The bending oflight rayscomes about justbecause theeffective speed ofthewaves is different inthematerials. Toremind youhow thatcomes about wehave drawn inFig. 31-2 several successive crests ofanelectric wave which arrives from a vacuum onto thesurface ofablock ofglass. Thearrow perpendicular tothewave crests indicates thedirection oftravel ofthewave. Now alloscillations inthewave must have thesame frequency. (We have seen thatdriven oscillations have the same frequency asthedriving source.) This means, also, thatthewave crests for thewaves onboth sides ofthesurface must have thesame spacing along 1/1esurface because they must travel together, sothatacharge sitting attheboundary will feelonlyonefrequency. Theshortest distance between crests ofthewave, however, isthewavelength which isthevelocity divided bythefrequency. Onthevacuum sideitisA0=21rc/w, andontheother sideitisA=21rv/w or21rc/wn, ifv=c/n isthevelocity ofthewave. From thefigure wecanseethattheonly wayforthe waves to“fit” properly attheboundary isforthewaves inthematerial tobe travelling atadifferent angle with respect tothesurface. From thegeometry of thefigure youcanseethat fora“fit” wemust have A0/sin 00=>1/sin 0,or sin00/sin 0=n,which isSnell’s law. Weshall, fortherestofourdiscussion, consider only whylight hasaneffective speed ofc/ninmaterial ofindex n,and nolonger worry, inthischapter, about thebending ofthelight direction. Wegoback now tothesituation shown inFig.31-1. Weseethatwhat we have todoistocalculate thefield produced atPbyalltheoscillating charges in theglass plate. Weshall callthispartofthefieldEa,anditisjustthesumwritten asthesecond term inEq.(31.2). When weaddittotheterm Es,duetothesource, wewillhave thetotal field atP. 31-2 This isprobably themost complicated thing thatwearegoing todothisyear, butitiscomplicated only inthatthere aremany pieces that have tobeputto- gether; each piece, however, isvery simple. Unlike other derivations where we say,“Forget thederivation, justlook attheanswer!,” inthiscase wedonot need theanswer somuch asthederivation. Inother words, thething tounder- stand nowisthephysical machinery fortheproduction oftheindex. Toseewhere wearegoing, letusfirstfindoutwhat the“correction field” E,would have tobeifthetotal field atPisgoing tolook likeradiation from the source thatisslowed down while passing through thethinplate. Iftheplate had noeffect onit,thefieldofawave travelling totheright (along thez-axis) would be E,=E0coso.1(t—z/c) (31.3) or,using theexponential notation, E,=E.,e‘““r"°>. (31.4) Now what would happen ifthewave travelled more slowly ingoing through theplate? Letuscallthethickness oftheplate Az.Iftheplate were notthere the wave would travel thedistance Azinthetime Az/c.Butifitappears totravel at thespeed c/nthen itshould take thelonger time nAz/c ortheadditional time At=(n—l)Az/c. After thatitwould continue totravel atthespeed cagain. Wecantakeintoaccount theextra delay ingetting through theplate byreplacing tinEq.(31.4) by(t-At)orby[t—(n—1)Az/c]. Sothewave after insertion oftheplate should bewritten E.1....1.... =E0e""“-‘"-"“’"-""‘. (31.5) Wecanalsowrite thisequation as Enftor plate :e—iw(n—1)Az/cE0e iw(t—z/c)’ which says thatthewave after theplate isobtained from thewave which could exist without theplate, i,e.,from Es,bymultiplying bythefactor e_"’*""*1)“/ ‘. Now weknow thatmultiplying anoscillating function likeembyafactor e“just saysthatwechange thephase oftheoscillation bytheangle 0,which is,ofcourse, what theextra delay inpassing through thethickness Azhasdone. Ithasretarded thephase bytheamount o.>(n—1)Az/c (retarded, because oftheminus signin theexponent). Wehave saidearlier thattheplate should addafield Eatotheoriginal field E,=E0e’*“(‘_‘/°’, butwehave found instead that theeffect oftheplate isto multiply thefield byafactor which shifts itsphase. However, thatisreally allright because wecangetthesame result byadding asuitable complex number. Itis particularly easy tofindtheright number toaddinthecase thatAzissmall, for youwillremember thatifxisasmall number then e”isnearly equal to(1+x). Wecanwrite, therefore, e-"‘"<"-‘>“’“ =1-io.>(n-l)AZ/C. (31.7) Using thisequality inEq.(31.6), wehave - ' — A -_ Enftnr ilatc =E0ewU_z/C) —E E0e1Iw(t z/c)-' c E. E. Thefirstterm isjustthefield from thesource, andthesecond term must justbe equal toEa,thefield produced totheright oftheplate bytheoscillating charges oftheplate——expressed here interms oftheindex ofrefraction n,anddepending, ofcourse, onthestrength ofthewave from thesource. ___i.i.i.__._- What wehave been doing iseasily visualized ifwelook atthecomplex number diagram inFig.31-3. Wefirstdraw thenumber E,(wechose some values forz andtsothatE,comes outhorizontal, butthisisnotnecessary). Thedelay dueto 31-3Imaginary Axis Angle Iw(n—l)Al/C E, V RealAx1s E Ea “WNW ‘E3 1~E 9 Fig. 31-3. Diagram forthe trans mitted wave ataparticular tandz. slowing down intheplate would delay thephase ofthisnumber, thatis,itwould rotate E,through anegative angle. Butthisisequivalent toadding thesmall vector Eaatroughly rightangles toE...Butthatisjustwhat thefactor —imeans inthesecond term ofEq.(31.8). Itsays thatifE,isreal, then Eaisnegative imaginary orthat, ingeneral, E,andEamake aright angle. 31-2 Thefieldduetothematerial Wenowhave toask:Isthefield Eaobtained inthesecond term ofEq.(31.8) thekind wewould expect from oscillating charges intheplate? Ifwecanshow thatitis,wewillthen have calculated what theindex nshould be![Since nisthe only nonfundamental number inEq.(3l.8).] Weturn now tocalculating what field Eathecharges inthematerial willproduce. (Tohelpyoukeep track ofthe many symbols wehave used uptonow, andwillbeusing intherestofourcalcula- tion, wehave putthem alltogether inTable 31-1.) Table 31-1 Symbols usedinthecalculations E,=field from thesource Ea=fieldproduced bycharges intheplate Az=thickness oftheplate z=perpendicular distance from theplate n=index ofrefraction w=frequency (angular) oftheradiation N=number ofcharges perunitvolume intheplate 17=number ofcharges perunitareaoftheplate q,=charge onanelectron m=mass ofanelectron wr)=resonant frequency ofanelectron bound inanatom Ifthesource S(ofFig.31-1) isfarofftotheleft,then thefield E,willhave thesame phase everywhere ontheplate, sowecanwrite thatintheneighborhood oftheplate E,=E.,e"<"'-Z/"> (31.9) Right attheplate, where z=0,wewillhave E,=E,e‘"‘(attheplate) (31.10) Each oftheelectrons intheatoms oftheplate willfeelthiselectric field and willbedriven upanddown (weassume thedirection ofE0isvertical) bytheelectric force qE.Tofindwhat motion weexpect fortheelectrons, wewillassume thatthe atoms arelittle oscillators, thatis,thattheelectrons arefastened elastically tothe atoms, which means thatifaforce isapplied toanelectron itsdisplacement from itsnormal position willbeproportional totheforce. You maythink thatthisisafunny model ofanatom ifyouhave heard about electrons whirling around inorbits. Butthat isjustanoversimplified picture. Thecorrect picture ofanatom, which isgiven bythetheory ofwave mechanics, saysthat, sofarasproblems involving light areconcerned, theelectrons behave as though theywere held bysprings. Soweshall suppose thattheelectrons have a linear restoring force which, together with their mass m,makes them behave like little oscillators, with aresonant frequency coo.Wehave already studied such os- cillators, andweknow thattheequation oftheir motion iswritten thisway: 2 114% +ofix)=F, (31.11) where Fisthedriving force. 31-4 Forourproblem, thedriving force comes from theelectric field ofthewave from thesource, soweshould use F=11.12.=q.E<>e““‘. (31.12) where qeistheelectric charge ontheelectron andforE,weusetheexpression E,=EOei“from (31.10). Ourequation ofmotion fortheelectron isthen dz .1.m<£ +wfix)=q,E()e ‘. (31.13) Wehave solved thisequation before, andweknow thatthesolution is x=><0e""‘, (31.14) where, bysubstituting in(31.13), wefindthat X0=--‘11E‘%. (31.15)m(w§ —0.12) sothat X=-—@- Jr’. (31.16)m(w% —(.02) Wehave what weneeded toknow—the motion oftheelectrons intheplate. And itisthesame forevery electron, except thatthemean position (the“zero” ofthe motion) is,ofcourse, different foreach electron. Now weareready tofindthefieldEathatthese atoms produce atthepoint P, because wehave already worked out(attheendofChapter 30)what field ispro- duced byasheet ofcharges thatallmove together. Referring back toEq.(30.19), weseethatthefield E,atPisjustanegative constant times thevelocity ofthe charges retarded intime theamount z/c. Diflerentiating xinEq.(31.16) toget thevelocity, andsticking intheretardation [orjust putting x0from (31.15) into (30.l8)] yields EE=-”—‘1*’lie 4- e‘"’<‘-='"’]- (31.17)a 260C m(w% _(02) Justasweexpected, thedriven motion oftheelectrons produced anextra wave which travels totheright (that iswhat thefactor e""“""’ says), andtheamplitude ofthiswave isproportional tothenumber ofatoms perunit area intheplate (thefactor 1))andalsoproportional tothestrength ofthesource field (thefactor E0). Then there aresome factors which depend ontheatomic properties (qe,m, and030),asweshould expect. Themost important thing, however, isthatthisformula (31.17) forEalooks very much liketheexpression forEathatwegotinEq.(31.8) bysaying thatthe original wave wasdelayed inpassing through amaterial withanindex ofrefraction n.Thetwoexpressions will,infact, beidentical if 2 (n-l)Az= (31.18)2€Ql’l’l((.0Q -'C0) Notice thatboth sides areproportional toAz,since 1;,which isthenumber of atoms perunitarea, isequal toNAZ,where Nisthenumber ofatoms perunit volume oftheplate. Substituting NAZfor1)andcancelling theAz,wegetourmain result, aformula fortheindex ofrefraction interms oftheproperties oftheatoms ofthematerial—and ofthefrequency ofthelight: 2 n=1+ (31.19)2e0m(w0 —w) This equation gives the“explanation” oftheindex ofrefraction thatwewished to obtain. 31-5 31-3 Dispersion Notice thatintheabove process wehave obtained something veryinteresting. Forwehave notonlyanumber fortheindex ofrefraction which canbecomputed from thebasic atomic quantities, butwehave also learned how theindex of refraction should vary with thefrequency wofthelight. This issomething we would never understand from thesimple statement that“light travels slower ina transparent material.” Westillhave theproblem, ofcourse, ofknowing howmany atoms perunitvolume there are,andwhat istheir natural frequency wo.Wedo notknow thisjustyet,because itisdifferent forevery different material, andwe cannot getageneral theory ofthatnow. Formulation ofageneral theory ofthe properties ofdifferent substances—their natural frequencies, and soon—is possible only with quantum atomic mechanics. Also, different materials have different properties anddifferent indexes, sowecannot expect, anyway, togeta general formula fortheindex which willapply toallsubstances. However, weshall discuss theformula wehave obtained, invarious possible circumstances. First ofall,formost ordinary gases (forinstance, forair,most colorless gases, hydrogen, helium, andsoon)thenatural frequencies oftheelectron oscillators correspond toultraviolet light. These frequencies arehigher than the frequencies ofvisible light, thatis,cooismuch larger than (0ofvisible light, andto afirstapproximation, wecandisregard 032incomparison with wfi.Then wefind thattheindex isnearly constant. Soforagas,theindex isnearly constant. This isalsotrueformost other transparent substances, likeglass. Ifwelook atour expression alittle more closely, however, wenotice thatas(.0rises, taking alittle bitmore away from thedenominator, theindex alsorises. Sonrises slowly with frequency. Theindex ishigher forbluelight than forredlight. That isthereason whyaprism bends thelight more inthebluethan inthered. Thephenomenon that theindex depends upon thefrequency iscalled the phenomenon ofdispersion, because itisthebasis ofthefactthatlight is“dispersed” byaprism intoaspectrum. Theequation fortheindex ofrefraction asafunction offrequency iscalled adispersion equation. Sowehave obtained adispersion equa- tion. (Inthepastfewyears “dispersion equations” have been finding anewusein thetheory Ofelementary particles.) Our dispersion equation suggests other interesting effects. Ifwehave a natural frequency wowhich liesinthevisible region, orifwemeasure theindex ofrefraction ofamaterial likeglass intheultraviolet, where wgetsnear wo,we seethatatfrequencies veryclose tothenatural frequency theindex cangetenor- mously large, because thedenominator cangotozero. Next, suppose thatwis greater than wq.This would occur, forexample, ifwetake amaterial likeglass, say,andshine x-ray radiation onit.Infact,since many materials which areopaque tovisible light, likegraphite forinstance, aretransparent tox-rays, wecanalso talkabout theindex ofrefraction ofcarbon forx-rays. Allthenatural frequencies ofthecarbon atoms would bemuch lower than thefrequency weareusing inthe x-rays, since x-ray radiation hasaveryhigh frequency. Theindex ofrefraction is thatgiven byourdispersion equation ifwesetwoequal tozero (weneglect <33in comparison with 0:2). Asimilar situation would occur ifwebeam radiowaves (orlight) onagasof freeelectrons. Intheupper atmosphere electrons areliberated from their atoms by ultraviolet light from thesunandthey situpthere asfreeelectrons. Forfree electrons wo=0(there isnoelastic restoring force). Setting we=0inourdisper- sionequation yields thecorrect formula fortheindex ofrefraction forradiowaves inthestratosphere, where Nisnowtorepresent thedensity offreeelectrons" (num- berperunitvolume) inthestratosphere. Butletuslook again attheequation, if webeam x-rays onmatter, orradiowaves (oranyelectric waves) onfreeelectrons theterm (oi?)—wz)becomes negative, andweobtain theresult thatnislessthan one. That means thattheeffective speed ofthewaves inthesubstance isfaster than clCanthatbecorrect? A Itiscorrect. Inspite ofthefactthatitissaidthatyoucannot send signals anyfaster than thespeed oflight, itisnevertheless truethattheindex ofrefraction ofmaterials ataparticular frequency canbeeither greater orlessthan l.This 31-6 justmeans that thephase shift which isproduced bythescattered light canbe either positive ornegative. Itcanbeshown, however, thatthespeed atwhich you cansend asignal isnotdetermined bytheindex atonefrequency, butdepends on what theindex isatmany frequencies. What theindex tellsusisthespeed atwhich thenodes (orcrests) ofthewave travel. Thenode ofawave isnotasignal byitself. Inaperfect wave, which hasnomodulations ofanykind, i.e.,which isasteady oscillation, youcannot really saywhen it“starts,” soyoucannot useitforatiming signal. Inorder tosend asignal youhave tochange thewave somehow, make a notch init,make italittle bitfatter orthinner. That means thatyouhave to have more than onefrequency inthewave, anditcanbeshown thatthespeed at which signals travel isnotdependent upon theindex alone, butupon thewaythat theindex changes with thefrequency. This subject wemust also delay (until Chapter 48). Then wewillcalculate foryoutheactual speed ofsignals through such apiece ofglass, andyouwillseethatitwillnotbefaster than thespeed of light, although thenodes, which aremathematical points, dotravel faster than thespeed oflight. Justtogiveaslight hintastohow thathappens, youwillnote thatthereal difficulty hastodowith thefactthattheresponses ofthecharges areopposite to thefield, i.e.,thesignhasgotten reversed. Thus inourexpression forx(Eq.31.16) thedisplacement ofthecharge isinthedirection opposite tothedriving field, because (033—0:2)isnegative forsmall 0.10. Theformula says that when the electric fieldispulling inonedirection, thecharge ismoving intheopposite direc- tion. How does thecharge happen tobegoing intheopposite direction? Itcertainly does notstart ofl'intheopposite direction when thefield isfirstturned on.When themotion firststarts there isatransient, which settles down after awhile, and only thenisthephase oftheoscillation ofthecharge opposite tothedriving field. And itisthen thatthephase ofthetransmitted field canappear tobeadvanced with respect tothesource wave. Itisthisadvance inphase which ismeant when wesaythatthe“phase velocity” orvelocity ofthenodes isgreater than c.In Fig.31-4 wegiveaschematic ideaofhow thewaves might look foracasewhere thewave issuddenly turned on(tomake asignal). You willseefrom thediagram thatthesignal (i.e., thestart ofthewave) isnotearlier forthewave which ends up with anadvance inphase. j StartE’(=1 1/ | I I Wave with no material t (bl Transmitted I042 withn>lM -v Fig. 31-4. Wave “signals.” delay01mm E (c) 1 I Transmitted wave ‘ wlth"<1 ' ncnotphase g':1'_"_S Letusnow look again atourdispersion equation. Weshould remark that ouranalysis oftherefractive index gives aresult thatissomewhat simpler than you would actually find innature. Tobecompletely accurate wemust addsome refinements. First, weshould expect thatourmodel oftheatomic oscillator should have some damping force (otherwise once started itwould oscillate forever, and wedonotexpect that tohappen). Wehave worked outbefore (Eq. 23.8) the motion ofadamped oscillator andtheresult isthatthedenominator inEq.(31.16), andtherefore in(31.19), ischanged from (Q3—wz)to(cu?)—032—l—ivw), where ‘Yisthedamping coeflicient. Weneed asecond modification totake intoaccount thefactthat there are several resonant frequencies foraparticular kind ofatom. Itiseasytofixupour 31-7 TlO tn, ‘U2 etc. it Fig. 31-5. Theindex ofrefraction as afunction offrequency.dispersion equation byimagining thatthere areseveral different kinds ofoscil- lators, butthateach oscillator actsseparately, andsowesimply addthecontri- butions ofalltheoscillators. Letussaythatthere areN),electrons perunitof volume, whose natural frequency iswkandwhose damping factor is'Y,,.We would then have forourdispersion equation n=1+-lg-Zj__NI¢___. (3120)250m 1,001%—-(.02-1-i'Yk(.o ' Wehave, finally, acomplete expression which describes theindex ofrefraction that isobserved formany substances.* Theindex described bythisformula varies with frequency roughly likethecurve shown inFig.31-5. Youwillnote thatsolong aswisnottooclose tooneoftheresonant frequen- cies, theslope ofthecurve ispositive. Such apositive slope iscalled “normal” dispersion (because itisclearly themost common occurrence). Very near the resonant frequencies, however, there isasmall range ofw’sforwhich theslope is negative. Such anegative slope isoften referred toas“anomalous” (meaning abnormal) dispersion, because itseemed unusual when itwasfirstobserved, long before anyone even knew there were ‘such things aselectrons. From ourpoint of view both slopes arequite “normal”! ‘ 31-4 Absorption Perhaps youhave noticed something alittle strange about thelastform (Eq. 31.20) weobtained forourdispersion equation. Because oftheterm i“/we putintotake account ofdamping, theindex ofrefraction isnow acomplex number! What does thatmean? Byworking outwhat therealandimaginary parts ofnarewecould write n=n’—in", (31.21) where n’andn”arerealnumbers. (We usetheminus signinfront ofthein" because thenn”willturnouttobeapositive number, asyoucanshow foryourself.) Wecanseewhat such acomplex index means bygoing back toEq.(31.6), which istheequation ofthewave after itgoes through aplate ofmaterial withan index n.Ifweputourcomplex nintothisequation, anddosome rearranging, we 8“ ,, ., .Eafterplate =€—w7I. Az/lc e-wtn -1)Az/c E-0ew(t—z/c) K 13' Thelastfactors, marked BinEq.(31.22), arejusttheform wehadbefore, and again describe awave whose phase hasbeen delayed bytheangle w(n’ —1)Az/c intraversing thematerial. Thefirstterm (A)isnewandisanexponential factor with arealexponent, because there were twoi’sthatcancelled. Also, theexponent isnegative, sothefactor isarealnumber lessthan one. Itdescribes adecrease inthemagnitude ofthefield and, asweshould expect, byanamount which is more thelarger Azis.Asthewave goes through thematerial, itisweakened. The material is“absorbing” partofthewave. Thewave comes outtheother sidewith lessenergy. Weshould notbesurprised atthis, because thedamping weputin fortheoscillators isindeed afriction force andmust beexpected tocause aloss ofenergy. Weseethattheimaginary part n”ofacomplex index ofrefraction represents anabsorption (or“attenuation”) ofthewave. Infact, n”issometimes referred toasthe“absorption index.” Wemayalsopoint outthatanimaginary parttotheindex ncorresponds to bending thearrow EainFig.31-3 toward theorigin. Itisclear whythetransmitted field isthen decreased. *Actually, although inquantum mechanics Eq.(31.20) isstillvalid, itsinterpretation issomewhat different. Inquantum mechanics even anatom with oneelectron, like hydrogen, hasseveral resonant frequencies. Therefore N),isnotreally thenumber of electrons having thefrequency wk,butisreplaced instead byNf},, where Nisthenumber ofatoms perunitvolume andfl(called theoscillator strength) isafactor thattells how strongly theatom exhibits each ofitsresonant frequencies wk. 31-8 Normally, forinstance asinglass, theabsorption oflight isvery small. This istobeexpected from ourEq.(31.20), because theimaginary part ofthe denominator, i‘Y;,w, ismuch smaller than theterm (wi—032). Butifthelight fre- quency wisveryclose towkthen theresonance term (wi—(.02)canbecome small compared with ivkw andtheindex becomes almost completely imaginary. The absorption ofthelight becomes thedominant effect. Itisjustthisefl'ect thatgives thedark lines inthespectrum oflight which wereceive from thesun. Thelight from thesolar surface haspassed through thesun’s atmosphere (aswellasthe earth's), andthelight hasbeen strongly absorbed attheresonant frequencies of theatoms inthesolar atmosphere. Theobservation ofsuch spectral lines inthesunlight allows ustotellthe resonant frequencies oftheatoms andhence thechemical composition ofthesun’s atmosphere. Thesame kind ofobservations tellusabout thematerials inthestars. From such measurements weknow thatthechemical elements inthesunandin thestars arethesame asthose wefindontheearth. 31-5 Theenergy carried byanelectric wave Wehave seen thattheimaginary part oftheindex means absorption. We shall now usethisknowledge tofindouthow much energy iscarried byalight wave. Wehave given earlier anargument that theenergy carried bylight is proportional toF,thetime average ofthesquare oftheelectric field inthewave. Thedecrease inEduetoabsorption must mean alossofenergy, which would go intosome friction oftheelectrons and, wemight guess, would endupasheat in thematerial. Ifweconsider thelight arriving onaunitarea, sayonesquare centimeter, of ourplate inFig.31-1, thenwecanwrite thefollowing energy equation (ifweassume thatenergy isconserved, aswedo!): Energy inpersec=energy outpersec+work done persec. (31.23) Forthefirstterm wecanwrite aE_§, where atistheasyetunknown constant of proportionality which relates theaverage value ofE2totheenergy being carried. Forthesecond term wemust includeithe part from theradiating atoms ofthe material, soweshould usea(E, +E.,)2, or(evaluating thesquare) a(b? —l— zfi+E)-Allofourcalculations have been made forathinlayer ofmaterial whose index isnottoofarfrom 1,sothatEawould always bemuch lessthan E,(just to make thecalculations easier). Inkeeping with ourapproximations, weshould, therefore, leave outtheterm EE,because itismuch smaller than m You may say: “Then youshould leave outm also, because itismuch smaller than E.” Itistruethati ismuch smaller than E,butwemust keep E orourapproxi- mation willbetheonethatwould apply ifweneglected thepresence ofthematerial completely! Onewayofchecking thatourcalculations areconsistent istoseethat wealways keep terms which areproportional toNAz,thearea density ofatoms inthematerial, butweleave outterms which areproportional to(NAz)2 orany higher power ofNAz. Ours iswhat should becalled a“low-density approxi- mation.” Inthesame spirit, wemight remark thatourenergygquation hasneglected theenergy inthereflected wave. ButthatisOKbecause thister1=n,\tQo, ispropor- tional to(NAz)2, since theamplitude ofthereflected wave isproportional to NAz. Forthelastterm inEq.(31.23) wewish tocompute therateatwhich the incoming wave isdoing work ontheelectrons. Weknow thatwork isforce times distance, sotherateofdoing work (also called power) istheforce times theveloc- ity.Itisreally F-V,butwedonotneed toworry about thedotproduct when the velocity andforce arealong thesame direction asthey arehere (except fora possible minus sign). Soforeach atom wetake ZIZEQ fortheaverage rate of 31-9 (0) 5 P-X» E=E‘ E=0 o \opaque screen (bl E‘Es TE=Es‘Ewall P ~,hole .S -)t- —‘\w0 Il (C) S ‘l<-I P -X pug I E“Es flE=5s*Eiw11”Eo10q'° |‘vtOll Fig. 31-6. Diffraction byascreen.doing work. Since_‘tl1ere areNAzatoms inaunitarea, thelastterm inEq.(31.23) should beNAzq,E,v. Ourenergy equation nowlooks like aff =(IE? —l—2otE_,Ea -1-NAZ q, (31.24) TheE?terms cancel, andwehave 2aE,E,, =NAZq,E—W. (31.25) Wenow goback toEq.(30.19), which tellsusthatforlarge z NA .,13,,=7% v(retbyz/c) (31.26) (recalling that1;=NAz). Putting Eq.(31.26) intotheleft-hand sideof(31.25), weget 2011% E,(at z)-v(ret byz/c).2e0c However, E,(at z)isE,(atatoms) retarded byz/c. Since theaverage isinde- pendent oftime, itisthesame now asretarded byz/c,orisE,(atatom) -v,the same average thatappears ontheright-hand sideof(31.25). Thetwosides are therefore equal if $=1,or0.=60¢. (31.27) Wehave discovered thatifenergy istobeconserved, theenergy carried inanelec- tricwave perunitarea andperunittime (orwhat wehave called theintensity) must begiven bye0cE2. Ifwecalltheintensity S,wehave _ intensity S= or =eocp, (31.28) energy/area/time where thebarmeans thetimeaverage. Wehave anicebonus result from ourtheory oftherefractive index! 31-6 Diffraction oflight byascreen Itisnow agood time totake upasomewhat different matter which wecan handle with themachinery ofthischapter. Inthelastchapter wesaidthatwhen youhave anopaque screen andthelight cancome through some holes, thedistribu- tionofintensity——the diffraction pattern—could beobtained byimagining instead thattheholes arereplaced bysources (oscillators) uniformly distributed overthe hole. Inother words, thediffracted wave isthesame asthough thehole were a newsource. Wehave toexplain thereason forthat, because theholeis,ofcourse, justwhere there arenosources, where there arenoaccelerating charges. Letusfirstask:“What isanopaque screen?” Suppose wehave acompletely opaque screen between asource Sandanobserver atP,asinFig.3l—6(a). Ifthe screen is“opaque” there isnofield atP.Why isthere nofield there? According tothebasic principles weshould obtain thefield atPasthefield E,ofthesource delayed, plus thefield from alltheother charges around. But, aswehave seen above, thecharges inthescreen willbesetinmotion bythefield E,,andthese motions generate anewfield which, ifthescreen isopaque, must exactly cancel thefield E,ontheback sideofthescreen. You say:“What amiracle thatitbal- ances exactly! Suppose itwasnotexactly right! ”Ifitwere notexactly right (re- member that thisopaque screen hassome thickness), thefield toward therear part ofthescreen would notbeexactly zero. So,notbeing zero, itwould set intomotion some other charges inthematerial ofthescreen, andthusmake alittle more field, trying togetthetotal balanced out. Soifwemake thescreen thick enough, there isnoresidual field, because there isenough opportunity tofinally getthething quieted down. Interms ofourformulas above wewould saythatthe 31-10 screen hasalarge andimaginary index, sothewave isabsorbed exponentially asit goes through. You know, ofcourse, thatathinenough sheet ofthemost opaque material, even gold, istransparent. Now letusseewhat happens with anopaque screen which hasholes init,as inFig.3l—6(b). What doweexpect forthefieldatP?ThefieldatPcanberepre- sented asasumoftwoparts—the fieldduetothesource Splusthefieldduetothe wall, i.e.,duetothemotions ofthecharges inthewalls. Wemight expect the motions ofthecharges inthewalls tobecomplicated, butwecanfindoutwhat fields theyproduce inarather simple way. Suppose thatwewere totakethesame screen, butplug uptheholes, asindi- cated inpart(c)ofthefigure. Weimagine thattheplugs areofexactly thesame material asthewall. Mind you, theplugs gowhere theholes were incase (b). Now letuscalculate thefield atP.Thefield atPiscertainly zero incase(c),but itisalsoequal tothefield from thesource plusthefield duetoallthemotions of theatoms inthewalls andintheplugs. Wecanwrite thefollowing equations: Case Eat P=Es + Ewalls Case (C): Exit P=0=Es'1"Ewell +Eplug» where theprimes refer tothecasewhere theplugs areinplace, butE,is,ofcourse, thesame inboth cases. Now ifwesubtract thetwoequations, weget Eat P= (Ewall T’Eivall) —Elling- Now iftheholes arenottoosmall (saymany wavelengths across), wewould not expect thepresence oftheplugs tochange thefields which arrive atthewalls except possibly foralittle bitaround theedges oftheholes. Neglecting thissmall effect, wecansetEm,“ =E(,,,,1| andobtain that Eat P: _E;IJlug- Wehave theresult thatthefieldatPwhen there areholes inascreen (case b)isthe same (except forsign) asthefieldthatisproduced bythatpartofacomplete opaque wallwhich islocated where theholes are! (The signisnottoointeresting, since we areusually interested inintensity which isproportional tothesquare ofthefield.) Itseems likeanamazing backwards-forwards argument. Itis,however, notonly true (approximately fornottoosmall holes), butuseful, andisthejustification fortheusual theory ofdiffraction. Thefield E,’,1,,g iscomputed inanyparticular case byremembering thatthe motion ofthecharges everywhere inthescreen isjustthatwhich willcancel out thefield E,ontheback ofthescreen. Once weknow these motions, weaddthe radiation fields atPduejusttothecharges intheplugs. Weremark again thatthistheory ofdiffraction isonly approximate, andwill begood only iftheholes arenottoosmall. Forholes which aretoosmall the E{,1,,g term willbesmall andthen thedifference between E(v,,11 andEwan (which difference wehave taken tobezero) maybecomparable toorlarger than thesmall E,Q1,,,, term, andourapproximation willnolonger bevalid. 31-ll 32 Radiation Damping. Light Scattering 32-1 Radiation resistance Inthelastchapter welearned thatwhen asystem isoscillating, energy is carried away, andwededuced aformula fortheenergy which isradiated byan oscillating system. Ifweknow theelectric field, then theaverage ofthesquare ofthefield times socistheamount ofenergy thatpasses persquare meter per second through asurface normal tothedirection inwhich theradiation isgoing: S=e(,c(E2). (32.1) Any oscillating charge radiates energy; forinstance, adriven antenna radiates energy. Ifthesystem radiates energy, theninorder toaccount fortheconservation ofenergy wemust findthatpower isbeing delivered along thewires which lead intotheantenna. That is,tothedriving circuit theantenna actslikearesistance, oraplace where energy canbe“lost” (theenergy isnotreally lost,itisreally radi- ated out,butsofarasthecircuit isconcerned, theenergy islost). Inanordinary resistance, theenergy which is“lost” passes intoheat; inthiscasetheenergy which is“lost” goes outintospace. Butfrom thestandpoint ofcircuit theory, without considering where theenergy goes, theneteffect onthecircuit isthesame—energy is“lost” from thatcircuit. Therefore theantenna appears tothegenerator as having aresistance, even though itmay bemade with perfectly good copper. Infact, ifitiswellbuilt itwillappear asalmost apure resistance, with very little inductance orcapacitance, because wewould liketoradiate asmuch energy as possible outoftheantenna. This resistance thatanantenna shows iscalled the radiation resistance. Ifacurrent Iisgoing totheantenna, then theaverage rateatwhich power is delivered totheantenna istheaverage ofthesquare ofthecurrent times there- sistance. Therateatwhich power isradiated bytheantenna isproportional to thesquare ofthecurrent intheantenna, ofcourse, because allthefields are proportional tothecurrents, andtheenergy liberated isproportional tothesquare ofthefield. Thecoefficient ofproportionality between radiated power and(I2) istheradiation resistance. Aninteresting question is,what isthisradiation resistance dueto? Letus takeasimple example: letussaythatcurrents aredriven upanddown inanan- tenna. Wefindthatwehave toputwork in,iftheantenna istoradiate energy. Ifwetake acharged body andaccelerate itupanddown itradiates energy; ifit were notcharged itwould notradiate energy. Itisonething tocalculate from the conservation ofenergy thatenergy islost,butanother thing toanswer thequestion, against whatforce arewedoing thework? That isaninterestingand verydifficult question which hasnever been completely andsatisfactorily answered forelectrons, although ithasbeen forantennas. What happens isthis: inanantenna, thefields produced bythemoving charges inonepart oftheantenna react onthemoving charges inanother part oftheantenna. Wecancalculate these forces andfind outhowmuch work theydo,andsofindtheright rulefortheradiation resistance. When wesay“We cancalculate——” thatisnotquite right—we cannot, because we havenotyetstudied thelawsofelectricity atshort distances; onlyatlarge distances doweknow what theelectric field is.Wesawtheformula (28.3), butatpresent itistoocomplicated forustocalculate thefields inside thewave zone. Ofcourse, since conservation ofenergy isvalid, wecancalculate theresult allright without knowing thefields atshort distances. (Asamatter offact, byusing thisargument backwards itturns outthat onecanfind theformula fortheforces atshort 32-132-1 Radiation resistance 32-2 Therateofradiation ofenergy 32-3 Radiation damping 32-4 Independent sources 32-5 Scattering oflight v ~“ Fig. 32-1. The area ofaspherical segment is21rrsin6-rd6.distances only byknowing thefield atvery large distances, byusing thelawsof conservation ofenergy, butweshall notgointothathere.) Theproblem inthecaseofasingle electron isthis: ifthere isonlyonecharge, what cantheforce acton? Ithasbeen proposed, intheoldclassical theory, that thecharge wasalittle ball, andthatonepartofthecharge acted ontheother part. Because ofthedelay intheaction across thetinyelectron, theforce isnotexactly inphase with themotion. That is,ifwehave theelectron standing still,weknow that“action equals reaction.” Sothevarious internal forces areequal, andthere isnonetforce. Butiftheelectron isaccelerating, then because ofthetime delay across it,theforce which isacting onthefront from theback isnotexactly the same astheforce ontheback from thefront, because ofthedelay intheeffect. This delay inthetiming makes foralackofbalance, so,asaneteffect, thething holds itself back byitsbootstraps! This model oftheorigin oftheresistance to acceleration, theradiation resistance ofamoving charge, hasruninto many difficulties, because ourpresent view oftheelectron isthatitisnota“little ball"; thisproblem hasnever been solved. Nevertheless wecancalculate exactly, of course, what thenetradiation resistance force must be,i.e.,how much lossthere must bewhen weaccelerate acharge, inspite ofnotknowing directly then1echa- nism ofhowthatforce works. 32-2 Therateofradiation ofenergy Now weshall calculate thetotal energy radiated byanaccelerating charge. Tokeep thediscussion general, weshall takethecaseofacharge accelerating any which way, butnonrelativistically. Atamoment when theacceleration is,say, vertical, weknow thattheelectric field thatisgenerated isthecharge multiplied bytheprojection oftheretarded acceleration, divided bythedistance. Soweknow theelectric field atanypoint, andwetherefore know thesquare oftheelectric field andthus theenergy e0cE2 leaving through aunitarea persecond. Thequantity socappears quite often inexpressions involving radiowave prop- agation. Itsreciprocal iscalled theimpedance ofavacuum, anditisaneasynumber toremember: ithasthevalue l/eoc=377ohms. Sothepower inwatts persquare meter isequal totheaverage ofthefield squared, divided by377. Using ourexpression (29.1) fortheelectric field, wefindthat q2a’2 sin26 isthepower persquare meter radiated inthedirection 6.Wenotice thatitgoes inversely asthesquare ofthedistance, aswesaidbefore. Now suppose wewanted thetotal energy radiated inalldirections: then wemust integrate (32.2) over all directions. First wemultiply bythearea, tofindtheamount thatflows within a little angle d6(Fig. 32-1). Weneed thearea ofaspherical section. Theway tothink ofitisthis: ifristheradius, then thewidth oftheannular segment is rd6,andthecircumference is21rrsin6,because rsin6istheradius ofthecircle. Sothearea ofthelittle piece ofthesphere is27rrsin0times rd6: dA=27rr2 sin6d6. (32.3) Bymultiplying theflux[(32.2), thepower persquare meter] bythearea insquare meters included inthesmall angle d6,wefindtheamount ofenergy that isliberated inthisdirection between 6and6+d6;then weintegrate thatover alltheangles 6from 0to180°: 22 "q(ll .2; P=[SdA=———, Isin‘ 6d6. (32.4)87T'€()C'i () Bywriting sin“6=(1—cosz 6)sin6itisnothard toshow thatf;sin“6d6= 4/3. Using thatfact, wefinally get 2a/2 P=-Q67re()c3 (325) 32-2 Thisexpression deserves some remarks. First ofall,since thevector a’hada certain direction, thea’2in(32.5) would bethesquare ofthevector a’,thatis, a’-a’,thelength ofthevector, squared. Secondly, theflux(32.2) wascalculated using theretarded acceleration; thatis,theacceleration atthetime atwhich the energy nowpassing through thesphere wasradiated. Wemight liketosaythat thisenergy wasinfactliberated atthisearlier time. This isnotexactly true; itis onlyanapproximate idea. Theexact time when theenergy isliberated canhot be defined precisely. Allwecanreally calculate precisely iswhat happens inacomplete motion, likeanoscillation orsomething, where theacceleration finally ceases. Then what wefindisthatthetotal energy fluxpercycle istheaverage ofaccelera- tionsquared, foracomplete cycle. This iswhat should really appear in(32.5). Or,ifitisamotion with anacceleration thatisinitially andfinally zero, then the totalenergy thathasflown outisthetime integral of(32.5). Toillustrate theconsequences offormula (32.5) when wehave anoscillating system, letusseewhat happens ifthedisplacement xofthecharge isoscillating sothattheacceleration ais—w2x0e”“. Theaverage oftheacceleration squared overacycle (remember thatwehave tobeverycareful when wesquare things that arewritten incomplex notation——it really isthecosine, andtheaverage ofcos2 wt isone-half) thusis (a’2) =%w4x§. Therefore242 P= (32.6) Theformulas wearenow discussing arerelatively advanced andmore or lessmodern; they date from thebeginning ofthetwentieth century, andthey are veryfamous. Because oftheir historical value, itisimportant forustobeable toreadabout them inolder books. Infact, theolder books alsoused asystem of units different from ourpresent mkssystem. However, allthese complications can bestraightened outinthefinal formulas dealing with electrons bythefollowing rule: Thequantity qf/41re0, where q,istheelectronic charge (incoulombs), has, historically, been written ase2.Itisveryeasytocalculate thateinthemkssystem isnumerically equal to1.5188 X10'“, because weknow that, numerically, qe=1.60206 X10"“) andl/41re0 =8.98748 X109. Therefore weshall often usetheconvenient abbreviation _1f e2—Era - (32.7) Ifweusetheabove numerical value ofeintheolder formulas andtreat them as though they were written inmks units, wewillgettheright numerical results. Forexample, theolder form of(32.5) isP=§e2a2/03. Again, thepotential energy ofaproton andanelectron atdistance risqf/41re0r ore2/r, with e=1.5188 X 10'“ mks. 32-3 Radiation damping Now thefactthatanoscillator loses acertain energy would mean thatifwe hadacharge ontheendofaspring (oranelectron inanatom) which hasanatural frequency wo,andwestart itoscillating andletitgo,itwillnotoscillate forever, even ifitisinempty space millions ofmiles from anything. There isnooil,no resistance, inanordinary sense; no“viscosity.” Butnevertheless itwillnot oscillate, aswemight once have said, “forever,” because ifitischarged itis radiating energy, andtherefore theoscillation willslowly dieout. How slowly? What istheQofsuch anoscillator, caused bytheelectromagnetic efiects, the so-called radiation resistance orradiation damping oftheoscillator? TheQofany oscillating system isthetotal energy content oftheoscillator atanytime divided bytheenergy lossperradian: W Q=Z?W$'32-3 Or(another waytowrite it),since dW/d¢ =(dW/dt)/(d¢/dt) =(dW/dt)/w, Q=%- (32.8) Ifforagiven Qthistellsushow theenergy oftheoscillation diesout,dW/dt = —(w/Q)W, which hasthesolution W=W0e“"‘/Q ifW0istheinitial energy (att =0). TofindtheQforaradiator, wegoback to(32.8) anduse(32.6) fordW/dt. Now what doweusefortheenergy Woftheoscillator? Thekinetic energy oftheoscillator is%mv2, andthemean kinetic energy ismw22:3/4. Butweremember thatforthetotal energy ofanoscillator, ontheaverage halfiskinetic andhalfis potential energy, andsowedouble ourresult, andfindforthetotal energy ofthe oscillator W=gmwzxfi. (32.9) What doweuseforthefrequency inourformulas? Weusethenatural frequency wobecause, forallpractical purposes, thatisthefrequency atwhich ouratom is radiating, andformweusetheelectron mass me. Then, making thenecessary divisions andcancellations, theformula comes down to l 41re2Q-3>\mec2- (32.10) (Inorder toseeitbetter andinamore historical form wewrite itusing ourab- breviation qf/41re0 =e2,andthefactor coo/C which wasleftover hasbeen written as21r/)\.) Since Qisdimensionless, thecombination e2/mecz must beaproperty only oftheelectron charge andmass, anintrinsic property oftheelectron, andit must bealength. Ithasbeen given aname, theclassical electron radius, because the early atomic models, which were invented toexplain theradiation resistance on thebasis oftheforce ofonepartoftheelectron acting ontheother parts, allneeded tohave anelectron whose dimensions were ofthisgeneral order ofmagnitude. However, thisquantity nolonger hasthesignificance thatwebelieve thattheelec- tronreally hassuch aradius. Numerically, themagnitude oftheradius is 2 to= =2.82><l0_15m. (32.11) Now letusactually calculate theQofanatom thatisemitting light—let us sayasodium atom. Forasodium atom, thewavelength isroughly 6000 angstroms, intheyellow partofthevisible spectrum, andthisisatypical wavelength. Thus Q=$5z5><10’, (32.12) sotheQofanatom isoftheorder 108. This means thatanatomic oscillator will oscillate for108radians orabout 107oscillations, before itsenergy fallsbyafactor l/e. The frequency ofoscillation oflight corresponding to6000 angstroms, 1/=c/A,isontheorder of1015 cycles/sec, andtherefore thelifetime, thetimeit takes fortheenergy ofaradiating atom todieoutbyafactor l/e,isontheorder of10-8 sec. Inordinary circumstances, freely emitting atoms usually take about thislong toradiate. This isvalid only foratoms which areinempty space, not being disturbed inanyway. Iftheelectron isinasolid andithastohitother atoms orother electrons, then there areadditional resistances anddiflerent damping. Theeffective resistance term "rintheresistance lawfortheoscillator canbe found from therelation 1/Q=V/wo, andweremember thatthesizeof'Ydeter- mines howwide theresonance curve is(Fig. 23-2). Thus wehave justcomputed thewidths ofspectral lines forfreely radiating atoms! Since )1=21rc/w, wefind that AX=21rcAw/wz =21rc'Y/wfi =21rc/Qwo =x/Q=41m,/3 =1.18><10*“m. (32.13) 32—4 32-4 Independent sources Inpreparation foroursecond topic, thescattering oflight, wemustnowdiscuss acertain feature ofthephenomenon ofinterference thatweneglected todiscuss previously. This isthequestion ofwhen interference does notoccur. Ifwehave twosources S1andS2,with amplitudes A1andA2,andwemake anobservation inacertain direction inwhich thephases ofarrival ofthetwosignals are¢1and¢2 (acombination oftheactual timeofoscillation andthedelayed time, depending on theposition ofobservation), then theenergy that wereceive canbefound by compounding thetwocomplex number vectors A1andA2,oneatangle ¢1and theother atangle 452(aswedidinChapter 30)andwefindthattheresultant energy isproportional to A3,=A?+/13+2A1A2cos(¢1 -<82). (32.14) Now ifthecross term 2AIA2cos(¢1—¢2)were notthere, then thetotal energy thatwould bereceived inagiven direction would simply bethesumoftheenergies, A?+AZ,thatwould beliberated byeach source separately, which iswhat we usually expect. That is,thecombined intensity oflight shining onsomething from twosources isthesumoftheintensities ofthetwolights. Ontheother hand, ifwehave things setjustright andwehave across term, itisnotsuch asum, because there isalsosome interference. Ifthere arecircumstances inwhich thisterm isof noimportance, then wewould saytheinterference isapparently lost. Ofcourse, innature itisalways there, butwemaynotbeabletodetect it. Letusconsider some examples. Suppose, first, that thetwosources are 7,000,000,000 wavelengths apart, notanimpossible arrangement. Then inagiven direction itistruethatthere isaverydefinite value ofthese phase differences. But, ontheother hand, ifwemove justahairinonedirection, afewwavelengths, which isnodistance atall(our eyealready hasahole initthatissolarge thatweare averaging theeffects over arange very wide compared with onewavelength) then wechange therelative phase, andthecosine changes very rapidly. Ifwetake the average oftheintensity overalittle region, then thecosine, which goesplus, minus, plus, minus, aswemove around, averages tozero. Soifweaverage overregions where thephase varies veryrapidly withposition, wegetnointerference. Another example. Suppose thatthetwosources aretwoindependent radio 0scillators—not asingle oscillator being fedbytwowires, which guarantees that thephases arekept together, buttwoindependent sources—and thatthey arenot precisely tuned atthesame frequency (itisvery hard tomake them atexactly the same frequency without actually wiring them together). Inthiscasewehave what wecalltwoindependent sources. Ofcourse, since thefrequencies arenotexactly equal, although they started inphase, oneofthem begins togetalittle ahead of theother, andpretty soon theyareoutofphase, andthenitgetsstillfurther ahead, andpretty soon they areinphase again. Sothephase difference between thetwo isgradually drifting with time, butifourobservation issocrude thatwecannot seethat little time, ifweaverage over amuch longer time, then although the intensity swells andfallslikewhat wecall“beats” insound, ifthese swellings and fallings aretoorapid forourequipment tofollow, thenagain thisterm averages out. Inother words, inanycircumstance inwhich thephase shift averages out,we getnointerference! Onefinds many books which saythattwodistinct light sources never interfere. This isnotastatement ofphysics, butismerely astatement ofthedegree ofsensi- tivity ofthetechnique oftheexperiments atthetime thebook waswritten. What happens inalight source isthatfirstoneatom radiates, thenanother atom radiates, andsoforth, andwehave justseen thatatoms radiate atrain ofwaves only for about 10's sec;after 1078 sec,some atom hasprobably taken over, then another atom takes over, andsoon.Sothephases canreally only staythesame forabout 10'” sec. Therefore, ifweaverage forvery much more than l0_8 sec,wedonot seeaninterference from twodifferent sources, because they cannot hold their phases steady forlonger than l0'8 sec.With photocells, veryhigh-speed detection 32-5 \ _\_./\\Incid entbeam _MOM (unpolarized) Scattered radiativon Fig. 32-2. Abeam ofradiation falls onanatom and causes the charges (electrons) intheatom tomove. The moving electrons inturn radiate in various directions.ispossible, andonecanshow thatthere isaninterference which varies with time, upanddown, inabout l0‘8 sec. Butmost detection equipment, ofcourse, does notlookatsuchfinetimeintervals, andthusseesnointerference. Certainly with theeye,which hasatenth-of-a-second averaging time, there isnochance whatever ofseeing aninterference between twodifferent ordinary sources. Recently ithasbecome possible tomake light sources which getaround this effect bymaking alltheatoms emit together intime. Thedevice which does thisis averycomplicated thing, andhastobeunderstood inaquantum-mechanical way. Itiscalled alaser, anditispossible toproduce from alaser asource inwhich the interference frequency, thetime atwhich thephase iskept constant, isverymuch longer than l0‘8 sec. Itcanbeoftheorder ofahundredth, atenth, oreven one second, andso,with ordinary photocells, onecanpick upthefrequency between twodifferent lasers. Onecaneasily detect thepulsing ofthebeats between two laser sources. Soon, nodoubt, someone willbeabletodemonstrate twosources shining onawall, inwhich thebeats aresoslow thatonecanseethewallgetbright anddark! Another casein which theinterference averages outisthatinwhich, instead of having only twosources, wehave many. Inthiscase, wewould write theexpression forAfsasthesumofawhole lotofamplitudes, complex numbers, squared, and wewould getthesquare ofeach one,alladded together, pluscross terms between every pair, andifthecircumstances aresuch thatthelatter average out,then there willbenoeffects ofinterference. Itmaybethatthevarious sources arelocated in such random positions that, although thephase difference between A2andA3is alsodefinite, itisvery different from thatbetween A1andA2,etc. Sowewould getawhole lotofcosines, many plus, many minus, allaveraging out. Soitisthatinmany circumstances wedonotseetheeffects ofinterference, butseeonly acollective, total intensity equal tothesumofalltheintensities. 32-5 Scattering oflight Theabove leads ustoaneffect which occurs inairasaconsequence ofthe irregular positions oftheatoms. When wewere discussing theindex ofrefraction, wesawthatanincoming beam oflight willmake theatoms radiate again. The electric field oftheincoming beam drives theelectrons upanddown, andthey radiate because oftheir acceleration. This scattered radiation combines togivea beam inthesame direction astheincoming beam, butofsomewhat different phase, andthisistheorigin oftheindex ofrefraction. Butwhat canwesayabout theamount ofre-radiated light insome other direc- tion? Ordinarily, iftheatoms arevery beautifully located inanicepattern, itis easy toshow thatwegetnothing inother directions, because weareadding alot ofvectors with their phases always changing, andtheresult comes tozero. But iftheobjects arerandomly located, then thetotal intensity inanydirection isthe sumoftheintensities thatarescattered byeach atom, aswehave justdiscussed. Furthermore, theatoms inagasareinactual motion, sothatalthough therelative phase oftwoatoms isadefinite amount now, later thephase would bequite differ- ent,andtherefore each cosine term willaverage out. Therefore, tofindouthow much light isscattered inagiven direction byagas,wemerely study theeffects of oneatom andmultiply theintensity itradiates bythenumber ofatoms. Earlier, weremarked thatthephenomenon ofscattering oflight ofthisnature istheorigin oftheblue ofthesky. Thesunlight goes through theair,andwhen welook toonesideofthesun—say at90°tothebeam—-we seeblue light; what wenowhave tocalculate ishowmuch light weseeandwhyitisblue. Iftheincident beam hastheelectric field E=E0el°" atthepoint where the atom islocated, weknow thatanelectron intheatom willvibrate upanddown in response tothisE(Fig. 32-2). From Eq.(23.8), theamplitude willbe i_ qeE0 _ 2 2 -m(o.>0 —w+lw"/)(32.15) 32-6 Wecould include thedamping andthepossibility thattheatom actslikeseveral oscillators ofdifferent frequency andsum over thevarious frequencies, butfor simplicity letusjusttakeoneoscillator andneglect thedamping. Then theresponse totheexternal electric field, which wehave already used inthecalculation ofthe index ofrefraction, issimply it=_-__"@E° - (32.16)m(w§ —(.02) Wecould now easily calculate theintensity oflight that isemitted invarious directions, using formula (32.2) andtheacceleration corresponding totheabove fr. Rather than dothis, however, weshall simply calculate thetotal amount of light scattered inalldirections, justtosavetime. Thetotal amount oflight energy persecond, scattered inalldirections bythesingle atom, isofcourse given by Eq.(32.7) So,putting together thevarious pieces andregrouping them, weget P=[(q§w“/l21r@o¢3)q§E§/m§(w2 —wf>)2] =(%@o¢E§)(81r/3)(q§/161r2@§m§¢4)[w4/(0)2 —w§)2l =(%6o¢'Ed)(31T’<2>/3)[w4/(‘"2 —w§)2] (32-17) forthetotal scattered power, radiated inalldirections. Wehave written theresult intheabove form because itisthen easy tore- member: First, thetotal energy thatisscattered isproportional tothesquare of theincident field. What does thatmean? Obviously, thesquare oftheincident fieldisproportional totheenergy which iscoming inpersecond. Infact,theenergy incident persquare meter persecond iseoctimes theaverage (E2) ofthesquare of theelectric field, andifE0isthemaximum value ofE,then (E2) =%E§. In other words, thetotal energy scattered isproportional totheenergy persquare meter thatcomes in;thebrighter thesunlight thatisshining inthesky,thebrighter theskyisgoing tolook. Next, what fraction oftheincoming light isscattered? Letusimagine a“tar- get”with acertain area, letussay0,inthebeam (notareal, material target, be- cause thiswould diffract light, andsoon;wemean animaginary area drawn in space). Thetotal amount ofenergy thatwould pass through thissurface 0'in agiven circumstance isproportional both totheincoming intensity andto0, andwould be P=(%e0cE§)o. (32.18) Now weinvent anidea: wesaythattheatom scatters atotal amount ofin- tensity which istheamount which would fallonacertain geometrical area, and wegivetheanswer bygiving thatarea. That answer, then, isindependent ofthe incident intensity; itgives theratio oftheenergy scattered totheenergy incident persquare meter. Inother words, theratio total energy scattered persecond__ isanarea.energy incident persquare meter persecond Thesignificance ofthisarea isthat, ifalltheenergy thatimpinged onthatarea were tobespewed inalldirections, then thatistheamount ofenergy thatwould bescattered bytheatom. This area iscalled across section forscattering; theidea ofcross section is used constantly, whenever some phenomenon occurs inproportion totheintensity ofabeam. Insuch cases onealways describes theamount ofthephenomenon bysaying what theeffective areawould have tobetopickupthatmuch ofthebeam. Itdoes notmean inanywaythatthisoscillator actually hassuch anarea. Ifthere were nothing present butafreeelectron shaking upanddown there would beno areadirectly associated withit,physically. Itismerely awayofexpressing the answer toacertain kind ofproblem; ittellsuswhat areatheincident beam would 32-7 have tohitinorder toaccount forthatmuch energy coming off.Thus, forourcase, 81rr§ co‘0, 3(Q2_w%)2 (32.19) (thesubscript sisfor“scattering”). Letuslook atsome examples. First, ifwegotoaverylownatural frequency wo,ortocompletely unbound electrons, forwhich wo=0,then thefrequency w cancels outandthecross section isaconstant. This low-frequency limit, orthe freeelectron cross section, isknown astheThompson scattering cross section. Itisanareawhose dimensions areapproximately 10"“ meter, more orless,ona side, i.e.,10-30 square meter, which israther small! Ontheother hand, ifwetakethecaseoflight intheair,weremember thatfor airthenatural frequencies oftheoscillators arehigher than thefrequency ofthe light thatweuse. This means that, toafirstapproximation, wecandisregard wz inthedenominator, andwefindthatthescattering isproportional tothefourth power ofthefrequency. That istosay,light which isofhigher frequency by,say, afactor oftwo, issixteen times more intensely scattered, which isaquite sizable difference. This means thatbluelight, which hasabout twice thefrequency ofthe reddish endofthespectrum, isscattered toafargreater extent than redlight. Thus when welook attheskyitlooks thatglorious blue thatweseeallthetime! There areseveral points tobemade about theabove results. Oneinteresting question is,why doweever seetheclouds? Where dotheclouds come from? Everybody knows itisthecondensation ofwater vapor. But, ofcourse, the water vapor isalready intheatmosphere before itcondenses, sowhydon’t wesee itthen? After itcondenses itisperfectly obvious. Itwasn’t there, now itisthere. Sothemystery ofwhere theclouds come from isnotreally such achildish mystery as“Where does thewater come from, Daddy?,” buthastobeexplained. Wehave justexplained thatevery atom scatters light, andofcourse thewater vapor willscatter light, too. Themystery iswhy, when thewater iscondensed into clouds, does itscatter such atremendously greater amount oflight? Consider what would happen if,instead ofasingle atom, wehadanagglom- erate ofatoms, saytwo, very close together compared with thewavelength ofthe light. Remember, atoms areonly anangstrom orsoacross, while thewavelength oflight issome 5000 angstroms, sowhen theyform aclump, afewatoms together, they canbevery close together compared with thewavelength oflight. Then when theelectric field acts, both oftheatoms willmove together. Theelectric field thatisscattered willthen bethesumofthetwoelectric fields inphase, i.e.,double theamplitude thatthere waswith asingle atom, andtheenergy which isscattered istherefore four times what itiswith asingle atom, nottwice! Solumps ofatoms radiate orscatter more energy than they doassingle atoms. Ourargument that thephases areindependent isbased ontheassumption thatthere isarealandlarge difference inphase between anytwoatoms, which istrueonly ifthey areseveral wavelengths apart andrandomly spaced, ormoving. Butifthey areright nextto each other, theynecessarily scatter inphase, andtheyhave acoherent interference which produces anincrease inthescattering. Ifwehave Natoms inalump, which isatinydroplet ofwater, theneachone willbedriven bytheelectric field inabout thesame wayasbefore (theeffect of oneatom ontheother isnotimportant; itisjusttogettheideaanyway) andthe amplitude ofscattering from each oneisthesame, sothetotal field which is scattered isN-fold increased. Theintensity ofthelight which isscattered isthen thesquare, orN2-fold, increased. Wewould have expected, iftheatoms were spread outinspace, only Ntimes asmuch asl,whereas wegetN2times asmuch as l!That istosay,thescattering ofwater inlumps ofNmolecules each isNtimes more intense than thescattering ofthesingle atoms. Soasthewater agglomerates thescattering increases. Does itincrease adinfinitum? No! When does this analysis begin tofail? How many atoms canweputtogether before wecannot drive thisargument anyfurther? Answer: Ifthewater drop getssobigthatfrom oneendtotheother isawavelength orso,then theatoms arenolonger allin 32-8 phase because theyaretoofarapart. Soaswekeep increasing thesizeofthedrop- letswegetmore andmore scattering, until such atime thatadrop getsabout the sizeofawavelength, andthenthescattering doesnotincrease anywhere nearly as rapidly asthedrop getsbigger. Furthermore, thebluedisappears, because forlong wavelengths thedrops canbebigger, before thislimit isreached, than they can beforshort wavelengths. Although theshort waves scatter more peratom than thelong waves, there isabigger enhancement fortheredendofthespectrum than fortheblueendwhen allthedrops arebigger than thewavelength, sothecolor is shifted from theblue toward thered. Now wecanmake anexperiment that demonstrates this. Wecanmake particles thatarevery small atfirst, andthen gradually grow insize. Weusea solution ofsodium thiosulfate (hypo) with sulphuric acid, which precipitates very finegrains ofsulphur. Asthesulphur precipitates, thegrains firststart verysmall, andthescattering isalittle bluish. Asitprecipitates more itgetsmore intense, and thenitwillgetwhitish astheparticles getbigger. Inaddition, thelight which goes straight through willhave theblue taken out. That iswhy thesunset isred,of course, because thelight thatcomes through alotofairtotheeyehashadalot ofbluelight scattered out,soitisyellow-red. Finally, there isoneother important feature which really belongs inthenext chapter, onpolarization, butitissointeresting thatwepoint itoutnow. This is thattheelectric field ofthescattered light tends tovibrate inaparticular direction. Theelectric field intheincoming light isoscillating insome way, andthedriven oscillator goes inthissame direction, andifwearesituated about atright angles tothebeam, wewillseepolarized light, thatistosay,light inwhich theelectric field isgoing only oneway. Ingeneral, theatoms canvibrate inanydirection at right angles tothebeam, butiftheyaredriven directly toward oraway from us,we donotseeit.Soiftheincoming light hasanelectric field which changes andos- cillates inanydirection, which wecallunpolarized light, then thelight which is coming outat90°tothebeam vibrates inonly onedirection! (SeeFig.32-3.) There isasubstance called polaroid which hastheproperty thatwhen light goes through it,only thepiece oftheelectric field which isalong oneparticular axiscangetthrough. Wecanusethistotestforpolarization, andindeed we findthelight scattered bythehypo solution tobestrongly polarized. 32-9Q \k> Electron moves ll plane .Lk AtomIncident beam + (unpolarized) Radiation scattered .1.kisplane polarized Fig. 32-3. Illustration oftheorigin of thepolarization ofradiation scattered atright angles totheincident beam. 33 Polarization 33-1 Theelectric vector oflight Inthischapter weshall consider those phenomena which depend onthefact thattheelectric field thatdescribes thelight isavector. Inprevious chapters we have notbeen concerned withthedirection ofoscillation oftheelectric field, except tonote that theelectric vector liesinaplane perpendicular tothedirection of propagation. Theparticular direction inthisplane hasnotconcerned us.We now consider those phenomena whose central feature istheparticular direction ofoscillation oftheelectric field. Inideally monochromatic light, theelectric field must oscillate atadefinite frequency, butsince thex-component andthey-component canoscillate independ- ently atadefinite frequency, wemust firstconsider theresultant effect produced bysuperposing twoindependent oscillations atright angles toeach other. What kind ofelectric field ismade upofanx-component anday-component which oscillate atthesame frequency? Ifoneadds toanx-vibration acertain amount of y-vibration atthesame phase, theresult isavibration inanewdirection inthe xy-plane. Figure 33-l illustrates thesuperposition ofdifferent amplitudes forthe x-vibration andthey-vibration. Buttheresultants shown inFig.33-1 arenotthe only possibilities; inallofthese cases wehave assumed thatthex-vibration and they-vibration areinphase, butitdoes nothave tobethatway. Itcould bethat thex-vibration andthey-vibration areoutofphase.33-1 Theelectric vector oflight 33-2 Polarization ofscattered light 33-3 Birefringence 33-4 Polarizers 33-5 Optical activity 33-6 Theintensity ofreflected light 33-7 Anomalous refraction Y Y Y Y Y Y x X . ‘ x x x = = = n E = 1 E =-1 By 1 Ey 1 By 1 Ky 0 y y !:x=o ax-Q !:x=1 nx=1 Ex:-1 xx=1 Fig. 33-l. Superposition ofx-vibrations andy-vibrations inphase. When thex-vibration andthey-vibration arenotinphase, theelectric field vector moves around inanellipse, andwecanillustrate thisinafamiliar way. If wehang aballfrom asupport byalong string, sothatitcanswing freely ina horizontal plane, itwillexecute sinusoidal oscillations. Ifweimagine horizontal x-andy-coordinates with their origin attherestposition oftheball, theballcan swing ineither thex-ory-direction with thesame pendulum frequency. By selecting theproper initial displacement andinitial velocity, wecansettheballin oscillation along either thex-axis orthey-axis, oralong anystraight lineinthe xy-plane. These motions oftheballareanalogous totheoscillations oftheelectric field vector illustrated inFig.33-1. Ineach instance, since thex-vibrations and they-vibrations reach their maxima andminima atthesame time, thex-andy-os- cillations areinphase. Butweknow thatthemost general motion oftheball ismotion inanellipse, which corresponds tooscillations inwhich thex-and y-directions arenotinthesame phase. Thesuperposition ofx-andy-vibrations which arenotinphase isillustrated inFig.33-2 foravariety ofangles between the phase ofthex-vibration andthatofthey-vibration. Thegeneral result isthatthe electric vector moves around anellipse. Themotion inastraight lineisaparticular 33-1 case corresponding toaphase difference ofzero (oranintegral multiple of1r); motion inacircle corresponds toequal amplitudes with aphase difference of90° (oranyoddintegral multiple of1r/2). InFig.33-2 wehave labeled theelectric fieldvectors inthex-andy-directions with complex numbers, which areaconvenient representation inwhich toexpress thephase difference. Donotconfuse therealandimaginary components ofthe complex electric vector inthisnotation with thex-andy-coordinates ofthefield. Thex-andy-coordinates plotted inFig.33-1 andFig. 33-2 areactual electric fields thatwecanmeasure. Therealandimaginary components ofacomplex electric field vector areonly amathematical convenience andhave nophysical significance. 0 b C d e =N , ¢<>=<¢, 1 cwwti 1 ¢ww¢; 1 certain; 1 =onme,1 cos(wt?“/4); @111/4 -sin mt;1 cos(ait-0,74), eh”/‘ -cos at;-1 h i =eoamt, 1 coaast; 1 coau1t;1 coaat;1 =-caifflfi-* /ll, em. sinwt;-1 -coa(a$4-3“/4); -cl,“/4 coome;1 Fig 33-2 Superposition ofx-vibrations andy-vibrations with equal amplitudes butvarious relative phases Thecomponents E,and E,areexpressed inboth real andcomplex notations. Now forsome terminology. Light islinearly polarized (sometimes called plane polarized) when theelectric field oscillates onastraight line; Fig. 33-1 illustrates linear polarization. When theendoftheelectric field vector travels in anellipse, thelight iselliptically polarized. When theendoftheelectric fieldvector travels around acircle, wehave circular polarization. Iftheendoftheelectric vector, when welook atitasthelight comes straight toward us,goes around inacounterclockwise direction, wecall itright-hand circular polarization. Figure 33—2(g) illustrates right-hand circular polarization, andFig.33-2(c) shows left-hand circular polarization. Inboth cases thelight iscoming outofthepaper. Our convention forlabeling left-hand and right-hand circular polarization is consistent with thatwhich isused today foralltheother particles inphysics which exhibit polarization (e.g., electrons). However, insome books onoptics the opposite conventions areused, soonemust becareful. Wehave considered linearly, circularly, andelliptically polarized light, which covers everything except forthecaseofunpolarized light. Now howcanthelight beunpolarized when weknow that itmust vibrate inoneoranother ofthese ellipses? Ifthelight isnotabsolutely monochromatic, orifthex-andy-phases arenotkept perfectly together, sothat theelectric vector first vibrates inone direction, then inanother, thepolarization isconstantly changing. Remember thatoneatom emits during l0—8 sec,andifoneatom emits acertain polarization, andthen another atom emits light with adifferent polarization, thepolarizations willchange every l0_8 sec. Ifthepolarization changes more rapidly than wecan detect it,thenwecallthelight unpolarized, because alltheeffects ofthepolarization average out. None oftheinterference effects ofpolarization would show upwith unpolarized light. Butasweseefrom thedefinition, light isunpolarized only if weareunable tofindoutwhether thelight ispolarized ornot. 33-2 33-2 Polarization ofscattered light Thefirstexample ofthepolarization effect thatwehavealready discussed is thescattering oflight. Consider abeam oflight, forexample from thesun,shining ontheair.Theelectric fieldwillproduce oscillations ofcharges intheair,andmo- tionofthese charges willradiate light withitsmaximum intensity inaplane normal tothedirection ofvibration ofthecharges. Thebeam from thesunisunpolarized, sothedirection ofpolarization changes constantly, andthedirection ofvibration ofthecharges intheairchanges constantly. Ifweconsider light scattered at90°, thevibration ofthecharged particles radiates totheobserver only when the vibration isperpendicular totheobserver’s lineofsight, andthen light willbe polarized along thedirection ofvibration. Soscattering isanexample ofonemeans ofproducing polarization. 33-3 Birefringence Another interesting effect ofpolarization isthefactthatthere aresubstances forwhich theindex ofrefraction isdifferent forlight linearly polarized inone direction andlinearly polarized inanother. Suppose thatwehadsome material which consisted oflong, nonspherical molecules, longer than they arewide, and suppose thatthese molecules were arranged inthesubstance with their long axes parallel. Then what happens when theoscillating electric field passes through this substance? Suppose thatbecause ofthestructure ofthemolecule, theelectrons inthesubstance respond more easily tooscillations inthedirection parallel tothe axes ofthemolecules than they would respond iftheelectric field tries topush them atright angles tothemolecular axis. Inthiswayweexpect adifferent response forpolarization inonedirection than forpolarization atright angles tothatdirec- tion. Letuscallthedirection oftheaxes ofthemolecules theoptic axis. When thepolarization isinthedirection oftheoptic axistheindex ofrefraction is different than itwould beifthedirection ofpolarization were atright angles toit. Such asubstance iscalled birefringent. Ithastworefrangibilities, i.e.,twoindexes ofrefraction, depending onthedirection ofthepolarization inside thesubstance. What kind ofasubstance canbebirefringent? Inabirefringent substance there must beacertain amount oflining up,foronereason oranother, ofunsymmetrical molecules. Certainly acubic crystal, which hasthesymmetry ofacube, cannot be birefringent. Butlong needlelike crystals undoubtedly contain molecules that areasymmetric, andoneobserves thiseffect very easily. Letusseewhat effects wewould expect ifwewere toshine polarized light through aplate ofabirefringent substance. Ifthepolarization isparallel tothe optic axis, thelight willgothrough with onevelocity; ifthepolarization isper- pendicular totheaxis, thelight istransmitted with adifferent velocity. Aninter- esting situation arises when, say,light islinearly polarized at45°totheoptic axis. Now the45°polarization, wehave already noticed, canberepresented asasuper- position ofthex-andthey-polarizations ofequal amplitude andinphase, as shown inFig. 33—2(a). Since thex-andy-polarizations travel with different velocities, their phases change atadifferent rateasthelight passes through the substance. So,although atthestart thex-andy-vibrations areinphase, inside thematerial thephase difference between x-andy-vibrations isproportional tothe depth inthesubstance. Asthelight proceeds through thematerial thepolarization changes asshown intheseries ofdiagrams inFig.33-2. Ifthethickness ofthe plate isjustright tointroduce a90°phase shift between thex-andy-polarizations, asinFig.33-2(c), thelight willcome outcircularly polarized. Such athickness iscalled aquarter-wave plate, because itintroduces aquarter-cycle phase difference between thex-andthey-polarizations. Iflinearly polarized light issentthrough twoquarter-wave plates, itwillcome outplane-polarized again, butatright angles totheoriginal direction, aswecanseefrom Fig.33-2(e). Onecaneasily illustrate thisphenomenon with apiece ofcellophane. Cello- phane ismade oflong, fibrous molecules, andisnotisotropic, since thefibers lie preferentially inacertain direction. Todemonstrate birefringence weneed a 33-3 CELLOPHANE t 0 .1’ . '—-> u. V 1 i 1 \‘\Poi_ARo|u/ Fig. 33-3. Anexperimental demon- stration ofthebirefringence ofcellophane. Theelectric vectors inthelight areindi- cated bythedotted lines. Thepass axes ofthepolaroid sheets andoptic axes of thecellophane areindicated byarrows. Theincident beam isunpolarized.beam oflinearly polarized light, andwecanobtain thisconveniently bypassing unpolarized light through asheet ofpolaroid. Polaroid, which wewilldiscuss later inmore detail, hastheuseful property thatittransmits light thatislinearly polarized parallel totheaxisofthepolaroid with very little absorption, butlight polarized inadirection perpendicular totheaxis ofthepolaroid isstrongly absorbed. When wepass unpolarized light through asheet ofpolaroid, only that partoftheunpolarized beam which isvibrating parallel totheaxisofthepolaroid getsthrough, sothatthetransmitted beam islinearly polarized. Thissame property ofpolaroid isalsouseful indetecting thedirection ofpolarization ofalinearly polarized beam, orindetermining whether abeam islinearly polarized ornot. One simply passes thebeam oflight through thepolaroid sheet androtates the polaroid intheplane normal tothebeam. Ifthebeam islinearly polarized, itwill notbetransmitted through thesheet when theaxisofthepolaroid isnormal to thedirection ofpolarization. Thetransmitted beam isonly slightly attenuated when theaxisofthepolaroid sheet isrotated through 90°. Ifthetransmitted in- tensity isindependent oftheorientation ofthepolaroid, thebeam isnotlinearly polarized. Todemonstrate thebirefringence ofcellophane, weusetwosheets ofpolaroid, asshown inFig.33-3. Thefirstgives usalinearly polarized beam which wepass through thecellophane andthen through thesecond polaroid sheet, which serves todetect anyeffect thecellophane may have hadonthepolarized light passing through it.Ifwefirstsettheaxesofthetwopolaroid sheets perpendicular toeach other andremove thecellophane, nolight willbetransmitted through thesecond polaroid. Ifwenowintroduce thecellophane between thetwopolaroid sheets, and rotate thesheet about thebeam axis, weobserve thatingeneral thecellophane makes itpossible forsome light topass through thesecond polaroid. However, there aretwoorientations ofthecellophane sheet, atright angles toeach other, which permit nolight topass through thesecond polaroid. These orientations in which linearly polarized light istransmitted through thecellophane with no effect onthedirection ofpolarization must bethedirections parallel andper- pendicular totheoptic axisofthecellophane sheet. Wesuppose thatthelight passes through thecellophane with twodifferent velocities inthese twodifferent orientations, butitistransmitted without changing thedirection ofpolarization. When thecellophane isturned halfway between these twoorientations, asshown inFig.33-3, weseethat thelight transmitted through thesecond polaroid isbright. Itjusthappens that ordinary cellophane used incommercial packaging is very close toahalf-wave thickness formost ofthecolors inwhite light. Such a sheet willturn theaxis oflinearly polarized light through 90°iftheincident linearly polarized beam makes anangle of45°with theoptic axis, sothatthebeam emerging from thecellophane isthenvibrating intheright direction topassthrough thesecond polaroid sheet. Ifweusewhite light inourdemonstration, thecellophane sheet willbeofthe proper half-wave thickness only foraparticular component ofthewhite light, andthetransmitted beam willhave thecolor ofthiscomponent. Thecolor trans- mitted depends onthethickness ofthecellophane sheet, andwecanvary the effective thickness ofthecellophane bytilting itsothatthelight passes through the cellophane atanangle, consequently through alonger path inthecellophane. As thesheet istilted thetransmitted color changes. With cellophane ofdifferent thicknesses onecanconstruct filters that willtransmit different colors. These filters have theinteresting property that they transmit onecolor when thetwo polaroid sheets have their axes perpendicular, andthecomplementary color when theaxes ofthetwopolaroid sheets areparallel. Another interesting application ofaligned molecules isquite practical. Certain plastics arecomposed ofvery long andcomplicated molecules alltwisted together. When theplastic issolidified verycarefully, themolecules arealltwisted inamass, sothatthere areasmany aligned inonedirection asanother, andso theplastic isnotparticularly birefringent. Usually there arestrains andstresses introduced when thematerial issolidified, sothematerial isnotperfectly homo- 33-4 geneous. However, ifweapply tension toapiece ofthisplastic material, itisas ifwewere pulling awhole tangle ofstrings, andthere willbemore strings preferen- tially aligned parallel tothetension than inanyother direction. Sowhen astress isapplied tocertain plastics, theybecome birefringent, andonecanseetheeffects ofthebirefringence bypassing polarized light through theplastic. Ifweexamine thetransmitted light through apolaroid sheet, patterns oflight anddark fringes will beobserved (incolor, ifwhite light isused). Thepatterns move asstress isapplied tothesample, andbycounting thefringes andseeing where most ofthem are,one candetermine what thestress is.Engineers usethisphenomenon asameans of finding thestresses inodd-shaped pieces thataredifficult tocalculate. Another interesting example ofawayofobtaining birefringence isbymeans ofaliquid substance. Consider aliquid composed oflong asymmetric molecules which carry aplusorminus average charge near theends ofthemolecule, sothat themolecule isanelectric dipole. Inthecollisions intheliquid themolecules willordinarily berandomly oriented, withasmany molecules pointed inonedirec- tionasinanother. Ifweapply anelectric field themolecules willtend tolineup, andthemoment theylineuptheliquid becomes birefringent. With twopolaroid sheets andatransparent cellcontaining such apolar liquid, wecandevise an arrangement with theproperty that light istransmitted only when theelectric field isapplied. Sowehave anelectrical switch forlight, which iscalled aKerr cell. This effect, thatanelectric field canproduce birefringence incertain liquids, iscalled theKerr eflect. 33-4 Polarizers Sofarwehave considered substances inwhich therefractive index isdifferent forlight polarized indifferent directions. Ofverypractical value arethose crystals andother substances inwhich notonly theindex, butalsothecoeflicient ofab- sorption, isdifferent forlight polarized indifferent directions. Bythesame argu- ments which supported theideaofbirefringence, itisunderstandable thatabsorp- tioncanvary with thedirection inwhich thecharges areforced tovibrate inan anisotropic substance. Tourmaline isanold, famous example andpolaroid is another. Polaroid consists ofathinlayer ofsmall crystals ofherapathite (asalt ofiodine andquinine), allaligned with their axes parallel. These crystals absorb light when theoscillations areinonedirection, andtheydonotabsorb appreciably when theoscillations areintheother direction. Suppose that wesend light into apolaroid sheet polarized linearly atan angle 0tothepassing direction. What intensity willcome through? This incident light canberesolved intoacomponent perpendicular tothepass direction which isproportional tosin0,andacomponent along thepass direction which ispro- portional tocos0.Theamplitude which comes outofthepolaroid isonly the cosine 0part; thesin0component isabsorbed. The amplitude which passes through thepolaroid issmaller than theamplitude which entered, byafactor cos0.Theenergy which passes through thepolaroid, i.e.,theintensity ofthe light, isproportional tothesquare ofcos0.Cos20,then, istheintensity transmitted when thelight enters polarized atanangle 0tothepass direction. Theabsorbed intensity, ofcourse, issin20. Aninteresting paradox ispresented bythefollowing situation. Weknow thatisisnotpossible tosend abeam oflight through twopolaroid sheets with their axes crossed atright angles. Butifweplace athird polaroid sheet between thefirsttwo, with itspassaxisat45°tothecrossed axes, some light istransmitted. Weknow thatpolaroid absorbs light, itdoes notcreate anything. Nevertheless, theaddition ofathird polaroid at45°allows more light togetthrough. The analysis ofthisphenomenon isleftasanexercise forthestudent. One ofthemost interesting examples ofpolarization isnotincomplicated crystals ordiflicult substances, butinoneofthesimplest andmost familiar of situations—the reflection oflight from asurface. Believe itornot,when light is reflected from aglass surface itmay bepolarized, andthephysical explanation of thisisvery simple. Itwasdiscovered empirically byBrewster thatlight reflected 33-5 ‘\ ~._“\‘\ ,..,.. ///////// *;_/////// \.\\I. "1 Fig. 33-4. Reflection oflinearly po- larized light atBrewster's angle. The polarization direction isindicated by dashed arrows; round dots indicate polarization normal tothepaper. .\O Q-—-9"‘< NIQ ‘~-":\‘<I~<mmI V z,+A Fig. 33-5. Amolecule with cishape that isnotsymmetric when reflected ina mirror. Abeam oflight, linearly polarized inthey-direction, falls onthemolecule.from asurface iscompletely polarized ifthereflected beam andthebeam refracted intothematerial form aright angle. Thesituation isillustrated inFig.33-4. If theincident beam ispolarized intheplane ofincidence, there willbenoreflection atall.Only iftheincident beam ispolarized normal totheplane ofincidence will itbereflected. Thereason isvery easy tounderstand. Inthereflecting material thelight ispolarized transversely, andweknow thatitisthemotion ofthecharges inthematerial which generates theemergent beam, which wecallthereflected beam. Thesource ofthisso-called reflected light isnotsimply thattheincident beam isreflected; ourdeeper understanding ofthisphenomenon tellsusthatthe incident beam drives anoscillation ofthecharges inthematerial, which inturn generates thereflected beam. From Fig.33-4 itisclear thatonlyoscillations normal tothepaper canradiate inthedirection ofreflection, andconsequently thereflected beam willbepolarized normal totheplane ofincidence. Iftheincident beam is polarized intheplane ofincidence, there willbenoreflected light. This phenomenon isreadily demonstrated byreflecting alinearly polarized beam from aflatpiece ofglass. Iftheglass isturned topresent diflerent angles of incidence tothepolarized beam, sharp attenuation ofthereflected intensity is observed when theangle ofincidence passes through Brewster’s angle. This attenuation isobserved onlyiftheplane ofpolarization liesintheplane ofincidence. Iftheplane ofpolarization isnormal totheplane ofincidence, theusual reflected intensity isobserved atallangles. 33-5 Optical activity Another most remarkable effect ofpolarization isobserved inmaterials composed ofmolecules which donothave reflection symmetry: molecules shaped something likeacorkscrew, orlikeagloved hand, oranyshape which, ifviewed through amirror, would bereversed inthesame waythataleft-hand glove reflects asaright-hand glove. Suppose allofthemolecules inthesubstance arethesame, i.e.,none isamirror image ofanyother. Suchia substance mayshow aninteresting effect called optical activity, whereby aslinearly polarized light passes through thesubstance, thedirection ofpolarization rotates about thebeam axis. Tounderstand thephenomenon ofoptical activity requires some calculation, butwecanseequalitatively how theeffect might come about, without actually carrying outthecalculations. Consider anasymmetric molecule intheshape of aspiral, asshown inFig.33-5. Molecules need notactually beshaped likea corkscrew inorder toexhibit optical activity, butthisisasimple shape which we shall take asatypical example ofthose that donothave reflection symmetry. When alight beam linearly polarized along they-direction fallsonthismolecule, theelectric field willdrive charges upanddown thehelix, thereby generating a current inthey-direction andradiating anelectric fieldEl,polarized inthey-direc- tion. However, iftheelectrons areconstrained tomove along thespiral, they must alsomove inthex-direction astheyaredriven upanddown. When acurrent isflowing upthespiral, itisalsoflowing intothepaper atz=21andoutofthe paper atz=21+A,ifAisthediameter ofourmolecular spiral. One might suppose thatthecurrent inthex-direction would produce nonetradiation, since thecurrents areinopposite directions onopposite sides ofthespiral. However, ifweconsider thex-components oftheelectric field arriving atz=22,wesee thatthefield radiated bythecurrent atz=21+Aandthefield radiated from z=z1arrive atz2separated intime bytheamount A/c, andthus separated in phase by1r+wA/c. Since thephase difference isnotexactly -rr,thetwofields donotcancel exactly, andweareleftwith asmall x-component intheelectric field generated bythemotion oftheelectrons inthemolecule, whereas thedriving electric fieldhadonly ay-component. This small x-component, added tothelarge y-component, produces aresultant field thatistilted slightly with respect tothe y-axis, theoriginal direction ofpolarization. Asthelight moves through the material, thedirection ofpolarization rotates about thebeam axis. Bydrawing a fewexamples andconsidering thecurrents thatwillbesetinmotion byanincident 33-6 electric field, onecanconvince himself thattheexistence ofoptical activity andthe signoftherotation areindependent oftheorientation ofthemolecules. Corn syrup isacommon substance which possesses optical activity. The phenomenon iseasily demonstrated with apolaroid sheet toproduce alinearly polarized beam, atransmission cellcontaining corn syrup, andasecond polaroid sheet todetect therotation ofthedirection ofpolarization asthelight passes through thecorn syrup. 33-6 Theintensity ofreflected light Letusnow consider quantitatively thereflection coeflicient asafunction of angle. Figure 33-6(a) shows abeam oflight striking aglass surface, where itis partly reflected andpartly refracted intotheglass. Letussuppose thattheincident beam, ofunitamplitude, islinearly polarized normal totheplane ofthepaper. Wewillcalltheamplitude ofthereflected wave b,andtheamplitude ofthere- fracted wave a.Therefracted andreflected waves will, ofcourse, belinearly polarized, andtheelectric field vectors oftheincident, reflected, andrefracted waves areallparallel toeach other. Figure 33—6(b) shows thesame situation, but nowwesuppose thattheincident wave, ofunitamplitude, ispolarized intheplane ofthepaper. Now letuscalltheamplitude ofthereflected andrefracted wave BandA,respectively. Wewish tocalculate how strong thereflection isinthetwosituations illus- trated inFig.33-6(a) and33—6(b). Wealready know thatwhen theangle between thereflected beam andrefracted beam isaright angle, there willbenoreflected wave inFig.33—6(b), butletusseeifwecannot getaquantitative answer—an exact formula forBandbasafunction oftheangle ofincidence, i. Theprinciple thatwemust understand isasfollows. Thecurrents thatare generated intheglass produce twowaves. First, theyproduce thereflected wave. Moreover, weknow thatifthere were nocurrents generated intheglass, thein- cident wave would continue straight intotheglass. Remember thatallthesources intheworld make thenetfield. Thesource oftheincident light beam produces a field ofunitamplitude which would move intotheglass along thedotted linein thefigure. This field isnotobserved, andtherefore thecurrents generated inthe glass must produce afield ofamplitude —lwhich moves along thedotted line. Using thisfact, wewillcalculate theamplitude oftherefracted waves, aandA. InFig.33—6(a) weseethatthefield ofamplitude bisradiated bythemotion ofcharges inside theglass which areresponding toafield ainside theglass, and thattherefore bisproportional toa.Wemight suppose thatsince ourtwofigures areexactly thesame, except forthedirection ofpolarization, theratio B/A would bethesame astheratio b/a.Thisisnotquite true, however, because inFig.33—6(b) thepolarization directions arenotallparallel toeach other, asthey areinFig. 33-6(a). Itisonly thecomponent ofAwhich isperpendicular toB,Acos(i+r), which iseffective inproducing B.Thecorrect expression fortheproportionality isthen b B 2" <3“) Now weuseatrick. Weknow thatinboth (a)and(b)ofFig.33-6 theelectric field intheglass must produce oscillations ofthecharges which generate afield of amplitude —l,polarized parallel totheincident beam, andmoving inthedirection ofthedotted line. Butweseefrom part(b)ofthefigure thatonly thecomponent ofAthatisnormal tothedashed linehastheright polarization toproduce this field, whereas inFig.33-6(a) thefullamplitude aiseffective, since thepolarization ofwave aisparallel tothepolarization ofthewave ofamplitude —1.Therefore wecanwrite Acos (i—r)__—_l———-id __l, (33.2) since thetwoamplitudes ontheleftsideofEq.(33.2) each produce thewave of amplitude —1. 33-7b -I 5 / , I"/ // K 1/ '1 Aa/ r ilri‘ —'_ i I Glass 1i~ Glass (.) (b) Fig. 33-6. Anincident wave ofunit amplitude isreflected andrefracted ata glass surface. Inla)theincident wave is linearly polarized normal totheplane of thepaper. Inlb)theincident wave is linearly polarized inthedirection shown bythedotted electric vector. Dividing Eq.(33.1) byEq.(33.2), weobtain B_cos(i+r) ( bcos(i-r)’ aresult which wecancheck against what wealready know. Ifweset(i+r)= 90°,Eq.(33.3) gives B=0,asBrewster saysitshould be,soourresults sofar areatleast notobviously wrong. Wehaveassumed unitamplitudes fortheincident waves, sothat |B|2/12 is thereflection coefficient forwaves polarized intheplane ofincidence, and|b|2/l2 isthereflection coefficient forwaves polarized normal totheplane ofincidence. Theratio ofthese tworeflection coeflicients isdetermined byEq.(33.3). Now weperform amiracle, andcompute notjusttheratio, buteach coefficient [Bl2andlbl2individually! Weknow from theconservation ofenergy thattheenergy intherefracted wave must beequal totheincident energy minus theenergy in thereflected wave, l—|B|2inonecase, 1—|b|2intheother. Furthermore, the energy which passes intotheglass inFig.33—6(b) istotheenergy which passes intotheglass inFig.33-6(a) astheratio ofthesquares oftherefracted amplitudes, |Al2/[a[2. One might askwhether wereally know how tocompute theenergy inside theglass, because, after all,there areenergies ofmotion oftheatoms in addition totheenergy intheelectric field. Butitisobvious thatallofthevarious contributions tothetotal energy willbeproportional tothesquare oftheamplitude oftheelectric field. Therefore wecanwrite33.3) 1-or_W _ ‘ ' Wenow substitute Eq.(33.2) toeliminate A/a from theexpression above, andexpress Binterms ofbbymeans ofEq.(33.3): 2cos2(i+r) 1_lblcos? (i—r)_ l 1-|b|2 _cos?(1-r)' (335) Thisequation contains only oneunknown amplitude, b.Solving forlb]2,weobtain 2 SIII2 —I‘) [bl- (33.6) and, with theaidof(33.3), 2_tan2 (i—r) lBl—tan? (i+r). (337) Sowehave found thereflection coeflicient |b]2foranincident wave polarized perpendicular totheplane ofincidence, andalsothereflection coeflicient |Bj2for anincident wave polarized intheplane ofincidence! Itispossible togoonwith arguments ofthisnature anddeduce thatbisreal. Toprove this,onemust consider acasewhere light iscoming from both sides of theglass surface atthesame time, asituation noteasy toarrange experimentally, butfuntoanalyze theoretically. Ifweanalyze thisgeneral case, wecanprove that bmust bereal, andtherefore, infact, thatb==i=sin(i—r)/sin (i+r).Itis even possible todetermine thesignbyconsidering thecase ofavery, very thin layer inwhich there isreflection from thefront andfrom theback surfaces, and calculating how much light isreflected. Weknow how much light should be reflected byathinlayer, because weknow howmuch current isgenerated, andwe have even worked outthefields produced bysuch currents. Onecanshow bythese arguments that '(i—r) tan(i—r)b=-2-, B=—-,?- 33.8sin(1+r) tan(1—l—r) () These expressions forthereflection coefficients asafunction oftheangles of incidence andrefraction arecalled Fresnel’s reflection formulas. Ifweconsider thelimit astheangles iandrgotozero, wefind, forthecaseof normal incidence, that B2zb2~(i—r)2/(i +r)2forboth polarizations, 33-8 since thesines arepractically equal totheangles, asarealsothetangents. Butwe know thatsini/sin r=n,andwhen theangles aresmall, i/r¢=n.Itisthuseasy toshow thatthecoefficient ofreflection fornormal incidence is _12B2 b2 ('1 ) == ' Itisinteresting tofindouthow much light isreflected atnormal incidence from thesurface ofwater, forexample. Forwater, nis4/3,sothatthereflection coefficient is(l/7)? ~2%. Atnormal incidence, only twopercent ofthelight isreflected from thesurface ofwater. 33-7 Anomalous refraction Thelastpolarization effect weshall consider wasactually oneofthefirst tobediscovered: anomalous refraction. Sailors visiting Iceland brought back to Europe crystals ofIceland spar(CaCO 3)which hadtheamusing property ofmak- inganything seen through thecrystal appear doubled, i.e.,astwoimages. This came totheattention ofHuygens, andplayed animportant roleinthediscovery ofpolarization. Asisoften thecase, thephenomena which arediscovered firstare thehardest, ultimately, toexplain. Itisonlyafter weunderstand aphysical concept thoroughly thatwecancarefully select those phenomena which most clearly and simply demonstrate theconcept. Anomalous refraction isaparticular case ofthesame birefringence thatwe considered earlier. Anomalous refraction comes about when theoptic axis, the long axisofourasymmetric molecules, isnotparallel tothesurface ofthecrystal. InFig.33-7 aredrawn twopieces ofbirefringent material, with theoptic axisas shown. Intheupper figure, theincident beam falling onthematerial islinearly polarized inadirection perpendicular totheoptic axisofthematerial. When this beam strikes thesurface ofthematerial, each point onthesurface actsasasource ofawave which travels intothecrystal with velocity 11L,thevelocity oflight in thecrystal when theplane ofpolarization isnormal totheoptic axis. Thewave- front isjusttheenvelope orlocus ofallthese little spherical waves, andthiswave- front moves straight through thecrystal andouttheother side. This isjustthe ordinary behavior wewould expect, andthisrayiscalled theordinary ray. Inthelower figure thelinearly polarized light falling onthecrystal hasits direction ofpolarization turned through 90°,sothattheoptic axisliesintheplane ofpolarization. When wenow consider thelittle waves originating atanypoint onthesurface ofthecrystal, weseethattheydonotspread outasspherical waves. Light travelling along theoptic axistravels with velocity vibecause thepolariza- tionisperpendicular totheoptic axis, whereas thelight travelling perpendicular totheoptic axistravels with velocity vubecause thepolarization isparallel tothe optic axis. Inabirefringent material 0“;évi,andinthefigure an<vi.Amore complete analysis willshow thatthewaves spread outonthesurface ofanellipsoid, withtheoptic axisasmajor axisoftheellipsoid. Theenvelope ofallthese elliptical waves isthewavefront which proceeds through thecrystal inthedirection shown. Again, attheback surface thebeam willbedeflected just asitwasatthefront surface, sothatthelight emerges parallel totheincident beam, butdisplaced from it.Clearly, thisbeam does notfollow Snell’s law, butgoes inanextraordinary direction. Itistherefore called theextraordinary ray. When anunpolarized beam strikes ananomalously refracting crystal, itis separated intoanordinary ray,which travels straight through inthenormal man- ner,andanextraordinary raywhich isdisplaced asitpasses through thecrystal. These twoemergent raysarelinearly polarized atright angles toeach other. That thisistruecanbereadily demonstrated with asheet ofpolaroid toanalyze the polarization oftheemergent rays. Wecanalsodemonstrate thatourinterpretation ofthisphenomenon iscorrect bysending linearly polarized light intothecrystal. Byproperly orienting thedirection ofpolarization oftheincident beam, wecan make thislight gostraight through without splitting, orwecanmake itgothrough without splitting butwith adisplacement. 33-9wave front I I I 0 > E x optic .\..... Q-oqmwsvefrout / 43.8.. _ Fig. 33-7. Theupper diagram shows thepath oftheordinary raythrough a doubly refracting crystal. The extraor- dinary ray isshown inthelower dia- gram. Theoptic axis liesintheplane of thepaper. \ / Fig.33-8. Two oppositely rotating vectorsof equal amplitude addtoproduce avector inafixed direction, butwith an oscillating amplitude. y E “". Fig.33-9. Acharge moving ina circle inresponse tocircularly polarized light.Wehave represented allthevarious polarization cases inFigs. 33-1 and33-2 assuperpositions oftwospecial polarization cases, namely xandyinvarious amounts andphases. Other pairs could equally wellhave been used. Polarization along anytwoperpendicular axes x’,y’inclined toxandywould serve aswell [forexample, anypolarization canbemade upofsuperpositions ofcases (a)and (e)ofFig.33-2]. Itisinteresting, however, thatthisideacanbeextended toother cases also. Forexample, anylinear polarization canbemade upbysuperposing suitable amounts atsuitable phases ofright andleftcircular polarizations [cases (c)and(g)ofFig.33-2], since twoequal vectors rotating inopposite directions addtogiveasingle vector oscillating inastraight line(Fig. 33-8). Ifthephase of oneisshifted relative totheother, thelineisinclined. Thus allthepictures of Fig.33-1 could belabeled “the superposition ofequal amounts ofright andleft circularly polarized light atvarious relative phases.” Astheleftslips behind the right inphase, thedirection ofthelinear polarization changes. Therefore optically active materials are,inasense, birefringent. Their properties canbedescribed by saying thattheyhave diflerent indexes forright- andleft-hand circularly polarized light. Superposition ofright andleftcircularly polarized light ofdifferent intensi- tiesproduces elliptically polarized light. Circularly polarized light hasanother interesting property-it carries angular momentum (about thedirection ofpropagation). Toillustrate this, suppose that such light fallsonanatom represented byaharmonic oscillator thatcanbedis- placed equally wellinanydirection intheplane xy.Then thex-displacement of theelectron willrespond totheE,component ofthefield, while they-component responds, equally, totheequal E,component ofthefield but90°behind inphase. That is,theresponding electron goes around inacircle, with angular velocity w, inresponse totherotating electric field ofthelight (Fig. 33-9). Depending on thedamping characteristics oftheresponse oftheoscillator, thedirection ofthe displacement aoftheelectron, andthedirection oftheforce q,Eonitneed notbe thesame butthey rotate around together. TheEmayhave acomponent atright angles toa,sowork isdone onthesystem andatorque -risexerted. Thework done persecond isrw.Over aperiod oftime Ttheenergy absorbed isra>T, while 1Tis theangular momentum delivered tothematter absorbing theenergy. Wesee therefore thatabeam ofright circularly polarized light containing atotal energy 8carries anangular momentum (with vector directed along thedirection ofprop- agation) 8/co. Forwhen thisbeam isabsorbed that angular momentum isde- livered totheabsorber. Left-hand circular light carries angular momentum ofthe opposite sign, -8/w. 33-10 34 Relativistic Effects inRadiation 34-1 Moving sources Inthepresent chapter weshall describe anumber ofmiscellaneous effects in connection with radiation, andthen weshall befinished with theclassical theory oflight propagation. Inouranalysis oflight, wehave gone rather farandinto considerable detail. The only phenomena ofanyconsequence associated with electromagnetic radiation thatwehave notdiscussed iswhat happens ifradiowaves arecontained inaboxwith reflecting walls, thesizeoftheboxbeing comparable toawavelength, oraretransmitted down alongtube. Thephenomena ofso-called cavity resonators and waveguides weshall discuss later; weshall first useanother physical example—sound—and then Weshall return tothissubject. Except for this, thepresent chapter isourlastconsideration oftheclassical theory oflight. Wecansummarize alltheeffects thatweshall nowdiscuss byremarking that they have todowith theeffects ofmoving sources. Wenolonger assume thatthe source islocalized, with allitsmotion being atarelatively lowspeed near afixed point. Werecall that thefundamental laws ofelectrodynamics saythat, atlarge distances from amoving charge, theelectric field isgiven bytheformula ___Qd2ere'_ E_ 41re0c2 dt2 (34'l) Thesecond derivative oftheunitvector eRIwhich points intheapparent direction ofthecharge, isthedetermining feature oftheelectric field. This unitvector does notpoint toward thepresent position ofthecharge, ofcourse, butrather inthe direction thatthecharge would seem tobe,iftheinformation travels only atthe finite speed cfrom thecharge totheobserver. Associated with theelectric field isamagnetic field, always atright angles totheelectric field andatright angles totheapparent direction ofthesource, given bytheformula B=—eR' XE/c. (34.2) Until now wehave considered only thecase inwhich motions arenonrela- tivistic inspeed, sothatthere isnoappreciable motion inthedirection ofthesource tobeconsidered. Now weshall bemore general andstudy thecasewhere themo- tionisatanarbitrary velocity, andseewhat different effects may beexpected in those circumstances. Weshall letthemotion beatanarbitrary speed, butofcourse weshall stillassume thatthedetector isvery farfrom thesource. Wealready know from ourdiscussion inChapter 28that theonly things that count ind2eR»/dt2 arethechanges inthedirection ofeg». Letthecoor- dinates ofthecharge be(x,y,z),with zmeasured along thedirection ofobserva- tion(Fig. 34-1). Atagiven moment intime, saythemoment 1',thethree compo- nents oftheposition arex('r), y(~r), andz(1'). Thedistance Risvery nearly equal toR(T) =R0+2(1). Now thedirection ofthevector eR1depends mainly on xandy,buthardly atallupon z:thetransverse components oftheunitvector are x/R andy/R, andwhen wedifferentiate these components wegetthings like R2inthedenominator: ¢i(x/R) _dx/dt _di_2c_. dz_R dzR2 34-134-1 Moving sources 34-2 Finding the“apparent” motion 34-3 Synchrotron radiation 34-4 Cosmic synchrotron radiation 34-5 Bremsstrahlung 34-6 TheDoppler effect 34-7 Thew,kfour-vector 34-8 Aberration 34-9 Themomentum oflight ’Q3T 5‘, zm re “““ ‘‘AIR0 0 1 Fig. 34—l. The path of0moving charge. The true position atthetime 1'isctT,buttheretarded position isatA. I'll)So,when wearefarenough away theonlyterms wehavetoworry about arethe variations ofxandy.Thus wetakeoutthefactor R0andget E=---_q E” 41re0c2R0 dtz’ 2I 13,,=-in %, (34.3) where R0isthedistance, more orless,toq;letustakeitasthedistance OPtothe origin ofthecoordinates (x,y,z).Thus theelectric field isaconstant multiplied byavery simple thing, thesecond derivatives ofthex-andy-coordinates. (We could putitmore mathematically bycalling xandythetransverse components of theposition vector rofthecharge, butthiswould notaddtotheclarity.) Ofcourse, werealize thatthecoordinates must bemeasured attheretarded time. Here wefindthatz(1')doesafiect theretardation. What time istheretarded time? Ifthetime ofobservation iscalled t(thetime atP)then thetimertowhich thiscorresponds atAisnotthetime t,butisdelayed bythetotal distance thatthe light hastogo,divided bythespeed oflight. Inthefirstapproximation, thisdelay isR0/c, aconstant (anuninteresting feature), butinthenext approximation we must include theefi"ects oftheposition inthez-direction atthetime 1,because ifqisalittle farther back, there isalittle more retardation. This isanefiect that wehave neglected before, anditistheonly change needed inorder tomake our results valid forallspeeds. What wemust nowdoistochoose acertain value oftandcalculate thevalue of1'from it,andthus findoutwhere xandyareatthat-r.These arethen the retarded xandy,which wecallx’andy’,whose second derivatives determine the field. Thus 1'isdetermined by I=1+%+and x’(l)=x(t), J/(I)=y(T)- (34-4) Now these arecomplicated equations, butitiseasy enough tomake ageometrical picture todescribe their solution. Thispicture willgiveusagood qualitative feeling forhow things work, butitstilltakes alotofdetailed mathematics todeduce theprecise results ofacomplicated problem. lift ,, ,,_\I. i /r 5 \._/ at O a - - - ToOBSERVER ° Fig. 34-2. Ageometrical solution of Eq.(34.5) tofindx'(t). 34-2 Finding the“apparent” motion Theabove equation hasaninteresting simplification. Ifwedisregard theun- interesting constant delay R0/c, which justmeans thatwemust change theorigin oftbyaconstant, then itsaysthat ct=01'+z(1'), x’=x('r), y’=y(1). (34.5) Now weneed tofindx’andy’asfunctions oft,not-r,andwecandothisinthe following way: Eq.(34.5) saysthatweshould take theactual motion andadda constant (thespeed oflight) times 7.What thatturns outtomean isshown in Fig.34-2. Wetake theactual motion ofthecharge (shown atleft)andimagine thatasitisgoing around itisbeing swept away from thepoint Patthespeed c (there arenocontractions fromrelativity oranything likethat;thisisjustamathe- matical addition ofthec-r). Inthiswaywegetanewmotion, inwhich theline- 34-2 of-sight coordinate isct,asshown attheright. (The figure shows theresult fora rather complicated motion inaplane, butofcourse themotion may notbein oneplane—it maybeeven more complicated than motion inaplane.) Thepoint isthatthehorizontal (i.e., line-of-sight) distance now isnolonger theoldz,but isz+cr,andtherefore isct.Thus wehave found apicture ofthecurve, x’ (and y’)against t!Allwehave todotofindthefieldistolook attheacceleration ofthiscurve, i.e.,todifferentiate ittwice. Sothefinal answer is:inorder tofind theelectric field foramoving charge, take themotion ofthecharge andtranslate itback atthespeed cto“open itout”; then thecurve, sodrawn, isacurve ofthe x’andy’positions ofthefunction oft.Theacceleration ofthiscurve gives the electric field asafunction oft.Or,ifwewish, wecannowimagine thatthiswhole “rigid” curve moves forward atthespeed cthrough theplane ofsight, sothatthe point ofintersection with theplane ofsight hasthecoordinates x’andy’.The acceleration ofthispoint makes theelectric field. This solution isjustasexact as theformula westarted with—it issimply ageometrical representation. x x’(t) Fig. 34-3. The x’(t) curve for a particle moving atconstant speed v= O.94c, acircle. 3O Ifthemotion isrelatively slow, forinstance ifwehave anoscillator justgoing upanddown slowly, then when weshoot thatmotion away atthespeed oflight, wewould get,ofcourse, asimple cosine curve, andthatgives aformula wehave been looking atforalong time: itgives thefieldproduced byanoscillating charge. Amore interesting example isanelectron moving rapidly, very nearly atthe speed oflight, inacircle. Ifwelook intheplane ofthecircle, theretarded x’(t) appears asshown inFig.34-3. What isthiscurve‘? IfWeimagine aradius vector from thecenter ofthecircle tothecharge, andifweextend thisradial linealittle bitpastthecharge, justashade ifitisgoing fast,then wecome toapoint onthe linethatgoes atthespeed oflight. Therefore, when wetranslate themotion back atthespeed oflight, thatcorresponds tohaving awheel with acharge onitrolling backward (without slipping) atthespeed c;thus wefindacurve which isvery close toacycloid—-it iscalled ahypocycloid. Ifthecharge isgoing very nearly at thespeed oflight, the“cusps” areverysharp indeed; ifitwent atexactly thespeed oflight, they would beactual cusps, infinitely sharp. “Infinitely sharp” isinter- esting; itmeans thatnear acusp thesecond derivative isenormous. Once ineach cycle wegetasharp pulse ofelectric field. Thisisnotatallwhat wewould getfrom anonrelativistic motion, where each time thecharge goes around there isan oscillation which isofabout thesame “strength” allthetime. Instead, there are very sharp pulses ofelectric field spaced attime intervals 1/To apart, where To istheperiod ofrevolution. These strong electric fields areemitted inanarrow cone inthedirection ofmotion ofthecharge. When thecharge ismoving away from P,there isvery little curvature and there isvery little radiated field inthe direction ofP. 34-3 Synchrotron radiation Wehave very fastelectrons moving incircular paths inthesynchrotron; they aretravelling atverynearly thespeed c,anditispossible toseetheabove radiation asactual light! Letusdiscuss thisinmore detail. Inthesynchrotron wehave electrons which goaround incircles inauniform magnetic field. First, letusseewhytheygoincircles. From Eq.(12.10), weknow thattheforce onaparticle inamagnetic field isgiven by F=qv><B, (34.6) 34-3Cl .. __. 0'49 C _. D \R FAn ‘\\\ a--- A _ --—_------>0 F B Fig. 34-4. Acharged particle moves inacircular (orhelical) path inauniform magnetic field.anditisatright angles both tothefield andtothevelocity. Asusual, theforce is equal totherateofchange ofmomentum with time. Ifthefieldisdirected upward outofthepaper, themomentum oftheparticle andtheforce onitareasshown inFig.34-4. Since theforce isatright angles tothevelocity, thekinetic energy, andtherefore thespeed, remains constant. Allthemagnetic fielddoes istochange thedirection ofmotion. Inashort time At,themomentum vector changes atright angles toitself byanamount Ap=FAt, andtherefore pturns through anangle A0=Ap/p =qvBAt/p, since [FI=qt/B. Butinthissame time theparticle has gone adistance As=vAt.Evidently, thetwolines ABandCDwillintersect ata point 0such that0A=0C=R,where As=RA0.Combining thiswith the previous expressions, wefindRA6/At =Rw=v=qvBR/p, from which wefind p=qBR (34.7) and w=qvB/p. (34.8) Since thissame argument canbeapplied during thenext instant, thenext, andso on,weconclude thattheparticle must bemoving inacircle ofradius R,withangu- larvelocity w. Theresult thatthemomentum oftheparticle isequal toacharge times the radius times themagnetic field isavery important lawthatisused agreat deal. Itisimportant forpractical purposes because ifwehave elementary particles which allhave thesame charge andweobserve them inamagnetic field, wecanmeasure theradii ofcurvature oftheir orbits and, knowing themagnetic field, thus deter- mine themomenta oftheparticles. Ifwemultiply both sides ofEq.(34.7) byc, andexpress qinterms oftheelectronic charge, wecanmeasure themomentum in units oftheelectron volt. Inthose units ourformula is pc(ev) =3Xl08(q/qe)BR, (34.9) where B,R,andthespeed oflight areallexpressed inthemks system, thelatter being 3X108,numerically. Themksunitofmagnetic field iscalled aweber persquare meter. There isan older unitwhich isstillincommon use,called agauss. Oneweber/m_2 isequal to104gauss. Togiveanideaofhowbigmagnetic fields are,thestrongest magnetic fieldthatonecanusually make inironisabout 1.5X104gauss; beyond that, the advantage ofusing iron disappears. Today, electromagnets wound with super- conducting wire areabletoproduce steady fields ofover 105gauss strength—that is,l0mksunits. Thefield oftheearth isafewtenths ofagauss attheequator. Returning toEq.(34.9), wecould imagine thesynchrotron running atabillion electron volts, sopcwould be109forabillion electron volts. (Weshall come back totheenergy injustamoment.) Then, ifwehadaBcorresponding to,say,10,000 gauss, which isagood substantial field, onemksunit, then weseethatRwould have tobe3.3meters. Theactual radius oftheCaltech synchrotron is3.7meters, thefield isalittle bigger, andtheenergy is1.5billion, butitisthesame idea. So now wehave afeeling forwhythesynchrotron hasthesizeithas. Wehave calculated themomentum, butweknow that thetotal energy, including therestenergy, isgiven byW=\/p202 +m2c4, andforanelectron therestenergy corresponding tomczis0.511 X106ev,sowhen pcis109evwe canneglect mc2, andsoforallpractical purposes W=pcwhen thespeeds are relativistic. Itispractically thesame tosaytheenergy ofanelectron isabillion electron volts astosaythemomentum times cisabillion electron volts. IfW= 109ev,itiseasy toshow thatthespeed differs from thespeed oflight bybutone partineight million! Weturn now totheradiation emitted bysuch aparticle. Aparticle moving onacircle ofradius 3.3meters, or20meters circumference, goes around once in roughly thetime ittakes light togo20meters. Sothewavelength thatshould be emitted bysuch aparticle would be20meters—in theshortwave radio region. Butbecause ofthepiling upefi"ect thatwehave been discussing (Fig. 34-3), and because thedistance bywhich wemust extend theradius toreach thespeed cis 34-4 only onepart ineight million oftheradius, thecusps ofthehypocycloid are enormously sharp compared with thedistance between them. The acceleration, which involves asecond derivative withrespect totime, getstwice the“compression factor" of8X10“because thetime scale isreduced byeight million twice inthe neighborhood ofthecusp. Thus wemight expect theeffective wavelength tobe much shorter, totheextent of64times 1012 smaller than 20meters, and that corresponds tothex-ray region. (Actually, thecusp itself isnottheentire determining factor; onemust alsoinclude acertain region about thecusp. This changes thefactor tothe3/2power instead ofthesquare, butstillleaves usabove theoptical region.) Thus, even though aslowly moving electron would have radiated 20-meter radiowaves, therelativistic effect cutsdown thewavelength so much thatwecanseeit!Clearly, thelight should bepolarized, with theelectric fieldperpendicular totheuniform magnetic field. Tofurther appreciate what wewould observe, suppose thatwewere totake such light (tosimplify things, because these pulses aresofarapart intime, we shall justtake onepulse) anddirect itonto adiffraction grating, which isalotof scattering wires. After thispulse comes away from thegrating, what dowesee? (Weshould seeredlight, blue light, andsoon,ifweseeanylight atall.) What dowesee? Thepulse strikes thegrating head-on, andalltheoscillators inthe grating, together, areviolently moved upandthen back down again, justonce. They then produce elTects invarious directions, asshown inFig. 34-5. Butthe point Piscloser tooneendofthegrating than totheother, soatthispoint the electric field arrives first from wire A,next from B,and soon;finally, thepulse from thelastwirearrives. Inshort, thesumofthe reflections from allthesuccessive wires isasshown inFig. 34—6(a); itisanelectric field which isaseries ofpulses, anditisverylikeasinewave whose wavelength isthedistance between thepulses, justasitwould beformonochromatic light striking thegrating! So,wegetcolored light allright. But,bythesame argument, willwenotgetlight from anykind ofa “pulse”? No. Suppose that thecurve were much smoother; then wewould add allthescattered waves together, separated byasmall time between them (Fig. 34—6b). Then weseethatthefieldwould notshake atall,itwould beaverysmooth curve, because each pulse does notvary much inthetime interval between pulses. The electromagnetic radiation emitted byrelativistic charged particles cir- culating inamagnetic fieldiscalled synchrotron radiation. Itissonamed forobvi- ousreasons, butitisnotlimited specifically tosynchrotrons, oreven toearthbound laboratories. ltisexciting andinteresting thatitalsooccurs innature!Z0 ‘Pulse from electron a0Q B- A‘ /Radiation scattered bygrating 2 P BA Fig. 34-5. The light which strikes a grating asasingle, sharp pulse isscat- tered invarious directions asdifferent colors. lllllltlt W/we(<1) (b) Fig. 34—6. Thetotal electric field due toaseries of(a)sharp pulses and (b) smooth pulses. Fig. 34—7. Thecrab nebula asseen inallcolors (nofilter). 34-5 la) (bl Fig. 34-8. The crab nebula asseen through ablue filter and apolaroid. (a)Electric vector vertical. (b)Electric vector horizontal. X B A A’B’¢_ ____._ i D1 D ct to) lb) Fig.34-9. Afast electron passing near anucleus radiates energy inthe direction ofitsmotion.34-4 Cosmic synchrotron radiation Intheyear 1054 theChinese andJapanese civilizations were among themost advanced intheworld; theywere conscious oftheexternal universe, andthey recorded, most remarkably, anexplosive bright starinthatyear. (Itisamazing thatnone oftheEuropean monks, writing allthebooks ofthemiddle ages, even bothered towrite thatastarexploded inthesky,buttheydidnot.) Today wemay takeapicture ofthatstar, andwhat weseeisshown inFig.34-7. Ontheoutside isabigmass ofredfilaments, which isproduced bytheatoms ofthethingas “ringing” attheir natural frequencies; thismakes abright linespectrum with different frequencies init.Theredhappens inthiscasetobeduetonitrogen. On theother hand, inthecentral region isamysterious, fuzzy patch oflight ina continuous distribution offrequency, i.e.,there arenospecial frequencies associated withparticular atoms. Yetthisisnotdust “litup”bynearby stars, which isone waybywhich onecangetacontinuous spectrum. Wecanseestars through it,so itistransparent, butitisemitting light. InFig.34-8 welook atthesame object, using light inaregion ofthespectrum which hasnobright spectral line,sothatweseeonlythecentral region. Butinthis case, also, polarizers have been putonthetelescope, andthetwoviews correspond totwoorientations 90°apart. Weseethatthepictures arediflerent! That istosay, thelight ispolarized. Thereason, presumably, isthatthere isalocal magnetic field, andmany veryenergetic electrons aregoing around inthatmagnetic field. Wehavejustillustrated howtheelectrons could goaround thefieldinacircle. Wecanaddtothis,ofcourse, anyuniform motion inthedirection ofthefield, since theforce, qvXB,hasnocomponent inthisdirection and, aswehave already remarked, thesynchrotron radiation isevidently polarized inadirection atright angles totheprojection ofthemagnetic field onto theplane ofsight. Putting these twofacts together, weseethatinaregion where onepicture is bright andtheother oneisblack, thelight must have itselectric fieldcompletely polarized inonedirection. This means thatthere isamagnetic fieldatright angles tothisdirection, while inother regions, where there isastrong emission intheother picture, themagnetic fieldmust betheother way. Ifwelookcarefully atFig.34-8, wemaynotice thatthere is,roughly speaking, ageneral setof“lines” thatgoone wayinonepicture andatright angles tothisintheother. Thepictures show akind offibrous structure. Presumably, themagnetic fieldlines willtend toextend rela- tively long distances intheir own direction, andso,presumably, there arelong regions ofmagnetic fieldwith alltheelectrons spiralling oneway, while inanother region thefield istheother wayandtheelectrons arealsospiralling thatway. What keeps theelectron energy sohighforsolongatime? After all,itis900 years since theexplosion—how canthey keep going sofast? How theymaintain theirenergy andhowthiswhole thing keeps going isstillnotthoroughly understood. 34-5 Bremsstrahlung Weshall next remark briefly ononeother interesting eflect ofavery fast- moving particle thatradiates energy. Theideaisvery similar totheonewehave justdiscussed. Suppose thatthere arecharged particles inapiece ofmatter and avery fastelectron, say,comes by(Fig. 34-9). Then, because oftheelectric field around theatomic nucleus theelectron ispulled, accelerated, sothatthecurve ofitsmotion hasaslight kink orbend init.Iftheelectron istravelling atvery nearly thespeed oflight, what istheelectric field produced inthedirection c? Remember ourrule: wetaketheactual motion, translate itbackwards atspeed c, andthatgives usacurve whose curvature measures theelectric field. Itwascoming toward usatthespeed v,sowegetabackward motion, with thewhole picture compressed intoasmaller distance inproportion asc-vissmaller than c.So, ifl—v/c<<1,there isaverysharp andrapid curvature atB’,andwhen wetake thesecond derivative ofthatwegetaveryhighfieldinthedirection ofthemotion. Sowhen very energetic electrons move through matter they spitradiation inafor- ward direction. This iscalled bremsstrahlung. Asamatter offact,thesynchrotron 34-6 isused, notsomuch tomake high-energy electrons (actually ifwecould getthem outofthemachine more conveniently wewould notsaythis) astomake very energetic photons-gamma rays—by passing theenergetic electrons through asolid tungsten “target,” andletting them radiate photons from thisbremsstrahlung effect. 34-6 TheDoppler effect Now wegoontoconsider some other examples oftheeffects ofmoving sources. Letussuppose thatthesource isastationary atom which isoscillating atoneof itsnatural frequencies, (.00.Then weknow thatthefrequency ofthelight wewould observe iswo.Butnow letustake another example, inwhich wehave asimilar oscillator oscillating with afrequency wl,andatthesame time thewhole atom, thewhole oscillator, ismoving along inadirection toward theobserver atvelocity v.Then theactual motion inspace, ofcourse, isasshown inFig.34—l0(a). Now weplay ourusual game, weaddcr;thatistosay,wetranslate thewhole curve backward andwefindthenthatitoscillates asinFig.34—lO(b). Inagiven amount oftime1,when theoscillator would have gone adistance vr,onthex’vs.ctdiagram itgoesadistance (c—v)r.Soalltheoscillations offrequency wlinthetimeA1are now found intheinterval A-r=(l—v/c)A-r;they aresquashed together, and asthiscurve comes byusatspeed c,wewillseelight ofahigher frequency, higher byjustthecompression factor (1-v/c). Thus weobserve ___21_..w_I_U/C (34.10) Wecan,ofcourse, analyze thissituation invarious other ways. Suppose that theatom were emitting, instead ofsinewaves, aseries ofpulses, pip,pip,pip,pip, atacertain frequency wl.Atwhat frequency would theybereceived byus? The firstonethatarrives hasacertain delay, butthenextoneisdelayed lessbecause in themeantime theatom moves closer tothereceiver. Therefore, thetime between the“pips” isdecreased bythemotion. Ifweanalyze thegeometry ofthesituation, wefindthatthefrequency ofthepipsisincreased bythefactor l/(1 —v/c). Isw=w0/(1—v/c), then, thefrequency thatwould beobserved ifwetook anordinary atom, which hadanatural frequency wo,andmoved ittoward the receiver atspeed v?No;aswewellknow, thenatural frequency 0.);ofamoving atom isnotthesame asthatmeasured when itisstanding still, because ofthe relativistic dilation intherateofpassage oftime. Thus ifwowere thetruenatural frequency, then themodified natural frequency wlwould be to,=0),,\/1-U2/C2. (34.11) Therefore theobserved frequency wis °’=w*°fl__../if/C2‘ (34.12) Theshift infrequency observed intheabove situation iscalled theDoppler eflect: ifsomething moves toward usthelight itemits appears more violet, andif itmoves away itappears more red. Weshall nowgivetwomore derivations ofthissame interesting andimportant result. Suppose, now, thatthesource isstanding stillandisemitting waves at frequency wo,while theobserver ismoving with speed vtoward thesource. After acertain period oftime ttheobserver willhave moved toanewposition, adistance vtfrom where hewasatt=0.How many radians ofphase willhehave seengo by?Acertain number, wot,went pastanyfixed point, andinaddition theobserver hasswept pastsome more byhisownmotion, namely anumber vtko(thenumber ofradians permeter times thedistance). Sothetotal number ofradians inthe time t,ortheobserved frequency, would bewl=wo+kov. Wehave made thisanalysis from thepoint ofview ofaman atrest; wewould liketoknow how itwould look tothemanwho ismoving. Here wehave toworry again about the difference inclock rateforthetwoobservers, andthistime thatmeans thatwe have todivide by\/1—v2/c2. Soifk0isthewave number, thenumber ofradians 34-7B B’ 2 oi (cl) (bl Fig. 34-10. Thex—zand x’— curves ofamoving oscillator.t permeter inthedirection ofmotion, andweisthefrequency, then theobserved frequency foramoving man is <3= (34.13) Forthecaseoflight, weknow thatkg=030/c.So,inthisparticular problem, theequation would read <3= (34.14) which looks completely unlike formula (34.12)! Isthefrequency thatwewould observe ifwemove toward asource different than thefrequency thatwewould seeifthesource moved toward us? Ofcourse not! Thetheory ofrelativity says thatthese twomust beexactly equal. Ifwewere expert enough mathematicians we would probably recognize that these twomathematical expressions areexactly equal! Infact, thenecessary equality ofthetwoexpressions isoneoftheways by which some people liketodemonstrate that relativity requires atime dilation, because ifwedidnotputthose square-root factors in,they would nolonger be equal. Since weknow about relativity, letusanalyze itinstillathird way, which may appear alittle more general. (Itisreally thesame thing, since itmakes no difference howwedoit!)According totherelativity theory there isarelationship between position andtime asobserved byoneman andposition andtime asseen byanother who ismoving relative tohim. Wewrote down those relationships long ago(Chapter 16). This istheLorentz transformation anditsinverse: x vt x'—vt’x-4. x=?_.\/1—212/c2 \/l—2:2/c2 t,=t+vx/c2 ’ I=t’—vx’/c2 _ (3415) \/1—v2/c2 \/1—v2/c2 Ifwewere standing stillontheground, theform ofawave would becos(wt—kx); allthenodes andmaxima andminima would follow thisform. Butwhat would a maninmotion, observing thesame physical wave, see? Where thefieldiszero, the positions ofallthenodes arethesame (when thefield iszero, everyone measures thefield aszero); thatisarelativistic invariant. Sotheform isthesame forthe other man too,except thatwemust transform itintohisframe ofreference: 2 I cos(wt —kx)=cos[w ti_vx,/C —kxl_vt \/1—v2/c2 V1-—v2/c2 Ifweregroup theterms inside thebrackets, weget 2 cos(wt —kx)=cos-w +kl) t’——i——-k+ W/c_x’ yl-122/0? \\/l—112/ofv v =cos[ w’ t’— k’ x']. (34.16) This isagain awave, acosine wave, inwhich there isacertain frequency w’,a constant multiplying t’,andsome other constant, k’,multiplying x’.Wecallk’the wave number, orthenumber ofwaves permeter, fortheother man. Therefore theother man willseeanewfrequency andanewwave number given by 03'= (34.17)\/ _U C kr= k+Q”/52. (34.18)\/1—v2c2 Ifwelook at(34.17), weseethatitisthesame formula (34.13), thatweobtained byamore physical argument. 34-8 34-7 Thew,kfour-vector Therelationships indicated inEqs.(34.17) and(34.18) areveryinteresting, because these saythatthenewfrequency w’isacombination oftheoldfrequency wandtheoldwave number k,andthatthenewwave number isacombination of theoldwave number andfrequency. Now thewave number istherateofchange ofphase with distance, andthefrequency istherateofchange ofphase with time, andinthese expressions weseeaclose analogy with theLorentz transformation of theposition andtime: ifwisthought ofasbeing liket,andkisthought ofasbeing likexdivided byc2,then theneww’willbeliket’,andthenewk’willbelike x’/c2. That istosay,under theLorentz transformation wandktransform thesame wayasdotandx.They constitute what wecallafour-vector; when aquantity has fourcomponents transforming liketime andspace, itisafour-vector. Everything seems allright, then, except foronelittle thing: wesaidthatafour-vector hasto havefour components; where aretheother twocomponents? Wehave seen that wandkareliketime andspace inonespace direction, butnotinalldirections, andsowemust next study theproblem ofthepropagation oflight inthree space dimensions, notjustinonedirection, aswehave been doing upuntil now. Suppose thatwehave acoordinate system, x,y,z,andawave which istravel- lingalong andwhose wavefronts areasshown inFig.34-ll.Thewavelength of thewave is)\,butthedirection ofmotion ofthewave does nothappen tobein thedirection ofoneoftheaxes. What istheformula forsuch awave? The answer isclearly cos(wt—ks),where k=21r/)\ andsisthedistance along the direction ofmotion ofthewave—the component ofthespatial position inthe direction ofmotion. Letusputitthisway: ifristhevector position ofapoint inspace, then sisr-ek,where e),isaunit vector inthedirection ofmotion. That is,sisjustrcos(r,ek),thecomponent ofdistance inthedirection ofmotion. Therefore ourwave iscos(wt—ke),-r). Now itturns outtobevery convenient todefine avector k,which iscalled thewave vector, which hasamagnitude equal tothewave number, 211-/x, andis pointed inthedirection ofpropagation ofthewaves: k=21re;,/A =kek. (34.19) Using thisvector, ourwave canbewritten ascos(wt—k-r),orascos(wt— k,,x—kyy——k,z). What isthesignificance ofacomponent ofk,sayk,,‘? Clearly, k,istherateofchange ofphase with respect tox.Referring toFig. 34-11, weseethatthephase changes aswechange x,justasifthere were awave along x,butofalonger wavelength. The“wavelength inthex-direction” islonger than anatural, truewavelength bythesecant oftheangle atbetween theactual direction ofpropagation andthex-axis: A,=A/cos oz. (34.20) Therefore therateofchange ofphase, which isproportional tothereciprocal of ).,,issmaller bythefactor cosa;thatisjusthow k,would vary—it would bethe magnitude ofk,times thecosine oftheangle between kandthex-axis! That, then, isthenature ofthewave vector thatweusetorepresent awave inthree dimensions. Thefour quantities w,k,,ky,k,transform inrelativity asa four-vector, where wcorresponds tothetime, andk,,,kg,k,correspond tothe x-,y-,andz-components ofthefour-vector. Inourprevious discussion ofspecial relativity (Chapter 17),welearned that there areways ofmaking relativistic dotproducts with four-vectors. Ifweusethe position vector x,,,where ustands forthefourcomponents (time andthree space ones), andifwecallthewave vector k,,,where theindex p.again hasfour values. time andthree space ones, then thedotproduct ofx,,andk,,iswritten Z’k,,x,. (seeChapter 17).This dotproduct isaninvariant, independent ofthecoordinate system; what isitequal to?Bythedefinition ofthisdotproduct infourdimensions, itis Z’k,,x,, =wt—-k,x—k,,y—k,z. (34.21) 34-9v X > L»‘a‘ X \\ Fig. 34-1 l.Aplane wave travelling inanoblique direction. S S F‘, 11/I,I _:r_.~~ -A V (cl (bl Fig. 34-12. Adistant source Sis viewed by(a)astationary telescope, and (b)alaterally moving telescope. El 5 v y 1 E F Fig. 34-l 3.Themagnetic force ona charge which isdriven bytheelectric field isinthedirection ofthelight beam.Weknow from ourstudy ofvectors thatZ’k,,x,, isinvariant under theLorentz transformation, since k,,isafour-vector. Butthisquantity isprecisely what appears inside thecosine foraplane wave, anditought tobeinvariant under aLorentz transformation. Wecannot have aformula with something thatchanges inside thecosine, since weknow thatthephase ofthewave cannot change when wechange thecoordinate system. 34-8 Aberration Inderiving Eqs. (34.17) and(34.18), wehave taken asimple example where k happened tobeinadirection ofmotion, butofcourse wecangeneralize ittoother cases also. Forexample, suppose there isasource sending outlight inacertain direction from thepoint ofview ofaman atrest,butwearemoving along onthe earth, say(Fig. 34-12). From which direction does thelight appear tocome? Tofindout,wewillhave towrite down thefourcomponents ofk,,andapply the Lorentz transformation. Theanswer, however, canbefound bythefollowing argument: wehave topoint ourtelescope atanangle toseethelight. Why? Because light iscoming down atthespeed c,andwearemoving sidewise atthe speed v,sothetelescope hastobetilted forward sothatasthelight comes down it goes “straight” down thetube. Itisvery easy toseethatthehorizontal distance isvtwhen thevertical distance isct,andtherefore, if6’istheangle oftilt,tan6’= 11/c. How nice! How nice, indeed—except foronelittle thing: 0’isnottheangle atwhich wewould have tosetthetelescope relative totheearth, because wemade ouranalysis from thepoint ofview ofa“fixed” observer. When wesaidthehori- zontal distance isvt,theman ontheearth would have found adifferent distance, since hemeasured with a“squashed” ruler. Itturns outthat, because ofthatcon- traction effect, tan0= "4/‘_. (34.22)\/l—112/c2 which isequivalent to sin0=v/c. (34.23) Itwillbeinstructive forthestudent toderive thisresult, using theLorentz trans- formation. Thiseffect, thatatelescope hastobetilted, iscalled aberration, andithasbeen observed. How canweobserve it?Who cansaywhere agiven starshould be? Suppose wedohave tolook inthewrong direction toseeastar; howdoweknow itisthewrong direction? Because theearth goes around thesun. Today wehave topoint thetelescope oneway; sixmonths later wehave totiltthetelescope the other way. That ishowwecantellthatthere issuch aneffect. 34-9 Themomentum oflight Now weturntoadifferent topic. Wehave never, inallourdiscussion ofthe past fewchapters, said anything about theeffects ofthemagnetic field that is associated with light. Ordinarily, theeffects ofthemagnetic field arevery small, butthere isoneinteresting andimportant effect which isaconsequence ofthe magnetic field. Suppose thatlight iscoming from asource andisacting ona charge anddriving thatcharge upanddown. Wewillsuppose thattheelectric field isinthex-direction, sothemotion ofthecharge isalsointhex-direction: it hasaposition xandavelocity v,asshown inFig.34-13. Themagnetic field isat right angles totheelectric field. Now astheelectric field actsonthecharge and moves itupanddown, what does themagnetic field do? Themagnetic field acts onthecharge (sayanelectron) only when itismoving; buttheelectron ismoving, itisdriven bytheelectric field, sothetwoofthem work together: While thething isgoing upanddown ithasavelocity andthere isaforce onit,Btimes vtimes q; butinwhich direction isthisforce? Itisinthedirection ofthepropagation oflight. Therefore, when light isshining onacharge anditisoscillating inresponse tothat 34-10 charge, there isadriving force inthedirection ofthelight beam. This iscalled radiation pressure orlight pressure. Letusdetermine howstrong theradiation pressure is.Evidently itisF=qvB or,since everything isoscillating, itisthetimeaverage ofthis,(F). From (34.2) the strength ofthemagnetic fieldisthesame asthestrength oftheelectric fielddivided byc,soweneed tofindtheaverage oftheelectric field, times thevelocity, times the charge, times l/c:(F)=q(vE)/c. Butthecharge qtimes thefieldEistheelectric force onacharge, andtheforce onthecharge times thevelocity isthework dW/dt being done onthecharge! Therefore theforce, the“pushing momentum,” thatis delivered persecond bythelight, isequal to1/ctimes theenergy absorbed from the light persecond! That isageneral rule, since wedidnotsayhowstrong theoscilla- torwas, orwhether some ofthecharges cancel out.Inanycircumstance where light isbeing absorbed, there isapressure. Themomentum thatthelight delivers isal- ways equal totheenergy thatisabsorbed, divided byc: (F)=i5Vc/i- (34.24) That light carries energy wealready know. Wenow understand thatitalso carries momentum, andfurther, thatthemomentum carried isalways 1/ctimes theenergy. When light isemitted from asource there isarecoil effect: thesame thing in reverse. Ifanatom isemitting anenergy Winsome direction, then there isa recoil momentum p=W/c. Iflight isreflected normally from amirror, weget twice theforce. That isasfarasweshall gousing theclassical theory oflight. Ofcourse we know thatthere isaquantum theory, andthatinmany respects light actslikea particle. Theenergy ofalight-particle isaconstant times thefrequency: W=hv=hw (34.25) Wenowappreciate thatlightalsocarries amomentum equal totheenergy divided byc,soitisalsotruethatthese effective particles, these photons, carry amomentum p=W/c =ttw/c =hk. (34.26) Thedirection ofthemomentum is,ofcourse, thedirection ofpropagation ofthe light. So,toputitinvector form, W=hat, p=mt. (34.27) Wealsoknow, ofcourse, thattheenergy andmomentum ofaparticle should form afour-vector. Wehave justdiscovered thatwandkform afour-vector. Therefore itisagood thing that(34.27) hasthesame constant inboth cases; itmeans thatthe quantum theory andthetheory ofrelativity aremutually consistent. Equation (34.27) canbewritten more elegantly asp,,=hk,,, arelativistic equation, foraparticle associated with awave. Although wehave discussed this only forphotons, forwhich k(themagnitude ofk)equals w/candp=W/c, the relation ismuch more general. Inquantum mechanics allparticles, notonly photons, exhibit wavelike properties, butthefrequency andwave number ofthe waves isrelated totheenergy andmomentum ofparticles by(34.27) (called the deBroglie relations) even when pisnotequal toW/c. Inthelastchapter wesawthatabeam ofright orleftcircularly polarized light alsocarries angular momentum inanamount proportional totheenergy 8of thewave. Inthequantum picture, abeam ofcircularly polarized light isregarded asastream ofphotons, each carrying anangular momentum ihalong thedirec- tionofpropagation. That iswhat becomes ofpolarization inthecorpuscular point ofview—the photons carry angular momentum likespinning riflebullets. Butthis “bullet” picture isreally asincomplete asthe“wave” picture, andweshallhave todiscuss these ideas more fully inalater chapter onQuantum Behavior. 34-11 35 Color Vision 35-1 Thehuman eyc Thephenomenon ofcolors depends partly onthephysical world. Wediscuss thecolors ofsoap films andsoonasbeing produced byinterference. Butalso, ofcourse, itdepends ontheeye,orwhat happens behind theeye,inthebrain. Physics characterizes thelight thatenters theeye,butafter that, oursensations are theresult ofphotochemical-neural processes andpsychological responses. There aremany interesting phenomena associated with vision which involve amixture ofphysical phenomena andphysiological processes, andthefullappreci- ation ofnatural phenomena, asweseethem, must gobeyond physics intheusual sense. Wemake noapologies formaking these excursions intoother fields, because theseparation offields, aswehave emphasized, ismerely ahuman convenience, andanunnatural thing. Nature isnotinterested inourseparations, andmany of theinteresting phenomena bridge thegaps between fields. InChapter 3wehave already discussed therelation ofphysics totheother sciences ingeneral terms, butnowwearegoing tolook insome detail ataspecific fieldinwhich physics andother sciences arevery, very closely interrelated. That areaisvision. Inparticular, weshall discuss color vision. Inthepresent chapter weshall discuss mainly theobservable phenomena ofhuman vision, andinthe nextchapter weshall consider thephysiological aspects ofvision, both inman and inother animals. Itallbegins with theeye;so,inorder tounderstand what phenomena wesee, some knowledge oftheeyeisrequired. Inthenext chapter weshall discuss in some detail howthevarious parts oftheeyework, andhowtheyareinterconnected withthenervous system. Forthepresent, weshall describe only briefly how the eyefunctions (Fig. 35-1). Light enters theeyethrough thecornea; wehave already discussed how itis bentandisimaged onalayer called theretina intheback oftheeye,sothatdifferent parts oftheretina receive light from different parts ofthevisual field outside. The retina isnotabsolutely uniform: there isaplace, aspot, inthecenter ofourfield ofviewwhich weusewhen wearetrying toseethings verycarefully, andatwhich we havethegreatest acuity ofvision; itiscalled thefovea ormacula. Thesideparts of theeye,aswecanimmediately appreciate from ourexperience inlooking atthings, arenotaseffective forseeing detail asisthecenter oftheeye. There isalsoaspot intheretina where thenerves carrying alltheinformation runout;thatisablind spot. There isnosensitive partoftheretina here, anditispossible todemonstrate thatifweclose, say,thelefteyeandlook straight atsomething, andthen move a finger oranother small object slowly outofthefieldofview itsuddenly disappears somewhere. Theonlypractical useofthisfactthatweknow ofisthatsome physiol- ogist became quite afavorite inthecourt ofaking ofFrance bypointing thisout tohim; intheboring sessions thathehadwith hiscourtiers, theking could amuse himself by“cutting offtheir heads” bylooking atoneandwatching another’s head disappear. Figure 35-2 shows amagnified view oftheinside oftheretina insomewhat schematic form. Indifferent parts oftheretina there aredifi'erent kinds ofstruc- tures. Theobjects thatoccur more densely near theperiphery oftheretina are called rods. Closer tothefovea, wefind, besides these rodcells, alsocone cells. Weshall describe thestructure ofthese cellslater. Aswegetclose tothefovea, the number ofcones increases, andinthefovea itself there areinfactnothing butcone cells, packed verytightly, sotightly thatthecone cells aremuch finer, ornarrower 35-1s \(1l\..-Il1i"~.'*.-i. / 1 C35-1 Thehuman eye 35-2 Color depends onintensity 35-3 Measuring thecolor sensation 35-4 Thechromaticity diagram 35-5 Themechanism ofcolor vision 35-6 Physiochemistry ofcolor vision OI'fl2O \.m / _\ Susp"ensory Cilicryligament \muscle 7..,- .».-'”'l‘\Vitreous humor A‘Choroid Retin.9 Sclera /‘“'”‘ ",1',Mciculalutea 1-Optic nerve Fig. 35-1. Theeye. " ’'' __,..._.-,_~:-__%_:._.r,_;q§- 'us ll3 It. ’—w;‘:r\¢'./ -,Qv->\'-R --'¢;r.¢'1-1..;"’.u-2'-wqlIf 4“:1,Qat°'m‘itF:_1":f.F 2'A}; I’\l>-vi-,__-¢_q¢?\.o‘T¢Y‘Z"B ‘A/l -‘A’‘air‘T - ' — _|I_, ._ .7-0_ Fig. 35-2. Thestructure oftheretina. (Light enters from below.) here than anywhere else. Sowemust appreciate thatweseewith thecones right inthemiddle ofthefield ofview, butaswegototheperiphery wehave theother cells, therods, Nowtheinteresting thing isthatintheretina eachofthecellswhich issensitive tolight isnotconnected byafiber directly totheoptic nerve, butis connected tomany other cells, which arethemselves connected toeach other. There areseveral kinds ofcells: there arecells thatcarry theinformation toward theoptic nerve, butthere areothers thataremainly interconnected “horizontally.” There areessentially fourkinds ofcells, butweshall notgointothese details now. Themain thing weemphasize isthatthelight signal isalready being “thought about.” That istosay,theinformation from thevarious cellsdoes notimmediately gotothebrain, spotforspot, butintheretina acertain amount oftheinformation hasalready been digested, byacombining oftheinformation from several visual receptors. Itisimportant tounderstand that some brain-function phenomena occur intheeyeitself. 35-2 Color depends onintensity One ofthemost striking phenomena ofvision isthedark adaptation ofthe eye. Ifwegointothedark from abrightly lighted room, wecannot seeverywell forawhile, butgradually things become more andmore apparent, andeventually wecansee something where wecould seenothing before. Iftheintensity ofthe light isvery low, thethings thatweseehave nocolor. Itisknown thatthisdark- adapted vision isamost entirely duetotherods, while thevision inbright light is duetothecones. Asaresult, there areanumber ofphenomena thatwecaneasily appreciate because ofthistransfer offunction from thecones androds together, tojusttherods. There aremany situations inwhich, ifthelight intensity were stronger, we could seecolor, andwewould findthese things quite beautiful. Oneexample is thatthrough atelescope wenearly always see“black andwhite” images offaint nebulae, butW.C.Miller oftheMt.Wilson andPalomar Observatories hadthe patience tomake color pictures ofsome ofthese objects. Nobody haseverreally seenthese colors withtheeye,buttheyarenotartificial colors, itismerely thatthe light intensity isnotstrong enough forthecones inoureyetoseethem. Among themore spectacular such objects aretheringnebula andtheCrab nebula. The former shows abeautiful blueinner part, withabright redouter halo, andthelatter shows ageneral bluish haze permeated bybright red-orange filaments. Inthebright light, apparently, therods areatvery lowsensitivity but,inthe dark, astimegoes ontheypickuptheir ability toseelight. Thevariations inlight intensity forwhich onecanadapt isover amillion toone. Nature does notdo allthiswithjustonekind ofcell, butshepasses herjobfrom bright-light-seeing cells, thecolor-seeing cells, thecones, tolow-intensity, dark-adapted cells, therods. Among theinteresting consequences ofthisshiftis,first, thatthere isnocolor, and second, thatthere isadifference intherelative brightness ofdifferently colored objects. Itturns outthattherods seebetter toward theblue than thecones do, andthecones cansee,forexample, deep redlight, while therodsfindthatabsolutely impossible tosee.Soredlight isblack sofarastherods areconcerned. Thus two pieces ofcolored paper, sayblueandred,inwhich theredmight beeven brighter than theblue ingood light, will, inthedark, appear completely reversed. Itisa verystriking effect. Ifweareinthedark andcanfindamagazine orsomething that hascolors and, before weknow forsurewhat thecolors are,wejudge thelighter anddarker areas, andifwethen carry themagazine intothelight, wemayseethis very remarkable shift between which wasthebrightest color andwhich wasnot. Thephenomenon iscalled thePurkinje eflect. InFig.35-3, thedashed curve represents thesensitivity oftheeyeinthedark, i.e.,using therods, while thesolid curve represents itinthelight. Weseethatthe peak sensitivity oftherods isinthegreen region andthatofthecones ismore in theyellow region. Ifthere isared-colored page (redisabout 650I‘l'l].L)wecansee itifitisbrightly lighted, butinthedark itisalmost invisible. 35-2 Ml W _/ A I Z9/b///fy8 ireVI8 /Tb/of 20 Fig. 35-3. Thespectral sensitivity of’1-/I l 1'Ii theeye. Dashed curve, rods; solid curve, o --r" 7% I0 66 40 E X0 I0 £ 40 W X0 Q $0 G0 E Q ¢°"e5- Wave/enyr/t inmu Another effect ofthefactthatrods take over inthedark, andthatthere are norods inthefovea, isthatwhen welook straight atsomething inthedark, our vision isnotquite asacute aswhen welook tooneside. Afaint starornebula can sometimes beseenbetter bylooking alittle toonesidethan directly atit,because wedonothave sensitive rods inthemiddle ofthefovea. Another interesting effect ofthefactthatthenumber ofcones decreases as wegofarther tothesideofthefield ofview isthateven inabright light color disappears astheobject goes fartooneside. Thewaytotestthatistolook in some particular fixed direction, letafriend walk infrom onesidewith colored cards, andtrytodecide what color they arebefore they areright infront ofyou. Onefinds thathecanseethatthecards arethere long before hecandetermine the color. When doing this,itisadvisable tocome infrom thesideopposite theblind spot, because itisotherwise rather confusing toalmost seethecolor, then not seeanything, then toseethecolor again. Another interesting phenomenon isthattheperiphery oftheretina isvery sensitive tomotion. Although wecannot seeverywellfrom thecorner ofoureye, ifalittle bugmoves andwedonotexpect anything tobemoving over there, weareimmediately sensitive toit.Weareall“wired up”tolook forsomething jiggling tothesideofthefield. 35-3 Measuring thecolor sensation Now wegotothecone vision, tothebrighter vision, andwecome tothe question which ismost characteristic ofcone vision, andthatiscolor. Aswe know, white light canbesplit byaprism intoawhole spectrum ofwavelengths which appear toustohave difierent colors; thatiswhat colors are,ofcourse: appearances. Any source oflight canbeanalyzed byagrating oraprism, and onecandetermine thespectral distribution, i.e.,the“amount” ofeach wavelength. Acertain light mayhave alotofblue, considerable red,verylittle yellow, andso on.That isallveryprecise inthesense ofphysics, butthequestion is,what color willitappear tobe? Itisevident thatthedifferent colors depend somehow upon thespectral distribution ofthelight, buttheproblem istofindwhat characteristics ofthespectral distribution produce thevarious sensations. Forexample, what do wehave todotogetagreen color? Weallknow thatwecansimply take apiece ofthespectrum which isgreen. Butisthattheonlywaytogetgreen, ororange, oranyother color? Isthere more than onespectral distribution which produces thesame apparent visual effect? Theanswer is,definitely yes.There isaverylimited number ofvisual effects, infactjustathree-dimensional manifold ofthem, asweshall shortly see, butthere isaninfinite number ofdifferent curves thatwecandraw forthelight thatcomes from different sources. Now thequestion wehave todiscuss is,under what conditions dodifferent distributions oflight appear asexactly thesame color totheeye? 35-3 Themost powerful psycho-physical technique incolor judgment istouse theeyeasanullinstrument. That is,wedonottrytodefine what consititutes a green sensation, ortomeasure inwhat circumstances wegetagreen sensation, because itturns outthat thisisextremely complicated. Instead, westudy the conditions under which two stimuli areindistinguishable. Then wedonot have todecide whether two people seethesame sensation indifferent cir- cumstances, butonly whether, ifforoneperson twosensations arethesame, they arealsothesame foranother. Wedonothave todecide whether, when onesees something green, what itfeels likeinside isthesame aswhat itfeels likeinside someone elsewhen heseessomething green; wedonotknow anything about that. Toillustrate thepossibilities, wemayuseaseries offourprojector lamps which have filters onthem, andwhose brightnesses arecontinuously adjustable over a wide range: onehasaredfilter andmakes aspot ofredlight onthescreen, the next onehasagreen filter andmakes agreen spot, thethird onehasabluefilter, andthefourth oneisawhite circle with ablack spot inthemiddle ofit.Now if weturn onsome redlight, andnext toitputsome green, weseethatinthearea ofoverlap itproduces asensation which isnotwhat wecallreddish green, buta newcolor, yellow inthisparticular case. Bychanging theproportions ofthered andthegreen, wecangothrough various shades oforange andsoforth. Ifwe have setitforacertain yellow, wecanalsoobtain thatsame yellow, notbymixing these twocolors butbymixing some other ones, perhaps ayellow filter with white light, orsomething likethat, togetthesame sensation. Inother words, itispossible tomake various colors inmore than onewaybymixing thelights from various filters. What wehave justdiscovered may beexpressed analytically asfollows. A particular yellow, forexample, canberepresented byacertain symbol Y,which is the“sum” ofcertain amounts ofred-filtered light (R)andgreen-filtered light (G). Byusing twonumbers, sayrandg,todescribe howbright the(R)and(G)are,we canwrite aformula forthisyellow: Y=rR+gG. (35.1) Thequestion is,canwemake allthedifferent colors byadding together twoor three lights ofdifferent, fixed colors? Letusseewhat canbedone inthatconnec- tion. Wecertainly cannot getallthedifferent colors bymixing onlyredandgreen, because, forinstance, blue never appears insuch amixture. However, byputting insome blue thecentral region, where allthree spots overlap, may bemade to appear tobeafairly nicewhite. Bymixing thevarious colors andlooking atthis central region, wefindthatwecangetaconsiderable range ofcolors inthatregion bychanging theproportions, andsoitisnotimpossible thatallthecolors canbe made bymixing these three colored lights. Weshall discuss towhat extent thisis true; itisinfactessentially correct, andweshall shortly seehow todefine the proposition better. Inorder toillustrate ourpoint, wemove thespots onthescreen sothatthey allfallontopofeach other, andthen wetrytomatch aparticular color which appears intheannular ringmade bythefourth lamp. What weonce thought was “white” coming from thefourth lamp nowappears yellowish. Wemaytrytomatch thatbyadjusting theredandgreen andblue asbestwecanbyakind oftrialand error, andwefindthatwecanapproach rather closely thisparticular shade of “cream” color. Soitisnothard tobelieve thatwecanmake allcolors. Weshall trytomake yellow inamoment, butbefore wedothat, there isonecolor thatmight bevery hard tomake. People who givelectures oncolor make allthe“bright” colors, buttheynever make brown, anditishard torecall everhaving seenbrown light. Asamatter offact, thiscolor isnever used foranystage effect, onenever seesaspotlight withbrown light; sowethink itmight beimpossible tomake brown. Inorder tofindoutwhether itispossible tomake brown, wepoint outthatbrown light ismerely something thatwearenotused toseeing without itsbackground. Asamatter offact, wecanmake itbymixing some redandyellow. Toprove thatwearelooking atbrown light, wemerely increase thebrightness oftheannular 35-4 background against which weseethevery same light, andweseethatthatis,in fact, what wecallbrown! Brown isalways adark color next toalighter back- ground. Wecaneasily change thecharacter ofthebrown. Forexample, ifwetake some green outwegetareddish brown, apparently achocolatey reddish brown, andifweputmore green intoit,inproportion, wegetthathorrible color which alltheuniforms oftheArmy aremade of,butthelight from thatcolor isnotso horrible byitself ;itisofyellowish green, butseen against alight background. Now weputayellow filter infront ofthefourth light andtrytomatch that. (The intensity must ofcourse bewithin therange ofthevarious lamps; wecannot match something which istoobright, because wedonothave enough power in thelamp.) Butwecanmatch theyellow; weuseagreen andredmixture, andput inatouch ofblue tomake iteven more perfect. Perhaps weareready tobelieve that, under good conditions, wecanmake aperfect match ofanygiven color. Now letusdiscuss thelaws ofcolor mixture. Inthefirstplace, wefound that different spectral distributions canproduce thesame color; next, wesawthat “any” color canbemade byadding together three special colors, red,blue, and green. Themost interesting feature ofcolor mixing isthis: ifwehave acertain light, which wemaycallX,andifitappears indistinguishable from Y,totheeye (itmay beadifferent spectral distribution, butitappears indistinguishable), we callthese colors “equal,” inthesense thattheeyeseesthem asequal, andwewrite X=Y. (35.2) Here isoneofthegreat laws ofcolor: iftwospectral distributions areindistinguish- able, andweaddtoeach oneacertain light, sayZ(ifwewrite X—l—Z,thismeans thatweshine both lights onthesame patch), andthenwetake Yandaddthesame amount ofthesame other light, Z,thenewmixtures arealsoindistinguishable." X+Z= Y-l—Z. (35.3) Wehave justmatched ouryellow; ifwenow shine pink light onthewhole thing, itwillstillmatch. Soadding anyother light tothematched lights leaves amatch. Inother words, wecansummarize allthese color phenomena bysaying thatonce wehave amatch between twocolored lights, seennext toeach other inthesame circumstances, then thismatch willremain, andonelight canbesubstituted for theother light inanyother color mixing situation. Infact, itturns out,anditis very important andinteresting, that thismatching ofthecolor oflights isnot dependent upon thecharacteristics oftheeyeatthemoment ofobservation: we know thatifwelook foralong time atabright redsurface, orabright redlight, andthen look atawhite paper, itlooks greenish, andother colors arealsodis- torted byourhaving looked solong atthebright red. Ifwenow have amatch between, say,twoyellows, andwelook atthem andmake them match, then we look atabright redsurface foralong time, andthen turn back totheyellow, it may notlook yellow anymore; Idonotknow what color itwilllook, butitwill notlook yellow. Nevertheless theyellows willstilllook matched, andso,asthe eyeadapts tovarious levels ofintensity, thecolor match stillworks, with the obvious exception ofwhen wegointotheregion where theintensity ofthelight getssolowthatwehave shifted from cones torods; then thecolor match isno longer acolor match, because weareusing adifferent system. Thesecond principle ofcolor mixing oflights isthis: anycolor atallcanbe made from three dififerent colors, inourcase, red,green, andbluelights. Bysuitably mixing thethree together wecanmake anything atall,aswedemonstrated with ourtwoexamples. Further, these laws arevery interesting mathematically. For those who areinterested inthemathematics ofthething, itturns outasfollows. Suppose thatwetakeourthree colors, which were red,green, andblue, butlabel them A,B,andC,andcallthem ourprimary colors. Then anycolor could be made bycertain amounts ofthese three: sayanamount aofcolor A,anamount bofcolor B,andanamount cofcolor Cmakes X: X=aA—l—bB+cC. (35.4) 35-5 Now suppose another color Yismade from thesame three colors: Y=a'A+b’B—l—c’C. (35.5) Then itturns outthatthemixture ofthetwolights (itisoneoftheconsequences ofthelaws thatwehave already mentioned) isobtained bytaking thesumofthe components ofXandY: Z=X+Y=(a+a’)A +(b—l—b')B +(c+c’)C. (35.6) Itisjustlikethemathematics oftheaddition ofvectors, where (a,b,c)arethe components ofonevector, and(a’,b’,c’)arethose ofanother vector, andthe newlight Zisthen the“sum” ofthevectors. This subject hasalways appealed to physicists andmathematicians. Infact, Schrodinger wrote awonderful paper on color vision inwhich hedeveloped thistheory ofvector analysis asapplied tothe mixing ofcolors. Now aquestion is,what arethecorrect primary colors touse? There isno such thing as“the” correct primary colors forthemixing oflights. There may be, forpractical purposes, three paints thataremore useful than others forgetting a greater variety ofmixed pigments, butwearenotdiscussing that matter now. Anythree diflerently colored lights whatsoever* canalways bemixed inthecorrect proportion toproduce anycolor whatsoever. Can wedemonstrate thisfantastic fact? Instead ofusing red,green, andblue, letususered,blue, andyellow inour projector. Canweusered,blue, andyellow tomake, say,green‘? Bymixing these three colors invarious proportions, wegetquite anarray of different colors, ranging over quite aspectrum. Butasamatter offact, after alot oftrialanderror, wefindthatnothing everlooks likegreen. Thequestion is,can wemake green‘? Theanswer isyes. How? Byprojecting some redonto thegreen, then wecanmake amatch with acertain mixture ofyellow andblue! Sowehave matched them, except thatwehadtocheat byputting theredontheother side. Butsince wehave some mathematical sophistication, wecanappreciate thatwhat we really showed wasnotthatXcould always bemade, say,ofred,blue, andyellow, butbyputting theredontheother sidewefound thatredplus Xcould bemade outofblueandyellow. Putting itontheother sideoftheequation, wecaninterpret thatasanegative amount, soifwewillallow thatthecoefficients inequations like (35.4) canbeboth positive andnegative, andifweinterpret negative amounts to mean thatwehave toaddthose totheother side, then anycolor canbematched by anythree, andthere isnosuch thing as“the” fundamental primaries. Wemay askwhether there arethree colors that come only with positive amounts forallmixings. Theanswer isno.Every setofthree primaries requires negative amounts forsome colors, andtherefore there isnounique waytodefine aprimary. Inelementary books theyaresaidtobered,green, andblue, butthat ismerely because with these awider range ofcolors isavailable without minus signs forsome ofthecombinations. 35-4 Thechromaticity diagram Now letusdiscuss thecombination ofcolors onamathematical level asa geometrical proposition. Ifanyonecolor isrepresented byEq.(35.4), wecanplot itasavector inspace byplotting along three axes theamounts a,b,andc,and then acertain color isapoint. Ifanother color isa’,b’,c’,thatcolor islocated somewhere else. Thesumofthetwo, asweknow, isthecolor which comes from adding these asvectors. Wecansimplify thisdiagram andrepresent everything onaplane bythefollowing observation: ifwehadacertain color light, andmerely doubled aandbandc,thatis,ifwemake them allstronger inthesame ratio, itis thesame color, butbrighter. Soifweagree toreduce everything tothesame light intensity, then wecanproject everything onto aplane, andthishasbeen done in Fig.35-4. Itfollows thatanycolor obtained bymixing agiven twoinsome pro- *Except, ofcourse, ifoneofthethree canbematched bymixing theother two. 35-6 portion willliesomewhere onalinedrawn between thetwopoints. Forinstance, afifty-fifty mixture would appear halfway between them, and1/4ofoneand3/4 oftheother would appear l/4ofthewayfrom onepoint totheother, andsoon. Ifweuseablueandagreen andared,asprimaries, weseethatallthecolors that wecanmake withpositive coeflicients areinside thedotted triangle, which contains almost allofthecolors thatwecaneversee,because allthecolors thatwecanever seeareenclosed intheoddly shaped area bounded bythecurve. Where didthis areacome from? Once somebody made averycareful match ofallthecolors that wecanseeagainst three special ones. Butwedonothave tocheck allcolors that wecansee,weonlyhave tocheck thepure spectral colors, thelines ofthespectrum. Anylight canbeconsidered asasumofvarious positive amounts ofvarious pure spectral colors—pure from thephysical standpoint. Agiven light willhave acer- tainamount ofred,yellow, blue, andsoon—spectral colors. Soifweknow how much ofeach ofourthree chosen primaries isneeded tomake each ofthese pure components, wecancalculate howmuch ofeach isneeded tomake ourgiven color. So,ifwefindoutwhat thecolor coeficients ofallthespectral colors areforany given three primary colors, then wecanwork outthewhole color mixing table. Anexample ofsuch experimental results formixing three lights together is given inFig.35-5. Thisfigure shows theamount ofeach ofthree different particular primaries, red,green andblue, which isrequired tomake each ofthespectral colors. Redisattheleftendofthespectrum, yellow isnext, andsoon,alltheway toblue. Notice thatatsome points minus signs arenecessary. Itisfrom such data thatitispossible tolocate theposition ofallofthecolors onachart, where thex-andthey-coordinates arerelated totheamounts ofthedifferent primaries thatareused. That isthewaythatthecurved boundary linehasbeen found. Itis thelocus ofthepure spectral colors. Now anyother color canbemade byadding spectral lines, ofcourse, andsowefindthatanything thatcanbeproduced by connecting onepartofthiscurve toanother isacolor thatisavailable innature. Thestraight lineconnects theextreme violet endofthespectrum with theextreme redend. Itisthelocus ofthepurples. Inside theboundary arecolors thatcanbe made with lights, andoutside itarecolors thatcannot bemade with lights, and nobody haseverseenthem (except, possibly, inafter-imagesl). 35-5 Themechanism ofcolor vision Now thenextaspect ofthematter isthequestion, whydocolors behave inthis way? Thesimplest theory, proposed byYoung andHelmholtz, supposes thatin theeyethere arethree difi"erent pigments which receive thelight andthatthese have different absorption spectra, sothatonepigment absorbs strongly, say,in thered,another absorbs strongly intheblue, another absorbs inthegreen. Then when weshine alight onthem wewillgetditferent amounts ofabsorptions inthe three regions, andthese three pieces ofinformation aresomehow maneuvered in thebrain orintheeye,orsomewhere, todecide what thecolor is.Itiseasy to demonstrate thatalloftherules ofcolor mixing would beaconsequence ofthis proposition. There hasbeen considerable debate about thething because thenext problem, ofcourse, istofindtheabsorption characteristics ofeach ofthethree pigments. Itturns out,unfortunately, thatbecause wecantransform thecolor coordinates inanymanner wewant to,wecanonly find allkinds oflinear combinations ofabsorption curves bythecolor-mixing experiments, butnotthe curves fortheindividual pigments. People have tried invarious ways toobtain a specific curve which does describe some particular physical property oftheeye. Onesuch curve iscalled abrightness curve, demonstrated inFig. 35-3. lnthis figure aretwocurves, oneforeyes inthedark, theother foreyes inthelight; thelatter isthecone brightness curve. This ismeasured byfinding what isthe smallest amount ofcolored light weneed inorder tobeabletojust seeit.This measures how sensitive theeyeisindifferent spectral regions. There isanother very interesting way tomeasure this. Ifwetake two colors and make them appear inanarea, byflickering back andforth from onetotheother, wesee aflicker ifthefrequency istoolow. However, asthefrequency increases, 35-7I I I I I l I Y I 300.8" - 2O _____.o/ / ’EUls / /(Ia°_ _ \\ 560 O.: \ — 500 \ _O, \5% \\\\\\\\//§O.4'— \ 59° - \ \$2200.5- so ,/R ..-,/ 100 O.2\'- ‘ O.I— / - 4.0B 9°‘A I I u 1 I O.l 0.2 0.5 0.4 0.5 0.6 0.7 IO Fig. 35-4. Thestandard chromaticity diagram. *£2 I!!!E!IIII5lIIIL!! IlllllflllllllllllIllllllflfllllllllll ,Illlllliillllllllll_,||||||l|“||'l\‘|||| 0Illllllllllllfilgl_,,,IIIIIIIIIIIEIIIII730 680 640 600 .760 .520 ‘W 440 400 WAI/ELEN6=T/-/ myCOEFFICIENTSonags Fig. 35-5. The color coefficients of pure spectral colors interms ofacertain setofstandard primary colors. gg a,_ 0.1 so ‘y, 01 no 0.2»- 400 0.!- 0 '4theflicker willultimately disappear atacertain frequency that depends onthe brightness ofthelight, letussayat16repetitions persecond. Now ifweadjust thebrightness ortheintensity ofonecolor against theother, there comes an intensity where theflicker at16cycles disappears. Togetflicker withthebrightness soadjusted, wehave togotoamuch lower frequency inorder toseeaflicker ofthe color. So,wegetwhat wecallaflicker ofthebrightness atahigher frequency and, atalower frequency, aflicker ofthecolor. Itispossible tomatch twocolors for “equal brightness” bythisflicker technique. Theresults arealmost, butnotexactly, thesame asthose obtained bymeasuring thethreshold sensitivity oftheeyefor seeing weak light bythecones. Most workers usetheflicker system asadefinition ofthebrightness curve. Now, ifthere arethree color-sensitive pigments intheeye,theproblem isto determine theshape oftheabsorption spectrum ofeach one. How? Weknow there arepeople who arecolor blind—eight percent ofthemale population, and one-half ofonepercent ofthefemale population. Most ofthepeople who are color blind orabnormal incolor vision have adifferent degree ofsensitivity than others toavariation ofcolor, buttheystillneed three colors tomatch. However, there aresome who arecalled dichromals, forwhom anycolor canbematched using only twoprimary colors. Theobvious suggestion, then, istosaythattheyare missing oneofthethree pigments. Ifwecanfindthree kinds ofcolor-blind dichro- mats who have different color-mixing rules, onekind should bemissing thered, another thegreen, andanother thebluepigmentation. Bymeasuring allthese types wecandetermine thethree curves! Itturns outthatthere arethree types ofdichro- matic color blindness; there aretwocommon types andathird very rare type, andfrom these three ithasbeen possible todeduce thepigment absorption spectra. 6| SO 0 ¢-- -s so I °.'_I co 0|‘... 06 0.1 PN o‘3xo'.4 dz do oir °o &T’°= d?!‘J3 03 $8 6‘! ‘ Fig. 35-6. Lociofcolors confused by Fig. 35-7. Loci ofcolors confused deuteranopes. byprotanopes. Figure 35-6 shows thecolor mixing ofaparticular typeofcolor-blind person called adeuteranope. Forhim, thelociofconstant colors arenotpoints, but certain lines, along each ofwhich thecolor appears tohimtobethesame. lfthe theory thatheismissing oneofthethree pieces ofinformation isright, allthese lines should intersect atapoint. Ifwecarefully measure onthisgraph, they do intersect perfectly. Obviously, therefore, thishasbeen made byamathematician anddoes notrepresent realdata! Asamatter offact, ifwelook atthelatest paper with realdata, itturns outthatinthegraph ofFig.35—6, thepoint offocus ofall thelines isnotexactly attheright place. Using thelines intheabove figure, we cannot find reasonable spectra; weneed negative andpositive absorptions in difl‘erent regions. Butusing thenewdata ofYustova, itturns outthateach ofthe absorption curves iseverywhere positive. 35-8 Figure 35-7 shows adifferent kind ofcolor blindness, thatoftheprotanope, which hasafocus neartheredendoftheboundary curve. Yustova getsapproxi- mately thesame position inthiscase. Using thethree different kinds ofcolor blindness, thethree pigment response curves have finally been determined, and areshown inFig.35-8. Finally? Perhaps. There isaquestion astowhether the three-pigment idea isright, whether color blindness results from lack ofone pigment, andeven whether thecolor-mix dataoncolor blindness areright. Difl"er- entworkers getdifferent results. This field isstillverymuch under development. 35-6 Physiochemistry ofcolor vision Now, what about checking these curves against actual pigments intheeye? Thepigments thatcanbeobtained from aretina consist mainly ofapigment called visual purple. Themost remarkable features ofthisare,first, thatitisintheeye ofalmost every vertebrate animal, andsecond, thatitsresponse curve fitsbeauti- fully withthesensitivity oftheeye,asseeninFig.35-9, inwhich areplotted onthe same scale theabsorption ofvisual purple andthesensitivity ofthedark-adapted eye. This pigment isevidently thepigment thatweseewith inthedark: visual purple isthepigment fortherods, andithasnothing todowith color vision. Thisfactwasdiscovered in1877. Even today itcanbesaidthatthecolor pigments ofthecones have never been obtained inatesttube. In1958itcould besaid that thecolor pigments hadnever been seen atall.Butsince thattime, twoof them have been detected byRushton byavery simple andbeautiful technique. Thetrouble is,presumably, thatsince theeyeissoweakly sensitive tobright light compared withlight oflowintensity, itneeds alotofvisual purple toseewith, butnotmuch ofthecolor pigments forseeing colors. Rushton’s ideaistoleave thepigment intheeye,andmeasure itanyway. What hedoes isthis. There isan instrument called anopthalmoscope forsending light intotheeyethrough thelens andthen focusing thelight thatcomes back out. With itonecanmeasure how much isreflected. Soonemeasures thereflection coelficient oflight which hasgone twice through thepigment (reflected byaback layer intheeyeball, andcoming outthrough thepigment ofthecone again). Nature isnotalways sobeautifully designed. Thecones areinterestingly designed sothatthelight thatcomes intothe cone bounces around andworks itswaydown intothelittle sensitive points atthe apex. Thelight goesright down intothesensitive point, bounces atthebottom and comes back outagain, having traversed aconsiderable amount ofthecolor-vision pigment; also, bylooking atthefovea, where there arenorods, oneisnotconfused byvisual purple. Butthecolor oftheretina hasbeen seenalong timeago: itisa sortoforangey pink; then there arealltheblood vessels, andthecolor ofthe material attheback, andsoon. How doweknow when wearelooking atthe pigment? Answer: First wetakeacolor-blind person, whohasfewer pigments and forwhom itistherefore easier tomake theanalysis. Second, thevarious pigments, likevisual purple, have anintensity change when theyarebleached bylight; when weshine light onthem theychange their concentration. So,while looking atthe absorption spectrum oftheeye,Rushton putanother beam inthewhole eye,which changes theconcentration ofthepigment, andhemeasured thechange inthe spectrum, andthedifference, ofcourse, hasnothing todowith theamount of blood orthecolor ofthereflecting layers, andsoon,butonly thepigment, andin thismanner Rushton obtained acurve forthepigment oftheprotanope eye,which isgiven inFig.35-10. Thesecond curve inFig.35-10 isacurve obtained with anormal eye. This wasobtained bytaking anormal eyeand, having already determined what one pigment was, bleaching theother oneintheredwhere thefirstoneisinsensitive. Redlight hasnoeffect ontheprotanope eye,butdoes inthenormal eye,andthus onecanobtain thecurve forthemissing pigment. Theshape ofonecurve fits beautifully with Yustova’s green curve, buttheredcurve isalittle bitdisplaced. Soperhaps wearegetting ontheright track. Orperhaps not——the latest work withdeuteranopes does notshow anydefinite pigment missing. 35-92.0 B G |.O R O 4 1. t F t HA) Fig. 35-8. The spectral sensitivity curves ofanormal trichromat's receptors. [um/'00:/fiarrdflsnqnfibne§§ggt3\\.fi-._l_._, 400 .500 coo Ware/my/b--m;4 Fig. 35-9. The sensitivity curve of thedark-adapted eye, compared with theabsorption curve ofvisual purple. Dgubu away tmblenmtty o,v54 = , 0 '59 o-cs -‘Nco: - ~I I , 0..H F:002 e . O-Ol rm = Fig. 35-10. Absorption spectrum of I thecolor pigment ofaprotanope color- lIo-1lr§ I_ = blind eye (squares) and anormal eye soo"If sso we 050 (dots). Color isnotaquestion ofthephysics ofthelight itself.Color isasensation, andthesensation fordifferent colors isdifferent indifferent circumstances. For instance, ifwehave apink light, made bysuperimposing crossing beams ofwhite light andredlight (allwecanmake with white andredispink, obviously), wemay show thatwhite light mayappear blue. Ifweplace anobject inthebeams, itcasts twoshadows—one illuminated bythewhite light alone andtheother bythered. Formost people the“white” shadow ofanobject looks blue, butifwekeep ex- panding thisshadow until itcovers theentire screen, weseethatitsuddenly appears white, notblue! Wecangetother effects ofthesame nature bymixing red,yellow, andwhite light. Red, yellow, andwhite light canproduce only orangey yellows, andsoon. Soifwemixsuch lights roughly equally, wegetonly orange light. Nevertheless, bycasting different kinds ofshadows inthelight, with various overlaps ofcolors, onegetsquite aseries ofbeautiful colors which arenotinthe light themselves (that isonly orange), butinoursensations. Weclearly seemany different colors thatarequite unlike the“physical” ones inthebeam. Itisvery important toappreciate thataretina isalready “thinking” about thelight; itis comparing what itseesinoneregion with what itseesinanother, although not consciously. What weknow ofhow itdoes thatisthesubject ofthenextchapter. BIBLIOGRAPHY Committee onCalorimetry, Optical Society ofAmerica, TheScience ofColor, Thomas Y.Crowell Company, NewYork, 1953. Hacm", S.,S.S1-ILAER, andM.H.Pnzauus, “Energy, Quanta, andVision,” Journal of General Physiology, 1942, 25,819-840. MORGAN, CLIFFORD andE1.1o'r STELLAR, Physiological Psychology, 2nded.,McGraw- HillBook Company, Inc., 1950. NUBERG, N.D.andE.N.YUSTOVA, “Researches onDichromatic Vision andthe Spectral Sensitivity oftheReceptors ofTrichromats,” presented atSymposium No.8, Visual Problems ofColour, Vol. II,National Physical Laboratory, Teddington, England, September 1957. Published byHerMajesty’s Stationery Office, London, 1958. RUS1-ITON, W.A.,“The Cone Pigments oftheHuman Fovea inColour Blind and Normal," presented atSymposium No. 8,Visual Problems ofColour, Vol. I,National Physical Laboratory, Teddington, England, September 1957. Published byHerMajesty’s Stationery Office, London, 1958. WOODWORTH, ROBERT S.,Experimental Psychology, Henry Holt andCompany, New York, 1938. Revised edition, 1954, byRobert S.Woodworth andH.Schlosberg. 35-10 36 Mechanisms ofSeeing 36-1 Thesensation ofcolor Indiscussing thesense ofsight, wehave torealize that(outside ofagallery ofmodern art!) onedoes notseerandom spots ofcolor orspots oflight. When welook atanobject weseeamanorathing; inother words, thebrain interprets what wesee. How itdoes that, nooneknows, anditdoes it,ofcourse, atavery high level. Although weevidently dolearn torecognize what aman looks like after much experience, there areanumber offeatures ofvision which aremore elementary butwhich alsoinvolve combining information from different parts of what wesee. Tohelp usunderstand how wemake aninterpretation ofanentire image, itisworth while tostudy theearliest stages oftheputting together ofin- formation from thedifferent retinal cells. Inthepresent chapter weshall concen- trate mainly onthataspect ofvision, although weshall alsomention anumber of sideissues aswegoalong. Anexample ofthefactthatwehave anaccumulation, atavery elementary level, ofinformation from several parts oftheeyeatthesame time, beyond our voluntary control orability tolearn, wasthatblue shadow which wasproduced bywhite light when both white andredwere shining onthesame screen. This effect atleast involves theknowledge thatthebackground ofthescreen ispink, even though, when wearelooking attheblue shadow, itisonly “white” light coming intoaparticular spot intheeye; somewhere, pieces ofinformation have been puttogether. Themore complete andfamiliar thecontext is,themore theeye willmake corrections forpeculiarities. Infact, Land hasshown thatifwemixthat apparent blue andtheredinvarious proportions, byusing twophotographic transparencies with absorption infront oftheredandthewhite indifferent pro- portions, itcanbemade torepresent arealscene, withrealobjects, rather faithfully. Inthiscasewegetalotofintermediate apparent colors too,analogous towhat we would getbymixing redandblue-green; itseems tobeanalmost complete setof colors, butifwelook veryhard atthem, theyarenotsovery good. Even so,itis surprising howmuch canbeobtained from justredandwhite. Themore thescene looks likearealsituation, themore oneisabletocompensate forthefactthatallthe light isactually nothing butpink! Another example istheappearance of“colors” inablack-and-white rotating disc, whose black andwhite areas areasshown inFig.36-1. When thediscis rotated, thevariations oflight anddark atanyoneradius areexactly thesame; it isonly thebackground thatisdifferent forthetwokinds of“stripes.” Yetoneof the“rings” appears colored with onecolor andtheother with another.* Noone yetunderstands thereason forthose colors, butitisclear thatinformation isbeing puttogether atavery elementary level, intheeyeitself, most likely. Almost allpresent-day theories ofcolor vision agree thatthecolor-mixing data indicate thatthere areonly three pigments inthecones oftheeye,andthatitis thespectral absorption inthose three pigments that fundamentally produces thecolor sense. Butthetotal sensation that isassociated with theabsorption characteristics ofthethree pigments acting together isnotnecessarily thesumof theindividual sensations. Weallagree thatyellow simply does notseem tobe reddish green; infactitmight beatremendous surprise tomost people todiscover thatlight is,infact, amixture ofcolors, because presumably thesensation oflight *Thecolors depend onspeed ofrotation, onthebrightness ofillumination, andto some extent onwholooks atthem andhowintently hestares atthem. 36-136-1 Thesensation ofcolor 36-2 Thephysiology oftheeye 36-3 Therodcells 36-4 Thecompound (insect) eye 36-5 Other eyes 36-6 Neurology ofvision Fig. 36-l. When adisc like the above isspun, colors appear inonly one ofthetwo darker "rings." Ifthespin direction isreversed, thecolors appear intheother ring. Neural Responses 18 ooo __.y-b-li,(p~1-2-I r-Q -lithe,-2}) I-N! IK‘(oOy§l)-l.(l9I9yI Fig. 36-2. Neural connections ac- cording toan"opponent" theory of color vision.isduetosome other process than asimple mixture likeachord inmusic, where thethree notes arethere atthesame time andifwelisten hard wecanhear them individually. Wecannot look hard andseetheredandthegreen. Theearliest theories ofvision said thatthere arethree pigments andthree kinds ofcones, each kind containing onepigment; thatanerve runs from each cone tothebrain, sothatthethree pieces ofinformation arecarried tothebrain; andtheninthebrain, anything canhappen. This, ofcourse, isanincomplete idea: itdoes nogood todiscover thattheinformation iscarried along theoptic nerve tothebrain, because wehave noteven started tosolve theproblem. Wemust askmore basic questions: Does itmake anydifference where theinformation isput together? Isitimportant thatitbecarried right upintothebrain intheoptic nerve, orcould theretina dosome analysis first? Wehave seen apicture ofthe retina asanextremely complicated thing with lotsofinterconnections (Fig. 35-2) anditmight make some analyses. Asamatter offact, people who study anatomy andthedevelopment ofthe eyehave shown thattheretina is,infact, thebrain: inthedevelopment oftheem- bryo, apiece ofthebrain comes outinfront, andlong fibers grow back, con- necting theeyestothebrain. Theretina isorganized injustthewaythebrain is organized and, assomeone hasbeautifully putit,“The brain hasdeveloped away tolook outupon theworld.” Theeyeisapiece ofbrain thatistouching light, so tospeak, ontheoutside. Soitisnotatallunlikely thatsome analysis ofthecolor hasalready been made intheretina. This gives usaveryinteresting opportunity. None oftheother senses involves such alarge amount ofcalculation, sotospeak, before thesignal getsintoanerve thatonecanmake measurements on.Thecalculations foralltherestofthesenses usually happen inthebrain itself, where itisverydiflicult togetatspecific places tomake measurements, because there aresomany interconnections. Here, with thevisual sense, wehave thelight, three layers ofcells making calculations, and theresults ofthecalculations being transmitted through theoptic nerve. So wehave thefirstchance toobserve physiologically how, perhaps, thefirstlayers ofthebrain work intheir firststeps. Itisthus ofdouble interest, notsimply interesting forvision, butinteresting tothewhole problem ofphysiology. Thefactthatthere arethree pigments does notmean thatthere must bethree kinds ofsensations. Oneoftheother theories ofcolor vision hasitthatthere are really opposing color schemes (Fig. 36-2). That is,oneofthenerve fibers carries alotofimpulses ifthere isyellow being seen, andlessthan usual forblue. Another nerve fiber carries green andredinformation inthesame way, andanother, white andblack. Inother words, inthistheory someone hasalready started tomake a guess astothesystem ofwiring, themethod ofcalculation. Theproblems wearetrying tosolve byguessing atthese firstcalculations are questions about theapparent colors thatareseenonapink background, what hap- pens when theeyeisadapted todifferent colors, andalsotheso-called psychological phenomena. Thepsychological phenomena areofthenature, forinstance, that white does not“feel” likeredandyellow andblue, andthistheory wasadvanced because thepsychologists saythatthere arefour apparent pure colors: “There are four stimuli which have aremarkable capacity toevoke psychologically simple blue, yellow, green, andredhues respectively. Unlike sienna, magenta, purple, or most ofthediscriminable colors, these simple hues areunmixed inthesense that none partakes ofthenature oftheother; specifically, blueisnotyellowish, reddish, orgreenish, andsoon;these arepsychologically primary hues.” That isapsycho- logical fact, so-called. Tofindoutfrom what evidence thispsychological factwas deduced, wemust search veryhard indeed through alltheliterature. Inthemodern literature allwefindonthesubject arerepeats ofthesame statement, orofoneby aGerman psychologist, whousesasoneofhisauthorities Leonardo daVinci, who, ofcourse, weallknow wasagreat artist. Hesays, “Leonardo thought there were fivecolors.” Then, looking stillfurther, wefind, inastillolder book, theevidence forthesubject. Thebook sayssomething likethis: “Purple isreddish-blue, orange isreddish-yellow, butcanredbeseenaspurplish-orange? Arenotredandyellow more unitary than purple ororange? Theaverage person, asked tostate which 36-2 colors areunitary, names red,yellow, andblue, these three, andsome observers addafourth, green. Psychologists areaccustomed toaccept thefour assalient hues.” Sothatisthesituation inthepsychological analysis ofthismatter: ifevery- body saysthere arethree, andsomebody saysthere arefour, andthey want itto befour, itwillbefour. That shows thedifficulty with psychological researches. Itisclear thatwehave such feelings, butitisverydifficult toobtain much informa- tionabout them. Sotheother direction togoisthephysiological direction, tofindoutexperi- mentally what actually happens inthebrain, theeye,theretina, orwherever, and perhaps todiscover thatsome combinations ofimpulses from various cells move along certain nerve fibers. Incidentally, primary pigments donothave tobein separate cells; onecould have cells inwhich aremixtures ofthevarious pigments, cellswiththeredandthegreen pigments, cellswith allthree (theinformation ofall three isthen white information), andsoon.There aremany ways ofhooking the system up,andwehave tofind outwhich waynature hasused. Itwould be hoped, ultimately, thatwhen weunderstand thephysiological connections wewill have alittle bitofunderstanding ofsome ofthose aspects ofthepsychology, sowe look inthatdirection. 36-2 Thephysiology oftheeye Webegin bytalking notonly about color vision, butabout vision ingeneral, justtoremind ourselves about theinterconnections intheretina, shown inFig. 35-2. Theretina isreally likethesurface ofthebrain. Although theactual picture through amicroscope isalittle more complicated looking than thissomewhat schematized drawing, bycareful analysis onecanseeallthese interconnections. There isnoquestion thatonepartofthesurface oftheretina isconnected toother parts, andthattheinformation thatcomes outonthelong axons, which produce theoptic nerve, arecombinations ofinformation from many cells. There arethree layers ofcells inthesuccession offunction: there areretinal cells, which arethe ones thatthelight afl'ects, anintermediate cellwhich takes information from a single orafewretinal cells andgives itoutagain toseveral cellsinathird layer of cellsandcarries ittothebrain. There areallkinds ofcross connections between cellsinthelayers. Wenow turn tosome aspects ofthestructure andperformance oftheeye (seeFig.35-1). Thefocusing ofthelight isaccomplished mainly bythecornea, bythefactthatithasacurved surface which “bends” thelight. This iswhywe cannot seeclearly under water, because wethen donothave enough difference between theindex ofthecornea, which is1.37, andthatofthewater, which is1.33. Behind thecornea iswater, practically, with anindex of1.33, andbehind thatisa lenswhich hasavery interesting structure: itisaseries oflayers, likeanonion, except thatitisalltransparent, andithasanindex of1.40inthemiddle and1.38 attheoutside. (Itwould beniceifwecould make optical glass inwhich wecould adjust theindex throughout, forthen wewould nothave tocurve itasmuch as wedowhen wehave auniform index.) Furthermore, theshape ofthecornea isnot thatofasphere. Aspherical lenshasacertain amount ofspherical aberration. Thecornea is“flatter” attheoutside than isasphere, injustsuch amanner that thespherical aberration islessforthecornea than itwould beifweputaspherical lensinthere! Thelight isfocused bythecornea-lens system onto theretina. As welook atthings thatarecloser andfarther away, thelenstightens andloosens andchanges thefocus toadjust forthedifferent distances. Toadjust forthetotal amount oflight there istheiris,which iswhat wecallthecolor oftheeye,brown orblue, depending onwho itis;astheamount oflight increases anddecreases, theirismoves inandout. Letusnow look attheneural machinery forcontrolling theaccommodation ofthelens, themotion oftheeye,themuscles which turntheeyeinthesocket, and theiris,shown schematically inFig.36-3. Ofalltheinformation thatcomes out oftheoptic nerve A,thegreat majority isdivided intooneoftwobundles (which wewilltalkabout later) andthence tothebrain. Butthere areafewfibers, of 36-3KgQ. '“~L\ -mym eag_, wk7/7A A J R 2 N Fig. 36-3. The neural interconnec- tions forthe mechanical operation of theeyes. Tlmpora|\ .¢6§ 6§§. llllrffi,-’~;}f\ rf1‘§,;’-;};\\\\\ \\\\‘§~‘E=.">"' \\t-»'i"’ Qgsll’“,\."(?%5*.({(E/f <l§e //ifi ii‘. ao @@@ Fig. 36-4. The neural connections from theeyes tothevisual cortex.interest tousnow, which donotrundirectly tothevisual cortex ofthebrain where we“see” theimages, butinstead gointothemid-brain H.These arethe fibers which measure theaverage light andmake adjustment fortheiris;or,ifthe image looks foggy, theytrytocorrect thelens; or,ifthere isadouble image, they trytoadjust theeyeforbinocular vision. Atanyrate, they gothrough themid- brain andfeedback intotheeye.AtKarethemuscles which runtheaccommoda- tionofthelens, andatLanother onethatruns intotheiris. Theirishastwo muscle systems. Oneisacircular muscle Lwhich, when itisexcited, pulls inand closes down theiris;itactsveryrapidly andthenerves aredirectly connected from thebrain through short axons intotheiris.Theopposite muscles areradial muscles, sothat, when thethings getdark andthecircular muscle relaxes, these radial muscles pullout. Here wehave, asinmany places inthebody, apairofmuscles which work inopposite directions, andinalmost every such casethenerve systems which control thetwoareverydelicately adjusted, sothatwhen signals aresentin totighten one, signals areautomatically sentintoloosen theother. Theirisisa peculiar exception: thenerves which make theiriscontract aretheones wehave already described, butthenerves which make theirisexpand come outfrom noone knows exactly where, godown intothespinal cord back ofthechest, into the thoracic sections, outofthespinal cord, upthrough theneck ganglia, andallthe wayaround andback upintothehead inorder toruntheother endoftheiris. Infact, thesignal goes through acompletely different nervous system, notthe central nervous system atall,butthesympathetic nervous system, soitisavery strange wayofmaking things go. Wehave already emphasized another strange thing about theeye,that the light-sensitive cellsareonthewrong side, sothatthelighthastogothrough several layers ofother cells before itgetstothereceptors—it isbuilt inside out! Sosome ofthefeatures arewonderful andsome areapparently stupid. Figure 36-4 shows theconnections oftheeyetothepartofthebrain which is most directly concerned with thevisual process. Theoptic nerve fibers runintoa certain areajustbeyond D,called thelateral geniculate, whereupon they runout toasection ofthebrain called thevisual cortex. Notice thatsome ofthefibers from each eyearesentover totheother sideofthebrain, sothepicture formed isin- complete. Theoptic nerves from theleftsideoftheright eyerunacross theoptic chiasma B,while theones ontheleftsideofthelefteyecome around andgothis same way. Sotheleftsideofthebrain receives alltheinformation which comes from theleftsideoftheeyeball ofeach eye,i.e.,ontheright sideofthevisual field, while theright sideofthebrain seestheleftsideofthevisual field. This is themanner inwhich theinformation from each ofthetwoeyes isputtogether inorder totellhowfaraway things are. This isthesystem ofbinocular vision. Theconnections between theretina andthevisual cortex areinteresting. Ifa spot intheretina isexcised ordestroyed inanyway, then thewhole fiber willdie, andwecanthereby findoutwhere itisconnected. Itturns outthat, essentially, theconnections areonetoone—for each spot intheretina there isonespot in thevisual cortex-and spots that arevery close together intheretina arevery close together inthevisual cortex. Sothevisual cortex stillrepresents thespatial arrangement oftherods andcones, butofcourse much distorted. Things which areinthecenter ofthefield, which occupy avery small part oftheretina, are expanded over many, many cells inthevisual cortex. Itisclear thatitisuseful to have things which areoriginally close together, stillclose together. The most remarkable aspect ofthematter, however, isthefollowing. Theplace where one would think itwould bemost important tohave things close together would be right inthemiddle ofthevisual field. Believe itornot,theup-and-down linein ourvisual field aswelook atsomething isofsuch anature thattheinformation from allthepoints ontheright sideofthatlineisgoing intotheleftsideofthebrain, andinformation from thepoints ontheleftsideisgoing intotheright sideofthe brain, andthewaythisareaismade, there isacutright down through themiddle, sothatthethings thatareveryclose together right inthemiddle areveryfarapart inthebrain! Somehow, theinformation hastogofrom onesideofthebrain to theother through some other channels, which isquite surprising. 36-4 Thequestion ofhowthisnetwork evergets“wired” together isveryinteresting. Theproblem ofhowmuch isalready wired andhowmuch islearned isanoldone. Itused tobethought long agothatperhaps itdoes nothave tobewired carefully atall,itisonly justroughly interconnected, andthen, byexperience, theyoung child learns thatwhen athing is“upthere” itproduces some sensation inthebrain. (Doctors always telluswhat theyoung child “feels,” buthow dotheyknow what achild feels attheageofone?) Thechild, attheageofone,supposedly seesthat anobject is“upthere,” getsacertain sensation, andlearns toreach “there,” because when hereaches “here,” itdoes notwork. That approach probably isnot correct, because wealready seethatinmany cases there arethese special detailed interconnections. More illuminating aresome most remarkable experiments done with asalamander. (Incidentally, with thesalamander there isadirect crossover connection, without theoptic chiasma, because theeyesareoneach sideofthehead andhave nocommon area. Salamanders donothave binocular vision.) Theexperi- ment isthis.Wecancuttheoptic nerve inasalamander andthenerve willgrow out from theeyesagain. Thousands andthousands ofcellfibers willthus re-establish themselves. Now, intheoptic nerve thefibers donotstayadjacent toeach other—it islikeagreat, sloppily made telephone cable, allthefibers twisting andturning, butwhen itgetstothebrain theyareallsorted outagain. When wecuttheoptic nerve ofthesalamander, theinteresting question is,willitevergetstraightened out? Theanswer isremarkable: yes. Ifwecuttheoptic nerve ofthesalamander andit grows back, thesalamander hasgood visual acuity again. However, ifwecutthe optic nerve andturntheeyeupside down andletitgrow back again, ithasgood visual acuity allright, butithasaterrible error: when thesalamander seesafly “uphere,” itjumps atit“down there,” anditnever learns. Therefore there issome mysterious waybywhich thethousands andthousands offibers findtheir right places inthebrain. This problem ofhowmuch iswired in,andhowmuch isnot,isanimportant problem inthetheory ofthedevelopment ofcreatures. Theanswer isnotknown butisbeing studied intensively. Thesame experiment inthecase ofagoldfish shows thatthere isaterrible knot, likeagreat scarorcomplication, intheoptic nerve where wecutit,butin spite ofallthisthefibers grow back totheir right places inthebrain. Inorder todothis,astheygrow intotheoldchannels oftheoptic nerve they must make several decisions about thedirection inwhich theyshould grow. How dothey dothis? There seem tobechemical clues thatdifferent fibers respond to differently. Think oftheenormous number ofgrowing fibers, each ofwhich isan individual differing insome wayfrom itsneighbors; inresponding towhatever thechemical clues are,itresponds inaunique enough waytofinditsproper place forultimate connection inthebrain! This isaninteresting—a fantastic-—thing. Itisoneofthegreat recently discovered phenomena ofbiology andisundoubtedly connected tomany older unsolved problems ofgrowth, organization, anddevelop- ment oforganisms, andparticularly ofembryos. Oneother interesting phenomenon hastodowith themotion oftheeye. The eyes must bemoved inorder tomake thetwoimages coincide indifferent cir- cumstances. These motions areofdifferent kinds: oneistofollow something, which requires thatboth eyesmust gointhesame direction, right orleft,andthe other istopoint them toward thesame place atvarious distances away, which requires thattheymust move oppositely. Thenerves going intothemuscles ofthe eyearealready wired upforjustsuch purposes. There isonesetofnerves which willpullthemuscles ontheinside ofoneeyeandtheoutside oftheother, andrelax theopposite muscles, sothatthetwoeyesmove together. There isanother center where anexcitation willcause theeyestomove intoward each other from parallel. Either eyecanbeturned outtothecorner iftheother eyemoves toward thenose, butitisimpossible consciously orunconsciously toturnboth eyesoutatthesame time, notbecause there arenomuscles, butbecause there isnowaytosend asignal toturn both eyes out,unless wehave hadanaccident orthere issomething the matter, forinstance ifanerve hasbeen cut. Although themuscles ofoneeyecan certainly steer thateyeabout, noteven aYogi isabletomove botheyesoutfreely 36-53 1 -C;-;: —\ \i as--5°,“ ~*°_\__ .t=on-_-; l_; : , : I/_,Cf a ; El. 5 '.\. , °oe" 5 Ft '3C2 \\‘_<l‘lllll\\itth\“*e » \_, /\ ' " ww- Fig. 36-5. Electron micrograph ofa rodcell. (iii! CIH3 C'H3 C\C/c\\/°\\/C\\/c\\ C/ \c/ \\c C C C O (II (i/C“: \c/\ cu, Fig. 36-6. The structure ofretinene.under voluntary control, because there does notseem tobeanywaytodoit.We arealready wired toacertain extent. This isanimportant point, because most of theearlier books onanatomy andpsychology, andsoon,donotappreciate ordo notemphasize thefactthat wearesocompletely wired already—they saythat everything isjustlearned. 36-3 Therodcells Letusnowexamine inmore detail what happens intherodcells. Figure 36-5 shows anelectron micrograph ofthemiddle ofarodcell(therodcellkeeps going upoutofthefield). There arelayer after layer ofplane structures, shown magnified attheright, which contain thesubstance rhodopsin (visual purple), thedye, orpigment, which produces theeffects ofvision intherods. Therhodopsin, which isthepigment, isabigprotein which contains aspecial group called retinene, which canbetaken offtheprotein, andwhich is,undoubtedly, themain cause of theabsorption oflight. Wedonotunderstand thereason fortheplanes, butitis verylikely thatthere issome reason forholding alltherhodopsin molecules parallel. Thechemistry ofthething hasbeen worked outtoalarge extent, butthere might besome physics toit.Itmay bethatallofthemolecules arearranged insome kind ofarowsothatwhen oneisexcited anelectron which isgenerated, say,may runallthewaydown tosome place attheendtogetthesignal out,orsomething. Thissubject isveryimportant, andhasnotbeenworked out.Itisafieldinwhich both biochemistry andsolid state physics, orsomething likeit,willultimately beused. This kind ofastructure, with layers, appears inother circumstances where light isimportant, forexample inthechloroplast inplants, where thelight causes photosynthesis. Ifwemagnify those, wefindthesame thing with almost thesame kind oflayers, butthere wehave chlorophyll, ofcourse, instead ofretinene. The chemical form ofretinene isshown inFig.36-6. Ithasaseries ofalternate double bonds along thesidechain, which ischaracteristic ofnearly allstrongly absorbing organic substances, likechlorophyll, blood, andsoon.This substance isimpossible forhuman beings tomanufacture intheir owncells—we have toeatit.Soweeatit intheform ofaspecial substance, which isexactly thesame asretinene except thatthere isahydrogen tiedontheright end; itiscalled vitamin A,andifwedo noteatenough ofit,wedonotgetasupply ofretinene, andtheeyebecomes what wecallnight blind, because there isthen notenough pigment intherhodopsin toseewith therods atnight. Thereason why such aseries ofdouble bonds absorbs light very strongly is alsoknown. Wemayjustgiveahint: Thealternating series ofdouble bonds is called aconjugated double bond; adouble bond means thatthere isanextra electron there, andthisextra electron iseasily shifted totheright orleft. When light strikes thismolecule, theelectron ofeach double bond isshifted over byonestep. All theelectrons inthewhole chain shift, likeastring ofdominoes falling over, and though each onemoves only alittle distance (wewould expect that, inasingle atom, wecould move theelectron only alittle distance), theneteffect isthesame asthough theoneattheendwasmoved over totheother end! Itisthesame as though oneelectron went thewhole distance back andforth, andso,inthismanner, wegetamuch stronger absorption under theinfluence oftheelectric field, than if wecould only move theelectron adistance which isassociated with oneatom. So,since itiseasytomove theelectrons back andforth, retinene absorbs light very strongly; thatisthemachinery ofthephysical-chemical endofit. 36-4 Thecompound (insect) eye Letusnowreturn tobiology. Thehuman eyeisnottheonly kind ofeye. In thevertebrates, almost alleyes areessentially likethehuman eye. However, in thelower animals there aremany other kinds ofeyes: eyespots, various eyecups, andother lesssensitive things, which wehave notime todiscuss. Butthere isone other highly developed eyeamong theinvertebrates, thecompound eyeoftheinsect. (Most insects having large compound eyes alsohave various additional simpler eyesaswell.) Abeeisaninsect whose vision hasbeen studied verycarefully. Itis 36-6 easytostudy theproperties ofthevision ofbeesbecause theyareattracted tohoney, andwecanmake experiments inwhich weidentify thehoney byputting itonblue paper orredpaper, andseewhich onethey come to.Bythismethod some very interesting things have been discovered about thevision ofthebee. Inthefirstplace, intrying tomeasure how acutely bees could seethecolor difference between twopieces of“white” paper, some researchers found theywere notvery good, andothers found they were fantastically good. Even ifthetwo pieces ofwhite paper were almost exactly thesame, thebees could stilltellthe difference. Theexperimenters used zincwhite foronepiece ofpaper andlead white fortheother, andalthough these look exactly thesame tous,thebeecould easily distinguish them, because they reflect adifferent amount intheultraviolet. Inthiswayitwasdiscovered thatthebee’s eyeissensitive over awider range of thespectrum than isourown. Oureyeworks from 7000 angstroms to4000 ang- stroms, from redtoviolet, butthebee’s canseedown to3000 angstroms intothe ultraviolet! This makes foranumber ofdifferent interesting effects. Inthefirst place, beescandistinguish between many flowers which touslook alike. Ofcourse, wemust realize thatthecolors offlowers arenotdesigned foroureyes, butforthe bee;theyaresignals toattract thebeestoaspecific flower. Weallknow thatthere aremany “white” flowers. Apparently white isnotvery interesting tothebees, because itturns outthat allofthewhite flowers have different proportions of reflection intheultraviolet; they donotreflect onehundred percent oftheultra- violet aswould atruewhite. Allthelight isnotcoming back, theultraviolet is missing, andthatisacolor, justas,forus,iftheblue ismissing, itcomes out yellow. So,alltheflowers arecolored forthebees. However, wealsoknow that redcannot beseenbybees. Thus wemight expect thatallredflowers should look black tothebee. NotsolAcareful study ofredflowers shows, first, thateven with ourowneyewecanseethatagreat majority ofredflowers have abluish tinge be- cause theyaremainly reflecting anadditional amount intheblue, which isthepart thatthebeesees. Furthermore, experiments alsoshow thatflowers vary intheir reflection oftheultraviolet over different parts ofthepetals, andsoon.Soifwe could seetheflowers asbeesseethem theywould beeven more beautiful andvaried! Ithasbeen shown, however, thatthere areafewredflowers which donotre- flectintheblueorintheultraviolet, andwould, therefore, appear black tothebee! Thiswasofquite some concern tothepeople whoworry about thismatter, because black does notseem likeaninteresting color, since itishard totellfrom adirty oldshadow. Itactually turned outthatthese flowers were notvisited bybees, these aretheflowers thatarevisited byhummingbirds, andhummingbirds canseethered! Another interesting aspect ofthevision ofthebeeisthatbeescanapparently tellthedirection ofthesunbylooking atapatch ofblue sky,without seeing the sunitself. Wecannot easily dothis. Ifwelook outthewindow attheskyandsee thatitisblue, inwhich direction isthesun? Thebeecantell,because thebeeis quite sensitive tothepolarization oflight, andthescattered light oftheskyis polarized.* There isstillsome debate about howthissensitivity operates. Whether itisbecause thereflections ofthelight aredifferent indifferent circumstances, or thebee’s eyeisdirectly sensitive, isnotyetknowntl Itisalsosaidthatthebeecannotice flicker upto200oscillations persecond, while weseeitonly upto20.Themotions ofbees inthehives arevery quick; thefeetmove and thewings vibrate, butitisvery hard forustoseethese motions with oureye. However, ifwecould seemore rapidly wewould beable toseethemotion. Itisprobably very important tothebeethatitseyehassuch arapid response. *Thehuman eyealsohasaslight sensitivity tothepolarization oflight, andonecan learn totellthedirection ofthesun! Thephenomenon that isinvolved here iscalled Haidinger’s brush; itisafaint, yellowish hourglass-like pattern seen atthecenter ofthe visual fieldwhen onelooks atabroad, featureless expanse using polarizing glasses. It canalsobeseenintheblueskywithout polarizing glasses ifonerotates hisheadbackand forth about theaxisofvision. TEvidence obtained since thislecture wasgiven indicates thattheeyeisdirectly sensi- tive. 36-7 @ 0-K IP93 Mb III In lof Fig. 36-7. The structure ofanom- matidium (asingle cellofacompound eye). 8 8/ r r Fig. 36-8. Schematic view ofpack- ingofommatidia intheeyeofabee. A0>88Bqf> I ‘\AD° -B/r Av. /Aad=x/a Bm _.._A__.... >8 Fig. 36-9. The optimum size foran ommatidium is5,...Now letusdiscuss thevisual acuity wecould expect from thebee. Theeyeof abeeisacompound eye,anditismade ofalarge number ofspecial cells called ommatidia, which arearranged conically onthesurface ofasphere (roughly) on theoutside ofthebee’s head. Figure 36-7 shows apicture ofonesuch ommatidium. Atthetopthere isatransparent area, akind of“lens,” butactually itismore like afilter orlight pipe tomake thelight come down along thenarrow fiber, which is where theabsorption presumably occurs. Outoftheother endofitcomes the nerve fiber. Thecentral fiber issurrounded onitssides bysixcells which, infact, have secreted thefiber. That isenough description forourpurposes; thepoint is thatitisaconical thing andmany canfitnexttoeach other allover thesurface of theeyeofthebee. Now letusdiscuss theresolution oftheeyeofthebee. Ifwedraw lines (Fig. 36-8) torepresent theommatidia onthesurface, which wesuppose isasphere of radius r,wemay actually calculate how wide each ommatidium isbyusing our brains, andassuming thatevolution isasclever asweare! Ifwehave averylarge ommatidium wedonothave much resolution. That is,onecellgetsapiece of information from onedirection, andtheadjacent cellgetsapiece ofinformation from another direction, andsoon,andthebeecannot seethings inbetween very well. Sotheuncertainty ofvisual acuity intheeyewillsurely correspond toan angle, theangle oftheendoftheommatidium relative tothecenter ofcurvature of theeye. (The eyecells, ofcourse, exist only atthesurface ofthesphere; inside thatisthehead ofthebee.) This angle, from oneommatidium tothenext, is,of course, thediameter oftheommatidia divided bytheradius oftheeyesurface: An,=6/r. (36.1) So,wemay say,“The finer wemake the8,themore thevisual acuity. Sowhy doesn’t thebeejustusevery, very fineommatidia?” Answer: Weknow enough physics torealize thatifwearetrying togetlight down intoanarrow slot, we cannot seeaccurately inagiven direction because ofthediflraction effect. The light thatcomes from several directions canenter and, duetodiffraction, wewill getlight coming inatangle A0,;such that A04 =X/8. (36.2) Now weseethatifwemake the6toosmall, then each ommatidium does not look inonly onedirection, because ofdiffraction! Ifwemake them toobig,each oneseesinadefinite direction, butthere arenotenough ofthem togetagood view ofthescene. Soweadjust thedistance dinorder tomake minimal thetotal effect ofthese two. Ifweaddthetwotogether, andfindtheplace where thesumhasa minimum (Fig. 36-9), wefindthat dA0+A0 l A L£fi'l)=°=?"E5’ <3“!which gives usadistance a=\/V. (36.4) Ifweguess thatrisabout 3millimeters, take thelight thatthebeeseesas4000 angstroms, andputthetwotogether andtake thesquare root, wefind a=(3><10-3 ><4><10-7)!/2m =3.5X10'5m =35ii. (36.5) Thebook saysthediameter is30a,sothatisrather good agreement! So,apparently, itreally works, andwecanunderstand what determines thesizeofthebee’s eye! Itisalsoeasytoputtheabove number back inandfindouthowgood thebee’s eye actually isinangular resolution; itisvery poor relative toourown. Wecansee things thatarethirty times smaller inapparent sizethan thebee; thebeehasa rather fuzzy out-of-focus image relative towhat wecansee. Nevertheless itisall right, anditisthebestthey cando.Wemight askwhythebees donotdevelop 36-8 agood eyelikeourown, with alensandsoon.There areseveral interesting rea- sons. Inthefirstplace, thebeeistoosmall; ifithadaneyelikeours, butonhis scale, theopening would beabout 30uinsizeanddiffraction would besoimpor- tantthatitwould notbeabletoseevery wellanyway. Theeyeisnotgood ifitis toosmall. Secondly, ifitwere asbigasthebee’s head, then theeyewould occupy thewhole head ofthebee. Thebeauty ofthecompound eyeisthatittakes upno space, itisjustaverythinlayer onthesurface ofthebee. Sowhen weargue that they should have done itourway, wemust remember thatthey hadtheir own problems! 36-5 Other eyes Besides thebees, many other animals canseecolor. Fish, butterflies, birds, andreptiles canseecolor, butitisbelieved that most mammals cannot. The primates canseecolor. Thebirds certainly seecolor, andthataccounts forthe colors ofbirds. There would benopoint inhaving such brilliantly colored males ifthefemales could notnotice it!That is,theevolution ofthesexual “whatever itis”thatthebirds have isaresult ofthefemale being abletoseecolor. Sonext time welook atapeacock andthink ofwhat abrilliant display ofgorgeous color itis,andhow delicate allthecolors are,andwhat awonderful aesthetic sense it takes toappreciate allthat, weshould notcompliment thepeacock, butshould compliment thevisual acuity andaesthetic sense ofthepeahen, because thatis what hasgenerated thebeautiful scene! Allinvertebrates have poorly developed eyes orcompound eyes, butallthe vertebrates have eyesvery similar toourown, with oneexception. Ifweconsider thehighest form ofanimal, weusually say,“Here weare!,” butifwetake aless prejudiced point ofview andrestrict ourselves totheinvertebrates, sothat we cannot include ourselves, andaskwhat isthehighest invertebrate animal, most zoologists agree thattheoctopus isthehighest animal! Itisvery interesting that, besides thedevelopment ofitsbrain anditsreactions andsoon,which arerather good foraninvertebrate, ithasalsodeveloped, independently, adifferent eye. It isnotacompound eyeoraneyespot—-it hasacornea, ithaslids, ithasaniris, ithasalens, ithastworegions ofwater, ithasaretina behind. Itisessentially the same astheeyeofthevertebrates! Itisaremarkable example ofacoincidence in evolution where nature hastwice discovered thesame solution toaproblem, with oneslight improvement. Intheoctopus italsoturns out,amazingly, thattheretina isapiece ofthebrain thathascome outinthesame wayinitsembryonic develop- ment asistrueforvertebrates, buttheinteresting thing which isdifferent isthatthe cells which aresensitive tolight areontheinside, andthecells which dothecal- culation areinback ofthem, rather than “inside out,” asinoureye. Sowesee, atleast, thatthere isnogood reason foritsbeing inside out. Theother timenature tried it,shegotitstraightened out! (SeeFig.36-10.) Thebiggest eyesintheworld arethose ofthegiant squid; they have been found uptol5inches indiameter! 36-6 Neurology ofvision Oneofthemain points ofoursubject istheinterconnection ofinformation from onepart oftheeyetotheother. Letusconsider thecompound eyeofthe horseshoe crab, onwhich considerable experimentation hasbeen done. First of all,wemust appreciate what kind ofinformation cancome along nerves. Anerve carries akind ofdisturbance which hasanelectrical effect thatiseasytodetect, a kind ofwavelike disturbance which runs down thenerve andproduces aneffect attheother end: along piece ofthenerve cell,called theaxon, carries theinfor- mation along, andacertain kind ofimpulse, called a“spike,” goes along ifitis excited atoneend. When onespike goes down thenerve, another cannot immedi- ately follow. Allthespikes areofthesame size, soitisnotthat wegethigher spikes when thething ismore strongly excited, butthatwegetmore spikes per second. Thesizeofthespike isdetermined bythefiber. Itisimportant toappreci- atethisinorder toseewhat happens next. 36-9’,.»*"""M ~'. “""””/,x-‘Ll 8 2~‘~ . Il K \ 'vi\ I1it|u(\‘§;, /;k,, H‘\Hll L /z I - ///’ F '\\ *-vim “_.xiii. .;'_-.‘.' ' ~ __, /‘-' "..“-‘_.' ,____._. ,_,.. /7://///If-.¢n auv vv Fig. 36-IO. Theeyeofanoctopus an7"’ \- ~" .4‘;-tn Fig. 36-l l.Thecompound eyeofthehorseshoe crab. la)Normal view. (b)Cross section. Figures 36-7, ll,I2,13reprinted withpermission from Goldsmith, Sensory Communications, W.A.Rosenblith, ed.Copyright I96l,Massachusetts lnstitute ofTechnology. Figure 36-11(a)shows thecompound eyeofthehorseshoe crab; itisnotvery much ofaneye,ithasonly about athousand ommatidia. Figure 36-1l(b)isa cross section through thesystem; onecanseetheommatidia, with thenerve fibers thatrunoutofthem andgointothebrain. Butnote thateven inahorseshoe crab there arelittle interconnections. They aremuch lesselaborate than inthehuman eye,anditgives usachance tostudy asimpler example. Letusnow look attheexperiments which have been done byputting fine electrodes intotheoptic nerve ofthehorseshoe crab, andshining light ononlyone oftheommatidia, which iseasy todowith lenses. Ifweturn alight onatsome instant to,andmeasure theelectrical pulses thatcome out,wefindthatthere isa slight delay andthen arapid series ofdischarges which gradually slow down toa uniform rate, asshown inFig.36-l2(a). When thelight goes out,thedischarge stops. Now itisvery interesting thatif,while ouramplifier isconnected tothis same nerve fiber, weshine light onadifferent ommatidium nothing happens; nosignal. I?‘ ~I‘"'1! ""'»‘l| '="' "" Ill, '" ~\“‘~\"»I/'~ ] (at Ei \\\\\ I.1 ‘\\E-‘if?-53,444/bums \i '°""l"!"" I Fig. 36-l2. Theresponse tolight of mm .. l'i‘' thenerve fibers oftheeyeofthehorse- shoe crab. Now wedoanother experiment: weshine thelight ontheoriginal ommatidium andgetthesame response, butifwenowturnlight onanother onenearby aswell, thepulses areinterrupted briefly andthen runatamuch lower rate(Fig. 36-12b). Therateofoneisinhibited bytheimpulses which arecoming outoftheother! Inother words, each nerve fiber carries theinformation from oneommatidium, buttheamount thatitcarries isinhibited bythesignals from theothers. So,for example, ifthewhole eyeismore orlessuniformly illuminated, theinformation coming from anyoneommatidium willberelatively weak, because itisinhibited bysomany. Infacttheinhibition isadditive—if weshine light onseveral nearby ommatidia theinhibition isvery great. Theinhibition isgreater when theom- matidia arecloser, andiftheommatidia arefarenough away from oneanother, inhibition ispractically zero. Soitisadditive anddepends onthedistance; hereis afirstexample ofinformation from different parts oftheeyebeing combined in theeyeitself. Wecansee,perhaps, ifwethink about itawhile, thatthisisadevice toenhance contrast attheedges ofobjects, because ifapart ofthescene islight andapartisblack, then theommatidia inthelighted areagiveimpulses thatare 36-10 inhibited byalltheother light intheneighborhood, soitisrelatively weak. On theother hand, anommatidium attheboundary which isgiven a“white” impulse isalsoinhibited byothers intheneighborhood, butthere arenotasmany ofthem, since some areblack; thenetsignal istherefore stronger. Theresult would bea curve, something likethat ofFig.36-13. Thecrab willseeanenhancement of thecontour. Thefactthatthere isanenhancement ofcontours haslong been known; in factitisaremarkable thing thathasbeen commented onbypsychologists many times. Inorder todraw anobject, wehave only todraw itsoutline. How used wearetolooking atpictures thathave onlytheoutline! What istheoutline? The outline isonlytheedge difference between light anddark oronecolor andanother. Itisnotsomething definite. Itisnot,believe itornot,thatevery object hasaline around it!There isnosuch line. Itisonlyinourownpsychological makeup that there isaline; wearebeginning tounderstand thereasons whythe“line” isenough ofacluetogetthewhole thing. Presumably ourowneyeworks insome similar manner—much more complicated, butsimilar. Finally, weshall briefly describe themore elaborate work, thebeautiful, advanced work thathasbeen done onthefrog. Doing acorresponding experiment onafrog, byputting very fine, beautifully built needlelike probes intotheoptic nerve ofafrog, onecanobtain thesignals that aregoing along oneparticular axon and,justasinthecase ofthehorseshoe crab, wefindthattheinformation does notdepend onjustonespot intheeye,butisasum ofinformation over several spots. Themost recent picture oftheoperation ofthefrog’s eyeisthefollowing. Onecanfindfourdifl'erent kinds ofoptic nerve fibers, inthesense thatthere are four different kinds ofresponses. These experiments were notdone byshining on-and-off impulses oflight, because thatisnotwhat afrogsees. Afrogjustsits there andhiseyesnever move, unless thelilypadisflopping back andforth, and inthatcase hiseyes wobble justright sothattheimage stays put. Hedoes not turn hiseyes. Ifanything moves inhisfield ofvision, likealittle bug(hehasto beabletoseesomething small moving inthefixed background), itturns outthat there arefourdifferent kinds offibers which discharge, whose properties aresum- marized inTable 36-1. Sustained edge detection, nonerasable, means thatifwe bring anobject with anedge intothefield ofview ofthefrog, then there arealot ofimpulses inthisparticular fiber while theobject ismoving, buttheydiedown to asustained impulse thatcontinues aslong astheedge isthere, even ifitisstanding still. Ifweturn outthelight, theimpulses stop. Ifweturn itonagain while the edge isstillinview, theystart again. They arenoterasable. Another kind offiber isvery similar, except thatiftheedge isstraight itdoes notwork. Itmust bea convex edge with dark behind it!How complicated must bethesystem ofinter- connections intheretina oftheeyeofthefroginorder forittounderstand thata convex surface hasmoved in!Furthermore, although thisfiber does sustain some- what, itdoes notsustain aslong astheother, andifweturnoutthelight andturn itonagain itdoes notbuild upagain. Itdepends onthemoving inoftheconvex Table 36-1 Types ofresponse inoptic nerve fibers ofafrog Type Speed Angular field 1.Sustained edge detection (nonerasable) 0.2-0.5 m/sec 1° 0.5m/sec 2°-3° 1-2m/sec 7°—l0° Upto15° Very large2.Convex edge detection (erasable) 3.Changing contrast detection 4.Dimming detection Upto%m/sec 5.Darkness detection ? ’ 36-11U R ~-Response atOmmotldlumO - /Illumination .. O0 Fig. 36-l3. The net response of horseshoe crab ommatidia near asharp change inillumination. 1»;I3 14 §1.._,_.....¢.Q......9.5...._..".1 -. . . . 3:-'-:3_;._ uhgs9'P9';-_= _~_-.-=¢_-::_-;=_-'=_-_-.___?— _ ._-_:—__-r_— >_:-t- —_-_,_;w_:-._-T-: -.—..-';-_- \' ‘I \*¥~ /’ ;l'__. =5"r,sal-1-Q1lIl::lfIL--an..-=.-»-'§*”~’Q..).~ . ._l__ l.if~f'*.~_ ‘lb __ .')"- ..".‘~.‘: 'II‘\'. .' A.-cg_< ._'..i---1 " -';"'-;1\.’ "-4- _.-.» \ _ 'é>"“.-_“,“ '"( | (>1 ‘, - I R P‘ -:. ...,‘- B,C ',,‘;.'",:'>_»( ,,4- '.vI.‘ _I __.‘.4- _‘_;_kl"=i."»‘-,i-:""':i';».- U " "' ll'.1"6 1 L e5 r Fig. 36-1 4. The tectum ofafrog.surface. Theeyeseesitmove inandremembers thatitisthere, butifwemerely turnoutthelight foramoment, itsimply forgets itandnolonger seesit. Another example ischange-in-contrast detection. Ifthere isanedge moving inoroutthere arepulses, butifthething stands stillthere arenopulses atall. Then there isadimming detector. Ifthelight intensity isgoing down it creates pulses, butifitstays down orstays up,theimpulse stops; itonly works while thelight isdimming. Then, finally, there areafewfibers which aredark detectors—a most amazing thing—they fireallthetime! Ifweincrease thelight, theyfirelessrapidly, butall thetime. Ifwedecrease thelight, theyfiremore rapidly, allthetime. Inthedark theyfirelikemad, perpetually saying, “Itisdark! Itisdark! Itisdark!” Now these responses seem toberather complicated toclassify, andwemight wonder whether perhaps theexperiments arebeing misinterpreted. Butitisvery interesting thatthese same classes areveryclearly separated intheanatomy ofthe frog! Byother measurements, after these responses hadbeen classified (afterwards, thatiswhat isimportant about this), itwasdiscovered thatthespeed ofthesignals onthedifferent fibers wasnotthesame, soherewasanother, independent wayto check which kind ofafiber wehave found! Another interesting question isfrom how biganarea isoneparticular fiber making itscalculations? Theanswer isdifferent forthedifferent classes. Figure 36-14 shows thesurface oftheso-called tectum ofafrog, where the nerves come intothebrain from theoptic nerve. Allthenerve fibers coming in from theoptic nerve make connections invarious layers ofthetectum. Thislayered structure isanalogous totheretina; thatispartly whyweknow thatthebrain and retina arevery similar. Now, bytaking anelectrode andmoving itdown insuc- cession through thelayers, wecanfindoutwhich kinds ofoptic nerves endwhere, andthebeautiful andwonderful result isthatthediflerent kinds offibers endin different layers! Thefirstones endinnumber 1type, thesecond innumber 2, thethrees andfives endinthesame place, anddeepest ofallisnumber four. (What acoincidence, they gotthenumbers almost intheright order! No,thatis whytheynumbered them thatway, thefirstpaper hadthenumbers inadiflerent order!) Wemaybriefly summarize what wehavejustlearned thisway: There arethree pigments, presumably. There may bemany different kinds ofreceptor cells con- taining thethree pigments indifferent proportions, butthere aremany cross connections which may permit additions andsubtractions through addition and reinforcement inthenervous system. Sobefore wereally understand color vision, wewillhave tounderstand thefinal sensation. Thissubject isstillanopen one,but these researches with microelectrodes andsoonwillperhaps ultimately giveus more information onhow weseecolor. BIBLIOGRAPHY Committee onCalorimetry, Optical Society ofAmerica, TheScience ofColor, Thomas Y.Crowell Company, New York, 1953. “Mechanisms ofVision,” 2ndSupplement toJournal ofGeneral Physiology, Vol. 43, No.6,Part 2,July 1960, Rockefeller Institute Press. SPECIFIC ARTICLESZ DEROBERTIS, E.,“Some Observations ontheUltrastructure andMorphogenesis ofPhotoreceptors," pp.1-15. Hunvrcn, L.M.andD.JAMESON, “Perceived Color, Induction Effects, and Opponent-Response Mechanisms," pp.63-80. ROSENBLITH, W.A.,ed.,Sensory Communication, Massachusetts Institute ofTech- nology Press, Cambridge, Mass., 1961. “Sight, Sense of,”Encyclopaedia Britannica, Vol. 20,1957, pp.628-635. 36-12 37 Quantum Behavior 37-1 Atomic mechanics Inthelastfewchapters wehave treated theessential ideas necessary foran understanding ofmost oftheimportant phenomena oflight—or electromagnetic radiation ingeneral. (Wehave leftafewspecial topics fornextyear. Specifically, thetheory oftheindex ofdense materials andtotal internal reflection.) What we have dealt with iscalled the“classical theory” ofelectric waves, which turns out tobeacompletely adequate description ofnature foralarge number ofefl"ects. Wehave nothadtoworry yetabout thefactthatlight energy comes inlumps or “photons.” Wewould liketotakeupasournextsubject theproblem ofthebehavior of relatively large pieces ofmatter—-their mechanical andthermal properties, for instance. Indiscussing these, wewillfindthatthe“classical” (orolder) theory fails almost immediately, because matter isreally made upofatomic-sized par- ticles. Still, wewilldealonly with theclassical part, because thatistheonly part thatwecanunderstand using theclassical mechanics wehave been learning. But weshall notbeverysuccessful. Weshall findthatinthecaseofmatter, unlike the caseoflight, weshall beindifliculty relatively soon. Wecould, ofcourse, con- tinuously skirt away from theatomic effects, butweshall instead interpose here a short excursion inwhich wewilldescribe thebasic ideas ofthequantum properties ofmatter, i.e.,thequantum ideas ofatomic physics, sothatyouwillhave some feeling forwhat itisweareleaving out. Forwewillhave toleave outsome im- portant subjects thatwecannot avoid coming close to. Sowewillgivenow theintroduction tothesubject ofquantum mechanics, butwillnotbeableactually togetintothesubject until much later. “Quantum mechanics” isthedescription ofthebehavior ofmatter inallits details and, inparticular, ofthehappenings onanatomic scale. Things onavery small scale behave likenothing thatyouhave anydirect experience about. They donotbehave likewaves, they donotbehave likeparticles, they donotbehave likeclouds, orbilliard balls, orweights onsprings, orlikeanything that you have everseen. Newton thought thatlight wasmade upofparticles, butthenitwasdiscovered, aswehave seenhere, thatitbehaves likeawave. Later, however (inthebeginning ofthetwentieth century) itwasfound thatlight didindeed sometimes behave like aparticle. Historically, theelectron, forexample, wasthought tobehave likea particle, andthen itwasfound thatinmany respects itbehaved likeawave. Soit really behaves likeneither. Now wehave given up.Wesay: “Itislikeneither.” There isonelucky break, however—electrons behave justlikelight. The quantum behavior ofatomic objects (electrons, protons, neutrons, photons, and soon)isthesame forall,they areall“particle waves,” orwhatever youwant to callthem. Sowhat welearn about theproperties ofelectrons (which weshall use forourexamples) willapply alsotoall“particles,” including photons oflight. Thegradual accumulation ofinformation about atomic andsmall-scale be- havior during thefirstquarter ofthiscentury, which gave some indications about how small things dobehave, produced anincreasing confusion which wasfinally resolved in1926 and1927 bySchrodinger, Heisenberg, andBorn. They finally obtained aconsistent description ofthebehavior ofmatter onasmall scale. We takeupthemain features ofthatdescription inthischapter. Because atomic behavior issounlike ordinary experience, itisvery difficult togetused toanditappears peculiar andmysterious toeveryone, both tothe 37-137-1 Atomic mechanics 37-2 Anexperiment withbullets 37-3 Anexperiment withwaves 37-4 Anexperiment withelectrons 37-5 Theinterference ofelectron waves 37-6 Watching theelectrons 37-7 First principles ofquantum mechanics 37-8 Theuncertainty principle novice andtotheexperienced physicist. Even theexperts donotunderstand it thewaythey would liketo,anditisperfectly reasonable thatthey should not, because allofdirect, human experience andofhuman intuition applies tolarge objects. Weknow how large objects willact,butthings onasmall scale justdo notactthatway. Sowehave tolearn about them inasortofabstract orimagi- native fashion andnotbyconnection with ourdirect experience. Inthischapter weshall tackle immediately thebasic element ofthemysterious behavior initsmost strange form. Wechoose toexamine aphenomenon which is impossible, absolutely impossible, toexplain inanyclassical way, andwhich has inittheheart ofquantum mechanics. Inreality, itcontains theonly mystery. Wecannot explain themystery inthesense of“explaining” howitworks. Wewill tellyouhowitworks. Intelling youhowitworks wewillhave toldyouabout the basic peculiarities ofallquantum mechanics. 37-2 Anexperiment withbullets Totrytounderstand thequantum behavior ofelectrons, weshall compare andcontrast their behavior, inaparticular experimental setup, with themore familiar behavior ofparticles likebullets, andwith thebehavior ofwaves like water waves. Weconsider firstthebehavior ofbullets intheexperimental setup shown diagrammatically inFig.37-1. Wehave amachine gunthatshoots astream ofbullets. Itisnotaverygood gun, inthatitsprays thebullets (randomly) overa fairly large angular spread, asindicated inthefigure. Infront ofthegunwehave awall(made ofarmor plate) thathasinittwoholes justabout bigenough toleta bullet through. Beyond thewallisabackstop (sayathick wallofwood) which will “absorb” thebullets when theyhitit.Infront ofthewallwehave anobject which weshall calla“detector” ofbullets. Itmight beaboxcontaining sand. Anybullet thatenters thedetector willbestopped andaccumulated. When wewish, wecan empty theboxandcount thenumber ofbullets that have been caught. The detector canbemoved back andforth (inwhat wewillcallthex-direction). With thisapparatus, wecanfindoutexperimentally theanswer tothequestion: “What istheprobability that abullet which passes through theholes inthewall will arrive atthebackstop atthedistance xfrom thecenter?” First, youshould realize that weshould talkabout probability, because wecannot saydefinitely where anyparticular bullet willgo.Abullet which happens tohitoneoftheholes maybounce offtheedges ofthehole, andmayendupanywhere atall.By“prob- ability” wemean thechance thatthebullet willarrive atthedetector, which wecan measure bycounting thenumber which arrive atthedetector inacertain time and thentaking theratio ofthisnumber tothetotalnumber thathitthebackstop during thattime. Or,ifweassume thatthegunalways shoots atthesame rateduring the measurements, theprobability wewant isjustproportional tothenumber that reach thedetector insome standard time interval. Forourpresent purposes wewould liketoimagine asomewhat idealized experiment inwhich thebullets arenotrealbullets, butareindestructible bullets—- theycannot break inhalf. Inourexperiment wefindthatbullets always arrive in lumps, andwhen wefindsomething inthedetector, itisalways onewhole bullet. Iftherateatwhich themachine gunfiresismade verylow,wefindthatatanygiven moment either nothing arrives, oroneandonly one—exactly one—bullet arrives atthebackstop. Also, thesizeofthelump certainly does notdepend ontherate offiring ofthegun. Weshall say:“Bullets always arrive inidentical lumps.” What wemeasure with ourdetector istheprobability ofarrival ofalump. Andwemeas- uretheprobability asafunction ofx.Theresult ofsuch measurements with this apparatus (wehave notyetdone theexperiment, sowearereally imagining the result) areplotted inthegraph drawn inpart(c)ofFig.37-1. Inthegraph weplot theprobability totheright andxvertically, sothatthex-scale fitsthediagram of theapparatus. Wecalltheprobability P12because thebullets may have come either through hole 1orthrough hole 2.You willnotbesurprised thatP12is large near themiddle ofthegraph butgetssmall ifxisvery large. You may wonder, however, whyP12hasitsmaximum value atx=0.Wecanunderstand 37-2 \\1*‘&°t’é5'5»= I xy-'”\\ , " \‘§_ e_-— —-— ———— ‘-U,l\'1\\ /£11GUN \ 2 . . BACKSTOP P+Fig. 37-l. Interference experiment WALL with bullets. (O) thisfactifwedoourexperiment again after covering uphole 2,andonce more while covering uphole l.When hole 2iscovered, bullets canpass only through hole 1,andwegetthecurve marked P1inpart (b)ofthefigure. Asyouwould expect, themaximum ofP1occurs atthevalue ofxwhich isonastraight linewith thegunandhole 1.When hole 1isclosed, wegetthesymmetric curve P2drawn inthefigure. P2istheprobability distribution forbullets thatpassthrough hole 2.Comparing parts (b)and(c)ofFig.37-1, wefindtheimportant result that P12 =P1—l—P2. (37.1) Theprobabilities justaddtogether. Theeffect with both holes open isthesumof theeffects with each hole open alone. Weshall callthisresult anobservation of “nointerference,” forareason thatyouwillseelater. Somuch forbullets. They come inlumps, andtheir probability ofarrival shows nointerference. l\s\DETECTO art... 4) , YW Fig. 37-2. Interference experiment WALL ABSQRBER I 11, with water waves. 12hf . (0) (bl (cl 37-3 Anexperiment withwaves Now wewish toconsider anexperiment with water waves. Theapparatus is shown diagrammatically inFig.37-2. Wehave ashallow trough ofwater. Asmall object labeled the“wave source” isjiggled upanddown byamotor andmakes circular waves. Totheright ofthesource wehave again awall with twoholes, andbeyond thatisasecond wall, which, tokeep things simple, isan“absorber,” sothatthere isnoreflection ofthewaves thatarrive there. This canbedone by building agradual sand “beach.” Infront ofthebeach weplace adetector which canbemoved back andforth inthex-direction, asbefore. Thedetector isnow a device which measures the“intensity” ofthewave motion. You canimagine a gadget which measures theheight ofthewave motion, butwhose scale iscalibrated inproportion tothesquare oftheactual height, sothatthereading isproportional totheintensity ofthewave. Ourdetector reads, then, inproportion totheenergy being carried bythewave—or rather, therateatwhich energy iscarried tothe detector. With ourwave apparatus, thefirstthing tonotice isthat theintensity can have anysize. Ifthesource justmoves avery small amount, then there isjusta little bitofwave motion atthedetector. When there ismore motion atthesource, 37-3 there ismore intensity atthedetector. Theintensity ofthewave canhave any value atall.Wewould notsaythatthere wasany“lumpiness” inthewave intensity. Now letusmeasure thewave intensity forvarious values ofx(keeping the wave source operating always inthesame way). Wegettheinteresting-looking .:urve marked I12inpart(c)ofthefigure. Wehave already worked outhow such patterns cancome about when we studied theinterference ofelectric waves. Inthiscasewewould observe thatthe original wave isdiffracted attheholes, andnewcircular waves spread outfrom each hole. Ifwecover onehole atatime andmeasure theintensity distribution atthe absorber wefindtherather simple intensity curves shown inpart(b)ofthefigure. I1istheintensity ofthewave from hole 1(which wefindbymeasuring when hole 2isblocked off)andI2istheintensity ofthewave from hole 2(seen when hole 1isblocked). Theintensity 112observed when both holes areopen iscertainly notthesum ofI1andI2.Wesaythat there is“interference” ofthetwowaves. Atsome places (where thecurve I12hasitsmaxima) thewaves are“inphase” andthewave peaks addtogether togivealarge amplitude and, therefore, alarge intensity. We saythatthetwowaves are“interfering constructively” atsuch places. There will besuch constructive interference wherever thedistance from thedetector toone hole isawhole number ofwavelengths larger (orshorter) than thedistance from thedetector totheother hole. Atthose places where thetwowaves arrive atthedetector with aphase differ- ence of1r(where theyare“out ofphase”) theresulting wave motion atthedetector willbethedifference ofthetwoamplitudes. Thewaves “interfere destructively,” andwegetalowvalue forthewave intensity. Weexpect such lowvalues wherever thedistance between hole 1andthedetector isdifferent from thedistance between hole2andthedetector byanoddnumber ofhalf-wavelengths. Thelowvalues of I12inFig.37-2 correspond totheplaces where thetwowaves interfere destructively. You willremember thatthequantitative relationship between I1,I2,andI12 canbeexpressed inthefollowing way: Theinstantaneous height ofthewater wave atthedetector forthewave from hole 1canbewritten as(therealpartof)h1e"‘“‘, where the“amplitude” h1is,ingeneral, acomplex number. The intensity is proportional tothemean squared height or,when weusethecomplex numbers, tolh1l2. Similarly, forhole 2theheight ish2e“" andtheintensity isproportional tolh2|2. When both holes areopen, thewave heights addtogive theheight (h1—l—h2)e““' andtheintensity |h1+h2|2. Omitting theconstant ofproportion- ality forourpresent purposes, theproper relations forinterfering waves are 1,=|h1|2, 12=122?, 112=|h1+i2,|2. (37.2) Youwillnotice thattheresult isquite different from thatobtained withbullets (Eq. 37.1). Ifweexpand |h1+h2|2weseethat [121+i.2|2=|i21|2+|h2|2+2|h1l|h2|cos 3, (37.3) where 5isthephase difference between h1andhg.Interms oftheintensities, we could write ___ I12 =I1-l-I2+2\/1112COS Thelastterm in(37.4) isthe“interference term.” Somuch forwater waves. The intensity canhave anyvalue, anditshows inteference. 37-4 Anexperiment withelectrons Now weimagine asimilar experiment with electrons. Itisshown diagram- matically inFig.37-3. Wemake anelectron gunwhich consists ofatungsten wire heated byanelectric current andsurrounded byametal boxwith ahole init.If thewire isatanegative voltage with respect tothebox, electrons emitted bythe wire willbeaccelerated toward thewalls andsome willpass through thehole. Alltheelectrons which come outofthegunwillhave (nearly) thesame energy. Infront ofthegunisagain awall (just athinmetal plate) with twoholes init. 37-4 DETECTOR /-_'- ' \\ WALL BllCKSTOP Fig. 37-3. Interference experiment with electrons. (O) Beyond thewallisanother plate which willserve asa“backstop.” Infront ofthe backstop weplace amovable detector. Thedetector rnight beageiger counter or, perhaps better, anelectron multiplier, which isconnected toaloudspeaker. Weshould sayright away thatyoushould nottrytosetupthisexperiment (asyoucould have done with thetwowehave already described). Thisexperiment hasnever been done injustthisway. Thetrouble isthattheapparatus would have tobemade onanimpossibly small scale toshow theeffects weareinterested in. Wearedoing a“thought experiment,” which wehave chosen because itiseasyto think about. Weknow theresults thatwould beobtained because there aremany experiments that have been done, inwhich thescale andtheproportions have been chosen toshow theeffects weshall describe. Thefirstthing wenotice with ourelectron experiment isthatwehear sharp “clicks” from thedetector (that is,from theloudspeaker). And all“clicks” are thesame. There areno“half-clicks.” Wewould alsonotice thatthe“clicks” come veryerratically. Something like: click .....click-click ...click ........click ....click-click ......click ..., etc.,justasyouhave, nodoubt, heard ageiger counter operating. Ifwecount theclicks which arrive inasufliciently long time—say formany minutes—and then count again foranother equal period, wefindthatthetwonumbers arevery nearly thesame. Sowecanspeak oftheaverage rateatwhich theclicks areheard (so-and-so-many clicks perminute ontheaverage). Aswemove thedetector around, therateatwhich theclicks appear isfaster orslower, butthesize(loudness) ofeach click isalways thesame. Ifwelower the temperature ofthewire intheguntherateofclicking slows down, butstilleach click sounds thesame. Wewould notice alsothatifweputtwoseparate detectors atthebackstop, oneortheother would click, butnever both atonce. (Except that once inawhile, ifthere were twoclicks veryclose together intime, ourearmight notsense theseparation.) Weconclude, therefore, thatwhatever arrives atthe backstop arrives in“lumps.” Allthe“1umps” arethesame size: only whole “lumps” arrive, andthey arrive oneatatime atthebackstop. Weshall say: “Electrons always arrive inidentical lumps.” Justasforourexperiment with bullets, wecannow proceed tofindexperi- mentally theanswer tothequestion: “What istherelative probability that an electron ‘lump’ willarrive atthebackstop atvarious distances xfrom thecenter?” Asbefore, weobtain therelative probability byobserving therateofclicks, holding theoperation ofthegunconstant. Theprobability thatlumps willarrive ata particular xisproportional totheaverage rateofclicks atthatx. Theresult ofourexperiment istheinteresting curve marked P12inpart (c) ofFig.37-3. Yes! That isthewayelectrons go. 37-5 Theinterference ofelectron waves Now letustrytoanalyze thecurve ofFig.37-3 toseewhether wecanunder- stand thebehavior oftheelectrons. Thefirstthing wewould sayisthatsince they come inlumps, each lump, which wemayaswellcallanelectron, hascome either through hole 1orthrough hole2.Letuswrite thisintheform ofa“Proposition”: 37-5~17JDl¢*¢2l"I still5.3HI\\\\ "rm1\M N_ ®X ll.1-2".-.11—'==5‘"I _R? (bl (cl Proposition A:Each electron either goesthrough hole1oritgoesthrough hole2. Assuming Proposition A,allelectrons thatarrive atthebackstop canbedi- vided intotwoclasses: (1)those thatcome through hole 1,and(2)those thatcome through hole2.Soourobserved curve must bethesumoftheeffects oftheelec- trons which come through hole 1andtheelectrons which come through hole 2. Letuscheck thisideabyexperiment. First, wewillmake ameasurement forthose electrons thatcome through hole 1.Weblock offhole2andmake ourcounts of theclicks from thedetector. From theclicking rate, wegetP1.Theresult ofthe measurement isshown bythecurve marked P1inpart(b)ofFig.37-3. Theresult seems quite reasonable. Inasimilar way, wemeasure P2,theprobability distribu- tionfortheelectrons thatcome through hole 2.Theresult ofthismeasurement isalsodrawn inthefigure. Theresult P12obtained with bothholes open isclearly notthesumofP1and P2,theprobabilities foreach holealone. Inanalogy with ourwater-wave experi- ment, wesay:“There isinterference.” Forelectrons: P12 séP1+P2. (37.5) How cansuch aninterference come about? Perhaps weshould say: “Well, thatmeans, presumably, thatitisnottruethatthelumps goeither through hole 1orhole2,because iftheydid,theprobabilities should add. Perhaps theygoina more complicated way. They split inhalfand...”Butno! They cannot, they always arrive inlumps ...“Well, perhaps some ofthem gothrough 1,andthen they goaround through 2,andthen around afewmore times, orbysome other complicated path ...then byclosing hole 2,wechanged thechance thatanelec- tron thatstarted outthrough hole 1would finally gettothebackstop ...”But notice! There aresome points atwhich very fewelectrons arrive when both holes areopen, butwhich receive many electrons ifweclose onehole, soclosing one hole increased thenumber from theother. Notice, however, thatatthecenter ofthepattern, P12ismore than twice aslarge asP1+P2.Itisasthough closing onehole decreased thenumber ofelectrons which come through theother hole. Itseems hard toexplain both eflects byproposing that theelectrons travel in complicated paths. Itisallquite mysterious. And themore youlook atitthemore mysterious itseems. Many ideas have been concocted totrytoexplain thecurve forP12in terms ofindividual electrons going around incomplicated ways through theholes. None ofthem hassucceeded. None ofthem cangettheright curve forP12in terms ofP1 andP2. Yet, surprisingly enough, themathematics forrelating P1andP2toP12is extremely simple. ForP12isjustlikethecurve 112ofFig.37-2, andthatwas simple. What isgoing onatthebackstop canbedescribed bytwocomplex numbers thatwecancall$1and<32(they arefunctions ofx,ofcourse). Theabsolute square of31gives theeffect with only hole lopen. That is,P1=|<l>1|2. Theeffect with only hole 2open isgiven by$2inthesame way. That is,P2=|<f>2]2. And the combined effect ofthetwoholes isjustP12 =I31+$2|2. Themathematics isthesame asthatwehadforthewater waves! (Itishard toseehow onecould getsuch asimple result from acomplicated game ofelectrons going back andforth through theplate onsome strange trajectory.) Weconclude thefollowing: Theelectrons arrive inlumps, likeparticles, and theprobability ofarrival ofthese lumps isdistributed likethedistribution of intensity ofawave. Itisinthissense thatanelectron behaves “sometimes likea particle andsometimes likeawave.” Incidentally, when wewere dealing with classical waves wedefined thein- tensity asthemean over time ofthesquare ofthewave amplitude, andweused complex numbers asamathematical trick tosimplify theanalysis. Butinquantum mechanics itturns outthattheamplitudes must berepresented bycomplex num- bers. Therealparts alone willnotdo.That isatechnical point, forthemoment, because theformulas look justthesame. 37-6 Since theprobability ofarrival through both holes isgiven sosimply, although itisnotequal to(P1+P2),thatisreally allthere istosay. Butthere arealarge number ofsubtleties involved inthefactthatnature does work thisway. We would liketoillustrate some ofthese subtleties foryounow. First, since thenum- berthatarrives ataparticular point isnotequal tothenumber thatarrives through 1plus thenumber that arrives through 2,aswewould have concluded from Proposition A,undoubtedly weshould conclude thatProposition Aisfalse. Itis nottruethattheelectrons goeither through hole 1orhole2.Butthatconclusion canbetested byanother experiment. 37-6 Watching theelectrons Weshall now trythefollowing experiment. Toourelectron apparatus we addavery strong light source, placed behind thewallandbetween thetwoholes, asshown inFig.37-4. Weknow thatelectric charges scatter light. Sowhen an electron passes, however itdoes pass, onitswaytothedetector, itwillscatter some light tooureye,andwecanseewhere theelectron goes. If,forinstance, anelectron were totakethepath viahole2thatissketched inFig.37-4, weshould seeaflash oflight coming from thevicinity oftheplace marked Ainthefigure. Ifanelectron passes through hole 1wewould expect toseeaflash from thevicinity oftheupper hole. Ifitshould happen thatwegetlight from both places atthesame time, because theelectron divides inhalf...Letusjustdotheexperiment! \\*Kis\\\\\\\\\\N.,1"-Q15I1.|en"r,..,_sounce t—'"| /, E3"-P. __ /-, ztacrnou ‘*GUN \\\\\\ Fig. 37-4. Adifferent electron ex- periment. (0) Here iswhat wesee:every time thatwehear a“c1ick” from ourelectron de- tector (atthebackstop), wealsoseeaflash oflight either near hole 1ornear hole 2,butnever both atonce! And weobserve thesame result nomatter where weput thedetector. From thisobservation weconclude thatwhen welook attheelectrons wefindthattheelectrons goeither through oneholeortheother. Experimentally, Proposition Aisnecessarily true. What, then, iswrong with ourargument against Proposition A?Why isn’t P12justequal toP1+P2? Back toexperiment! Letuskeep track oftheelectrons andfindoutwhat they aredoing. Foreach position (x-location) ofthedetector wewillcount theelectrons thatarrive andalsokeep track ofwhich holetheywent through, bywatching fortheflashes. Wecankeep track ofthings thisway: whenever wehear a“click” wewillputacount inColumn 1ifweseetheflash near hole 1,andifweseetheflash near hole 2,wewillrecord acount inColumn 2. Every electron which arrives isrecorded inoneoftwoclasses: those which come through 1andthose which come through 2.From thenumber recorded inColumn 1wegettheprobability P1thatanelectron willarrive atthedetector viahole 1; andfrom thenumber recorded inColumn 2wegetP2,theprobability that an electron willarrive atthedetector viahole2.Ifwenowrepeat such ameasurement formany values ofx,wegetthecurves forP1andP2shown inpart(b)ofFig.37-4. Well, thatisnottoosurprising! WegetforP1something quite similar to what wegotbefore forP1byblocking offhole2;andP2issimilar towhat wegot byblocking hole 1.Sothere isnotanycomplicated business likegoing through both holes. When wewatch them, theelectrons come through justaswewould 37-7 > > P+ , File'1’2 (bl (cl expect them tocome through. Whether theholes areclosed oropen, those which weseecome through hole 1aredistributed inthesame waywhether hole2isopen orclosed. Butwait! What dowehave nowforthetotal probability, theprobability that anelectron willarrive atthedetector byanyroute? Wealready have thatinforma- tion. Wejustpretend thatwenever looked atthelight flashes, andwelump to- gether thedetector clicks which wehave separated into thetwocolumns. We must justaddthenumbers. Fortheprobability thatanelectron willarrive atthe backstop bypassing through either hole, wedofindP12=P1—|-P2. That is, although wesucceeded inwatching which hole ourelectrons come through, we nolonger gettheoldinterference curve P12, butanew one, P12, showing no interference! Ifweturn outthelight P12isrestored. Wemust conclude thatwhen welookattheelectrons thedistribution ofthem onthescreen isdifferent than when wedonotlook. Perhaps itisturning onour light source thatdisturbs things‘? Itmust bethattheelectrons arevery delicate, andthelight, when itscatters offtheelectrons, gives them ajoltthatchanges their motion. Weknow thattheelectric field ofthelight acting onacharge willexert aforce onit.Soperhaps weshould expect themotion tobechanged. Anyway, thelight exerts abiginfluence ontheelectrons. Bytrying to“watch” theelectrons wehave changed their motions. That is,thejoltgiven totheelectron when the photon isscattered byitissuch astochange theelectron’s motion enough sothat ifitmight have gone towhere P12wasatamaximum itwillinstead land where P12wasaminimum; thatiswhywenolonger seethewavy interference effects. You may bethinking: “Don’t usesuch abright source! Turn thebrightness down! Thelight waves willthen beweaker andwillnotdisturb theelectrons so much. Surely, bymaking thelight dimmer anddimmer, eventually thewave willbeweak enough thatitwillhave anegligible efiect.” O.K. Let’s tryit.The firstthing weobserve isthattheflashes oflight scattered from theelectrons as theypassbydoes notgetweaker. Itisalways thesame-sized flash. Theonly .hing that happens asthelight ismade dimmer isthat sometimes wehear a“click” from thedetector butseenoflash atall.Theelectron hasgone bywithout being “seen.” What weareobserving isthatlight alsoactslikeelectrons, weknew that itwas“wavy,” butnow wefindthatitisalso“lumpy.” Italways arrives-or is scattered—in lumps thatwecall“photons.” Asweturn down theintensity of thelight source wedonotchange thesizeofthephotons, only therateatwhich they areemitted. That explains why, when oursource isdim, some electrons get bywithout being seen. There didnothappen tobeaphoton around atthetime theelectron went through. Thisisallalittle discouraging. Ifitistruethatwhenever we“see” theelectron weseethesame-sized flash, then those electrons weseearealways thedisturbed ones. Letustrytheexperiment with adimlight anyway. Now whenever wehear aclick inthedetector wewillkeep acount inthree columns: inColumn (1)those electrons seen byhole 1,inColumn (2)those electrons seen byhole 2,andin Column (3)those electrons notseenatall.When wework upourdata (computing theprobabilities) wefindthese results: Those “seen byhole l”have adistribution likeP1;those “seen byhole 2”have adistribution likeP2(sothatthose “seen by either hole 1or2”have adistribution likeP12); andthose “not seenatall”have a “wavy” distribution justlikeP12ofFig.37-3! Iftheelectrons arenotseen, we have interference! That isunderstandable. When wedonotseetheelectron, nophoton disturbs it,andwhen wedoseeit,aphoton hasdisturbed it.There isalways thesame amount ofdisturbance because thelight photons allproduce thesame-sized effects andtheeflect ofthephotons being scattered isenough tosmear outanyinter- ference efl'ect. Isthere notsome way wecanseetheelectrons without disturbing them? Welearned inanearlier chapter that themomentum carried bya“photon” isinversely proportional toitswavelength (p=h/)1). Certainly thejoltgiven totheelectron when thephoton isscattered toward oureyedepends onthe momentum thatphoton carries. Aha! Ifwewant todisturb theelectrons only 37-8 slightly weshould nothave lowered theintensity ofthelight, weshould have lowered itsfrequency (thesame asincreasing itswavelength). Letususelight of aredder color. Wecould even useinfrared light, orradiowaves (like radar), and “see” where theelectron went with thehelp ofsome equipment thatcan“see” light ofthese longer wavelengths. Ifweuse“gentler” light perhaps wecanavoid disturbing theelectrons somuch. Letustrytheexperiment with longer waves. Weshall keep repeating ourex- periment, each time with light ofalonger wavelength. Atfirst, nothing seems to change. Theresults arethesame. Then aterrible thing happens. You remember thatwhen wediscussed themicroscope wepointed outthat, duetothewave nature ofthelight, there isalimitation onhowclose twospots canbeandstillbeseen astwoseparate spots. This distance isoftheorder ofthewavelength oflight. So now, when wemake thewavelength longer than thedistance between ourholes, weseeabigfuzzy flash when thelight isscattered bytheelectrons. Wecanno longer tellwhich holetheelectron went through! Wejustknow itwent somewhere! And itisjustwithlight ofthiscolor thatwefindthatthejolts given totheelectron aresmall enough sothatP{2begins tolook likeP12—that webegin togetsome interference effect. And itisonlyforwavelengths much longer than theseparation ofthetwoholes (when wehave nochance atalloftelling where theelectron went) thatthedisturbance duetothelight getssufiiciently small thatweagain getthe curve P12shown inFig.37-3. Inourexperiment wefindthatitisimpossible toarrange thelight insuch a waythatonecantellwhich hole theelectron went through, andatthesame time notdisturb thepattern. Itwassuggested byHeisenberg thatthethen newlaws of nature could only beconsistent ifthere were some basic limitation onourexperi- mental capabilities notpreviously recognized. Heproposed, asageneral principle, hisuncertainty principle, which wecanstate interms ofourexperiment asfollows: “Itisimpossible todesign anapparatus todetermine which holetheelectron passes through, thatwillnotatthesame time disturb theelectrons enough todestroy the interference pattern.” Ifanapparatus iscapable ofdetermining which holetheelec- tron goes through, itcannot besodelicate thatitdoes notdisturb thepattern in anessential way. Noonehaseverfound (oreven thought of)awayaround the uncertainty principle. Sowemust assume thatitdescribes abasic characteristic ofnature. Thecomplete theory ofquantum mechanics which wenow usetodescribe atoms and, infact, allmatter depends onthecorrectness oftheuncertainty prin- ciple. Since quantum mechanics issuch asuccessful theory, ourbelief inthe uncertainty principle isreinforced. Butifawayto“beat” theuncertainty principle were ever discovered, quantum mechanics would give inconsistent results and would have tobediscarded asavalid theory ofnature. “Well,” yousay,“what about Proposition A‘?Itistrue, orisitnottrue, thattheelectron either goes through hole loritgoes through hole 2?” Theonly answer thatcanbegiven isthatwehave found from experiment thatthere isa certain special waythatwehave tothink inorder thatwedonotgetintoincon- sistencies. What wemust say(toavoid making wrong predictions) isthefollowing. Ifonelooks attheholes or,more accurately, ifonehasapiece ofapparatus which iscapable ofdetermining whether theelectrons gothrough hole Iorhole 2,then onecansaythatitgoes either through hole 1orhole 2.But,when onedoes not trytotellwhich waytheelectron goes, when there isnothing intheexperiment to disturb theelectrons, then onemay notsaythatanelectron goes either through hole 1orhole2.Ifonedoes saythat, andstarts tomake anydeductions from the statement, hewillmake errors intheanalysis. This isthelogical tightrope on which wemust walk ifwewish todescribe nature successfully. -ii Ifthemotion ofallmatter—as wellaselectrons—must bedescribed interms ofwaves, what about thebullets inourfirstexperiment? Why didn’t weseean interference pattern there‘? Itturns outthatforthebullets thewavelengths were so tinythattheinterference patterns became veryfine. Sofine,infact, thatwith any 37-9 x Q2 R2(smoothed) (0) lb) Fig. 37-5. Interference pattern with bullets: (a)actual (schematic), (b)ob- served.detector offinite sizeonecould notdistinguish theseparate maxima andminima. What wesawwasonly akind ofaverage, which istheclassical curve. InFig.37-5 wehave tried toindicate schematically what happens with large-scale objects. Part (a)ofthefigure shows theprobability distribution onemight predict for bullets, using quantum mechanics. Therapid wiggles aresupposed torepresent theinterference pattern onegetsforwaves ofveryshort wavelength. Anyphysical detector, however, straddles several wiggles oftheprobability curve, sothatthe measurements show thesmooth curve drawn inpart (b)ofthefigure. 37-7 First principles ofquantum mechanics Wewillnow write asummary ofthemain conclusions ofourexperiments. Wewill, however, puttheresults inaform which makes them trueforageneral class ofsuch experiments. Wecanwrite oursummary more simply ifwefirst define an“ideal experiment” asoneinwhich there arenouncertain external influences, i.e.,nojiggling orother things going onthatwecannot take intoac- count. Wewould bequite precise ifwesaid: “Anideal experiment isoneinwhich alloftheinitial andfinal conditions oftheexperiment arecompletely specified.” What wewillcall“anevent” is,ingeneral, justaspecific setofinitial andfinal conditions. (For example: “anelectron leaves thegun, arrives atthedetector, and nothing elsehappens”) Now foroursummary. SUMMARY (1)Theprobability ofanevent inanideal experiment isgiven bythesquare of theabsolute value ofacomplex number ¢which iscalled theprobability amplitude. P=probability, ¢=probability amplitude, (37.6) P=|¢|”- (2)When anevent canoccur inseveral alternative ways, theprobability ampli- tude fortheevent isthesum oftheprobability amplitudes foreach way considered separately. There isinterference. ¢=¢1+¢2, P=l¢1+¢2I2- (37-7) (3)Ifanexperiment isperformed which iscapable ofdetermining whether oneor another alternative isactually taken, theprobability oftheevent isthesum oftheprobabilities foreach alternative. Theinterference islost. P=P1—l—P2. (37.8) Onemight stillliketoask:“How does itwork? What isthemachinery behind thelaw?” Noonehasfound anymachinery behind thelaw. Noonecan“explain” anymore than wehave just“explained.” Noonewillgiveyouanydeeper repre- sentation ofthesituation. Wehave noideas about amore basic mechanism from which these results canbededuced. Wewould like toemphasize avery important diflerence between classical and quantum mechanics. Wehave been talking about theprobability thatanelectron willarrive inagiven circumstance. Wehave implied thatinourexperimental arrangement (oreven inthebestpossible one) itwould beimpossible topredict exactly what would happen. Wecanonly predict theodds! This would mean, if itwere true, thatphysics hasgiven upontheproblem oftrying topredict exactly what willhappen inadefinite circumstance. Yes! physics hasgiven up. Wedo notknow howtopredict what would happen inagiven circumstance, andwebelieve now thatitisimpossible, thattheonly thing thatcanbepredicted istheprob- ability ofdiflerent events. Itmust berecognized thatthisisaretrenchment inour earlier ideal ofunderstanding nature. Itmay beabackward step, butnoone hasseenawaytoavoid it. 37-10 Wemake now afewremarks onasuggestion thathassometimes been made totrytoavoid thedescription wehave given: “Perhaps theelectron hassome kind ofinternal works-—some inner variables—that wedonotyetknow about. Perhaps thatiswhywecannot predict what willhappen. Ifwecould look more closely at theelectron wecould beabletotellwhere itwould endup.” Sofarasweknow, thatisimpossible. Wewould stillbeindifliculty. Suppose wewere toassume that inside theelectron there issome kind ofmachinery thatdetermines where itis going toendup.That machine must alsodetermine which hole itisgoing togo through onitsway. Butwemust notforget thatwhat isinside theelectron should notbedependent onwhat wedo,andinparticular upon whether weopen orclose oneoftheholes. Soifanelectron, before itstarts, hasalready made upitsmind (a)which hole itisgoing touse,and(b)where itisgoing toland, weshould find P1forthose electrons thathave chosen hole 1,P2forthose thathave chosen hole 2,andnecessarily thesumP1+P2forthose thatarrive through thetwoholes. There seems tobenowayaround this. Butwehave verified experimentally that thatisnotthecase. And noonehasfigured awayoutofthispuzzle. Soatthe present time wemust limit ourselves tocomputing probabilities. Wesay“atthe present time,” butwesuspect very strongly thatitissomething thatwillbewith usforever—-that itisimpossible tobeat thatpuzzle—that thisisthewaynature really is. 37-8 Theuncertainty principle This isthewayHeisenberg stated theuncertainty principle originally: Ifyou make themeasurement onanyobject, andyoucandetermine thex-component of itsmomentum with anuncertainty Ap,youcannot, atthesame time, know its x-position more accurately than Ax=h/Ap. Theuncertainties intheposition andmomentum atanyinstant must have their product greater than Planck’s constant. This isaspecial caseoftheuncertainty principle thatwasstated above more generally. Themore general statement wasthatonecannot design equipment inanywaytodetermine which oftwoalternatives istaken, without, atthesame time, destroying thepattern ofinterference. Letusshow foroneparticular casethatthekind ofrelation given byHeisen- berg must betrueinorder tokeep from getting intotrouble. Weimagine amodifi- cation oftheexperiment ofFig.37-3, inwhich thewallwith theholes consists ofa plate mounted onrollers sothatitcanmove freely upanddown (inthex-direction), asshown inFig.37-6. Bywatching themotion oftheplate carefully wecantryto tellwhich holeanelectron goesthrough. Imagine what happens when thedetector isplaced atx=0.Wewould expect thatanelectron which passes through hole 1 must bedeflected downward bytheplate toreach thedetector. Since thevertical component oftheelectron momentum ischanged, theplate must recoil with an equal momentum intheopposite direction. Theplate willgetanupward kick. Iftheelectron goes through thelower hole, theplate should feeladownward kick. Itisclear thatforevery position ofthedetector, themomentum received bythe plate willhave adifferent value foratraversal viahole 1than foratraversal via hole2.So!Without disturbing theelectrons atall,butjustbywatching theplate, wecantellwhich path theelectron used. Now inorder todothisitisnecessary toknow what themomentum ofthe screen is,before theelectron goes through. Sowhen wemeasure themomentum after theelectron goes by,wecanfigure outhowmuch theplate’s momentum has changed. Butremember, according totheuncertainty principle wecannot atthe same time know theposition oftheplate with anarbitrary accuracy. Butifwedo notknow exactly where theplate iswecannot sayprecisely where thetwoholes are. They willbeinadifferent place forevery electron thatgoes through. This means thatthecenter ofourinterference pattern willhave adifferent location foreach electron. Thewiggles oftheinterference pattern willbesmeared out.Weshall show quantitatively inthenextchapter thatifwedetermine themomentum oftheplate sufliciently accurately todetermine from therecoil measurement which hole was used, then theuncertainty inthex-position oftheplate will, according totheun- 37-11ROLLERS '0 <3";/v |'o r—-| ,:-1-’ 7‘-Elm“:2 '=;'--------Ii__l€‘\\:\-\\ ///T\ ztggkaou \ 223 4? mononr-"Reel '7»‘ RQ.LERS WALL BACKSTOP Fig. 37-6. Anexperiment inwhich therecoil ofthewall ismeasured. certainty principle, beenough toshift thepattern observed atthedetector upand down inthex-direction about thedistance from amaximum toitsnearest minimum. Such arandom shift isjustenough tosmear outthepattern sothatnointerference isobserved. Theuncertainty principle “protects” quantum mechanics. Heisenberg recog- nized thatifitwere possible tomeasure themomentum andtheposition simultane- ously with agreater accuracy, thequantum mechanics would collapse. Sohe proposed thatitmust beimpossible. Then people satdown andtried tofigure out ways ofdoing it,andnobody could figure outawaytomeasure theposition and themomentum ofanything—a screen, anelectron, abilliard ball, anything-—with anygreater accuracy. Quantum mechanics maintains itsperilous butaccurate existence. 37-12 38 The Relation ofWave and Particle Viewpoints 38-1 Probability wave amplitudes Inthischapter weshall discuss therelationship ofthewave andparticle viewpoints. Wealready know, from thelastchapter, thatneither thewave view- point northeparticle viewpoint iscorrect. Usually wehave tried topresent things accurately, oratleast precisely enough thattheywillnothave tobechanged when welearn more—it may beextended, butitwillnotbechanged! Butwhen wetry totalkabout thewave picture ortheparticle picture, both areapproximate, and both willchange. Therefore what welearn inthischapter willnotbeaccurate ina certain sense; itisakind ofhalf-intuitive argument thatwillbemade more precise later, butcertain things willbechanged alittle bitwhen weinterpret them correctly inquantum mechanics. Thereason fordoing such athing, ofcourse, isthatwe arenotgoing togodirectly intoquantum mechanics, butwewant tohave atleast some ideaofthekinds ofeffects thatwewillfind. Furthermore, allourexperiences arewith waves andwith particles, andsoitisrather handy tousethewave and particle ideas togetsome understanding ofwhat happens ingiven circumstances before weknow thecomplete mathematics ofthequantum-mechanical amplitudes. Weshall trytoillustrate theweakest places aswegoalong, butmost ofitisvery nearly correct—it isjustamatter ofinterpretation. First ofall,weknow thatthenewwayofrepresenting theworld inquantum mechanics—the newframework—is togiveanamplitude forevery event thatcan occur, andiftheevent involves thereception ofoneparticle then wecangivethe amplitude tofindthatoneparticle atdiflerent places andatdiflerent times. The probability offinding theparticle isthen proportional totheabsolute square of theamplitude. Ingeneral, theamplitude tofindaparticle indifferent places at different times varies with position andtime. Inaspecial case theamplitude varies sinusoidally inspace and time like ei(‘”’_"") (donotforget that these amplitudes arecomplex numbers, notreal numbers) andinvolves adefinite frequency wandwave number k.Then itturns outthatthiscorresponds toaclassical limiting situation where wewould have believed thatwehave aparticle whose energy Ewasknown andisrelated tothe frequency by E=hw, (38.1) andwhose momentum pisalsoknown andisrelated tothewave number by p=hk. (38.2) This means thattheidea ofaparticle islimited. Theidea ofaparticle—its location, itsmomentum, etc.—which weusesomuch, isincertain ways unsatis- factory. Forinstance, ifanamplitude tofindaparticle atdifferent places isgiven bye““"""' ”,whose absolute square isaconstant, thatwould mean thattheprob- ability offinding aparticle isthesame atallpoints. That means wedonotknow where itis-—it canbeanywhere—there isagreat uncertainty initslocation. Ontheother hand, iftheposition ofaparticle ismore orlesswellknown and wecanpredict itfairly accurately, then theprobability offinding itindifferent places must beconfined toacertain region, whose length wecallAx.Outside this region, theprobability iszero. Now thisprobability istheabsolute square ofan amplitude, andiftheabsolute square iszero, theamplitude isalsozero, sothat 38-138-1 Probability wave amplitudes 38-2 Measurement ofposition and momentum 38-3 Crystal diffraction 38-4 Thesizeofanatom 38-5 Energy levels 38-6 Philosophical implications -Ax—i—> Fig. 38-1. Awave packet oflength Ax. C _i-> '_’_’l_Ao_______.IB‘ ‘W Fig. 38-2. Diffraction ofparticles passing through aslit.wehave awave train whose length isAx(Fig. 38-1), andthewavelength (the distance between nodes ofthewaves inthetrain) ofthatwave train iswhat corre- sponds totheparticle momentum. Here weencounter astrange thing about waves; averysimple thing which has nothing todowith quantum mechanics strictly. Itissomething that anybody whoworks with waves, even ifheknows noquantum mechanics, knows: namely, wecannot define aunique wavelength forashort wave train. Such awave train does nothave adefinite wavelength; there isanindefiniteness inthewave number that isrelated tothefinite length ofthetrain, andthus there isanindefiniteness in themomentum. 38-2 Measurement ofposition andmomentum Letusconsider twoexamples ofthisidea—to seethereason why there isan uncertainty intheposition and/orthemomentum, ifquantum mechanics isright. Wehave alsoseenbefore thatifthere were notsuch athing—if itwere possible to measure theposition andthemomentum ofanything simultaneously—we would have aparadox; itisfortunate thatwedonothave such aparadox, andthefact thatsuch anuncertainty comes naturally from thewave picture shows thatevery- thing ismutually consistent. Here isoneexample which shows therelationship between theposition and themomentum inacircumstance thatiseasy tounderstand. Suppose wehave a single slit,andparticles arecoming from very faraway with acertain energy—so that they areallcoming essentially horizontally (Fig. 38-2). Wearegoing to concentrate onthevertical components ofmomentum. Allofthese particles have acertain horizontal momentum po,say,inaclassical sense. So,intheclassical sense, thevertical momentum PU’before theparticle goes through thehole, is definitely known. Theparticle ismoving neither upnordown, because itcame from asource thatisfaraway-and sothevertical momentum isofcourse zero. But nowletussuppose thatitgoes through aholewhose width isB.Then after ithas come outthrough thehole, weknow theposition vertically—the yposition—with considerable accuracy—namely 1B. That is,theuncertainty inposition, Ay,is oforder B.Now wemight alsowant tosay,since weknow themomentum is absolutely horizontal, thatAp,,iszero; butthatISwrong. Weonce knew themo- mentum washorizontal, butwedonotknow itanymore. Before theparticles passed through thehole, wedidnotknow their vertical positions. Now thatwe have found thevertical position byhaving theparticle come through thehole, we have lostourinformation onthevertical momentum! Why? According tothe wave theory, there isaspreading out,ordiffraction, ofthewaves after they go through theslit,justasforlight. Therefore there isacertain probability that particles coming outoftheslitarenotcoming exactly straight. Thepattern is spread outbythediffraction effect, andtheangle ofspread, which wecandefine astheangle ofthefirstminimum, isameasure oftheuncertainty inthefinal angle. How does thepattern become spread? Tosayitisspread means thatthere is some chance fortheparticle tobemoving upordown, thatis,tohave acomponent ofmomentum upordown. Wesaychance andparticle because wecandetect this diffraction pattern with aparticle counter, andwhen thecounter receives the particle, sayatCinFig.38-2, itreceives theentire particle, sothat, inaclassical sense, theparticle hasavertical momentum, inorder togetfrom theslituptoC. Togetarough ideaofthespread ofthemomentum, thevertical momentum p,,hasaspread which isequal topoA0,where p0isthehorizontal momentum. And how bigisA0inthespread-out pattern? Weknow thatthefirstminimum occurs atanangle A0such thatthewaves from oneedge oftheslithave totravel onewavelength farther than thewaves from theother side—we worked thatout before (Chapter 30). Therefore A6is)\/B, andsoAp,,inthisexperiment ispox/B. Note thatifwemake Bsmaller andmake amore accurate measurement ofthe position oftheparticle, thediffraction pattern getswider. Remember, when we closed theslitsontheexperiment with themicrowaves, wehadmore intensity farther out. Sothenarrower wemake theslit,thewider thepattern gets, andthe 38-2 more isthelikelihood thatwewould findthattheparticle hassidewise momentum. Thus theuncertainty inthevertical momentum isinversely proportional tothe uncertainty ofy.Infact, weseethattheproduct ofthetwoisequal top0>\. ButA isthewavelength andpoisthemomentum, andinaccordance with quantum mechanics, thewavelength times themomentum isPlanck’s constant h.Sowe obtain therulethattheuncertainties inthevertical momentum andinthevertical position have aproduct oftheorder h: AyAp,, zh. (38.3) Wecannot prepare asystem inwhich weknow thevertical position ofaparticle andcanpredict how itwillmove vertically with greater certainty than given by (38.3). Thatis,theuncertainty inthevertical momentum mustexceed h/Ay, where Ayistheuncertainty inourknowledge oftheposition. Sometimes people sayquantum mechanics isallwrong. When theparticle arrived from theleft,itsvertical momentum waszero. And now thatithasgone through theslit,itsposition isknown. Both position andmomentum seem to beknown with arbitrary accuracy. Itisquite truethatwecanreceive aparticle, andonreception determine what itsposition isandwhat itsmomentum would have hadtohave been tohave gotten there. That istrue, butthatisnotwhat the uncertainty relation (38.3) refers to.Equation (38.3) refers tothepredictability ofasituation, notremarks about thepast. Itdoes nogood tosay“Iknew what themomentum wasbefore itwent through theslit,andnowIknow theposition,” because nowthemomentum knowledge islost. Thefactthatitwent through the slitnolonger permits ustopredict thevertical momentum. Wearetalking about apredictive theory, notjustmeasurements after thefact. Sowemust talkabout what wecanpredict. Now letustakethething theother wayaround. Letustakeanother example ofthesame phenomenon, alittle more quantitatively. Intheprevious example wemeasured themomentum byaclassical method. Namely, weconsidered the direction andthevelocity andtheangles, etc.,sowegotthemomentum byclassical analysis. Butsince momentum isrelated towave number, there exists innature stillanother waytomeasure themomentum ofaparticle—photon orotherwise— which hasnoclassical analog, because itusesEq.(38.2). Wemeasure thewave- lengths ofthewaves. Letustrytomeasure momentum inthisway. Suppose wehave agrating with alarge number oflines (Fig. 38-3), andsend abeam ofparticles atthegrating. Wehave often discussed thisproblem: ifthe particles have adefinite momentum, then wegetavery sharp pattern inacertain direction, because oftheinterference. And wehave alsotalked about how accu- rately wecandetermine thatmomentum, thatistosay,what theresolving power ofsuch agrating is.Rather than derive itagain, werefer toChapter 30,where wefound thattherelative uncertainty inthewavelength thatcanbemeasured with agiven grating is1/Nm, where Nisthenumber oflines onthegrating and mistheorder ofthediflraction pattern. That is, Ah/A =l/Nm. (38.4) Now formula (38.4) canberewritten as Ax/x2 =1/Nmx =1/L, (38.5) where Listhedistance shown inFig.38-3. This distance isthedifference between thetotal distance thattheparticle orwave orwhatever itishastotravel ifitis reflected from thebottom ofthegrating, andthedistance thatithastotravel if itisreflected from thetopofthegrating. That is,thewaves which form thediffrac- tionpattern arewaves which come from different parts ofthegrating. Thefirst ones thatarrive come from thebottom endofthegrating, from thebeginning of thewave train, andtherestofthem come from later parts ofthewave train, coming from different parts ofthegrating, until thelastonefinally arrives, andthatinvolves apoint inthewave train adistance Lbehind thefirstpoint. Soinorder thatwe shall have asharp lineinourspectrum corresponding toadefinite momentum, 38-3NmX=L <_ \\/ / / -— / / /\\ <-— / \/ \ / \ \\ Fig. 38-3. Determination ofmomen tumbyusing adiffraction grating. dsin0 ‘ dsin8ldr\ 9 ’ Fig.38-4. Scattering ofwaves by crystal planes.with anuncertainty given by(38.4), wehave tohave awave train ofatleast length L.Ifthewave train istooshort wearenotusing theentire grating. Thewaves which form thespectrum arebeing reflected from only avery short sector ofthe grating ifthewave train istooshort, andthegrating willnotwork right—we will findabigangular spread. Inorder togetanarrower one,weneed tousethewhole grating, sothatatleast atsome moment thewhole wave train isscattering simul- taneously from allparts ofthegrating. Thus thewave train must beoflength L inorder tohave anuncertainty inthewavelength lessthan thatgiven by(38.5). Incidentally, Ax/x2 =A(l/A) =Ak/21r. (38.6) Therefore Ak=21r/L, (38.7) where Listhelength ofthewave train. This means thatifwehave awave train whose length islessthan L,theun- certainty inthewave number must exceed 21r/L. Ortheuncertainty inawave number times thelength ofthewave train—we willcallthatforamoment Ax— exceeds 21r. WecallitAxbecause thatistheuncertainty inthelocation ofthe particle. Ifthewave train exists onlyinafinite length, then thatiswhere wecould findtheparticle, within anuncertainty Ax.Now thisproperty ofwaves, thatthe length ofthewave train times theuncertainty ofthewave number associated with itisatleast 21r,isaproperty thatisknown toeveryone whostudies them. Ithas nothing todowith quantum mechanics. Itissimply thatifwehave afinite train, wecannot count thewaves initvery precisely. Letustryanother waytoseethe reason forthat. Suppose thatwehave afinite train oflength L;then because ofthewayithas todecrease attheends, asinFig. 38-1, thenumber ofwaves inthelength Lis uncertain bysomething like*1. Butthenumber ofwaves inLiskL/2-rr. Thus k isuncertain, andweagain gettheresult (38.7), aproperty merely ofwaves. The same thing works whether thewaves areinspace andkisthenumber ofradians percentimeter andListhelength ofthetrain, orthewaves areintime andwis thenumber ofoscillations persecond andTisthe“length” intime thatthewave train comes in.That is,ifwehave awave train lasting only foracertain finite time T,then theuncertainty inthefrequency isgiven by Aw=21r/T. (38.8) Wehave tried toemphasize thatthese areproperties ofwaves alone, andtheyare wellknown, forexample, inthetheory ofsound. Thepoint isthatinquantum mechanics weinterpret thewave number as being ameasure ofthemomentum ofaparticle, with therulethatp=hk,so thatrelation (38.7) tellsusthatApwh/Ax. This, then, isalimitation oftheclassi- calidea ofmomentum. (Naturally, ithastobelimited insome ways ifweare going torepresent particles bywaves!) Itisnicethatwehave found arulethat gives ussome ideaofwhen there isafailure ofclassical ideas. 38-3 Crystal diffraction Next letusconsider thereflection ofparticle waves from acrystal. Acrystal isathick thing which hasawhole lotofsimilar atoms—we willinclude some com- plications later—in anicearray. Thequestion ishow tosetthearray sothatwe getastrong reflected maximum inagiven direction foragiven beam of,say,light (x-rays), electrons, neutrons, oranything else. Inorder toobtain astrong reflection, thescattering from alloftheatoms must beinphase. There cannot beequal num- bersinphase andoutofphase, orthewaves willcancel out. Thewaytoarrange things istofind theregions ofconstant phase, aswehave already explained; theyareplanes which make equal angles with theinitial andfinal directions (Fig. 38-4). Ifweconsider twoparallel planes, asinFig.38-4, thewaves scattered from the twoplanes willbeinphase provided thedifference indistance travelled byawave- 38—4 I front isanintegral number ofwavelengths. This difference canbeseen tobe 2dsin6,where distheperpendicular distance between theplanes. Thus the condition forcoherent reflection is 2dsin0=n)\ (n=l,2,...). (38.9) If,forexample, thecrystal issuch thattheatoms happen tolieonplanes obey- ingcondition (38.9) with n=1,then there willbeastrong reflection. If,onthe other hand, there areother atoms ofthesame nature (equal indensity) halfway between, then theintermediate planes willalsoscatter equally strongly andwill interfere with theothers andproduce noeffect. Sodin(38.9) must refer toad- jacent planes; wecannot takeaplane fivelayers farther back andusethisformula! Asamatter ofinterest, actual crystals arenotusually assimple asasingle kind ofatom repeated inacertain way. Instead, ifwemake atwo-dimensional analog, theyaremuch likewallpaper, inwhich there issome kind offigure which repeats allover thewallpaper. By“figure” wemean, inthecaseofatoms, some arrangement—calcium andacarbon andthree oxygens, etc.,forcalcium carbonate, andsoon—which may involve arelatively large number ofatoms. Butwhatever itis,thefigure isrepeated inapattern. This basic figure iscalled aunitcell. Thebasic pattern ofrepetition defines what wecallthelattice type; thelattice typecanbeimmediately determined bylooking atthereflections andseeing what their symmetry is.Inother words, where wefindanyreflections atalldetermines thelattice type, butinorder todetermine what isineach oftheelements ofthe lattice onemust take into account theintensity ofthescattering atthevarious directions. Which directions scatter depends onthetypeoflattice, buthowstrongly each scatters isdetermined bywhat isinside each unitcell,andinthatwaythe structure ofcrystals isworked out. Two photographs ofx-ray diflraction patterns areshown inFigs. 38-5 and 38-6; theyillustrate scattering from rock saltandmyoglobin, respectively. Incidentally, aninteresting thing happens ifthespacings ofthenearest planes arelessthan A/2. Inthiscase (38.9) hasnosolution forn.Thus ifAisbigger than twice thedistance between adjacent planes then there isnosidediffraction pattern, andthelight-or whatever itiswillgoright through thematerial with- outbouncing offorgetting lost. Sointhecase oflight, where Aismuch bigger than thespacing, ofcourse itdoes gothrough andthere isnopattern ofreflection from theplanes ofthecrystal. This factalsohasaninteresting consequence inthecaseofpiles which make neutrons (these areobviously particles, foranybody’s moneyl). Ifwetake these neutrons andletthem into along block ofgraphite, theneutrons diffuse and work their wayalong (Fig. 38-7). They diffuse because they arebounced bythe atoms, butstrictly, inthewave theory, they arebounced bytheatoms because ofdiffraction from thecrystal planes. Itturns outthatifwetakeaverylong piece ofgraphite, theneutrons thatcome outthefarendarealloflong wavelength! Infact,ifoneplots theintensity asafunction ofwavelength, wegetnothing except forwavelengths longer than acertain minimum (Fig. 38-8). Inother words, we cangetvery slow neutrons thatway. Only theslowest neutrons come through; theyarenotdiffracted orscattered bythecrystal planes ofthegraphite, butkeep going right through likelight through glass, andarenotscattered outthesides. There aremany other demonstrations ofthereality ofneutron waves andwaves ofother particles. 38-4 Thesizeofanatom Wenow consider another application oftheuncertainty relation, Eq.(38.3). Itmust notbetaken tooseriously; theidea isright buttheanalysis isnotvery accurate. Theideahastodowith thedetermination ofthesizeofatoms, andthe factthat, classically, theelectrons would radiate light andspiral inuntil theysettle down right ontopofthenucleus. Butthatcannot beright quantum-mechanically because then wewould know where each electron wasandhowfastitwasmoving. 38-5Figure 38-5 Figure 38-6 SHORT-X NEUTRONS // -‘ —-tone-xPlLE: GRAPHITE _‘NEUTRONS \\SHORT-X NEUTRONS Fig. 38-7. Diffusion ofpile neutrons through graphite block. Ki.Xmlnntensty Fig. 38-8. Intensity ofneutrons outof graphite rodasfunction ofwavelength. Suppose wehave ahydrogen atom, andmeasure theposition oftheelectron; wemust notbeabletopredict exactly where theelectron willbe,orthemomentum spread willthen turn outtobeinfinite. Every time welook attheelectron, itis somewhere, butithasanamplitude tobeindifl'erent places sothere isaprob- ability ofitbeing found indifferent places. These places cannot allbeatthe nucleus; weshallsuppose there isaspread inposition oforder a.Thatis,thedis- tance oftheelectron from thenucleus isusually about a.Weshall determine aby minimizing thetotal energy oftheatom. Thespread inmomentum isroughly h/abecause oftheuncertainty relation, so thatifwetrytomeasure themomentum oftheelectron insome manner, such asbyscattering x-rays offitandlooking fortheDoppler effect from amoving scatterer, wewould expect nottogetzeroevery time—-the electron isnotstanding still—but themomenta must beoftheorder p~h/a. Then thekinetic energy isroughly —§mv2 =p2/2m =h2/2ma2. (Inasense, thisisakind ofdimensional analysis tofindoutinwhat waythekinetic energy depends upon Planck’s constant, upon m,andupon thesizeoftheatom. Weneed nottrust ouranswer towithin factors like2,1r,etc.Wehave noteven defined averyprecisely.) Now thepotential energy isminus e2overthedistance fromthecenter, say—e2/a, where, weremem- ber,e2isthecharge ofanelectron squared, divided by41re0. Now thepoint is thatthepotential energy isreduced ifagetssmaller, butthesmaller ais,thehigher themomentum required, because oftheuncertainty principle, andtherefore the higher thekinetic energy. Thetotalenergy is E=h2/2ma2 —e2/a. (38.10) Wedonotknow what ais,butweknow thattheatom isgoing toarrange itself tomake some kind ofcompromise sothattheenergy isaslittle aspossible. In order tominimize E,wedifferentiate with respect toa,setthederivative equal to zero, andsolve fora.Thederivative ofEis dE/da =—h2/maa +e2/a2, (38.11) andsetting dE/da =0gives forathevalue ao=h2/me2 =0.528 angstrom =0.528 XlO“1°meter. (38.12) This particular distance iscalled theBohr radius, andwehave thus learned that atomic dimensions areoftheorder ofangstroms, which isright: This ispretty good—in fact,itisamazing, since until nowwehave hadnobasis forunderstanding thesizeofatoms! Atoms arecompletely impossible from theclassical point of view, since theelectrons would spiral intothenucleus. Now ifweputthevalue (38.12) foraninto(38.10) tofindtheenergy, itcomes out E0=—e2/2a0 =—me“/2h2 =-13.6 ev. (38.13) What does anegative energy mean‘? Itmeans thattheelectron haslessenergy when itisintheatom than when itisfree. Itmeans itisbound. Itmeans ittakes energy tokicktheelectron out;ittakes energy oftheorder of13.6evtoionize a hydrogen atom. Wehave noreason tothink thatitisnottwoorthree times this— orhalfofthis—or (1/1r)times this,because wehave used such asloppy argument. However, wehave cheated, wehave used alltheconstants insuch awaythatit happens tocome outtheright number! This number, 13.6electron volts, iscalled aRydberg ofenergy; itistheionization energy ofhydrogen. Sowenowunderstand whywedonotfallthrough thefloor. Aswewalk, our shoes with their masses ofatoms push against thefloor with itsmass ofatoms. Inorder tosquash theatoms closer together, theelectrons would beconfined toa smaller space and, bytheuncertainty principle, their momenta would have tobe higher ontheaverage, andthatmeans highenergy; theresistance toatomic com- pression isaquantum-mechanical effect andnotaclassical effect. Classically, 38-6 wewould expect thatifweweretodraw alltheelectrons andprotons closer together, theenergy would bereduced stillfurther, andthebestarrangement ofpositive and negative charges inclassical physics isallontopofeach other. This waswell known inclassical physics andwasapuzzle because oftheexistence oftheatom. Ofcourse, theearly scientists invented some ways outofthetrouble—but never mind, wehave theright wayout,now! (Maybe.) Incidentally, although wehave noreason tounderstand itatthemoment, ina situation where there aremany electrons itturns outthattheytrytokeep away from each other. Ifoneelectron isoccupying acertain space, then another does notoccupy thesame space. More precisely, there aretwospincases, sothattwo cansitontopofeach other, onespinning onewayandonetheother way. Butafter thatwecannot putanymore there. Wehave toputothers inanother place, and thatistherealreason thatmatter hasstrength. Ifwecould putalltheelectrons inthesame place itwould condense even more than itdoes. Itisthefactthatthe electrons cannot allgetontopofeach other that makes tables andeverything elsesolid. Obviously, inorder tounderstand theproperties ofmatter, wewillhave touse quantum mechanics andnotbesatisfied with classical mechanics. 38-5 Energy levels Wehave talked about theatom initslowest possible energy condition, but itturns outthattheelectron candoother things. Itcanjiggle andwiggle ina more energetic manner, andsothere aremany different possible motions forthe atom. According toquantum mechanics, inastationary condition there canonly bedefinite energies foranatom. Wemake adiagram (Fig. 38-9) inwhich weplot theenergy vertically, andwemake ahorizontal lineforeach allowed value ofthe energy. When theelectron isfree, i.e.,when itsenergy ispositive, itcanhave any energy; itcanbemoving atanyspeed. Butbound energies arenotarbitrary. The atom must have oneoranother outofasetofallowed values, such asthose in Fig.38-9. Nowletuscalltheallowed values oftheenergy E0,E1,E2,E3.Ifanatom is initially inoneofthese “excited states,” E1,E2,etc., itdoes notremain inthat stateforever. Sooner orlateritdrops toalower stateandradiates energy inthe form oflight. Thefrequency ofthelight thatisemitted isdetermined byconserva- tionofenergy plus thequantum-mechanical understanding thatthefrequency of thelight isrelated totheenergy ofthelight by(38.1). Therefore thefrequency of thelight which isliberated inatransition from energy E3toenergy E1(forex- ample) is (.031=(E3—E1)/ft. (38.14) This, then, isacharacteristic frequency oftheatom anddefines aspectral emission line. Another possible transition would befrom E3toE0. That would have a different frequency (.030 =(E3 — Another possibility isthatiftheatom were excited tothestate E1itcould drop to theground state E0,emitting aphoton offrequency 6010 =(E1 -' Thereason webring upthree transitions istopoint outaninteresting relationship. Itiseasytoseefrom (38.14), (38.15), and(38.16) that W30=wsi-l"0110- (38-17) Ingeneral, ifwefmdtwospectral lines, weshall expect tofindanother lineatthe sumofthefrequencies (orthedifference inthefrequencies), andthatallthelines canbeunderstood byfinding aseries oflevels such thatevery linecorresponds to thedifference inenergy ofsome pairoflevels. This remarkable coincidence in 38-7Energy1 E0 Fig. 38-9. Energy diagram foran atom, showing several possible transitions. spectral frequencies wasnoted before quantum mechanics wasdiscovered, anditis called theRitz combination principle. This isagain amystery from thepoint of view ofclassical mechanics. Letusnotbelabor thepoint thatclassical mechanics isafailure intheatomic domain; weseem tohave demonstrated thatpretty well. Wehave already talked about quantum mechanics asbeing represented by amplitudes which behave likewaves, with certain frequencies andwave numbers. Letusobserve how itcomes about from thepoint ofview ofamplitudes thatthe atom hasdefinite energy states. Thisissomething wecannot understand from what hasbeen saidsofar,butweareallfamiliar with thefactthatconfined waves have definite frequencies. Forinstance, ifsound isconfined toanorgan pipe, orany- thing likethat, then there ismore than onewaythatthesound canvibrate, but foreach such waythere isadefinite frequency. Thus anobject inwhich thewaves areconfined hascertain resonance frequencies. Itistherefore aproperty ofwaves inaconfined space—-a subject which wewilldiscuss indetail with formulas later on-that they exist only atdefinite frequencies. And since thegeneral relation exists between frequencies oftheamplitude andenergy, wearenotsurprised to finddefinite energies associated with electrons bound inatoms. 38-6 Philosophical implications Letusconsider briefly some philosophical implications ofquantum mechanics. Asalways, there aretwoaspects oftheproblem: oneisthephilosophical implica- tion forphysics, and theother istheextrapolation ofphilosophical matters to other fields. When philosophical ideas associated with science aredragged into another field, they areusually completely distorted. Therefore weshall confine ourremarks asmuch aspossible tophysics itself. First ofall,themost interesting aspect istheideaoftheuncertainty principle; making anobservation affects thephenomenon. ,Ithasalways been known that making observations aflects aphenomenon, butthepoint isthattheeffect cannot bedisregarded orminimized ordecreased arbitrarily byrearranging theapparatus. When welook foracertain phenomenon wecannot helpbutdisturb itinacertain minimum way, andthedisturbance isnecessary fortheconsistency oftheviewpoint. The observer wassometimes important inprequantum physics, butonly ina rather trivial sense. Theproblem hasbeen raised: ifatreefallsinaforest andthere isnobody there tohear it,does itmake anoise? Arealtreefalling inarealforest makes asound, ofcourse, even ifnobody isthere. Even ifnooneispresent tohear it,there areother traces left. Thesound willshake some leaves, andifwewere careful enough wemight findsomewhere thatsome thorn hadrubbed against a leafandmade atinyscratch thatcould notbeexplained unless weassumed the leafwere vibrating. Soinacertain sense wewould have toadmit thatthere isa sound made. Wemight ask:wasthere asensation ofsound? No,sensations have todo,presumably, with consciousness. And whether ants areconscious and whether there were antsintheforest, orwhether thetreewasconscious, wedonot know. Letusleave theproblem inthatform. Another thing that people have emphasized since quantum mechanics was developed istheideathatweshould notspeak about those things which wecannot measure. (Actually relativity theory alsosaidthis.) Unless athing canbedefined bymeasurement, ithasnoplace inatheory. And since anaccurate value ofthe momentum ofalocalized particle cannot bedefined bymeasurement ittherefore hasnoplace inthetheory. Theideathatthisiswhat wasthematter with classical theory isafalse position. Itisacareless analysis ofthesituation. Justbecause we cannot measure position andmomentum precisely does notapriori mean thatwe cannot talkabout them. Itonly means thatweneed nottalkabout them. The situation inthesciences isthis: Aconcept oranideawhich cannot bemeasured orcannot bereferred directly toexperiment may ormay notbeuseful. Itneed notexist inatheory. Inother words, suppose wecompare theclassical theory of theworld with thequantum theory oftheworld, andsuppose thatitistrueex- perimentally thatwecanmeasure position andmomentum only imprecisely. The question iswhether theideas oftheexact position ofaparticle andtheexact 38-8 momentum ofaparticle arevalid ornot. Theclassical theory admits theideas; thequantum theory does not. This does notinitself mean thatclassical physics iswrong. When thenewquantum mechanics wasdiscovered, theclassical people- which included everybody except Heisenberg, Schrodinger, and Born—said: “Look, your theory isnotanygood because youcannot answer certain questions like: what istheexact position ofaparticle?, which hole does itgothrough?, andsome others.” Heisenberg’s answer was: “Idonotneed toanswer such ques- tions because youcannot asksuch aquestion experimentally.” Itisthatwedo nothaveto.Consider twotheories (a)and(b);(a)contains anideathatcannot be checked directly butwhich isused intheanalysis, andtheother, (b),does not contain theidea. Ifthey disagree intheir predictions, onecould notclaim that (b)isfalse because itcannot explain thisidea thatisin(a),because thatidea is oneofthethings thatcannot bechecked directly. Itisalways good toknow which ideas cannot bechecked directly, butitisnotnecessary toremove them all.Itis nottruethatwecanpursue science completely byusing onlythose concepts which aredirectly subject toexperiment. Inquantum mechanics itself there isawave function amplitude, there isa potential, andthere aremany constructs thatwecannot measure directly. Thebasis ofascience isitsability topredict. Topredict means totellwhat willhappen inan experiment thathasnever been done. How canwedothat? Byassuming thatwe know what isthere, independent oftheexperiment. Wemust extrapolate the experiments toaregion where they have notbeen done. Wemust take ourcon- cepts andextend them toplaces where theyhave notyetbeen checked. Ifwedo notdothat, wehave noprediction. Soitwasperfectly sensible fortheclassical physicists togohappily along andsuppose that theposition—which obviously means something forabaseball—meant something alsoforanelectron. Itwas notstupidity. Itwasasensible procedure. Today wesaythatthelawofrelativity issupposed tobetrueatallenergies, butsomeday somebody maycome along and sayhow stupid wewere. Wedonotknow where weare“stupid” until we“stick ourneck out,” andsothewhole ideaistoputourneck out. And theonlywayto findoutthatwearewrong istofindoutwhat ourpredictions are. Itisabsolutely necessary tomake constructs. Wehave already made afewremarks about theindeterminacy ofquantum mechanics. That is,thatweareunable nowtopredict what willhappen inphysics inagiven physical circumstance which isarranged ascarefully aspossible. If wehave anatom thatisinanexcited state andsoisgoing toemit aphoton, we cannot saywhen itwillemit thephoton. Ithasacertain amplitude toemit the photon atanytime, andwecanpredict only aprobability foremission; wecannot predict thefuture exactly. Thishasgiven risetoallkinds ofnonsense andquestions onthemeaning offreedom ofwill, andoftheideathattheworld isuncertain. Ofcourse wemust emphasize thatclassical physics isalsoindeterminate, ina sense. Itisusually thought thatthisindeterminacy, thatwecannot predict the future, isanimportant quantum-mechanical thing, andthisissaidtoexplain the behavior ofthemind, feelings offreewill, etc. Butiftheworld were classical—if thelaws ofmechanics were classical-it isnotquite obvious thatthemind would notfeelmore orlessthesame. Itistrueclassically thatifweknew theposition and thevelocity ofevery particle intheworld, orinaboxofgas,wecould predict ex- actly what would happen. And therefore theclassical world isdeterministic. Suppose, however, thatwehave afinite accuracy anddonotknow exactly where justoneatom is,saytoonepartinabillion. Then asitgoes along ithitsanother atom, andbecause wedidnotknow theposition better than toonepartinabillion, wefindanevenlarger error intheposition after thecollision. Andthatisamplified, ofcourse, inthenextcollision, sothatifwestart with only atinyerror itrapidly magnifies toaverygreat uncertainty. Togiveanexample: ifwater fallsoveradam, itsplashes. Ifwestand nearby, every now andthen adrop willland onournose. This appears tobecompletely random, yetsuch abehavior would bepredicted bypurely classical laws. Theexact position ofallthedrops depends upon the precise wigglings ofthewater before itgoes over thedam. How‘? Thetiniest irregularities aremagnified infalling, sothatwegetcomplete randomness. Ob- 38-9 viously, wecannot really predict theposition ofthedrops unless weknow the motion ofthewater absolutely exactly. Speaking more precisely, given anarbitrary accuracy, nomatter howprecise, onecanfindatime long enough thatwecannot make predictions valid forthat longatime. Now thepoint isthatthislength oftimeisnotverylarge. Itisnot thatthetimeismillions ofyears iftheaccuracy isonepartinabillion. Thetime goes, infact, only logarithmically with theerror, anditturns outthatinonlya very, verytinytimeweloseallourinformation. Iftheaccuracy istaken tobeone partinbillions andbillions andbillions—no matter howmany billions wewish, provided wedostopsomewhere—then wecanfindatimelessthanthetimeit took tostate theaccuracy-after which wecannolonger predict what isgoing tohappen! Itistherefore notfairtosaythatfrom theapparent freedom and indeterminacy ofthehuman mind, weshould have realized thatclassical “deter- ministic” physics could noteverhope tounderstand it,andtowelcome quantum mechanics asarelease from a“completely mechanistic” universe. Foralready in classical mechanics there wasindeterminability from apractical point ofview. 38-10 39 The Kinetic Theory ofGases 39-1 Properties ofmatter With thischapter webegin anewsubject which willoccupy usforsome time. Itisthefirstpartoftheanalysis oftheproperties ofmatter from thephysical point ofview, inwhich, recognizing thatmatter ismade outofagreat many atoms, or elementary parts, which interact electrically andobey thelaws ofmechanics, we trytounderstand whyvarious aggregates ofatoms behave thewaythey do. Itisobvious thatthisisadifficult subject, andweemphasize atthebeginning thatitisinfactanextremely difficult subject, andthatwehave todeal with it differently than wehave dealt with theother subjects sofar. Inthecase ofme- chanics andinthecaseoflight, wewere abletobegin with aprecise statement of some laws, likeNewton’s laws, ortheformula forthefield produced byanac- celerating charge, from which awhole host ofphenomena could beessentially understood, andwhich would produce abasis forourunderstanding ofmechanics andoflight from thattime on.That is,wemay learn more later, butwedonot learn different physics, weonly learn better methods ofmathematical analysis todealwith thesituation. Wecannot usethisapproach effectively instudying theproperties ofmatter. Wecandiscuss matter only inamost elementary way; itismuch toocomplicated asubject toanalyze directly from itsspecific basic laws, which arenone other than thelaws ofmechanics andelectricity. Butthese areabittoofaraway from the properties wewish tostudy; ittakes toomany steps togetfrom Newton’s laws to theproperties ofmatter, andthese steps are,inthemselves, fairly complicated. Wewillnow start totake some ofthese steps, butwhile many ofouranalyses willbequite accurate, they willeventually getlessandlessaccurate. Wewill have only arough understanding oftheproperties ofmatter. Oneofthereasons thatwehave toperform theanalysis soimperfectly isthat themathematics ofitrequires adeep understanding ofthetheory ofprobability; wearenotgoing towant toknow where every atom isactually moving, butrather, howmany move hereandthere ontheaverage, andwhat theodds arefordifferent eflects. Sothissubject involves aknowledge ofthetheory ofprobability, andour mathematics isnotyetquite ready andwedonotwant tostrain ittoohard. Secondly, andmore important from aphysical standpoint, theactual behavior oftheatoms isnotaccording toclassical mechanics, butaccording toquantum mechanics, andacorrect understanding ofthesubject cannot beattained until we understand quantum mechanics. Here, unlike thecaseofbilliard balls andauto- mobiles, thedifference between theclassical mechanical laws andthequantum- mechanical laws isvery important andvery significant, sothatmany things that wewilldeduce byclassical physics willbefundamentally incorrect. Therefore there willbecertain things tobepartially unlearned; however, weshall indicate inevery casewhen aresult isincorrect, sothatwewillknow justwhere the“edges” are. One ofthereasons fordiscussing quantum mechanics inthepreceding chapters wastogiveanideaastowhy, more orless,classical mechanics isincorrect inthevarious directions. Why dowedealwith thesubject now atall? Why notwait ahalfayear, or ayear, until weknow themathematics ofprobability better, andwelearn alittle quantum mechanics, andthen wecandoitinamore fundamental way? The answer isthatitisadifficult subject, andthebestwaytolearn istodoitslowly! Thefirstthing todoistogetsome idea, more orless,ofwhat ought tohappen in 39-139-1 Properties ofmatter 39-2 Thepressure ofagas 39-3 Compressibility ofradiation 39-4 Temperature andkinetic energy 39-5 Theideal gaslaw diflerent circumstances, andthen, later, when weknow thelaws better, wewill formulate them better. Anyone whowants toanalyze theproperties ofmatter inarealproblem might want tostart bywriting down thefundamental equations andthen trytosolve them mathematically. Although there arepeople whotrytousesuch anapproach, these people arethefailures inthisfield; therealsuccesses come tothose whostart from aphysical point ofview, people whohave arough ideawhere they aregoing andthen begin bymaking theright kind ofapproximations, knowing what isbig andwhat issmall inagiven complicated situation. These problems aresocompli- cated thateven anelementary understanding, although inaccurate andincomplete, isworth while having, andsothesubject willbeonethatweshall goover again andagain, each time with more andmore accuracy, aswegothrough ourcourse inphysics. Another reason forbeginning thesubject right now isthatwehave already used many ofthese ideas in,forexample, chemistry, andwehave even heard of some ofthem inhigh school. Itisinteresting toknow thephysical basis for these things. Asaninteresting example, weallknow thatequal volumes ofgases, atthe same pressure andtemperature, contain thesame number ofmolecules. Thelaw ofmultiple proportions, thatwhen twogases combine inachemical reaction the volumes needed always stand insimple integral proportions, wasunderstood ulti- mately byAvogadro tomean thatequal volumes have equal numbers ofatoms. Now whydotheyhave equal numbers ofatoms? Canwededuce from Newton’s laws thatthenumber ofatoms should beequal? Weshall address ourselves to thatspecific matter inthischapter. Insucceeding chapters, weshall discuss various other phenomena involving pressures, volumes, temperature, andheat. Weshall alsofindthatthesubject canbeattacked from anonatomic point of view, andthatthere aremany interrelationships oftheproperties ofsubstances. Forinstance, when wecompress something, itheats; ifweheat it,itexpands. There isarelationship between these twofacts which canbededuced independently ofthemachinery underneath. This subject iscalled thermodynamics. Thedeepest understanding ofthermodynamics comes, ofcourse, from understanding theactual machinery underneath, andthat iswhat weshall do:weshall take theatomic viewpoint from thebeginning anduseittounderstand thevarious properties of matter andthelaws ofthermodynamics. Letus,then, discuss theproperties ofgases from thestandpoint ofNewton’s laws ofmechanics. 39-2 Thepressure ofagas First, weknow thatagasexerts apressure, andwemust clearly understand what thisisdueto.Ifourearswere afewtimes more sensitive, wewould hear a perpetual rushing noise. Evolution hasnotdeveloped theeartothat point, because itwould beuseless ifitwere somuch more sensitive—we would hear a perpetual racket. Thereason isthattheeardrum isincontact with theair,andair isalotofmolecules inperpetual motion andthese bang against theeardrums. Inbanging against theeardrums they make anirregular tattoo—boom, boom, boom—which wedonothear because theatoms aresosmall, andthesensitivity oftheearisnotquite enough tonotice it.Theresult ofthisperpetual bombardment istopush thedrum away, butofcourse there isanequal perpetual bombardment ofatoms ontheother sideoftheeardrum, sothenetforce onitiszero. Ifwewere totake theairaway from oneside, orchange therelative amounts ofaironthe twosides, theeardrum would then bepushed onewayortheother, because the amount ofbombardment ononesidewould begreater than ontheother. We sometimes feelthisuncomfortable efl'ect when wegouptoofastinanelevator or anairplane, especially ifwealsohave abadcold(when wehave acold, inflammation closes thetube which connects theairontheinside oftheeardrum with theoutside airthrough thethroat, sothatthetwopressures cannot readily equalize). 39-2 Inconsidering how toanalyze thesituation quantitatively, weimagine that wehave avolume ofgasinabox, atoneendofwhich isapiston which canbe moved (Fig. 39-1). Wewould liketofindoutwhat force onthepiston results from thefactthatthere areatoms inthisbox. Thevolume oftheboxisV,andasthe atoms move around inside theboxwith various velocities they bang against the piston. Suppose there isnothing, avacuum, ontheoutside ofthepiston. What of it?Ifthepiston were leftalone, andnobody held onto it,each time itgotbanged itwould pickupalittle momentum anditwould gradually getpushed outofthe box. Soinorder tokeep itfrom being pushed outofthebox, wehave tohold it withaforce F.Theproblem is,howmuch force? Onewayofexpressing theforce istotalkabout theforce perunitarea: ifAistheareaofthepiston, thentheforce onthepiston willbewritten asanumber times thearea. Wedefine thepressure, then, asequal totheforce thatwehave toapply onapiston, divided bythearea ofthepiston: P=F/A. (39.1) Tomake sureweunderstand theidea (wehave toderive itforanother purpose anyway), thedifferential work dWdone onthegasincompressing itbymoving thepiston inadifferential amount —dx would betheforce times thedistance thatwecompress it,which, according to(39.1), would bethepressure times the area, times thedistance, which isequal tominus thepressure times thechange inthevolume: dW =F(—dx) =—PA dx=—PdV. (39.2) (The areaAtimes thedistance dxisthevolume change.) Theminus signisthere because, aswecompress it,wedecrease thevolume; ifwethink about itwecan seethatifagasiscompressed, work isdone onit. How much force dowehave toapply tobalance thebanging ofthemolecules? Thepiston receives from each collision acertain amount ofmomentum. Acertain amount ofmomentum persecond willpour intothepiston, anditwillstarttomove. Tokeep itfrom moving, wemust pour back intoitthesame amount ofmomentum persecond from ourforce. Ofcourse, theforce istheamount ofmomentum per second thatwemust pour in.There isanother waytoputit:ifweletgoofthe piston itwillpickupspeed because ofthebombardments; with each collision we getalittle more speed, andthespeed thusaccelerates. Therateatwhich thepiston picks upspeed, oraccelerates, isproportional totheforce onit.Soweseethat theforce, which wealready have saidisthepressure times thearea, isequal tothe momentum persecond delivered tothepiston bythecolliding molecules. Tocalculate themomentum persecond iseasy-we candoitintwoparts: first, wefindthemomentum delivered tothepiston byoneparticular atom ina collision with thepiston, then wehave tomultiply bythenumber ofcollisions per second thattheatoms have with thewall. Theforce willbetheproduct ofthese twofactors. Now letusseewhat thetwofactors are:Inthefirstplace, weshall suppose thatthepiston isaperfect “reflector” fortheatoms. Ifitisnot,thewhole theory iswrong, andthepiston willstart toheat upandthings willchange, but eventually, when equilibrium hassetin,thenetresult isthatthecollisions are eflectively perfectly elastic. Ontheaverage, every particle thatcomes inleaves withthesame energy. Soweshall imagine thatthegasisinasteady condition, and welosenoenergy tothepiston because thepiston isstanding still. Inthose cir- cumstances, ifaparticle comes inwith acertain speed, itcomes outwith thesame speed and, wewillsay,with thesame mass. Ifvisthevelocity ofanatom, andv,isthex-component ofv,thenmu,isthe x-component ofmomentum “in”; butwealsohave anequal component ofmo- mentum “out,” andsothetotal momentum delivered tothepiston bytheparticle, inonecollision, is2mv,, because itis“reflected.” Now, weneed thenumber ofcollisions made bytheatoms inasecond, orin acertain amount oftime dt;then wedivide bydt.How many atoms arehitting? Letussuppose thatthere areNatoms inthevolume V,orn=N/Vineach unit volume. Tofindhow many atoms hitthepiston, wenote that, given acertain 39-3a0 0.0 <oa0 0I\\\\\\\\‘> 7" l<i— Ii>!dll<—- Fig. 39-l. Atoms ofagasinabox with africtionless piston. amount oftime t,ifaparticle hasacertain velocity toward thepiston itwillhit during thetime t,provided itisclose enough. Ifitistoofaraway, itgoes only partwaytoward thepiston inthetime t,butdoes notreach thepiston. Therefore itisclear thatonly those molecules which arewithin adistance v,tfrom thepiston aregoing tohitthepiston inthetime t.Thus thenumber ofcollisions inatime tis equal tothenumber ofatoms which areintheregion within adistance v,,t,and since theareaofthepiston isA,thevolume occupied bytheatoms which aregoing tohitthepiston isv,,tA. Butthenumber ofatoms thataregoing tohitthepiston isthatvolume times thenumber ofatoms perunitvolume, nv,,tA. Ofcourse we donotwant thenumber thathitinatimet,wewant thenumber thathitpersecond, sowedivide bythetime t,togetnv,,A. (This time tcould bemade veryshort; if wefeelwewant tobemore elegant, wecallitdt,thendifferentiate, butitisthesame thing.) Sowefindthattheforce is F=nu,/1 -2mv,,. (39.3) See,theforce isproportional tothearea, ifwekeep theparticle density fixed as wechange thearea! Thepressure isthen P=znmvi. (39.4) Now wenotice alittle trouble with thisanalysis: First, allthemolecules do nothave thesame velocity, andthey donotmove inthesame direction. So,all thevf’saredifferent! Sowhat wemust do,ofcourse, istotakeanaverage ofthe v§’s,since each onemakes itsown contribution. What wewant isthesquare of 22,,averaged over allthemolecules: P=nmoi). (39.5) Did weforget toinclude thefactor 2?No; ofalltheatoms, only half are headed toward thepiston. Theother halfareheaded theother way, andifwetake (bf), weareaveraging thenegative v,,’ssquared, aswell asthepositive v,,’s. So when wejusttake (vi),without looking, wearegetting twice asmuch aswewant. Theaverage ofvi,forpositive 11,,isequal totheaverage ofviforall1),,times one half. Now astheatoms bounce around, itisclear thatthere isnothing special about the“x-direction”; theatoms may alsobemoving upanddown, back andforth, inandout. Therefore itisgoing tobetruethat (vi), theaverage motion ofthe atoms inonedirection, andtheaverage intheother twodirections, areallgoing tobeequal: (vi)=(vi)=(vii (39-6) Itisonly amatter ofrather tricky mathematics tonotice, therefore, thatthey are each equal toone-third oftheir sum, which isofcourse thesquare ofthemagnitude ofthevelocity: <»i>=an+vi+vi)=<02)/1 <39-1) Thishastheadvantage thatwedonothave toworry about anyparticular direction, andsowewrite ourpressure formula again inthisform: P=(§)n(mv2/2). (39.3) Thereason wewrote thelastfactor as(mv2/2) isthatthisisthekinetic energy of thecenter-of-mass motion ofthemolecule. Wefind, therefore, that PV=N(%)(mv2/2). (39.9) With thisequation wecancalculate howmuch thepressure is,ifweknow thespeeds. Asaverysimple example letustakehelium gas,oranyother gas,likemercury vapor, orpotassium vapor ofhigh enough temperature, orargon, inwhich allthe molecules aresingle atoms, forwhich wemay suppose that there isnointernal 39-4 motion intheatom. Ifwehadacomplex molecule, there might besome internal motion, mutual vibrations, orsomething. Wesuppose thatwemaydisregard that; thisisactually aserious matter thatwewillhave tocome back to,butitturns out tobeallright. Wesuppose thattheinternal motion oftheatoms canbedisre- garded, andtherefore, forthispurpose, thatthekinetic energy ofthecenter-of-mass motion isalltheenergy there is.Soforamonatomic gas,thekinetic energy isthe total energy. Ingeneral, wearegoing tocallUthetotal energy (itissometimes called thetotal internal energy—we may wonder why, since there isnoexternal energy toagas), i.e.,alltheenergy ofallthemolecules inthegas,ortheobject, whatever itis. Foramonatomic gaswewillsuppose thatthetotal energy Uisequal toanum- berofatoms times theaverage kinetic energy ofeach, because wearedisregarding anypossibility ofexcitation ormotion inside theatoms themselves. Then, inthese circumstances, wewould have PV=§U. (39.10) Incidentally, wecanstophereandfindtheanswer tothefollowing question: Suppose thatwetakeacanofgasandcompress thegasslowly, howmuch pressure doweneed tosqueeze thevolume down? Itiseasy tofindout,since thepressure is§theenergy divided byV.Aswesqueeze itdown, wedowork onthegasandwe thereby increase theenergy U.Sowearegoing tohave some kind ofadifferential equation: Ifwestart outinagiven circumstance withacertain energy andacertain volume, wethen know thepressure. Now westart tosqueeze, butthemoment we do,theenergy Uincreases andthevolume Vdecreases, sothepressure goes up. So,wehave tosolve adifferential equation, andwewillsolve itinamoment. Wemust firstemphasize, however, thataswearecompressing thisgas,weare supposing thatallthework goes intoincreasing theenergy oftheatoms inside. Wemayask,“Isn’t thatnecessary? Where elsecould itgo?”Itturns outthatit cangoanother place. There arewhat wecall“heat leaks” through thewalls: thehot(i.e.,fast-moving) atoms thatbombard thewalls, heatthewalls, andenergy goes away. Weshall suppose forthepresent thatthisisnotthecase. Forsomewhat wider generality, although wearestillmaking some veryspecial assumptions about ourgas,weshallwrite, notPV=%U,but PV=(1-1)U. (39.11) Itiswritten (‘Y—l)times Uforconventional reasons, because wewilldealwith afewother cases later where thenumber infront ofUwillnotbe§,butwillbea different number. So,inorder todothething ingeneral, wecallit‘Y—1,because people have been calling itthatforalmost onehundred years. This 'Y,then, is1%, because §—1is§foramonatomic gaslikehelium. Wehave already noticed that when wecompress agasthework done is —PdV. Acompression inwhich there isnoheat energy added orremoved is called anadiabaticcompression, from theGreek a(not) +dia(through) +bainein (togo). (The word adiabatic isused inphysics inseveral ways, anditissometimes hard toseewhat iscommon about them.) That is,foranadiabatic compression allthework done goes intochanging theinternal energy. That isthekey-that there arenoother losses ofenergy—for then wehave PdV =—dU. Butsince U=PV/(7 —1),wemaywrite dU=(PdV+VdP)/(‘Y -1). (39.12) Sowehave PdV =—(PdV+ VdP)/(‘Y —1),or,rearranging theterms, 'YPdV =—VdP, or ('YdV/V) +(dP/P) =0. (39.13) Fortunately, assuming thatVisconstant, asitisforamonatomic gas,wecaninte- grate this: itgives 7lnV+lnP=lnC,where lnCistheconstant ofintegration. Ifwetaketheexponential ofboth sides, wegetthelaw PV" =C(aconstant). (39.14) 39-5 Inother words, under adiabatic conditions, where thetemperature rises aswe compress because noheat isbeing lost, thepressure times thevolume tothe% power isaconstant foramonatomic gas! Although wederived ittheoretically, thisis,infact, thewaymonatomic gases behave experimentally. 39-3 Compressibility ofradiation Wemay giveoneother example ofthekinetic theory ofagas,onewhich is notused inchemistry somuch, butisused inastronomy. Wehave alarge number ofphotons inaboxinwhich thetemperature isveryhigh. (The boxis,ofcourse, thegasinavery hotstar. Thesunisnothotenough; there arestilltoomany atoms, butatstillhigher temperatures incertain veryhotstars, wemayneglect the atoms andsuppose thattheonly objects thatwehave intheboxarephotons.) Now then, aphoton hasacertain momentum p.(We always findthatwearein terrible trouble when wedokinetic theory: pisthepressure, butpisthemomentum; visthevolume, butvisthevelocity; Tisthetemperature, butTisthekinetic energy orthetime orthetorque; onemust keep one’s wits about one!) This pismomentum, itisavector. Going through thesame analysis asbefore, itis thex-component ofthevector pwhich generates the“kick,” andtwice thex-com- ponent ofthevector pisthemomentum which isgiven inthekick. Thus 2p,re- places 2mv,,, andinevaluating thenumber ofcollisions, 0,,isstill1),,sowhen we getallthewaythrough, wefindthatthepressure inEq.(39.4) is,instead, P=2np,,v,,. (39.15) Then, intheaveraging, itbecomes ntimes theaverage ofp,,v,, (thesame factor of 2)and, finally, putting intheother twodirections, wefind PV=N(p-v)/3. (39.16) This checks with theformula (39.9), because themomentum ismv;itisalittle more general, thatisall.Thepressure times thevolume isthetotal number of atoms times §(p-v),averaged. Now, forphotons, what isp-v?Themomentum andthevelocity areinthe same direction, andthevelocity isthespeed oflight, sothisisthemomentum of each oftheobjects, times thespeed oflight. Themomentum times thespeed of light ofevery photon isitsenergy: E=pc,sothese terms aretheenergies of each ofthephotons, andweshould, ofcourse, take anaverage energy, times the number ofphotons. Sowehave §oftheenergy inside thegas: PV=U/3(photon gas). (39.17) Forphotons, then, since wehave %infront, ("I—1)in(39.11) is§,orY=%,and wehave discovered thatradiation inaboxobeys thelaw PV4/3 =C. (39.18) Soweknow thecompressibility ofradiation! That iswhat isused inananalysis ofthecontribution ofradiation pressure inastar, thatishow wecalculate it,and how itchanges when wecompress it.What wonderful things arealready within ourpower! 39-4 Temperature andkinetic energy Sofarwehave notdealt with temperature; wehave purposely been avoiding thetemperature. Aswecompress agas,weknow thattheenergy ofthemolecules increases, andweareused tosaying thatthegasgetshotter; wewould liketo understand what thishastodowith thetemperature. Ifwetrytodotheexperi- ment, notadiabatically butatwhat wecallconstant temperature, what arewedoing? Weknow thatifwetaketwoboxes ofgasandletthem sitnext toeach other long enough, even ifatthestart they were atwhat wecalldifferent temperatures, they 39-6 willintheendcome tothesame temperature. Now what does thatmean? That means thatthey gettoacondition thatthey would gettoifweleftthem alone long enough! What wemean byequal temperature isjustthat—the finalcondition when things have been sitting around interacting with each other long enough. Letusconsider, now, what happens ifwehave twogases incontainers sepa- rated byamovable piston asinFig.39-2 (just forsimplicity weshall take two monatomic gases, sayhelium andneon). Incontainer (1)theatoms have mass m1,velocity v1,andthere aren1perunitvolume, andintheother container the atoms have mass m2,velocity v2,there aren2atoms perunitvolume. What are theconditions forequilibrium? Obviously, thebombardment from theleftsidemust besuch thatitmoves thepiston totheright andcompresses theother gasuntil itspressure builds up, andthething willthus slosh back andforth, andwillgradually come torestat aplace where thepressures areequal onboth sides. Sowecanarrange thatthe pressures areequal; thatjustmeans thattheinternal energies perunitvolume are equal, orthatthenumbers ntimes theaverage kinetic energies oneach sideare equal. What wehave totrytoprove, eventually, isthatthenumbers themselves areequal. Sofar,allweknow isthat thenumbers times thekinetic energies areequal, nifmivi/2l ="2(m2v%/2), from (39.8), because thepressures areequal. Wemust realize thatthisisnotthe only condition over thelong run, butsomething elsemust happen more slowly asthetruecomplete equilibrium corresponding toequal temperatures setsin. Toseetheidea, suppose thatthepressure ontheleftsidewere developed by having avery high density butalowvelocity. Byhaving alarge nandasmall v, wecangetthesame pressure asbyhaving asmall nandalarge v.Theatoms may bemoving slowly butbepacked nearly solidly, orthere maybefewer buttheyare hitting harder. Will itstaylikethatforever? Atfirstwemight think so,butthen wethink again andfindwehave forgotten oneimportant point. That is,thatthe intermediate piston does notreceive asteady pressure; itwiggles, justlikethe eardrum thatwewere firsttalking about, because thebangings arenotabsolutely uniform. There isnotaperpetual, steady pressure, butatattoo—the pressure varies, andsothething jiggles. Suppose thattheatoms ontheright sidearenot jiggling much, butthose ontheleftarefewandfarbetween andvery energetic. Thepiston will, now andthen, getabigimpulse from theleft,andwillbedriven against theslow atoms ontheright, giving them more speed. (Aseach atom collides with thepiston, iteither gains orloses energy, depending upon whether thepiston ismoving onewayortheother when theatom strikes it.)So,asaresult ofthecollisions, thepiston finds itself jiggling, jiggling, jiggling, andthisshakes theother gas-it gives energy totheother atoms, andtheybuild upfaster motions, until theybalance thejiggling thatthepiston isgiving tothem. Thesystem comes tosome equilibrium where thepiston ismoving atsuch amean square speed that itpicks upenergy from theatoms atabout thesame rateasitputs energy back intothem. Sothepiston picks upacertain mean irregularity inspeed, anditis ourproblem tofindit.When wedofindit,wecansolve ourproblem better, be- cause thegases willadjust their velocities until therateatwhich they aretrying topour energy intoeach other through thepiston willbecome equal. Itisquite difficult tofigure outthedetails ofthepiston inthisparticular cir- cumstance; although itisideally simple tounderstand, itturns outtobealittle harder toanalyze. Before weanalyze that, letusanalyze another problem in which wehave aboxofgasbutnowwehave twodifferent kinds ofmolecules init, having masses m1andmg,velocities v1and02,andsoforth; there isnow amuch more intimate relationship. IfalloftheNo.2molecules arestanding still, that condition isnotgoing tolast,because theygetkicked bytheNo.lmolecules and sopickupspeed. Iftheyareallgoing much faster than theNo.1molecules, then maybe thatwillnotlasteither—they willpasstheenergy back totheNo.1mole- cules. Sowhen both gases areinthesame box, theproblem istofindtherulethat determines therelative speeds ofthetwo. 39-7 n\\\\\\\\0o oV o o ll) (2) Fig.39-2. Atoms oftwo different monatomic gases are separated bya movable piston. '1 Vi > ya <—— "z Fig. 39-3. Acollision between un- equal molecules, viewed intheCMsystem.This isstillavery difficult problem, butwewillsolve itasfollows. First we consider thefollowing sub-problem (again thisisoneofthose cases where—never mind thederivation—in theendtheresult isvery simple toremember, butthe derivation isjustingenious). Letussuppose thatwehave twomolecules, ofdiffer- entmass, colliding, andthatthecollision isviewed onthecenter-of-mass (CM) system. Inorder toremove acomplication, welook atthecollision intheCM. Asweknow from thelaws ofcollision, bytheconservation ofmomentum anden- ergy, after themolecules collide theonly way they canmove issuch thateach maintains itsownoriginal speed—and theyjustchange their direction. Sowehave anaverage collision thatlooks likethatinFig.39-3. Suppose, foramoment, that wewatch allthecollisions with theCMatrest. Suppose weimagine thatthey are allinitially moving horizontally. Ofcourse, after thefirstcollision some ofthem aremoving atanangle. Inother words, ifthey were allgoing horizontally, then atleast some would later bemoving vertically. Now insome other collision, they would becoming infrom another direction, andthen they would betwisted at stillanother angle. Soeven ifthey were completely organized inthebeginning, they would getsprayed around atallangles, andthen thesprayed ones would get sprayed some more, andsprayed some more, andsprayed some more. Ultimately, what willbethedistribution? Answer: Itwillbeequally likely tofindanypair moving inanydirection inspace. After thatfurther collisions could notchange the distribution. They areequally likely togoinalldirections, buthowdowesaythat? There isofcourse nolikelihood thattheywillgoinanyspecific direction, because aspecific direction istooexact, sowehave totalkabout perunit“something.” Theideais that anyarea onasphere centered atacollision point willhave justasmany molecules going through itasgothrough anyother equal area onthesphere. Sotheresult ofthecollisions willbetodistribute thedirections sothatequal areas onasphere willhave equal probabilities. Incidentally, ifwejustwant todiscuss theoriginal direction andsome other direction anangle 0from it,itisaninteresting property thatthediflerential area ofasphere ofunitradius issin0d0times 21r,andthatisthesame asthediffer- ential ofcos0.Sowhat itmeans isthatthecosine oftheangle 0between anytwo directions isequally likely tobeanything from —lto+1. Next, wehave toworry about theactual case, where wedonothave the collision intheCMsystem, butwehave twoatoms which arecoming together with vector velocities v1andv2.What happens now? Wecananalyze thiscollision with thevector velocities v1andv2inthefollowing way: Wefirstsaythatthere isacertain CM; thevelocity oftheCMisgiven bythe“average” velocity, with weights proportional tothemasses, sothevelocity oftheCMisvCM =(m1v1+ m2v2)/ (m1+m2). Ifwewatch thiscollision intheCM system, then weseea collision justlikethatinFig.39-3, with acertain relative velocity wcoming in. Therelative velocity isjustv1—v2.Now theideaisthat, first, thewhole CMis moving, andintheCMthere isarelative velocity w,andthemolecules collide andcome offinsome newdirection. Allthishappens while theCMkeeps right on moving, without anychange. Now then, what isthedistribution resulting from this? From ourprevious argument weconclude this: that atequilibrium, alldirections forWareequally likely, relative tothedirection ofthemotion oftheCM.*There willbenoparticular correlation, intheend,between thedirection ofthemotion oftherelative velocity andthatofthemotion oftheCM. Ofcourse, ifthere were, thecollisions would spray itabout, soitisallsprayed around. Sothecosine oftheangle between w andv¢Miszero ontheaverage. That is, (W'VCM> = "This argument, which wastheoneused byMaxwell, involves some subtleties. Al- though theconclusion iscorrect, theresult doesnotfollow purely from theconsiderations ofsymmetry thatweusedbefore, since, bygoing toareference frame moving through thegas,wemayfindadistorted velocity distribution. Wehavenotfound asimple proof ofthisresult. 39-8 Butw-v01/1canbeexpressed interms ofv1andv2aswell: (Vi—V2)'(miV1 +m2V2) mi+W12 2_ 2 _ _:(mivi m2v22n;l'_}Emni2 m1)(Vi V2)_ (39.20) First, letuslook atthev1-v2;what istheaverage ofv1-v2?That is,what is theaverage ofthecomponent ofvelocity ofonemolecule inthedirection ofan- other? Surely there isjustasmuch likelihood offinding anygiven molecule moving onewayasanother. Theaverage ofthevelocity v2inanydirection iszero. Certainly, then, inthedirection ofv1,v2haszero average. So,theaverage ofv1-v2is zero! Therefore, weconclude thattheaverage ofm11)?must beequal totheaverage ofmgvg. That is,theaverage kinetic energy ofthetwomust beequal: gimp? =%m2v§. (39.21) Ifwehave twokinds ofatoms inagas,itcanbeshown, andwepresume tohave shown it,thattheaverage ofthekinetic energy ofoneisthesame astheaverage of thekinetic energy oftheother, when theyareboth inthesame gasinthesame box inequilibrium. That means thattheheavy ones willmove slower than thelight ones; thisiseasily shown byexperimentation with “atoms” ofdifferent masses inanairtrough. Nowwewould liketogoonestepfurther, andsaythatifwehavetwodifferent gases separated inabox, they willalsohave equal average kinetic energy when they have finally come toequilibrium, even though they arenotinthesame box. Wecanmake theargument inanumber ofways. Onewayistoargue thatifwe have afixed partition with atinyholeinit(Fig. 39-4) sothatonegascould leak outthrough theholes while theother could not,because themolecules aretoo big,andthese hadattained equilibrium, thenweknow thatinonepart, where they aremixed, theyhave thesame average kinetic energy, butsome come through the holewithout lossofkinetic energy, sotheaverage kinetic energy inthepure gas andinthemixture must bethesame. That isnottoosatisfactory, because maybe there arenoholes, forthiskind ofmolecule, thatseparate onekind from theother. Letusnow goback tothepiston problem. Wecangive anargument which shows thatthekinetic energy ofthispiston must alsobe%m2v§. Actually, thatwould bethekinetic energy duetothepurely horizontal motion ofthepiston, so,forgetting itsupanddown motion, itwillhave tobethesame as%m2v§z. Likewise, from theequilibrium ontheother side, wecanprove that thekinetic energy ofthepiston is%m1v§z. Although thisisnotinthemiddle ofthegas,but isononesideofthegas,wecanstillmake theargument, although itisalittle more difficult, thattheaverage kinetic energy ofthepiston andofthegasmolecules areequal asaresult ofallthecollisions. Ifthisstilldoes notsatisfy us,wemay make anartificial example bywhich theequilibrium isgenerated byanobject which canbehitonallsides. Suppose thatwehave ashort rodwith aballoneach endsticking through thepiston, ona frictionless sliding universal joint. Each ballisround, likeoneofthemolecules, andcanbehitonallsides. This whole object hasacertain total mass, m. Now, wehave thegasmolecules with mass m1andmass mgasbefore. The result ofthecollisions, bytheanalysis thatwasmade before, isthat thekinetic energy ofmbecause ofcollisions withthemolecules ononesidemust beém122%,on theaverage. Likewise, because ofthecollisions with molecules ontheother side, ithastobe%m2v§ ontheaverage. So,therefore, both sides have tohave the same kinetic energy when they areinthermal equilibrium. So,although weonly proved itforamixture ofgases, itiseasily extended tothecasewhere there are twodifferent, separate gases atthesame temperature. Thus when wehave twogases atthesame temperature, themean kinetic energy oftheCMmotions areequal. Themean molecular kinetic energy isaproperty only ofthe“temperature.” Being aproperty ofthe“temperature,” andnotofthegas,wecanuseitasadefini- tionofthetemperature. Themean kinetic energy ofamolecule isthus some 39-9W'Vcivi= --.0- .O- _-0.0-_ _O 0 0.0 , 0 Fig. 39-4. Two gases inaboxwitha semipermeable membrane. function ofthetemperature. Butwhoistotelluswhat scale touseforthetempera- ture? Wemayarbitrarily define thescale oftemperature sothatthemean energy islinearly proportional tothetemperature. Thebestwaytodoitwould beto callthemean energy itself “thetemperature.” That would bethesimplest possible function. Unfortunately, thescale oftemperature hasbeen chosen differently, soinstead ofcalling ittemperature directly weuseaconstant conversion factor between theenergy ofamolecule andadegree ofabsolute temperature called a degree Kelvin. Theconstant ofproportionality isk=1.38 X10*“ joule for every degree Kelvin.* SoifTisabsolute temperature, ourdefinition saysthatthe mean molecular kinetic energy is5%kT.(The %isputinasamatter ofconvenience, soastogetridofitsomewhere else.) Wepoint outthatthekinetic energy associated withthecomponent ofmotion inanyparticular direction isonly %kT. Thethree independent directions thatare involved make it%kT. 39-5 Theideal gaslaw Now, ofcourse, wecanputourdefinition oftemperature into Eq.(39.9) andsofindthelawforthepressure ofgases asafunction ofthetemperature: itis thatthepressure times thevolume isequal tothetotal number ofatoms times the universal constant k,times thetemperature: PV=NkT. (39.22) Furthermore, atthesame temperature andpressure andvolume, thenumber of atoms isdetermined; ittooisauniversal constant! Soequal volumes ofdifferent gases, atthesame pressure andtemperature, have thesame number ofmolecules, because ofNewton’s laws. That isanamazing conclusion! Inpractice, when dealing with molecules, because thenumbers aresolarge, thechemists have artificially chosen aspecific number, avery large number, and called itsomething else. They have anumber which they callamole. Amole is merely ahandy number. Why they didnotchoose 1024 objects, soitwould come outeven, isahistorical question. They happened tochoose, fortheconvenient number ofobjects onwhich they standardize, N11=6.02 X1023 objects, and thisiscalled amole ofobjects. Soinstead ofmeasuring thenumber ofmolecules inunits, theymeasure interms ofnumbers ofmoles.’t Interms ofN0wecanwrite thenumber ofmoles, times thenumber ofatoms inamole, times kT,andifwe want to,wecantake thenumber ofatoms inamole times k,which isamole’s worth ofk,andcallitsomething else,andwedo—we callitR.Amole’s worth of kis8.3l7joules: R=Nok =8.3l7j -mole“ '°K_1. Thus wealsofindthegas lawwritten asthenumber ofmoles (also called N)times RT,orthenumber of atoms, times kT: PV=NRT. (39.23) Itisthesame thing, justadifl'erent scale formeasuring numbers. Weuselasa unit, andchemists use6X1023 asaunit! Wenowmake onemore remark about ourgaslaw,andthathastodowiththe lawforobjects other than monatomic molecules. Wehave dealt only with the CMmotion oftheatoms ofamonatomic gas. What happens ifthere areforces present? First, consider thecase thatthepiston isheld byahorizontal spring, andthere areforces onit.Theexchange ofjiggling motion between atoms and piston atanymoment does notdepend onwhere thepiston isatthatmoment, of course. Theequilibrium conditions arethesame. Nomatter where thepiston is, itsspeed ofmotion must besuch thatitpasses energy tothemolecules injust *Thecentigrade scale isjustthisKelvin scale withazerochosen at273.16 °K,so T=273.16 +centigrade temperature. 'lWhat thechemists callmolecular weights arethemasses ingrams ofamole ofa molecule. Themole isdefined sothatthemass ofamole ofcarbon atoms ofisotope 12 (i.e.,having 6protons and6neutrons inthenucleus) isexactly 12grams. 39-10 theright way. Soitmakes nodiflerence about thespring. Thespeed atwhich thepiston hastomove, ontheaverage, isthesame. Soourtheorem, thatthemean value ofthekinetic energy inonedirection is%kT, istruewhether there areforces present ornot. Consider, forexample, adiatomic molecule composed ofatoms m,1andmg. What wehave proved isthatthemotion oftheCMofpartAandthatofpartB aresuch that (%m,1v§,) =(%mBv§;) =%kT. I-low canthisbe,ifthey areheld together? Although they areheld together, when they arespinning andturning inthere, when something hitsthem, exchanging energy with them, theonlything thatcounts ishowfast they aremoving. That alone determines how fastthey exchange energy incollisions. Attheparticular instant, theforce isnotanes- sential point. Therefore thesame principle isright, even when there areforces. Letusprove, finally, thatthegaslawisconsistent alsowith adisregard ofthe internal motion. Wedidnotreally include theinternal motions before; wejust treated amonatomic gas. Butweshall nowshow thatanentire object, considered asasingle body oftotal mass M,hasavelocity oftheCMsuch that sMv?;M =%kT. (39.24) Inother words, wecanconsider either theseparate pieces orthewhole thing! Letusseethereason forthat: Themass ofthediatomic molecule isM=m,1—l— mB,andthevelocity ofthecenter ofmass isequal tovCM =(m,1v_.1 +mBvB)/ M. Now weneed (vim). Ifwesquare vCM, weget 2_mivi +Zmamsva -vs+mivfevoivi — M2 ' Now wemultiply -§Mandtake theaverage, andthusweget LM2__m,1%kT —l—2m,.1mB(v,.1~vB) —l—mB%kT2 UCM — M =%1<T+-————2'""’"‘j,<,"‘ “>- (We have used thefactthat(m,1 +mg)/M =1.)Now what is(VA-vg)? (It hadbetter bezero!) Tofindout,letususeourassumption thattherelative velocity, w=v,1—v1;isnotanymore likely topoint inonedirection than inanother— thatis,thatitsaverage component inanydirection iszero. Thus weassume that (W'vOM> =0- Butwhat isw~vCM? Itis _ _(VA—VB)'(mAvA —l—MBVB)w VCM — M __mAUi -l"(ma —mA)(VA 'VB) —71131132 _ — M Therefore, since (mAv§1) =<l'nBl)%>, thefirstandlastterms cancel outonthe average, andweareleftwith (mg -m,4)(v_.1 'V3> =0. Thus ifm_.1¢mB,wefindthat(v,.1-V13)=0,andtherefore thatthebodily motion oftheentire molecule, regarded asasingle particle ofmass M,hasakinetic energy, ontheaverage, equal to%kT. Incidentally, wehave alsoproved atthesame time thattheaverage kinetic energy oftheinternal motions ofthediatomic molecule, disregarding thebodily motion oftheCM, is%kT! For,thetotal kinetic energy oftheparts ofthemolecule is%m,1v§1 +-§—mBv§,, whose average is%kT+%kT, or3kT. The kinetic energy ofthecenter-of-mass motion is%kT, sotheaverage kinetic energy oftherotational andvibratory motions ofthetwoatoms inside themolecule isthedifference, §kT. 39-11 Thetheorem concerning theaverage energy oftheCMmotion isgeneral: foranyobject considered asawhole, with forces present orno,forevery inde- pendent direction ofmotion thatthere is,theaverage kinetic energy inthatmotion is%kT. These “independent directions ofmotion” aresometimes called the degrees offreedom ofthesystem. Thenumber ofdegrees offreedom ofamolecule composed ofratoms is3r,since each atom needs three coordinates todefine its position. Theentire kinetic energy ofthemolecule canbeexpressed either asthe sum ofthekinetic energies oftheseparate atoms, orasthesum ofthekinetic energy oftheCMmotion plus thekinetic energy oftheinternal motions. The latter cansometimes beexpressed asasum ofrotational kinetic energy ofthe molecule andvibrational energy, butthisisanapproximation. Our theorem, applied tother-atom molecule, saysthatthemolecule willhave, ontheaverage, 3rkT/2 joules ofkinetic energy, ofwhich %kTiskinetic energy ofthecenter-of-mass motion oftheentire molecule, andtherest,%(r—1)kT, isinternal vibrational and rotational kinetic energy. 39-12 40 The Principles ofStatistical Mechanics 40-1 Theexponential atmosphere Wehave discussed ‘some oftheproperties oflarge numbers ofintercolliding atoms. Thesubject iscalled kinetic theory, adescription ofmatter from thepoint ofview ofcollisions between theatoms. Fundamentally, weassert thatthegross properties ofmatter should beexplainable interms ofthemotion ofitsparts. Welimit ourselves forthepresent toconditions ofthermal equilibrium, that is,toasubclass ofallthephenomena ofnature. Thelaws ofmechanics which apply justtothermal equilibrium arecalled statistical mechanics, andinthissection wewant tobecome acquainted with some ofthecentral theorems ofthissubject. Wealready have oneofthetheorems ofstatistical mechanics, namely, the mean value ofthekinetic energy foranymotion attheabsolute temperature T is3}-kTforeachindependent motion, i.e.,foreachdegree offreedom. Thattellsus something about themean square velocities oftheatoms. Ourobjective nowis tolearn more about thepositions oftheatoms, todiscover how many ofthem aregoing tobeindifferent places atthermal equilibrium, andalsotogointoa little more detail onthedistribution ofthevelocities. Although wehave themean square velocity, wedonotknow how toanswer aquestion such ashow many of them aregoing three times faster than therootmean square, orhowmany ofthem aregoing one-quarter oftheroot mean square speed. Orhave they allthesame speed exactly? So,these arethetwoquestions thatweshall trytoanswer: How arethemole- cules distributed inspace when there areforces acting onthem, andhowarethey distributed invelocity? Itturns outthatthetwoquestions arecompletely independent, andthatthe distribution ofvelocities isalways thesame. Wealready received ahintofthelatter factwhen wefound thattheaverage kinetic energy isthesame, —1=kTperdegree of freedom, nomatter what forces areacting onthemolecules. Thedistribution of thevelocities ofthemolecules isindependent oftheforces, because thecollision rates donotdepend upon theforces. Letusbegin withanexample: thedistribution ofthemolecules inanatmosphere likeourown, butwithout thewinds andother kinds ofdisturbance. Suppose that wehave acolumn ofgasextending toagreat height, andatthermal equilibrium- unlike ouratmosphere, which asweknow getscolder aswegoup. Wecould remark thatifthetemperature differed atdiflerent heights, wecould demonstrate lack ofequilibrium byconnecting arodtosome balls atthebottom (Fig. 40-1), where theywould pickup1-kTfrom themolecules there andwould shake, viathe rod, theballs atthetopandthose would shake themolecules atthetop. So, ultimately, ofcourse, thetemperature becomes thesame atallheights inagravita- tional field. Ifthetemperature isthesame atallheights, theproblem istodiscover by what lawtheatmosphere becomes tenuous aswegoup.IfNisthetotal number ofmolecules inavolume Vofgasatpressure P,then weknow PV=NkT, or P=nkT, where n=N/Visthenumber ofmolecules perunitvolume. Inother words, ifweknow thenumber ofmolecules perunitvolume, weknow thepressure, andviceversa: they areproportional toeach other, since thetemperature iscon- stant inthisproblem. Butthepressure isnotconstant, itmust increase asthe altitude isreduced, because ithastohold, sotospeak, theweight ofallthegas above it.That isthecluebywhich wemaydetermine howthepressure changes withheight. Ifwetakeaunitareaatheight h,thenthevertical force from below, 40-140-1 Theexponential atmosphere 40-2 TheBoltzmann law 40-3 Evaporation ofaliquid 40-4 Thedistribution of’molecular speeds 40-5 Thespecific heats ofgases 40-6 Thefailure ofclassical physics 0\§\\\.:~_\¥>'s>:\.\\<\\>\>\°:,0\.._ S’h+dh l. Fig.40-1. The pressure atheight h must exceed that ath—l—dhbythe weight oftheintervening gas.lleclnnill toreqmlizi temperature O.6- H2 0.6- nth) n(Cl 0.4“ Oz O2“ O l I l 20 40 60 80 HEIGHT (Kilometer) Fig.40-2. The normalized density asafunction ofheight intheearth's gravitational field foroxygen and for hydrogen, atconstant temperature.onthisunitarea, isthepressure P.Thevertical force perunitareapushing down ataheight h+dhwould bethesame, intheabsence ofgravity, buthereitisnot, because theforce from below must exceed theforce from above bytheweight of gasinthesection between handh+dh.Now mgistheforce ofgravity oneach molecule, where gistheacceleration duetogravity, andndhisthetotal number of molecules intheunitsection. Sothisgives usthediflerential equation P;,+d;, — P),=dP=—mgn dh.Since P=nkT, andTisconstant, wecaneliminate either Porn,sayP,andget @__%an‘ kr” forthedifferential equation, which tellsushowthedensity goes down aswegoup inenergy. Wethushave anequation fortheparticle density n,which varies with height, butwhich hasaderivative which isproportional toitself. Now afunction which hasaderivative proportional toitself isanexponential, andthesolution ofthis diflerential equation is n=n.,@"'"""”“”. (40.1) Here theconstant ofintegration, no,isobviously thedensity ath=O(which can bechosen anywhere), andthedensity goes down exponentially with height. Note thatifwehave different kinds ofmolecules with different masses, they godown with different exponentials. Theones which were heavier would decrease with altitude faster than thelight ones. Therefore wewould expect thatbecause oxygen isheavier than nitrogen, aswegohigher andhigher inanatmosphere with nitrogen andoxygen theproportion ofnitrogen would increase. This does not really happen inourownatmosphere, atleast atreasonable heights, because there issomuch agitation which mixes thegases back together again. Itisnotan isothermal atmosphere. Nevertheless, there isatendency forlighter materials, likehydrogen, todominate atvery great heights intheatmosphere, because the lowest masses continue toexist, while theother exponentials have alldied out (Fig. 40-2). 40-2 TheBoltzmann law Here wenote theinteresting factthatthenumerator intheexponent ofEq. (40.1) isthepotential energy ofanatom. Therefore wecanalsostate thisparticular lawas:thedensity atanypoint isproportional to e—(the potential energy ofeach atom/kT) u That may beanaccident, i.e.,may betrueonly forthisparticular case ofa uniform gravitational field. However, wecanshow thatitisamore general prop- osition. Suppose thatthere were some kind offorce other than gravity acting onthemolecules inagas. Forexample, themolecules maybecharged electrically, andmaybeacted onbyanelectric field oranother charge thatattracts them. Or, because ofthemutual attractions oftheatoms foreach other, orforthewall, or forasolid, orsomething, there issome force ofattraction which varies with position andwhich actsonallthemolecules. Now suppose, forsimplicity, that themolecules areallthesame, andthattheforce actsoneach individual one, so thatthetotal force onapiece ofgaswould besimply thenumber ofmolecules times theforce oneach one. Toavoid unnecessary complication, letuschoose a coordinate system with thex-axis inthedirection oftheforce, F. Inthesame manner asbefore, ifwetake twoparallel planes inthegas,sepa- rated byadistance dx,then theforce oneach atom, times thenatoms percm3 (thegeneralization oftheprevious nmg), times dx,must bebalanced bythepressure change: Fndx=dP=kTdn.Or,toputthislawinaform which willbeuseful touslater, F=kT5'2(lnn). (40.2) 40-2 Forthepresent, observe that-Fdxisthework wewould dointaking amolecule from xtox+dx,andifFcomes from apotential, i.e.,ifthework done canbe represented byapotential energy atall,then thiswould alsobethedifference in thepotential energy (P.E.). Thenegative differential ofpotential energy isthe work done, Fdx,andwefindthatd(lnn)=—d(P.E.)/kT, or,after integrating, n=(constant)e“P'E‘/kT. (40.3) Therefore what wenoticed inaspecial caseturns outtobetrueingeneral. (What ifFdoes notcome from apotential? Then (40.2) hasnosolution atall.Energy canbegenerated, orlostbytheatoms running around incyclic paths forwhich thework done isnotzero, andnoequilbrium canbemaintained atall.Thermal equilibrium cannot exist iftheexternal forces ontheatoms arenotconservative.) Equation (40.3), known asBolt2mann’s law,isanother oftheprinciples ofstatistical mechanics: thattheprobability offinding molecules inagiven spatial arrangement varies exponentially with thenegative ofthepotential energy ofthatarrangement, divided bykT. This, then, could tellusthedistribution ofmolecules: Suppose thatwehada positive ioninaliquid, attracting negative ionsaround it,howmany ofthem would beatdifferent distances? Ifthepotential energy isknown asafunction ofdistance, then theproportion ofthem atdifferent distances isgiven bythislaw,andsoon, through many applications. 40-3 Evaporation ofaliquid Inmore advanced statistical mechanics onetries tosolve thefollowing im- portant problem. Consider anassembly ofmolecules which attract each other, andsuppose thattheforce between anytwo, sayiandj,depends only ontheir separation r,-,-,andcanberepresented asthederivative ofapotential function V(r,~,-). Figure 40-3 shows aform such afunction might have. Forr>r0,the energy decreases asthemolecules come together, because they attract, andthen theenergy increases very sharply asthey come stillcloser together, because they repel strongly, which ischaracteristic oftheway molecules behave, roughly speaking. Now suppose wehave awhole boxfullofsuch molecules, andwewould like toknow how they arrange themselves ontheaverage. The answer ise“P'E‘”‘T. Thetotal potential energy inthiscasewould bethesumoverallthepairs, supposing thattheforces areallinpairs (there maybethree-body forces inmore complicated things, butinelectricity, forexample, thepotential energy isallinpairs). Then the probability forfinding molecules inanyparticular combination ofr,-y’swillbe proportional to eXp[- v(t,-,-)/la].7-1] Now, ifthetemperature isvery high, sothatkT>>|V(r0)|, theexponent is relatively small almost everywhere, andtheprobability offinding amolecule is almost independent ofposition. Letustake thecase ofjust twomolecules: thee"P'E'/H would betheprobability offinding them atvarious mutual distances r.Clearly, where thepotential goes most negative, theprobability islargest, and where thepotential goes toward infinity, theprobability isalmost zero, which occurs forvery small distances. That means thatforsuch atoms inagas,there is nochance thattheyareontopofeach other, since theyrepel sostrongly. Butthere isagreater chance offinding them perunitvolume atthepoint rothan atanyother point. How much greater, depends onthetemperature. Ifthetemperature is very large compared with thedifference inenergy between r=roandr=oo, theexponential isalways nearly unity. Inthiscase, where themean kinetic energy (about kT)greatly exceeds thepotential energy, theforces donotmake much diflerence. Butasthetemperature falls, theprobability offinding themolecules atthepreferred distance r0gradually increases relative totheprobability offinding them elsewhere and, infact, ifkTismuch lessthan lV(r0)|, wehave arelatively 40-3P.E. Vlfl Ff r Fig.40-3. Apotential-energy func- tion fortwo molecules, which depends only ontheir separation. h=h h=O Fig.40-4. Only those molecules moving upath=0with sufficient velocity canarrive atheight h.large positive exponent inthatneighborhood. Inother words, inagiven volume they aremuch more likely tobeatthedistance ofminimum energy than farapart. As thetemperature falls, theatoms falltogether, clump inlumps, andreduce to liquids, andsolids, andmolecules, andasyouheat them uptheyevaporate. The requirements forthedetermination ofexactly how things evaporate, exactly how things should happen inagiven circumstance, involve thefollowing. First, todiscover thecorrect molecular-force lawV(r), which must come from something else, quantum mechanics, say,orexperiment. But, given thelawof force between themolecules, todiscover what abillion molecules aregoing todo merely consists ofstudying thefunction e—2Vi'i/'°T. Surprisingly enough, since itis such asimple function andsuch aneasy idea, given thepotential, thelabor is enormously complicated; thedifficulty isthetremendous number ofvariables. Inspite ofsuch difficulties, thesubject isquite exciting andinteresting. Itis often called anexample ofa“many-body problem,” anditreally hasbeen avery interesting thing. Inthatsingle formula must becontained allthedetails, for example, about thesolidification ofgas,ortheforms ofthecrystals thatthesolid cantake, andpeople have been trying tosqueeze itout,butthemathematical difliculties arevery great, notinwriting thelaw,butindealing with soenormous anumber ofvariables. That then, isthedistribution ofparticles inspace. That istheendofclassical statistical mechanics, practically speaking, because ifweknow theforces, wecan, inprinciple, findthedistribution inspace, andthedistribution ofvelocities is something thatwecanwork outonce andforall,andisnotsomething thatis diflerent forthedifferent cases. The great problems areingetting particular information outofourformal solution, andthatisthemain subject ofclassical statistical mechanics. 40-4 Thedistribution ofmolecular speeds Now wegoontodiscuss thedistribution ofvelocities, because sometimes itis interesting oruseful toknow how many ofthem aremoving atdiflerent speeds. Inorder todothat, wemaymake useofthefacts which wediscovered with regard tothegasintheatmosphere. Wetake ittobeaperfect gas,aswehave already assumed inwriting thepotential energy, disregarding theenergy ofmutual attrac- tionoftheatoms. Theonly potential energy thatweincluded inourfirstexample wasgravity. Wewould, ofcourse, have something more complicated ifthere were forces between theatoms. Thus weassume thatthere arenoforces between theatoms and, foramoment, disregard collisions also, returning later tothe justification ofthis. Now wesawthatthere arefewer molecules attheheight h than there areattheheight 0;according toformula (40.1), they decrease expo- nentially with height. How canthere befewer atgreater heights? After all,do notallthemolecules which aremoving upatheight 0arrive ath?Nol, because some ofthose which aremoving upat0aregoing tooslowly, andcannot climb thepotential hilltoh.With thatclue, wecancalculate howmany must bemoving atvarious speeds, because from (40.1) weknow howmany aremoving with less than enough speed toclimb agiven distance h.Those arejusttheones thataccount forthefactthatthedensity athislower than at0. Now letusputthatideaalittle more precisely: letuscount howmany mole- cules arepassing from below toabove theplane h=0(bycalling itheight =0, wedonotmean thatthere isafloor there; itisjustaconvenient label, andthere is gasatnegative h).These gasmolecules aremoving around inevery direction, but some ofthem aremoving through theplane, andatanymoment acertain number persecond ofthem arepassing through theplane from below toabove withdifferent velocities. Now wenote thefollowing: ifwecalluthevelocity which isjustneeded togetuptotheheight h(kinetic energy mu2/2 =mgh), then thenumber of molecules persecond which arepassing upward through thelower plane ina vertical direction with velocity component greater than uisexactly thesame as thenumber which pass through theupper plane with anyupward velocity. Those molecules whose vertical velocity does notexceed ucannot getthrough theupper 40-4 plane. Sotherefore weseethat Number passing h=0withii,>u=number passing h=hwith11,>0. Butthenumber which passthrough hwith anyvelocity greater than 0islessthan thenumber which passthrough thelower height with anyvelocity greater than O, because thenumber ofatoms isgreater; thatisallweneed. Weknow already that thedistribution ofvelocities isthesame, after theargument wemade earlier about thetemperature being constant allthewaythrough theatmosphere. So,since the velocity distributions arethesame, anditisjustthatthere aremore atoms lower down, clearly thenumber n>0(h), passing with positive velocity atheight h, andthenumber n>0(0), passing with positive velocity atheight 0,areinthesame ratio asthedensities atthetwoheights, which ise""""/'°T. Butn>o(h) =n>,,(0), andtherefore wefindthat ">u(0) =e—mqh/kT =e—'m'u2/2l€T9 ">0(0) since %mu2 =mgh. Thus, inwords, thenumber ofmolecules perunitarea per second passing theheight 0with az-component ofvelocity greater than uis e—’"“2/ “Ttimes thetotal number thatarepassing through theplane with velocity greater than zero. Now thisisnotonly trueatthearbitrarily chosen height 0,butofcourse itis trueatanyother height, andthus thedistributions ofvelocities areallthesame! (The final statement does notinvolve theheight h,which appeared only inthe intermediate argument.) Theresult isageneral proposition thatgives usthedistri- bution ofvelocities. Ittellsusthatifwedrillalittle holeinthesideofagaspipe, avery tinyhole, sothatthecollisions arefewandfarbetween, i.e.,arefarther apart than thediameter ofthehole, then theparticles which arecoming outwill have different velocities, butthefraction ofparticles which come outatavelocity greater than uise‘””‘2/ 2”. Now wereturn tothequestion about theneglect ofcollisions: Why does it notmake anydifference? Wecould have pursued thesame argument, notwith a finite height h,butwithaninfinitesimal height h,which issosmall thatthere would benoroom forcollisions between 0andh.Butthatwasnotnecessary: theargu- ment isevidently based onananalysis" oftheenergies involved, theconservation ofenergy, andinthecollisions thatoccur there isanexchange ofenergies among themolecules. However, wedonotreally carewhether wefollow thesame mole- culeifenergy ismerely exchanged with another molecule. Soitturns outthateven iftheproblem isanalyzed more carefully (and itismore difficult, naturally, todoa rigorous job), itstillmakes nodifference intheresult. Itisinteresting thatthevelocity distribution wehave found isjust n>u (Xe—kirietic energy/kT. This wayofdescribing thedistribution ofvelocities, bygiving thenumber of molecules thatpass agiven area with acertain minimum z-component, isnotthe most convenient wayofgiving thevelocity distribution. Forinstance, inside the gas, onemore often wants toknow how many molecules aremoving with a z-component ofvelocity between twogiven values, andthat, ofcourse, isnotdi- rectly given byEq.(40.4). Wewould liketostate ourresult inthemore con- ventional form, even though what wealready have written isquite general. Note thatitisnotpossible tosaythatanymolecule hasexactly some stated velocity; none ofthem hasavelocity exactly equal tol.7962899173 meters persecond. Soinorder tomake ameaningful statement, wehave toaskhow many aretobe found insome range ofvelocities. Wehave tosayhowmany have velocities between 1.796 and1.797, andsoon.Onmathematical terms, letf(u)dubethefraction of allthemolecules which have velocities between uandu+duor,what isthesame thing (ifduisinfinitesimal), allthathave avelocity uwith arange du.Figure 40-5 shows apossible form forthefunction f(u), andtheshaded part, ofwidth duand mean height f(u), represents thisfraction f(u)du.That is,theratio oftheshaded 40-5ll") 1|-‘I Fig. 40-5. Avelocity distribution function. Theshaded area isffuldu, the fraction ofparticles having velocities within arange duabout u. area tothetotal area ofthecurve istherelative proportion ofmolecules with velocity uwithin du.Ifwedefine f(u)sothatthefraction having avelocity inthis range isgiven directly bytheshaded area, then thetotal area must be100percent ofthem, thatis, /°°f(u)du =1. (40.5) Now wehave only togetthisdistribution bycomparing itwith thetheorem wederived before. First weask,what isthenumber ofmolecules passing through anarea persecond with avelocity greater than u,expressed interms off(u)? Atfirstwemight think itismerely theintegral of f(u)du,butitisnot,because wewant thenumber thatarepassing theareapersecond. Thefaster ones pass more often, sotospeak, than theslower ones, andinorder toexpress how many pass, youhave tomultiply bythevelocity. (We discussed thatintheprevious chapter when wetalked about thenumber ofcollisions.) Inagiven time tthe total number which pass through thesurface isallofthose which have been able toarrive atthesurface, andthenumber which arrive come from adistance ut. Sothenumber ofmolecules which arrive isnotsimply thenumber which arethere, butthenumber thatarethere perunit volume, multiplied bythedistance that they sweep through inracing fortheareathrough which they aresupposed togo, andthatdistance isproportional tou.Thus weneed theintegral ofulimesf(u) du, aninfinite integral with alower limit u,andthismust bethesame aswefound before, namely e_"‘“2/2”, with aproportionality constant which wewillgetlater: I:uf(u) du=const -e_'"“2/2”. (40.6) Now ifwedifierentiate theintegral with respect tou,wegetthething thatis inside theintegral, i.e.,theintegrand (with aminus sign, since uisthelower limit), andifwedifferentiate theother side, wegetutimes thesame exponential (and some constants). Theu’scancel andwefind f(u)du =Ce_'"“2/2” du. (40.7) Weretain theduonboth sides asareminder thatitisadistribution, andittells what theproportion isforvelocity between uandu+du. Theconstant Cmust besodetermined thattheintegral isunity, according to Eq.(40.5). Now wecanprove* that £:°e“"2 dx=\/Tr. Using thisfact, itiseasytofindthatC=\/m/21rkT. Since velocity andmomentum areproportional, wemaysaythatthedistribu- tion ofmomenta isalso proportional toe“K"E‘/'°T perunit momentum range. Itturns outthatthistheorem istrueinrelativity too,ifitisinterms ofmomentum, while ifitisinvelocity itisnot,soitisbesttolearn itinmomentum instead ofin velocity: f(p)dP=Ce'K‘E'”°T dz» (40-8) Sowefindthat theprobabilities ofdifferent conditions ofenergy, kinetic and potential, areboth given bye“°"°'gY/"T, averyeasything toremember andarather beautiful proposition. *Togetthevalue oftheintegral, let I=Ila-12 dx. Then no m 0°“O 12=f_me"2dx- f_m8-Way =f_wj_w e—<="+~’>dy dx, which isadouble integral over thewhole xy-plane. Butthiscanalsobewritten inpolar coordinates as Go 0° I2=IOe*'2-21rrdr =1rf0 e-‘dt =1r. 40-6 Sofarwehave, ofcourse, only thedistribution ofthevelocities “vertically.” Wemight want toask,what istheprobability thatamolecule ismoving inanother direction? Ofcourse these distributions areconnected, andonecanobtain the complete distribution from theonewehave, because thecomplete distribution depends only onthesquare ofthemagnitude ofthevelocity, notupon thez-com- ponent. Itmust besomething thatisindependent ofdirection, andthere isonly onefunction involved, theprobability ofdifferent magnitudes. Wehave the distribution ofthez-component, andtherefore wecangetthedistribution ofthe other components from it.Theresult isthattheprobability isstillproportional to e"'K"E'/"T, butnow thekinetic energy involves three parts, mvf/2, mv;/2, and mvf/2, summed intheexponent. Orwecanwrite itasaproduct: f(v,,, 121,,22,)du,,du,,du, 2 2 2 I e—mUz/2kT _e—mvu/2kT _e-—m1)z/ZICT dvz dz)” dvz. You canseethatthisformula must beright because, first, itisafunction only of v2,asrequired, andsecond, theprobabilities ofvarious values ofv,obtained by integrating over all1),,and1),,isjust(40.7). Butthisonefunction (40.9) cando both those things! 40-5 Thespecific heats ofgases Now weshall look atsome ways totest thetheory, and toseehow successful istheclassical theory ofgases. Wesawearlier thatifUistheinternal energy ofNmolecules, then PV=NkT =(7—l)Uholds, sometimes, for some gases, maybe. Ifitisamonatomic gas,weknow thisisalso equal to% ofthekinetic energy ofthecenter-of-mass motion oftheatoms. Ifitisamonatomic gas,then thekinetic energy isequal totheinternal energy, andtherefore V— l=§.Butsuppose itis,say,amore complicated molecule, thatcanspin and vibrate, andletussuppose (itturns outtobetrueaccording toclassical mechanics) thattheenergies oftheinternal motions arealsoproportional tokT. Then ata given temperature, inaddition tokinetic energy kT,ithasinternal vibrational or rotational energy. Sothetotal Uincludes notjusttheinternal kinetic energy, but also therotational energy, andwegetadifferent value ofV.Technically, the bestwaytomeasure Visbymeasuring thespecific heat, which isthechange in energy with temperature. Wewillreturn tothatapproach later. Forourpresent purposes, wemay suppose Visfound experimentally from thePV" curve for adiabatic compression. Letusmake acalculation of‘Yforsome cases. First, foramonatomic gas Uisthetotal energy, thesame asthekinetic energy, andweknow already that Vshould be%.Foradiatomic gas,wemaytake, asanexample, oxygen, hydrogen iodide, hydrogen, etc., andsuppose that thediatomic gascanberepresented as twoatoms heldtogether bysome kind offorce liketheoneofFig.40—3. Wemay alsosuppose, anditturns outtobequite true, thatatthetemperatures thatare ofinterest forthediatomic gas,thepairs ofatoms tend strongly tobeseparated byr0,thedistance ofpotential minimum. Ifthiswere nottrue, iftheprobability were notstrongly varying enough tomake thegreat majority sitnear thebottom, wewould have toremember thatoxygen gasisamixture ofO2andsingle oxygen atoms inanontrivial ratio. Weknow thatthere are,infact,veryfewsingle oxygen atoms, which means thatthepotential energy minimum isvery much greater in magnitude than kT,aswehave seen. Since they areclustered strongly around r0,theonly partofthecurve thatisneeded isthepartnear theminimum, which may beapproximated byaparabola. Aparabolic potential implies aharmonic oscillator, andinfact, toanexcellent approximation, theoxygen molecule canbe represented astwoatoms connected byaspring. Now what isthetotal energy ofthismolecule attemperature T?Weknow thatforeach ofthetwoatoms, each ofthekinetic energies should be%kT, sothe kinetic energy ofboth ofthem is%kT+%kT. Wecanalsoputthisinadifi'erent way: thesameg plusgcanalsobelooked ataskinetic energy ofthecenter ofmass 40-7 Table 40-1 Values ofthespecific heatratio, 'Y, forvarious gases Gas T(°C) ‘Y He —180 1.660 Kr 19 1.68 Ar 15 1.668 H2 100 1.404 Oz 100 1.399 HI 100 1.40 BI2 300 1.32 I2 185 1.30 NH3 15 1.310 C2H6 15 1.22 Y |.eX _°_H: —-x--0; K|. - _ |.2es———————— —— |.2 'l.s..a......'....a*wTEMPERATURE (‘Cl Fig. 40-6. Experimental values of 7asafunction oftemperature forhydro- gen and oxygen. Classical theory predicts ‘Y=1.286, independent of temperature.(%),kinetic energy ofrotation (§),andkinetic energy ofvibration (§). Weknow thatthekinetic energy ofvibration is%,since there isjustonedimension involved andeach degree offreedom has%kT. Regarding therotation, itcanturn about either oftwoaxes, sothere aretwoindependent motions. Weassume thatthe atoms aresome kind ofpoints, andcannot spin about thelinejoining them; thisissomething tobear inmind, because ifwegetadisagreement, maybe thatis where thetrouble is.Butwehave onemore thing, which isthepotential energy of vibration; howmuch isthat? Inaharmonic oscillator theaverage kinetic energy andaverage potential energy areequal, andtherefore thepotential energy of vibration is%kT, also. Thegrand total ofenergy isU=%kT, orkTis%Uper atom. That means, then, that‘Yis%instead of§,i.e.,'Y=1.286. Wemaycompare these numbers with therelevant measured values shown in Table 40-1. Looking firstathelium, which isamonatomic gas,wefindvery nearly 1§-,andtheerror isprobably experimental, although atsuchalowtemperature there maybesome forces between theatoms. Krypton andargon, both monatomic, agree alsowithin theaccuracy oftheexperiment. Weturn tothediatomic gases andfindhydrogen with 1.404, which does not agree with thetheory, 1.286. Oxygen, 1.399, isvery similar, butagain notin agreement. Hydrogen iodide again issimilar at1.40. Itbegins tolook asthough theright answer is1.40, butitisnot,because ifwelook further atbromine wesee 1.32, andatiodine wesee1.30. Since 1.30isreasonably close to1.286, iodine maybesaidtoagree rather well, butoxygen isfaroff.Soherewehave adilemma. Wehave itright foronemolecule, wedonothave itright foranother molecule, andwemay need tobepretty ingenious inorder toexplain both. Letuslook further atastillmore complicated molecule with large numbers ofparts, forexample, C2H6, which isethane. Ithaseight different atoms, andthey areallvibrating androtating invarious combinations, sothetotal amount of internal energy must beanenormous number ofkT’s, atleast l2kT forkinetic energy alone, andW’—1must bevery close tozero, orValmost exactly 1.In fact,itislower, but1.22isnotsomuch lower, andishigher than the1%;calculated from thekinetic energy alone, anditisjustnotunderstandable! Furthermore, thewhole mystery isdeep, because thediatomic molecule cannot bemade rigid byalimit. Even ifwemade thecouplings stiffer indefinitely, although itmight notvibrate much, itwould nevertheless keep vibrating. Thevibrational energy inside isstillkT,since itdoes notdepend onthestrength ofthecoupling. Butifwecould imagine absolute rigidity, stopping allvibration toeliminate a variable, then wewould getU=%kTand7=1.40forthediatomic case. This looks good forH2orO2. Ontheother hand, wewould stillhave problems, because ‘Yforeither hydrogen oroxygen varies with temperature! From the measured values shown inFig.40—6, weseethatforH2,'Yvaries from about 1.6 at-—l85°C to1.3at2000°C. Thevariation ismore substantial inthecase of hydrogen than foroxygen, butnevertheless, even inoxygen, 7tends definitely to goupaswegodown intemperature. 40-6 Thefailure ofclassical physics So,allinall,wemight saythatwehave some difliculty. Wemight trysome force lawother than aspring, butitturns outthatanything elsewillonly make 1/higher. Ifweinclude more forms ofenergy, Vapproaches unity more closely, contradicting thefacts. Alltheclassical theoretical things thatonecanthink of willonly make itworse. Thefactisthatthere areelectrons ineach atom, andwe know from their spectra thatthere areinternal motions; each oftheelectrons should have atleast %kTofkinetic energy, andsomething forthepotential energy, so when these areadded in,Vgetsstillsmaller. Itisridiculous. Itiswrong. Thefirstgreat paper onthedynamical theory ofgases wasbyMaxwell in 1859. Onthebasis ofideas wehave been discussing, hewasable accurately to explain agreat many known relations, such asBoyle’s law, thediflusion theory, theviscosity ofgases, andthings weshall talkabout inthenextchapter. Helisted allthese great successes inafinal summary, andattheendhesaid, “Finally, by 40-8 establishing anecessary relation between themotions oftranslation androtation (heistalking about the%kTtheorem) ofallparticles notspherical, weproved that asystem ofsuch particles could notpossibly satisfy theknown relation between thetwospecific heats.” Heisreferring to‘Y(which weshall seelater isrelated to twoways ofmeasuring specific heat), andhesaysweknow wecannot gettheright answer. Tenyears later, inalecture, hesaid, “Ihave now putbefore youwhat I consider tobethegreatest difficulty yetencountered bythemolecular theory.” These words represent thefirstdiscovery thatthelaws ofclassical physics were wrong. This wasthefirst indication that there wassomething fundamentally impossible, because arigorously proved theorem didnotagree with experiment. About 1890, Jeans wastotalkabout thispuzzle again. Oneoften hears itsaidthat physicists atthelatter part ofthenineteenth century thought they knew allthe significant physical laws andthatalltheyhadtodowastocalculate more decimal places. Someone mayhave saidthatonce, andothers copied it.Butathorough reading oftheliterature ofthetime shows theywere allworrying about something. Jeans saidabout thispuzzle thatitisaverymysterious phenomenon, anditseems asthough asthetemperature falls, certain kinds ofmotions “freeze out.” Ifwecould assume thatthevibrational motion, say,didnotexist atlowtem- perature anddidexist athightemperature, then wecould imagine thatagasmight exist atatemperature sufficiently lowthat vibrational motion does notoccur, so'Y=1.40, orahigher temperature atwhich itbegins tocome in,so‘Yfalls. Thesame might beargued fortherotation. Ifwecaneliminate therotation, say it“freezes out” atsufficiently lowtemperature, then wecanunderstand thefact thattheVofhydrogen approaches 1.66aswegodown intemperature. How can weunderstand such aphenomenon? Ofcourse thatthese motions “freeze out” cannot beunderstood byclassical mechanics. Itwasonly understood when quan- tummechanics wasdiscovered. Without proof, wemay state theresults forstatistical mechanics ofthe quantum-mechanical theory. Werecall that according toquantum mechanics, a system which isbound byapotential, forthevibrations, forexample, willhave a discrete setofenergy levels, i.e.,states ofdiflerent energy. Now thequestion is: how isstatistical mechanics tobemodified according toquantum-mechanical theory? Itturns out,interestingly enough, thatalthough most problems aremore difficult inquantum mechanics than inclassical mechanics, problems instatistical mechanics aremuch easier inquantum theory! Thesimple result wehave inclassi- calmechanics, that n=n0e"e“‘”gY/"T, becomes thefollowing very important theorem: lftheenergies ofthesetofmolecular states arecalled, say,E0,E1,E2, ...,E,-,...,then inthermal equilibrium theprobability offinding amolecule intheparticular state ofhaving energy E,isproportional toe*E1'/"T. That gives theprobability ofbeing invarious states. Inother words, therelative chance, the probability, ofbeing instate E1relative tothechance ofbeing instate E0,is P1 e——E;llcT 8 which, ofcourse, isthesame as H1 =l’l()€_(E1_E0)/kT, since P1=n1/N andP0=n0/N. Soitislesslikely tobeinahigher energy state than inalower one. Theratio ofthenumber ofatoms intheupper state tothe number inthelower state iseraised tothepower (minus theenergy difference, over kT)—a very simple proposition. Now itturns outthatforaharmonic oscillator theenergy levels areevenly spaced. Calling thelowest energy E0=0(itactually isnotzero, itisalittle ditTer- ent,butitdoes notmatter ifweshift allenergies byaconstant), thefirstoneis then E1=hw,andthesecond oneislhw, andthethird oneis3fi<.o, andsoon. Now letusseewhat happens. Wesuppose wearestudying thevibrations ofa diatomic molecule, which weapproximate asaharmonic oscillator. Letusask 40-9 what istherelative chance offinding amolecule instate E1instead ofinstate E0. Theanswer isthatthechance offinding itinstate E1,relative tothatoffinding itin state E0,goes down ase_”°’/'°T. Now suppose thatkTismuch lessthan hw,and wehave alow-temperature circumstance. Then theprobability ofitsbeing instate E1isextremely small. Practically alltheatoms areinstate E0.Ifwechange the temperature butstillkeep itverysmall, then thechance ofitsbeing instate E1= ho.)remains infinitesimal——the energy oftheoscillator remains nearly zero; itdoes notchange with temperature solong asthetemperature ismuch lessthan hw. Alloscillators areinthebottom state, andtheir motion iseffectively “frozen”; there isnocontribution ofittothespecific heat. Wecanjudge, then, from Table 40-1, thatat100°C, which is373degrees absolute, kTismuch lessthan thevibra- tional energy intheoxygen orhydrogen molecules, butnotsointheiodine mole- cule. Thereason forthedifference isthataniodine atom isveryheavy, compared with hydrogen, andalthough theforces may becomparable iniodine andhydro- gen, theiodine molecule issoheavy that thenatural frequency ofvibration is verylowcompared with thenatural frequency ofhydrogen. With hmhigher than kTatroom temperature forhydrogen, butlower foriodine, only thelatter, iodine, exhibits theclassical vibrational energy. Asweincrease thetemperature ofagas, starting from avery lowvalue ofT,with themolecules almost allintheir lowest state, they gradually begin tohave anappreciable probability tobeinthesecond state, andthen inthenext state, andsoon.When theprobability isappreciable formany states, thebehavior ofthegasapproaches thatgiven byclassical physics, because thequantized states become nearly indistinguishable from acontinuum ofenergies, andthesystem canhave almost anyenergy. Thus, asthetemperature rises, weshould again gettheresults ofclassical physics, asindeed seems tobethe caseinFig.40-6. Itispossible toshow inthesame waythattherotational states ofatoms arealsoquantized, butthestates aresomuch closer together thatin ordinary circumstances kTis bigger than thespacing. Then many levels areexcited, andtherotational kinetic energy inthesystem participates intheclassical way. Theoneexample where thisisnotquite trueatroom temperature isforhydrogen. This isthefirsttime thatwehave really deduced, bycomparison with experi- ment, thatthere wassomething wrong with classical physics, andwehave looked foraresolution ofthedifficulty inquantum mechanics inmuch thesame wayas itwasdone originally. Ittook 30or40years before thenext difliculty wasdis- covered, andthat hadtodoagain with statistical mechanics, butthistime the mechanics ofaphoton gas. That problem wassolved byPlanck, intheearly years ofthiscentury. 40-10 41 The Brownian Movement 41-1 Equipartition ofenergy TheBrownian movement wasdiscovered in1827byRobert Brown, abotanist. While hewasstudying microscopic life,henoticed little particles ofplant pollens jiggling around intheliquid hewaslooking atinthemicroscope, andhewaswise enough torealize thatthese were notliving, butwerejustlittle pieces ofdirtmoving around inthewater. Infacthehelped todemonstrate thatthishadnothing todo with lifebygetting from theground anoldpiece ofquartz inwhich there was some water trapped. Itmust have been trapped formillions andmillions ofyears, butinside hecould seethesame motion. What oneseesisthatverytinyparticles arejiggling allthetime. This waslater proved tobeoneoftheeffects ofmolecular motion, andwecan understand itqualitatively bythinking ofagreat push ballonaplaying field, seen from agreat distance, with alotofpeople underneath, allpushing theball invarious directions. Wecannot seethepeople because weimagine thatweare toofaraway, butwecanseetheball, andwenotice thatitmoves around rather irregularly. Wealsoknow, from thetheorems thatwehave discussed inprevious chapters, thatthemean kinetic energy ofasmall particle suspended inaliquid oragaswillbe%kTeven though itisveryheavy compared with amolecule. Ifit isvery heavy, that means thatthespeeds arerelatively slow, butitturns out, actually, thatthespeed isnotreally soslow. Infact, wecannot seethespeed of such aparticle veryeasily because although themean kinetic energy is%kT, which represents aspeed ofamillimeter orsopersecond foranobject amicron ortwo indiameter, thisisvery hard toseeeven inamicroscope, because theparticle continuously reverses itsdirection anddoes notgetanywhere. How faritdoes get wewilldiscuss attheendofthepresent chapter. This problem wasfirstsolved by Einstein atthebeginning ofthepresent century. Incidentally, when wesaythat themean kinetic energy ofthisparticle is %kT, weclaim tohave derived thisresult from thekinetic theory, that is,from Newton’s laws. Weshall findthatwecanderive allkinds ofthings—marvelous things-—from thekinetic theory, anditismost interesting thatwecanapparently getsomuch from solittle. Ofcourse wedonotmean thatNewton’s lawsare“little” —they areenough todoit,really--what wemean isthatwedidnotdovery much. How dowegetsomuch out? Theanswer isthatwehave been perpetually making acertain important assumption, which isthat ifagiven system isinthermal equilibrium atsome temperature, itwill also beinthermal equilibrium with anything elseatthesame temperature. Forinstance, ifwewanted toseehow a particle would move ifitwasreally colliding with water, wecould imagine that there wasagaspresent, composed ofanother kind ofparticle, little finepellets that(wesuppose) donotinteract with water, butonly hittheparticle with “hard” collisions. Suppose theparticle hasaprong sticking outofit;allourpellets have todoishittheprong. Weknow allabout thisimaginary gasofpellets attempera- tureT-—it isanideal gas. Water iscomplicated, butanideal gasissimple. Now, ourparticle hastobeinequilibrium with thegasofpellets. Therefore, themean motion oftheparticle must bewhat wegetforgaseous collisions, because ifit were notmoving attheright speed relative tothewater but, say,wasmoving faster, that would mean that thepellets would pick upenergy from itandget hotter than thewater. Butwehadstarted them atthesame temperature, andwe assume thatifathing isonce inequilibrium, itstays inequilibrium—-parts ofit donotgethotter andother parts colder, spontaneously. 41-l41-1 Equipartition ofenergy 41-2 Thermal equilibrium of radiation 41-3 Equipartition andthequantum oscillator 41-4 Therandom walk S 8 0 I a L t (0) (bl Fig.41-1. (a)Asensitive light-beam galvanometer. Light from asource Lis reflected from asmall mirror onto a scale. (b)Aschematic record ofthe reading ofthescale asafunction of thetime. C L C L R R M lb) Fig.41-2. Ahigh-Q resonant circuit. (a)Actual circuit, attemperature T. (b)Artificial circuit, with anideal (noise- less) resistance and a“noise generator" G.This proposition istrueandcanbeproved from thelaws ofmechanics, but theproof isverycomplicated andcanbeestablished onlybyusing advanced mechanics. Itismuch easier toprove inquantum mechanics than itisinclassical mechanics. Itwasproved firstbyBoltzmann, butfornowwesimply take ittobe true, andthen wecanargue thatourparticle hastohave %kTofenergy ifitishit with artificial pellets, soitalsomust have §kTwhen itisbeing hitwith water at thesame temperature andwetake away thepellets; soitis%kT. Itisastrange lineofargument, butperfectly valid. Inaddition tothemotion ofcolloidal particles forwhich theBrownian move- ment wasfirstdiscovered, there areanumber ofother phenomena, both inthe laboratory andinother situations, where onecanseeBrownian movement. Ifwe aretrying tobuild themost delicate possible equipment, sayavery small mirror onathinquartz fiber foravery sensitive ballistic galvanometer (Fig. 41-1), the mirror does notstayput,butjiggles allthetime-—all thetime—so thatwhen we shine alight onitandlook attheposition ofthespot, wedonothave aperfect instrument because themirror isalways jiggling. Why? Because theaverage kinetic energy ofrotation ofthismirror hastobe,ontheaverage, %kT. What isthemean-square angle over which themirror willwobble? Suppose wefindthenatural vibration period ofthemirror bytapping ononesideand seeing howlong ittakes tooscillate back andforth, andwealsoknow themoment ofinertia, I.Weknow theformula forthekinetic energy ofrotation~it isgiven byEq.(19.8): T=%Iw2. That isthekinetic energy, andthepotential energy that goes with itwillbeproportional tothesquare oftheangle—it isV=@102. But,ifweknow theperiod t0andcalculate from thatthenatural frequency w0= 21r/t0, then thepotential energy isV=%Io.>§t92. Now weknow thattheaverage kinetic energy is%kT, butsince itisaharmonic oscillator theaverage potential energy isalso%kT. Thus %Iw%<62> =%kT.()r (e2)=kr/101%,. (41.1) Inthiswaywecancalculate theoscillations ofagalvanometer mirror, andthereby findwhat thelimitations ofourinstrument willbe.Ifwewant tohave smaller oscillations, wehave tocool themirror. Aninteresting question is,where to coolit.This depends upon where itisgetting its“kicks” from. Ifitisthrough the fiber, wecool itatthetop—if themirror issurrounded byagasandisgetting hit mostly bycollisions inthegas,itisbetter tocool thegas. Asamatter offact,if weknow where thedamping oftheoscillations comes from, itturns outthatthatis always thesource ofthefluctuations also, apoint which wewillcome back to. Thesame thing works, amazingly enough, inelectrical circuits. Suppose that wearebuilding averysensitive, accurate amplifier foradefinite frequency andhave aresonant circuit (Fig. 41-2) intheinput soastomake itvery sensitive tothis certain frequency, likearadio receiver, butareally good one. Suppose wewish togodown tothevery lowest limit ofthings, sowetake thevoltage, sayoffthe inductance, andsend itintotherestoftheamplifier. Ofcourse, inanycircuit likethis,there isacertain amount ofloss. Itisnotaperfect resonant circuit, but itisaverygood oneandthere isalittle resistance, say(weputtheresistor insowe canseeit,butitissupposed tobesmall). Now wewould liketofindout:How much does thevoltage across theinductance fluctuate? Answer: Weknow that %LI2 isthe“kinetic energy”——the energy associated withacoilinaresonant circuit (Chapter 25).Therefore themean value of%LI2 isequal to%kT—that tellsuswhat thermscurrent isandwecanfindoutwhat thermsvoltage isfrom thermscurrent. Forifwewant thevoltage across theinductance theformula isV1,=iwLi, and themean absolute square voltage ontheinductance is(Vi) =L2w§(12), and putting in%L(I2) =%kT, weobtain (Vi)=Lw§kT. (41.2) Sonow wecandesign circuits andtellwhen wearegoing togetwhat iscalled Johnson noise, thenoise associated with thermal fluctuations! 41-2 Where dothefluctuations come from thistime? They come again from the resist0r—they come from thefactthat theelectrons intheresistor arejiggling around because theyareinthermal equilibrium with thematter intheresistor, and they make fluctuations inthedensity ofelectrons. They thus make tinyelectric fields which drive theresonant circuit. Electrical engineers represent theanswer inanother way. Physically, the resistor iseffectively thesource ofnoise. However, wemayreplace therealcircuit having anhonest, true physical resistor which ismaking noise, byanartificial circuit which contains alittle generator thatisgoing torepresent thenoise, and now theresistor isotherwise ideal—no noise comes from it.Allthenoise isin theartificial generator. And soifweknew thecharacteristics ofthenoise generated byaresistor, ifwehadtheformula forthat, thenwecould calculate what thecircuit isgoing todoinresponse tothat noise. So,weneed aformula forthenoise fluctuations. Now thenoise thatisgenerated bytheresistor isatallfrequencies, since theresistor byitself isnotresonant. Ofcourse theresonant circuit only “listens” tothepart thatisnear theright frequency, buttheresistor hasmany diflerent frequencies init.Wemaydescribe howstrong thegenerator is,asfollows: Themean power thattheresistor would absorb ifitwere connected directly across thenoise generator would be(E2)/R, ifEwere thevoltage from thegenerator. Butwewould liketoknow inmore detail how much power there isatevery fre- quency. There isvery little power inanyonefrequency; itisadistribution. Let P(w) dwbethepower thatthegenerator would deliver inthefrequency range dw intotheverysame resistor. Then wecanprove (weshall prove itforanother case, butthemathematics isexactly thesame) thatthepower comes out P(w) dw=(2/1r)kTdw, (41.3) andisindependent oftheresistance when putthisway. 41-2 Thermal equilibrium ofradiation Now wegoontoconsider astillmore advanced andinteresting proposition thatisasfollows. Suppose wehave acharged oscillator likethose wewere talking about when wewere discussing light, letussayanelectron oscillating upanddown inanatom. Ifitoscillates upanddown, itradiates light. Now suppose thatthis oscillator isinaverythingasofother. atoms, andthatfrom time totimetheatoms collide with it.Then inequilibrium, after along time, thisoscillator willpick up energy such thatitskinetic energy ofoscillation is%kT, andsince itisaharmonic oscillator, itsentire energy ofmotion willbecome kT.That is,ofcourse, awrong description sofar,because theoscillator carries electric charge, andifithasan energy kTitisshaking upanddown andradiating light. Therefore itisimpossible tohave equilibrium ofrealmatter alone without thecharges initemitting light, andaslight isemitted, energy flows away, theoscillator loses itskTastime goes on,andthus thewhole gaswhich iscolliding with theoscillator gradually cools off.And thatis,ofcourse, thewayahotstove cools onacoldnight byradiating thelight intothesky,because theatoms arejiggling their charge andtheycontin- ually radiate, andslowly, because ofthisradiation, thejiggling motion slows down. Ontheother hand, ifweenclose thewhole thing inaboxsothatthelight does notgoaway toinfinity, then wecaneventually getthermal equilibrium. Wemay either putthegasinaboxwhere wecansaythatthere areother radiators inthe boxwalls sending light back or,totake anicer example, wemaysuppose thebox hasmirror walls. Itiseasier tothink about thatcase. Thus weassume thatallthe radiation that goes outfrom theoscillator keeps running around inthebox. Then, ofcourse, itistruethattheoscillator starts toradiate, butpretty soon it canmaintain itskTofkinetic energy inspite ofthefactthatitisradiating, because itisbeing illuminated, wemaysay,byitsownlight reflected from thewalls ofthe box. That is,after awhile there isagreat dealoflight rushing around inthebox, andalthough theoscillator isradiating some, thelight comes back andreturns some oftheenergy thatwasradiated. 41-3 Weshall nowdetermine howmuch lightthere must beinsuch aboxattempera- tureTinorder thattheshining ofthelight onthisoscillator willgenerate just enough energy toaccount forthelight itradiated. Letthegasatoms bevery fewandfarbetween, sothat wehave anideal oscillator with noresistance except radiation resistance. Then weconsider thatat thermal equilibrium theoscillator isdoing twothings atthesame time. First, it hasamean energy kT,andwecalculate howmuch radiation itemits. Second, this radiation should beexactly theamount thatwould result because ofthefactthat thelight shining ontheoscillator isscattered. Since there isnowhere elsetheenergy cango,thiseffective radiation isreally justscattered light from thelight thatis inthere. Thus wefirstcalculate theenergy thatisradiated bytheoscillator persecond, iftheoscillator hasacertain energy. (Weborrow from Chapter 32onradiation resistance anumber ofequations without going back over their derivation.) The energy radiated perradian divided bytheenergy oftheoscillator iscalled 1/Q (Eq. 32.8): 1/Q =(dW/dt)/w0 W.Using thequantity ‘Y,thedamping constant, thiscanalsobewritten as1/Q=“//(.00, where 0:0isthenatural frequency ofthe oscillator-—if gamma isverysmall, Qisverylarge. Theenergy radiated persecond isthen dW _o.>0W _w0W’Y _ Theenergy radiated persecond isthus simply gamma times theenergy ofthe oscillator. Now theoscillator should have anaverage energy kT,soweseethat gamma kTistheaverage amount ofenergy radiated persecond: (dW/dt) =YkT. (41.5) Now weonly have toknow what gamma is.Gamma iseasily found from Eq. (32.12). Itis 2 '1=%=‘%’_°:’°. (41.6) where r0=e2/mc2 istheclassical electron radius, andwehave setA=21rc/w0. Ourfinal result fortheaverage rateofradiation oflight near thefrequency w0istherefore dW_2r0w?,kT Ti"5T‘ <4‘-7) Next weaskhow much light must beshining ontheoscillator. Itmust be enough thattheenergy absorbed from thelight (and thereupon scattered) isjust exactly thismuch. Inother words, theemitted light isaccounted forasscattered light from thelight thatisshining ontheoscillator inthecavity. Sowemust now calculate howmuch light isscattered from theoscillator ifthere isacertain amount —unknown—of radiation incident onit.LetI(w)dwbetheamount oflight energy there isatthefrequency w,within acertain range dw(because there isnolight at exactly acertain frequency; itisspread allover thespectrum). SoI(o.>)isacertain spectral distribution which wearenowgoing tofind—it isthecolor ofafurnace at temperature Tthatweseewhen weopen thedoor andlook inthehole. Now how much light isabsorbed? Weworked outtheamount ofradiation absorbed from agiven incident light beam, andwecalculated itinterms ofacross section. Itisjustasthough wesaidthatallofthelight thatfallsonacertain cross section isabsorbed. Sothetotal amount that isre-radiated (scattered) istheincident intensity I(w)dwmultiplied bythecross section 0'. Theformula forthecross section which wederived (Eq. 31.19) didnothave thedamping included. Itisnothard togothrough thederivation again andputin theresistance term which weneglected. Ifwedothat, andcalculate thecross section thesame way, weget 2 w4 0 (_81rr0 ‘,,-——3 ————-F_wg)2 +v2w2 (41.8) 41-4 Now, asafunction offrequency, 0,isofsignificant sizeonly forcoverynear tothenatural frequency w0. (Remember thattheQforaradiating oscillator is about 108.) Theoscillator scatters very strongly when wisequal tow0,andvery weakly forother values ofco.Therefore wecanreplace wby(.00and(.02—1.13 by2w0(w —co0),andweget 21rr0o.>% "8r <4‘-9) Now thewhole curve islocalized near w=w0. (Wedonotreally have tomake anyapproximations, butitismuch easier todotheintegrals ifwesimplify the equation abit.) Now wemultiply theintensity inagiven frequency range bythe cross section ofscattering, togettheamount ofenergy scattered intherange dw. Thetotal energy scattered isthen theintegral ofthisforallw.Thus id-5% =1 I((.o)o',(w) dot ° (41.10) =/up 21rr%)w%1I(w) dw _ 03l(<4>—¢°0)2+Y2/4] Now wesetdW,/dt =3'YkT. Why three? Because when wemade our analysis ofthecross section inChapter 32,weassumed thatthepolarization was such thatthelight could drive theoscillator. Ifwehadused anoscillator which could move only inonedirection, andthelight, say,waspolarized inthewrong way, itwould notgiveanyscattering. Sowemust either average thecross section ofanoscillator which cangoonly inonedirection, over alldirections ofincidence andpolarization ofthelight or,more easily, wecanimagine anoscillator which willfollow thefieldnomatter which waythefield ispointing. Such anoscillator, which canoscillate equally inthree directions, would have 3kTaverage energy because there are3degrees offreedom inthatoscillator. Soweshould use3vkT because ofthe3degrees offreedom. Now wehave todotheintegral. Letussuppose thattheunknown spectral distribution I(w)ofthelight isasmooth curve anddoes notvaryverymuch across thevery narrow frequency region where asispeaked (Fig. 41-3). Then theonly significant contribution comes when toisvery close tow0,within anamount gamma, which isvery small. Sotherefore, although I(w)maybeanunknown and complicated function, theonly place where itisimportant isnear co=w0,and there wemayreplace thesmooth curve byaflatone—a “constant”-—at thesame height. Inother words, wesimply take I(w)outside theintegral signandcallit 1(w0). Wemayalsotaketherestoftheconstants outinfront oftheintegral, and what wehave leftis §w3w%1(w0) [0( =3vkT. (41.11) Now, theintegral should gofrom 0tooo,but0issofarfrom w0thatthecurve is allfinished bythattime, sowegoinstead tominus oo—it makes nodiflerence and itismuch easier todotheintegral. Theintegral isaninverse tangent function of theform jdx/(x2 +a2). Ifwelook itupinabook weseethat itisequal to 1r/a. Sowhat itcomes toforourcaseis211-/Y. Therefore weget,with some re- arranging, 921111(0),)=5 (41.12)’Il'2I'()OJQ Then wesubstitute theformula (41.6) forgamma (donotworry about writing w0;since itistrueofany(1)0,wemayjustcallitw)andtheformula forI(w)then comes out 20.»kT And thatgives usthedistribution oflight inahotfurnace. Itiscalled theblack- 41-5Itu) I((u,,) ---- -- --~---- K to(4)0-Y Q04-Y we Fig.41-3. Thefactors intheintegrand (41.10). Thepeak istheresonance curve 1/(<1: —w()l2 -1-‘Y2/4. Toagood ap- proximation the factor I(w) can be replaced byl(w()). Ilia) 2T0 /’\\ To \ \ \ \ \ "X ‘\ u ‘IMO ll lVl$lILE l UV lX'IlYl ’ Fig.41-4. The blackbody intensity distribution attwotemperatures, accord- ingtoclassical physics (solid curves). The dashed curves show theactual distribution.body radiation. Black, because theholeinthefurnace thatwelook atisblack when thetemperature iszero. Inside aclosed boxattemperature T,(41.13) isthedistribution ofenergy of theradiation, according toclassical theory. First, letusnotice aremarkable feature ofthat expression. Thecharge oftheoscillator, themass oftheoscillator, all properties specific totheoscillator, cancel out,because once wehave reached equilib- rium with oneoscillator, wemust beatequilibrium with anyother oscillator ofadiflerent mass, orwewillbeintrouble. Sothisisanimportant kind ofcheck ontheproposition thatequilibrium does notdepend onwhat weareinequilibrium with, butonly onthetemperature. Now letusdraw apicture ofthe1(w) curve (Fig. 41-4). Ittellsushow much light wehave atdiflerent frequencies. Theamount ofintensity thatthere isinourbox, perunitfrequency range, goes, aswesee,asthesquare ofthefrequency, which means thatifwehave abox atanytemperature atall,andifwelook atthex-rays thatarecoming out,there willbealotofthem! Ofcourse weknow thisisfalse. When weopen thefurnace andtake alook atit,wedonotburn oureyesoutfrom x-rays atall.Itiscompletely false. Further- more, thetotal energy inthebox, thetotal ofallthisintensity summed over all frequencies, would bethearea under thisinfinite curve. Therefore, something is fundamentally, powerfully, andabsolutely wrong. Thus wastheclassical theory absolutely incapable ofcorrectly describing the distribution oflight from ablackbody, justasitwasincapable ofcorrectly de- scribing thespecific heats ofgases. Physicists went back andforth overthisderiva- tionfrom many different points ofview, andthere isnoescape. This isthepre- diction ofclassical physics. Equation (41.13) iscalled Rayleigh’s law,anditisthe prediction ofclassical physics, andisobviously absurd. 41-3 Equipartition andthequantum oscillator Thedifliculty above wasanother part ofthecontinual problem ofclassical physics, which started with thedifliculty ofthespecific heatofgases, andnowhas been focused onthedistribution oflight inablackbody. Now, ofcourse, atthe time that theoreticians studied thisthing, there were also many measurements oftheactual curve. And itturned outthatthecorrect curve looked likethedashed curves inFig.41-4. That is,thex-rays were notthere. Ifwelower thetemperature, thewhole curve goes down inproportion toT,according totheclassical theory, buttheobserved curve alsocutsofl"sooner atalower temperature. Thus thelow- frequency endofthecurve isright, butthehigh-frequency endiswrong. Why? When SirJames Jeans wasworrying about thespecific heats ofgases, henoted thatmotions which have high frequency are“frozen out” asthetemperature goes toolow. That is,ifthetemperature istoolow, ifthefrequency istoohigh, the oscillators donothavekTofenergy ontheaverage. Now recall howourderivation of(41.13) worked: Italldepends ontheenergy ofanoscillator atthermal equilib- rium. What thekTof(41.5) was, andwhat thesame kTin(41.13) is,isthemean energy ofaharmonic oscillator offrequency (.0attemperature T.Classically, this iskT,butexperimentally, no!—not when thetemperature istoolowortheoscillator frequency istoohigh. And sothereason thatthecurve fallsoflisthesame reason thatthespecific heats ofgases fail. Itiseasier tostudy theblackbody curve than it isthespecific heats ofgases, which aresocomplicated, therefore ourattention is focused ondetermining thetrue blackbody curve, because thiscurve isacurve which correctly tellsus,atevery frequency,/what theaverage energy ofharmonic oscillators actually isasafunction oftemperature. Planck studied thiscurve. Hefirst determined theanswer empirically, by fitting theobserved curve with anicefunction thatfitted very well. Thus hehad anempirical formula fortheaverage energy ofaharmonic oscillator asafunction offrequency. Inother words, hehadtheright formula instead ofkT,andthen by fiddling around hefound asimple derivation foritwhich involved avery peculiar assumption. That assumption wasthattheharmonic oscillator cantakeupenergies onlyhwatatime. Theideathattheycanhave anyenergy atallisfalse. Ofcourse, thatwasthebeginning oftheendofclassical mechanics. 41-6 Theveryfirstcorrectly determined quantum-mechanical formula willnow be derived. Suppose thatthepermitted energy levels ofaharmonic oscillator were equally spaced athw0apart, sothattheoscillator could takeononlythese different energies (Fig. 41-5). Planck made asomewhat more complicated argument than theonethatisbeing given here, because thatwasthevery beginning ofquantum mechanics andhehadtoprove some things. Butwearegoing totakeitasafact (which hedemonstrated inthiscase) thattheprobability ofoccupying alevel of energy EisP(E) =ae_E/'°7'. Ifwegoalong with that, wewillobtain theright result. Suppose now thatwehave alotofoscillators, andeach isavibrator offre- quency w0.Some ofthese vibrators willbeinthebottom quantum state, some will beinthenextone,andsoforth. What wewould liketoknow istheaverage energy ofallthese oscillators. Tofindout,letuscalculate thetotal energy ofalltheoscilla- torsanddivide bythenumber ofoscillators. That willbetheaverage energy per oscillator inthermal equilibrium, andwillalsobetheenergy thatisinequilibrium with theblackbody radiation andthatshould goinEq.(41.13) inplace ofkT. Thus weletN0bethenumber ofoscillators that areintheground state (the lowest energy state); N1thenumber ofoscillators inthestate E1;N2thenumber thatareinstate E2;andsoon.According tothehypothesis (which wehave not proved) thatinquantum mechanics thelawthatreplaced theprobability e_P"E"/"T ore_K'E"”‘T inclassical mechanics isthattheprobability goes down ase_AE/kT, where AEistheexcess energy, weshall assume thatthenumber N1thatareinthe firststate willbethenumber N0thatareintheground state, times e"“"/"T. Simi- larly, N2,thenumber ofoscillators inthesecond state, isN2=N0e_2"”/"T. To simplify thealgebra, letuscalle—"“’”'T =x.Then wesimply have N1=N0x, N2 =NQXZ, ...,Nn =N0x". Thetotal energy ofalltheoscillators must firstbeworked out. Ifanoscillator isintheground state, there isnoenergy. Ifitisinthefirststate, theenergy ishw0, andthere areN1ofthem. SoN1hw, orhwN0x ishow much energy wegetfrom those. Those thatareinthesecond state have 2hw0, andthere areN2ofthem, soN2-2hw =2hwN0x2 ishow much energy weget,andsoon.Then weadd italltogether togetE10,=N0hw(0 +x+2x2+3x3+...). And now, howmany oscillators arethere? Ofcourse, N0isthenumber that areintheground state, N1inthefirststate, andsoon,andweaddthem together: N101 =N0(l +x—l—x2+x3+...).Thus theaverage energy is 2 3<E):Em =N0hr.o(0+x+2x +3x -l--~-)_ (4114) Nmt N0(l-l-X-l-X2-l-"') Now thetwosums which appear hereweshall leave forthereader toplaywithand have some funwith. When weareallfinished summing andsubstituting forxin thesum, weshould get—if wemake nomistakes inthesum— hw(E)=FT» (41.15) This, then, wasthefirstquantum-mechanical formula everknown, oreverdiscussed, anditwasthebeautiful culmination ofdecades ofpuzzlement. Maxwell knew that there wassomething wrong, andtheproblem was, what wasright? Here isthequantitative answer ofwhat isright instead ofkT.This expression should, ofcourse, approach kTasco—>0orasT—>oo.Seeifyoucanprove thatitdoes— learn how todothemathematics. This isthefamous cutoff factor thatJeans waslooking for,andifweuseit instead ofkTin(41.13), weobtain forthedistribution oflight inablack box it/.2“dwI((.o)dw _n_2c2(eM/kT _1) (41.16) Weseethatforalarge w,even though wehave 0:3inthenumerator, there isane raised toatremendous power inthedenominator, sothecurve comes down again anddoes not“blow up”—we donotgetultraviolet light andx-rays where wedo notexpect them! 41-7AZ E4IMm P‘IAoxp(-4M:/kT) if— E_.,-Sflu 15-A¢11p(-am/arr) L E2I21110 P2IAup('2hw/kT) iii E1Iflu: P,IAup(-1w/KT) —l— EDI O PolA Fig.41-5. The energy levels ofa harmonic oscillator areequally spaced: En=rrtiw. Sac B Fig.41-6. Arandom walk of36 steps oflength l.How farisS36from B? Ans: about 6!ontheaverage.One might complain thatinourderivation of(41.16) weused thequantum theory fortheenergy levels oftheharmonic oscillator, buttheclassical theory in determining thecross section 0",.Butthequantum theory oflightinteracting with a harmonic oscillator gives exactly thesame result asthatgiven bytheclassical theory. That, infact,iswhywewere justified inspending somuch time onouranalysis of theindex ofrefraction andthescattering oflight, using amodel ofatoms like little oscillators-—the quantum formulas aresubstantially thesame. Now letusreturn totheJohnson noise inaresistor. Wehave already re- marked thatthetheory ofthisnoise power isreally thesame theory asthatofthe classical blackbody distribution. Infact, rather amusingly, wehave already saidthatiftheresistance inacircuit were notarealresistance, butwere anantenna (anantenna actslikearesistance because itradiates energy), aradiation resistance, itwould beeasyforustocalculate what thepower would be.Itwould bejustthe power thatruns intotheantenna from thelight thatisallaround, andwewould getthesame distribution, changed byonly oneortwofactors. Wecansuppose thattheresistor isagenerator with anunknown power spectrum P(w). Thespec- trum isdetermined bythefact that thissame generator, connected toaresonant circuit ofanyfrequency, asinFig.41-2(b), generates intheinductance avoltage ofthemagnitude given inEq.(41.2). One isthus ledtothesame integral asin (41.10), andthesame method works togiveEq.(41.3). Forlowtemperatures the kTin(41.3) must ofcourse bereplaced by(41.15). Thetwotheories (blackbody radiation andJohnson noise) arealsoclosely related physically, forwemay of course connect aresonant circuit toanantenna, sotheresistance Risapure radiation resistance. Since (41.2) does notdepend onthephysical origin ofthe resistance, weknow thegenerator Gforarealresistance andforradiation resist- ance isthesame. What istheorigin ofthegenerated power P(w) iftheresistance R isonly anideal antenna inequilibrium with itsenvironment attemperature T‘? Itistheradiation I(w)inthespace attemperature Twhich impinges ontheantenna and,as“received signals,” makes aneffective generator. Therefore onecandeduce adirect relation ofP(w) andI(w), leading then from (41.13) to(41.3). Allthethings wehave been talking about—the so-called Johnson noise and Planck’s distribution, andthecorrect theory oftheBrownian movement which we areabout todescribe—are developments ofthefirstdecade orsoofthiscentury. Now with those points andthathistory inmind, wereturn totheBrownian move- ment. 41-4 Therandom walk Letusconsider howtheposition ofajiggling particle should change withtime, forvery long times compared with thetime between “kicks.” Consider alittle Brownian movement particle which isjiggling about because itisbombarded on allsides byirregularly jiggling water molecules. Query: After agiven length of time, howfaraway isitlikely tobefrom where itbegan? This problem wassolved byEinstein andSmoluchowski. Ifweimagine thatwedivide thetime intolittle intervals, letussayahundredth ofasecond orso,then after thefirsthundredth of asecond itmoves here, andinthenext hundredth itmoves some more, inthe next hundredth ofasecond itmoves somewhere else, andsoon.Interms ofthe rateofbombardment, ahundredth ofasecond isavery long time. Thereader mayeasily verify thatthenumber ofcollisions asingle molecule ofwater receives inasecond isabout 1014, soinahundredth ofasecond ithas1012collisions, which isalot!Therefore, after ahundredth ofasecond itisnotgoing toremember what happened before. Inother words, thecollisions areallrandom, sothatone“step” isnotrelated totheprevious “step.” Itislikethefamous drunken sailor problem: thesailor comes outofthebarandtakes asequence ofsteps, buteach stepis chosen atanarbitrary angle, atrandom (Fig. 41-6). Thequestion is:After along time, where isthesailor? Ofcourse wedonotknow! Itisimpossible tosay. What dowemean—he isjustsomewhere more orlessrandom. Well then, onthe average, where ishe? Ontheaverage, howfaraway from thebarhashegone? Wehave already answered thisquestion, because once wewere discussing the 41-8 superposition oflight from awhole lotofdifferent sources atdiflerent phases, andthatmeant adding alotofarrows atdifferent angles (Chapter 32). There we discovered thatthemean square ofthedistance from oneendtotheother ofthe chain ofrandom steps, which wastheintensity ofthelight, isthesum ofthe intensities oftheseparate pieces. And so,bythesame kind ofmathematics, we canprove immediately thatifRNisthevector distance from theorigin after Nsteps, themean square ofthedistance from theorigin isproportional tothenumber N ofsteps. That is,(Riv) =NL2, where Listhelength ofeach step. Since thenum- berofsteps isproportional tothetime inourpresent problem, themean square distance isproportional tothetime: (R2)=at. (41.17) This does notmean thatthemean distance isproportional tothetime. Ifthemean distahce were proportional tothetime itwould mean thatthedrifting isatanice uniform velocity. Thesailor ismaking some relatively sensible headway, butonly such thathismean square distance isproportional totime. That isthecharacter- isticofarandom walk. Wemayshow veryeasily thatineach successive stepthesquare ofthedistance increases, ontheaverage, byL2.Forifwewrite RN=R,v_1 +L,wefindthat R€Vis RN'RN=R13=Rl\l—1+2R~_.-L+L2. andaveraging over many trials, wehave (Riv) =(R§;_1) +L2,since (RN_1 -L) =0.Thus. byinduction, R5,=NL2. (41.18) Now wewould liketocalctllate thecoefficient orinEq.(41.17), andtodoso wemust addafeature. Wearegoing tosuppose thatifwewere toputaforce onthisparticle (having nothing todowith theBrownian movement—we are taking asideissue forthemoment), thenitwould react inthefollowing wayagainst theforce. First, there would beinertia. Letmbethecoefficient ofinertia, the eflective mass oftheobject (not necessarily thesame astherealmass ofthereal particle, because thewater hastomove around theparticle ifwepullonit).Thus ifwetalkabout motion inonedirection, there isaterm likem(d2x/dt2) ononeside. And next, wewant alsotoassume thatifwekept asteady pullontheobject, there would beadrag onitfrom thefluid, proportional toitsvelocity. Besides theinertia ofthefluid, there isaresistance toflowduetotheviscosity andthecomplexity of thefluid. Itisabsolutely essential thatthere besome irreversible losses, something likeresistance, inorder thatthere befluctuations. There isnowaytoproduce the kTunless there arealsolosses. Thesource ofthefluctuations isveryclosely related tothese losses. What themechanism ofthisdrag is,wewilldiscuss soon—we shall talkabout forces thatareproportional tothevelocity andwhere theycome from. Butletussuppose fornow thatthere issuch aresistance. Then theformula for themotion under anexternal force, when wearepulling onitinanormal manner, is d2 dmfi+,1%=Fm. (41.19) Thequantity ucanbedetermined directly from experiment. Forexample, wecan watch thedrop fallunder gravity. Then weknow thattheforce ismg,anduismg divided bythespeed offallthedrop ultimately acquires. Orwecould putthe drop inacentrifuge andseehowfastitsediments. Orifitischarged, wecanput anelectric fieldonit.Souisameasurable thing, notanartificial thing, anditis known formany types ofcolloidal particles, etc. Now letususethesame formula inthecasewhere theforce isnotexternal, butisequal totheirregular forces oftheBrownian movement. Weshall then try todetermine themean square distance thattheobject goes. Instead oftaking the distances inthree dimensions, letustakejustonedimension, andfindthemean ofx2,justtoprepare ourselves. (Obviously themean ofx2isthesame asthemean ofy2isthesame asthemean of22,andtherefore themean square ofthedistance 41-9 isjust3times what wearegoing tocalculate.) Thex-component oftheirregular forces is,ofcourse, justasirregular asanyother component. What istherateof change ofx2? Itisd(x2)/dt =2x(dx/dt), sowhat wehave tofindistheaverage oftheposition times thevelocity. Weshall show thatthisisaconstant, andthat therefore themean square radius willincrease proportionally tothetime, andat what rate. Now ifwemultiply Eq.(41.19) byx,mx(d2x/dt2) +;.tx(dx/dt) =xF,,. Wewant thetime average ofx(dx/dt), soletustake theaverage ofthewhole equation, andstudy thethree terms. Now what about xtimes theforce? Ifthe particle happens tohave gone acertain distance x,then, since theirregular force is completely irregular anddoes notknow where theparticle started from, thenext impulse canbeinanydirection relative tox.Ifxispositive, there isnoreason why theaverage force should alsobeinthatdirection. Itisjustaslikely tobeone wayastheother. Thebombardment forces arenotdriving itinadefinite direction. Sotheaverage value ofxtimes Fiszero. Ontheother hand, fortheterm mx(d2x/dt2) wewillhave tobealittle fancy, andwrite thisas d2x d[x(dx/dt)] (dX)2 1 .1 '""Ifi=’”T-'" T" Thus weputinthese twoterms andtake theaverage ofboth. Soletusseehow much xtimes thevelocity should be.Now xtimes thevelocity hasamean that doesnotchange withtime, because when itgetstosome position ithasnore- membrance ofwhere itwasbefore, sothings arenolonger changing with time. Sothisquantity, ontheaverage, iszero. Wehave leftthequantity mv2, andthat istheonly thing weknow: mv2/2 hasamean value %kT. Therefore wefindthat <mx —l—/.4<x =(xF,) d _‘<m"2> -l"ga (X2) =0,implies O1‘ d(x2) _kTT_27- (41.20) Therefore theobject hasamean square distance (R2), attheendofacertain amount oft,equal to (R2)=skri. (41.21) And sowecanactually determine howfartheparticles go!Wefirstmust determine how they react toasteady force, how fastthey drift under aknown force (to find11),andthen wecandetermine how farthey gointheir random motions. This equation wasofconsiderable importance historically, because itwasoneof thefirstways bywhich theconstant kwasdetermined. After all,wecanmeasure 11.,thetime, howfartheparticles go,andwecantakeanaverage. Thereason that thedetermination ofkwasimportant isthat inthelawPV=RTforamole, weknow thatR,which canalsobemeasured, isequal tothenumber ofatoms ina mole times k.Amole wasoriginally defined assoandsomany grams ofoxygen— 16(now carbon isused), sothenumber ofatoms inamole wasnotknown, orig- inally. Itis,ofcourse, avery interesting andimportant problem. How bigare atoms? How many arethere? Sooneoftheearliest determinations ofthenumber ofatoms wasbythethedetermination ofhowfaradirty little particle would move ifwewatched itpatiently under amicroscope foracertain length oftime. And thus Boltzmann’s constant kandtheAvogadro number N0were determined be- cause Rhadalready been measured. 41-10 42 Applications ofKinetic Theory 42-1 Evaporation Inthischapter weshall discuss some further applications ofkinetic theory. Intheprevious chapter weemphasized one particular aspect ofkinetic theory, namely, thattheaverage kinetic energy inanydegree offreedom ofamolecule or other object is%kT. Thecentral feature ofwhat weshall nowdiscuss, ontheother hand, isthefactthattheprobability offinding aparticle indifferent places, per unitvolume, varies ase'1°““'“““‘1 "“""‘~”/"T; weshall make anumber ofapplications ofthis. Thephenomena which wewant tostudy arerelatively complicated: aliquid evaporating, orelectrons inametal coming outofthesurface, orachemical re- action inwhich there arealarge number ofatoms involved. insuch cases itisno longer possible tomake from thekinetic theory anysimple andcorrect statements, because thesituation istoocomplicated. Therefore, thischapter, except where otherwise emphasized, isquite inexact. Theidea tobeemphasized isonly that wecanunderstand, from thekinetic theory, more orlesshow things ought tobe- have. Byusing thermodynamic arguments, orsome empirical measurements of certain critical quantities, wecangetamore accurate representation ofthephe- nomena. However, itisvery useful toknow even only more orlesswhy something behaves asitdoes, sothatwhen thesituation isanewone,oronethatwehave not yetstarted toanalyze, wecansay,more orless,what ought tohappen. Sothis discussion ishighly inaccurate butessentially right—-right inidea, butalittle bit simplified, letussay,inthespecific details. Thefirstexample thatweshall consider istheevaporation ofaliquid. Suppose wehave aboxwith alarge volume, partially filled with liquid inequilibrium and with thevapor atacertain temperature. Weshall suppose thatthemolecules of thevapor arerelatively farapart, andthat inside theliquid, themolecules are packed close together. Theproblem istofindouthowmany molecules there are inthevapor phase, compared with thenumber there areintheliquid. How dense isthevapor atagiven temperature, andhowdoes itdepend onthetemperature? Letussaythatnequals thenumber ofmolecules perunitvolume inthevapor. That number, ofcourse, varies with thetemperature. Ifweaddheat, wegetmore evaporation. Now letanother quantity, 1/Va, equal thenumber ofatoms per unitvolume intheliquid: Wesuppose thateach molecule intheliquid occupies a certain volume, sothatifthere aremore molecules ofliquid, thenalltogether they occupy abigger volume. Thus ifVaisthevolume occupied byonemolecule, the number ofmolecules inaunitvolume isaunitvolume divided bythevolume of each molecule. Furthermore, wesuppose thatthere isaforce ofattraction between themolecules tohold them together intheliquid. Otherwise wecannot understand whyitcondenses. Thus suppose thatthere issuchaforce andthatthere isanenergy ofbinding ofthemolecules intheliquid which islostwhen theygointothevapor. That is,wearegoing tosuppose that, inorder totakeasingle molecule outofthe liquid intothevapor, acertain amount ofwork Whastobedone. There isa certain difference, W,intheenergy ofamolecule intheliquid from what itwould have ifitwere inthevapor, because wehave topullitaway from theother molecules which attract it. Now weusethegeneral principle thatthenumber ofatoms perunitvolume intwodifferent regions isn2/n1 =e*‘EFE1‘/"T. Sothenumber nperunitvolume inthevapor, divided bythenumber 1/V,, perunitvolume intheliquid, isequal to nV,,=Q-“"”, (42.1) 42-142-1 Evaporation 42-2 Thermionic emission 42-3 Thermal ionization 42-4 Chemical kinetics 42-5 Einstein’s laws ofradiation because thatisthegeneral rule. Itisliketheatmosphere inequilibrium under gravity, where thegasatthebottom isdenser than thatatthetopbecause ofthework mghneeded toliftthegasmolecules totheheight I1.Intheliquid, themolecules aredenser than inthevapor because wehave topullthem outthrough theenergy “hill” W,andtheratio ofthedensities ise'W/kT. This iswhat wewanted todeduce——that thevapor density varies asetothe minus some energy orother overkT.Thefactors infront arenotreally interesting tous,because inmost cases thevapor density isvery much lower than theliquid density. Inthose circumstances, where wearenotnear thecritical point where they arealmost thesame, butwhere thevapor density ismuch lower than the liquid density, then thefactthatnisverymuch lessthan l/V,,isoccasioned bythe factthatWisverymuch greater than kT.Soformulas such as(42.1) areinteresting only when Wisvery much bigger than kT,because inthose circumstances, since weareraising etominus atremendous amount, ifwechange Talittle bit,that tremendous power changes abit,andthechange produced intheexponential factor isvery much more important than anychange thatmight occur inthefactors out infront. Why should there beanychanges insuch factors asVa? Because ours wasanapproximate analysis. After all,there isnotreally adefinite volume for each molecule; aswechange thetemperature, thevolume V,,does notstayconstant —the liquid expands. There areother little features likethat, andsotheactual situation ismore complicated. There areslowly varying temperature-dependent factors allover theplace. Infact, wemight saythat Witself varies slightly with temperature, because atahigher temperature, atadifierent molecular volume, there would bedifferent average attractions, andsoon.So,while wemight think that ifwehave aformula inwhich everything varies inanunknown way with temperature then wehave noformula atall,ifwerealize thattheexponent W/kT is,ingeneral, verylarge, weseethatinthecurve ofthevapor density asafunction oftemperature most ofthevariation isoccasioned bytheexponential factor, and ifwetake Wasaconstant andthecoefiicient l/Va asnearly constant, itisagood approximation forshort intervals along thecurve. Most ofthevariation, inother words, isofthegeneral nature e_W/"T. Itturns outthat there aremany, many phenomena innature which are characterized byhaving toborrow anenergy from somewhere, andinwhich the central feature ofthetemperature variation isetotheminus theenergy over kT. This isauseful factonly when theenergy islarge compared with kT,sothatmost ofthevariation iscontained inthevariation ofthekTandnotintheconstant and inother factors. Now letusconsider another wayofobtaining asomewhat similar result for theevaporation, butlooking atitinmore detail. Toarrive at(42.1), wesimply applied arulewhich isvalid atequilibrium, butinorder tounderstand things better, there isnoharm intrying tolook atthedetails ofwhat isgoing on.Wemayalso describe what isgoing oninthefollowing way: themolecules thatareinthevapor continually bombard thesurface oftheliquid; when they hitit,they maybounce offorthey may getstuck. There isanunknown factor forthat—maybe 50-50, maybe 10to90—we donotknow. Letussaytheyalways getstuck—we canana- lyzeitover again later ontheassumption thattheydonotalways getstuck. Then atagiven moment there willbeacertain number ofatoms which arecondensing onto thesurface oftheliquid. Thenumber ofcondensing molecules, thenumber thatarrive onaunitarea, isthenumber nperunitvolume times thevelocity v. This velocity ofthemolecules isrelated tothetemperature, because weknow that %mv2 isequal to%kTontheaverage. Sovissome kind ofamean velocity. Of course weshould integrate over theangles andgetsome kind ofanaverage, butit isroughly proportional totheroot-mean-square velocity, within some factor. Thus NC=nv (42.2) isthenumber which arrive perunitarea andarecondensing. Atthesame time, however, theatoms intheliquid arejiggling about, and from time totime oneofthem getskicked out. Now wehave toestimate howfast they getkicked out. Theidea willbethat atequilibrium thenumber that are kicked outpersecond andthenumber thatarrive persecond areequal. 42-2 How many getkicked out? Inorder togetkicked out,aparticular molecule hastohave acquired byaccident anexcess energy over itsneighbors—a consider- ableexcess energy, because itisattracted verystrongly bytheother molecules inthe liquid. Ordinarily itdoes notleave because itissostrongly attracted, butinthe collisions sometimes oneofthem getsanextra energy byaccident. And thechance thatitgetstheextra energy Wwhich itneeds inourcaseisverysmall ifW>>kT. Infact, e_W/"T isthechance thatanatom haspicked upmore than thismuch en- ergy. That isthegeneral principle inkinetic theory: inorder toborrow anexcess energy Wover theaverage, theodds areetotheminus theenergy thatwehave to borrow, over kT.Now suppose thatsome molecules have borrowed thisenergy. Wenowhave toestimate how many leave thesurface persecond. Ofcourse, just because amolecule hasthenecessary energy does notmean thatitwillactually evaporate, since itmaybeburied toodeeply inside theliquid or,even ifitisnear thesurface, itmaybetravelling inthewrong direction. Thenumber thataregoing toleave aunitarea persecond isgoing tobesomething likethis: thenumber of atoms there arenear thesurface, perunitarea, divided bythetime ittakes oneto escape, multiplied bytheprobability e'W/'°T thatthey areready toescape inthe sense thatthey have enough energy. Weshall suppose thateach molecule atthesurface oftheliquid occupies a certain cross-sectional area A.Then thenumber ofmolecules perunit area of liquid surface willbel/A. And now, howlong does ittakeamolecule toescape? Ifthemolecules have acertain average speed v,andhave tomove, say,onemolec- ulardiameter D,thethickness ofthefirstlayer, then thetime ittakes togetacross thatthickness isthetime needed toescape, ifthemolecule hasenough energy. Thetime willbeD/tn. Thus thenumber evaporating should beapproximately /v,=(1/A)(v/D)e"W/kT. (42.3) Now thearea ofeach atom times thethickness ofthelayer isapproximately the same asthevolume Vaoccupied byasingle atom. Andso,inorder togetequilib- rium, wemust have N,=Ne,or nv=(v/V,)e-W”"". (42.4) Wemaycancel thev‘s,since they areequal; even though oneisthevelocity ofa molecule inthevapor andtheother isthevelocity ofanevaporating molecule, these arethesame, because weknow their mean kinetic energy (inonedirection) is%kT. Butonemayobject, “No! No!These aretheespecially fast-moving ones; these aretheones that have picked upexcess energy.” Notreally, because the moment they start topullaway from theliquid, they have tolosethat excess energy against thepotential energy. So,astheycome tothesurface theyareslowed down tothevelocity v!Itisthesame asitwasinourdiscussion ofthedistribution ofmolecular velocities intheatmosphere—at thebottom, themolecules hada certain distribution ofenergy. Theones thatarrive atthetophave thesame distri- bution ofenergy, because theslow ones didnotarrive atall,andthefastones were slowed down. Themolecules thatareevaporating have thesame distribution of energy astheones inside—a rather remarkable fact. Anyway, itisuseless totryto argue soclosely about ourformula because ofother inaccuracies, such astheprob- ability ofbouncing back rather than entering theliquid, andsoon.Thus wehave arough ideaoftherateofevaporation andcondensation, andwesee,ofcourse, thatthevapor density nvaries inthesame wayasbefore, butnowwehave under- stood itinsome detail rather than justasanarbitrary formula. This deeper understanding permits ustoanalyze some things. Forexample, suppose thatwewere topump away thevapor atsuch agreat ratethatweremoved thevapor asfastasitformed (ifwehadvery good pumps andtheliquid was evaporating very slowly), how fastwould evaporation occur ifwemaintained a liquid temperature T?Suppose that wehave already experimentally measured theequilibrium vapor density, sothatweknow, atthegiven temperature, how many molecules perunitvolume areinequilibrium with theliquid. Now wewould liketoknow howfastitwillevaporate. Even though wehave used only arough analysis sofarastheevaporation part ofitisconcerned, thenumber ofvapor 42-3 molecules arriving wasnotdone sobadly, aside from theunknown factor ofre- flection coeflicient. Sotherefore wemay usethefactthatthenumber thatare leaving, atequilibrium, isthesame asthenumber thatarrive. True, thevapor is being swept away andsothemolecules areonlycoming out,butifthevapor were leftalone, itwould attain theequilibrium density atwhich thenumber thatcome back would equal thenumber thatareevaporating. Therefore, wecaneasily see thatthenumber thatarecoming ofl"thesurface persecond isequal totheunknown reflection coeflicient Rtimes thenumber that would come down tothesurface persecond were thevapor stillthere, because thatishowmany would balance the evaporation atequilibrium: N,=rll)R=(vR/V,)e-W/”. (42.5) Ofcourse, thenumber ofmolecules thathittheliquid from thevapor iseasy to calculate, since wedonotneed toknow asmuch about theforces aswedowhen weareworrying about how they gettoescape through theliquid surface; itis much easier tomake theargument theother way. 42-2 Thermionic emission Wemay giveanother example ofavery practical situation thatissimilar to theevaporation ofaliquid—so similar that itisnotworth making aseparate analysis. Itisessentially thesame problem. Inaradio tube there isasource of electrons, namely aheated tungsten filament, andapositively charged plate to attract theelectrons. Anyelectron thatescapes from thesurface ofthetungsten is immediately swept away totheplate. That isourideal “pump,” which is“pump- ing” theelectrons away allthetime. Now thequestion is:I-low many electrons persecond canwegetoutofapiece oftungsten, andhowdoes thatnumber vary with temperature‘? Theanswer tothatproblem isthesame as(42.5), because it turns outthatinapiece ofmetal, electrons areattracted totheions, ortoatoms, ofthemetal. They areattracted, ifwemaysayitcrudely, tothemetal. Inorder togetanelectron outofapiece ofmetal, ittakes acertain amount ofenergy or work topullitout. This work varies with thediflerent kinds ofmetal. Infact,it varies even with thecharacter ofthesurface ofagiven kind ofmetal, butthetotal work may beafewelectron volts, which, incidentally, istypical oftheenergy involved inchemical reactions. Wecanremember thelatter factbyremembering thatthevoltage inachemical celllikeaflashlight battery, which isproduced by chemical reactions, isabout onevolt. How canwefindouthowmany electrons come outpersecond? Itwould be quite difficult toanalyze theeffects ontheelectrons going out; itiseasier to analyze thesituation theother way. So,wecould start outbyimagining thatwedid notdraw theelectrons away, andthattheelectrons were likeagas,andcould come back tothemetal. Then there would beacertain density ofelectrons at equilibrium which would, ofcourse, begiven byexactly thesame formula as(42.1), where Vaisthevolume perelectron inthemetal, roughly, andWisequal toq,¢, where ¢istheso-called workfunction, orthevoltage needed topullanelectron ofl thesurface. This would tellushow many electrons would have tobeinthe surrounding space andstriking themetal inorder tobalance theones thatare coming out. And thusitiseasytocalculate howmany arecoming outifwesweep away allofthem, because thenumber thatarecoming outisexactly equal tothe number thatwould begoing inwith theabove density ofelectron “vapor.” In other words, theanswer isthatthecurrent ofelectricity thatcomes inperunit area isequal tothecharge oneach times thenumber that arrive persecond per unit area, which isthenumber perunit volume times thevelocity, aswehave seen many times: I=qenv =(qev/Va)e-q‘¢/kT. (42.6) Now oneelectron voltcorresponds tokTatatemperature ofll,600 degrees. The filament ofthetube may beoperating atatemperature of,say,1100 degrees, so theexponential factor issomething likee_1°; when wechange thetemperature a 42-4 little bit,theexponential factor changes alot.Thus, again, thecentral feature of theformula isthee““¢"’”°T. Asamatter offact,thefactor infront isquite wrong- itturns outthatthebehavior ofelectrons inametal isnotcorrectly described by theclassical theory, butbyquantum mechanics, butthisonly changes thefactor in front alittle. Actually, noonehaseverbeen abletogetthething straightened out verywell, even though many people have used thehigh-class quantum-mechanical theory fortheir calculations. Thebigproblem is,does Wchange slightly with temperature? Ifitdoes, onecannot distinguish aWchanging slowly with tempera- turefrom adifferent coefficient infront. That is,ifWchanged linearly, say,with temperature, sothat W=W0+akT, then wewould have e_w/fr :e_<w,, +.,/tr)/fr :e-me-W0/kT Thus alinearly temperature-dependent Wisequivalent toashifted “constant.” Itisreally quite difficult andusually fruitless totrytoobtain thecoefficient in thefront accurately. 42-3 Thermal ionization Now wegoontoanother example ofthesame idea; always thesame idea. This hastodowith ionization. Suppose thatinagaswehave awhole lotofatoms which areintheneutral state, say,butthegasishotandtheatoms canbecome ionized. Wewould liketoknow howmany ionsthere areinagiven circumstance ifwehave acertain density ofatoms perunitvolume atacertain temperature. Again weconsider aboxinwhich there areNatoms which canhold electrons. (Ifanelectron hascome offanatom, itiscalled anion,andiftheatom isneutral, wesimply callitanatom.) Then suppose that, atanygiven moment, thenumber ofneutral atoms isna,thenumber ofionsisn,-,andthenumber ofelectrons isne, allperunit volume. The problem is:What istherelationship ofthese three numbers? Inthefirstplace, wehave twoconditions orconstraints onthenumbers. For instance, aswevary different conditions, likethetemperature andsoon,nu—l—n,- would remain constant, because thiswould besimply thenumber Nofatomic nuclei thatareinthebox. Ifwekeep thenumber ofnuclei perunitvolume fixed, andchange, say,thetemperature, then astheionization proceeded some atoms would turntoions, butthetotal number ofatoms plusionswould beunchanged. That is,na+n,=N.Another condition isthatiftheentire gasistobeelectri- cally neutral (and ifweneglect double ortriple ionization), thatmeans thatthe number ofionsisequal tothenumber ofelectrons atalltimes, orn,-=n,,.These aresubsidiary equations thatsimply express theconservation ofcharge andthe conservation ofatoms. These equations aretrue, andweultimately willusethem when weconsider arealproblem. Butwewant toobtain another relationship between thequantities. Wecandothisasfollows. Weagain usetheideathatittakes acertain amount ofenergy tolifttheelectron outoftheatom, which wecalltheionization energy, andwewould write itasW,inorder tomake alloftheformulas look thesame. So weletWequal theenergy needed topullanelectron outofanatom andmake an ion. Now weagain saythatthenumber offreeelectrons perunitvolume inthe “vapor” isequal tothenumber ofbound electrons perunitvolume intheatoms, times etotheminus theenergy difference between being bound andbeing free, over kT.That isthebasic equation again. How canwewrite it?Thenumber of freeelectrons perunitvolume would, ofcourse, bene,because thatisthedefinition ofne.Now what about thenumber ofelectrons perunitvolume thatarebound toatoms? Thetotal number ofplaces thatwecould puttheelectrons isapparently n,,+n,-,andwewillsuppose thatwhen theyarebound each oneisbound within a certain volume Va.Sothetotal amount ofvolume which isavailable toelectrons which would bebound is(na—l—n,-)V,,, sowemight want towrite ourformula as Fla -W/kTn=——————— e .C (nu +ni)Va 42-5 Theformula iswrong, however, inoneessential feature, which isthefollowing: when anelectron isalready onanatom, another electron cannot come tothat volume anymore! Inother words, allthevolumes ofallthepossible sitesarenot really available fortheoneelectron which istrying tomake upitsmind whether ornottobeinthevapor orinthecondensed position, because inthisproblem there isanextra feature thatwhen oneelectron iswhere another electron is,itisnot allowed togo—it isrepelled. Forthatreason, itcomes outthatweshould count only thatpartofthevolume which isavailable foranelectron tositonornot. That is,those which arealready occupied donotcount inthetotal available volume, buttheonly volume which isallowed isthatoftheions, where there arevacant places fortheelectron togo.Then, inthose circumstances, wefindthatanicer waytowrite ourformula is =%@-W/”. (42.7) This formula iscalled theSaha ionization equation. Now letusseeifwecanunder- stand qualitatively why aformula likethisisright, byarguing about thekinetic things thatarehappening. First, every once inawhile anelectron comes toanionandtheycombine to make anatom. And also, every once inawhile, anatom getsintoacollision and breaks upintoanionandanelectron. Now those tworates must beequal. How fastdoelectrons andionsfindeach other? Therateiscertainly increased ifthe number ofelectrons perunitvolume isincreased. Itisalsoincreased ifthenumber ofionsperunitvolume isincreased. That is,thetotal rateatwhich recombination isoccurring iscertainly proportional tothenumber ofelectrons times thenumber ofions. Now thetotal rateatwhich ionization isoccurring duetocollisions must bedependent linearly onhow many atoms there aretoionize. And sotherates willbalance when there issome relationship between theproduct n,n,-andthe number ofatoms, na.Thefactthatthisrelationship happens tobegiven bythis particular formula, where Wistheionization energy, isofcourse alittle bitmore information, butwecaneasily understand that theformula would necessarily involve theconcentrations oftheelectrons, ions, andatoms inthecombination nen,-/na toproduce aconstant independent ofthen’s,anddependent only on temperature, theatomic cross sections, andother constant factors. Wemay also note that, since theequation involves thenumbers perunit volume, ifwewere todotwoexperiments with agiven total number Nofatoms plusions, thatis,acertain fixed number ofnuclei, butusing boxes with different volumes,,the n’swould allbesmaller inthelarger box. Butsince theratio nen,-/n,, stays thesame, thetotal number ofelectrons andionsmust begreater inthelarger box. Toseethis, suppose thatthere areNnuclei inside aboxofvolume V,and thatafraction fofthem areionized. Then n,=fN/V=n,~,andna=(1—f)N/V. Then ourequation becomes f2 {Y _e-—W/kT T1_fV____Va . (42.8) Inother words, ifwetakeasmaller andsmaller density ofatoms, ormake thevol- ume ofthecontainer bigger andbigger, thefraction fofelectrons andionsmust increase. That ionization, justfrom “expansion” asthedensity goes down, isthe reason whywebelieve thatatverylowdensities, such asinthecold space between thestars, there maybeionspresent, even though wemight notunderstand itfrom thepoint ofview oftheavailable energy. Although ittakes many, many kTof energy tomake them, there areionspresent. Why canthere beionspresent when there issomuch space around, while if weincrease thedensity, theionstend todisappear? Answer: Consider anatom. Every once inawhile, light, oranother atom, oranion,orwhatever itisthat maintains thermal equilibrium, strikes it.Very rarely, because ittakes such a terrific amount ofexcess energy, anelectron comes offandanionisleft. Now thatelectron, ifthespace isenormous, wanders andwanders anddoes notcome near anything foryears, perhaps. Butonce inaverygreat while, itdoes come back 42-6 toanionandthey combine tomake anatom. Sotherateatwhich electrons are coming outfrom theatoms isveryslow. Butifthevolume isenormous, anelectron which hasescaped takes solong tofindanother iontorecombine with thatits probability ofrecombination isvery, very small; thus, inspite ofthelarge excess energy needed, there maybeareasonable number ofelectrons. 42-4 Chemical kinetics Thesame situation thatwehave justcalled “ionization” isalsofound ina chemical reaction. Forinstance, iftwoobjects AandBcombine intoacompound AB,then ifwethink about itforawhile weseethatABiswhat wehave called an atom, Biswhat wecallanelectron, andAiswhat wecallanion. With these substitutions theequations ofequilibrium areexactly thesame inform: FL”=¢@"W”“T. (42.9)nAB This formula, ofcourse, isnotexact, since the“constant” cdepends onhowmuch volume isallowed fortheAandBtocombine, andsoon,butbythermodynamic arguments onecanidentify what themeaning oftheWintheexponential factor is,anditturns outthatitisveryclose totheenergy needed inthereaction. Suppose that wetried tounderstand thisformula asaresult ofcollisions, much inthewaythatweunderstood theevaporation formula, byarguing about how many electrons came offandhow many ofthem came back perunittime. Suppose that AandBcombine inacollision every once inawhile toform a compound AB. And suppose thatthecompound ABisacomplicated molecule which jiggles around andishitbyother molecules, andfrom time totime itgets enough energy toexplode andbreak upagain intoAandB. Now itactually turns out,inchemical reactions, thatiftheatoms come to- gether withtoosmall anenergy, even though energy maybereleased inthereaction A+B—>AB,thefactthatAandBmay touch each other does notnecessarily make thereaction start. Itusually isrequired thatthecollision berather hard, in fact, togetthereaction togoatall—a “soft” collision between AandBmaynot doit,even though energy may bereleased intheprocess. Soletussuppose that itisvery common inchemical reactions that, inorder forAandBtoform AB, theycannot justhiteach other, buttheyhave tohiteach other withsufifcient energy. This energy iscalled theactivation energy—the energy needed to“activate” the reaction. CallA*theactivation energy, theexcess energy needed inacollision in order that thereaction may really occur. Then therateR;atwhich AandB produce ABwould involve thenumber ofatoms ofAtimes thenumber ofatoms ofB,times therateatwhich asingle atom would strike acertain cross section 0,4B,times afactor e'A*/ kT,which istheprobability thattheyhave enough energy: Rf =I’l_4I'l3f)0'AB€~A*/kT. (42.10) Now wehave tofindtheopposite rate, RT.There isacertain chance thatABwill flyapart. Inorder toflyapart, itnotonly must have theenergy Wwhich itneeds inorder togetapart atallbut,justasitwashard forAandBtocome together, so there isakind ofhillthatAandBhave toclimb overtogetapart again; theymust have notonly enough energy justtogetready topullapart, butacertain excess. Itislikeclimbing ahilltogetintoadeep valley; theyhave toclimb thehillcoming inandthey have toclimb outofthevalley andthen over thehillcoming back (Fig. 42-1). Thus therateatwhich ABgoes toAandBwillbeproportional tothe number n_4Bthatarepresent, times eT‘W+"*’/"T: R,=c’n..,,e-‘W+“*>”‘T. (42.11) Thec’willinvolve thevolume ofatoms andtherateofcollisions, which wecan work out,aswedidthecaseofevaporation, with areas andtimes andthicknesses; butweshall notdothis. Themain feature ofinterest tousisthatwhen these two 42-7E _ _pA_ A+B Ti“ W AB T Fig.42-1. The energy relationship forthereaction A—l—B->AB. rates areequal, theratio ofthem isequal tounity. This tellsusthatnAn);/n,4B= ce"W”‘T, asbefore, where cinvolves thecross sections, velocities, andother factors independent ofthen’s. Theinteresting thing isthattherateofthereaction alsovaries ase_°°““/"7', although theconstant isnotthesame asthatwhich governs theconcentrations; theactivation energy A*isquite different from theenergy W.Wgoverns thepro- portions ofA,B,andABthatwehave inequilibrium, butifwewant toknow how fastA+Bgoes toAB,thatisnotaquestion ofequilibrium, andhere adifl'erent energy, theactivation energy, governs therateofreaction through anexponential factor. Furthermore, A*isnotafundamental constant likeW.Suppose thatatthe surface ofthewall—or atsome other place—A andBcould temporarily stick there insuch away thatthey could combine more easily. Inother words, wemight finda“tunnel” through thehill,orperhaps alower hill. Bytheconservation of energy, when weareallfinished wehave stillmade ABoutofAandB,sothe energy difference Wwillbequite independent ofthewaythereaction occurred, buttheactivation energy A*willdepend verymuch onthewaythereaction occurs. This iswhytherates ofchemical reactions areverysensitive tooutside conditions. Wecanchange theratebyputting inasurface ofadifl'erent kind, wecanputitin a“diflerent barrel” anditwillgoatadifferent rate, ifitdepends onthenature of thesurface. Orifweputinathird kind ofobject itmaychange therateverymuch; some things produce enormous changes inratesimply bychanging theA*alittle bit—they arecalled catalysts. Areaction might practically notoccur atallbecause A*istoobigatthegiven temperature, butwhen weputinthisspecial stuff, the catalyst, then thereaction goes very fastindeed, because A*isreduced. Incidentally, there issome trouble with such areaction, AplusB,making AB, because wecannot conserve both energy andmomentum when wetrytoputtwo objects together tomake onethatismore stable. Therefore, weneed atleast a third object C,sotheactual reaction ismuch more complicated. Theforward ratewould involve theproduct nAn3nC, anditmight seem that ourformula is going wrong, butno!When welook attherateatwhich ABgoes theother way, wefindthatitalsoneeds tocollide with C,sothere isannABncinthereverse rate; then@’scancel outintheformula fortheequilibrium concentrations. Thelawof equilibrium, (42.9), which wefirstwrote down isabsolutely guaranteed tobetrue, nomatter what themechanism ofthereaction maybe! 42-5 Einstein’s laws ofradiation Wenowturntoaninteresting analogous situation having todowith theblack- body radiation law. Inthelastchapter weworked outthedistribution lawforthe radiation inacavity thewayPlanck did,considering theradiation from anoscilla- tor.Theoscillator hadtohave acertain mean energy, andsince itwasoscillating, itwould radiate andwould keep pumping radiation intothecavity until itpiled upenough radiation tobalance theabsorption andemission. Inthat way we found that theintensity ofradiation atfrequency wwasgiven bytheformula hwadwI(w)dw Tr2c2(eM/W _1) (42.12) This result involved theassumption thattheoscillator which wasgenerating the radiation haddefinite, equally spaced energy levels. Wedidnotsaythatlight had tobeaphoton oranything likethat. There wasnodiscussion about how, when an atom goes from onelevel toanother, theenergy must come outinoneunitofen- ergy, hm,intheform oflight. Planck’s original idea wasthat thematter was quantized butnotthelight: material oscillators cannot take upjustanyenergy, buthave totake itinlumps. Furthermore, thetrouble with thederivation isthat itwaspartially classical. Wecalculated therate ofradiation from an oscillator according toclassical physics; then weturned around andsaid, “No, thisoscillator hasalotofenergy levels.” Sogradually, inorder tofind the right result, thecompletely quantum-mechanical result, there wasaslow develop- 42-8E l!s I ment which culminated inthequantum mechanics of1927. Butinthemeantime, there wasanattempt byEinstein toconvert Planck’s viewpoint thatonlyoscillators ofmatter were quantized, totheideathatlight wasreally photons andcould be considered inacertain wayasparticles with energy hw.Furthermore, Bohr had pointed outthatanysystem ofatoms hasenergy levels, buttheyarenotnecessarily equally spaced likePlanck‘s oscillator. And soitbecame necessary torederive or atleast rediscuss theradiation lawfrom amore completely quantum-mechanical viewpoint. Einstein assumed that Planck’s final formula wasright, andheused that formula toobtain some new information, previously unknown, about theinter- action ofradiation with matter. Hisdiscussion went asfollows: Consider anytwo ofthemany energy levels ofanatom, saythemthlevel andthenthlevel (Fig. 42-2). Now Einstein proposed thatwhen such anatom haslight oftheright frequency shining onit,itcanabsorb thatphoton oflight andmake atransition from state ntostate m,andthattheprobability thatthisoccurs persecond depends upon the twolevels, ofcourse, butisproportional tohowintense thelight isthatisshining onit.Letuscalltheproportionality constant B,,,,,, merely toremind usthatthis isnotauniversal constant ofnature, butdepends ontheparticular pairoflevels: some levels areeasy toexcite; some levels arehard toexcite. Now what isthe formula going tobefortherateofemission from mton?Einstein proposed that thismust have twoparts toit.First, even ifthere were nolight present, there would besome chance thatanatom inanexcited state would falltoalower state, emitting aphoton; thiswecallspontaneous emission. Itisanalogous totheidea thatanoscillator with acertain amount ofenergy, even inclassical physics, does notkeep thatenergy, butloses itbyradiation. Thus theanalog ofspontaneous radiation ofaclassical system isthatiftheatom isinanexcited state there isa certain probability Am, which depends onthelevels again, forittogodown from m ton,andthisprobability isindependent ofwhether light isshining ontheatom or not. Butthen Einstein went further, andbycomparison with theclassical theory andbyother arguments, concluded thatemission wasalsoinfluenced bythepres- ence oflight—that when light oftheright frequency isshining onanatom, ithas anincreased rateofemitting aphoton thatisproportional totheintensity ofthe light, with aproportionality constant B,,,,,. Later, ifwededuce thatthiscoefficient iszero, then wewillhave found thatEinstein waswrong. Ofcourse wewillfind hewasright. Thus Einstein assumed thatthere arethree kinds ofprocesses: anabsorption proportional totheintensity oflight, anemission proportional totheintensity of light, called induced emission orsometimes stimulated emission, andaspontaneous emission independent oflight. Now suppose thatwehave, inequilibrium attemperature T,acertain number ofatoms N,,inthestate nandanother number Nminthestate m.Then thetotal number ofatoms thataregoing from ntomisthenumber thatareinthestate n times theratepersecond that, ifoneisinn,itgoesuptom.Sowehave aformula forthenumber thataregoing from ntompersecond: R,,_,,,, =N,,B,,,,,I(w). (42.13) Thenumber thatwillgofrom mtonisexpressed inthesame manner, asthenum- berNmthatareinm,times thechance persecond thateach onegoes down ton. This time ourexpression is Rm—>n =Nm[Amn + BmnI(w)]' Now weshall suppose thatinthermal equilibrium thenumber ofatoms going up must equal thenumber coming down. That isoneway, atleast, inwhich the number willbesure tostayconstant ineach level.* Sowetake these tworates *This isnottheonly wayonecanarrange tokeep thenumbers ofatoms inthevarious levels constant, butitistheway itactually works. That every process must, inthermal equilibrium, bebalanced byitsexact opposite iscalled theprinciple ofdetailed balancing. 42-9m Spontaneous ominion *|"°'9'l°" Lfnduccd millioni——n Fig.42-2. Transitions between two energy levels ofonatom. h Blue m Red, laser light n Hg.42—3. Byexdfing,say bybhe light, ahigher state h,which may emit a photon leaving atoms instate m,the number inthisstate mbecomes sufficiently large tostart laser action.tobeequal atequilibrium. Butwehave oneother piece ofinformation: weknow how large N,,,iscompared with N,,-the ratio ofthose twoise“‘E»=“E"”"T. Now Einstein assumed thattheonlylight which iseffective inmaking thetransition from ntomisthelight which hasthefrequency corresponding totheenergy difference, soEm-En=ho)inallourformulas. Thus Nm=N,,e-WT. (42.15) Thus ifwesetthetworates equal: N,,B,,,,,I(w) =Nm[Amn +B,,,,,I(w)], and divide byNm,weget B,,,1(<»)e"“/"T =A,,,,,+B,,,,,I(w). (42.16) From thisequation, wecancalculate I(w). Itissimply AmnI(w) Bmeww _BM (42.17) ButPlanck hasalready toldusthattheformula must be(42.12). Therefore we candeduce something: First, thatBM,must equal B,,,,,, since otherwise wecannot getthe(e"‘””°T —1).SoEinstein discovered some things thathedidnotknow how tocalculate, namely thattheinduced emission probability andtheabsorption prob- ability must beequal. This isinteresting. And furthermore, inorder for(42.17) and(42.12) toagree, A,,,,,/B,,,,, mustbehas/1&2. (42.18) Soifweknow, forinstance, theabsorption rateforagiven level, wecandeduce thespontaneous emission rateandtheinduced emission rate, oranycombination. This isasfarasEinstein oranyone elsecould gousing such arguments. To actually compute theabsolute spontaneous emission rateortheother rates for anyspecific atomic transition, ofcourse, requires aknowledge ofthemachinery oftheatom, called quantum electrodynamics, which wasnotdiscovered until eleven years later. This work ofEinstein wasdone in1916. Thepossibility ofinduced emission has,today, found interesting applications. Ifthere islight present, itwilltend toinduce thedownward transition. Thetransi- tionthen adds itshwtotheavailable light energy, ifthere were some atoms sitting intheupper state. Now wecanarrange, bysome nonthermal method, tohave a gasinwhich thenumber inthestate misverymuch greater than thenumber inthe state n.This isfaroutofequilibrium, andsoisnotgiven bytheformula e_”“’/"T, which isforequilibrium. Wecaneven arrange itsothatthenumber intheupper state isvery large, while thenumber inthelower state ispractically zero. Then light which hasthefrequency corresponding totheenergy difference Em—E,, willnotbestrongly absorbed, because there arenotmany atoms instate nto absorb it.Ontheother hand, when thatlight ispresent, itwillinduce theemission from thisupper state! So,ifwehadalotofatoms intheupper state, there would beasortofchain reaction, inwhich, themoment theatoms began toemit, more would becaused toemit, andthewhole lotofthem would dump down together. This iswhat iscalled alaser, or,inthecaseofthefarinfrared, amaser. Various tricks canbeused toobtain theatoms instate m.There maybehigher levels towhich theatoms cangetifweshine inastrong beam oflight ofhigh frequency. From these high levels, they may trickle down, emitting various pho- tons, until they allgetstuck inthestate m.Ifthey tend tostayinthestate m without emitting, thestate iscalled metastable. And then they arealldumped down together byinduced emissions. One more technical point-if weputthis system inanordinary box, itwould radiate insomany difl'erent directions spon- taneously, compared with theinduced effect, that wewould stillbeintrouble. Butwecanenhance theinduced eflect, increase itsefficiency, byputting nearly perfect mirrors oneach sideofthebox, sothatthelight which isemitted gets another chance, andanother chance, andanother chance, toinduce more emission. Although themirrors arealmost onehundred percent reflecting, there isaslight amount oftransmission ofthemirror, andalittle light getsout. Intheend, of course, from theconservation ofenergy, allthelight goes outinaniceuniform straight direction which makes thestrong light beams thatarepossible today with lasers. 42-10 43 Diffusion 43-1 Collisions between molecules Wehave considered sofaronly themolecular motions inagaswhich isin thermal equilibrium. Wewant nowtodiscuss what happens when things arenear, butnotexactly in,equilibrium. Inasituation farfrom equilibrium, things are extremely complicated, butinasituation very close toequilibrium wecaneasily work outwhat happens. Toseewhat happens, wemust, however, return tothe kinetic theory. Statistical mechanics andthermodynamics dealwiththeequilibrium situation, butaway from equilibrium wecanonly analyze what occurs atom by atom, sotospeak. Asasimple example ofanonequilibrium circumstance, weshall consider thediffusion ofions inagas. Suppose thatinagasthere isarelatively small concentration ofions—electrically charged molecules. Ifweputanelectric field onthegas,then each ionwillhave aforce onitwhich isdifferent from theforces ontheneutral molecules ofthegas. Ifthere were noother molecules present, an ionwould have aconstant acceleration until itreached thewall ofthecontainer. Butbecause ofthepresence oftheother molecules, itcannot dothat; itsvelocity increases only until itcollides with amolecule andloses itsmomentum. Itstarts again topickupmore speed, butthen itloses itsmomentum again. Theneteffect isthatanionworks itswayalong anerratic path, butwith anetmotion inthedi- rection oftheelectric force. Weshall seethattheionhasanaverage “drift” with amean speed which isproportional totheelectric field—the stronger thefield, thefaster itgoes. While thefield ison,andwhile the.ion ismoving along, itis, ofcourse, notinthermal equilibrium, itistrying togettoequilibrium, which isto besitting attheendofthecontainer. Bymeans ofthekinetic theory wecancom- pute thedrift velocity. Itturns outthat with ourpresent mathematical abilities wecannot really compute precisely what willhappen, butwecanobtain approximate results which exhibit alltheessential features. Wecanfindouthowthings willvarywithpressure, with temperature, andsoon,butitwillnotbepossible togetprecisely thecorrect numerical factors infront ofalltheterms. Weshall, therefore, inourderivations, notworry about theprecise value ofnumerical factors. They canbeobtained only byavery much more sophisticated mathematical treatment. Before weconsider what happens innonequilibrium situations, weshall need tolook alittle closer atwhat goes oninagasinthermal equilibrium. Weshall need toknow, forexample, what theaverage time between successive collisions ofamolecule is. Anymolecule experiences asequence ofcollisions with other moleculcs—in a random way, ofcourse. Aparticular molecule will, inalong period oftime T, have acertain number, N,ofhits. Ifwedouble thelength oftime, there willbe twice asmany hits. Sothenumber ofcollisions isproportional tothetime T.We would liketowrite itthisway: N=T/'r (43.1) Wehave written theconstant ofproportionality as1/-r,where 1-willhave thedi- mensions ofatime. Theconstant Tistheaverage timebetween collisions. Suppose, forexample, thatinanhour there are60collisions; then 1'isoneminute. We would saythat~r(one minute) istheaverage timebetween thecollisions. Wemayoften wish toaskthefollowing question: “What isthechance thata molecule willexperience acollision during thenext small interval oftime dt?” Theanswer, wemayintuitively understand, isdt/1'. Butletustrytomake amore 43-143-1 Collisions between molecules 43-2 Themean freepath 43-3 Thedriftspeed 43-4 Ionic conductivity 43-5 Molecular diffusion 43-6 Thermal conductivity convincing argument. Suppose thatthere were averylarge number Nofmolecules. How many willhave collisions inthenextinterval oftimedt?Ifthere isequilibrium, nothing ischanging ontheaverage with time. SoNmolecules waiting thetime dt willhave thesame number ofcollisions asonemolecule waiting forthetime Ndt. That number weknow isNdt/T. Sothenumber ofhitsofNmolecules isNdt/1 inatime dt,andthechance, orprobability, ofahitforanyonemolecule isjust 1/Naslarge, or(1/N)(N dt/1) =dt/1, asweguessed above. That istosay,the fraction ofthemolecules which willsufler acollision inthetimedtisdt/er. Totake anexample, if1'isoneminute, then inonesecond thefraction ofparticles which willsuffer collisions isl/60. What thismeans, ofcourse, isthat l/60 ofthe molecules happen tobeclose enough towhat they aregoing tohitnext that their collisions willoccur inthenext minute. When wesaythat1',themean time between collisions, isoneminute, wedo notmean thatallthecollisions willoccur attimes separated byexactly oneminute. Aparticular particle does nothave acollision, wait oneminute, andthen have another collision. The times between successive collisions arequite variable. Wewillnotneed itforourlater work here, butwemaymake asmall diversion to answer thequestion: “What arethetimes between collisions?” Weknow thatfor thecase above, theaverage time isoneminute, butwemight liketoknow, for example, what isthechance thatwegetnocollision fortwominutes? Weshall findtheanswer tothegeneral question: “What istheprobability thatamolecule willgoforatime twithout having acollision?” Atsome arbitrary instant—that wecallt=0—we begin towatch aparticular molecule. What is thechance that itgetsbyuntil twithout colliding with another molecule? To compute theprobability, weobserve what ishappening toallN0molecules ina container. After wehave waited atime t,some ofthem willhave hadcollisions. WeletN(t)bethenumber thathave nothadcollisions uptothetime t.N(t)is,of course, lessthan N0.WecanfindN(t)because weknow howitchanges withtime. Ifweknow thatN(t) molecules have gotbyuntil t,then N(t—l—dt),thenumber which getbyuntil t—l—dt,islessthan N(t) bythenumber thathave collisions in dt.Thenumber thatcollide indtwehave written above interms ofthemean time -rasdN=N(t)dt/'r. Wehave theequation N(t+dz)=N(t)-N(t)§- (43.2) Thequantity ontheleft-hand side, N(t+dt),canbewritten, according tothe definitions ofcalculus, asN(t) +(dN/dt) dt. Making this substitution, Eq. (43.2) yields dN(t) _ N(t) T"-T‘ (43-3) Thenumber thatarebeing lostintheinterval dtisproportional tothenumber that arepresent, andinversely proportional tothemean life1-.Equation (43.3) iseasily integrated ifwerewrite itas dN(t)__g_Wm_T (43.4) Each sideisaperfect diflerential, sotheintegral is lnN(t) =—t/'r —l—(aconstant), (43.5) which saysthesame thing as N(t) =(constant)e_’/T. (43.6) Weknow thattheconstant must bejustN0,thetotal number ofmolecules present, since allofthem start att=0towait fortheir “next” collision. Wecanwrite ourresult as N(t)=Noe-”’. (43.7) 43-2 Ifwewish theprobability ofnocollision, P(t), wecangetitbydividing N(t) by N0,so P(t) =e_”’. (43.8) Ourresult is:theprobability thataparticular molecule survives" atime twithout a collision ise_”’, where Tisthemean time between collisions. Theprobability starts outatl(orcertainty) fort=0,andgetslessastgetsbigger andbigger. Theprobability thatthemolecule avoids acollision foratime equal to'rise_1= 0.37...Thechance islessthan one-half thatitwillhave agreater than average time between collisions. That isallright, because there areenough molecules which go collision-free fortimes much longer than themean timebefore colliding, sothatthe average time canstillbe1. Weoriginally defined 1'astheaverage time between collisions. Theresult we have obtained inEq.(43.7) alsosaysthatthemean time from anarbitrary starting instant tothenextcollision isalso1'.Wecandemonstrate thissomewhat surprising factinthefollowing way. Thenumber ofmolecules which experience their next collision intheinterval dtatthetime tafter anarbitrarily chosen starting time is N(t)dt/'r. Their “time until thenextcollision” is,ofcourse, justt.The“average time until thenextcollision” isobtained intheusual way: Average time until thenextcollision =L/tL125! -N0 0 T Using N(t) obtained in(43.7) andevaluating theintegral, wefindindeed that1' istheaverage time from anyinstant until thenext collision. 43-2 Themean freepath Another wayofdescribing themolecular collisions istotalknotabout the time between collisions, butabout howfartheparticle moves between collisions. Ifwesaythattheaverage time between collisions is1',andthatthemolecules have amean velocity v,wecanexpect that theaverage distance between collisions, which weshall callI,isjusttheproduct of1'andv.This distance between collisions isusually called themean freepath: Mean freepathl =rv. (43.9) Inthischapter weshall bealittle careless about what kindofaverage wemean inanyparticular case. Thevarious possible averages—the mean, theroot-mean- square, etc.—are allnearly equal anddiffer byfactors which areneartoone. Since adetailed analysis isrequired toobtain thecorrect numerical factors anyway, we need notworry about which average isrequired atanyparticular point. Wemay also warn thereader that thealgebraic symbols weareusing forsome ofthe physical quantities (e.g., Iforthemean freepath) donotfollow agenerally accepted convention, mainly because there isnogeneral agreement. Justasthechance thatamolecule willhave acollision inashort time dtis equal todt/1', thechance thatitwillhave acollision ingoing adistance dxisdx/l. Following thesame lineofargument used above, thereader canshow thatthe probability thatamolecule willgoatleast thedistance xbefore having itsnext collision ise""I. Theaverage distance amolecule goesbefore colliding withanother molecule—- themean freepath l—will depend onhow many molecules there arearound and onthe“size” ofthemolecules, i.e.,howbigatarget theyrepresent. Theeflective “size” ofatarget inacollision weusually describe bya“collision cross section,” thesame ideathatisused innuclear physics, orinlight-scattering problems. Consider amoving particle which travels adistance dxthrough agaswhich hasnoscatterers (molecules) perunit volume (Fig. 43-1). Ifwelook ateach unitofarea perpendicular tothedirection ofmotion ofourselected particle, we willfindthere nodxmolecules. Ifeach onepresents aneffective collision area or, asitisusually called, “collision cross section,” 0,,thenthetotalareacovered by thescatterers isacno dx. 43-3collision oreo is0;, -.,‘~¢_¢; unit area i. o"/t Q 0 °<> Q 00,, ooa o 069 Total number ofmolecules Total oreo ooveredic o'n¢dx isn,,d: Fig.43-1. Collision cross section. By“collision cross section” wemean theareawithin which thecenter ofour particle must belocated ifitistocollide with aparticular molecule. Ifmolecules were little spheres (aclassical picture) wewould expect that0,=1r(r1 +r2)2, where r1andr2aretheradii ofthetwocolliding objects. Thechance thatour particle willhave acollision istheratio oftheareacovered byscattering molecules tothetotal area, which wehave taken tobeone. Sotheprobability ofacollision in going adistance dxisjust0,n0 dx: Chance ofacollision indx=0,n0 dx. (43.10) Wehave seenabove thatthechance ofacollision indxcanalsobewritten in terms ofthemean freepath lasdx/l. Comparing thiswith (43.10), wecanrelate themean freepath tothecollision cross section: §=0,n0, (43.11) which iseasier toremember ifwewrite itas 0,n0l =1. (43.12) This formula canbethought ofassaying thatthere should beonecollision, ontheaverage, when theparticle goes through adistance Iinwhich thescattering molecules could justcover thetotal area. Inacylindrical volume oflength Ianda base ofunitarea, there aren0lscatterers; ifeach onehasanarea0,thetotal area covered isn0l0,, which isjustoneunitofarea. Thewhole areaisnotcovered, of course, because some molecules arepartly hidden behind others. That iswhy some molecules gofarther than lbefore having acollision. Itisonlyontheaverage thatthemolecules have acollision bythetimetheygothedistance I.From measure- ments ofthemean freepath lwecandetermine thescattering cross section 0,,and compare theresult withcalculations based onadetailed theory ofatomic structure. Butthatisadifferent subject! Sowereturn totheproblem ofnonequilibrium states. 43-3 Thedriftspeed Wewant todescribe what happens toamolecule, orseveral molecules, which aredifferent insome wayfrom thelarge majority ofthemolecules inagas. We shall refer tothe“majority” molecules asthe“background” molecules, andwe shall callthemolecules which aredifferent from thebackground molecules “special” molecules or,forshort, theS-molecules. Amolecule could bespecial forany number ofreasons: Itmight beheavier than thebackground molecules. Itmight beadifferent chemical. Itmight have anelectric charge—i.e., beanioninaback- ground ofuncharged molecules. Because oftheir different masses orcharges the S-molecules may have forces onthem which arediflerent from theforces onthe background molecules. Byconsidering what happens tothese S-molecules wecan understand thebasic effects which come intoplayinasimilar wayinmany different phenomena. Tolistafew: thediflusion ofgases, electric currents inbatteries, sedimentation, centrifugal separation, etc. Webegin byconcentrating onthebasic process: anS-molecule inaback- ground gasisacted onbysome specific force F(which might be,e.g.,gravitational orelectrical) andinaddition bythenot-so-specific forces duetocollisions with the background molecules. Wewould liketodescribe thegeneral behavior ofthe S-molecule. What happens toit,indetail, isthatitdarts around hither andyonas itcollides over andover again with other molecules. Butifwewatch itcarefully weseethatitdoes make some netprogress inthedirection oftheforce F.Wesay thatthere isadriftsuperposed onitsrandom motion. Wewould liketoknow what thespeed ofitsdrift is—its drift velocity—due totheforce F. Ifwestart toobserve anS-molecule atsome instant wemayexpect thatitis somewhere between twocollisions. Inaddition tothevelocity itwasleftwithafter itslastcollision itispicking upsome velocity component duetotheforce F.Ina 43-4 short time (ontheaverage, inatime1)itwillexperience acollision andstart outon anewpiece ofitstrajectory. Itwillhave anewstarting velocity, butthesame ac- celeration from F. Tokeep things simple forthemoment, weshall suppose that after each collision ourS-molecule getsacompletely “fresh” start. That is,thatitkeeps no remembrance ofitspastacceleration byF.This might beareasonable assumption ifourS-molecule were much lighter than thebackground molecules, butitis certainly notvalid ingeneral. Weshall discuss later animproved assumption. Forthemoment, then, ourassumption isthat theS-molecule leaves each collision with avelocity which may beinanydirection with equal likelihood. Thestarting velocity willtake itequally inalldirections andwillnotcontribute toanynetmotion, soweshall notworry further about itsinitial velocity after a collision. Inaddition_ toitsrandom motion, each S-molecule willhave, atany moment, anadditional velocity inthedirection oftheforce F,which ithaspicked upsince itslastcollision. What istheaverage value ofthispartofthevelocity? Itisjusttheacceleration F/m (where misthemass oftheS-molecule) times the average time since thelastcollision. Now theaverage time since thelastcollision must bethesame astheaverage time untilthenextcollision, which wehave called -r,above. Theaverage velocity from F,ofcourse, isjustwhat iscalled thedrift velocity, sowehave therelation F4.1....-5 (43.13) This basic relation istheheart ofoursubject. There may besome complication indetermining what 1-is,butthebasic process isdefined byEq.(43.13). You willnotice thatthedrift velocity isproportional totheforce. There is, unfortunately, nogenerally used name fortheconstant ofproportionality. Differ- entnames have been used foreach different kind offorce. Ifinanelectrical prob- lemtheforce iswritten asthecharge times theelectric field, F=qE,then thecon- stant ofproportionality between thevelocity andtheelectric field Eisusually called the“mobility.” Inspite ofthepossibility ofsome confusion, weshall use theterm mobility fortheratio ofthedrift velocity totheforce foranyforce. We write Urlrift =MF (43-14) ingeneral, andweshall callitthemobility. Wehave from Eq.(43.13) that ]J.=1'/m. (43.15) Themobility isproportional tothemean time between collisions (there arefewer collisions toslow itdown) andinversely proportional tothemass (more inertia means lessspeed picked upbetween collisions). Togetthecorrect numerical coefficient inEq.(43.13), which iscorrect as given, takes some care. Without intending toconfuse, weshould stillpoint out thatthearguments have asubtlety which canbeappreciated onlybyacareful and detailed study. Toillustrate thatthere aredifficulties, inspite ofappearances, we shall make over again theargument which ledtoEq.(43.13) inareasonable but erroneous way(and thewayonewillfindinmany textbooksl). Wemight have said: Themean time between collisions is1'.After acollision theparticle starts outwith arandom velocity, butitpicks upanadditional velocity between collisions, which isequal totheacceleration times thetime. Since ittakes thetime1-toarrive atthenextcollision itgetsthere withthevelocity (F/m)*r. Atthe beginning ofthecollision ithadzerovelocity. Sobetween thetwocollisions ithas, ontheaverage, avelocity one-half ofthefinal velocity, sothemean drift velocity is%F'r/m. (Wrong!) This result iswrong andtheresult inEq.(43.13) isright, although thearguments may sound equally satisfactory. Thereason thesecond result iswrong issomewhat subtle, andhastodowiththefollowing: Theargument ismade asthough allcollisions were separated bythemean time 1.Thefactis thatsome times areshorter andothers arelonger than themean. Short times occur more often butmake lesscontribution tothedrift velocity because they have less 43-5 | b--Lt. 1 I o ° O 0 0 met{l\ 0o ° o O L} ° Q Area A ° o Q ° 0 Gaswith NIions L, O perunitvolume + ° o _ Insulator Tobattery with voltage V Fig.43-2. Electric current from an ionized gas.chance “toreally getgoing.” Ifonetakes proper account ofthedistribution of freetimes between collisions, onecanshow thatthere should notbethefactor Q thatwasobtained from thesecond argument. Theerror wasmade intrying to relate byasimple argument theaverage final velocity totheaverage velocity itself. This relationship isnotsimple, soitisbest toconcentrate onwhat iswanted: theaverage velocity itself. Thefirstargument wegave determines theaverage velocity directly—and correctly! Butwecanperhaps seenowwhyweshall notin general trytogetallofthecorrect numerical coefficients inourelementary deriva- tions! Wereturn nowtooursimplifying assumption thateachcollision knocks out allmemory ofthepast motion——that afresh start ismade after each collision. Suppose ourS-molecule isaheavy object inabackground oflighter molecules. Then ourS-molecule willnotloseits“forward” momentum ineach collision. It would take several collisions before itsmotion was“randomized” again. We should assume, instead, that ateach c0llision—in each time 1'ontheaverage —itloses acertain fraction ofitsmomentum. Weshall notwork outthedetails, butjuststate thattheresult isequivalent toreplacing -r,theaverage collision time, byanew—and longer—-'r which corresponds totheaverage “forgetting time,” i.e.,theaverage timetoforget itsforward momentum. With such aninterpretation ofrwecanuseourformula (43.15) forsituations which arenotquite assimple aswefirstassumed. 43-4 Ionic conductivity Wenowapply ourresults toaspecial case. Suppose wehave agasinavessel inwhich there arealsosome ions—atoms ormolecules with anetelectric charge. Weshow thesituation schematically inFig.43-2. Iftwoopposite walls ofthe container aremetallic plates, wecanconnect them totheterminals ofabattery andthereby produce anelectric field inthegas. Theelectric field willresult ina force ontheions, sothey willbegin todrift toward oneortheother oftheplates. Anelectric current willbeinduced, andthegaswith itsions willbehave likea resistor. Bycomputing theionflow from thedrift velocity wecancompute the resistance. Weask, specifically: How does theflow ofelectric current depend onthevoltage difference Vthatweapply across thetwoplates? Weconsider thecasethatourcontainer isarectangular boxoflength band cross-sectional areaA(Fig. 43-2). Ifthepotential difference, orvoltage, from one plate totheother isV,theelectric fieldEbetween theplates isV/b. (Theelectric potential isthework done incarrying aunitcharge from oneplate totheother. Theforce onaunitcharge isE.IfEisthesame everywhere between theplates, which isagood enough approximation fornow, thework done onaunitcharge isjustEb,soV=Eb.) Thespecial force onanionofthegasisqE,where qis thecharge ontheion. Thedrift velocity oftheionisthen ptimes thisforce, or Vlldi-ifl:=MF=ME=uq3- (43-16) Anelectric current Iistheflow ofcharge inaunittime. Theelectric current to oneoftheplates isgiven bythetotal charge oftheionswhich arrive attheplate in aunitoftime. Iftheionsdrift toward theplate with thevelocity vdm, thenthose which arewithin adistance (vd,,;,- T)willarrive attheplate inthetime T.If there aren,-ionsperunitvolume, thenumber which reach theplate inthetime Tis(n,--A-vdm, -T).Each ioncarries thecharge q,sowehave that Charge collected inT=qn,-Avd,,f,T. (43.17) Thecurrent Iisthecharge collected inTdivided byT,so I=q7l,'Al)d,-ift. 43-6 Substituting 220,0, from (43.16), wehave A1=,tq2n.-3V. (43.19) Wefindthatthecurrent isproportional tothevoltage, which isjusttheform of Ohm’s law, andtheresistance Ristheinverse oftheproportionality constant: 1 Ai=,u.q2n;-5- (43.20) Wehave arelation between theresistance andthemolecular properties n,-,q,and u,which depends inturn onmand1-.Ifweknow n,-andqfrom atomic measure- ments, ameasurement ofRcould beused todetermine ju,andfrom ualsoT. 43-5 Molecular diffusion Weturnnowtoadifferent kind ofproblem, andadifferent kind ofanalysis: thetheory ofdiffusion. Suppose that wehave acontainer ofgasinthermal equilibrium, andthatweintroduce asmall amount ofadifferent kind ofgasat some place inthecontainer. Weshall calltheoriginal gasthe“background” gas andthenewonethe“special” gas. Thespecial gaswillstart tospread outthrough thewhole container, butitwillspread slowly because ofthepresence oftheback- ground gas. This slow spreading-out process iscalled diffusion. Thediffusion is controlled mainly bythemolecules ofthespecial gasgetting knocked about bythe molecules ofthebackground gas. After alarge number ofcollisions, thespecial molecules endupspread outmore orlessevenly throughout thewhole volume. We must becareful nottoconfuse diffusion ofagaswith thegross transport thatmay occur duetoconvection currents. Most commonly, themixing oftwogases occurs byacombination ofconvection anddiffusion. Weareinterested now only inthe case thatthere areno“wind” currents. Thegasisspreading only bymolecular motions, bydiffusion. Wewish tocompute how fastdiffusion takes place. Wenow compute thenetflow ofmolecules ofthe“special” gasduetothe molecular motions. There willbeanetflow only when there issome nonuniform distribution ofthemolecules, otherwise allofthemolecular motions would average togivenonetflow. Letusconsider firsttheflow inthex-direction. Tofindthe flow, weconsider animaginary plane surface perpendicular tothex-axis andcount thenumber ofspecial molecules thatcross thisplane. Toobtain thenetflow, we must count aspositive those molecules which cross inthedirection ofpositive x andsubtract from thisnumber thenumber which cross inthenegative x-direction. Aswehave seen many times, thenumber which cross asurface areainatime AT isgiven bythenumber which start theinterval ATinavolume which extends the distance vATfrom theplane. (Note thatv,here, istheactual molecular velocity, notthedrift velocity.) Weshall simplify ouralgebra bygiving oursurface oneunitofarea. Then thenumber ofspecial molecules which passfrom lefttoright (taking the+x-direc- tiontotheright) isn_vAT,where n_isthenumber ofspecial molecules perunit volume totheleft(within afactor of2orso,butweareignoring such factorsl). Thenumber which cross from right toleftis,similarly, n+vAT,where n+isthe number density ofspecial molecules ontheright-hand sideoftheplane. Ifwe callthemolecular current J,bywhich wemean thenetflow ofmolecules perunit areaperunittime, wehave J=n_vATA—Tn+v AT, (43.21) OI‘ J=(n_-n+)v. (43.22) What shall weuseforn_andn+? When wesay“the density ontheleft,” howfartotheleftdowemean? Weshould choose thedensity attheplace from which themolecules started their “flight,” because thenumber which start such 43-7 trips isdetermined bythenumber present atthatplace. Sobyn_weshould mean thedensity adistance totheleftequal tothemean freepath I,andbyn+,thedensity atthedistance ltotheright ofourimaginary surface. Itisconvenient toconsider thatthedistribution ofourspecial molecules in space isdescribed byacontinuous function ofx,y,andzwhich weshall callna. Byn,,(x, y,z)wemean thenumber density ofspecial molecules inasmall volume element centered on(x,y,z).Interms ofn,wecanexpress thedifference (n+— n_)as d,, d,(n+-n_)=gzAx=-5;-21. (43.23) Substituting thisresult inEq.(43.22) andneglecting thefactor of2,weget dnJ,=—lvd): (43.24) Wehave found thattheflow ofspecial molecules isproportional tothederivative ofthedensity, ortowhat issometimes called the“gradient” ofthedensity. Itisclear thatwehave made several rough approximations. Besides various factors oftwowehave leftout,wehave used vwhere weshould have used 21,,and wehave assumed that n+andn_refer toplaces attheperpendicular distance I from oursurface, whereas forthose molecules which donottravel perpendicular tothesurface element, lshould correspond totheslant distance from thesurface. Allofthese refinements canbemade; theresult ofamore careful analysis shows thattheright-hand sideofEq.(43.24) should bemultiplied by1/3. Soabetter answer is J,=— (43.25) Similar equations canbewritten forthecurrents inthey-andz-directions. Thecurrent J,andthedensity gradient dn,/dx canbemeasured bymacroscopic observations. Their experimentally determined ratio iscalled the“diffusion co- efficient,” D.That is, dn,J,-—D-5)? (43.26) Wehave been abletoshow thatforagasweexpect D=glv. (43.27) Sofarinthischapter wehave considered twodistinct processes: mobility, thedrift ofmolecules dueto“outside” forces; anddiffusion, thespreading deter- mined only bytheinternal forces, therandom collisions. There is,however, a relation between them, since they both depend basically onthethermal motions, andthemean freepath lappears inboth calculations. If,inEq.(43.25), wesubstitute I=orand1'=pm,wehave J,=—31;mv2,u %. (43.28) Butmvzdepends only onthetemperature. Werecall that %mv2=gkr, (43.29)SO dn,J,-—;tkTa-;c-- (43.30) WefindthatD,thediffusion coeflicient, isjustkTtimes /.t,themobility coefficient: D=].l.kT. (43.31) And itturns outthatthenumerical coefficient in(43.31) isexactly right—no extra factors have tobethrown intoadjust forourrough assumptions. Wecanshow, 43-8 infact, that (43.31) must always becorrect—even incomplicated situations (forexample, thecaseofasuspension inaliquid) where thedetails ofoursimple calculations would notapply atall. Toshow that(43.31) must becorrect ingeneral, weshall derive itinadifl'erent way, using only ourbasic principles ofstatistical mechanics. Imagine asituation inwhich there isagradient of“special” molecules, andwehave adiffusion current proportional tothedensity gradient, according toEq.(43.26). Wenow apply a force fieldinthex-direction, sothateach special molecule feelstheforce F.Accord- ingtothedefinition ofthemobility juthere willbeadrift velocity given by Udrgft =,U.F. Byourusual arguments, thedrift current (thenetnumber ofmolecules which pass aunitofareainaunitoftime) willbe ',(ll‘lft =ndvflflfli! (43-33) or Jdrift ="aI~lF- (43-34) Wenowadjust theforce Fsothatthedriftcurrent duetoFjustbalances thediffu- sion, sothatthere isnonetflow ofourspecial molecules. Wehave J,+Jdm, =0, or d1.1)7,4;=Mr. (43.35) Under the“balance” conditions wefindasteady (with time) gradient ofdensity given by dn.,_midiU; ---D - (43.36) Butnotice! Wearedescribing anequilibrium condition, soourequilibrium laws ofstatistical mechanics apply. According tothese laws theprobability of finding amolecule atthecoordinate xisproportional toeTU/kT, where Uisthe potential energy. Interms ofthenumber density n,,,thismeans that n,=n0e“U/H. (43.37) Ifwedifferentiate (43.37) with respect tox,wefind dlla _ _U/101' L dU E — —l’l()€ ' 9 or 4”“__BaQ2E?“kTdx' (4349) Inoursituation, since theforce Fisinthex-direction, thepotential energy Uis just—Fx, and—dU/dx =F.Equation (43.39) then gives dn,_n,F-6,;-W- (43.40) [This isjustexactly Eq.(40.2), from which wededuced e'U/'°T inthefirstplace, sowehave come inacircle]. Comparing (43.40) with (43.36), wegetexactly Eq. (43.31). Wehave shown thatEq.(43.31), which gives thediffusion current interms ofthemobility, hasthecorrect coefficient andisverygenerally true. Mobility and diffusion areintimately connected. This relation wasfirstdeduced byEinstein. 43-6 Thermal conductivity Themethods ofthekinetic theory thatwehave been using above canbeused alsotocompute thethermal conductivity ofagas. Ifthegasatthetopofacon- tainer ishotter thanthegasatthebottom, heatwillflowfrom thetoptothebottom. (Wethink ofthetopbeing hotter because otherwise convection currents would be 43-9 setupandtheproblem would nolonger beoneofheat conduction.) Thetransfer ofheatfrom thehotter gastothecolder gasisbythediffusion ofthe“hot” mole- cules—those with more energy—downward andthediffusion ofthe“cold” mole- cules upward. Tocompute theflowofthermal energy wecanaskabout theenergy carried downward across anelement ofareabythedownward-moving molecules, andabout theenergy carried upward across thesurface bytheupward-moving molecules. Thedifference willgiveusthenetdownward flow ofenergy. Thethermal conductivity Kisdefined astheratio oftherateatwhich thermal energy iscarried across aunitsurface area, tothetemperature gradient: 1dQ dT‘Z-E" ——K-aE' Since thedetails ofthecalculations arequite similar tothose wehave done above inconsidering theflowofelectric current inanionized gas,weshall leave itasan exercise forthereader toshow that knlv where ('Y—1)kT istheaverage energy ofamolecule atthetemperature T. Ifweuseourrelation nl0,=1,theheat conductivity canbewritten as K=TE?‘ 2;:' Wehave arather surprising result. Weknow thattheaverage velocity ofgas molecules depends onthetemperature butnotonthedensity. Weexpect 0,to depend onlyonthesizeofthemolecules. Sooursimple result saysthatthethermal conductivity K(and therefore therateofflowofheatinanyparticular circumstance) isindependent ofthedensity ofthegas! Thechange inthenumber of“carriers” ofenergy with achange indensity isjustcompensated bythelarger distance the “carriers” cangobetween collisions. Onemay ask: “Istheheatflow independent ofthegasdensity inthelimit as thedensity goestozero? When there isnogasatall?” Certainly not! Theformula (43.43) wasderived, aswere alltheothers inthischapter, under theassumption thatthemean freepath between collisions ismuch smaller than anyofthedimen- sions ofthecontainer. Whenever thegasdensity issolowthatamolecule hasafair chance ofcrossing from onewall ofitscontainer totheother without having a collision, none ofthecalculations ofthischapter apply. Wemust insuch cases go back tokinetic theory andcalculate again thedetails ofwhat willoccur. 43-10 44 The Laws ofThermodynamics 44-1 Heat engines; thefirstlaw Sofarwehave been discussing theproperties ofmatter from theatomic point ofview, trying tounderstand roughly what willhappen ifwesuppose thatthings aremade ofatoms obeying certain laws. However, there areanumber ofrelation- ships among theproperties ofsubstances which canbeworked outwithout con- sideration ofthedetailed structure ofthematerials. Thedetermination ofthe relationships among thevarious properties ofmaterials, without knowing their internal structure, isthesubject ofthermodynamics. Historically, thermodynamics wasdeveloped before anunderstanding oftheinternal structure ofmatter was achieved. Togiveanexample: weknow from thekinetic theory thatthepressure ofagas iscaused bymolecular bombardment, andweknow thatifweheat agas,sothat thebombardment increases, thepressure must increase. Conversely, ifthepiston inacontainer ofthegasismoved inward against theforce ofbombardment, the energy ofthemolecules bombarding thepiston willincrease, andconsequently thetemperature willincrease. So,ontheonehand, ifweincrease thetemperature atagiven volume, weincrease thepressure. Ontheother hand, ifwecompress thegas,wewillfindthatthetemperature willrise. From thekinetic theory, one canderive aquantitative relationship between these twoeffects, butinstinctively onemight guess thattheyarerelated insome necessary fashion which isindepend- entofthedetails ofthecollisions. Letusconsider another example. Many people arefamiliar with thisinterest- ingproperty ofrubber: Ifwetakearubber band andpullit,itgetswarm. Ifone puts itbetween hislips, forexample, and pulls itout, hecanfeeladistinct warming, andthiswarming isreversible inthesense thatifherelaxes therubber band quickly while itisbetween hislips, itisdistinctly cooled. That means that when westretch arubber band itheats, andwhen werelease thetension oftheband itcools. Now ourinstincts might suggest thatifweheated aband, itmight pull: thatthefactthatpulling aband heats itmight imply thatheating aband should cause ittocontract. And, infact, ifweapply agasflame toarubber band holding aweight, wewillseethattheband contracts abruptly (Fig. 44-1). Soitistruethat when weheat arubber band itpulls, andthisfactisdefinitely related tothefact thatwhen werelease thetension ofit,itcools. Theinternal machinery ofrubber thatcauses these effects isquite complicated. Wewilldescribe itfrom amolecular point ofview tosome extent, although our main purpose inthischapter istounderstand therelationship ofthese effects independently ofthemolecular model. Nevertheless, wecanshow from themolec- ularmodel thattheeffects areclosely related. Onewaytounderstand thebehavior ofrubber istorecognize thatthissubstance consists ofanenormous tangle of long chains ofmolecules, akind of“molecular spaghetti,” with oneextra compli- cation: between thechains there arecross-1inks—like spaghetti thatissometimes welded together where itcrosses another piece ofspaghetti—a grand tangle. When wepulloutsuch atangle, some ofthechains tend tolineupalong thedirec- tionofthepull. Atthesame time, thechains areinthermal motion, sothey hit each other continually. Itfollows thatsuch achain, ifstretched, would notby itself remain stretched, because itwould behitfrom thesides bytheother chains andother molecules, andwould tend tokink upagain. Sotherealreason whya rubber band tends tocontract isthis: when onepulls itout,thechains arelength- wise, andthethermal agitations ofthemolecules onthesides ofthechains tend 44-144-1 Heat engines; thefirstlaw 44-2 Thesecond law 44-3 Reversible engines 44-4 Theefliciency ofanideal engine 44-5 Thethermodynamic temperature 44-6 Entropy ///////////////// §~>————<»- Fig.44-1. Theheated rubber band. ‘E Fig. engine.?~\>\/4?VA\ /////////////////////N 44—2. The rubber-bond heattokink thechains up,andmake them shorten. Onecanthen appreciate thatifthe chains areheld stretched andthetemperature isincreased, sothatthevigor ofthe bombardment onthesides ofthechains isalsoincreased, thechains tend topull in,andtheyareabletopullastronger weight when heated. If,after being stretched foratime, arubber band isallowed torelax, each chain becomes soft. andthe molecules striking itloseenergy asthey pound intotherelaxing chain. Sothe temperature falls. Wehave seen how these twoprocess, contraction when heated andcooling during relaxation, canberelated bythekinetic theory, butitwould beatremendous challenge todetermine from thetheory theprecise relationship between thetwo. Wewould have toknow howmany collisions there were each second andwhat the chains look like,andwewould have totakeaccount ofallkinds ofother complica- tions. Thedetailed mechanism issocomplex thatwecannot, bykinetic theory, really determine exactly what happens; still, adefinite relation between thetwo effects weobserve canbeworked outwithout knowing anything about theinternal machinery! Thewhole subject ofthermodynamics depends essentially upon thefollowing kind ofconsideration: because arubber band is“stronger” athigher temperatures than itisatlower temperatures, itought tobepossible toliftweights, andto move them around, andthustodowork with heat. Infact, wehave already seen experimentally thataheated rubber band canliftaweight. Thestudy oftheway thatonedoes work -with heat isthebeginning ofthescience ofthermodynamics. Canwemake anengine which usestheheating efi"ect onarubber band todowork’? Onecanmake asillylooking engine thatdoes justthis. Itconsists ofabicycle wheel inwhich allthespokes arerubber bands (Fig. 44-2). Ifoneheats therubber bands ononesideofthewheel with apairofheatlamps, theybecome “stronger” than therubber bands ontheother side. Thecenter ofgravity ofthewheel will bepulled tooneside, away from thebearing, sothatthewheel turns. Asitturns, cool rubber bands move toward theheat, andtheheated bands move away from theheat andcool, sothatthewheel turns slowly solong astheheat isapplied. Theefficiency ofthisengine isextremely low. Four hundred watts ofpower pour intothetwolamps, butitisjustpossible toliftaflywithsuch anengine! Aninter- esting question, however, iswhether wecangetheat todothework inmore efiicient ways. Infact, thescience ofthermodynamics began with ananalysis, bythegreat engineer Sadi Carnot, oftheproblem ofhow tobuild thebestandmost efiicient engine, andthisconstitutes oneofthefewfamous cases inwhich engineering has contributed fundamentally tophysical theory. Another example that comes tomind isthemore recent analysis ofinformation theory byClaude Shannon. These twoanalyses, incidentally, turn outtobeclosely related. Now thewayasteam engine ordinarily operates isthatheat from afireboils some water, andthesteam soformed expands andpushes onapiston which makes awheel goaround. Sothesteam pushes thepiston—what then? Onehastofinish thejob: astupid waytocomplete thecycle would betoletthesteam escape into theair,forthen onehastokeep supplying water. Itischeaper—more efi’1cient— toletthesteam gointoanother box,where itiscondensed bycoolwater, andthen pump thewater back intotheboiler, sothatitcirculates continuously. Heat is thus supplied totheengine andconverted intowork. Now would itbebetter to usealcohol? What property should asubstance have sothatitmakes thebest possible engine? That wasthequestion towhich Carnot addressed himself, and oneoftheby-products wasthediscovery ofthetype ofrelationship thatwehave justexplained above. Theresults ofthermodynamics areallcontained implicitly incertain appar- ently simple statements called thelawsofthermodynamics. Atthetimewhen Carnot lived, thefirstlawofthermodynamics, theconservation ofenergy, wasnotknown. Carnot’s arguments were socarefully drawn, however, that they arevalid even though thefirstlawwasnotknown inhistime! Some time afterwards, Clausius made asimpler derivation thatcould beunderstood more easily than Carnot’s very subtle reasoning. Butitturned outthatClausius assumed, nottheconserva- 44-2 tionofenergy ingeneral, butthatheat wasconserved according tothecaloric theory, which waslater shown tobefalse. Soithasoften been saidthatCarnot’s logic waswrong. Buthislogic wasquite correct. Only Clausius’ simplified ver- sion, thateverybody read, wasincorrect. Theso-called second lawofthermodynamics wasthus discovered byCarnot before thefirstlaw! Itwould beinteresting togiveCarnot’s argument thatdidnot usethefirstlaw, butweshall notdosobecause wewant tolearn physics, not history. Weshall usethefirstlawfrom thestart, inspite ofthefactthatagreat dealcanbedone without it. Letusbegin bystating thefirstlaw,theconservation ofenergy: ifonehasa system andputs heat intoit,anddoes work onit,then itsenergy isincreased by theheatputinandthework done. Wecanwrite thisasfollows: Theheat Qput intothesystem, plusthework Wdone onthesystem, istheincrease intheenergy Uofthesystem; thelatter energy issometimes called theinternal energy: Change inU=Q+W. (44.1) Thechange inUcanberepresented asadding alittle heatAQandadding alittle work AW: AU=AQ—l—AW, (44.2) which isadifferential form ofthesame law. Weknow thatvery well, from an earlier chapter. 44-2 Thesecond law Now, what about thesecond lawofthermodynamics? Weknow thatifwe dowork against friction, say,thework losttousisequal totheheat produced. Ifwedowork inaroom attemperature T,andwedothework slowly enough, the room temperature does notchange much, andwehave converted work intoheat atagiven temperature. What about thereverse possibility? Isitpossible toconvert theheat back into work atagiven temperature? The second lawofthermo- dynamics asserts thatitisnot. Itwould bevery convenient tobeabletoconvert heatintowork merely byreversing aprocess likefriction. Ifweconsider only the conservation ofenergy, wemight think thatheatenergy, such asthatinthevibra- tional motions ofmolecules, might provide agoodly supply ofuseful energy. ButCarnot assumed thatitisimpossible toextract theenergy ofheat atasingle temperature. Inother words, ifthewhole world were atthesame temperature, onecould notconvert anyofitsheatenergy intowork: while theprocess ofmaking work gointoheatcantake place atagiven temperature, onecannot reverse itto getthework back again. Specifically, Carnot assumed thatheatcannot betaken inatacertain temperature andconverted intowork withnoother change inthe system orthesurroundings. That lastphrase isvery important. Suppose wehave acanofcompressed airatacertain temperature, andwelettheairexpand. Itcandowork; itcanmake hammers go,forexample. Itcools offalittle intheexpansion, butifwehadabig sea,liketheocean, atagiven temperature—a heatreservoir—we could warm itup again. Sowehave taken theheat outofthesea,andwehave done work with the compressed air.ButCarnot wasnotwrong, because wedidnotleave everything as itwas. Ifwerecompress theairthatweletexpand, wewillfindwearedoing extra work, andwhen wearefinished wewilldiscover thatwenotonly gotno work outofthesystem attemperature T,butweactually putsome in.Wemust talkonlyabout situations inwhich thenetresult ofthewhole process istotakeheat away andconvert itintowork, justasthenetresult oftheprocess ofdoing work against friction istotakework andconvert itintoheat. Ifwemove inacircle, we canbring thesystem back precisely toitsstarting point, with thenetresult that wedidwork against friction andproduced heat. Can wereverse theprocess? Turn aswitch, sothateverything goesbackwards, sothefriction does work against us,andcools thesea? According toCarnot: no! Soletussuppose thatthisis impossible. 44—3 W Fig.44-3. Heat engine. Fig.44-4. Reversible heat transfer.Ifitwere possible itwould mean, among other things, thatwecould take heatoutofacold body andputitintoahotbody atnocost, asitwere. Now we know itisnatural thatahotthing canwarm upacool thing; ifwesimply puta hotbody andacoldonetogether, andchange nothing else,ourexperience assures usthatitisnotgoing tohappen thatthehotonegetshotter, andthecoldonegets colder! Butifwecould obtain work byextracting theheat outoftheocean, say, orfrom anything elseatasingle temperature, then thatwork could beconverted back intoheatbyfriction atsome other temperature. Forinstance, theother arm ofaworking machine could berubbing something thatisalready hot. Thenet result would betotake heat from a“cold” body, theocean, andtoputitintoa hotbody. Now, thehypothesis ofCarnot, thesecond lawofthermodynamics, is sometimes stated asfollows: heatcannot, ofitself, fiowfrom acoldtoahotobject. But,aswehave justseen, these twostatements areequivalent: first, thatonecan- notdevise aprocess whose onlyresult istoconvert heattowork atasingle temper- ature, andsecond, thatonecannot make heat flow byitself from acold toahot place. Weshall mostly usethefirstform. Carnot’s analysis ofheat engines isquite similar totheargument thatwe gave about weight-lifting engines inourdiscussion oftheconservation ofenergy inChapter 4.Infact, thatargument waspatterned after Carnot’s argument about heat engines, andsothepresent treatment willsound very much thesame. Suppose webuild aheatengine thathasa“boiler” somewhere atatempera- tureT1.Acertain heat Q1istaken from theboiler, thesteam engine does some work W,anditthendelivers some heat Q2intoa“condenser” atanother tempera- tureT2(Fig. 44-3). Carnot didnotsayhowmuch heat, because hedidnotknow thefirstlaw,andhedidnotusethelawthat Q2wasequal toQ1because hedid notbelieve it.Although everybody thought that, according tothecaloric theory, theheats Q1andQ2would have tobethesame, Carnot didnotsaytheywere the same—-that ispartofthecleverness ofhisargument. Ifwedousethefirstlaw,we findthattheheat delivered, Q2,istheheat Q1thatwasputinminus thework W thatwasdone: Q2=Q1—W. (44.3) (Ifwehave some kind ofcyclic process where water ispumped back into the boiler after itiscondensed, wewillsaythatwehave heat Q1absorbed andwork Wdone, during each cycle, foracertain amount ofwater thatgoes around the cycle.) Now weshall build another engine, andseeifwecannot getmore work from thesame amount ofheatbeing delivered atthetemperature T1,with thecondenser stillatthetemperature T2.Weshall usethesame amount ofheat Q1from the boiler, andweshall trytogetmore work than wedidoutofthesteam engine, perhaps byusing another fluid, such asalcohol. 44-3 Reversible engines Now wemust analyze ourengines. Onething isclear: wewilllosesomething iftheengines contain devices inwhich there isfriction. Thebestengine willbea frictionless engine. Weassume, then, thesame idealization thatwedidwhen we studied theconservation ofenergy; thatis,aperfectly frictionless engine. Wemust alsoconsider theanalog offrictionless motion, “frictionless” heat transfer. Ifweputahotobject atahigh temperature against acold object, sothat theheatflows, then itisnotpossible tomake thatheatflow inareverse direction byavery small change inthetemperature ofeither object. Butwhen wehave a practically frictionless machine, ifwepush itwith alittle force oneway, itgoes thatway, andifwepush itwith alittle force theother way, itgoes theother way. Weneed tofindtheanalog offrictionless motion: heattransfer whose direction we canreverse with onlyatinychange. Ifthedifference intemperature isfinite, thatis impossible, butifonemakes sure thatheat flows always between twothings at essentially thesame temperature, with justaninfinitesimal difierence tomake it flowinthedesired direction, theflowissaidtobereversible (Fig. 44-4). Ifweheat 44-4 theobject ontheleftalittle, heatwillflow totheright; ifwecool italittle, heat willflow totheleft. Sowefindthattheideal engine isaso-called reversible engine, inwhich every process isreversible inthesense that,byminor changes, infinitesimal changes, wecanmake theengine gointheopposite direction. That means that nowhere inthemachine must there beanyappreciable friction, andnowhere in themachine must there beanyplace where theheatofthereservoirs, ortheflame oftheboiler, isindirect contact with something definitely cooler orwarmer. Letusnowconsider anidealized engine inwhich alltheprocesses arereversible. Toshow thatsuch athing ispossible inprinciple, wewillgiveanexample ofan engine cycle which may ormay notbepractical, butwhich isatleast reversible, inthesense ofCarnot’s idea. Suppose thatwehave agasinacylinder equipped withafrictionless piston. Thegasisnotnecessarily aperfect gas.Thefluid doesnot even have tobeagas,buttobespecific letussaywedohave aperfect gas. Also, suppose thatwehave twoheatpads, T1andT2-—great bigthings thathave definite temperatures, T1andT2.Wewillsuppose inthiscasethatT1ishigher than T2. Letusfirstheatthegasandatthesame time expand it,while itisincontact with theheat padatT1.Aswedothis, pulling thepiston outvery slowly astheheat flows intothegas,wewillmake sure thatthetemperature ofthegasnever gets very farfrom T1.Ifwepullthepiston outtoofast,thetemperature ofthegaswill falltoomuch below T1andthen theprocess willnotbequite reversible, butif wepullitoutslowly enough, thetemperature ofthegaswillnever depart much from T1.Ontheother hand, ifwepush thepiston back slowly, thetemperature would beonly infinitesimally higher than T1,andtheheatwould pour back. We seethat such anisothermal (constant-temperature) expansion, done slowly and gently enough, isareversible process. Tounderstand what wearedoing, weshall useaplot(Fig. 44-6) ofthepressure ofthegasagainst itsvolume. Asthegasexpands, thepressure falls. Thecurve marked (1)tells ushow thepressure andvolume change ifthetemperature is kept fixed atthevalue T1. Foranideal gasthiscurve would bePV=NkT1. During anisothermal expansion thepressure falls asthevolume increases until westopatthepoint b.Atthesame time, acertain heat Q1must flowintothegas from thereservoir, forifthegaswere expanded without being incontact with the reservoir itwould cooloff,aswealready know. Having completed theisothermal expansion, stopping atthepoint b,letustakethecylinder away from thereservoir andcontinue theexpansion. This time wepermit noheat toenter thecylinder. Again weperform theexpansion slowly, sothere isnoreason why wecannot reverse it,andweagain assume there isnofriction. Thegascontinues toexpand andthetemperature falls, since there isnolonger anyheat entering thecylinder. Weletthegasexpand, following thecurve marked (2),until thetemperature fallstoT2,atthepoint marked c.This kind ofexpansion, made without adding heat, iscalled anadiabatic expansion. Foranideal gas,wealready know that curve (2)hastheform PV" =constant, where Visaconstant greater than l,so thattheadiabatic curve hasamore negative slope than theisothermal curve. The gascylinder hasnow reached thetemperature T2,sothatifweputitontheheat padattemperature T2there willbenoirreversible changes. Now weslowly com- press thegaswhile itisincontact with thereservoir atT2,following thecurve marked (3)(Fig. 44-5, Step2).Because thecylinder isincontact withthereservoir, thetemperature does notrise,butheat Q2flows from thecylinder intothereservoir atthetemperature T2.Having compressed thegasisothermally along curve (3) tothepoint d,weremove thecylinder from theheat padattemperature T2and compress itstillfurther, without letting anyheat flow out. Thetemperature will rise,andthepressure willfollow thecurve marked (4).Ifwecarry outeach step properly, wecanreturn tothepoint aattemperature T1where westarted, and repeat thecycle. Weseethatonthisdiagram wehave carried thegasaround acomplete cycle, andduring onecycle wehave putQ1inattemperature T1,andhave removed Q2 attemperature T2.Now thepoint isthatthiscycle isreversible, sothatwecould represent allthesteps theother way around. Wecould have gone backwards instead offorwards: wecould have started atpoint a,attemperature T1,expanded 44-5V T??? I Ti T2 Step (1) Iiathn-all expansion It11, absorb heat Q1 Step(2)Milbatlc expuuton; tQpex-lure run nu1'1z12 sap(3)1|at.hor—l ctlwellloa at1'2,deliver hut.Q2 Step (lo)Adiubattc cusp-anion, tune:-ntm-e risen h-an T2toT1 Fig.44-5. Steps inCarnot cycle PressureVI//I W, W,Ti T2 VII; [Wm Qi TI T2 7///I Wm '7/7;//I Ti T2 O $\“‘\"T ..\\\‘X/Z\(T \ mg: T=T| / Useful bd /\Work T=T m\\ 2‘(Sic W \ \\\ \\Volume Fig. 44-6. TheCarnot cycle. litW’-W Uuehll Hark 01-W 0,-w’ Fig. 44-7. Reversible engine Abeing driven backwards byengine B.along thecurve (4),expanded further atthetemperature F2,absorbing heat Q2, andsoon,going around thecycle backward. Ifwegoaround thecycle inonedirec- tion, wemust dowork onthegas;ifwegointheother direction, thegasdoes work onus. Incidentally, itiseasy tofindoutwhat thetotal amount ofwork is,because thework during anyexpansion isthepressure times thechange involume, jPdV. Onthisparticular diagram, wehave plotted Pvertically andVhorizontally. So ifwecallthevertical distance yandthehorizontal distance x,thisisjydx-—in other words, thearea under thecurve. Sothearea under each ofthenumbered curves isameasure ofthework done byoronthegasinthecorresponding step. Itiseasy toseethatthenetwork done istheshaded area ofthepicture. Now thatwehave given asingle example ofareversible machine, weshall suppose thatother such engines arealsopossible. Letusassume thatwehave a reversible engine Awhich takes Q1atT1,does work W,anddelivers some heatat T2.Now letusassume wehave anyother engine B,made byman, already designed ornotyetinvented, made ofrubber bands, steam, orwhatever, reversible ornot, which isdesigned sothatittakes inthesame amount ofheat Q1atT1,andrejects theheat atthelower temperature T2(Fig. 44-7). Assume thatengine Bdoes some work, W’.Now weshall show thatW’isnotgreater than W—that noengine candomore work than areversible one. Why’? Suppose that, indeed, W’were bigger than W.Then wecould take theheat Q1outofthereservoir atT1,and with engine Bwecould dowork W’anddeliver some heat tothereservoir atT2; wedonotcarehowmuch. That done, wecould savesome ofthework W’,which is supposed tobegreater than W;wecould useapartofit,W,andsavetheremainder, W’—-W,foruseful work. With thework Wwecould runengine Abackwards because itisareversible engine. Itwillabsorb some heat from thereservoir at T2anddeliver Q1back tothereservoir atT1. After thisdouble cycle, thenet result would bethatwewould have puteverything back thewayitwasbefore, and wewould have done some excess work, namely W’—W,andallwewould have done would betoextract energy from thereservoir atT2!Wewere careful tore- store theheat Q1tothereservoir atT1.Sothatreservoir canbesmall and“inside” ourcombined machine A+B,whose neteffect istherefore toextract anetheat W’—Wfrom thereservoir atT2andconvert itintowork. Buttoobtain useful work from areservoir atasingle temperature withnoother changes isimpossible according toCarnot’s postulate; itcannot bedone. Therefore noengine which absorbs agiven amount ofheatfrom ahigher temperature T1anddelivers itatthe temperature T2candomore work than areversible engine operating under the same temperature conditions. Now suppose thatengine Bisalsoreversible. Then, ofcourse, notonly must W’benotgreater than W,butnowwecanreverse theargument andshow thatW cannot begreater than W’. So,ifboth engines arereversible they must both do thesame amount ofwork, andwethus come toCarnot’s brilliant conclusion: thatifanengine isreversible, itmakes nodifference howitisdesigned, because the amount ofwork onewillobtain iftheengine absorbs agiven amount ofheat at temperature T1anddelivers heat atsome other temperature T2does notdepend onthedesign oftheengine. Itisaproperty oftheworld, notaproperty ofaparticu- larengine. Ifwecould findoutwhat thelawisthatdetermines how much work weobtain when weabsorb theheat Q1atT1anddeliver heatatT2,thisquantity would bea universal thing, independent ofthesubstance. Ofcourse ifweknew theproperties ofaparticular substance, wecould work itoutandthensaythatallother substances must givethesame amount ofwork inareversible engine. That isthekeyidea, the cluebywhich wecanfindtherelationship between howmuch, forinstance, arub- berband contracts when weheatit,andhowmuch itcools when weletitcontract. Imagine thatweputthatrubber band inareversible machine, andthatwemake it goaround areversible cycle. Thenetresult, thetotal amount ofwork done, is that universal function, that great function which isindependent ofsubstance. Soweseethat asubstance’s properties must belimited inacertain way; one 44-6 cannot make upanything hewants, orhewould beabletoinvent asubstance which hecould usetoproduce more than themaximum allowable work when hecarried itaround areversible cycle. This principle, thislimitation, istheonlyrealrulethat comes outofthethermodynamics. 44-4 Theefliciency ofanideal engine Now weshall trytofindthelawwhich determines thework Wasafunction ofQ1,T1,andT2.Itisclear that Wisproportional toQ1,forifweconsider two reversible engines inparallel, both working together andboth double engines, the combination isalsoareversible engine. Ifeach oneabsorbed heat Q1,thetwoto- gether absorb 2Q1andthework done is2W,andsoon.Soitisnotunreasonable thatWis proportional toQ1. Now thenext important stepistofindthisuniversal law. Wecan,andwill, dosobystudying areversible engine with theoneparticular substance whose laws weknow, aperfect gas. Itisalsopossible toobtain therulebyapurely logical argument, using noparticular substance atall. This isoneofthevery beautiful pieces ofreasoning inphysics andwearereluctant nottoshow itto you, soforthose who would liketoseeitweshall discuss itinjustamoment. Butfirstweshall usethemuch lessabstract andsimpler method ofdirect calcula- tionforaperfect gas. Weneed only obtain formulas forQ1andQ2(forWisjustQ1-Q2), the heats exchanged with thereservoirs during theisothermal expansion orcon- traction. Forexample, how much heat Q1isabsorbed from thereservoir at temperature T1during theisothermal expansion [marked (l)in Fig.44-6] from point a,atpressure pa,volume Va,temperature T1,topoint bwith pressure p1,,volume V1,,andthesame temperature T1?Foraperfect gaseach molecule hasanenergy thatdepends only onthetemperature, andsince thetemperature andthenumber ofmolecules arethesame ataandatb,theinternal energy isthesame. There is nochange inU;allthework done bythegas, W=£bpdV, during theexpansion isenergy Q1taken from thereservoir. During theexpansion, pV=NkT1, or NkT1p=7- O1‘ b b dVQ1=pdV= NkT17 (44.4) OI‘ VQ1=NkT11117: istheheattaken from thereservoir atT1.Inthesame way, forthecompression at T2[curve (3)ofFig.44-6] theheat delivered tothereservoir atT2is Q2=NkT2ln-;§~ (44.5) Tofinish ouranalysis weneed only findarelation between V,/Vd and V1,/V,,. This wedobynoting that(2)isanadiabatic expansion from btoc,during which pV" isaconstant. Since pV=NkT, wecanwrite thisas(pV)V"_1 =const or, interms ofTandV,asTV’_' =const, or T11/Z-1 =T21/Z-1. (44.6) Likewise, since (4),theexpansion from dtoa,isalsoadiabatic, wefind T1V,I—1 =T21/1-1. (44.6a) 44-7 \\Q2; //////I/-1-I ta-»Q»i'__‘2% T: WIS ws I; Q: //)//////// E Fig.44-8. Engines land 2together areequivalent toengine 3.Ifwedivide thisequation bytheprevious one, wefindthat V1,/Va must equal Vc/Vd,sotheln’sin(44.4) and(44.5) areequal, andthat 2_Q.T11-T: (44.7) This istherelation wewere seeking. Although proved foraperfect gasengine, weknow itmust betrueforanyreversible engine atall. Now weshall seehow thisuniversal lawcould alsobeobtained bylogical argument, without knowing theproperties ofanyspecific substances, asfollows. Suppose that wehave three engines andthree temperatures, letussayT1,T2, andT3.Letoneengine absorb heat Q1from thetemperature T1anddoacertain amount ofwork W13, andletitdeliver heat Q3tothetemperature T3(Fig. 44-8). Letanother engine runbackwards between T2andT3. Suppose thatweletthe second engine beofsuch asizethatitwillabsorb thesame heat Q3,anddeliver theheat Q2.Wewillhave toputacertain amount ofwork, W32,intoit—negative because theengine isrunning backwards. When thefirstmachine goes through a cycle, itabsorbs heat Q1anddelivers Q3atthetemperature T3;then thesecond machine takes thesame heat Q3outofthereservoir atthetemperature T3and delivers itinto thereservoir attemperature T2. Therefore thenetresult ofthe twomachines intandem istotake theheat Q1from T1,anddeliver Q2atT2. Thetwomachines arethus equivalent toathird one, which absorbs Q1atT1, does work W12, and delivers heat Q2atT2,because W12 =W13 —W32, asonecanimmediately show from thefirstlaw,asfollows: W13 —W32 =(Q1 '-Q3)"(Q2 —Q3)=Q1-Q2=W12- (44-3) Wecannow obtain thelaws which relate theefiiciencies oftheengines, because there clearly must besome kind ofrelationship between theefficiencies ofengines running between thetemperatures T1andT3,andbetween T2andT3,andbe- tween T1andT2. Wecanmake theargument very clear inthefollowing way: Wehave just seenthatwecanalways relate theheatabsorbed atT1totheheat delivered atT2 byfinding theheat delivered atsome other temperature T3. Therefore wecan getalltheengines’ properties ifweintroduce astandard temperature, analyzing everything with thatstandard temperature. Inother words, ifweknew theeffi- ciency ofanengine running between acertain temperature Tandacertain arbi- trary standard temperature, then wecould work outtheefliciency foranyother difference intemperature. Because weassume weareusing onlyreversible engines, wecanwork from theinitial temperature down tothestandard temperature and back uptothefinal temperature again. Weshall define thestandard temperature arbitrarily asonedegree. Weshall alsoadopt aspecial symbol fortheheatwhich isdelivered atthisstandard temperature: weshall callitQS.Inother words, when areversible engine absorbs theheat Qattemperature T,itwilldeliver, attheunit temperature, aheat Q3. Ifoneengine, absorbing heat Q1atT1,delivers theheat Q3atonedegree, andifanengine absorbing heat Q2attemperature T2willalso deliver thesame heat Q,gatonedegree, thenitfollows thatanengine which absorbs heat Q1atthetemperature T1willdeliver heat Q2ifitrunsbetween T1andT2,as wehave already proved byconsidering engines running between three tempera- tures. Soallwereally have todoistofindhow much heat Q1weneed toputin atthetemperature T1inorder todeliver acertain amount ofheat Q5attheunit temperature. Ifwediscover that, wehave everything. Theheat Q,ofcourse, is afunction ofthetemperature T.Itiseasytoseethattheheatmust increase asthe temperature increases, forweknow thatittakes work torunanengine backwards anddeliver heatatahigher temperature. Itisalsoeasytoseethattheheat Q1must beproportional toQ5.Sothegreat lawissomething likethis: foragiven amount ofheat Q5delivered atonedegree from anengine running attemperature T degrees, theheat Qabsorbed must bethatamount Q5times some increasing func- tionofthetemperature: Q=Qsf(T)- (44-9) 44-8 44-5 Thethermodynamic temperature Atthisstage wearenotgoing totrytofindtheformula fortheabove in- creasing function ofthetemperature interms ofourfamiliar mercury temperature scale, butinstead weshall define temperature byanewscale. Atonetime “the temperature” wasdefined arbitrarily bydividing theexpansion ofwater intoeven degrees ofacertain size. Butwhen onethen measures temperature with amercury thermometer, onefinds that thedegrees arenolonger even. Butnow wecan make adefinition oftemperature which isindependent ofanyparticular substance. Wecanusethatfunction f(T), which does notdepend onwhat device weuse,be- cause theefficiency ofthese reversible engines isindependent oftheir working sub- stances. Since thefunction wefound isrising with temperature, wewilldefine the function itself asthetemperature, measured inunits ofthestandard one-degree temperature, asfollows: Q=ST, (44.10) where Q3=S-1°. (44.11) This means thatwecantellhow hotanobject isbyfinding outhowmuch heatis absorbed byareversible engine working between thetemperature oftheobject andtheunittemperature (Fig. 44-9). Ifseven times more heat istaken outofa boiler than isdelivered ataone-degree condenser, thetemperature oftheboiler willbecalled seven degrees, andsoforth. So,bymeasuring how much heat is absorbed atdifferent temperatures, wedetermine thetemperature. Thetempera- turedefined inthiswayiscalled theabsolute thermodynamic temperature, andit isindependent ofthesubstance. Weshall usethisdefinition exclusively from now on.* Now weseethatwhen wehave twoengines, oneworking between T1and onedegree, theother working between T2andonedegree, delivering thesame heat atunittemperature, then theheats absorbed must berelated by Q__Q -T_11-s_-if (44.12) Butthatmeans thatifwehave asingle engine running between T1andT2,then theresult ofthewhole analysis, thegrand finale, isthat Q1istoT1asQ2istoT2, iftheengine absorbs energy Q1attemperature T1anddelivers heat Q2attemper- ature T2.Whenever theengine isreversible, thisrelationship between theheats must follow. That isallthere istoit:thatisthecenter oftheuniverse ofthermo- dynamics. Ifthisisallthere istothermodynamics, whyisitconsidered such adifficult subject‘? Indoing aproblem involving agiven mass ofsome substance, thecondi- tionofthesubstance atanymoment canbedescribed bytelling what itstempera- tureisandwhat itsvolume is.Ifweknow thetemperature andvolume ofasub- stance, andthat thepressure issome function ofthetemperature andvolume, thenweknow theinternal energy. Onecould say,“Idonotwant todoitthatway. Tellmethetemperature andthepressure, andIwilltellyouthevolume. Ican think ofthevolume asafunction oftemperature andpressure, andtheinternal energy asafunction oftemperature andpressure, andsoon.” That iswhythermo- dynamics ishard, because everyone usesadiflerent approach. Ifwecould only sit down once anddecide onourvariables, andstick tothem, itwould befairly easy. Now westart tomake deductions. JustasF9maisthecenter oftheuni- verse inmechanics, anditgoes onandonandonafter that, inthesame waythe principle justfound isallthere istothermodynamics. Butcanonemake conclu- sions outofit? *Wehave previously defined ourscale oftemperature inadifferent way, namely by stating thatthemean kinetic energy ofamolecule inaperfect gasisproportional tothe temperature, orthattheperfect gaslawsayspVisproportional toT.Isthisnewdefinition equivalent? Yes,since thefinalresult (44.7) derived from thegaslawisthesame adthat derived here. Weshall discuss thispoint again inthenext chapter. 44-9/////4y///// 051' T a,=s-1° 777777!7777—r"" Fig. 44-9. Absolute thermodynamic temperature.w-o-s-1‘ Webegin. Toobtain ourfirstconclusion, weshall combine both laws, the lawofconservation ofenergy andthislawwhich relates theheats Q2and Q1, andwecaneasily obtain theefiiciency ofareversible engine. From thefirstlaw, wehave W=Q1—Q2.According toournewprinciple, _T2Q2—T1Q1, sothework becomes W=Q1(1- =Q1 (44.13) which tellsustheefficiency oftheengine—how much work wegetoutofsomuch heat. Theefliciency ofanengine isproportionalto thedifference inthetempera- tures between which theengine runs, divided bythehigher temperature: . _W_T1—~T2EffiClCI1Cy —-E; —T ' Theefficiency cannot begreater than unity andtheabsolute temperature cannot belessthan zero, absolute zero. So,since T2must bepositive, theefficiency is always lessthan unity. That isourfirstconclusion. 44-6 Entropy Equation (44.7) or(44.12) canbeinterpreted inaspecial way. Working always with reversible engines, aheat Q1attemperature T1is“equivalent” to Q2atT2ifQ1/T1=Q2/T2, inthesense thatasoneisabsorbed theother is delivered. This suggests thatifwecallQ/Tsomething, wecansay:inareversible process asmuch Q/T isabsorbed asisliberated; there isnogain orlossofQ/T. This Q/T iscalled entropy, andwesay“there isnonetchange inentropy ina reversible cycle.” IfTis1°,then theentropy isQ/1°or,aswesymbolized it, Q3/1°=S.Actually, Sistheletter usually used forentropy, anditisnumerically equal totheheat (which wehave called Q5)delivered toa1°-reservoir (entropy is notitself aheat, itisheatdivided byatemperature, hence itismeasured injoules perdegree). Now itisinteresting that besides thepressure, which isafunction ofthe temperature andthevolume, andtheinternal energy, which isafunction of temperature andvolume, wehave found another quantity which isafunction of thecondition, i.e.,theentropy ofthesubstance. Letustrytoexplain how we compute it,andwhat wemean when wecallita“function ofthecondition.” Consider thesystem intwodiflerent conditions, much aswehadintheexperiment where wedidtheadiabatic andisothermal expansions. (Incidentally, there isno need thataheatengine have only tworeservoirs, itcould have three orfourdiffer- enttemperatures atwhich ittakes inanddelivers heats, andsoon.) Wecan move around onapVdiagram allover theplace, andgofrom onecondition to another. Inother words, wecould saythegasisinacertain condition a,andthen itgoes over tosome other condition, b,andwewillrequire thatthistransition, made from atob,bereversible. Now suppose thatallalong thepath from atob wehave little reservoirs atdifferentltemperatures, sothattheheat dQremoved from thesubstance ateach little stepisdelivered toeach reservoir atthetempera- ture corresponding ‘tothat point onthepath. Then letusconnect allthese reservoirs, byreversible heatengines, toasingle reservoir attheunittemperature. When wearefinished carrying thesubstance from atob,weshall bring allthe reservoirs back totheir original condition. Any heat dQthathasbeen absorbed from thesubstance attemperature Thasnow been converted byareversible machine, andacertain amount ofentropy dShasbeen delivered attheunit temperature asfollows: dS=dQ/T. (44.15) Letuscompute thetotal amount ofentropy which hasbeen delivered. The entropy difference, ortheentropy needed togofronra tobbythisparticular 44-10 reversible transformation, isthetotal entropy, thetotal oftheentropy taken out ofthelittle reservoirs, anddelivered attheunittemperature: b Sb—S11=/ (44.16) Thequestion is,does theentropy difference depend upon thepath taken? There ismore than onewaytogofrom atob.Remember thatintheCarnot cycle we could gofrom atocinFig.44-6 byfirstexpanding isothermally andthen adi- abatically; orwecould firstexpand adiabatically andthen isothermally. Sothe question iswhether theentropy change which occurs when wegofrom atobin Fig.44-10 isthesame ononeroute asitisonanother. Itmust bethesame, be- cause ifwewent alltheway around thecycle, going forward ononepath and backward onanother, wewould have areversible engine, andthere would beno lossofheat tothereservoir atunittemperature. Inatotally reversible cycle, no heat must betaken from thereservoir attheunit temperature, sotheentropy needed togofrom atobisthesame over onepath asitisover another. Itis independent ofpath, anddepends only ontheendpoints. Wecan, therefore, say thatthere isacertain function, which wecalltheentropy ofthesubstance, that depends only onthecondition, i.e.,only onthevolume andtemperature. Wecanfindafunction S(V,T)which hastheproperty thatifwecompute the change inentropy, asthesubstance ismoved along anyreversible path, interms oftheheat rejected atunittemperature, then AS= (44.17) where dQistheheat removed from thesubstance attemperature T.This total entropy change isthedifference between theentropy calculated attheinitial and final points: AS=S(V1,,T1,) -S(V11,T,1) =fbd%- (44.18) This expression does notcompletely define theentropy, butrather only thed1fler- ence ofentropy between twodifferent conditions. Only ifwecanevaluate the entropy foronespecial condition canwereally define Sabsolutely. Foralong time itwasbelieved thatabsolute entropy meant nothing—that only differences could bedefined—but finally Nernst proposed what hecalled theheattheorem, which isalsocalled thethird lawofthermodynamics. Itisvery simple. Wewillsaywhat itis,butwewillnotexplain why itistrue. Nernst’s postulate states simply that theentropy ofanyobject atabsolute zero iszero. Weknow ofonecase ofTandV,namely T=0,where Siszero; andsowecan gettheentropy atanyother point. Togiveanillustration ofthese ideas, letuscalculate theentropy ofaperfect gas. Inanisothermal (and therefore reversible) expansion, jdQ/T isQ/T, since Tisconstant. Therefore (from 44.4) thechange inentropy is S(V,1,T) -s(V1,,T)=Nklnb soS(V, T)=NklnVplussome function ofTonly. How does Sdepend onT? Weknow thatforareversible adiabatic expansion, noheat isexchanged. Thus theentropy does notchange even though Vchanges, provided thatTchanges also, such thatTVl_' =constant. Canyouseethatthisimplies that S(V,T) =Nk[lnV+ 7—i—flnT] +a, where aissome constant independent ofboth VandT?[aiscalled thechemical constant. Itdepends onthegasinquestion, andmaybedetermined experimentally from theNernst theorem bymeasuring theheatliberated incooling andcondensing 44-llTemperatureb Reservoirs a dW Enqinu if-1 ‘O Volumc Fig. 44—lO. Change inentropy dur- ingareversible transformation. 45'5r5o b TemperatureDA5- Sb-$0 Total Entropy Change -O Volumc Fig.44-1 l.Change inentropy ina completely reversible cycle. thegasuntil itisbrought toasolid (orforhelium, aliquid) at0°,byintegrating jdQ/T. Itcanalsobedetermined theoretically bymeans ofPlanck’s constant andquantum mechanics, butweshall notstudy itinthiscourse.] Now weshall remark onsome oftheproperties oftheentropy ofthings. We firstremember thatifwegoalong areversible cycle from atob,then theentropy ofthesubstance willchange byS1,—S,1.And weremember thataswegoalong thepath, theentropy-the heat delivered atunittemperature-—increases accord- ingtotheruledS=dQ/T, where dQistheheat weremove from thesubstance when itstemperature isT. Wealready know that ifwehave areversible cycle, thetotal entropy of everything isnotchanged, because theheat Q1absorbed atT1andtheheat Q2 delivered atT2correspond toequal andopposite changes inentropy, sothatthe netchange intheentropy iszero. Soforareversible cycle there isnochange in theentropy ofanything, including thereservoirs. This rulemay look likethe conservation ofenergy again, butitisnot; itapplies only toreversible cycles. Ifweinclude irreversible cycles there isnolawofconservation ofentropy. Weshall givetwoexamples. First, suppose thatwedoirreversible work on anobject byfriction, generating aheat Qonsome object attemperature T.The entropy isincreased byQ/T. Theheat Qisequal tothework, andthus when wedoacertain amount ofwork byfriction against anobject whose temperature isT,theentropy ofthewhole world increases byW/T. Another example ofirreversibility isthis: Ifweputtogether twoobjects thatareatdifferent temperatures, sayT1andT2,acertain amount ofheatwillflow from onetotheother byitself. Suppose, forinstance, weputahotstone incold water. Then when acertain heatAQistransferred from T1toT2,howmuch does theentropy ofthehotstone change? Itdecreases byAQ/T 1.How much does the water entropy change? Itincreases byAQ/T2. Theheatwill,ofcourse, flowonly from thehigher temperature T1tothelower temperature T2,sothatAQispositive ifT1isgreater than T2.Sothechange inentropy ofthewhole world ispositive, anditisthedifference ofthetwofractions: AS=%— (44.19) Sothefollowing proposition istrue: inanyprocess thatisirreversible, the entropy ofthewhole world isincreased. Only inreversible processes does the entropy remain constant. Since noprocess isabsolutely reversible, there isalways atleast asmall gainintheentropy; areversible process isanidealization inwhich wehave made thegain ofentropy minimal. Unfortunately, wearenotgoing toenter into thefield ofthermodynamics very far. Ourpurpose isonly toillustrate theprincipal ideas involved andthe reasons why itispossible tomake such arguments, butwewillnotusethermo- dynamics very much inthiscourse. Thermodynamics isused very often byengi- neers and, particularly, bychemists. Sowemust learn ourthermodynamics in practice inchemistry orengineering. Because itisnotworth while duplicating everything, weshall justgive some discussion oftheorigin ofthetheory, rather than much detail forspecial applications. Thetwolaws ofthermodynamics areoften stated thisway: First law: theenergy oftheuniverse isalways constant. Second law: theentropy oftheuniverse isalways increasing. That isnotaverygood statement ofthesecond law; itdoes notsay,forexample, thatinareversible cycle theentropy stays thesame, anditdoes notsayexactly what theentropy is.Itisjustaclever wayofremembering thetwolaws, butit does notreally tellusexactly where westand. Wehave summarized thelaws dis- cussed inthischapter inTable 44-l. Inthenextchapter weshall apply these laws todiscover therelationship between theheat generated intheexpansion ofa rubber band, andtheextra tension when itisheated. 44-12 Table 44-1 Summary ofthelawsofthermodynamics First law: Heat putintoasystem +Work done onasystem =Increase ininternal energy of thesystem: dQ+dW=dU. Second law: Aprocess whose onlynetresult istotakeheatfrom areservoir andconvert ittowork isimpossible. Noheatengine taking heatQ1from T1anddelivering heatQ2atT2candomore work than areversible engine, forwhich W=Q1—Q2=Q1(T%-Z!£)- 1 Theentropy ofasystem isdefined thisway: (a)IfheatAQisadded reversibly toasystem attemperature T,theincrease inentropy ofthesystem isAS=AQ/T. (b)AtT=0,S=0(third law). Inareversible change, thetotalentropy ofallparts ofthesystem (including reservoirs) doesnotchange. Inirreversible change, thetotalentropy ofthesystem always increases. 44-13 45 Illustrations ofThermodynamics 45-1 Internal energy Thermodynamics isarather difficult andcomplex subject when wecome to apply it,anditisnotappropriate forustogoveryfarintotheapplications inthis course. Thesubject isofverygreat importance, ofcourse, toengineers andchem- ists,andthose who areinterested inthesubject canlearn about theapplications inphysical chemistry orinengineering thermodynamics. There arealso good reference books, such asZemansky’s Heat andThermodynamics, where onecan learn more about thesubject. IntheEncyclopedia Britannica, fourteenth edition, onecanfindexcellent articles onthermodynamics andthermochemistry, andin thearticle onchemistry, thesections onphysical chemistry, vaporization, liquefi- cation ofgases, andsoon. Thesubject ofthermodynamics iscomplicated because there aresomany different ways ofdescribing thesame thing. Ifwewish todescribe thebehavior of agas,wecansaythatthepressure depends onthetemperature andonthevolume, orwecansaythat thevolume depends onthetemperature andthepressure. Orwith respect totheinternal energy U,wemight saythatitdepends onthe temperature andvolume, ifthose arethevariables wehave chosen—but wemight alsosaythatitdepends onthetemperature andthepressure, orthepressure and thevolume, andsoon. Inthelastchapter wediscussed another function of temperature andvolume, called theentropy S,andwecanofcourse construct as many other functions ofthese variables aswelike: U—TSisafunction oftem- perature andvolume. Sowehave alarge number ofdifl'erent quantities which can befunctions ofmany different combinations ofvariables. Tokeepthesubject simple inthischapter, weshalldecide atthestarttouse temperature andvolume astheindependent variables. Chemists usetemperature andpressure, because they areeasier tomeasure andcontrol inchemical experi- ments, butweshall usetemperature andvolume throughout thischapter, except inoneplace where weshall seehowtomake thetransformation intothechemists’ system ofvariables. Weshall first, then, consider only onesystem ofindependent variables: temperature andvolume. Secondly, weshall discuss onlytwodependent functions: theinternal energy andthepressure. Alltheother functions canbederived from these, soitisnotnecessary todiscuss them. With these limitations, thermo- dynamics isstillafairly difficult subject, butitisnotquite soimpossible! First weshall review some mathematics. Ifaquantity isafunction oftwo variables, theidea ofthederivative ofthequantity requires alittle more careful thought than forthecasewhere there isonly onevariable. What dowemean by thederivative ofthepressure with respect tothetemperature? The pressure change accompanying achange inthetemperature depends partly, ofcourse, on what happens tothevolume while Tischanging. Wemust specify thechange inV before theconcept ofaderivative with respect toThasaprecise meaning. We might ask,forexample, fortherateofchange ofPwith respect toTifVisheld constant. This ratio isjusttheordinary derivative thatweusually write asdP/dT. Wecustomarily useaspecial symbol, 6P/6T, toremind usthatPdepends onan- other variable VaswellasonT,andthatthisother variable isheld constant. We shall notonly usethesymbol 6tocallattention tothefactthattheother variable isheld constant, butweshall alsowrite thevariable that isheld constant asa subscript, (GP/6T)v. Since wehave only twoindependent variables, thisnotation isredundant, butitwillhelp uskeep ourwits about usinthethermodynamic jungle ofpartial derivatives. 45-145-1 Internal energy 45-2 Applications 45-3 TheClausius-Clapeyron equationFI l ll l l l 1 ll11 1 1 l l! =11 Letussuppose thatthefunction f(x,y)depends onthetwoindependent vari- ables xandy.By(Elf/ox)” wemean simply theordinary derivative, obtained inthe usual way, ifwetreat yasaconstant: 6f f(X+Ax,y) —f(X,y)_— =limitX1,1 A2:->0 AX (.1)Similarly, wedefine (E)=111,111 .6y441,40 Ar Forexample, iff(x,y)=x2+yx,then (6f/6x),, =2x—l—y,and(6f/6}’): =x. Wecanextend thisidea tohigher derivatives: 62f/8y2 or62f/6y6x. Thelatter symbol indicates thatwefirstdifferentiate fwith respect tox,treating yasacon- stant, then differentiate theresult with respect toy,treating xasaconstant. The actual order ofdifferentiation isimmaterial: 62f/6x6y =62f/6y6x. Wewillneed tocompute thechange Afinf(x,y)when xchanges tox+Ax andychanges toy+Ay.Weassume throughout thefollowing thatAxandAy areinfinitesimally small: Af=f(x+Any+Av)—f(x,y) =ffx+Any+Ar)—f(X,y +Av)+f(X,y +Av)—f(X.J11 Thelastequation isthefundamental relation that expresses Afinterms ofAx andAy. Asanexample oftheuseofthisrelation, letuscalculate thechange inthe internal energy U(T, V)when thetemperature changes from TtoT+ATandthe volume changes from VtoV+AV. Using Eq.(45.1), wewrite AU=AT +AV(‘;_;>T- (45.2) Inourlastchapter wefound another expression forthechange AUintheinternal energy when aquantity ofheatAQwasadded tothegas: AU=AQ —PAV. (45.3) Incomparing Eqs. (45.2) and(45.3) onemight atfirstbeinclined tothink that P=(0U/6V)T, butthisisnotcorrect. Toobtain thecorrect relation, letusfirst suppose thatweaddaquantity ofheat AQtothegaswhile keeping thevolume constant, sothat AV=0.With AV=0,Eq.(45.3) tells usthat AU=AQ, andEq.(45.2) tells usthat AU=(6U/élT)V AT,sothat (BU/8T)V =AQ/AT. Theratio AQ/AT, theamount ofheat onemust putintoasubstance inorder to change itstemperature byonedegree with thevolume heldconstant, iscalled the specific heatatconstant volume andisdesignated bythesymbol CV.Bythisargu- ment wehave shown that 6U(W), -CV. (45.4) Now letusagain addaquantity ofheat AQtothegas,butthistime wewill hold Tconstant andallow thevolume tochange byAV. Theanalysis inthiscase ismore complex, butwecancalculate AUbytheargument ofCarnot, making use oftheCarnot cycle weintroduced inthelastchapter. The pressure-volume diagram fortheCarnot cycle isshown inFig. 45-1. Aswehave already shown, thetotal amount ofwork done bythegasinareversible cycle isAQ(AT/T), where AQistheamount ofheat energy added tothegasasit expands isothermally attemperature Tfrom volume VtoV+AV,andT—AT isthefinal temperature reached bythegasasitexpands adiabatically onthe second legofthecycle. Now wewillshow thatthiswork done isalsogiven by 45-2 theshaded area inFig.45-1. Inanycircumstances, thework done bythegasis jPdV,andispositive when thegasexpands andnegative when thegasiscom- pressed. lfweplotPvs.V.thevariation ofPandVisrepresented byacurve which gives thevalue ofPcorresponding toaparticular value ofV.Asthevolume changes from onevalue toanother, thework done bythegas,theintegral jPdV, istheareaunder thecurve connecting theinitial andfinal values ofV.When we apply thisideatotheCarnot cycle, weseethataswegoaround thecycle, paying attention tothesignofthework done bythegas,thenetwork done bythegas isjusttheshaded area inFig.45-1. Now wewant toevaluate theshaded area geometrically. Thecycle wehave used inFig.45-1 differs from thatused intheprevious chapter inthatwenow suppose thatATandAQareinfinitesimally small. Weareworking between adia- batic lines andisothermal lines that arevery close together, andthefigure de- scribed bytheheavy lines inFig.45-1 willapproach aparallelogram asthein- crements ATandAQapproach zero. Theareaofthisparallelogram isjustAVAP, where AVisthechange involume asenergy AQisadded tothegasatconstant temperature, andAPisthechange inpressure asthetemperature changes byAT atconstant volume. One caneasily show that theshaded area inFig.45-1 is given byAVAPbyrecognizing thattheshaded area isequal tothearea enclosed bythedotted lines inFig.45-2, which inturndiffers from therectangle bounded byAPandAVonly bytheaddition andsubtraction oftheequal triangular areas inFig.45-2. Now letussummarize theresults ofthearguments wehave developed sofar: Work done bythegas=shaded area =AVAP =AQ OI‘ LT-(heat needed tochange VbyAV),1,,,,111,,1T T >(45.5) =AV- (change inPwhen Tchanges byAT)1,1,,,,,1,,,,11/ or All;-(heat needed tochange VbyAV)T =T(6P/6T)v. Equation (45.5) expresses theessential result ofCarnot’s argument. Thewhole of thermodynamics canbededuced from Eq.(45.5) andtheFirst Law, which is stated inEq.(45.3). Equation (45.5) isessentially theSecond Law, although it wasoriginally deduced byCarnot inaslightly different form, since hedidnot useourdefinition oftemperature. Now wecanproceed tocalculate (6U/6V);-. Byhowmuch would theinternal energy Uchange ifwechanged thevolume byAV? First, Uchanges because heat isputin,andsecond, Uchanges because work isdone. Theheatputinis 6PAQ=T57.VAV, according toEq.(45.5), andthework done onthesubstance is—PAV. There- forethechange AUininternal energy hastwopieces: 6PAU-T(fi>VAV —PAV. (45.6) Dividing both sides byAV,wefindfortherateofchange ofUwith Vatconstant T 6U 6P(W),_1(,,),_P. 1.1.1. Inourthermodynamics, inwhich TandVaretheonly variables andPandUare theonly functions, Eqs. (45.3) and(45.7) arethebasic equations from which all theresult ofthesubject canbededuced. 45-3AV PRESSUREg|>§*0 -1 T-AT VOLUME Fig.45-1. Pressure-volume diagram foraCarnot cycle. Thecurves marked Tand T——ATareisothermal lines; the steeper curves areadiabatic lines. AV isthevolume change asheat AQ is added tothegasatconstant temperature T.APisthepressure change atconstant volume asthegastemperature ischanged from TtoT—AT. V '0 l> ___"l-- \ V Fig. 45-2. Shaded area =area enclosed bydashed lines =area of rectangle =APAV.l l 21 45-2 Applications Now letusdiscuss themeaning ofEq.(45.7) andseewhyitanswers the questions which weproposed inourlastchapter. Weconsidered thefollowing problem: inkinetic theory itisobvious thatanincrease intemperature leads to anincrease inpressure. because ofthebombardments oftheatoms onapiston. Forthesame physical reason, when weletthepiston move back, heat istaken outofthegasand, inorder tokeep thetemperature constant, heatwillhave tobe putback in.Thegascools when itexpands, andthepressure riseswhen itisheated. There must besome connection between these twophenomena, andthiscon- nection isgiven explicitly inEq.(45.7). Ifwehold thevolume fixed andincrease thetemperature, thepressure rises atarate(6P/6T)1,. Related tothatfactisthis: ifweincrease thevolume, thegaswillcool unless wepour some heat intomain- tainthetemperature constant, and(6U/6 V)Ttellsustheamount ofheat needed tomaintain thetemperature. Equation (45.7) expresses thefundamental inter- relationship between these twoeffects. That iswhat wepromised wewould find when wecame tothelaws ofthermodynamics. Without knowing theinternal mechanism ofthegas,andknowing only thatwecannot make perpetual motion ofthesecond type, wecandeduce therelationship between theamount ofheat needed tomaintain aconstant temperature when thegasexpands, andthepressure change when thegasisheated! Now thatwehave theresult wewanted foragas,letusconsider therubber band. When westretch arubber band, wefindthatitstemperature falls, andwhen weheat arubber band, wefindthatitpulls itself in.What istheequation that gives thesame relation forarubber band asEq.(45.3) gives forgas? Fora rubber band thesituation willbesomething likethis: when heat AQisputin,the internal energy ischanged byAUandsome work isdone. Theonly difference will bethatthework done bytherubber band is—FALinstead ofPAV, where F istheforce ontheband, andListhelength oftheband. Theforce Fisafunction oftemperature andoflength oftheband. Replacing PAV inEq.(45.3) by—FAL, weget AU=AQ+FAL (45.8) Comparing Eqs. (45.3) and(45.8), weseethat therubber band equation isob- tained byamere substitution ofoneletter foranother. Furthermore, ifwesub- stitute LforV,and—-FforP,allofourdiscussion oftheCarnot cycle applies to therubber band. Wecanimmediately deduce, forinstance, that theheat AQ needed tochange thelength byALisgiven bytheanalog toEq.(45.5): AQ= —T(6F/6T)1, AL. This equation tells usthat ifwekeep thelength ofarubber band fixed andheat theband, wecancalculate how much theforce willincrease interms oftheheat needed tokeep thetemperature constant when theband is stretched alittle bit. Soweseethatthesame equation applies toboth gasanda rubber band. Infact, ifonecanwrite AU=AQ+AAB, where AandBrepre- sentdifferent quantities, force andlength, pressure andvolume, etc.,onecanapply theresults obtained foragasbysubstituting AandBforPandV.Forexample. consider theelectric potential difference, or“voltage.” Einabattery andthe charge AZthatmoves through thebattery. Weknow thatthework done ina reversible electric cell,likeastorage battery, isEAZ. (Since weinclude noPAV term inthework, werequire thatourbattery maintain aconstant volume.) Let usseewhat thermodynamics cantellusabout theperformance ofabattery. If wesubstitute EforPandZforVinEq.(45.6), weobtain AU 6EK2——T(fi)Z —E. (45.9) Equation (45.9) saysthattheinternal energy Uischanged when acharge AZmoves through thecell. Why isAU/AZ notsimply thevoltage Eofthebattery? The answer isthat arealbattery gets warm when charge moves through thecell. Theinternal energy ofthebattery ischanged, first, because thebattery didsome work ontheoutside circuit, andsecond, because thebattery isheated. There- 45-4 markable thing isthat thesecond part canagain beexpressed interms ofthe wayinwhich thebattery voltage changes with temperature. Incidentally, when thecharge moves through thecell,chemical reactions occur, andEq.(45.9) sug- gests anifty wayofmeasuring theamount ofenergy required toproduce achemical reaction. Allweneed doisconstruct acellthatworks onthereaction, measure thevoltage, andmeasure how much thevoltage changes with temperature when wedraw nocharge from thebattery! Now wehave assumed thatthevolume ofthebattery canbemaintained con- stant, since wehave omitted thePAV term when wesetthework done bythe battery equal toEAZ. Itturns outthatitistechnically quite difficult tokeep the volume constant. Itismuch easier tokeep thecellatconstant atmospheric pressure. Forthatreason, thechemists donotlikeanyoftheequations wehave written above: they prefer equations which describe performance under constant pressure. Wechose atthebeginning ofthischapter touseVandTasindependent variables. Thechemists prefer PandT,andwewillnow consider howtheresults wehave obtained sofarcanbetransformed intothechemists’ system ofvariables. Remember thatinthefollowing treatment confusion caneasily setinbecause we areshifting gears from TandVtoTandP. Westarted inEq.(45.3) with AU=AQ—PAV; PAV may bereplaced by EAZ orAAB. Ifwecould somehow replace thelastterm, PAV, byVAP, then wewould have interchanged VandP,andthechemists would behappy. Well, aclever man noticed thatthedifferential oftheproduct PVisd(PV) =PdV + VdP, andifheadded thisequality toEq.(45.3), heobtained Awm=PM~+mP AU =AQ-PAV A(U+PV)=AQ+VAP Inorder thattheresult look likeEq.(45.3), wedefine U—l—PVtobesomething new, called theenthalpy, H,andwewrite AH=AQ+VAP. Now weareready totransform ourresults into chemists’ language with thefollowing rules: U—->H,P—>—V,V—>P.Forexample, thefundamental relationship thatchemists would useinstead ofEq.(45.7) is an_av_ fairTin). V~Itshould now beclear how onetransforms tothechemists’ variables TandP. Wenow goback toouroriginal variables: fortheremainder ofthischapter, T andVaretheindependent variables. Now letusapply theresults wehave obtained toanumber ofphysical situa- tions. Consider firsttheideal gas. From kinetic theory weknow thattheinternal energy ofagasdepends only onthemotion ofthemolecules andthenumber of molecules. Theinternal energy depends onT,butnotonV.Ifwechange V,but keep Tconstant, Uisnotchanged. Therefore (6U/6V)T=0,andEq.(45.7) tells usthatforanideal gas T(g-1;)‘, -P=0. (45.10) Equation (45.10) isadifferential equation that cantellussomething about P. Wetake account ofthepartial derivatives inthefollowing way: Since thepartial derivative isatconstant V,wewillreplace thepartial derivative byanordinary derivative andwrite explicitly, toremind us,“constant V.”Equation (45.10) then becomes T2-,1}: —P=0; const V, (45.11) which wecanintegrate toget lnP=lnT+const; const V, P=const XT; const V. (45.12) 45-5 PRESSLRE % LIQUID LIQUIDmoVAPOR T-AT VAPOR VOLU ME Fig. 45-3. Isothermal lines for a condensable vapor compressed ina cylinder. Attheleft, thesubstance isin theliquid phase. Attheright, thesub- stance isvaporized. lnthecenter, both liquid and vapor are present inthe cylinder. PRESSUREAP l T-AT \ Vi. Va l l VOLUME Fig. 45-4. Pressure-volume diagram foraCarnot cycle with acondensable vapor inthecylinder. Attheleft, the substance isinthe liquid state. A quantity ofheat Lisadded attempera- ture Ttovaporize theliquid. Thevapor expands adiabatically asTchanges to T—AT.Weknow thatforanideal gasthepressure isequal to RTP-7, (45.13) which isconsistent with (45.12), since VandRareconstants. Why didwebother togothrough thiscalculation ifwealready knew theresults? Because wehave been using twoindependent definitions oftemperature! Atonestage weassumed thatthekinetic energy ofthemolecules wasproportional tothetemperature, an assumption thatdefines onescale oftemperature which wewillcalltheideal gas scale. TheTinEq.(45.13) isbased onthegasscale. Wealsocalltemperatures measured onthegasscale kinetic temperatures. Later, wedefined thetemperature inasecond waywhich wascompletely independent ofanysubstance. From argu- ments based ontheSecond Lawwedefined what wemight callthe“grand thermo- dynamic absolute temperature” T,theTthat appears inEq.(45.12). What we proved here isthat thepressure ofanideal gas(defined asoneforwhich the internal energy does notdepend onthevolume) isproportional tothegrand thermodynamic absolute temperature. Wealso know that thepressure ispro- portional tothetemperature measured onthegasscale. Therefore wecandeduce that thekinetic temperature isproportional tothe“grand thermodynamic ab- solute temperature.” That means, ofcourse, thatifwewere sensible wecould make twoscales agree. Inthisinstance, atleast, thetwoscales have been chosen sothattheycoincide; theproportionality constant hasbeen chosen tobe1.Most ofthetime man chooses trouble forhimself, butinthiscasehemade them equal! 45-3 TheClausius-Clapeyron equation Thevaporization ofaliquid isanother application oftheresults wehave derived. Suppose wehave some liquid inacylinder, such thatwecancompress it bypushing onthepiston, andweaskourselves, “Ifwekeep thetemperature constant, howdoes thepressure vary with volume?” Inother words, wewant to draw anisothermal lineontheP-Vdiagram. Thesubstance inthecylinder isnot theideal gasthatweconsidered earlier; now itmay beintheliquid orthevapor phase, orboth may bepresent. Ifweapply sufficient pressure, thesubstance willcondense toaliquid. Now ifwesqueeze stillharder, thevolume changes verylittle, andourisothermal linerises rapidly with decreasing volume, asshown attheleftinFig.45-3. Ifweincrease thevolume bypulling thepiston out,thepressure drops until wereach thepoint atwhich theliquid starts toboil, andthen vapor starts toform. Ifwepullthepiston outfarther, allthathappens isthatmore liquid vaporizes. When there ispart liquid andpart vapor inthecylinder, thetwophases arein equilibrium—liquid isevaporating andvapor iscondensing atthesame rate. If wemake more room forthevapor, more vapor isneeded tomaintain thepressure, soalittle more liquid evaporates, butthepressure remains constant. Onthe flatpartofthecurve inFig.45-3 thepressure does notchange, andthevalue of thepressure here iscalled thevapor pressure attemperature T.Aswecontinue to increase thevolume, there comes atime when there isnomore liquid toevaporate. Atthisjuncture, ifweexpand thevolume further, thepressure willfallasforan ordinary gas,asshown attheright oftheP-Vdiagram. Thelower curve inFig. 45-3 istheisothermal lineataslightly lower temperature T—AT. Thepressure intheliquid phase isslightly reduced because liquid expands with anincrease in temperature (for most substances, butnotforwater near thefreezing point) and, ofcourse, thevapor pressure islower atthelower temperature. Wewillnowmake acycle outofthetwoisothermal lines byconnecting them (say byadiabatic lines) attheends oftheflatsections, asshown inFig.45-4. Thelittle jiggle inthelower right-hand corner ofthefigure willmake little differ- ence andwewillneglect it.Wearegoing tousetheargument ofCarnot, which tellsusthattheheatadded tothesubstance inchanging itfrom aliquid toavapor isrelated tothework done bythesubstance asitgoes around thecycle. Let 45-61 l l uscallLtheheat needed tovaporize thesubstance inthecylinder. Asinthe argument immediately preceding Eq.(45.5), weknow that L(AT/T) =work done bythesubstance. Asbefore, thework done bythesubstance istheshaded area, which isapproximately AP(V(; —V1,), where APisthedifference invapor pressure atthetwotemperatures TandT-—AT,VGisthevolume ofthegas,and V1,isthevolume oftheliquid, both volumes measured atthevapor pressure. Setting these twoexpressions fortheareaequal, wegetLAT/T =AP(VG —V1,), or Equation (45.14) gives therelationship between therateofchange ofvapor pres- sure with temperature andtheamount ofheat required toevaporate theliquid. This relationship wasdeduced byCarnot, butitiscalled theClausius-Clapeyron equation. Now letuscompare Eq.(45.14) with theresults deduced from kinetic theory. Usually V1;isvery much larger than V1,.SoVG—VLzVG=RT/P permole. Ifwefurther assume thatLisaconstant, independent oftemperature—not avery good approximation—then wewould have 6P/8T =L/(RT2P). The solution ofthisdifferential equation is P=conste_L/RT. (45.15) Letuscompare thiswith thepressure variation with temperature thatwededuced earlier from kinetic theory. Kinetic theory indicated thepossibility, atleast roughly, thatthenumber ofmolecules ofvapor above aliquid would be n= e-—(UG—U],)/RT’ VA where U1;—U1,istheinternal energy permole intheliquid minus theinternal energy permole inthegas,i.e.,theenergy needed tovaporize amole ofliquid. Equation (45.15) from thermodynamics andEq.(45.16) from kinetic theory are veryclosely related because thepressure isnkT, buttheyarenotexactly thesame. However, theywillturnouttobeexactly thesame ifweassume L—UG=const, instead ofL=const. Ifweassume L—U3=const, independent oftempera- ture, then theargument leading toEq.(45.15) willproduce Eq.(45.16). This comparison shows theadvantages anddisadvantages ofthermodynamics over kinetic theory: First ofall,Eq.(45.14) obtained bythermodynamics isexact, while Eq.(45.16) canonly beapproximated, forinstance, ifUisnearly constant, andifthemodel isright. Second, wemay notunderstand correctly how thegas goes intotheliquid; nevertheless, Eq.(45.14) isright, while (45.16) isonlyapproxi- mate. Third, although ourtreatment applies toagascondensing intoaliquid, the argument istrueforanyother change ofstate. Forinstance, thesolid-to-liquid transition hasthesame kind ofcurve asthatshown inFigs. 45-3 and45-4. Intro- ducing thelatent heatformelting, M/mole, theformula analogous toEq.(45.14) then is(8P,,,e11/8T),, =M/[T(V“q -V1,1,1111)]. Although wemaynotunderstand thekinetic theory ofthemelting process, wenevertheless have acorrect equation. However, when wecanunderstand thekinetic theory, wehave another advantage. Equation (45.14) isonly adifferential relationship, andwehave nowayofobtain- ingtheconstants ofintegration. Inthekinetic theory wecanobtain theconstants alsoifwehave agood model thatdescribes thephenomenon completely. Sothere areadvantages anddisadvantages toeach. When knowledge isweak andthesitua- tioniscomplicated, thermodynamic relations arereally themost powerful. When thesituation isverysimple andatheoretical analysis canbemade, then itisbetter totrytogetmore information from theoretical analysis. Onemore example: blackbody radiation. Wehave discussed aboxcontaining radiation andnothing else. Wehave talked about theequilibrium between the oscillator andtheradiation. Wealsofound thatthephotons hitting thewall of theboxwould exert thepressure P,andwefound PV=U/3, where Uisthe 45-7 totalenergy ofallthephotons andVisthevolume ofthebox. Ifwesubstitute U=3PVinthebasic Eq.(45.7), wefind 6U 8P(W21 _31>_r<5T)V _P. (45.17) Since thevolume ofourboxisconstant, wecanreplace (6P/6T )1,bydP/dT to obtain anordinary differential equation wecanintegrate: lnP=4lnT—l—const, orP=const XT4.Thepressure ofradiation varies asthefourth power ofthe temperature, andtheenergy content oftheradiation, U/V=P/3, alsovaries as T4.Itisusual towrite U/V=(40/c)T4, where cisthespeed oflight and0'isa constant. Itisnotpossible togetafrom thermodynamics alone. Here isagood example ofitspower, anditslimitations. Toknow thatU/Vgoes asT4isagreat deal, buttoknow how bigU/Vactually isatanytemperature requires thatwego intothekind ofdetail thatonly acomplete theory cansupply. Forblackbody radiation wehave such atheory andwecanderive anexpression fortheconstant 0'inthefollowing manner. LetI(w)dwbetheintensity distribution, theenergy flow through 1m2inone second withfrequency between atandw+dw.Theenergy density distribution = energy/volume =I(w)dw/c is L;=total energy density =/ energy density cuandor+dwat=O =/TI(w) do)_ 0 c From ourearlier discussions, weknow that hwaI )Z 4*? - (w n_2c2(ert.»/tr _1) Substituting thisexpression forI(w)inourequation forU/V,weget g_ 1I”hwsdw _1/“T233 0em./kT_l Ifwesubstitute x=hw/kT, theexpression becomes '17Th31r2c3 11E1—'U(kT)‘*f’flax l This integral isjustsome number thatwecanget,approximately, bydrawing a curve andtaking theareabycounting squares. Itisroughly 6.5.Themathemati- cians among uscanshow that theintegral isexactly 1r‘/l5.* Comparing this expression with U/V=(40/c)T4, wefind k41r2 watts1 L = TT8 mi 4 GT60h3c2 5'67 X10 (meter)? (degree)4 ‘Since (ex—1)_‘ =e"+e_2" +...,theintegral is M W Zf e‘""’x3dx. rt-1 O Butfre-"Zdx=l/n,anddifferentiating withrespect tonthree times gives frx3e-'"' dx=6/n4, sotheintegral is6(1+fi-+{-1-+...)andagood estimate comes from adding thefirstfewterms. InChapter 50wewillfindawaytoshow thatthesumofthe reciprocal fourth powers oftheintegers is,infact,1r‘/90. 45-8 lfwemake asmall hole inourbox, how much energy willflow persecond through theholeofunitarea? Togofrom energy density toenergy flow, wemulti- plytheenergy density U/Vbyc.Wealsomultiply by3,which arises asfollows: first, afactor of2,because onlytheenergy which isflowing outescapes; andsecond, another factor -2-,because energy which approaches thehole atanangle tothe normal islesseffective ingetting through theholebyacosine factor. Theaverage value ofthecosine is2.Itisclear now whywewrite U/V =(40/c)T4: sothat wecanultimately saythatthefluxfrom asmall hole isaT4perunitarea. 45-9 46 Ratchet and pawl 46-1 How aratchet works Inthischapter wediscuss theratchet andpawl, avery simple device which allows ashaft toturn only oneway. Thepossibility ofhaving something turn only onewayrequires some detailed andcareful analysis, andthere aresome very interesting consequences. Theplan ofthediscussion came about inattempting todevise anelementary explanation, from themolecular orkinetic point ofview, forthefactthatthere is amaximum amount ofwork which canbeextracted from aheat engine. Of course wehave seentheessence ofCarnot’s argument, butitwould benicetofind anexplanation which iselementary inthesense thatwecanseewhat ishappening physically. Now, there arecomplicated mathematical demonstrations which follow from Newton’s laws todemonstrate thatwecangetonly acertain amount ofwork outwhen heatflows from oneplace toanother, butthere isgreat difficulty inconverting thisintoanelementary demonstration. Inshort, wedonotunder- stand it,although wecanfollow themathematics. InCarnot’s argument, thefactthat more than acertain amount ofwork cannot beextracted ingoing from onetemperature toanother isdeduced from another axiom, which isthatifeverything isatthesame temperature, heatcannot beconverted towork bymeans ofacyclic process. First, letusback upandtry tosee,inatleast oneelementary example, why thissimpler statement istrue. Letustrytoinvent adevice which willviolate theSecond Law ofThermo- dynamics, thatis,agadget which willgenerate work from aheat reservoir with everything atthesame temperature. Letussaywehave aboxofgasatacertain temperature, andinside there isanaxlewith vanes init.(SeeFig.46-1 buttake T1=T2=T,say.) Because ofthebombardments ofgasmolecules onthevane, thevane oscillates andjiggles. Allwehave todoistohook onto theother endof theaxleawheel which canturnonly oneway-the ratchet andpawl. Then when theshaft tries tojiggle oneway, itwillnotturn, andwhen itjiggles theother, it willturn. Then thewheel willslowly turn, andperhaps wemight even tieaflea onto astring hanging from adrum ontheshaft, andlifttheflea! Now letusask ifthisispossible. According toCarnot’s hypothesis, itisimpossible. Butifwe justlook atit.wesee,primafacie, thatitseems quite possible. Sowemust look more closely. Indeed, ifwelook attheratchet andpawl, weseeanumber of complications. First. ouridealized ratchet isassimple aspossible, buteven so,there isapawl, andthere must beaspring inthepawl. Thepawl must return after coming offa tooth, sothespring isnecessary. Another feature ofthisratchet andpawl, notshown inthefigure, isquite essential. Suppose thedevice were made ofperfectly elastic parts. After thepawl islifted offtheendofthetooth andispushed back bythespring, itwillbounce against thewheel andcontinue tobounce. Then, when another fluctuation came, thewheel could turntheother way, because thetooth could getunderneath during themoment when thepawl wasup!Therefore anessential partoftheirreversibility ofourwheel isadamping ordeadening mechanism which stops thebouncing. When thedamping happens, ofcourse, theenergy thatwasinthepawl goes into thewheel andshows upasheat. So,asitturns, thewheel willgethotter andhotter. Tomake thething simpler, wecanputagasaround thewheel totakeupsome of theheat. Anyway, letussaythegaskeeps rising intemperature, along with the wheel. Willitgoonforever? No! Thepawl andwheel, both atsome temperature 46-146-1 How aratchet works 46-2 Theratchet asanengine 46-3 Reversibility inmechanics 46-4 Irreversibility 46-5 Order andentropy T2 I 1‘K I i _7 Fig.46-l. The ratchet and pawl machine. T,alsohave Brownian motion. This motion issuch that, every once inawhile, by accident, thepawl liftsitself upandover atooth justatthemoment when the Brownian motion onthevanes istrying toturn theaxle backwards. And as things gethotter, thishappens more often. So,thisisthereason thisdevice does notwork inperpetual motion. When thevanes getkicked, sometimes thepawl liftsupandgoes overtheend. Butsome- times, when ittries toturn theother way, thepawl hasalready lifted duetothe fluctuations ofthemotions onthewheel side, andthewheel goes back theother way! Thenetresult isnothing. Itisnothard todemonstrate that when the temperature onboth sides isequal, there willbenonetaverage motion ofthe wheel. Ofcourse thewheel willdoalotofjiggling thiswayandthatway, butit willnotdowhat wewould like, which istoturnjustoneway. Letuslook atthereason. Itisnecessary todowork against thespring in order toliftthepawl tothetopofatooth. Letuscallthisenergy e,andlet0be theangle between theteeth. Thechance thatthesystem canaccumulate enough energy, e,togetthepawl over thetopofthetooth, ise"‘/"T. Buttheprobability thatthepawl willaccidentally beupisalsoe_‘”‘T. Sothenumber oftimes that thepawl isupandthewheel canturn backwards freely isequal tothenumber oftimes thatwehave enough energy toturn itforward when thepawl isdown. Wethus geta“balance,” andthewheel willnotgoaround. 46-2 Theratchet asanengine Letusnow gofurther. Take theexample where thetemperature ofthe vanes isT1andthetemperature ofthewheel, orratchet, isT2,andT2islessthan T1. Because thewheel iscold andthefluctuations ofthepawl arerelatively in- frequent, itwillbevery hard forthepawl toattain anenergy e.Because ofthe high temperature T1,thevanes willoften attain theenergy e,soourgadget will goinonedirection, asdesigned. Wewould nowliketoseeifitcanliftweights. Onto thedrum inthemiddle wetieastring, andputaweight, such asourflea,onthestring. WeletLbethe torque duetotheweight. IfLisnottoogreat, ourmachine willlifttheweight because theBrownian fluctuations make itmore likely tomove inonedirection than theother. Wewant tofindhowmuch weight itcanlift,howfastitgoesaround, andsoon. First weconsider aforward motion, theusual wayonedesigns aratchet to run. Inorder tomake onestepforward, how much energy hastobeborrowed from thevane end? Wemust borrow anenergy etoliftthepawl. Thewheel turns through anangle 0against atorque L,sowealsoneed theenergy L0.Thetotal amount ofenergy thatwehave toborrow isthus e+L0.Theprobability that wegetthisenergy isproportional toe_(‘+”)”‘T1. Actually, itisnotonly aques- tionofgetting theenergy, butwealsowould liketoknow thenumber oftimes persecond ithasthisenergy. The probability persecond isproportional to e_(‘+L"”"T1, andweshall calltheproportionality constant l/1'. Itwillcancel outintheendanyway. When aforward step happens, thework done onthe weight isL0.Theenergy taken from thevane ise+L0.Thespring getswound upwith energy e,then itgoes clatter, clatter, bang, andthisenergy goes intoheat. Alltheenergy taken outgoes tolifttheweight andtodrive thepawl, which then fallsback andgives heat totheother side. Now welook attheopposite case, which isbackward motion. What happens here? Togetthewheel togobackwards allwehave todoissupply theenergy to liftthepawl high enough sothattheratchet willslip. This isstillenergy e.Our probability persecond forthepawl toliftthishigh isnow (l/-r)e*""T2. Our proportionality constant isthesame, butthistime kT2shows upbecause ofthe different temperature. When thishappens, thework isreleased because thewheel slips backward. Itloses onenotch, soitreleases work L0.Theenergy taken from theratchet system ise,andtheenergy given tothegasatT1onthevane sideis L6—l—e.Ittakes alittle thinking toseethereason forthat. Suppose thepawl has lifted itself upaccidentally byafluctuation. Then when itfallsback andthespring 46-2 Table 46-1 Summary ofoperation ofratchet andpawl. l Forward: Need energy e+L6from vane. Rate =-e*f”+9/"TiT Takes from vane L0+e Does work L0 Gives toratchet e l Backward: Needs energy e forpawl. Rate =-e“”‘T2T Takes from ratchet e Releases work L6 same asabove with signreversed. Gives tovane L0+e . . L0lfsystem isreversible, rates areequal, hence LL =-5-- T1 T2 Heat toratchet = e _ Hence g= Heat from vane L0+e Q1 T1 pushes itdown against thetooth, there isaforce trying toturnthewheel, because thetooth ispushing onaninclined plane. This force isdoing work, andsoisthe force duetotheweights. Soboth together make upthetotal force, andallthe energy which isslowly released appears atthevane endasheat. (Ofcourse it must, byconservation ofenergy, butonemust becareful tothink thething through!) Wenotice that allthese energies areexactly thesame, butreversed. So,depending upon which ofthese tworates isgreater, theweight iseither slowly lifted orslowly released. Ofcourse, itisconstantlyjiggling around, going upfora while anddown forawhile, butwearetalking about theaverage behavior. Suppose thatforaparticular weight therates happen tobeequal. Then we addaninfinitesimal weight tothestring. Theweight willslowly godown, and work willbedone onthemachine. Energy willbetaken from thewheel andgiven tothevanes. lfinstead wetake offalittle bitofweight, then theimbalance is theother way. Theweight islifted, andheat istaken from thevane andputinto thewheel. Sowehave theconditions ofCarnot’s reversible cycle, provided that theweight isjustsuch thatthese twoareequal. This condition isevidently that (e—l—L0)/T1 =e/T2. Letussaythatthemachine isslowly lifting theweight. Energy Q1istaken from thevanes andenergy Q2isdelivered tothewheel, and these energies areintheratio (e+L9)/e. Ifwearelowering theweight, wealso have Q1/Q2 =(e—l—L0)/e. Thus (Table 46-1) wehave Q1/Q2 =T1/Ta Furthermore, thework wegetoutistotheenergy taken from thevane asL0 istoL0+e,hence as(T1-T2)/T1. Weseethat ourdevice cannot extract more work than this,operating reversibly. This istheresult thatweexpected from Carnot’s argument, andthemain result ofthislecture. However, wecanuseour device tounderstand afewother phenomena, even outofequilibrium, andthere- forebeyond therange ofthermodynamics. Letusnowcalculate howfastourone-way device would turnifeverything were atthesame temperature andwehung aweight onthedrum. lfwepullvery, very hard, ofcourse, there areallkinds ofcomplications. Thepawl slips over the ratchet, orthespring breaks, orsomething. Butsuppose wepullgently enough thateverything works nicely. Inthose circumstances, theabove analysis isright fortheprobability ofthewheel going forward andbackward, ifweremember 46-3 (I) L‘ Fig.46—2. Angular velocity ofthe ratchet usufunction oftorque.thatthetwotemperatures areequal. Ineach stepanangle 0isobtained, sothe angular velocity is0times theprobability ofoneofthese jumps persecond. It goes forward with probability (1/'r)e“(°+L°)”°T and backward with probability (1/'r)e_‘/ "T,sothatfortheangular velocity wehave w=(0/,r)e—(e+L9>/kT _e—¢/l¢T =(e/¢)e-"’°T(e-L”/'” -1). (46.1) Ifweplotwagainst L,wegetthecurve shown inFig.46—2. Weseethatitmakes a great difference whether Lispositive ornegative. IfLincreases inthepositive range, which happens when wetrytodrive thewheel backward, thebackward velocity approaches aconstant. AsLbecomes negative, wreally “takes off” forward, since etoatremendous power isverygreat! Theangular velocity thatwasobtained from different forces isthusvery un- symmetrical. Going onewayitiseasy: wegetalotofangular velocity foralittle force. Going theother way, wecanputonalotofforce, andyetthewheel hardly goesaround. Wefindthesame thing inanelectrical rectifier. Instead oftheforce, wehave theelectric field, andinstead oftheangular velocity, wehave theelectric current. Inthecaseofarectifier, thevoltage isnotproportional toresistance, andthe situation isunsymmetrical. Thesame analysis thatwemade forthemechanical rectifier willalsowork foranelectrical rectifier. Infact, thekind offormula we obtained above istypical ofthecurrent-carrying capacities ofrectifiers asafunc- tionoftheir voltages. Now letustake alltheweights away, andlook attheoriginal machine. If T2were lessthan T1,theratchet would goforward, asanybody would believe. Butwhat ishard tobelieve, atfirstsight, istheopposite. IfT2isgreater than T1, theratchet goes around theopposite way! Adynamic ratchet with lotsofheat initruns itself backwards, because theratchet pawl isbouncing. Ifthepawl, for amoment, isontheincline somewhere, itpushes theinclined plane sideways. Butitisalways pushing onaninclined plane, because ifithappens toliftuphigh enough togetpast thepoint ofatooth, then theinclined plane slides by,andit comes down again onaninclined plane. Soahotratchet andpawl isideally built togoaround inadirection exactly opposite tothatforwhich itwasoriginally designed! Inspite ofallourcleverness oflopsided design, ifthetwotemperatures are exactly equal there isnomore propensity toturn onewaythan theother. The moment welook atit,itmaybeturning onewayortheother, butinthelong run, itgetsnowhere. Thefactthat itgetsnowhere isreally thefundamental deep principle onwhich allofthermodynamics isbased. 46-3 Reversibility inmechanics What deeper mechanical principle tellsusthat, inthelong run,ifthetempera- tureiskept thesame everywhere, ourgadget willturn neither totheright norto theleft? Weevidently have afundamental proposition thatthere isnowayto design amachine which, lefttoitself, willbemore likely tobeturning oneway than theother after along enough time. Wemust trytoseehowthisfollows from thelaws ofmechanics. Thelaws ofmechanics gosomething likethis: themass times theacceleration istheforce, andtheforce oneach particle issome complicated function ofthe positions ofalltheother particles. There areother situations inwhich forces depend onvelocity, such asinmagnetism, butletusnotconsider thatnow. We take asimpler case, such asgravity, where forces depend only onposition. Now suppose thatwehave solved oursetofequations andwehave acertain motion x(t)foreach particle. Inacomplicated enough system, thesolutions arevery complicated, andwhat happens with time turns outtobevery surprising. Ifwe write down anyarrangement weplease fortheparticles, wewillseethisarrange- ment actually occur ifwewaitlong enough! Ifwefollow oursolution foralong 46-4 enough time, ittrieseverything thatitcando,sotospeak. This isnotabsolutely necessary inthesimplest devices, butwhen systems getcomplicated enough, with enough atoms, ithappens. Now there issomething elsethesolution cando.If wesolve theequations ofmotion, wemay getcertain functions such as t+I2+1“.Weclaim thatanother solution would be—t+t2—t3.Inother words, ifwesubstitute ——teverywhere fortthroughout theentire solution, wewill once again getasolution ofthesame equation. This follows from thefactthat ifwesubstitute —tfortintheoriginal differential equation, nothing ischanged, since only second derivatives with respect totappear. This means thatifwehave acertain motion, then theexact opposite motion isalsopossible. Inthecomplete confusion which comes ifwewaitlong enough, itfinds itself going onewaysome- times, anditfinds itself going theother waysometimes. There isnothing more beautiful about oneofthemotions than about theother. Soitisimpossible to design amachine which, inthelong run,ismore likely tobegoing onewaythan theother, ifthemachine issufficiently complicated. Onemight think upanexample forwhich thisisobviously untrue. lfwetake awheel, forinstance, andspinitinempty space, itwillgothesame wayforever. Sothere aresome conditions, liketheconservation ofangular momentum, which violate theabove argument. Thisjustrequires thattheargument bemade with a little more care. Perhaps thewalls take uptheangular momentum, orsomething likethat, sothat wehave nospecial conservation laws. Then, ifthesystem is complicated enough, theargument istrue. Itisbased onthefactthatthelaws of mechanics arereversible. Forhistorical interest, wewould liketoremark onadevice invented by Maxwell, who firstworked outthedynamical theory ofgases. Hesupposed the following situation: Wehave twoboxes ofgasatthesame temperature, with a little hole between them. Atthehole sitsalittle demon (who may beamachine ofcoursel). There isadoor onthehole, which canbeopened orclosed bythe demon. Hewatches themolecules coming from theleft. Whenever heseesafast molecule, heopens thedoor. When heseesaslow one,heleaves itclosed. Ifwe want himtobeanextra special demon, hecanhave eyesattheback ofhishead, anddotheopposite tothemolecules from theother side. Heletstheslow ones through totheleft,andthefastthrough totheright. Pretty soon theleftsidewill getcold andtheright sidehot. Then, aretheideas ofthermodynamics violated because wecould have such ademon? Itturns out,ifwebuild afinite-sized demon, thatthedemon himself getsso warm thathecannot seevery wellafter awhile. Thesimplest possible demon, as anexample, would beatrapdoor heldover theholebyaspring. Afastmolecule comes through, because itisabletoliftthetrapdoor. Theslow molecule cannot getthrough, andbounces back. Butthisthing isnothing butourratchet andpawl inanother form, andultimately themechanism willheat up. Ifweassume that thespecific heat ofthedemon isnotinfinite, itmust heat up.Ithasbutafinite number ofinternal gears andwheels, soitcannot getridoftheextra heat thatit getsfrom observing themolecules. Soon itisshaking from Brownian motion so much that itcannot tellwhether itiscoming orgoing, much lesswhether the molecules arecoming orgoing, soitdoes notwork. 46-4 Irreversibility Areallthelaws ofphysics reversible? Evidently not! Justtrytounscramble anegg! Run amoving picture backwards, andittakes only afewminutes for everybody tostart laughing. Themost natural characteristic ofallphenomena istheir obvious irreversibility. Where does irreversibility come from? Itdoes notcome from Newton’s laws. Ifweclaim thatthebehavior ofeverything isultimately tobeunderstood interms ofthelaws ofphysics, andifitalso turns outthat alltheequations have the fantastic property that ifweputt=—twehave another solution, then every phenomenon isreversible. How then does itcome about innature onalarge scale thatthings arenotreversible? Obviously there must besome law, some 46-5 obscure butfundamental equation, perhaps inelectricity, maybe inneutrino physics, inwhich itdoes matter which waytime goes. Letusdiscuss thatquestion now. Wealready know oneofthose laws, which says that theentropy isalways increasing. Ifwehave ahotthing andacold thing, theheatgoes from hottocold. Sothelawofentropy isonesuch law. But weexpect tounderstand thelawofentropy from thepoint ofview ofmechanics. Infact, wehave justbeen successful inobtaining alltheconsequences oftheargu- ment thatheat cannot fiow backwards byitself from justmechanical arguments, andwethereby obtained anunderstanding oftheSecond Law. Apparently we cangetirreversibility from reversible equations. Butwasitonly amechanical argument thatweused? Letuslook intoitmore closely. Since ourquestion hastodowith theentropy, ourproblem istotrytofinda microscopic description ofentropy. Ifwesaywehave acertain amount ofenergy insomething, likeagas,then wecangetamicroscopic picture ofit,andsay thatevery atom hasacertain energy. Allthese energies added together giveus thetotal energy. Similarly, maybe every atom hasacertain entropy. Ifweadd everything up,wewould have thetotal entropy. Itdoes notwork sowell, but letusseewhat happens. Asanexample, wecalculate theentropy difference between agasatacertain temperature atonevolume, andagasatthesame temperature atanother volume. Weremember, from Chapter 44,thatwehad, forthechange inentropy, Inthepresent case, theenergy ofthegasisthesame before andafter expansion, since thetemperature does notchange. Sowehave toaddenough heattoequal thework done bythegasor,foreach little change involume, dQ=PdV. Putting thisinfordQ,weget V’dV V2NkTdV..=/._=/_L V,T V,VT _ V2 asweobtained inChapter 44.Forinstance, ifweexpand thevolume byafactor of2,theentropy change isNkln2. Letusnowconsider another interesting example. Suppose wehave aboxwith abarrier inthemiddle. Ononesideisneon (“black” molecules), andontheother, argon (“white” molecules). Now wetake outthebarrier, andletthem mix. How much hastheentropy changed? Itispossible toimagine that instead ofthe barrier wehave apiston, with holes initthatletthewhites through butnotthe blacks, andanother kind ofpiston which istheother wayaround. Ifwemove onepiston toeach end, weseethat, foreach gas,theproblem isliketheonewe justsolved. Sowegetanentropy change ofNkln2,which means thattheentropy hasincreased bykln2permolecule. The2hastodowith theextra room that themolecule has,which israther peculiar. Itisnotaproperty ofthemolecule itself, butofhowmuch room themolecule hastorunaround in.This isastrange situation, where entropy increases butwhere everything hasthesame temperature andthesame energy! Theonly thing thatischanged isthatthemolecules are distributed differently. Wewellknow thatifwejustpullthebarrier out,everything willgetmixed upafter along time, duetothecollisions, thejiggling, thebanging, andsoon. Every once inawhile awhite molecule goes toward ablack, andablack onegoes toward awhite, andmaybe they pass. Gradually thewhites worm their way, by accident, across into thespace ofblacks, andtheblacks worm their way, by accident, into thespace ofwhites. Ifwewait long enough wegetamixture. 46-6 Clearly, thisisanirreversible process intherealworld, andought toinvolve an increase intheentropy. Here wehave asimple example ofanirreversible process which iscompletely composed ofreversible events. Every time there isacollision between anytwo molecules, theygoofi"incertain directions. Ifwetook amoving picture ofacolli- sioninreverse, there would benothing wrong with thepicture. Infact, onekind ofcollision isjustaslikely asanother. Sothemixing iscompletely reversible, and yetitisirreversible. Everyone knows thatifwestarted with white andwith black, separated, wewould getamixture within afewminutes. Ifwesatandlooked at itforseveral more minutes, itwould notseparate again butwould staymixed. Sowehave anirreversibility which isbased onreversible situations. Butwealso seethereason now. Westarted with anarrangement which is,insome sense, ordered. Duetothechaos ofthecollisions, itbecomes disordered. Itisthechange from anordered arrangement toadisordered arrangement which isthesource of theirreversibility. Itistruethatifwetook amotion picture ofthis,andshowed itbackwards, wewould seeitgradually become ordered. Someone would say, “That isagainst thelaws ofphysics!” Sowewould runthefilmover again, andwewould look at every collision. Every onewould beperfect, andevery onewould beobeying thelaws ofphysics. Thereason, ofcourse, isthatevery molecule’s velocities are justright, soifthepaths areallfollowed back, theygetback totheir original condi- tion. Butthatisavery unlikely circumstance tohave. Ifwestart with thegasin nospecial arrangement, justwhites andblacks, itwillnever getback. 46-5 Order andentropy S0wenow have totalkabout what wemean bydisorder andwhat wemean byorder. Itisnotaquestion ofpleasant order orunpleasant disorder. What is different inourmixed andunmixed cases isthefollowing. Suppose wedivide the space intolittle volume elements. Ifwehave white andblack molecules, howmany ways could wedistribute them among thevolume elements sothatwhite isonone side, andblack ontheother? Ontheother hand, how many ways could wedis- tribute them with norestriction onwhich goes where? Clearly, there aremany more ways toarrange them inthelatter case. Wemeasure “disorder” bythe number ofways thattheinsides canbearranged, sothatfrom theoutside itlooks thesame. Thelogarithm ofthatnumber ofways istheentropy. Thenumber of ways intheseparated caseisless,sotheentropy isless, orthe“disorder” isless. Sowith theabove technical definition ofdisorder wecanunderstand the proposition. First, theentropy measures thedisorder. Second, theuniverse al- ways goes from “order” to“disorder,” soentropy always increases. Order isnot order inthesense thatwelikethearrangement, butinthesense thatthenumber ofdifferent ways wecanhook itup,andstillhave itlook thesame from theoutside, isrelatively restricted. Inthecase where wereversed ourmotion picture ofthe gasmixing, there wasnotasmuch disorder aswethought. Every single atom had exactly thecorrect speed anddirection tocome outright! Theentropy wasnot high after all,even though itappeared so. What about thereversibility oftheother physical laws? When wetalked about theelectric field which comes from anaccelerating charge, itwassaidthat wemust take theretarded field. Atatime tandatadistance rfrom thecharge, wetake thefield duetotheacceleration atatime t—r/c,nott+r/c. Soit looks, atfirst, asifthelawofelectricity isnotreversible. Very strangely, however, thelaws weused come from asetofequations called Maxwell’s equations, which are,infact, reversible. Furthermore, itispossible toargue thatifwewere touse only theadvanced field, thefield duetothestate ofafiairs att+r/c,anddoit absolutely consistently inacompletely enclosed space, everything happens exactly thesame wayasifweuseretarded fields! This apparent irreversibility inelectricity, atleast inanenclosure, isthus notanirreversibility atall.Wehave some feeling forthatalready, because weknow thatwhen wehave anoscillating charge which generates fields which arebounced from thewalls ofanenclosure weultimately 46-7 gettoanequilibrium inwhich there isnoone-sidedness. Theretarded field ap- proach isonly aconvenience inthemethod ofsolution. Sofarasweknow, allthefundamental laws ofphysics, likeNewton’s equa- tions, arereversible. Then where does irreversibility come from? Itcomes from order going todisorder, butwedonotunderstand thisuntil weknow theorigin oftheorder. Why isitthatthesituations wefindourselves inevery dayarealways outofequilibrium? Onepossible explanation isthefollowing. Look again atour boxofmixed white andblack molecules. Now itispossible, ifwewaitlongenough, bysheer, grossly improbable, butpossible, accident, thatthedistribution ofmole- cules getstobemostly white ononesideandmostly black ontheother. After that, astimes goes onandaccidents continue, they getmore mixed upagain. Thus onepossible explanation ofthehigh degree oforder inthepresent-day world isthatitisjustaquestion ofluck. Perhaps ouruniverse happened tohave hadafluctuation ofsome kind inthepast, inwhich things gotsomewhat separated, andnow they arerunning back together again. This kind oftheory isnotun- symmetrical, because wecanaskwhat theseparated gaslooks likeeither alittle inthefuture oralittle inthepast. Ineither case, weseeagrey smear attheinter- face, because themolecules aremixing again. Nomatter which wayweruntime, thegasmixes. Sothistheory would saytheirreversibility isjustoneoftheacci- dents oflife. Wewould liketoargue thatthisisnotthecase. Suppose wedonotlook at thewhole boxatonce, butonly atapiece ofthebox. Then, atacertain moment, suppose wediscover acertain amount oforder. Inthislittle piece, white andblack areseparate. What should wededuce about thecondition inplaces where wehave notyetlooked‘? Ifwereally believe thattheorder arose from complete disorder byafluctuation, wemust surely take themost likely fluctuation which could produce it,andthemost likely condition isnotthattherestofithasalsobecome disentangled! Therefore, from thehypothesis thattheworld isafluctuation, all ofthepredictions arethatifwelook atapart oftheworld wehave never seen before, wewillfinditmixed up,andnotlikethepiece wejustlooked at.Ifour order were duetoafluctuation, wewould notexpect order anywhere butwhere we have justnoticed it. Now weassume theseparation isbecause thepastoftheuniverse wasreally ordered. Itisnotduetoafluctuation, butthewhole thing used tobewhite and black. This theory now predicts that there willbeorder inother places——the order isnotduetoafluctuation, butduetoamuch higher ordering atthebeginning oftime. Then wewould expect tofindorder inplaces where wehave notyet looked. Theastronomers, forexample, have only looked atsome ofthestars. Every daythey turn their telescopes toother stars, andthenew stars aredoing the same thing astheother stars. Wetherefore conclude thattheuniverse isnota fluctuation, andthat theorder isamemory ofconditions when things started. This isnottosaythatweunderstand thelogic ofit.Forsome reason, theuniverse atonetimehadaverylowentropy foritsenergy content, andsince thentheentropy hasincreased. Sothatisthewaytoward thefuture. That istheorigin ofallir- reversibility, thatiswhat makes theprocesses ofgrowth anddecay, thatmakes us remember thepast andnotthefuture, remember thethings which arecloser to thatmoment inthehistory oftheuniverse when theorder washigher than now, andwhy wearenotable toremember things where thedisorder ishigher than now, which wecallthefuture. So,aswecommented inanearlier chapter, the entire universe isinaglass ofwine, ifwelook atitclosely enough. Inthiscase theglass ofwine iscomplex, because there iswater andglass andlight andevery- thing else. Another delight ofoursubject ofphysics isthat even simple andidealized things, liketheratchet andpawl, work only because theyarepartoftheuniverse. Theratchet andpawl works inonly onedirection because ithassome ultimate contact with therestoftheuniverse. Iftheratchet andpawl were inaboxand isolated forsome suflicient time, thewheel would benomore likely togooneway than theother. Butbecause wepulluptheshades andletthelight out,because 46-8 wecooloffontheearth andgetheatfrom thesun,theratchets andpawls that wemake canturnoneway. This one-wayness isinterrelated withthefactthatthe ratchet ispartoftheuniverse. Itispartoftheuniverse notonlyinthesense that itobeys thephysical lawsoftheuniverse, butitsone-way behavior istiedtothe one-way behavior oftheentire universe. Itcannot becompletely understood until themystery ofthebeginnings ofthehistory oftheuniverse arereduced stillfurther from speculation toscientific understanding. 46~9 47 Sound. The wave equation 47-1 Waves Inthischapter weshall discuss thephenomenon ofwaves. This isaphenom- enon which appears inmany contexts throughout physics, andtherefore our attention should beconcentrated onitnotonlybecause oftheparticular example considered here, which issound, butalsobecause ofthemuch wider application oftheideas inallbranches ofphysics. Itwaspointed outwhen westudied theharmonic oscillator thatthere arenot onlymechanical examples ofoscillating systems butelectrical ones aswell. Waves arerelated tooscillating systems, except thatwave oscillations appear notonly as time-oscillations atoneplace, butpropagate inspace aswell. Wehave really already studied waves. When westudied light, inlearning about theproperties ofwaves inthatsubject, wepaid particular attention totheinterfer- ence inspace ofwaves from several sources atdifferent locations andallatthe same frequency. There aretwoimportant wave phenomena thatwehave notyet discussed which occur inlight, i.e.,electromagnetic waves, aswellasinanyother form ofwaves. Thefirstofthese isthephenomenon ofinterference intimerather than interference inspace. Ifwehave twosources ofsound which have slightly different frequencies andifwelisten toboth atthesame time, then sometimes the waves come with thecrests together andsometimes with thecrest andtrough to- gether (seeFig.47-1). Therising andfalling ofthesound thatresults isthephenom- enon ofbeats or,inother words, ofinterference intime. Thesecond phenomenon involves thewave patterns which result when thewaves areconfined within agiven volume andreflect back andforth from walls. These efl"ects could have been discussed, ofcourse, forthecaseofelectro- magnetic waves. Thereason fornothaving done thisisthatbyusing oneexample wewould notgenerate thefeeling thatweareactually learning about many diflerent subjects atthesame time. Inorder toemphasize thegeneral applicability ofwaves beyond electrodynamics, weconsider here adifl'erent example, inparticular sound waves. Other examples ofwaves arewater waves consisting oflong swells thatwe seecoming intotheshore, orthesmaller water waves consisting ofsurface tension ripples. Asanother example, there aretwokinds ofelastic waves insolids; a compressional (orlongitudinal) wave inwhich theparticles ofthesolid oscillate back andforth along thedirection ofpropagation ofthewave (sound waves ina gasareofthiskind), andatransverse wave inwhich theparticles ofthesolid os- cillate inadirection perpendicular tothedirection ofpropagation. Earthquake waves contain elastic waves ofboth kinds, generated byamotion atsome place in theearth’s crust. Stillanother example ofwaves isfound inmodern physics. These arewaves which givetheprobability amplitude offinding aparticle atagiven place—the “matter waves” which wehave already discussed. Their frequency isproportional totheenergy andtheir wave number isproportional tothemomentum. They are thewaves ofquantum mechanics. Inthischapter weshall consider onlywaves forwhich thevelocity isindepend- entofthewavelength. This is,forexample, thecaseforlight inavacuum. The speed oflight isthen thesame forradiowaves, blue light, green light, orforany other wavelength. Because ofthisbehavior, when webegan todescribe thewave phenomenon wedidriotnotice atfirstthatwehadwave propagation. Instead, we saidthatifacharge ismoved atoneplace, theelectric field atadistance xwas 47-147-1 Waves 47-2 Thepropagation ofsound 47-3 Thewave equation 47-4 Solutions ofthewave equation 47-5 Thespeed ofsound Fig. 47-1. Interference intime of twosound sources with slightly different frequencies, resulting inbeats. '<-—- cti—>| ..1\ r"\I \\ /I‘ > , I \ I \\// Fig.47-2. The solid curve shows what theelectric field might belikecit some instant oftime andthedashed curve shows what theelectric field iscitutime tloter.proportional totheacceleration, notatthetimet,butattheearlier timet—x/c. Therefore ifwewere topicture theelectric field inspace atsome instant oftime, asinFig.47-2, theelectric field atatime tlater would have moved thedistance ct,asindicated inthefigure. Mathematically, wecansaythatintheone-dimen- sional example wearetaking, theelectric field isafunction ofx—ct.Wesee thatatt=0,itissome function ofx.Ifweconsider alater time, weneed onlyto increase xsomewhat togetthesame value oftheelectric field. Forexample, ifthe maximum fieldoccurred atx=3attimezero, thentofindthenewposition of themaximum fieldattimetweneed x—ct=3 orx=3+ct. Weseethatthiskind offunction represents thepropagation ofawave. Such afunction, f(x—ct),then represents awave. Wemay summarize this description ofawave bysaying simply that f(x—ct)=f(x+Ax—c(t+At)), when Ax=cAt. There is,ofcourse, another possibility, i.e.,thatinstead ofa source totheleftasindicated inFig.47-2, wehave asource ontheright, sothat thewave propagates toward negative x.Then thewave would bedescribed by g(x+ct). There istheadditional possibility thatmore thanonewave exists inspace at thesame time, andsotheelectric fieldisthesumofthetwofields, eachonepropa- gating independently. This behavior ofelectric fields maybedescribed bysaying thatiff1(x—ct)isawave, andiff2(x —ct)isanother wave, then their sumis alsoawave. This iscalled theprinciple ofsuperposition. Thesame principle is valid insound. Wearefamiliar withthefactthatifasound isproduced, wehearwithcomplete fidelity thesame sequence ofsounds aswasgenerated. Ifwehadhighfrequencies travelling faster than lowfrequencies, ashort, sharp noise would beheard asa succession ofmusical sounds. Similarly, ifredlighttravelled faster than blue light, aflashofwhite lightwould beseenfirstasred,thenaswhite. andfinally as blue. Wearefamiliar withthefactthatthisisnotthecase. Both sound and light travel withaspeed inairwhich isverynearly independent offrequency. Examples ofwave propagation ‘forwhich thisindependence isnottruewillbe considered inChapter 48. Inthecaseoflight(electromagnetic waves) wegavearulewhich determined theelectric field atapoint asaresult oftheacceleration ofacharge. Onemight expect nowthatwhat weshould doisgivearulewhereby some quality oftheair, saythepressure, isdetermined atagiven distance from asource interms ofthe source motion, delayed bythetravel time ofthesound. Inthecase oflight this procedure wasacceptable because allthatweknew wasthatacharge atoneplace exerts aforce onanother charge atanother place. Thedetails ofpropagation from theoneplace totheother were notabsolutely essential. Inthecaseofsound, however, weknow thatitpropagates through theairbetween thesource andthe hearer, anditiscertainly anatural question toaskwhat, atanygiven moment, thepressure oftheairis.Wewould like,inaddition, toknow exactly howtheair moves. Inthecase ofelectricity wecould accept arule, since wecould saythat wedonotyetknow thelawsofelectricity, butwecannot make thesame remark withregard tosound. Wewould notbesatisfied witharulestating howthesound pressure moves through theair,because theprocess ought tobeunderstandable as aconsequence ofthelawsofmechanics. Inshort, sound isabranch ofmechanics, andsoitistobeunderstood interms ofNewton’s laws. Thepropagation of sound from oneplace toanother ismerely aconsequence ofmechanics andthe properties ofgases, ifitpropagates inagas,oroftheproperties ofliquids orsolids, ifitpropagates through such mediums. Later weshall derive theproperties oflight anditswave propagation inasimilar wayfrom thelaws ofelectrody- namics. 47-2 47-2 Thepropagation ofsound Weshall giveaderivation oftheproperties ofthepropagation ofsound between thesource andthereceiver asaconsequence ofNewton's laws, andweshall notconsider theinteraction with thesource andthereceiver. Ordinarily weem- phasize aresult rather than aparticular derivation ofit.Inthischapter wetake theopposite view. Thepoint here, inacertain sense, isthederivation itself.This problem ofexplaining newphenomena interms ofoldones, when weknow the laws oftheoldones, isperhaps thegreatest artofmathematical physics. The mathematical physicist hastwoproblems: oneistofindsolutions, given theequa- tions, andtheother istofindtheequations which describe anewphenomenon. Thederivation here isanexample ofthesecond kind ofproblem. Weshall take thesimplest example here—the propagation ofsound inone dimension. Tocarry outsuch aderivation itisnecessary firsttohave some kind ofunderstanding ofwhat isgoing on.Fundamentally what isinvolved isthatif anobject ismoved atoneplace intheair,weobserve thatthere isadisturbance which travels through theair. Ifweaskwhat kind ofdisturbance, wewould say thatwewould expect thatthemotion oftheobject produces achange ofpressure. Ofcourse, iftheobject ismoved gently. theairmerely flows around it,butwhat we areconcerned with isarapid motion. sothatthere isnotsuflicient time forsuch aflow. Then, with themotion. theairiscompressed andachange ofpressure is produced which pushes onadditional air. This airisinturn compressed, which leads again toanextra pressure, andawave ispropagated. Wenowwant toformulate such aprocess. Wehave todecide what variables weneed. Inourparticular problem wewould need toknow how much theair hasmoved, sothattheairdisplacement inthesound wave iscertainly onerelevant variable. Inaddition wewould liketodescribe howtheairdensity changes asitis displaced. Theairpressure alsochanges, sothisisanother variable ofinterest. Then, ofcourse, theairhasavelocity, sothatweshall have todescribe thevelocity oftheairparticles. Theairparticles alsohave accelerations——but aswelistthese many variables wesoon realize thatthevelocity andacceleration would beknown ifweknew howtheairdisplacement varies with time. Aswesaid, weshall consider thewave inonedimension. Wecandothisif wearesufficiently farfrom thesource thatwhat wecallthewavefronts arevery nearly planes. Wethusmake ourargument simpler bytaking theleast complicated example. Weshall then beabletosaythatthedisplacement, X,depends only on xandt,andnotonyandz.Therefore thedescription oftheairisgiven byx(x,t). Isthisdescription complete? Itwould appear tobefarfrom complete, for weknow none ofthedetails ofhowtheairmolecules aremoving. They aremoving inalldirections, andthisstate ofaffairs iscertainly notdescribed bymeans ofthis function X(x,t).From thepoint ofview ofkinetic theory, ifwehave ahigher density ofmolecules atoneplace andalower density adjacent tothatplace, the molecules would move away from theregion ofhigher density totheoneoflower density, soastoequalize thisdifference. Apparently wewould notgetanoscillation andthere would benosound. What isnecessary togetthesound wave isthis situation: asthemolecules rush outoftheregion ofhigher density andhigher pressure, they givemomentum tothemolecules intheadjacent region oflower density. Forsound tobegenerated, theregions overwhich thedensity andpressure change must bemuch larger than thedistance themolecules travel before colliding with other molecules. Thisdistance isthemean freepath. andthedistance between pressure crests andtroughs must bemuch larger than this. Otherwise themolecules would move freely from thecrest tothetrough andimmediately smear outthewave. Itisclear that wearegoing todescribe thegasbehavior onascale large compared with themean freepath, andsotheproperties ofthegaswillnotbe described interms oftheindividual molecules. Thedisplacement, forexample, willbethedisplacement ofthecenter ofmass ofasmall element ofthegas,and thepressure ordensity willbethepressure ordensity inthisregion. Weshall callthepressure Pandthedensity p,andthey willbefunctions ofxandt.We must keep inmind thatthisdescription isanapproximation which isvalid only when these gasproperties donotvary toorapidly with distance. 47-3 mi X(x,t) __., Ij[om VOLUME [NEW VOI-WE l | X1-X7—f)x, (1+£iF+l‘X'(x+AT,n lI__él__ ‘<—-— xrn./u,t) —-—>J Fig.47-3. The displacement ofthe ciircitxisX(x,tl, and atx+Axitis Xlx+Ax,t).Theoriginal volume ofthe dirfor0unitcireci oftheplane wove is Ax; the new volume isAx—l—X(x+ Ax,1)—Xlx,1).47-3 Thewave equation Thephysics ofthephenomenon ofsound waves thusinvolves three features: I.Thegasmoves andchanges thedensity. II.Thechange indensity corresponds toachange inpressure. III.Pressure inequalities generate gasmotion. Letusconsider IIfirst. Foragas,aliquid, orasolid, thepressure issome function ofthedensity. Before thesound wave arrives, wehave equilibrium, withapressure P0andacorresponding density p0.Apressure Pinthemedium isconnected to thedensity bysome characteristic relation P=f(p)and, inparticular, theequilib- rium pressure P0isgiven byP0=f(p0).Thechanges ofpressure insound from theequilibrium value areextremely small. Aconvenient unitformeasuring pressure isthebar,where lbar=105n/m2. Thepressure oflstandard atmosphere isvery nearly lbar: latm =1.0133 bars. Insound weusealogarithmic scale ofin- tensities since thesensitivity oftheearisroughly logarithmic. This scale isthe decibel scale, inwhich theacoustic pressure level forthepressure amplitude P isdefined as I(acoustic pressure level) =20log10(P/PM) indb, (47.1) where thereference pressure Pm;=2Xl0_1° bar. Apressure amplitude of P=l03P,,., =2Xl0_7 bar* corresponds toamoderately intense sound of 60decibels. Weseethatthepressure changes insound areextremely small com- pared with theequilibrium, ormean, pressure oflatm. Thedisplacements and thedensity changes arecorrespondingly extremely small. Inexplosions wedonot have such small changes; theexcess pressures produced canbegreater than latm. These large pressure changes leadtoneweffects which weshall consider later. Insound wedonotoften consider acoustic intensity levels over 100db; 120dbisalevel which ispainful totheear. Therefore, forsound, ifwewrite P=Po+Pt, P=Po+pt, (47-2) weshall always have thepressure change P,very small compared with Poandthe density change p,very small compared with po.Then Po+Pa=f(P0+Pt)=f(Po) +Ptf’(Po), (47-3) where P0=f(p0)andf'(p0)stands forthederivative off(p)evaluated atp=po. Wecantakethesecond stepinthisequality onlybecause p,isverysmall. Wefindin thiswaythattheexcess pressure P,isproportional totheexcess density pe,andwe may calltheproportionality factor KI P,=xpe, where K=f’(p0) =(dP/dp)O. (II) (47.4) Therelation weneeded forIIisthisvery simple one. Letusnowconsider I.Weshall suppose thattheposition ofaportion ofair undisturbed bythesound wave isxandthedisplacement atthetime tduetothe sound isx(x,t),sothatitsnewposition isx+X(x,1),asinFig.47—3. Now the undisturbed position ofanearby portion ofairisx+Ax.anditsnewposition isx+Ax+X(x+Ax,t).Wecannowfindthedensity changes inthefollowing way. Since wearelimiting ourselves toplane waves, wecantake aunit area perpendicular tothex-direction, which isthedirection ofpropagation ofthesound wave. Theamount ofair,perunitarea, inAxisthen p0Ax,where p0istheun- disturbed, orequilibrium, airdensity. This air,when displaced bythesound wave, now liesbetween x+X(x,t)andx+Ax+X(x+Ax,t),sothatwehave the same matter inthisinterval that wasinAxwhen undisturbed. Ifpisthenew density, then p0Ax=p[x+Ax+X(x—l—Ax,t)—x—X(x,t)]. (47.5) *With thischoice ofPM,thePisnotthepeakpressure inthesound wave butthe“root- mean-square" pressure, which is1/(2)1/2times thepeak pressure. 47-4 Since Axissmall, wecanwrite X(x+Ax,t)—X(x,t)=(6x/6x) Ax. This derivative isapartial derivative, since xdepends onthetimeaswellasonx. Ourequation then is p0Ax=pAx+Ax) (47.6) of 6pt=(pt+p.)g+pt+p... (41.1) Now insound waves allchanges aresmall sothatp,issmall, Xissmall, and6X/6x isalsosmall. Therefore intherelation thatwehave justfound, 3X 19X Pe=_P0 E _Pe5’ wecanneglect p,6x/6x compared with p06x/6x. Thus wegettherelation we needed forI: p.=-1». <1) (41.9) This equation iswhat wewould expect physically. Ifthedisplacements vary with x,then there willbedensity changes. Thesignisalsoright: ifthedisplacement X increases with x,sothattheairisstretched out,thedensity must godown. Wenow need thethird equation, which istheequation ofthemotion pro- duced bythepressure. Ifweknow therelation between theforce andthepressure, wecanthen gettheequation ofmotion. Ifwetakeathinslabofairoflength Ax andofunitarea perpendicular tox,then themass ofairinthisslabisp0Axand ithastheacceleration 82X/6t2, sothemass times theacceleration forthisslabof matter isp0Ax(62X/6t2). (Itmakes nodiflerence forsmall Axwhether theaccelera- tion62X/8t2 isevaluated atanedge oftheslaboratsome intermediate position.) Ifnow wefindtheforce onthismatter foraunitareaperpendicular tox,itwill then beequal topoAx(62X/6t2). Wehave theforce inthe+x-direction, atx,of amount P(x, t)perunitarea, andwehave theforce intheopposite direction, at x+Ax,ofamount P(x+Ax,t)perunitarea (Fig. 47—4): 6P 8P,P(x, t)—P(x—l—Ax,t)=—5Ax=—E;Ax, (47.10) since Axissmall andsince theonly partofPwhich changes istheexcess pressure Pe.Wenow have III: 62x_ 6P. andsowehave enough equations tointerconnect things andreduce down toone variable, saytoX.Wecaneliminate P,from IIIbyusing II,sothatweget 62x_ dp,,pgW —-—K 5 s andthen wecanuseItoeliminate pg.Inthiswaywefindthatp0cancels outand thatweareleftwith 62x azx Weshall callof=K,sothatwecanwrite 2afx 1axax? =g Tfl" This isthewave equation which describes thebehavior ofsound inmatter. 47-5P(x,1): I:P(x+Ax,t) _'___Al '- Fig.47-4. The net force inthe positive x-direction produced bythepres- sure ucting onunitarea perpendicular toxis—(6P/6x) Ax. 47-4 Solutions ofthewave equation Wenowcanseewhether thisequation really does describe theessential properties ofsound waves inmatter. Wewant todeduce thatasound pulse, or disturbance, willmove withaconstant speed. Wewant toverify thattwodifferent pulses canmove through each other——the principle ofsuperposition. Wealso want toverify thatsound cangoeither totheright ortotheleft.Allthese properties should becontained inthisoneequation. Wehave remarked that anyplane-wave disturbance which moves with a constant velocity vhastheformf(x—vt).Now wehave toseewhether X(x,t)= f(x—-vt)isasolution ofthewave equation. When wecalculate 6X/6x, wegetthe derivative ofthefunction, 6X/6x =f'(x—vt).Differentiating oncemore, wefind 62x6—fl=f”(x —vt). (47.15) Thedifferentiation ofthissame function with respect totgives —vtimes the derivative ofthefunction, or6X/8t =——vf’(x —vt),and thesecond time de- rivative is 2 %=v2f”(x -vt). (47.16) Itisevident thatf(x—vt)willsatisfy thewave equation provided thewavevelocity visequal toc,,. Wefind, therefore, from thelaws ofmechanics that anysound disturbance propagates with thevelocity c,,and inaddition wefind that C.=K“=<dP/dot”. andsowehave related thewave velocity toaproperty ofthemedium. Ifweconsider awave travelling intheopposite direction, sothatX(x,t)= g(x+vt),itiseasytoseethatsuch adisturbance alsosatisfies thewave equation. Theonly difference between such awave andonetravelling from lefttoright is inthesign ofv,butwhether wehave x+viorx—vtasthevariable inthe function does notaflect thesignof62X/éttz, since itinvolves only v2.Itfollows thatwehave asblution forwaves propagating ineither direction with speed c,. Anextremely interesting question isthatofsuperposition. Suppose onesolu- tionofthewave equation hasbeen found, sayX1.This means thatthesecond de- rivative ofX1with respect toxisequal toI/cftimes thesecond derivative ofX1 with respect tot.Now anyother solution X2hasthissame property. Ifwesuper- posethese twosolutions, wehave X(x,I)=X1(X,I)+X2(X,I), (47-17) andwewish toverify thatX(x,t)isalsoawave, i.e.,thatXsatisfies thewave equa- tion. Wecaneasily prove thisresult, since wehave 62x 62x1 62x2—=— 1 47.186x2 6x2 +6x2 ( ) and, inaddition, 62x 62x, 62x2 W=W+Tn‘ (47-19> Itfollows that 62X/6x2 =(1/cf)62X/Gt’, sowehave verified theprinciple of superposition. Theproof oftheprinciple ofsuperposition follows from thefact thatthewave equation islinear inX. Wecannow expect that aplane light wave propagating inthex-direction, polarized sothattheelectric fieldisinthey-direction, willsatisfy thewave equation a’-’E,,_1a2E,, 47-6 where cisthespeed oflight. Thiswave equation isoneoftheconsequences of Maxwell’s equations. Theequations ofelectrodynamics willlead tothewave equation forlight justastheequations ofmechanics lead tothewave equation forsound. 47-5 Thespeed ofsound Ourdeduction ofthewave equation forsound hasgiven usaformula which connects thewave speed with therateofchange ofpressure with thedensity atthe normal pressure: dPcf=(710), - (47.21) Inevaluating thisrateofchange, itisessential toknow howthetemperature varies. Inasound wave, wewould expect thatintheregion ofcompression thetemperature would beraised, andthatintheregion ofrarefaction thetemperature would be lowered. Newton wasthefirsttocalculate therateofchange ofpressure with density, andhesupposed thatthetemperature remained unchanged. Heargued that theheat wasconducted from oneregion totheother sorapidly that the temperature could notriseorfall. Thisargument gives theisothermal speed of sound, anditiswrong. Thecorrect deduction wasgiven later byLaplace, whoput forward theopposite idea—that thepressure andtemperature change adiabatically inasound wave. Theheatflowfrom thecompressed region totherarefied region is negligible solongasthewavelength islongcompared withthemean freepath. Under thiscondition theslight amount ofheat flow inasound wave does not affect thespeed, although itgives asmall absorption ofthesound energy. Wecan expect correctly thatthisabsorption increases asthewavelength approaches the mean freepath, butthese wavelengths aresmaller byfactors ofabout amillion thanthewavelengths ofaudible sound. Theactual variation ofpressure withdensity inasound wave istheonethat allows noheatflow. Thiscorresponds totheadiabatic variation, which wefound tobePV“’=const, where Vwasthevolume. Since thedensity pvaries inversely withV,theadiabatic connection between Pandpis P=const p", (47.22) from which wegetdP/dp ='YP/p. Wethenhave forthespeed ofsound the relation CZ=%- (47.23) Wecanalso write cf='YPV/pV and make useoftherelation PV=NkT. Further, weseethatpVisthemass ofgas,which canalsobeexpressed asNm, orasp,where misthemass ofamolecule auditisthemolecular weight. Inthis waywefindthat cf=inf=151, (47.24) from which itisevident thatthespeed ofsound depends onlyonthegastemperature andnotonthepressure orthedensity. Wealsohave observed that kT=31;m(v2), (47.25) where (112)isthemean square ofthespeed ofthemolecules. Itfollows that cf=(‘Y/3)(v2), or .Y1/2 c,=<5) v,,.,. (47.26) This equation states thatthespeed ofsound issome number which isroughly l/(3)1’ 2times some average speed, v,,,,,ofthemolecules (thesquare root ofthe 47-7 mean square velocity). Inother words, thespeed ofsound isofthesame order of magnitude asthespeed ofthemolecules, andisactually somewhat lessthan this average speed. Ofcourse wecould expect such aresult, because adisturbance likeachange inpressure is,after all,propagated bythemotion ofthemolecules. However, such anargument does nottellustheprecise propagation speed; itcould have turned outthat sound wascarried primarily bythefastest molecules, orbythe slowest molecules. Itisreasonable andsatisfying thatthespeed ofsound isroughly %_;oftheaverage molecular speed v,,,,. 47-8 48 Beats 48-1 Adding twowaves Some time agowediscussed inconsiderable detail theproperties oflight waves andtheir interference—that is,theeffects ofthesuperposition oftwowaves from diflerent sources. Inallthese analyses weassumed thatthefrequencies ofthe sources were allthesame. Inthischapter weshall discuss some ofthephenomena which result from theinterference oftwosources which have difierent frequencies. Itiseasytoguess what isgoing tohappen. Proceeding inthesame wayaswe have done previously, suppose wehave twoequal oscillating sources ofthesame frequency whose phases aresoadjusted, say,thatthesignals arrive inphase at some point P.Atthatpoint, ifitislight, thelight isvery strong; ifitissound, it isvery loud; orifitiselectrons, many ofthem arrive. Ontheother hand, ifthe arriving signals were 180°outofphase, wewould getnosignal atP,because the netamplitude there isthen aminimum. Now suppose that someone twists the “phase knob” ofoneofthesources andchanges thephase atPback andforth, say, firstmaking it0°andthen 180°, andsoon.Ofcourse, wewould thenfindvariations inthenetsignal strength. Now wealsoseethatifthephase ofonesource isslowly changing relative tothat oftheother inagradual, uniform manner, starting at zero, going uptoten,twenty, thirty, forty degrees, andsoon,then what wewould measure atPwould beaseries ofstrong andweak “pulsations,” because when the phase shifts through 360° theamplitude returns toamaximum. Ofcourse, to saythatonesource isshifting itsphase relative toanother atauniform rateisthe same assaying thatthenumber ofoscillations persecond isslightly different for thetwo. Soweknow theanswer: ifwehave twosources atslightly different frequencies weshould find, asanetresult, anoscillation with aslowly pulsating intensity. That isallthere really istothesubject! Itisvery easy toformulate thisresult mathematically also. Suppose, forex- ample, thatwehave twowaves, andthatwedonotworry forthemoment about allthespatial relations, butsimply analyze what arrives atP.From onesource, let ussay,wewould have coswlt,andfrom theother source, cosw21,where thetwo w’sarenotexactly thesame. Ofcourse theamplitudes maynotbethesame, either, butwecansolve thegeneral problem later; letusfirsttake thecase where the amplitudes areequal. Then thetotal amplitude atPisthesumofthese twocosines. Ifweplottheamplitudes ofthewaves against thetime, asinFig.48-1, weseethat CoslO1rt48-1 Adding twowaves 48-2 Beat notes andmodulation 48-3 Side bands 48-4 Localized wave trains 48-5 Probability amplitudes for particles 48-6 Waves inthree dimensions 48-7 Normal modes Cos81rt \\ /Z__\\ /, §\ Fig.48-1. Thesuperposition oftwo /\\ / \ \ / \\ // \ cosine waves with frequencies inthe rutio 8:10. Theprecise repetition ofthe pattern within each "beat" isnottypical ofthegeneral ccise. 48-1/ / \ / / \ / \ // \ / \ / where thecrests coincide wegetastrong wave, andwhere atrough andcrest coincide wegetpractically zero, andthen when thecrests coincide again wegeta strong wave again. Mathematically, weneed only toaddtwocosines andrearrange theresult somehow. There exist anumber ofuseful relations among cosines which arenot difficult toderive. Ofcourse weknow that e'i(t1“l'b) =eitleib, andthate“hasarealpart, cosa,andanimaginary part, sina.Ifwetakethereal partofe““+"), wegetcos(a+b).Ifwemultiply out: e‘“e“’ =(cosa+isina)(cosb +isinb), wegetcosacosb—sinasinb,plus some imaginary parts. Butwenow need only therealpart, sowehave cos(a+b)=cosacosb—sinasinb. (48.2) Now ifwechange thesignofb,since thecosine does notchange signwhile the sinedoes, thesame equation, fornegative b,is cos(a—b)=cosacosb+sinasinb. (48.3) Ifweaddthese twoequations together, welosethesines andwelearn thatthe product oftwocosines ishalfthecosine ofthesum, plus halfthecosine ofthe difference: cosacos b=%cos(a+b)+%cos(a—b). (48.4) Now wecanalsoreverse theformula andfindaformula forcosor+cosBif wesimplyleta=a+bandB=a—b. That is,a= §(a+/3)andb= §(a—,8),sothat cosa +cos/3 =2cos%(a+,8)cos%(a-B). (48.5) Now wecananalyze ourproblem. Thesumofcoswltandcos(021is coscult+cosw2t=2cos %(w1 +w2)tcos%(w1 —w2)t. (48.6) Now letussuppose thatthetwofrequencies arenearly thesame, sothat%(w1+(02) istheaverage frequency, andismore orlessthesame aseither. Butwl——(1)2is much smaller than color(.02because, aswesuppose, (.01and(1)2arenearly equal. That means that wecanrepresent thesolution bysaying that there isahigh- frequency cosine wave more orlessliketheones westarted with, butthatits“size” isslowly changing—its “size” ispulsating with afrequency which appears tobe %(w1 ——L02). Butisthisthefrequency atwhich thebeats areheard? Although (48.6) saysthattheamplitude goesascos%(w1 -(1:2),what itisreally telling usis that thehigh-frequency oscillations arecontained between twoopposed cosine curves (shown dotted inFig.48-1). Onthisbasis onecould saythattheamplitude varies atthefrequency %(w1 —(.02),butifwearetalking about theintensity of thewave wemust think ofitashaving twice thisfrequency. That is,themodulation oftheamplitude, inthesense ofthestrength ofitsintensity, isatfrequency wl— (02,although theformula tellsusthatwemultiply byacosine wave athalfthat frequency. Thetechnical basis forthedifference isthatthehigh frequency-wave hasalittle different phase relationship inthesecond half-cycle. Ignoring thissmall complication, wemay conclude thatifweaddtwowaves offrequency wland(.02,wewillgetanetresulting wave ofaverage frequency §(w1 +w2)which oscillates instrength with afrequency wl—L02. Ifthetwoamplitudes aredifferent, wecandoitalloveragain bymultiplying thecosines bydifferent amplitudes A1andA2,anddoalotofmathematics, re- arranging, andsoon,using equations like(48.2)—(48.5). However, there areother, easier ways ofdoing thesame analysis. Forexample, weknow thatitismuch 48-2 easier towork withexponentials than with sines andcosines andthatwecanrepre- sentA1coswltastherealpartofA1e“"1‘. Theother wave would similarly bethe realpart ofA2e“"1‘. Ifweaddthetwo, wegetA1ei“1’ +A2ei"’2’. Ifwethen factor outtheaverage frequency, wehave Aleiwlt +A2eiw2t =e1/2i(w1+w2)t[A 1el/ 2i(w1—w2)t +A2e—1/21Z(w1—w2)t]_ Again wehave thehigh-frequency wave with amodulation atthelower frequency. 48-2 Beat notes andmodulation Ifwearenow asked fortheintensity ofthewave ofEq.(48.7), wecaneither take theabsolute square oftheleftside, oroftheright side. Letustake theleft side. Theintensity then is 1=A?+/15+2.41/12 cos(0)1~(02);. (48.8) Weseethattheintensity swells andfallsatafrequency wl—(.02,varying between thelimits (A1+A2)2 and(A1—A2)2. IfA1¢A2,theminimum intensity isnotzero. Onemore waytorepresent thisideaisbymeans ofadrawing, likeFig.48-2. Wedraw avector oflength A1,rotating atafrequency wl,torepresent oneofthe waves inthecomplex plane. Wedraw another vector oflength A2,going around atafrequency (.02,torepresent thesecond wave. Ifthetwofrequencies areexactly equal, their resultant isoffixed length asitkeeps revolving, andwegetadefinite, fixed intensity from thetwo. Butifthefrequencies areslightly different, thetwo complex vectors goaround atdifferent speeds. Figure 48-3 shows what the situation looks likerelative tothevector A1e"°’1‘. WeseethatA2isturning slowly away from A1,andsotheamplitude thatwegetbyadding thetwoisfirststrong, andthen, asitopens out,when itgetstothe180° relative position theresultant getsparticularly weak, andsoon.Asthevectors goaround, theamplitude ofthe sumvector getsbigger andsmaller, andtheintensity thuspulsates. Itisarelatively simple idea, andthere aremany different ways ofrepresenting thesame thing. Theeffect isveryeasytoobserve experimentally. Inthecaseofacoustics, we may arrange twoloudspeakers driven bytwoseparate oscillators, oneforeach loudspeaker, sothatthey each make atone. Wethus receive onenote from one source andadifferent note from theother source. Ifwemake thefrequencies exactly thesame, theresulting effect willhave adefinite strength atagiven space location. Ifwethen de-tune them alittle bit,wehear some variations inthe intensity. Thefarther they arede-tuned, themore rapid arethevariations of sound. Theearhassome trouble following variations more rapid than tenorso persecond. Wemayalsoseetheeffect onanoscilloscope which simply displays thesum ofthecurrents tothetwospeakers. Ifthefrequency ofpulsing isrelatively low,we simply seeasinusoidal wave train whose amplitude pulsates, butaswemake the pulsations more rapid weseethekind ofwave shown inFig.48-1. Aswegoto greater frequency differences, the“bumps” move closer together. Also, ifthe amplitudes arenotequal andwemake onesignal stronger than theother, thenwe getawave whose amplitude does noteverbecome zero, justasweexpect. Every- thing works thewayitshould, both acoustically andelectrically. Theopposite phenomenon occurs tool Inradio transmission using so-called amplitude modulation (AM), thesound isbroadcast bytheradio station asfollows: theradio transmitter hasanACelectric oscillation which isataveryhighfrequency, forexample 800kilocycles persecond, inthebroadcast band. Ifthiscarrier signal isturned on,theradio station emits awave which isofuniform amplitude at800,000 oscillations asecond. Thewaythe“information” istransmitted, theuseless kind ofinformation about what kind ofcartobuy, isthatwhen somebody talks intoa microphone theamplitude ofthecarrier signal ischanged instepwiththevibrations ofsound entering themicrophone. 48-3A u)l= (4)2-= UJ K2 Fig.48-2. Theresultant oftwocom- plex vectors ofequal frequency. 3 4 2 6 9 7 e Fig.48-3. Theresultant oftwocom- plex vectors ofunequal frequency, as seen intherotating frame ofreference of onevector. Nine successive positions of theslowly rotating vector areshown. rilllllnrllllllmrtllllllllllllllllllllllllllFig.48-4. Amodulated carrier wave. lnthisschematic sketch, 0.),/cum =5. Inanactual radiowave, 0),/w,,, ~100. We-(Du Bk UITU 7) Fig.48-5. The frequency spectrum ofacarrier wave cu,modulated bya single cosine wave w,,,.Ifwetake asthesimplest mathematical casethesituation where asoprano is singing aperfect note, with perfect sinusoidal oscillations ofhervocal cords, then wegetasignal whose strength isalternating asshown inFig.48—4. The audiofrequency alternation isthen recovered inthereceiver; wegetridofthe carrier wave andjustlook attheenvelope which represents theoscillations ofthe vocal cords, orthesound ofthesinger. The loudspeaker then makes corre- sponding vibrations atthesame frequency intheair,andthelistener isthen essentially unable totellthedifference, sothey say. Because ofanumber ofdis- tortions andother subtle effects, itis,infact,possible totellwhether wearelistening toaradio ortoarealsoprano; otherwise theideaisasindicated above. 48-3 Side bands Mathematically, themodulated wave described above would beexpressed as S=(1+bcos wmt) cosw,t, (48.9) where wcrepresents thefrequency ofthecarrier andwmisthefrequency ofthe audio tone. Again weuseallthose theorems about thecosines, orwecanuse e“;itmakes nodifference-—-it iseasier with ei’,butitisthesame thing. We then get S=coswct+ébcos(we+w,,,)t +{vbcos(w,—w,,,)t. (48.10) So,from another point ofview, wecansaythattheoutput wave ofthesystem consists ofthree waves added insuperposition: first, theregular wave atthefre- quency we,thatis,atthecarrier frequency, andthen twonewwaves attwonew frequencies. Oneisthecarrier frequency plusthemodulation frequency, andthe other isthecarrier frequency minus themodulation frequency. If,therefore, we make some kind ofplot oftheintensity being generated bythegenerator asa function offrequency, wewould findalotofintensity atthefrequency ofthecarrier, naturally, butwhen asinger started tosing, wewould suddenly alsofindintensity proportional tothestrength ofthesinger, b2,atfrequency wc+wmandw,—wm, asshown inFig.48-5. These arecalled sidebands; when there isamodulated signal from thetransmitter, there aresidebands. Ifthere ismore than onenote atthesame time, saymmandcum’,there aretwoinstruments playing; orifthere is anyother complicated cosine wave, then, ofcourse, wecanseefrom themathematics thatwegetsome more waves thatcorrespond tothefrequencies w,:l:wml. Therefore, when there isacomplicated modulation thatcanberepresented asthesum ofmany cosines,* wefindthattheactual transmitter istransmitting over arange offrequencies, namely thecarrier frequency plusorminus themaxi- mum frequency thatthemodulation signal contains. Although atfirstwemight believe thataradio transmitter transmits only at thenominal frequency ofthecarrier, since there arebig, superstable crystal oscillators inthere, andeverything isadjusted tobeatprecisely 800kilocycles, themoment someone announces thatthey areat800kilocycles, hemodulates the 800kilocycles, andsotheyarenolonger precisely at800kilocycles! Suppose that theamplifiers aresobuilt thatthey areabletotransmit over agood range ofthe ear’s sensitivity (theearcanhear upto20,000 cycles persecond, butusually radio transmitters andreceivers donotwork beyond 10,000, sowedonothear the highest parts), then, when theman speaks, hisvoice may contain frequencies ranging up,say,to10,000 cycles, sothetransmitter istransmitting frequencies *Aslight sideremark: Inwhat circumstances canacurve berepresented asasumof alotofcosines? Answer: Inallordinary circumstances, except forcertain cases the mathematicians candream up.Ofcourse, thecurve must have only onevalue atagiven point, anditmust notbeacrazy curve which jumps aninfinite number oftimes inan infinitesimal distance, orsomething likethat. Butaside from such restrictions any reasonable curve (onethatasinger isgoing tobeabletomake byshaking hervocal cords) canalways becompounded byadding cosine waves together. 48-4 which mayrange from 790to810kilocycles persecond. Now ifthere were another station at795kc/sec, there would bealotofconfusion. Also, ifwemade our receiver sosensitive thatitpicked uponly 800, anddidnotpick upthe10kilo- cycles oneither side, wewould nothear what theman wassaying, because the information would beonthese other frequencies! Therefore itisabsolutely es- sential tokeep thestations acertain distance apart, sothattheir sidebands donot overlap and, also, thereceiver must notbesoselective thatitdoes notpermit reception ofthesidebands aswellasofthemain nominal frequency. Inthecase ofsound, thisproblem does notreally cause much trouble. Wecanhear over a 120 kc/sec range, andwehave usually from 500to1500 kc/sec inthebroadcast band, sothere isplenty ofroom forlotsofstations. Thetelevision problem ismore difficult. Astheelectron beam goes across thefaceofthepicture tube, there arevarious little spots oflight anddark. That “light” and“dark” isthe“signal.” Now ordinarily thebeam scans overthewhole picture, 500lines, approximately, inathirtieth ofasecond. Letusconsider thatthe resolution ofthepicture vertically andhorizontally ismore orlessthesame, so thatthere arethesame number ofspots perinch along ascan line. Wewant to beabletodistinguish dark from light, dark from light, dark from light, over, say, 500lines. Inorder tobeabletodothiswithcosine waves, theshortest wavelength needed thus corresponds toawavelength, from maximum tomaximum, ofone 250th ofthescreen size. Sowehave 250X500X30pieces ofinformation per second. Thehighest frequency thatwearegoing tocarry, therefore, isclose to 4megacycles persecond. Actually, tokeep thetelevision stations apart, wehave tousealittle bitmore than this,about 6mc/sec; partofitisused tocarry thesound signal, andother information. So,television channels are6megacycles persecond wide. Itcertainly would notbepossible totransmit TVonan800kc/sec carrier, since wecannot modulate atahigher frequency than thecarrier. Atanyrate, thetelevision band starts at54megacycles. Thefirsttransmission channel, which ischannel 2(l),hasafrequency range from 54to60mc/sec, which is6mc/sec wide. “But,” onemight say,“wehave justproved thatthere were sidebands onboth sides, andtherefore itshould betwice thatwide.” Itturns out thattheradio engineers arerather clever. Ifweanalyze themodulation signal using notjustcosine terms, butcosine andsineterms, toallow forphase differences, wethen seethatthere isadefinite, invariant relationship between thesideband on thehigh-frequency sideandthesideband onthelow-frequency side. What we mean isthatthere isnonewinformation onthatother sideband. Sowhat isdone istosuppress onesideband, andthereceiver iswired inside such thattheinforma- tionwhich ismissing isreconstituted bylooking atthesingle sideband andthe carrier. Single side-band transmission isaclever scheme fordecreasing theband widths needed totransmit information. 48-4 Localized wave trains Thenextsubject weshall discuss istheinterference ofwaves inboth space and time. Suppose thatwehave twowaves travelling inspace. Weknow, ofcourse, thatwecanrepresent awave travelling inspace bye““"_"”). This might be,for example, thedisplacement inasound wave. This isasolution ofthewave equation provided that(02=k2c2, where cisthespeed ofpropagation ofthewave. In thiscasewecanwrite itase”°(’—°‘>, which isofthegeneral form f(x—ct).There- forethismust beawave which istravelling atthisvelocity, w/k, andthat isc andeverything isallright. Now wewant toaddtwosuch waves together. Suppose wehave awave that istravelling with onefrequency, andanother wave travelling with another fre- quency. Weleave tothereader toconsider thecase where theamplitudes are different; itmakes norealdifference. Thus wewant toadde“"’1‘“"1") +e““2"'°=”>. Wecanaddthese bythesame kind ofmathematics weused when weadded signal waves. Ofcourse, ifcisthesame forboth, thisiseasy, since itisthesame aswhat wedidbefore: e'iw1(t—:c/c) +e1'w2(t—:c/c) =eiwl t’+eiugt’, 48-5 except thatt’=t—-x/cisthevariable instead oft.Sowegetthesame kind of mo/dulations, naturally, butwesee,ofcourse, thatthose modulations aremoving along with thewave. Inother words, ifweadded twowaves, butthese waves were notjustoscillating, butalsomoving inspace, then theresultant wave would move along also, atthesame speed. Now wewould liketogeneralize thistothecaseofwaves inwhich therelation- shipbetween thefrequency andthewave number kisnotsosimple. Example: material having anindex ofrefraction. Wehave already studied thetheory ofthe index ofrefraction inChapter 31,where wefound thatwecould write k=nw/c, where nistheindex ofrefraction. Asaninteresting example, forx-rays wefound thattheindex nis 2 n=1- (48.12) Weactually derived amore complicated formula inChapter 31,butthisoneisas good asany, asanexample. Incidentally, weknow thateven when wandkarenotlinearly proportional, theratio to/kiscertainly thespeed ofpropagation fortheparticular frequency and wave number. Wecallthisratio thephase velocity; itisthespeed atwhich thephase, orthenodes ofasingle wave, would move along: 11,,= (48.13) This phase velocity, forthecase ofx-rays inglass, isgreater than thespeed of light invacuum (since nin48.12 islessthan l),andthat isabitbothersome, because wedonotthink wecansend signals faster than thespeed oflight! What wearegoing todiscuss now istheinterference oftwowaves inwhich toandkhave adefinite formula relating them. Theabove formula fornsaysthat kisgiven asadefinite function ofw.Tobespecific, inthisparticular problem, the formula forkinterms ofwis k=9-1, (48.14)c we where a=Nqf/2e0m, aconstant. Atanyrate, foreach frequency there isadefinite wave number, andwewant toaddtwosuch waves together. LetusdoitjustaswedidinEq.(48.7): ei(w1t-klx) +ei(w2t—k2x) =el/2t'[(o.'l-|-<-:2)!-(kl-}-Ic2):z:] X {e1/2i[(<o1—w2)t—(k1-k2)a:] + e—1/2i[(w1—a:2)t—(k1—k2)z]} _ Sowehave amodulated wave again, awave which travels withthemean frequency andthemean wave number, butwhose strength isvarying with aform which de- pends onthedifference frequency andthedifference wave number. Now letustakethecasethatthedifference between thetwowaves isrelatively small. Letussuppose thatweareadding twowaves whose frequencies arenearly equal; then (L01+(02)/2 ispractically thesame aseither oneofthew’s,and similarly for(kl+k2)/2. Thus thespeed ofthewave, thefastoscillations, the nodes, isstillessentially m/k. Butlook, thespeed ofpropagation ofthemodulation isnotthesame! How much dowehave tochange xtoaccount foracertain amount oft?Thespeed ofthismodulation wave istheratio UM=_- (48.16) Thespeed ofmodulation issometimes called thegroup velocity. Ifwetake the casethatthedifference infrequency isrelatively small, andthedifference inwave number isthen alsorelatively small, then thisexpression approaches, inthelimit, 44,=3‘-1;? (48.17) 48-6 Inother words, fortheslowest modulation, theslowest beats, there isadefinite speed atwhich theytravel which isnotthesame asthephase speed ofthewaves- what amysterious thing! Thegroup velocity isthederivative ofwwithrespect tok,andthephase velocity isw/k. Letusseeifwecanunderstand why. Consider twowaves, again ofslightly different wavelength, asinFig.48-1. They areoutofphase, inphase, outof phase, andsoon.Now these waves represent, really, thewaves inspace travelling withslightly different frequencies also. Nowbecause thephase velocity, thevelocity ofthenodes ofthese twowaves, isnotprecisely thesame, something newhappens. Suppose weridealong withoneofthewaves andlookattheother one;iftheyboth went atthesame speed, then theother wave would stayright where itwasrelative tous,asweridealong onthiscrest. Werideonthatcrest andright opposite us weseeacrest; ifthetwovelocities areequal thecrests stayontopofeachother. Butitisnotsothatthetwovelocities arereally equal. There isonly asmall differ- ence infrequency andtherefore only asmall difference invelocity, butbecause of thatdifference invelocity, asweridealong theother wave moves slowly forward, say,orbehind, relative toourwave. Soastimegoeson,whathappens tothenode? Ifwemove onewave train justashade forward, thenode moves forward (orback- ward) aconsiderable distance. Thatis,thesumofthese twowaves hasanenvelope, andasthewaves travel along, theenvelope rides onthem atadifferent speed. Thegroup velocity isthespeed atwhich modulated signals would betransmitted. Ifwemade asignal, i.e.,some kindofchange inthewave thatonecould recog- nizewhen helistened toit,akind ofmodulation, then that modulation would travel atthegroup velocity, provided thatthemodulations were relatively slow. (When they arefast,itismuch more difficult toanalyze.) Now wemayshow (atlong last), thatthespeed ofpropagation ofx-rays ina block ofcarbon isnotgreater than thespeed oflight, although thephase velocity isgreater than thespeed oflight. Inorder todothat, wemust finddw/dk, which wegetbydifferentiating (48.14): dk/dw =1/c+a/w2c.Thegroup velocity, therefore, isthereciprocal ofthis,namely, 11,,=T+%M . (48.18) which issmaller thanclSoalthough thephases cantravel faster thanthespeed of light, themodulation signals travel slower, andthatistheresolution oftheapparent paradox! Ofcourse, ifwehave thesimple casethatw=kc,then dw/dk isalso c.Sowhen allthephases have thesame velocity, naturally thegroup hasthesame velocity. 48-5 Probability amplitudes forparticles Letusnowconsider onemore example ofthephase velocity which isextremely interesting. Ithastodowithquantum mechanics. Weknow thattheamplitude tofindaparticle ataplace can,insome circumstances, vary inspace andtime, let ussayinonedimension, inthismanner: -p=AW’-"’>, (48.19) where toisthefrequency, which isrelated totheclassical ideaoftheenergy through E=hw,andkisthewave number, which isrelated tothemomentum through p=hk.Wewould saytheparticle hadadefinite momentum pifthewave number were exactly k,that is,aperfect wave which goes onwith thesame amplitude everywhere. Equation (48.19) gives theamplitude, andifwetake theabsolute square, wegettherelative probability forfinding theparticle asafunction ofposi- tionandtime. This isaconstant, which means thattheprobability isthesame to findaparticle anywhere. Now suppose, instead, thatwehaveasituation where we know thattheparticle ismore likely tobeatoneplace than atanother. Wewould 48-7 ...¢1tlllllllllllI1v....-- ""'ll|llllllll"""" ‘Fig. 48-6. Alocalized wave train.represent such asituation byawave which hasamaximum anddiesoutoneither side(Fig. 48-6). (Itisnotquite thesame asawave like(48.1) which hasaseries of maxima, butitispossible, byadding several waves ofnearly thesame toandk together, togetridofallbutonemaximum.) Now inthose circumstances, since thesquare of(48.19) represents thechance offinding aparticle somewhere, weknow thatatagiven instant theparticle is most likely tobenear thecenter ofthe“lump,” where theamplitude ofthewave ismaximum. Ifnowwewaitafewmoments, thewaves willmove, andafter some timethe“lump” willbesomewhere else. Ifweknew thattheparticle originally was situated somewhere, classically, wewould expect thatitwould later beelsewhere asamatter offact,because ithasaspeed, after all,andamomentum. Thequantum theory, then, willgointothecorrect classical theory fortherelationship ofmo- mentum, energy, andvelocity onlyifthegroup velocity, thevelocity ofthemodula- tion, isequal tothevelocity thatwewould obtain classically foraparticle ofthe same momentum. Itisnownecessary todemonstrate thatthisis,orisnot,thecase. According totheclassical theory, theenergy isrelated tothevelocity through anequation like 2 E=——”i-_- 48.20\/1-02/c2 ( ) Similarly, themomentum is mvP=* - (48.21) \/l—02/02 That istheclassical theory, andasaconsequence oftheclassical theory, byeliminat- ingv,wecanshow that E2—p202 =m2c4. That isthefour-dimensional grand result thatwehave talked andtalked about, thatp,,p,, =m2;thatistherelation between energy andmomentum intheclassical theory. Now thatmeans, since these E’sandp’saregoing tobecome w’sandk’s, bysubstitution ofE=hwandp=hk,thatforquantum mechanics itisnecessary that 22 '17:’-8218=m2c2. (48.22) This, then, istherelationship between thefrequency andthewave number ofa quantum-mechanical amplitude wave representing aparticle ofmass m.From thisequation wecandeduce thatwis w=c\/k2 —l—m2c2/h2. Thephase velocity, w/k, ishere again faster than thespeed oflight! Now letuslook atthegroup velocity. Thegroup velocity should bedw/dk, thespeed atwhich themodulations move. Wehave todifferentiate asquare root, which isnotverydifficult. Thederivative is is= odk \/k2 _|_m2c2/;L2 Now thesquare root is,after all,w,sowecould write thisasdw/dk =c2k/w. Further, k/wisp/E, so C2 2 "4=Y”- Butfrom (48.20) and(48.21), C21)/E =v,thevelocity oftheparticle, according toclassical mechanics. Soweseethatwhereas thefundamental quantum-mechani- calrelationship E=hmandp=hk,fortheidentification ofwandkwith the classical Eandp,only produces theequation 0:2—kzcz =m2c“/h2, now we also understand therelationships (48.20) and(48.21) which connected Eandp 48-8 tothevelocity. Ofcourse thegroup velocity must bethevelocity oftheparticle iftheinterpretation isgoing tomake anysense. Ifwethink theparticle isover hereatonetime, andthentenminutes laterwethink itisoverthere, asthequantum mechanics said, thedistance traversed bythe“lump,” divided bythetime interval, must be,classically, thevelocity oftheparticle. 48-6 Waves inthree dimensions Weshall now bring ourdiscussion ofwaves toaclose with afewgeneral remarks about thewave equation. These remarks areintended togivesome view ofthefuture—not thatwecanunderstand everything exactly justnow, butrather to seewhat things aregoing tolook likewhen westudy waves alittle more. First of all,thewave equation forsound inonedimension was &_Lf§ax2_c2az2’ where cisthespeed ofwhatever thewave is—in thecaseofsound, itisthesound speed; inthecaseoflight, itisthespeed oflight. Weshowed thatforasound wave thedisplacements would propagate themselves atacertain speed. Buttheexcess pressure alsopropagates atacertain speed, andsodoes theexcess density. Sowe should expect thatthepressure would satisfy thesame equation, asindeed itdoes. Weshall leave ittothereader toprove thatitdoes. Hint: peisproportional tothe rateofchange ofXwithrespect tox.Therefore ifwedifferentiate thewave equation with respect tox,wewillimmediately discover that6x/6x satisfies thesame equa- tion. That istosay,p,satisfies thesame equation. ButP,isproportional tope, andtherefore P,does too. Sothepressure, thedisplacements, everything, satisfy thesame wave equation. Usually oneseesthewave equation forsound written interms ofpressure instead ofinterms ofdisplacement, because thepressure isascalar andhasno direction. Butthedisplacement isavector andhasdirection, anditisthuseasier toanalyze thepressure. Thenextmatter wediscuss hastodowith thewave equation inthree dimen- sions. Weknow thatthesound wave solution inonedimension ise““‘”"’), with to=kc,,butwealso know that inthree dimensions awave would berepre- sented bye"(°"_'°»"_'°1/"“'°=‘), where, inthiscase, w2=kzcf, which is,ofcourse, (kf,+kj+kf)cf. Now what wewant todoistoguess what thecorrect wave equa- tioninthree dimensions is.Naturally, forthecaseofsound thiscanbededuced by going through thesame dynamic argument inthree dimensions thatwemade in onedimension. Butweshall notdothat; instead wejustwrite down what comes out:theequation forthepressure (ordisplacement, oranything) is 62F, 8211* a2P, 162F e _ e_ 6x2+6y2+622_cf612 (4823) That thisistruecanbeverified bysubstituting ine“"”'l"‘). Clearly, every timewe differentiate with respect tox,wemultiply by——ik,,. Ifwedifferentiate twice, it isequivalent tomultiplying by—k§, sothefirstterm would become —kfP,, for thatwave. Similarly, thesecond term becomes —k§P,, andthethird term becomes —kfP,. Ontheright, weget—(w2/cf)P,. Then, ifwetake away theP,’sand change thesign, weseethattherelationship between kandwistheonethatwewant. Working backwards again, wecannot resist writing down thegrand equation which corresponds tothedispersion equation (48.22) forquantum-mechanical waves. If¢represents theamplitude forfinding aparticle atposition x,y,z,at thetime t,then thegreat equation ofquantum mechanics forfreeparticles isthis: 62¢ 82¢ 62¢ 102¢_m2c2 dx?+fly?+622_c2612_h2¢' (4824) First ofall,therelativity character ofthisexpression issuggested bytheappearance 48-9 ofx,y,zandtinthenicecombination relativity usually involves. Second, itisa wave equation which, ifwetryaplane wave, would produce asaconsequence that —k2 +m2/C2 =m2c2/h2, which istheright relationship forquantum mechanics. There isstillanother great thing contained inthewave equation: thefactthatany superposition ofwaves isalso asolution. Sothisequation contains allofthe quantum mechanics andtherelativity thatwehave been discussing sofar,atleast solong asitdeals with asingle particle inempty space with noexternal potentials orforces onit! 48-7 Normal modes Now weturntoanother example ofthephenomenon ofbeats which israther curious andalittle different. Imagine twoequal pendulums which have, between them, arather weak spring connection. They aremade asnearly aspossible the same length. Ifwepulloneaside andletgo,itmoves back andforth, anditpulls ontheconnecting spring asitmoves back andforth, andsoitreally isamachine forgenerating aforce which hasthenatural frequency oftheother pendulum. Therefore, asaconsequence ofthetheory ofresonance, which westudied before, when weputaforce onsomething atjusttheright frequency, itwilldrive it.So, sureenough, onependulum moving back andforth drives theother. However, in thiscircumstance there isanewthing happening, because thetotal energy ofthe system isfinite, sowhen onependulum pours itsenergy intotheother todrive it, itfinds itself gradually losing energy, until, ifthetiming isjustright along with the speed, itloses allitsenergy andisreduced toastationary condition! Then, of course, itistheother pendulum ballthathasalltheenergy andthefirstonewhich hasnone, andastime goes onweseethatitworks alsointheopposite direction, andthattheenergy ispassed back intothefirstball; thisisaveryinteresting and amusing phenomenon. Wesaid, however, that thisisrelated tothetheory of beats, andwemust nowexplain howwecananalyze thismotion from thepoint of viewofthetheory ofbeats. Wenote thatthemotion ofeither ofthetwoballs isanoscillation which has anamplitude which changes cyclically. Therefore themotion ofoneoftheballs ispresumably analyzable inadifferent way, inthatitisthesumoftwooscillations, present atthesame time buthaving twoslightly different frequencies. Therefore it ought tobepossible tofindtwoother motions inthissystem, andtoclaim that what wesawwasasuperposition ofthetwosolutions, because thisisofcourse a linear system. Indeed, itiseasy tofindtwoways thatwecould start themotion, each oneofwhich isaperfect, single-frequency motion—absolutely periodic. Themotion thatwestarted with before wasnotstrictly periodic, since itdidnot last; soon oneballwaspassing energy totheother andsochanging itsamplitude; butthere areways ofstarting themotion sothatnothing changes and, ofcourse, assoon asweseeitweunderstand why. Forexample, ifwemade both pendulums gotogether, then, since theyareofthesame length andthespring isnotthendoing anything, they willofcourse continue toswing likethatforalltime, assuming no friction andthateverything isperfect. Ontheother hand, there isanother possible motion which also hasadefinite frequency: thatis,ifwemove thependulums oppositely, pulling them aside exactly equal distances, then again they would be inabsolutely periodic motion. Wecanappreciate thatthespring justadds alittle totherestoring force thatthegravity supplies, thatisall,andthesystem justkeeps oscillating ataslightly higher frequency than inthefirst case. Why higher? Because thespring ispulling, inaddition tothegravitation, anditmakes thesystem alittle “stiffer,” sothatthefrequency ofthismotion isjustashade higher than that oftheother. Thus thissystem hastwoways inwhich itcanoscillate with unchanging amplitude: itcaneither oscillate inamanner inwhich both pendulums gothe same wayandoscillate allthetime atonefrequency, ortheycould goinopposite directions ataslightly higher frequency. Now theactual motion ofthething, because thesystem islinear, canbe represented asasuperposition ofthetwo. (The subject ofthischapter, remember, 48-10 istheeffects ofadding twomotions with different frequencies.) Sothink what would happen ifwecombined these twosolutions. Ifatt=0thetwomotions arestarted withequal amplitude andinthesame phase, thesumofthetwomotions means thatoneball, having been impressed onewaybythefirstmotion andthe other waybythesecond motion, isatzero, while theother ball, having been dis- placed thesame wayinboth motions, hasalarge amplitude. Astime goes on, however, thetwobasic motions proceed independently, sothephase ofonerelative totheother isslowly shifting. That means, then, thatafter asufficiently longtime, when thetime isenough that onemotion could have gone “900%” oscillations, while theother went only “900,” therelative phase would bejustreversed with respect towhat itwasbefore. That is,thelarge-amplitude motion willhave fallen tozero, andinthemeantime, ofcourse, theinitially motionless ballwillhave attained fullstrength! Soweseethatwecould analyze thiscomplicated motion either bytheidea thatthere isaresonance andthatonepasses energy totheother, orelsebythe superposition oftwoconstant-amplitude motions attwodifferent frequencies. 48-ll 49 Modes 49-1 Thereflection ofwaves This chapter willconsider some oftheremarkable phenomena which area result ofconfining waves insome finite region. Wewillbeledfirsttodiscover a fewparticular facts about vibrating strings, forexample, andthen thegeneraliza- tionofthese facts willgiveusaprinciple which isprobably themost far-reaching principle ofmathematical physics. Ourfirstexample ofconfining waves willbetoconfine awave atoneboundary. Letustakethesimple example ofaone-dimensional wave onastring. Onecould equally wellconsider sound inonedimension against awall, orother situations ofasimilar nature, buttheexample ofastring willbesufficient forourpresent purposes. Suppose thatthestring isheld atoneend, forexample byfastening it toan“infinitely solid” wall. This canbeexpressed mathematically bysaying that thedisplacement yofthestring attheposition x=0must bezero, because the enddoes notmove. Now ifitwere notforthewall, weknow thatthegeneral solution forthemotion isthesum oftwofunctions, F(x—ct)andG(x+ct), thefirstrepresenting awave travelling oneway inthestring, andthesecond a wave travelling theother wayinthestring: y=F(x—ct)+G(x—l—ct) (49.1) isthegeneral solution foranystring. Butwehave next tosatisfy thecondition thatthestring does notmove atoneend. Ifweputx=0inEq.(49.1) andex- amine yforanyvalue oft,wegety=F(-ct) +G(-l-ct). Now ifthisistobe zero foralltimes, itmeans thatthefunction G(ct) must be—F(—ct). Inother words, Gofanything must be—Fofminus thatsame thing. Ifthisresult isput back intoEq.(49.1), wefindthatthesolution fortheproblem is y=F(x—ct)—-F(—x —ct). (49.2) Itiseasytocheck thatwewillgety-0ifwesetx=0. Figure 49-1 shows awave travelling inthenegative x-direction near x=0, andahypothetical wave travelling intheother direction reversed insignandon theother sideoftheorigin. Wesayhypothetical because, ofcourse, there isno string tovibrate onthatsideoftheorigin. Thetotal motion ofthestring istobe regarded asthesumofthese twowaves intheregion ofpositive x.Astheyreach theorigin, they willalways cancel atx=0,andfinally thesecond (reflected) wave willbetheonly onetoexist forpositive xanditwill, ofcourse, betravelling intheopposite direction. These results areequivalent tothefollowing statement: ifawave reaches theclamped endofastring, itwillbereflected with achange in sign. Such areflection canalways beunderstood byimagining thatwhat iscoming totheendofthestring comes outupside down from behind thewall. Inshort, if weassume thatthestring isinfinite andthatwhenever wehave awave going one waywehave another onegoing theother waywith thestated symmetry, thedis- placement atx=0willalways bezero anditwould make nodifference ifwe clamped thestring there. Thenext point tobediscussed isthereflection ofaperiodic wave. Suppose thatthewave represented byF(x—ct)isasinewave andhasbeen reflected; then thereflected wave —F(—x —ct)isalsoasinewave ofthesame frequency, but travelling intheopposite direction. This situation can bemost simply described byusing thecomplex function notation: F(x—ct)=e““(‘""/‘) and 49-]49-1 Thereflection ofwaves 49-2 Confined waves, withnatural frequencies 49-3 Modes intwodimensions 49-4 Coupled pendulums 49-5 Linear systems F(xtvt)Fixedem\ __\_‘-~ _> I’. \\_/' -F(—X-0-Vt) _.>7‘/ \_--4 I“\I \rdz ‘~_’- **** -4 I\4,’ v-\I ‘ ,_I \4/ \_ ‘\\ l4_\ z \_-—\1 <—— \_¢ »__> Fig.49-1. Reflection ofaWave asa superposition oftwotravelling waves. F(—x —ct)=e""’(‘+‘/‘). Itcanbeseen that ifthese aresubstituted in(49.2) andifxissetequal to0,theny=0forallvalues oft,soitsatisfies thenecessary condition. Because oftheproperties ofexponentials, thiscanbewritten ina simpler form: y=e""’t(e"""’”‘/° —em‘/°) =—2iei“tsin (wx/c). (49.3) There issomething interesting andnewhere, inthatthissolution tellsusthat ifwelook atanyfixed x,thestring oscillates atfrequency cu.Nomatter where this point is,thefrequency isthesame! Butthere aresome places, inparticular wher- ever sin(wx/c) =0,where there isnodisplacement atall. Furthermore, ifat anytime twetakeasnapshot ofthevibrating string, thepicture willbeasinewave. However, thedisplacement ofthissinewave willdepend upon thetime t.From inspection ofEq.(49.3) wecanseethatthelength ofonecycle ofthesinewave is equal tothewavelength ofeither ofthesuperimposed waves: >.=24¢/8. (49.4) Thepoints where there isnomotion satisfy thecondition sin(wx/c) =0,which means that(wx/c) =0,1r,21r,...,nrr...These points arecalled nodes. Between anytwosuccessive nodes, every point moves upanddown sinusoidally, butthe pattern ofmotion stays fixed inspace. This isthefundamental characteristic of what wecallamode. Ifonecanfindapattern ofmotion which hastheproperty thatatanypoint theobject moves perfectly sinusoidally, andthatallpoints move atthesame frequency (though some willmove more than others), then wehave what iscalled amode. 49-2 Confined waves, withnatural frequencies Thenext interesting problem istoconsider what happens ifthestring isheld atboth ends, sayatx=0andx=L.Wecanbegin withtheideaofthereflection ofwaves, starting withsome kind ofabump moving inonedirection. Astimegoes on.wewould expect thebump togetnear oneend,andastime goes stillfurther it willbecome akind oflittle wobble, because itiscombining with thereversed- image bump which iscoming from theother side. Finally theoriginal bump will disappear andtheimage bump willmove intheother direction torepeat theprocess attheother end. This problem hasaneasy solution, butaninteresting question iswhether wecanhave asinusoidal motion (thesolution justdescribed isperiodic, butofcourse itisnotsinusoidal] yperiodic). Letustrytoputasinusoidally periodic wave onastring. Ifthestring istiedatoneend, weknow itmust look likeour earlier solution (49.3). Ifitistiedattheother end, ithastolook thesame atthe other end. Sotheonly possibility forperiodic sinusoidal motion isthatthesine wave must neatly fitintothestring length. Ifitdoes notfitintothestring length, then itisnotanatural frequency atwhich thestring cancontinue tooscillate. In short, ifthestring isstarted with asinewave shape thatjustfitsin,then itwill continue tokeep thatperfect shape ofasinewave andwilloscillate harmonically atsome frequency. Mathematically, wecanwrite sinkxfortheshape, where kisequal tothe factor (w/c) inEqs. (49.3) and(49.4), andthisfunction willbezero atx=0. However, itmust alsobezero attheother end. Thesignificance ofthisisthatk isnolonger arbitrary, aswasthecase forthehalf-open string. With thestring closed atboth ends, theonly possibility isthatsin(kL) =0,because thisisthe only condition thatwillkeep both ends fixed. Now inorder forasinetobezero, theangle must beeither 0,1r,21r,orsome other integral multiple of1r.Theequation kL=n1r (49.5) will,therefore, giveanyoneofthepossible k’s,depending onwhat integer isputin. Foreach ofthek’sthere isacertain frequency w,which, according to(49.3), is simply o.>=kc=n1rc/L. (49.6) 49-2 Sowehave found thefollowing: thatastring hasaproperty thatitcanhave sinusoidal motions, butonly atcertain frequencies. This isthemost important characteristic ofconfined waves. Nomatter how complicated thesystem is,it always turns outthat there aresome patterns ofmotion which have aperfect sinusoidal timedependence, butwithfrequencies thatareaproperty oftheparticu- larsystem andthenature ofitsboundaries. Inthecaseofthestring wehave many different possible frequencies, each one, bydefinition, corresponding toamode, because amode isapattern ofmotion which repeats itself sinusoidally. Figure 49-2 shows thefirstthree modes forastring. Forthefirstmode thewavelength )1is2L.This canbeseen ifonecontinues thewave outtox=2Ltoobtain one complete cycle ofthesinewave. Theangular frequency wis21rcdivided bythe wavelength, ingeneral, andinthiscase, since Ais2L,thefrequency is1rc/L, which isinagreement with (49.6) with n=1.Letuscallthefirstmode frequency w1. Now thenextmode shows twoloops with onenode inthemiddle. Forthismode thewavelength, then, issimply L.Thecorresponding value ofkistwice asgreat andthefrequency istwice aslarge; itis2w1.Forthethird mode itis3w1,andsoon. Soallthedifferent frequencies ofthestring aremultiples, l,2,3,4,andsoon,of thelowest frequency w1. Returning nowtothegeneral motion ofthestring, itturns outthatanypossible motion canalways beanalyzed byasserting thatmore than onemode isoperating atthesame time. Infact, forgeneral motion aninfinite number ofmodes must beexcited atthesame time. Togetsome ideaofthis,letusillustrate what happens when there aretwomodes oscillating atthesame time: Suppose thatwehave the firstmode oscillating asshown bythesequence ofpictures inFig.49-3, which illustrates thedeflection ofthestring forequally spaced time intervals extending through halfacycle ofthelowest frequency. Now, atthesame time, wesuppose thatthere isanoscillation ofthesecond mode also. Figure 49-3 alsoshows asequence ofpictures ofthismode, which at thestart is90°outofphase with thefirstmode. This means thatatthestart ithas nodisplacement, butthetwohalves ofthestring have oppositely directed velocities. Now werecall ageneral principle relating tolinear systems: ifthere areanytwo solutions, thentheirsumisalsoasolution. Therefore athird possible motion of thestring would beadisplacement obtained byadding thetwosolutions shown in Fig.49-3. Theresult, alsoshown inthefigure, begins tosuggest theideaofabump running back andforth between theends ofthestring, although with only two modes wecannot make avery good picture ofit;more modes areneeded. This result is,infact, aspecial case ofagreat principle forlinear systems: Anymotion atallcanbeanalyzed byassuming thatitisthesumofthemotions ofallthedifferent modes, combined with appropriate amplitudes andphases. The importance oftheprinciple derives from thefactthat each mode isvery simple—it isnothing butasinusoidal motion intime. Itistruethateventhegeneral motion ofastring isnotreally very complicated, butthere areother systems, for example thewhipping ofanairplane wing, inwhich themotion ismuch more complicated. Nevertheless, even with anairplane wing, wefindthere isacertain particular wayoftwisting which hasonefrequency andother ways oftwisting that have other frequencies. Ifthese modes canbefound, then thecomplete motion canalways beanalyzed asasuperposition ofharmonic oscillations (except when thewhipping isofsuch degree thatthesystem cannolonger beconsidered aslinear). 49-3 Modes intwodimensions Thenextexample tobeconsidered istheinteresting situation ofmodes intwo dimensions. Uptothispoint wehave talked onlyabout one-dimensional situations —astretched string orsound waves inatube. Ultimately weshould consider three dimensions, butaneasier stepwillbethattotwodimensions. Consider fordefinite- nessarectangular rubber drumhead which isconfined soastohave nodisplace- ment anywhere ontherectangular edge, andletthedimensions oftherectangle 49-3v“ ‘ \\ ’/ I \_\ I,’ yr/’~\1’ \\ I’ ‘ , \ \ 1 I \\ /I \\ —P I y /’_‘\I \ I \ 1 \ \ ' \ / I \ 1/ \ /\ 1 \I \a \¢ Fig. 49-2. Thefirstthree modes ofa vibrating string. 2-3.... .02-3 ult--E "-‘\ Q-3Jlr\ lg-\>“4~' "L‘? \/W"\_/ —FinsrMODE __ ___SECOND MODE comeosrrc wave Fig. 49-3. Two modes combine to give atravelling wave. 7 Clompod Edges b / Wave 9H.[¢“"‘+“"'] 71 0 9 O it Fig. 49-4. Vibrating rectangular plate.beaandb,asshown inFig.49-4. Now thequestion is,what arethecharacteristics ofthepossible motion? Wecanstart with thesame procedure used forthestring. Ifwehadnoconfinement atall,wewould expect waves travelling along with some kind ofwave motion. Forexample, (e’”")(e_"”=""""1/l’) would represent asinewave travelling insome direction which depends ontherelative values ofk,andk,,. Now howcanwemake thex-axis, thatis,theliney=0,anode? Using theideas developed fortheone-dimensional string, wecanimagine another wave repre- sented bythecomplex function (—e‘“‘)(e_”‘»"‘“°u”). Thesuperposition ofthese waves willgivezero displacement aty=0regardless ofthevalues ofxandt. (Although these functions aredefined fornegative ywhere there isnodrumhead tovibrate, thiscanbeignored, since thedisplacement istruly zero aty=0.) Inthiscasewecanlook upon thesecond function asthereflected wave. However, wewant anodal lineaty=baswellasaty=O.How dowedo that? Thesolution isrelated tosomething wedidwhen studying reflection from crystals. These waves which cancel each other aty=0willdothesame aty=b onlyif2bsin0isanintegral multiple of)\,where 0istheangle shown inFig.49-4: m>\=2bsin0, m=0,1,2,... (49.7) Now inthesame waywecanmake they-axis anodal linebyadding twomore functions —(ef“‘)(e+”‘1”+”‘v”) and -l—(e"‘"‘)(e""”%"_"'°i1”), each representing a reflection ofoneoftheother twowaves from thex=0line. Thecondition fora nodal lineatx=aissimilar totheonefory=b.Itisthat2acos0must alsobe anintegral multiple ofA: n)\=2acos0. (49.8) Then thefinalresult isthatthewaves bouncing about intheboxproduce astanding- wave pattern, thatis,adefinite mode. Sowemust satisfy theabove twoconditions ifwearetohave amode. Letus firstfindthewavelength. This canbeobtained byeliminating theangle 0from (49.7) and(49.8) toobtain thewavelength interms ofa,b,nandm.Theeasiest waytodothatistodivide both sides oftherespective equations by2band2a, square them, andaddthetwoequations together. Theresult issinz0+cos2 0= l=(n>\/2a)2 —l—(m>\/2b)2, which canbesolved forA: l n2 m2F=Q+133- (49.9) Inthiswaywehave determined thewavelength interms oftwointegers, andfrom thewavelength weimmediately getthefrequency w,because, asweknow, the frequency isequal to21rcdivided bythewavelength. This result isinteresting andimportant enough thatweshould deduce itbya purely mathematical analysis instead ofbyanargument about thereflections. Letusrepresent thevibration byasuperposition offourwaves chosen sothatthe four lines x=0,x=a,y=0,andy=bareallnodes. Inaddition weshall require thatallwaves have thesame frequency, sothattheresulting motion will represent amode. From ourearlier treatment oflight reflection weknow that (ei“')(e_“‘¢‘+”°v”) represents awave travelling inthedirection indicated inFig. 49-4. Equation (49.6), thatis,k=w/c, stillholds, provided k2=kg+kj. (49.l0) Itisclear from thefigure thatk,=kcos0andk,,=ksin6. Now ourequation forthedisplacement, say¢>,oftherectangular drumhead takes onthegrand form ¢= [eiWf][e(—ik1Z-l-ikyy) _ e(-I-ik,;z+il1:uy) _ e(—1'lcx:|:—1'lcuy) + e(-l-ilCz2—‘l'lCU1/)1 (49.lla) Although thislooks rather amess, thesumofthese things now isnotvery hard. 49-4 Table 49-1 Mdh m n 2 / 0esape (w/wo) wwt) + 1 1 1.25 1.12 _i__[____ + — 1 2 2.00 1.41 ____._l___.i__ _:_l:_1___ + —l- — 1 3 3.25 1.80 :___l__Ji_ --------- —- 2 1 4.25 2.06 + lI +:l __+__+ ——--- 2 2 5.00 2.24 €_Zl_i_ Theexponentials canbecombined togivesinefunctions, sothatthedisplacement turns outtobe ¢=[-4sink,,xsink1,y][eM]. (49111)) Inother words, itisasinusoidal oscillation, allright, with apattern thatisalso sinusoidal inboth thex-andthey-direction. Ourboundary conditions areof course satisfied atx=0andy=0.Wealsowant ¢tobezerowhen x=aand when y=b.Therefore wehave toputintwoother conditions: k,,amust bean integral multiple of1r,andk,,bmust beanother integral multiple of1r.Since we have seen that k,=kcos 6and k,,=ksin 0,weimmediately getequations (49.7) and(49.8) andfrom these thefinal result (49.9). Now letustake asanexample arectangle whose width istwice theheight. Ifwetakea=2banduseEqs. (49.4) and(49.9), wecancalculate thefrequencies ofallofthemodes:2 <8”= 4'" (49.12) Table 49-1 listsafewofthesimple modes andalsoshows their shape inaqualitative way. Themost important point tobeemphasized about thisparticular caseisthat thefrequencies arenotmultiples ofeach other, noraretheymultiples ofanynum- ber. Theideathatthenatural frequencies areharmonically related isnotgenerally true. Itisnottrueforasystem ofmore than onedimension, norisittrueforone- dimensional systems which aremore complicated than astring with uniform density andtension. Asimple example ofthelatter isahanging chain inwhich thetension ishigher atthetopthan atthebottom. Ifsuch achain issetinharmonic oscillation, there arevarious modes andfrequencies, butthefrequencies arenot simple multiples ofanynumber, norarethemode shapes sinusoidal. Themodes ofmore complicated systems arestillmore elaborate. Forexample, inside themouth wehave acavity above thevocal cords, andbymoving the tongue andthelips,andsoforth, wemake anopen-ended pipe oraclosed-ended pipe ofdifferent diameters andshapes; itisaterribly complicated resonator, but 49-5to -l>~+3to Y x Fig.49-5. Two coupled pendulums.itisaresonator nevertheless. Now when onetalks withthevocal cords, theyare made toproduce some kind oftone. Thetone israther complicated andthere are many sounds coming out,butthecavity ofthemouth further modifies thattone because ofthevarious resonant frequencies ofthecavity. Forinstance, asinger cansingvarious vowels, a,oro,oroo,andsoforth, atthesame pitch, butthey sound different because thevarious harmonics areinresonance inthiscavity to different degrees. Theverygreat importance oftheresonant frequencies ofacavity inmodifying thevoice sounds canbedemonstrated byasimple experiment. Since thespeed ofsound goesasthereciprocal ofthesquare rootofthedensity, the speed ofsound maybevaried byusing different gases. Ifoneuseshelium instead ofair,sothatthedensity islower, thespeed ofsound ismuch higher, andallthe frequencies ofacavity willberaised. Consequently ifonefillsone’s lungs with helium before speaking, thecharacter ofhisvoice willbedrastically altered even though thevocal cords may stillbevibrating atthesame frequency. 49-4 Coupled pendulums Finally weshould emphasize thatnotonly domodes exist forcomplicated continuous systems, butalsoforverysimple mechanical systems. Agood example isthesystem oftwocoupled pendulums discussed inthepreceding chapter. In thatchapter itwasshown thatthemotion could beanalyzed asasuperposition of twoharmonic motions withdifferent frequencies. Soeventhissystem canbean- alyzed interms ofharmonic motions ormodes. Thestring hasaninfinite number of modes andthetwo-dimensional surface alsohasaninfinite number ofmodes. In asense itisadouble infinity, ifweknow how tocount infinities. Butasimple mechanical thing which hasonly twodegrees offreedom, andrequires only two variables todescribe it,hasonly twomodes. Letusmake amathematical analysis ofthese twomodes forthecasewhere the pendulums areofequal length. Letthedisplacement ofonebex,andthedisplace- ment oftheother bey,asshown inFig.49-5. Without aspring, theforce onthe firstmass isproportional tothedisplacement ofthat mass, because ofgravity. There would be,ifthere werenospring, acertain natural frequency woforthisone alone. Theequation ofmotion without aspring would be 2 m%=—mw§x. (49.13) Theother pendulum would swing inthesame wayifthere were nospring. In addition totheforce ofrestoration duetogravitation, there isanadditional force pulling thefirstmass. That force depends upon theexcess distance ofxoveryand isproportional tothat difl'erence, soitissome constant which depends onthe geometry, times (x-y).Thesame force inreverse sense actsonthesecond mass. Theequations ofmotion thathave tobesolved aretherefore dz dzmE;=—mw§x -k(x-y),mé=—mw§y -k(y-X).(49.14) Inorder tofindamotion inwhich both ofthemasses move atthesame fre- quency, wemust determine how much each mass moves. Inother words, pendulum xandpendulum ywilloscillate atthesame frequency, buttheir ampli- tudes must have certain values, AandB,whose relation isfixed. Letustrythis solution: x=.46“, y=Ba“. (49.15) Ifthese aresubstituted inEqs.(49.14) andsimilar terms arecollected, theresults are <w2—w%—%)A =—%B, (w2—wg——%)B =-$4(49.16) 49-6 Theequations aswritten have hadthecommon factor e*"‘removed andhave been divided bym. Now weseethatwehavetwoequations forwhat looks liketwounknowns. Butthere really arenottwounknowns, because thewhole sizeofthemotion is something thatwecannot determine from these equations. Theabove equations candetermine onlytheratio ofAtoB,buttheymust bothgivethesame ratio. The necessity forboth ofthese equations tobeconsistent isarequirement thatthe frequency besomething very special. Inthisparticular casethiscanbeworked outrather easily. Ifthetwoequa- tions aremultiplied together, theresult is 2 2 (82-8?,-5)AB=(5)AB. (49.17)m "1 Theterm ABcanberemoved from both sides unless AandBarezero, which means there isnomotion atall.Ifthere ismotion, then theother terms must beequal, giving aquadratic equation tosolve. Theresult isthat there aretwopossible frequencies : “’%=°’g’ ‘"3=“3+ (4918) Furthermore, ifthese values offrequency aresubstituted back intoEq.(49.16), wefindthatforthefirstfrequency A=B,andforthesecond frequency A=—B. These arethe“mode shapes,” ascanbereadily verified byexperiment. Itisclear thatinthefirstmode, where A=B,thespring isnever stretched, andboth masses oscillate atthefrequency wo,asthough thespring were absent. Intheother solution, where A=-B,thespring contributes arestoring force and raises thefrequency. Amore interesting caseresults ifthependulums have different lengths. Theanalysis isvery similar tothatgiven above, andisleftasanexercise forthereader. 49-5 Linear systems Now letussummarize theideas discussed above, which areallaspects ofwhat isprobably themost general andwonderful principle ofmathematical physics. If wehave alinear system whose characterjis independent ofthetime, thenthemotion does nothave tohave anyparticular simplicity, andinfactmay beexceedingly complex, butthere arevery special motions, usually aseries ofspecial motions, inwhich thewhole pattern ofmotion varies exponentially with thetime. Forthe vibrating systems thatwearetalking about now, theexponential isimaginary, and instead ofsaying “exponentially” wemight prefer tosay“sinusoidally” with time. However, onecanbemore general andsaythatthemotions willvaryexponentially with thetime invery special modes, with very special shapes. Themost general motion ofthesystem canalways berepresented asasuperposition ofmotions involving each ofthediflerent exponentials. Thisisworth stating again forthecaseofsinusoidal motion: alinear system need notbemoving inapurely sinusoidal motion, i.e.,atadefinite single frequency, butnomatter howitdoesmove, thismotion canberepresented asasuperposition ofpure sinusoidal motions. Thefrequency ofeach ofthese motions isacharacter- isticofthesystem, andthepattern orwaveform ofeach motion isalsoacharacter- isticofthesystem. Thegeneral motion inanysuch system canbecharacterized bygiving thestrength andthephase ofeach ofthese modes, andadding them all together. Another wayofsaying thisisthatanylinear vibrating system isequivalent toasetofindependent harmonic oscillators, with thenatural frequencies corre- sponding tothemodes. Weconclude thischapter byremarking ontheconnection ofmodes with quantum mechanics. Inquantum mechanics thevibrating object, orthething that varies inspace, istheamplitude ofaprobability function thatgives theprobability offinding anelectron, orsystem ofelectrons, inagiven configuration. This ampli- tude function canvary inspace andtime, andsatisfies, infact, alinear equation. 49-7 Butinquantum mechanics there isatransformation, inthatwhat wecallfrequency oftheprobability amplitude isequal, intheclassical idea, toenergy. Therefore wecantranslate theprinciple stated above tothiscasebytaking theword frequency andreplacing itwithenergy. Itbecomes something likethis: aquantum-mechanical system, forexample anatom, need nothave adefinite energy, justasasimple mechanical system does nothave tohave adefinite frequency; butnomatter how thesystem behaves, itsbehavior canalways berepresented asasuperposition of states ofdefinite energy. Theenergy ofeach state isacharacteristic oftheatom, andsoisthepattern ofamplitude which determines theprobability offinding particles indifl'erent places. Thegeneral motion canbedescribed bygiving the amplitude ofeach ofthese different energy states. This istheorigin ofenergy levels inquantum mechanics. Since quantum mechanics isrepresented bywaves, inthecircumstance inwhich theelectron does nothave enough energy toulti- mately escape from theproton, they areconfined waves. Like theconfined waves ofastring, there aredefinite frequencies forthesolution ofthewave equation for quantum mechanics. The quantum-mechanical interpretation isthat these are definite energies. Therefore aquantum-mechanical system, because itisrepresented bywaves, canhave definite states offixed energy; examples aretheenergy levels of various atoms. 49-8 50 Harmonics 50-1 Musical tones Pythagoras issaidtohave discovered thefactthattwosimilar strings under thesame tension anddiffering only inlength, when sounded together give an effect thatispleasant totheearifthelengths ofthestrings areintheratio oftwo small integers. Ifthelengths areasoneistotwo, they then correspond tothe octave inmusic. Ifthelengths areastwoistothree, they correspond tothein- terval between CandG,which iscalled afifth. These intervals aregenerally accepted as“pleasant” sounding chords. Pythagoras wassoimpressed bythisdiscovery thathemade itthebasis ofa school—Pythagoreans they were called—which held mystic beliefs inthegreat powers ofnumbers. Itwasbelieved thatsomething similar would befound out about theplanets-or “spheres.” Wesometimes hear theexpression: “the music ofthespheres.” Theideawasthatthere would besome numerical relationships between theorbits oftheplanets orbetween other things innature. People usually think thatthisisjustakind ofsuperstition heldbytheGreeks. Butisitsodifferent from ourown scientific interest inquantitative relationships? Pythagoras’ dis- covery wasthefirstexample, outside geometry, ofanynumerical relationship in nature. Itmust have been very surprising tosuddenly discover thatthere wasa factofnature thatinvolved asimple numerical relationship. Simple measurements oflengths gave aprediction about something which hadnoapparent connection to geometry-the production ofpleasant sounds. This discovery ledtotheextension thatperhaps agood toolforunderstanding nature would bearithmetic andmathe- matical analysis. Theresults ofmodern science justify thatpoint ofview. Pythagoras could only have made hisdiscovery bymaking anexperimental observation. Yetthisimportant aspect does notseem tohave impressed him. Ifithad, physics might have hadamuch earlier start. (Itisalways easy tolook back atwhat someone elsehasdone andtodecide what heshould have done!) Wemight remark onathird aspect ofthisvery interesting discovery: thatthe discovery hadtodowithtwonotes thatsound pleasant totheear.Wemayquestion whether weareanybetter offthan Pythagoras inunderstanding whyonly certain sounds arepleasant toourear. Thegeneral theory ofaesthetics isprobably no further advanced nowthan inthetime ofPythagoras. Inthisonediscovery ofthe Greeks, there arethethree aspects: experiment, mathematical relationships, and aesthetics. Physics hasmade great progress ononly thefirst twoparts. This chapter willdealwithourpresent-day understanding ofthediscovery ofPythagoras. Among thesounds thatwehear, there isonekind thatwecallnoise. Noise corresponds toasortofirregular vibration oftheeardrum thatisproduced bythe irregular vibration ofsome object intheneighborhood. Ifwemake adiagram to indicate thepressure oftheairontheeardrum (and, therefore, thedisplacement ofthedrum) asafunction oftime, thegraph which corresponds toanoise might look likethatshown inFig.50-1(a). (Such anoise might correspond roughly to thesound ofastamped foot.) Thesound ofmusic hasadiflerent character. Music ischaracterized bythepresence ofmore-or-less sustained tones—or musical “notes.” (Musical instruments maymake noises aswell!) Thetone maylastfora relatively short time, aswhen akeyispressed onapiano, oritmay besustained almost indefinitely, aswhen aflute player holds along note. What isthespecial character ofamusical note from thepoint ofview ofthe pressure intheair? Amusical notediffers from anoise inthatthere isaperiodicity initsgraph. There issome uneven shape tothevariation oftheairpressure with 50-150-1 Musical tones 50-2 TheFourier series 50-3 Quality andconsonance 50-4 TheFourier coefficients 50-5 Theenergy theorem 50-6 Nonlinear responses APRESSURE TIME> /\/'\ \/ tn)Auo|$E APRESSURE TIME |-r—-l lb)AMUSICAL tom: Fig. 50-1. Pressure asafunction of time for(a)anoise, and (blamusical tone. time, andtheshape repeats itself over andover again. Anexample ofapressure- time function thatwould correspond toamusical note isshown inFig.50-l(b). Musicians willusually speak ofamusical tone interms ofthree character- istics: theloudness, thepitch, andthe“quality.” The “loudness” isfound to correspond tothemagnitude ofthepressure changes. The“pitch” corresponds to theperiod oftime foronerepetition ofthebasic pressure function. (“Low” notes have longer periods than “high” notes.) The“quality” ofatone hastodo with thediflerences wemay stillbeable tohear between twonotes ofthesame loudness andpitch. Anoboe, aviolin, orasoprano arestilldistinguishable even when theysound notes ofthesame pitch. Thequality hastodowith thestructure oftherepeating pattern. Letusconsider, foramoment, thesound produced byavibrating string. If wepluck thestring, bypulling ittoonesideandreleasing it,thesubsequent motion willbedetermined bythemotions ofthewaves wehave produced. Weknow that these waves willtravel inboth directions, andwillbereflected attheends. They willslosh back andforth foralong time. Nomatter howcomplicated thewave is, however, itwillrepeat itself. Theperiod ofrepetition isjustthetime Trequired forthewave totravel twofulllengths ofthestring. Forthatisjustthetime re- quired foranywave, once started, toreflect offeach endandreturn toitsstarting position, andbeproceeding intheoriginal direction. Thetime isthesame for waves which start outineither direction. Each point onthestring will,then, return toitsstarting position after oneperiod, andagain oneperiod later, etc. The sound wave produced must alsohave thesame repetition. Weseewhyaplucked string produces amusical tone. 50-2 TheFourier series Wehave discussed inthepreceding chapter another way oflooking at themotion ofavibrating system. Wehave seenthatastring hasvarious natural modes ofoscillation, andthat anyparticular kind ofvibration that may beset upbythestarting conditions canbethought ofasacombination—in suitable proportions—of several ofthenatural modes, oscillating together. Forastring wefound that thenormal modes ofoscillation hadthefrequencies wo,2w0, 3w0, ...Themost general motion ofaplucked string, therefore, iscomposed of thesumofasinusoidal oscillation atthefundamental frequency wo,another atthe second harmonic frequency Zwo, another atthethird harmonic 3w0,etc.Now the fundamental mode repeats itself every period T1=21r/wo. Thesecond harmonic mode repeats itself every T2=21r/2w11. Italsorepeats itself every T1=2T2, after twoofitsperiods. Similarly, thethird harmonic mode repeats itself after a time T1which is3ofitsperiods. Weseeagain why aplucked string repeats its whole pattern with aperiodicity ofT1.Itproduces amusical tone. Wehave been talking about themotion ofthestring. Butthesound, which is themotion oftheair,isproduced bythemotion ofthestring, soitsvibrations too must becomposed ofthesame harmonics—though wearenolonger thinking about thenormal modes oftheair.Also, therelative strength oftheharmonics may be different intheairthan inthestring, particularly ifthestring is“coupled” tothe airviaasounding board. Theefficiency ofthecoupling totheairisdifferent for different harmonics. Ifweletf(t)represent theairpressure asafunction oftime foramusical tone [such asthatinFig.50—1(b)], then weexpect thatf(t)canbewritten asthesumof anumber ofsimple harmonic functions oftime—like coswt—for eachofthevarious harmonic frequencies. Iftheperiod ofthevibration isT,thefundamental angular frequency willbew=21r/T, andtheharmonics willbe2w,3w,etc. There isoneslight complication. Foreach frequency wemayexpect thatthe starting phases willnotnecessarily bethesame forallfrequencies. Weshould, therefore, usefunctions likecos(wt+¢).Itis,however, simpler touseinstead both thesineandcosine functions foreach frequency. Werecall that cos(wt+¢)=(cos¢coswt—sin¢sinwt) (50.1) 50-2 andsince ¢isaconstant, anysinusoidal oscillation atthefrequency wcanbe written asthesumofatermwithcoswtandanother termwithsinwt. Weconclude, then, thatanyfunction f(t)thatisperiodic withtheperiod T canbewritten mathematically as f(t)=flo +a1coswt+b1sinwt +a2cos2wt+b2sin2wt +a3cos3wt+b3sin3wt +... +... (50.2) where w=21r/T andthea’sandb’sarenumerical constants which tellushow much ofeachcomponent oscillation ispresent intheoscillation f(t). Wehave added the“zero-frequency” termaosothatourformula willbecompletely general, although itisusually zero foramusical tone. Itrepresents ashift oftheaverage value (that is,the“zero” level) ofthesound pressure. With itourformula can take care ofanycase. Theequality ofEq.(50.2) isrepresented schematically in Fig.50-2. (Theamplitudes, a,,andb,,,oftheharmonic functions must besuitably chosen. They areshown schematically andwithout anyparticular scale inthe figure.) Theseries (50.2) iscalled theFourier series forf(t). Wehave said thatanyperiodic function canbemade upinthisway. We should correct thatandsaythatanysound wave, oranyfunction weordinarily encounter inphysics, canbemade upofsuch asum. Themathematicians can invent functions which cannot bemade upofsimple harmonic functions-for instance, afunction thathasa“reverse twist” sothatithastwovalues forsome values oft!Weneed notworry about such functions here. 50-3 Quality andconsonance Now weareabletodescribe what itisthatdetermines the“quality” ofa musical tone. Itistherelative amounts ofthevarious harmonics—-the values of thea’sandb’s.Atonewithonlythefirstharmonic isa“pure” tone. Atone withmany strong harmonics isa“rich” tone. Aviolin produces adifferent pro- portion ofharmonics thandoesanoboe. Wecan“manufacture” various musical tones ifweconnect several “oscilla- tors” toaloudspeaker. (An oscillator usually produces anearly pure simple harmonic function.) Weshould choose thefrequencies oftheoscillators tobew, 2w,3w,etc.Then byadjusting thevolume control oneachoscillator, wecanadd inanyamount wewish ofeach harmonic—thereby producing tones ofdifferent quality. Anelectric organ works inmuch thisway. The“keys” select thefrequency ofthefundamental oscillator andthe“stops” areswitches thatcontrol therelative proportions oftheharmonics. Bythrowing these switches, theorgan canbemade tosound likeaflute, oranoboe, oraviolin. Itisinteresting thattoproduce such “artificial” tones weneed onlyoneoscilla- torforeachfrequency—-we donotneedseparate oscillators forthesineandcosine components. Theearisnotvery sensitive totherelative phases oftheharmonics. Itpays attention mainly tothetotal ofthesineandcosine parts ofeach frequency. Ouranalysis ismore accurate than isnecessary toexplain thesubjective aspect of music. Theresponse ofamicrophone orother physical instrument doesdepend onthephases, however, andourcomplete analysis maybeneeded totreat such cases. The“quality” ofaspoken sound alsodetermines thevowel sounds thatwe recognize inspeech. Theshape ofthemouth determines thefrequencies ofthe natural modes ofvibration oftheairinthemouth. Some ofthese modes areset intovibration bythesound waves from thevocal chords. Inthisway,theampli- tudes ofsome oftheharmonics ofthesound areincreased withrespect toothers. When wechange theshape ofourmouth, harmonics ofdifferent frequencies are given preference. These effects account forthediflerence between an“e—e—e” sound andan“a-a-a” sound. 50-3t(tl 7I rrjirif+ I + '+b‘ + etc. + ate. Fig. 50-2. Any periodic function f(t) isequal toasum ofsimple harmonic functions. Weallknow thataparticular vowel sound—say “e-e-e”—still “sounds like” thesame vowel whether wesay(orsing) itatahigh oralowpitch. From the mechanism wedescribe, wewould expect thatparticular frequencies areemphasized when weshape ourmouth foran“e—e—e,” andthattheydonotchange aswechange thepitch ofourvoice. Sotherelation oftheimportant harmonics tothefunda- mental—that is,the“quality”—changes aswechange pitch. Apparently themech- anism bywhich werecognize speech isnotbased onspecific harmonic relation- ships. What should wesaynow about Pythagoras’ discovery? Weunderstand that twosimilar strings with lengths intheratio of2to3willhave fundamental fre- quencies intheratio 3to2.Butwhy should they “sound pleasant” together? Perhaps weshould take ourclue from thefrequencies oftheharmonics. The second harmonic ofthelower shorter string willhave thesame frequency asthe third harmonic ofthelonger string. (Itiseasy toshow—or tobelieve—that a plucked string produces strongly theseveral lowest harmonics.) Perhaps weshould make thefollowing rules. Notes sound consonant when theyhave harmonics withthesame frequency. Notes sound dissonant iftheir upper harmonics have frequencies near toeach other butfarenough apart thatthere are rapid beats between thetwo. Why beats donotsound pleasant, andwhyunison oftheupper harmonics does sound pleasant, issomething thatwedonotknow how todefine ordescribe. Wecannot sayfrom thisknowledge ofwhat sounds good, what ought, forexample, tosmell good. Inother words, ourunderstanding ofitisnotanything more general than thestatement thatwhen theyareinunison theysound good. Itdoes notpermit ustodeduce anything more than theproper- tiesofconcordance inmusic. Itiseasy tocheck ontheharmonic relationships wehave described bysome simple experiments with apiano. Letuslabel the3successive C’snear themiddle ofthekeyboard byC,C’,andC”,andtheG’sjustabove byG,G’,andG”. Then thefundamentals willhave relative frequencies asfollows: c-2G-3 c'-4 G’—6 c"-8 G”-12 These harmonic relationships canbedemonstrated inthefollowing way: Suppose wepress C’slowly—so that itdoes notsound butwecause thedamper tobe lifted. Ifwethen sound C,itwillproduce itsown fundamental andsome second harmonic. Thesecond harmonic willsetthestrings ofC’intovibration. lfwe now release C(keeping C’pressed) thedamper willstop thevibration oftheC strings, andwecanhear (softly) thenote C’asitdiesaway. Inasimilar way, the third harmonic ofCcancause avibration ofG’.Orthesixth ofC(now getting much weaker) cansetupavibration inthefundamental ofG”. Asomewhat diflerent result isobtained ifwepress Gquietly andthen sound C’.Thethird harmonic ofC’willcorrespond tothefourth harmonic ofG,so onlythefourth harmonic ofGwillbeexcited. Wecanhear (ifwelisten closely) thesound ofG”,which istwooctaves above theGwehave pressed! Itiseasyto think upmany more combinations forthisgame. Wemay remark inpassing thatthemajor scale canbedefined justbythe condition that thethree major chords (F-A-C); (C—E—G); and(G-B—D) each represent tone sequences with thefrequency ratio (4:5:6). These ratios—plus thefactthatanoctave (C—C’, B—B’, etc.) hastheratio 1:2—determine thewhole scale forthe“ideal” case, orforwhat iscalled “just intonation.” Keyboard in- struments likethepiano arenotusually tuned inthismanner, butalittle “fudging” isdone sothatthefrequencies areapproximately correct forallpossible starting tones. Forthistuning, which iscalled “tempered,” theoctave (still 1:2)isdivided into12equal intervals forwhich thefrequency ratio is(2)” 12.Afifth nolonger hasthefrequency ratio 3/2,but27'12=1.499, which isapparently close enough formost ears. 50-4 Wehave stated aruleforconsonance interms ofthecoincidence ofharmonics. Isthiscoincidence perhaps thereason thattwonotes areconsonant? Oneworker hasclaimed thattwopure tones—tones carefully manufactured tobefreeofhar- monics—do notgivethesensations ofconsonance ordissonance astherelative frequencies areplaced atorneartheexpected ratios. (Such experiments arediflicult because itisdifficult tomanufacture pure tones, forreasons thatweshall seelater.) Wecannot stillbecertain whether theearismatching harmonics ordoing arith- metic when wedecide thatwelikeasound. 50-4 TheFourier coefficients Letusreturn nowtotheideathatanynote-that is,aperiodic sound—can be represented byasuitable combination ofharmonics. Wewould liketoshow how wecanfindoutwhat amount ofeach harmonic isrequired. Itis,ofcourse, easy tocompute f(t), using Eq.(50.2), ifwearegiven allthecoefficients aandb.The question nowis,ifwearegiven f(t)howcanweknow what thecoefficients ofthe various harmonic terms should be? (Itiseasytomake acake from arecipe; but canwewrite down therecipe ifwearegiven acake?) Fourier discovered thatitwasnotreally verydifficult. Theterm a1,iscertainly easy. Wehave already saidthatitisjusttheaverage value off(t)over oneperiod (from t=0tot=T).Wecaneasily seethatthisisindeed so.Theaverage value ofasineorcosine function overoneperiod iszero. Over two,orthree, oranywhole number ofperiods, itisalsozero. Sotheaverage value ofalloftheterms onthe right-hand sideofEq.(50.2) iszero, except fora0.(Recall thatwemust choose w=21r/T.) Now theaverage ofasumisthesumoftheaverages. Sotheaverage off(t)is justtheaverage ofa0.Buta0isaconstant, soitsaverage isjustthesame asits value. Recalling thedefinition ofanaverage, wehave 1 T a1,=TA f(t)dt. (50.3) Theother coefficients areonly alittle more difficult. Tofindthem wecanuse atrick discovered byFourier. Suppose wemultiply both sides ofEq.(50.2) by some harmonic function-—say bycos7wt. Wehave then f(t)-cos7wt=ao-cos7wt +a1coswt~cos7wt+b1sinwt -cos7wt +a2cos2wt-cos7wt+b2sin2wt-cos7wt +... +... +a7cos7wt~cos7wt+b7sin7wt'cos7wt +... +... (5()_4) Now letusaverage both sides. Theaverage ofaocos7wtover thetime Tispro- portional totheaverage ofacosine over 7whole periods. Butthatisjustzero. Theaverage ofalmost alloftherestoftheterms isalsozero. Letuslook atthe a1term. Weknow, ingeneral, that cosAcosB=%cos(A+B)+%cos (A—B). (50.5) Thea1term becomes %a1(cos 8wt+cos6wt). (50.6) Wethushave twocosine terms, onewith 8fullperiods inTandtheother with 6. They bothaverage tozero. Theaverage ofthea1term istherefore zero. Forthea2term, wewould finda2cos9wtanda2cosSwt,each ofwhich also averages tozero. Forthea9term, wewould findcosl6wt andcos(-2wt). But cos(—2wt) isthesame ascos2wt,soboth ofthese have zero averages. Itisclear 50-5 thatalloftheaterms willhave azeroaverage except one. Andthatoneisthe a7term. Forthisonewehave §a7(cos l4wt+cos0). (50.7) Thecosine ofzero isone,anditsaverage, ofcourse, isone. Sowehave theresult thattheaverage ofalloftheaterms ofEq.(50.4) equals %a7. Thebterms areeven easier. When wemultiply byanycosine term likecosnwt, wecanshow bythesame method thatallofthebterms havetheaverage value zero. WeseethatFourier’s “trick” hasacted likeasieve. When wemultiply by cos7wtandaverage, allterms drop outexcept a7,andwefindthat Average [f(t)~cos7wt]=a1/2, (50.8) O1‘ T a7=%‘/0f(t)-66478141. (50.9) Weshall leave itforthereader toshow thatthecoefficient bycanbeobtained bymultiplying Eq.(50.2) bysin7wtandaveraging both sides. Theresult is T2 .b7=T/Q) f(t)-SlI17¢uldl. (50.10) Nowwhatistruefor7weexpect istrueforanyinteger. Sowecansummarize ourproof andresult inthefollowing more elegant mathematical form. Ifmand nareintegers other than zero, andifw=27r/T, then '8 I./sinnwtcosmwtdt =0. (50.11)0 T II./ d= - 0cosnwtcosmwt t Olfngsm_ (50.12) T T/2ifn=m. III. /sinnwtsinmwtdt=o IV. f(t) =ao+2:ancosnwt+i:b,,sinnwt. (50.13) n—l n=1 1' v.ao=%/0f(t)-dt. (50.14) 1' an= f(t)-cosnwtdt. (50.15)T6 T b,,=%.]-f(t)-sinnwtdt. (50.16)0 Inearlier chapters itwasconvenient tousetheexponential notation forrepre- senting simple harmonic motion. Instead ofcoswtweusedReel“",therealpart oftheexponential function. Wehave used cosine andsinefunctions inthis chapter because itmade thederivations perhaps alittleclearer. Ourfinalresult of Eq.(50.13) can,however, bewritten inthecompact form f(t)=ReZa,,e"""‘, (50.17)n=0 where 21,,isthecomplex number an—ib,,(with bo=0).Ifwewishtousethe same notation throughout, wecanwrite also 2T -6,,=T/0f(t)e_”“"dt (n21). (50.18) 50-6 Wenowknow howto“analyze” aperiodic wave intoitsharmonic compon- ents. Theprocedure iscalled Fourier analysis, andtheseparate terms arecalled Fourier components. Wehave notshown, however, thatonce wefindallofthe Fourier components andaddthem together, wedoindeed getback ourf(t). The mathematicians have shown, forawide class offunctions, infactforallthatare ofinterest tophysicists, thatifwecandotheintegrals wewillgetbackf(t). There isoneminor exception. Ifthefunction f(t)isdiscontinuous, i.e.,ifitjumps suddenly from onevalue toanother, theFourier sum willgiveavalue atthebreakpoint halfway between theupper andlower values atthediscontinuity. Soifwehave the strange function f(t)=0,03t<to,andf(t)=1fortogt5T,theFourier sumwillgivetheright value everywhere except atto,where itwillhave thevalue % instead of1.Itisrather unphysical anyway toinsist thatafunction should be zerouptoto,butlright atto.Soperhaps weshould make the“rule” forphysicists that anydiscontinuous function (which canonly beasimplification ofareal physical function) should bedefined with halfway values atthediscontinuities. Then anysuch function-with anyfinite number ofsuch jumps—as well asall other physically interesting functions, aregiven correctly bytheFourier sum. Asanexercise, wesuggest thatthereader determine theFourier series for thefunction shown inFig.50-3. Since thefunction cannot bewritten inanexplicit algebraic form, youwillnotbeabletodotheintegrals from zerotoTintheusual way. Theintegrals areeasy, however, ifweseparate them into twoparts: the integral from zero toT/2(over which f(t)=l)andtheintegral from T/2toT (over which f(t)=-1). Theresult should be f(t)=ér(sinwt+%sin3wt +gsin5851+ --), (50.19) where w=27r/T.Wethus findthatoursquare wave (with theparticular phase chosen) hasonly oddharmonics, andtheir amplitudes areininverse proportion totheir frequencies. Letuscheck thatEq.(50.19) doesindeed giveusbackf(t)forsome value oft. Letuschoose t=T/4, orwt=7r/2. Wehave 4 :_ > ~sL)|b-IE.CI f(t)=;(S111g+ésin 5%’+--) (50.20) 4 l 1 1-7_<l—§+5-7+--) (50.21) Theseries* hasthevalue 7r/4, andwefindthatf(t)=1. 50-5 Theenergy theorem Theenergy inawave isproportional tothesquare ofitsamplitude. Fora. . . . T2wave ofcomplex shape, theenergy inoneperiod Wlllbeproportional tolof(t)dt. Wecanalsorelate thisenergy totheFourier coefficients. Wewrite T T ea no faf2(t)dt =/0[(1,+Zancosnwt +26.,sinnwjzdr. (50.22)'n==1 ‘ll’-=1 When weexpand thesquare ofthebracketed term wewillgetallpossible cross terms, such asa5cosSwt-b7cos7wt. Wehave shown above, however, [Eqs. (50.11) and(50.l2)] thattheintegrals ofallsuch terms over oneperiod iszero. IThe series canbeevaluated inthefollowing way. First weremark that* jo[dx/(1 +x2)]=tan—1x. Second, weexpand theintegrand inaseries 1/(1+x2)= 1—-x2+x4—x6+...Weintegrate theseries term byterm (from zero tox)to obtain tan” x=1-x3/3 +x5/5 —x7/7 +...Settingx =1,we have thestated result, since tan” 1=1r/4. 50-7f(t) 9| 1‘N ___‘-1.-. -| L-_____. Fig. 50-3. Square-wave function f(1)= +1forO <t<1/2, f(t)=—-lforT/2 <t<T. hm ‘Hm / / Kin X; / / 10) LINEAR (bl NONLINEAR ‘wt’ “fin ‘wt- Kulnf "furl Fig. 50-4. Linear and nonlinear re sponses. Mn nouiuuu ' t re/1-Lmzan \\ Fig. 50-5. The response ofanon- linear device totheinput coswt. A linear response isshown forcomparison.Wehave leftonly thesquare terms likea§cosf 5wt. Theintegral ofanycosine squared orsinesquared over oneperiod isequal toT/2, soweget T T/0f2(r)dr= Ta%+5(ai+a§+ ---+bi+b§+ ---) _ 2 T0° 2 2_Tao+52(6,,+6,). (50.23)n1 This equation iscalled the“energy theorem,” andsaysthatthetotal energy ina wave isjustthesumoftheenergies inalloftheFourier components. Forexample, applying thistheorem totheseries (50.19), since [f(t)]2 =1weget T_Z.(i)’(1+i+i+L...)_2 7r 325272 ’ sowelearn thatthesum ofthesquares ofthereciprocals oftheoddintegers is 7r2/8.Inasimilar way, byfirstobtaining theFourier series forthefunction and using theenergy theorem, wecanprove that 1+1/24 —l—1/34 +~--is7r‘*/90, aresult weneeded inChapter 45. 50-6 Nonlinear responses Finally, inthetheory ofharmonics there isanimportant phenomenon which should beremarked upon because ofitspractical importance--that ofnonlinear effects. Inallthesystems thatwehave been considering sofar,wehave supposed thateverything waslinear, thattheresponses toforces, saythedisplacements or theaccelerations, were always proportional totheforces. Orthatthecurrents in thecircuits were proportional tothevoltages, andsoon.Wenowwish toconsider cases where there isnotastrict proportionality. Wethink, atthemoment, ofsome device inwhich theresponse, which wewillcallx,,1,,atthetime t,isdetermined bytheinput x,,,atthetime t.Forexample, x1,,might betheforce andx,,,,,might bethedisplacement. Orx1,,might bethecurrent andxou,thevoltage. Ifthedevice islinear, wewould have xout(t) =KX1n(t). (50-24) where Kisaconstant independent oftandofx1,,. Suppose, however, thatthe device isnearly, butnotexactly, linear, sothatwecanwrite X6680) =KfX1n(l) +¢X§1(l)], (50-25) where eissmall incomparison with unity. Such linear andnonlinear responses are shown inthegraphs ofFig.50-4. Nonlinear responses have several important practical consequences. We shall discuss some ofthem now. First weconsider what happens ifweapply a pure tone attheinput. Weletx1,,=coswt.Ifweplotx,,,,,asafunction oftime wegetthesolid curve shown inFig.50-5. Thedashed curve gives, forcomparison, theresponse ofalinear system. Weseethat theoutput isnolonger acosine function. Itismore peaked atthetopandflatter atthebottom. Wesaythatthe output isdistorted. Weknow, however, that such awave isnolonger apure tone, thatitwillhave harmonics. Wecanfind what theharmonics are. Using x,,,=coswtwith Eq.(50.25), wehave x,,,,,=K(coswt+666$’<61). (50.26) From theequality cos2 0=%(l-cos20),wehave x0111 =K(cos wt—l—5—5cos2wt) - (50.27) Theoutput hasnotonly acomponent atthefundamental frequency, that was present attheinput, butalsohassome ofitssecond harmonic. There hasalso 50-8 appeared attheoutput aconstant term K(e/2), which corresponds totheshift of theaverage value, shown inFig.50-5. Theprocess ofproducing ashift ofthe average value iscalled rectification. Anonlinear response willrectify andwillproduce harmonics ofthefrequencies atitsinput. Although thenonlinearity weassumed produced onlysecond harmon- ics,nonlinearities ofhigher order—those which have terms likexi,andx1‘,1,for example-—will produce harmonics higher than thesecond. Another effect which results from anonlinear response ismodulation. Ifour input function contains two(ormore) pure tones, theoutput willhave notonly their harmonics, butstillother frequency components. Letx1,,=Acosw1t+ Bcos (4)26,where now w1andw2arenotintended tobeinaharmonic relation. Inaddition tothelinear term (which isKtimes theinput) weshall have acompo- nent intheoutput given by X0“,=Ke(Acosw1t+Bcosw2t)2 (50.28) =Ke(A2 cos2 w1t+B2cos2wzt+2ABcosw1tcosw2t). (50.29) Thefirsttwoterms intheparentheses ofEq.(50.29) arejustthose which gave theconstant terms andsecond harmonic terms wefound above. Thelastterm is new. Wecanlook atthisnew“cross term” ABcosw1tcoswgtintwoways. First, ifthetwofrequencies arewidely different ‘(forexample, ifw1ismuch greater than (.02)wecanconsider thatthecross term represents acosine oscillation ofvarying amplitude. That is,wecanthink ofthefactors inthisway: ABcosw1tcosw2t=C(t)cosw1t, (50.30) with C(t) =ABcos(ugt. (50.31) Wesaythattheamplitude ofcosw1ismodulated with thefrequency 032. Alternatively, wecanwrite thecross term inanother way: ABcosw1tcos w2t=ATI2[cos(w1+w2)t+cos(w1-w2)t]. (50.32) Wewould now saythattwonewcomponents have been produced, oneatthesum frequency (w1+w2),another atthedifference frequency (w1—w2). Wehave twodifferent, butequivalent, ways oflooking atthesame result. Inthespecial case thatw1>><02,wecanrelate these twodifferent views byre- marking thatsince (w1+(.02)and(w1-(.02)arenear toeach other wewould expect toobserve beats between them. Butthese beats have justtheeffect of modulating theamplitude oftheaverage frequency w1byone-half thedifference frequency 2w2. Wesee,then, whythetwodescriptions areequivalent. Insummary, wehave found thatanonlinear response produces several effects: rectification, generation ofharmonics, and modulation, orthegeneration of components with sumanddifference frequencies. Weshould notice thatallthese effects (Eq. 50.29) areproportional notonly tothenonlinearity coeflicient e,butalsototheproduct oftwoamplitudes—either A2,B2,orAB. Weexpect these effects tobemuch more important forstrong signals than forweak ones. Theefl'ects wehave been describing have many practical applications. First, with regard tosound, itisbelieved thattheearisnonlinear. This isbelieved to account forthefactthatwith loud sounds wehave thesensation thatwehear harmonics andalsosumanddifference frequencies even ifthesound waves contain only pure tones. Thecomponents which areused insound-reproducing equipment—amplifiers, loudspeakers, etc.—always have some nonlinearity. They produce distortions in thesound——they generate harmonics, etc.—which were notpresent intheoriginal sound. These newcomponents areheard bytheearandareapparently objection- able. Itisforthisreason that“Hi-Fi” equipment isdesigned tobeaslinear as 50-9 possible. (Why thenonlinearities oftheeararenot“objectionable” inthesame way, orhow weeven know thatthenonlinearity isintheloudspeaker rather than intheearisnotclear!) Nonlinearities arequite necessary, andare,infact, intentionally made large incertain parts ofradio transmitting andreceiving equipment. InanAMtrans- mitter the“voice” signal (with frequencies ofsome kilocycles persecond) is combined with the“carrier” signal (with afrequency ofsome megacycles per second) inanonlinear circuit called amodulator, toproduce themodulated oscillation thatistransmitted. Inthereceiver, thecomponents ofthereceived signal arefedtoanonlinear circuit which combines thesumanddifference frequencies ofthemodulated carrier togenerate again thevoice signal. When wediscussed thetransmission oflight, weassumed thattheinduced oscillations ofcharges were proportional totheelectric field ofthelight—that the response waslinear. That isindeed avery good approximation. Itisonly within thelastfewyears thatlight sources have been devised (lasers) which produce an intensity oflight strong enough sothatnonlinear effects canbeobserved. Itis now possible togenerate harmonics oflight frequencies. When astrong redlight passes through apiece ofglass, alittle bitofblue light—second harmonic- comes out! 50-l0 51 Waves 51-1 Bowwaves Although wehave finished ourquantitative analyses ofwaves, thisadded chapter onthesubject isintended togive some appreciation, qualitatively, for various phenomena thatareassociated with waves, which aretoocomplicated to analyze indetail here. Since wehave been dealing with waves forseveral chapters, more properly thesubject might becalled “some ofthemore complex phenomena associated with waves.” Thefirsttopic tobediscussed concerns theeffects thatareproduced bya source ofwaves which ismoving faster thanthewave velocity, orthephase velocity. Letusfirstconsider waves thathave adefinite velocity, likesound andlight. If wehave asource ofsound which ismoving faster than thespeed ofsound, then something likethishappens: Suppose atagiven moment asound wave isgenerated from thesource atpoint x1inFig.51-1; then, inthenext moment, asthesource moves tox2,thewave from x1expands byaradius r1smaller than thedistance thatthesource moves; and, ofcourse, another wave starts from x2. When the sound source hasmoved stillfarther, tox3,andawave isstarting there, thewave from x2hasnow expanded tor2,andtheonefrom x1hasexpanded tor3.Of course thething isdone continuously, notinsteps, andtherefore, wehave aseries ofwave circles with acommon tangent linewhich goes through thecenter ofthe source. Weseethatinstead ofasource generating spherical waves, asitwould ifit were standing still,itgenerates awavefront which forms acone inthree dimensions, orapairoflines intwodimensions. Theangle ofthecone isvery easy tofigure out. Inagiven amount oftime thesource moves adistance, sayx3-x1,pro- portional tov,thevelocity ofthesource. Inthemeantime thewavefront hasmoved outadistance r3,proportional tocw,thespeed ofthewave. Therefore itisclear thatthehalf-angle ofopening hasasineequal totheratio ofthespeed ofthewaves, divided bythespeed ofthesource, andthissinehasasolution only ifcwisless than 0,orthespeed oftheobject isfaster than thespeed ofthewave: . c,,,_s1n0 -7 (51.1) Incidentally, although weimplied that itisnecessary tohave asource of sound, itturns out,veryinterestingly, thatonce theobject ismoving faster than the speed ofsound, itwillmake sound. That is,itisnotnecessary thatithave acertain tone vibrational character. Any object moving through amedium faster than the speed atwhich themedium carries waves willgenerate waves oneach side, auto- matically, justfrom themotion itself. This issimple inthecase ofsound, butit alsooccurs inthecaseoflight. Atfirstonemight think nothing canmove faster than thespeed oflight. However, light inglass hasaphase velocity lessthan the speed oflight inavacuum, anditispossible toshoot acharged particle ofvery high energy through ablock ofglass such thattheparticle velocity isclose tothe speed oflight inavacuum, while thespeed oflight intheglass may beonly § thespeed oflight inthevacuum. Aparticle moving faster than thespeed oflight inthemedium willproduce aconical wave oflight with itsapex atthesource, likethewave wake from aboat (which isfrom thesame efl'ect, asamatter of fact). Bymeasuring thecone angle, wecandetermine thespeed oftheparticle. This isused technically todetermine thespeeds ofparticles asoneofthemethods ofdetermining their energy inhigh-energy research. Thedirection ofthelight is allthatneeds tobemeasured. 51-151-1 Bowwaves 51-2 Shock waves 51-3 Waves insolids 51-4 Surface waves €‘n'=&‘?)>4 Fig. 51-l. Theshock wave front lies onacone with apex atthesource and half-angle 0=sin“! v/c,,,. Pressurec hcw 2| ClFig.51-2. Ashock wave induced in agasbyaprojectile moving faster than sound. Thislight issometimes called Cerenkov radiation, because itwasfirstobserved byCerenkov. How intense thislight should bewasanalyzed theoretically byFrank andTamm. The1958 Nobel Prize forphysics wasawarded jointly toallthree for thiswork. Thecorresponding circumstances inthecase ofsound areillustrated inFig. 51-2, which isaphotograph ofanobject moving through agasataspeed greater than thespeed ofsound. Thechanges inpressure produce achange inrefractive index, andwith asuitable optical system theedges ofthewaves canbemade visible. Weseethattheobject moving faster than thespeed ofsound does, indeed, produce aconical wave. Butcloser inspection reveals thatthesurface isactually curved. Itisstraight asymptotically, butitiscurved near theapex, andwehave nowtodiscuss howthatcanbe,which brings ustothesecond topic ofthischapter. °6 iit2>t, O ta’1'2 O Fig.51-3. Wavefront "snapshots" at Distcnce successive instants intime. 51-2 Shock waves Wave speed often depends ontheamplitude, andinthecaseofsound thespeed depends upon theamplitude inthefollowing way. Anobject moving through the airhastomove theairoutoftheway, sothedisturbance produced inthiscase issome kind ofapressure step, with thepressure higher behind thewavefront than intheundisturbed region notyetreached bythewave (running along atthenormal speed, say). Buttheairthatisleftbehind, after thewavefront passes, hasbeen compressed adiabatically, andtherefore thetemperature isincreased. Now the speed ofsound increases with thetemperature, sothespeed intheregion behind thejump isfaster than intheairinfront. That means thatanyother disturbance thatismade behind thisstep, saybyacontinuous pushing ofthebody, oranyother disturbance, willridefaster than thefront, thespeed increasing withhigher pressure. Figure 51-3 illustrates thesituation, with some little bumps ofpressure added to thepressure contour toaidvisualization. Weseethatthehigher pressure regions attherearovertake thefront astime goes on,until ultimately thecompressional wave develops asharp front. Ifthestrength isvery high, “ultimately” means right away; ifitisrather weak, ittakes alongtime; itmaybe,infact,thatthesound isspreading anddying outbefore ithastime todothis. Thesounds wemakeintalking areextrerneiy weakrelative tothea*t'i'n'08'p‘iiei‘1t pressure—only lpartinamillion orso.Butforpressure changes oftheorder ofl atmosphere, thewave velocity increases byabout twenty percent, andthewavefront sharpens upatacorrespondingly high rate. Innature nothing happens infinitely ggpidly, presumably, andwhat wecalla“sharp” front has,actually, averyslight thickness; itisnotinfinitely steep. Thedistances overwhich itisvarying areofthe 51-2 order ofonemean freepath, inwhich thetheory ofthewave equation begins to failbecause wedidnotconsider thestructure ofthegas. Now, referring again toFig.51-2, weseethatthecurvature canbeunderstood ifweappreciate thatthepressures near theapex arehigher than theyarefarther back, andsotheangle 0isgreater. That is,thecurve istheresult ofthe factthatthe speed depends upon thestrength ofthewave. Therefore thewave from anatomic bomb explosion travels much faster thanthespeed ofsound forawhile, until itgets sofaroutthatitisweakened tosuch anextent from spreading thatthepressure bump issmall compared with atmospheric pressure. Thespeed ofthebump then approaches thespeed ofsound inthegasintowhich itisgoing. (Incidentally, it always turns outthatthespeed oftheshock ishigher than thespeed ofsound in thegasahead, butislower than thespeed ofsound inthegasbehind. That is, impulses from theback willarrive atthefront, butthefront rides intothemedium inwhich itisgoing faster than thenormal speed ofsignals. Soonecannot tell, acoustically, thattheshock iscoming until itistoolate. Thelight from thebomb arrives first, butonecannot tellthattheshock iscoming until itarrives, because there isnosound signal coming ahead ofit.) This isaveryinteresting phenomenon, thispiling upofwaves, andthemain point onwhich itdepends isthatafter awave ispresent, thespeed oftheresulting wave should behigher. Another example ofthesame phenomenon isthefollowing. Consider water flowing inalong channel with finite width andfinite depth. Ifa piston, orawallacross thechannel, ismoved along thechannel fastenough, water piles up,likesnow before asnow plow. Now suppose thesituation isasshown in Fig.51-4, withasudden stepinwater height somewhere inthechannel. ltcanbe demonstrated thatlong waves inachannel travel faster indeeper water than they doinshallow water. Therefore anynewbumps orirregularities inenergy supplied bythepiston runoffforward andpileupatthefront. Again, ultimately what we have isjustwater withasharp front, theoretically. However, asFig.51-4 shows, there arecomplications. Pictured isawave coming upachannel; thepiston isat thefarright endofthechannel. Atfirstitmight have appeared likeawell-behaved wave, asonemight expect, butfarther along thechannel, ithasbecome sharper and sharper until theevents pictured occurred. There isaterrible churning atthe surface, asthepieces ofwater falldown, butitisessentially averysharp risewith nodisturbance ofthewater ahead. Actually water ismuch more complicated than sound. However,just toillus- trate apoint, wewilltrytoanalyze thespeed ofsuch aso-called bore, inachannel. Thepoint hereisnotthatthisisofanybasic importance forourpurposes—it is notagreat generalization-it isonly toillustrate thatthelaws ofmechanics that wealready know arecapable ofexplaining thephenomenon. Imagine, foramoment, thatthewater does look something likeFig.5l—5(a), thatwater atthehigher height h2ismoving withavelocity v,andthatthefront is moving withvelocity uintoundisturbed water which isatheight h1.Wewould like todetermine thespeed atwhich thefront moves. Inatime Atavertical plane initially atx1moves adistance 0Attox2,while thefront ofthewave hasmoved uAt. Now weapply theequations ofconservation ofmatter andmomentum. First, theformer: Perunitchannel width, weseethattheamount h2vAtofmatter that hasmoved pastx1(shown shaded) iscompensated bytheother shaded region, which amounts to(h2—h1)uAt.So,dividing byAt,vh2=u(h2 —h1). That does notyetgiveusenough, because although wehave h2andh1,wedonotknow either uor1»;wearetrying togetboth ofthem. Now thenextstepistouseconservation ofmomentum. Wehave notdiscussed theproblems ofwater pressure, oranything inhydrodynamics, butitisclear any- waythatthepressure ofwater atagiven depth isjustenough toholdupthecolumn ofwater above it.Therefore thepressure ofwater isequal top,thedensity ofwater, times g,times thedepth below thesurface. Since thepressure increases linearly with depth, theaverage pressure over theplane atx1,say,is2pgh2,which isalso theaverage force perunitwidth andperunitheight pushing theplane toward x2. Sowemultiply byanother h2togetthetotal force which isacting onthewater 51-3Figure 5l-4 I I I |-vat1 p<—UAt 1 1 I XI *2 Fig.51-5. Two cross sections ofa bore inachannel, with lbataninterval Atlater than (a)xU pushing from theleft. Ontheother hand, there ispressure inthewater onthe right also, exerting anopposite force ontheregion inquestion, which is,bythe same kind ofanalysis, %pgh2. Now wemust balance theforces against therateof change ofthemomentum. Thus wehave tofigure outhowmuch more momentum there isinsituation (b)inFig.51-5 than there wasin(a).Weseethattheadditional mass thathasacquired thespeed visjustph2u At—ph2v At(perunitwidth), and multiplying thisbyvgives theadditional momentum tobeequated totheimpulse FAt: (ph2uAt—phzvAtlv=(fpghi -ipghi) A1- Ifweeliminate vfrom thisequation bysubstituting vh2-=u(h2—h1),already found, andsimplify, wegetfinally thatu2=gh2(h1 +h2)/2h 1. Iftheheight difference isvery small, sothath1andh2arenearly equal, this saysthatthevelocity = Aswewillseelater, thatisonly trueprovided the wavelength ofthewave islonger than thedepth ofthechannel. Wecould alsodotheanalogous thing forsound waves—including thecon- servation ofinternal energy, nottheconservation ofentropy, because theshock is irreversible. Infact, ifonechecks theconservation ofenergy inthebore problem, onefinds thatenergy isnotconserved. Iftheheight difference issmall, itisalmost perfectly conserved, butassoon astheheight difference becomes veryappreciable, there isanetlossofenergy. Thisismanifested asthefalling water andthechurning shown inFig.51-4. Inshock waves there isacorresponding apparent lossofenergy, from the point ofview ofadiabatic reactions. Theenergy inthesound wave, behind the shock, goesintoheating ofthegasafter shock passes, corresponding tochurning of thewater inthebore. Inworking itout,three equations forthesound caseturn outtobenecessary forsolution, andthetemperature behind theshock isnotthe same asthetemperature infront, aswehave seen. Ifwetrytomake abore thatisupside down (h2<h1),then wefindthatthe energy losspersecond isnegative. Since energy isnotavailable from anywhere, thatbore cannot then maintain itself ;itisunstable. Ifwewere tostart awave of thatsort, itwould flatten out,because thespeed dependence onheight thatresulted insharpening inthecasewediscussed would now have theopposite effect. 51-3 Waves insolids Thenext kind ofwaves tobediscussed arethemore complicated waves in solids. Wehave already discussed sound waves ingasandinliquid, andthere isa direct analog toasound wave inasolid. Ifasudden push isapplied toasolid, itis compressed. Itresists thecompression, andawave analogous tosound isstarted. However there isanother kind ofwave thatispossible inasolid, andwhich isnot possible inafluid. Ifasolid isdistorted bypushing itsideways (called shearing), thenittriestopullitself back. That isbydefinition what distinguishes asolid from aliquid: ifwedistort aliquid (internally), hold itaminute sothatitcalms down, andthenletgo,itwillstaythatway, butifwetakeasolid andpush it,likeshearing apiece of“Jello,” andletitgo,itfliesback andstarts ashear wave, travelling in thesame waythecompressions travel. Inallcases, theshear wave speed isless than thespeed oflongitudinal waves. Theshear waves aresomewhat more anal- ogous, sofarastheir polarizations areconcerned, tolight waves. Sound hasno polarization, itisjustapressure wave. Light hasacharacteristic orientation per- pendicular toitsdirection oftravel. Inasolid, thewaves areofboth kinds. First, there isacompression wave, analogous tosound, thatruns atonespeed. Ifthesolid isnotcrystalline, thena shear wave polarized inanydirection willpropagate atacharacteristic speed. (Ofcourse allsolids arecrystalline, butifweuseablock made upofmicrocrystals ofallorientations, thecrystal anisotropies average out.) Another interesting question concerning sound waves isthefollowing: What happens ifthewavelength inasolid getsshorter, andshorter, andshorter? How short canitget? Itisinteresting thatitcannot getanyshorter than thespace 51-4 between theatoms, because ifthere issupposed tobeawave inwhich onepoint goesupandthenextdown, etc.,theshortest possible wavelength isclearly the atom spacing. Interms ofthemodes ofoscillation, wesaythatthere arelongi- tudinal modes, andtransverse modes, long wave modes, short wave modes. As weconsider wavelengths comparable tothespacing between theatoms, then the speeds arenolonger constant; there isadispersion effect where thevelocity isnot independent ofthewave number. But, ultimately, thehighest mode oftransverse waves would bethatinwhich every atom isdoing theopposite ofneighboring atoms. Now from thepoint ofview ofatoms, thesituation islikethetwopendulums thatwewere talking about, forwhich there aretwomodes, oneinwhich theyboth gotogether, andtheother inwhich they goapart. Itispossible toanalyze the solid waves another way, interms ofasystem ofcoupled harmonic oscillators, like anenormous number ofpendulums, withthehighest mode such thattheyoscillate oppositely, andlower modes with dilferent relationships ofthetiming. The shortest wavelengths aresoshort that they arenotusually available technically. However they areofgreat interest because, inthetheory ofthermo- dynamics ofasolid, theheat properties ofasolid, forexample specific heats, can beanalyzed interms oftheproperties oftheshort sound waves. Going tothe extreme ofsound waves ofever shorter wavelength, onenecessarily comes to theindividual motions oftheatoms; thetwothings arethesame ultimately. Avery interesting example ofsound waves inasolid, both longitudinal and transverse, arethewaves thatareinthesolid earth. Who makes thenoises wedo notknow, butinside theearth, from time totime, there areearthquakes——some rock slides pastsome other rock. That islikealittle noise. S0waves likesound waves start outfrom such asource verymuch longer inwavelength than oneusu- allyconsiders insound waves, butstilltheyaresound waves, andtheytravel around intheearth. Theearth isnothomogeneous, however, andtheproperties ofpressure, density, compressibility, andsoon,change with depth, andtherefore thespeed varies with depth. Then thewaves donottravel instraight lines——there isakind ofindex ofrefraction andthey goincurves. Thelongitudinal waves andthe transverse waves have different speeds, sothere aredifferent solutions forthediffer- entspeeds. Therefore ifweplace aseismograph atsome location andwatch the waythething jiggles after there hasbeen anearthquake somewhere else,thenwedo notjustgetanirregular jiggling. Wemight getajiggling, andaquieting down, and then another jiggling~what happens depends upon thelocation. Ifitwere close enough, wewould firstreceive longitudinal waves from thedisturbance, andthen, afewmoments later, transverse waves, because they travel more slowly. By measuring thetime difference between thetwo, wecantellhowfaraway theearth- quake is,ifweknow enough about thespeeds andcomposition oftheinterior regions involved. Anexample ofthebehavior pattern ofwaves intheearth isshown inFig.5l—6. Thetwokinds ofwaves arerepresented bydifferent symbols. Ifthere were anearth- quake attheplace marked “source,” thetransverse waves andlongitudinal waves would arrive atdifferent times atthestation bythemost direct routes, andthere would alsobereflections atdiscontinuities, resulting inother paths andtimes. It turns outthatthere isacore intheearth which does notcarry transverse waves. Ifthestation isopposite thesource, transverse waves stillarrive, butthetiming is notright. What happens isthatthetransverse wave comes tothecore, andwhen- everthetransverse waves come toasurface which isoblique, between twomaterials, twonewwaves aregenerated, onetransverse andonelongitudinal. Butinside the core oftheearth, atransverse wave isnotpropagated (oratleast, there isnoevi- dence forit,only foralongitudinal wave); itcomes outagain inboth forms and comes tothestation. Itisfrom thebehavior ofthese earthquake waves thatithasbeen determined thattransverse waves cannot bepropagated within theinner circle. This means thatthecenter oftheearth isliquid inthesense thatitcannot propagate transverse waves. Theonlywayweknow what isinside theearth isbystudying earthquakes. So,byusing alarge number ofobservations ofmany earthquakes atdifferent 51-5/"’P’P“\‘ sounc: -.»’."s —-- r.-'" STITION:-._.=a_*,_='.»-_.-..:_,q_‘j?\_E¢_E\..-" __ /*s»r§'" ‘ ,/1"-L‘I1 //"Q)4”at\\ JPKP \‘\\rxxP. )\. \h‘\/ ' / \\§\ //\ _._/ \~'_' vxvnxr 1 '“ V IRANSVERSE (SI LONoiYuD\nAL \v.K) \ v v v-¢-...._...-.. 0 o Fig. 51-6. Schematic oftheearth, showing paths oflongitudinal andtrans- verse sound waves. consumer ' Q: r?‘IW “ ""r.s.s. l 5L X,1a lllOB5 EARTHYDE k E;,jl if-1’- _,-:1‘.1. uiiIEit 056-v-iiiI”992'1 2%>32:1" B37 :1? Y0! 7474:/' 10 670—§I an-3" Half?S0 %-£0 I006=4O|5ABE|.LA .,OO_~-t -»2.. 09so 3,’"es[41stations, thedetails have been worked out—the speed, thecurves, etc.areallknown. Weknow what thespeeds ofvarious kinds ofwaves areatevery depth. Knowing that, therefore, itispossible tofigure outwhat thenormal modes oftheearth are, because weknow thespeed ofpropagation ofsound waves—in other words, the elastic properties ofboth kinds ofwaves atevery depth. Suppose theearth were distorted intoanellipsoid andletgo. Itisjustamatter ofsuperposing waves travelling around intheellipsoid todetermine theperiod andshapes inafreemode. Wehave figured outthatifthere isadisturbance, there arealotofmodes, from the lowest, which isellipsoidal, tohigher modes with more structure. N C0>4EIlEN€E " /ILl;muon INMINUYKS 'E5 ‘L_'0III:P(CYll_338 El‘§5=1:11“K4 4§§la4-Q[.35Q33,,iE-its-‘I1:ilkI;’*: '1L 1 iii?‘— __QisI—"~=< _,_-F _-=‘?A~—LFig.51-7. Power versus frequency asdetected atseismographs inFlafia, Peru, and Isabella, California. The coherence isameasure ofthecouplingr"’r,s,s, 2El!éilll l Wm ‘ M between thestations. [From Benioff, Press and Smith, J.Geoph. Research 66,605 (l96ll]. 55 4oo—-—~~-~- -——i-----—-- -—__z-----~ —------—roumzn ANALYSIS E;l$lBELLA srmun |msooo Mm sso svur s,MODE r'“F,s,r,s,s,s,r, l :2 'soo Q: ~-—-I F\U)noizso—-—~~l HA6“YUUE.,,8 -auttzu.5‘1x-—euu_A moi _T_z1__iAl.)I00 ~'— J 50 [___O8lB0 00182 OOIB4 OOI86 0.0188 OOISO COB? FREQUENCY INCYCLES PER MINUTE Fig.51-8. High-resolution analysis of oneoftheseismograph records, showing spectral doublet...\...Iv1 N z.9 ': an-9 '1' 2 |§‘a[|_|_‘‘D _ ,_,_,___,,AA. .._,,_4_. .. ... . . ‘. ‘'o ous 00¢ ous nos om om oos on on mz 0| 'll ms ms "E°""°' ‘"mu“ PE"""“"E rnsouzucv lNcvctcs PKR smut: TheChilean earthquake ofMay 1960 made aloud enough “noise” thatthe signals went around theearth many times, andnewseismographs ofgreat delicacy were made justintime todetermine thefrequencies ofthefundamental modes of theearth andtocompare them withthevalues thatwere calculated from thetheory ofsound withtheknown velocities, asmeasured from theindependent earthquakes. Theresult ofthisexperiment isillustrated inFig.51-7, which isaplotofthestrength ofthesignal versus thefrequency ofitsoscillation (aFourier analysis). Note that atcertain particular frequencies there ismuch more being received than atother frequencies; there arevery definite maxima. These arethenatural frequencies of theearth, because these arethemain frequencies atwhich theearth canoscillate. Inother words, iftheentire motion oftheearth ismade upofmany different modes, wewould expect toobtain, foreach station, irregular bumpings which indicate a superposition ofmany frequencies. Ifweanalyze thisinterms offrequencies, we should beabletofindthecharacteristic frequencies oftheearth. Thevertical dark lines inthefigure arethecalculated frequencies, andwefindaremarkable agree- ment, anagreement duetothefactthatthetheory ofsound isright fortheinside oftheearth. Avery curious point isrevealed inFig. 51-8, which shows avery careful measurement, with better resolution ofthelowest mode, theellipsoidal mode of theearth. Note thatitisnotasingle maximum, butadouble one,54.7minutes and 53.1minutes-—slightly different. Thereason forthetwodifferent frequencies was notknown atthetime thatitwasmeasured, although itmay have been found in themeantime. There areatleast twopossible explanations: Onewould bethat there maybeasymmetry intheearth’s distribution, which would result intwosimi- larmodes. Another possibility, which iseven more interesting, isthis: Imagine thewaves going around theearth intwodirections from thesource. Thespeeds willnotbeequal because ofeffects oftherotation oftheearth intheequations of motion, which have notbeen taken intoaccount inmaking theanalysis. Motion inarotating system ismodified byCoriolis forces, andthese may cause theob- served splitting. Regarding themethod bywhich these quakes have been analyzed. what is obtained ontheseismograph isnotacurve ofamplitude asafunction offrequency, butdisplacement asafunction oftime, always avery irregular tracing. Tofind theamount ofallthedifferent sinewaves foralldifferent frequencies, weknow that thetrick istomultiply thedata byasinewave ofagiven frequency andintegrate, i.e.,average it,andintheaverage allother frequencies disappear. Thefigures were thus plots oftheintegrals found when thedata were multiplied bysine waves ofdifferent cycles perminute, andintegrated. 51-6 51-4 Surface waves Now, thenextwaves ofinterest, thatareeasily seenbyeveryone andwhich areusually used asanexample ofwaves inelementary courses, arewater waves. Asweshall soon see,they aretheworst possible example, because they areinno respects likesound andlight; theyhave allthecomplications thatwaves canhave. Letusstart with long water waves indeep water. Iftheocean isconsidered in- finitely deep andadisturbance ismade onthesurface, waves aregenerated. All kinds ofirregular motions occur, butthesinusoidal typemotion, withaverysmall disturbance, might look likethecommon smooth ocean waves coming intoward theshore. Now withsuchawave, thewater, ofcourse, ontheaverage, isstanding still, butthewave moves. What isthemotion, isittransverse orlongitudinal? Itmust beneither; itisnottransverse, norisitlongitudinal. Although thewater at agiven place isalternately trough orhill,itcannot simply bemoving upanddown, bytheconservation ofwater. That is,ifitgoesdown, where isthewater going togo? Thewater isessentially incompressible. Thespeed ofcompression of waves—that is,sound inthewater——is much, much higher, andwearenotcon- sidering thatnow. Since water isincompressible onthisscale, asahillcomes down thewater must move away from theregion. What actually happens isthat particles ofwater near thesurface move approximately incircles. When smooth swells arecoming, aperson floating inatirecanlookatanearby object andsee itgoing inacircle. Soitisamixture oflongitudinal andtransverse, toaddtothe confusion. Atgreater depths inthewater themotions aresmaller circles until, reasonably fardown, there isnothing leftofthemotion (Fig. 51-9). Awater wave Wave crest Fig.51-9. Deep-water waves are __ _I A __ \ formed from particles moving incircles. ,”‘'’ll‘ ‘’\‘X’ \,"‘Y’F'7It , Note thesystematic phase shifts from \_» ~-/\_/\_/\_//~_ ’\/“ \- \_/ ~_/‘xx’ circle tocircle. How would afloating water molecules move in obiect move? circular orbits when Wave trough wave passes hy Tofindthevelocity ofsuch waves isaninteresting problem: itmust besome combination ofthedensity ofthewater, theacceleration ofgravity, which isthe restoring force thatmakes thewaves, andpossibly ofthewavelength andofthe depth. Ifwetakethecasewhere thedepth goestoinfinity, itwillnolonger depend onthedepth. Whatever formula wearegoing togetforthevelocity ofthephases ofthewaves must combine thevarious factors tomake theproper dimensions, andifwetrythisinvarious ways, wefindonly onewaytocombine thedensity, g,andAinorder tomake avelocity, namely, \/5, which does notinclude theden- sityatall.Actually, thisformula forthephase velocity isnotexactly right, but acomplete analysis ofthedynamics, which wewillnotgointo, shows that the factors areaswehave them, except for\/E: v,,1,.,,,, =\/g)\/21r (forgravity waves). Itisinteresting thatthelong waves gofaster than theshort waves. Thus ifaboat makes waves farout,because there issome sports-car driver inamotorboat travelling by,then after awhile thewaves come toshore with slow sloshings at firstandthenmore andmore rapid sloshings, because thefirstwaves thatcome are long. Thewaves getshorter andshorter asthetime goes on,because thevelocities goasthesquare rootofthewavelength. Onemayobject, “That isnotright, wemust look atthegroup velocity inorder tofigure itout!” Ofcourse thatistrue. Theformula forthephase velocity does nottelluswhat isgoing toarrive first; what tellsusisthegroup velocity. Sowe havetowork outthegroup velocity, anditisleftasaproblem toshow ittobe one-half ofthephase velocity, assuming thatthevelocity goes asthesquare root ofthewavelength, which isallthatisneeded. Thegroup velocity alsogoes asthe square rootofthewavelength. How canthegroup velocity gohalfasfastasthe phase? Ifonelooks atthebunch ofwaves thataremade byaboattravelling 51-7 roo- L V,cm/sec(Io -1 E____._ u_ 5_ X,cm Fig.5l—l l.Phase velocity vs.wave- length forwater.along, following aparticular crest, hefinds that itmoves forward inthegroup andgradually getsweaker anddiesoutinthefront, andmystically andmysteriously aweak oneintheback works itswayforward andgetsstronger. Inshort, thewaves aremoving through thegroup while thegroup isonly moving athalfthespeed thatthewaves aremoving. l 4 Fig.5l-l0. Thewake ofaboat. Because thegroup velocities andphase velocities arenotequal, thenthewaves thatareproduced byanobject moving through arenolonger simply acone, but itismuch more interesting. WecanseethatinFig.51-10, which shows thewaves produced byanobject moving through thewater. Note thatitisquite different than what wewould have forsound, inwhich thevelocity isindependent ofwave- length, where wewould have wavefronts only along thecone, travelling outward. Instead ofthat, wehave waves intheback withfronts moving parallel tothemotion oftheboat, andthen wehave little waves onthesides atother angles. This entire pattern ofwaves can,with ingenuity, beanalyzed byknowing only this: thatthe phase velocity isproportional tothesquare root ofthewavelength. Thetrick is thatthepattern ofwaves isstationary relative tothe(constant-velocity) boat; any other pattern would getlostfrom theboat. Thewater waves thatwehave been considering sofarwere longwaves inwhich theforce ofrestoration isduetogravitation. Butwhen waves getvery short in thewater, themain restoring force iscapillary attraction, i.e.,theenergy ofthe surface, thesurface tension. Forsurface tension waves, itturns outthatthephase velocity is Uphase =\/21rT/>\p (forripples), where Tisthesurface tension andpthedensity. Itistheexact opposite: thephase velocity ishigher, theshorter thewavelength, when thewavelength getsverysmall. When wehave both gravity andcapillary action, aswealways do,wegetthecom- bination ofthese twotogether: vphase = + where k=21r/)\ isthewave number. Sothevelocity ofthewaves ofwater is really quite complicated. Thephase velocity asafunction ofthewavelength is shown inFig.51-11; forvery short waves itisfast, forvery long waves itisfast, andthere isaminimum speed atwhich thewaves cango.Thegroup velocity can becalculated from theformula: itgoes to%thephase velocity forripples and% thephase velocity forgravity waves. Totheleftoftheminimum thegroup velocity ishigher than thephase velocity; totheright, thegroup velocity islessthan the 51-8 phase velocity. There areanumber ofinteresting phenomena associated with these facts. Inthefirstplace, since thegroup velocity isincreasing sorapidly as thewavelength goes down, ifwemake adisturbance there willbeaslowest endof thedisturbance going attheminimum speed with thecorresponding wavelength, andtheninfront, going athigher speed, willbeashortwave andaverylongwave. Itisveryhard toseethelongones, butitiseasytoseetheshort onesinawater tank. Soweseethattheripples often used toillustrate simple waves arequite inter- esting andcomplicated; theydonothave asharp wavefront atall,asisthecasefor simple waves likesound andlight. Themain wave haslittle ripples which runout ahead. Asharp disturbance inthewater does notproduce asharp wave because ofthedispersion. First come theveryfinewaves. Incidentally, ifanobject moves through thewater atacertain speed, arather complicated pattern results, because allthedifferent waves aregoing atdifferent speeds. Onecandemonstrate thiswith atrayofwater andseethatthefastest ones arethefinecapillary waves. There are slowest waves, ofacertain kind, which gobehind. Byinclining thebottom, one seesthatwhere thedepth islower, thespeed islower. Ifawave comes inatan angle tothelineofmaximum slope, itbends andtends tofollow thatline. Inthis wayonecanshow various things, andweconclude thatwaves aremore compli- cated inwater than inair. Thespeed oflongwaves inwater withcirculational motions isslower when the depth isless,faster indeep water. Thus aswater comes toward abeach where the depth lessens, thewaves goslower. Butwhere thewater isdeeper, thewaves are faster, sowegettheeffects ofshock waves. This time, since thewave isnotso simple, theshocks aremuch more contorted, andthewave over-curves itself, in thefamiliar wayshown inFig.51-12. This iswhat happens when waves come into theshore, andtherealcomplexities innature arewellrevealed insuch acircum- st..nce. Noonehasyetbeen abletofigure outwhat shape thewave should take asitbreaks. Itiseasyenough when thewaves aresmall, butwhen onegetslarge andbreaks, then itismuch more complicated. Fig. 5l-l2. Awater wave. Aninteresting feature about capillary waves canbeseen inthedisturbances made byanobject moving through thewater. From thepoint ofview oftheobject itself, thewater isflowing past, andthewaves which ultimately sitaround itare always thewaves which have justtheright speed tostaystillwith theobject inthe water. Similarly, around anobject inastream, with thestream flowing by,the pattern ofwaves isstationary, andatjusttheright wavelengths togoatthesame speed asthewater going by.Butifthegroup velocity islessthan thephase velocity, then thedisturbances propagate outbackwards inthestream, because thegroup velocity isnotquite enough tokeep upwith thestream. Ifthegroup velocity is faster than thevelocity ofthephase, thepattern ofwaves willappear infront of theobject. Ifonelooks closely atobjects inastream, onecanseethat there are little ripples infront andlong “slurps” intheback. 51-9 Another interesting feature ofthissortcanbeobserved inpouring liquids. Ifmilkispoured fastenough outofabottle, forinstance, alarge number oflines canbe.seen crossing bothways intheoutgoing stream. They arewaves starting from thedisturbance attheedges andrunning out,much likethewaves about an object inastream. There areeffects from both sides which produce thecrossed pattern. Wehaveinvestigated some oftheinteresting properties ofwaves andthevari- ouscomplications ofdependence ofphase velocity onwavelength, thespeed of thewaves ondepth, andsoforth, thatproduce thereally complex, andtherefore interesting, phenomena ofnature. 51-10 52 Symmetry inPhysical Laws 52-1 Symmetry operations Thesubject ofthischapter iswhat wemay callsymmetry inphysical laws. Wehave already discussed certain features ofsymmetry inphysical laws incon- nection withvector analysis (Chapter ll),thetheory ofrelativity (Chapter 16),and rotation (Chapter 20). Why should webeconcerned with symmetry? Inthefirstplace, symmetry is fascinating tothehuman mind, andeveryone likes objects orpatterns thatarein some waysymmetrical. Itisaninteresting factthatnature often exhibits certain kinds ofsymmetry intheobjects wefindintheworld around us.Perhaps the most symmetrical object imaginable isasphere, andnature isfullofspheres- stars, planets, water droplets inclouds. Thecrystals found inrocks exhibit many different kinds ofsymmetry, thestudy ofwhich tellsussome important things about thestructure ofsolids. Even theanimal andvegetable worlds show some degree of symmetry, although thesymmetry ofaflower orofabeeisnotasperfect oras fundamental asisthatofacrystal. Butourmain concern here isnotwith thefactthattheobjects ofnature are often symmetrical. Rather, wewish toexamine some oftheeven more remarkable symmetries oftheuniverse—the symmetries thatexist inthebasic laws themselves which govern theoperation ofthephysical world. First, what issymmetry? How canaphysical lawbe“symmetrical”? The problem ofdefining symmetry isaninteresting oneandwehavealready noted that Weyl gave agood definition, thesubstance ofwhich isthatathing issymmetrical ifthere issomething wecandotoitsothatafterwehavedoneit,itlooks thesame asitdidbefore. Forexample, asymmetrical vaseisofsuch akind thatifwereflect orturnit,itwilllook thesame asitdidbefore. Thequestion wewish toconsider here iswhat wecandotophysical phenomena, ortoaphysical situation inan experiment, andyetleave theresult thesame. Alistoftheknown operations under which various physical phenomena remain invariant isshown inTable 52-1. 52-2 Symmetry inspace andtime Thefirstthing wemight trytodo,forexample, istotranslate thephenomenon inspace. Ifwedoanexperiment inacertain region, andthen build another ap- paratus atanother place inspace (ormove theoriginal oneover) then, whatever went oninoneapparatus, inacertain order intime, willoccur inthesame wayif wehave arranged thesame condition, with alldueattention totherestrictions that wementioned before: thatallofthosefeatures oftheenvironment which make it notbehave thesame wayhave alsobeen moved over—we talked about how to define how much weshould include inthose circumstances, andweshall notgo intothose details again. Inthesame way, wealsobelieve today thatdisplacement intimewillhave no eflect onphysical laws. (That is,asfarasweknow today—all ofthese things are asfarasweknow today!) That means thatifwebuild acertain apparatus andstart itatacertain time, sayonThursday at10:00 a.m., andthenbuild thesame appara- tusandstart it,say,three days later inthesame condition, thetwoapparatuses will gothrough thesame motions inexactly thesame wayasafunction oftime no matter what thestarting time, provided again, ofcourse, thattherelevant features oftheenvironment arealsomodified appropriately intime. That symmetry means, 52-152-1 Symmetry operations 52-2 Symmetry inspace andtime 52-3 Symmetry andconservation laws 52-4 Mirror reflections 52-5 Polar andaxial vectors 52-6 Which hand isright? 52-7 Parity isnotconserved! 52-8 Antimatter 52-9 Broken symmetries Table 52-1 Symmetry Operations Translation inspace Translation intime Rotation through afixed angle Uniform velocity inastraight line(Lorentz transformation) Reversal oftime Reflection ofspace Interchange ofidentical atoms oridentical particles Quantum-mechanical phase Matter-antimatter (charge conjugation) ofcourse, thatifonebought General Motors stock three months ago,thesame thing would happen toitifhebought itnow! Wehave towatch outforgeographical differences too, forthere are,of course, variations inthecharacteristics oftheearth’s surface. So,forexample, ifwemeasure themagnetic field inacertain region andmove theapparatus to some other region, itmaynotwork inprecisely thesame waybecause themagnetic field isdifferent, butwesaythatisbecause themagnetic fieldisassociated withthe earth. Wecanimagine thatifwemove thewhole earth andtheequipment, itwould make nodifference intheoperation oftheapparatus. Another thing thatwediscussed inconsiderable detail wasrotation inspace: ifweturn anapparatus atanangle itworks justaswell, provided weturnevery- thing elsethatisrelevant along with it.Infact, wediscussed theproblem ofsym- metry under rotation inspace insome detail inChapter ll,andweinvented a mathematical system called vector analysis tohandle itasneatly aspossible. Onamore advanced level wehadanother symmetry—the symmetry under uniform velocity inastraight line. That istosay—a rather remarkable effect—that ifwehave apiece ofapparatus working acertain wayandthen takethesame ap- paratus andputitinacar,andmove thewhole car,plusalltherelevant surround- ings, atauniform velocity inastraight line,then sofarasthephenomena inside thecarareconcerned there isnodifference: allthelaws ofphysics appear thesame. Weeven know how toexpress thismore technically, andthatisthatthemathe- matical equations ofthephysical laws must beunchanged under aLorentz trans- formation. Asamatter offact,itwasastudy oftherelativity problem thatconcen- trated physicists’ attention most sharply onsymmetry inphysical laws. Now theabove-mentioned symmetries have allbeen ofageometrical nature, time andspace being more orlessthesame, butthere areother symmetries ofa different kind. Forexample, there isasymmetry which describes thefactthatwe canreplace oneatom byanother ofthesame kind; toputitdifferently, there are atoms ofthesame kind. Itispossible tofindgroups ofatoms suchthatifwechange apair around, itmakes nodifference-the atoms areidentical. Whatever one atom ofoxygen ofacertain typewilldo,another atom ofoxygen ofthattypewill do.Onemaysay,“That isridiculous, thatisthedefinition ofequal types!” That maybemerely thedefinition, butthen westilldonotknow whether there areany “atoms ofthesame type”; thefactisthatthere aremany, many atoms ofthesame type. Thus itdoes mean something tosaythatitmakes nodifference ifwereplace oneatom byanother ofthesame type. Theso-called elementary particles ofwhich theatoms aremade arealsoidentical particles intheabove sense—all electrons arethesame; allprotons arethesame; allpositive pions arethesame; and soon. After such along listofthings thatcanbedone without changing thephe- nomena, onemight think wecould dopractically anything; soletusgivesome examples tothecontrary, justtoseethedifference. Suppose thatweask: “Are thephysical laws symmetrical under achange ofscale?” Suppose webuild a certain piece ofapparatus, andthen build another apparatus fivetimes bigger in every part, willitwork exactly thesame way? Theanswer is,inthiscase, no! Thewavelength oflight emitted, forexample, bytheatoms inside oneboxof sodium atoms andthewavelength oflight emitted byagasofsodium atoms five times involume isnotfivetimes longer, butis.infactexactly thesame astheother. Sotheratio ofthewavelength tothesizeoftheemitter willchange. Another example: weseeinthenewspaper, every once inawhile pictures ofa great cathedral made with little matchsticks—a tremendous work ofartbysome retired fellow who keeps gluing matchsticks together. Itismuch more elaborate andwonderful than anyrealcathedral. Ifweimagine thatthiswooden cathedral were actually built onthescale ofarealcathedral, weseewhere thetrouble is; itwould notlast—the whole thing would collapse because ofthefactthatscaled-up matchsticks arejustnotstrong enough. “Yes,” onemight say,“but wealsoknow thatwhen there isaninfluence from theoutside, italsomust bechanged inpro- portionl” Wearetalking about theability oftheobject towithstand gravitation. Sowhat weshould doisfirsttotakethemodel cathedral ofrealmatchsticks and 52-2 therealearth, andthen weknow itisstable. Then weshould takethelarger cathe- dralandtake abigger earth. Butthen itiseven worse, because thegravitation is increased stillmore! Today, ofcourse, weunderstand thefactthatphenomena depend onthescale onthegrounds thatmatter isatomic innature, andcertainly ifwebuilt anappara- tusthatwassosmall there were onlyfiveatoms init,itwould clearly besomething wecould notscale upanddown arbitrarily. Thescale ofanindividual atom isnot atallarbitrary—it isquite definite. Thefactthatthelawsofphysics arenotunchanged under achange ofscale wasdiscovered byGalileo. Herealized thatthestrengths ofmaterials were notin exactly theright proportion totheir sizes, andheillustrated thisproperty thatwe were justdiscussing, about thecathedral ofmatchsticks, bydrawing twobones, thebone ofonedog, intheright proportion forholding uphisweight, andthe imaginary bone ofa“super dog” that would be,say,tenorahundred times bigger—that bone wasabig,solid thing with quite different proportions. Wedo notknow whether heever carried theargument quite totheconclusion thatthe laws ofnature must have adefinite scale, buthewassoimpressed with thisdis- covery thatheconsidered ittobeasimportant asthediscovery ofthelaws of motion, because hepublished them both inthesame volume, called “On Two New Sciences.” Another example inwhich thelaws arenotsymmetrical, thatweknow quite well, isthis: asystem inrotation atauniform angular velocity does notgivethe same apparent laws asonethatisnotrotating. Ifwemake anexperiment and thenputeverything inaspace shipandhave thespace shipspinning inempty space, allalone ataconstant angular velocity, theapparatus willnotwork thesame way because, asweknow, things inside theequipment willbethrown totheoutside, andsoon,bythecentrifugal orcoriolis forces, etc.Infact,wecantellthattheearth isrotating byusing aso-called Foucault pendulum, without looking outside. Next wemention avery interesting symmetry which isobviously false, i.e., reversibility intime. Thephysical laws apparently cannot bereversible intime, because, asweknow, allobvious phenomena areirreversible onalarge scale: “The moving finger writes, andhaving writ, moves on.” Sofaraswecantell,this irreversibility isduetotheverylarge number ofparticles involved, andifwecould seetheindividual molecules, wewould notbeabletodiscern whether themachinery wasworking forward orbackwards. Tomake itmore precise: webuild asmall apparatus inwhich weknow what alltheatoms aredoing, inwhich wecanwatch them jiggling. Now webuild another apparatus likeit,butwhich starts itsmotion inthefinal condition oftheother one, with allthevelocities precisely reversed. Itwillthengothrough thesame motions, butexactly inreverse. Putting itanother way: ifwetake amotion picture, with sufficient detail, ofalltheinner works ofa piece ofmaterial andshine itonascreen andrunitbackwards, nophysicist will beabletosay,“That isagainst thelawsofphysics, thatisdoing something wrong! ” Ifwedonotseeallthedetails, ofcourse, thesituation willbeperfectly clear. If weseetheeggsplattering onthesidewalk andtheshell cracking open, andsoon, then wewillsurely say,“That isirreversible, because ifwerunthemoving picture backwards theeggwillallcollect together andtheshell willgoback together, and thatisobviously ridiculous!” Butifwelook attheindividual atoms themselves, thelaws look completely reversible. This is,ofcourse, amuch harder discovery tohave made, butapparently itistruethatthefundamental physical laws, ona microscopic andfundamental level, arecompletely reversible intime! 52-3 Symmetry andconservation laws Thesymmetries ofthephysical laws areveryinteresting atthislevel, butthey turn out,intheend, tobeeven more interesting andexciting when wecome to quantum mechanics. Forareason which wecannot make clear atthelevel ofthe present discussion—a factthat most physicists stillfindsomewhat staggering, a most profound andbeautiful thing, isthat, inquantum mechanics, foreach of therules ofsymmetry there isacorresponding conservation law; there isadefinite 52-3 connection between thelaws ofconservation andthesymmetries ofphysical laws. Wecanonly state thisatpresent, without anyattempt atexplanation. Thefact, forexample, thatthelaws aresymmetrical fortranslation inspace when weaddtheprinciples ofquantum mechanics, turns outtomean thatmo- mentum isconserved. That thelaws aresymmetrical under translation intime means, inquantum mechanics, thatenergy isconserved. Invariance under rotation through afixed angle inspace corresponds tothe conservation ofangular momentum. These connections arevery interesting and beautiful things, among themost beautiful andprofound things inphysics. Incidentally, there areanumber ofsymmetries which appear inquantum mechanics which have noclassical analog, which have nomethod ofdescription in classical physics. Oneofthese isasfollows: Ifitistheamplitude forsome process orother, weknow thattheabsolute square ofitistheprobability thattheprocess willoccur. Now ifsomeone elsewere tomake hiscalculations, notwiththistl/,but with aWwhich differs merely byachange inphase (letAbesome constant, and multiply e“times theold¢),theabsolute square ofit’,which istheprobability of theevent, isthen equal totheabsolute square ofit: v’=‘I/6“; ll!/I2=l\//l2- (52-1) Therefore thephysical laws areunchanged ifthephase ofthewave function is shifted byanarbitrary constant. That isanother symmetry. Physical laws must beofsuch anature thatashift inthequantum-mechanical phase makes nodiffer- ence. Aswehave justmentioned, inquantum mechanics there isaconservation lawforevery symmetry. Theconservation lawwhich isconnected with thequan- tum-mechanical phase seems tobetheconservation ofelectrical charge. This is altogether averyinteresting business! 52-4 Mirror reflections Now thenext question, which isgoing toconcern usformost oftherestof thischapter, isthequestion ofsymmetry under reflection inspace. Theproblem isthis: Arethephysical laws symmetrical under reflection? Wemay putitthis way: Suppose webuild apiece ofequipment, letussayaclock, with lotsofwheels andhands andnumbers; itticks, itworks, andithasthings wound upinside. Welook attheclock inthemirror. How itlooks inthemirror isnotthequestion. Butletusactually build another clock which isexactly thesame asthefirstclock looks inthemirror—every time there isascrew with aright-hand thread inone, weuseascrew with aleft-hand thread inthecorresponding place oftheother; where oneismarked “2”ontheface, wemark a“S”onthefaceoftheother; each coiled spring istwisted onewayinoneclock andtheother wayinthemirror- image clock; when weareallfinished, wehave twoclocks, both physical, which bear toeach other therelation ofanobject anditsmirror image, although they areboth actual, material objects, weemphasize. Now thequestion is:Ifthetwo clocks arestarted inthesame condition, thesprings wound tocorresponding tight- nesses, willthetwoclocks tickandgoaround, forever after, asexact mirror images? (This isaphysical question, notaphilosophical question.) Ourintuition about the laws ofphysics would suggest thatthey would. Wewould suspect that, atleast inthecaseofthese clocks, reflection inspace isoneofthesymmetries ofphysical laws, thatifwechange everything from “right” to“left” andleave itotherwise thesame, wecannot tellthedifference. Letus, then, suppose foramoment thatthisistrue. Ifitistrue. thenitwould beimpossible todistinguish “right” and“left” byanyphysical phenomenon, justasitis,for example, impossible todefine aparticular absolute velocity byaphysical phe- nomenon. Soitshould beimpossible, byanyphysical phenomenon, todefine absolutely what wemean by“right” asopposed to“left,” because thephysical laws should besymmetrical. Ofcourse, theworld doesnothavetobesymmetrical. Forexample, using what wemay call“geography,” surely “right” canbedefined. Forinstance, westand 52-4 inNew Orleans andlook atChicago, andFlorida istoourright (when ourfeet areonthegroundl). Sowecandefine “right” and“left” bygeography. Ofcourse, theactual situation inanysystem does nothave tohave thesymmetry thatweare talking about; itisaquestion ofwhether thelawsaresymmetrica1—in other words, whether itisagainst thephysical laws tohave asphere liketheearth with “left- handed dirt” onitandaperson likeourselves standing looking atacitylike Chicago from aplace likeNew Orleans, butwitheverything theother wayaround, soFlorida isontheother side. Itclearly seems notimpossible, notagainst the physical laws, tohave everything changed leftforright. Another point isthatourdefinition of“right” should notdepend onhistory. Aneasywaytodistinguish right from leftistogotoamachine shop andpickupa screw atrandom. Theodds areithasaright-hand thread—not necessarily, but itismuch more likely tohave aright-hand thread than aleft-hand one. This isa question ofhistory orconvention, orthewaythings happen tobe,andisagain notaquestion offundamental laws. Aswecanwellappreciate, everyone could have started outmaking left-handed screws! Sowemust trytofindsome phenomenon inwhich “right hand” isinvolved fundamentally. The next possibility wediscuss isthefactthat polarized light rotates itsplane ofpolarization asitgoes through, say,sugar water. Aswesaw inChapter 33,itrotates, letussay,totheright inacertain sugar solution. That isawayofdefining “right-hand,” because wemaydissolve some sugar inthewater andthenthepolarization goes totheright. Butsugar hascome from living things, andifwetrytomake thesugar artificially, then wediscover thatitdoesnotrotate theplane ofpolarization! Butifwethen take that same sugar which ismade artificially andwhich does notrotate theplane ofpolarization, andputbacteria init(they eatsome ofthesugar) andthen filter outthebacteria, wefindthatwe stillhave sugar left(almost halfasmuch aswehadbefore), andthistime itdoes rotate theplane ofpolarization, buttheother way! Itseems very confusing, but iseasily explained. Fig.52—l. (alL-alanine (left), andlb) D-alanine (right). Take another example: Oneofthesubstances which iscommon toallliving creatures andthatisfundamental tolifeisprotein. Proteins consist ofchains of amino acids. Figure 52-1 shows amodel ofanamino acidthatcomes outofa protein. This amino acidiscalled alanine, andthemolecular arrangement would look likethatinFig.52-1(a) ifitcame outofaprotein ofarealliving thing. On theother hand, ifwetrytomake alanine from carbon dioxide, ethane, andam- monia (and wecanmake it,itisnotacomplicated molecule), wediscover thatwe aremaking equal amounts ofthismolecule andtheoneshown inFig.52-l(b)! Thefirstmolecule, theonethatcomes from theliving thing, iscalled L-alanine. Theother one, which isthesame chemically, inthat ithasthesame kinds of atoms andthesame connections oftheatoms, isa“right-hand” molecule, com- pared with the“left-hand” L-alanine, anditiscalled D-alanine. Theinteresting thing isthatwhen wemake alanine athome inalaboratory from simple gases, wegetanequal mixture ofboth kinds. However, theonly thing thatlifeusesis L-alanine. (This isnotexactly true. Here andthere inliving creatures there isa special useforD-alanine, butitisveryrare. Allproteins useL-alanine exclusively.) Now ifwemake both kinds, andwefeedthemixture tosome animal which likes to“eat,” oruseup,alanine, itcannot useD-alanine, soitonlyusestheL-alanine; thatiswhat happened tooursugar—after thebacteria eatthesugar thatworks 52-5 wellforthem, only the“wrong” kind isleft! (Left-handed sugar tastes sweet, but notthesame asright-handed sugar.) Soitlooks asthough thephenomena oflifepermit adistinction between “right” and“left,” orchemistry permits adistinction, because thetwomolecules arechemically different. Butno,itdoes not! Sofarasphysical measurements can bemade, such asofenergy, therates ofchemical reactions, andsoon,thetwo kinds work exactly thesame wayifwemake everything elseinamirror image too. Onemolecule willrotate light totheright, andtheother willrotate ittotheleft inprecisely thesame amount, through thesame amount offluid. Thus, sofaras physics isconcerned, these twoamino acids areequally satisfactory. Sofaraswe understand things today, thefundamentals oftheSchrodinger equation have it thatthetwomolecules should behave inexactly corresponding ways, sothatone istotheright astheother istotheleft. Nevertheless, inlifeitisalloneway! Itispresumed thatthereason forthisisthefollowing. Letussuppose, for example, thatlifeissomehow atonemoment inacertain condition inwhich all theproteins insome creatures have left-handed amino acids, andalltheenzymes arelopsided—every substance intheliving creature islopsided—it isnotsymmet- rical. Sowhen thedigestive enzymes trytochange thechemicals inthefood from onekind toanother, onekind ofchemical “fits” intotheenzyme, buttheother kind does not(like Cinderella andtheslipper, except thatitisa“left foot” that wearetesting). Sofarasweknow, inprinciple, wecould build afrog, forexample, inwhich every molecule isreversed, everything islikethe“left-hand” mirror image ofarealfrog; wehave aleft-hand frog. This left-hand frogwould goonallright forawhile, buthewould findnothing toeat,because ifheswallows afly,his enzymes arenotbuilt todigest it.Theflyhasthewrong “kind” ofamino acids (unless wegivehimaleft-hand fly). Soasfarasweknow, thechemical andlife processes would continue inthesame manner ifeverything were reversed. Iflifeisentirely aphysical andchemical phenomenon, thenwecanunderstand thattheproteins areallmade inthesame corkscrew only from theideathatatthe very beginning some living molecules, byaccident, gotstarted andafewwon. Somewhere, once, oneorganic molecule waslopsided inacertain way, andfrom thisparticular thing the“right” happened toevolve inourparticular geography; aparticular historical accident wasone-sided, andeversince then thelopsidedness haspropagated itself. Once having arrived atthestate thatitisinnow, ofcourse, itwillalways continue—all theenzymes digest theright things, manufacture the right things: when thecarbon dioxide andthewater vapor, andsoon,gointhe plant leaves, theenzymes thatmake thesugars make them lopsided because the enzymes arelopsided. Ifanynewkind ofvirus orliving thing were tooriginate at alater time, itwould survive onlyifitcould “eat” thekind ofliving matter already present. Thus it,too,must beofthesame kind. There isnoconservation ofthenumber ofright-handed molecules. Once started, wecould keep increasing thenumber ofright-handed molecules. So thepresumption is,then, thatthephenomena inthecase oflifedonotshow alack ofsymmetry inphysical laws, butdoshow, onthecontrary, theuniversal nature andthecommonness ofultimate origin ofallcreatures onearth, inthesense de- scribed above. 52-5 Polar andaxial vectors Now wegofurther. Weobserve thatinphysics there arealotofother places where wehave “right” and“left” hand rules. Asamatter offact,when welearned about vector analysis welearned about theright-hand rules wehave touseinorder togettheangular momentum, torque, magnetic field, andsoon,tocome outright. Theforce onacharge moving inamagnetic field, forexample, isF=qvXB. Inagiven situation, inwhich weknow F,v,andB,isn’t thatequation enough to define right-handedness? Asamatter offact, ifwegoback andlook atwhere the vectors came from, weknow thatthe“right-hand rule” wasmerely aconvention; itwasatrick. Theoriginal quantities, liketheangular momenta andtheangular velocities, andthings ofthiskind, were notreally vectors atall! They areall 52-6 somehow associated with acertain plane, anditisjustbecause there arethree dimensions inspace thatwecanassociate thequantity withadirection perpendicu- lartothat plane. Ofthetwopossible directions, wechose the“right-hand” direction. Soifthelaws ofphysics aresymmetrical, weshould findthatifsome demon were tosneak intoallthephysics laboratories andreplace theword “right” for “left” inevery book inwhich “right-hand rules” aregiven, andinstead wewere to useall“left-hand rules,” uniformly, then itshould make nodifference whatever inthephysical laws. Letusgiveanillustration. There aretwokinds ofvectors. There are“honest” vectors, forexample astepArinspace. Ifinourapparatus there isapiece hereand something elsethere, then inamirror apparatus there willbetheimage piece and theimage something else, andifwedraw avector from the“piece” tothe“some- thing else,” onevector isthemirror image oftheother (Fig. 52-2). Thevector arrow changes itshead, justasthewhole space turns inside out;such avector wecalla polar vector. Buttheother kind ofvector, which hastodowith rotations, isofadifferent nature. Forexample, suppose thatinthree dimensions something isrotating as shown inFig.52-3. Then ifwelook atitinamirror, itwillberotating asindicated, namely, asthemirror image oftheoriginal rotation. Now wehave agreed torepre- sentthemirror rotation bythesame rule, itisa“vector” which, onreflection, does notchange about asthepolar vector does, butisreversed relative tothepolar vectors andtothegeometry ofthespace; such avector iscalled anaxial vector. Now ifthelawofreflection symmetry isright inphysics, then itmust betrue thattheequations must besodesigned thatifwechange thesignofeach axial vector andeach cross-product ofvectors, which would bewhat corresponds to reflection, nothing willhappen. Forinstance, when wewrite aformula which says thattheangular momentum isL=rXp,thatequation isallright, because if wechange toaleft-hand coordinate system, wechange thesignofL,butpandr donotchange; thecross-product signischanged, since wemust change from a right-hand ruletoaleft-hand rule. Asanother example, weknow thattheforce onacharge moving inamagnetic field isF=qvXB,butifwechange from a right- toaleft-handed system, since Fandvareknown tobepolar vectors thesign change required bythecross-product must becancelled byasignchange inB, which means thatBmust beanaxial vector. Inother words, ifwemake such a reflection, Bmust goto—B. Soifwechange ourcoordinates from right toleft, wemust alsochange thepoles ofmagnets from north tosouth. Letusseehowthatworks inanexample. Suppose thatwehave twomagnets, asinFig.52-4. Oneisamagnet with thecoils going around acertain way, and with current inagiven direction. Theother magnet looks likethereflection of thefirstmagnet inamirror—the coilwillwind theother way, everything that happens inside thecoilisexactly reversed, andthecurrent goes asshown. Now, from thelaws fortheproduction ofmagnetic fields, which wedonotknow yet officially, butwhich wemost likely learned inhigh school, itturns outthatthe magnetic field isasshown inthefigure. Inonecasethepole isasouth magnetic pole, while intheother magnet thecurrent isgoing theother wayandthemagnetic fieldisreversed—it isanorth magnetic pole. Soweseethatwhen wegofrom right toleftwemust indeed change from north tosouth! Never mind changing north tosouth; these tooaremere conventions. Let ustalkabout phenomena. Suppose, now, thatwehave anelectron moving through onefield, going intothepage. Then, ifweusetheformula fortheforce, vXB (remember thecharge isminus), wefindthattheelectron willdeviate intheindi- cated direction according tothephysical law. Sothephenomenon isthatwehave acoilwith acurrent going inaspecified sense andanelectron curves inacertain way—that isthephysics—never mind how welabel everything. Now letusdothesame experiment withamirror: wesend anelectron through inacorresponding direction andnowtheforce isreversed, ifwecalculate itfrom thesame rule, andthatisvery good because thecorresponding motions arethen mirror images! 52-7O O Fig.52-2. Astep inspace and its mirror image. o’ UP Fig. 52-3. Arotating wheel and its mirror image. Note that theangular velocity "vector" isnot reversed in direction. -ll--ti-Fig. 52-4. Amagnet and itsmirror image. 52-6 Which hand isright? Sothefactofthematter isthatinstudying anyphenomenon there arealways tworight-hand rules, oraneven number ofthem, andthenetresult isthatthe phenomena always look symmetrical. Inshort, therefore, wecannot tellright from leftifwealsoarenotabletotellnorth from south. However, itmayseem thatwe cantellthenorth pole ofamagnet. Thenorth pole ofacompass needle, forex- ample, isonethatpoints tothenorth. Butofcourse thatisagain alocal property thathastodowith geography oftheearth; thatisjustliketalking about inwhich direction isChicago, soitdoes notcount. Ifwehave seen compass needles, we may have noticed thatthenorth-seeking pole isasortofbluish color. Butthatis justduetotheman who painted themagnet. These arealllocal, conventional criteria. However, ifamagnet were tohave theproperty thatifwelooked atitclosely enough wewould seesmall hairs growing onitsnorth pole butnotonitssouth pole, ifthatwere thegeneral rule, orifthere were anyunique waytodistinguish thenorth from thesouth pole ofamagnet, then wecould tellwhich ofthetwo cases weactually had, andthatwould betheendofthelawofreflection symmetry. Toillustrate thewhole problem stillmore clearly, imagine thatwewere talking toaMartian, orsomeone veryfaraway, bytelephone. Wearenotallowed tosend himanyactual samples toinspect; forinstance, ifwecould send light, wecould send himright-hand circularly polarized light andsay,“That isright-hand light— justwatch thewayitisgoing.” Butwecannot givehimanything, wecanonlytalk tohim. Heisfaraway, orinsome strange location, andhecannot seeanything wecansee. Forinstance, wecannot say,“Look atUrsa major; now seehow those stars arearranged. What wemean by‘right’ is...”Weareonly allowed totelephone him. Now wewant totellhimallabout us.Ofcourse, firstwestart defining num- bers, andsay,“Tick, tick, two,tick, tick, tick, three ...,”sothatgradually hecan understand acouple ofwords, andsoon.After awhile wemaybecome veryfamil- iarwith thisfellow, andhesays, “What doyouguys look like?” Westart tode- scribe ourselves, andsay,“Well, wearesixfeettall.” Hesays, “Wait aminute, what issixfeet?” Isitpossible totellhimwhat sixfeetis?Certainly! Wesay, “You know about thediameter ofhydrogen atoms—we arel7,000,000,000 hydrogen atoms high!” That ispossible because physical laws arenotvariant under change ofscale, andtherefore wecandefine anabsolute length. And sowe define thesizeofthebody, andtellhimwhat thegeneral shape is—it hasprongs with fivebumps sticking outontheends, andsoon,andhefollows usalong, and wefinish describing howwelook ontheoutside, presumably without encountering anyparticular difliculties. Heiseven making amodel ofusaswegoalong. Hesays, “My, youarecertainly veryhandsome fellows; nowwhat isontheinside?” Sowe start todescribe thevarious organs ontheinside, andwecome totheheart, and wecarefully describe theshape ofit,andsay,“Now puttheheart ontheleftside.” Hesays, “Duhhh—the leftside?” Now ourproblem istodescribe tohimwhich sidetheheart goes onwithout hisever seeing anything thatwesee,andwithout ourever sending anysample tohimofwhat wemean by“right”—-no standard right-handed object. Canwedoit? 52-7 Parity isnotconserved! Itturns outthatthelaws ofgravitation, thelaws ofelectricity andmagnetism, nuclear forces, allsatisfy theprinciple ofreflection symmetry, sothese laws, or anything derived from them, cannot beused. Butassociated with themany par- ticles thatarefound innature there isaphenomenon called betadecay, orweak decay. Oneoftheexamples ofweak decay, inconnection withaparticle discovered inabout 1954, posed astrange puzzle. There wasacertain charged particle which disintegrated into three 1r-mesons, asshown schematically inFig. 52-5. This particle wascalled, forawhile, a'r-meson. Now inFig.52-5 wealsoseeanother particle which disintegrates intotwomesons; onemust beneutral, from thecon- 52-8 servation ofcharge. Thisparticle wascalled a0-meson. Soontheonehand we have aparticle called a'r,which disintegrates intothree 1r-mesons, anda0,which disintegrates intotwo1r-mesons. Now itwassoon discovered thatthe1andthe 6arealmost equal inmass; infact, within theexperimental error, they areequal. Next, thelength oftime ittook forthem todisintegrate into three 1r’sandtwo 1r’swasfound tobealmost exactly thesame; they livethesame length oftime. Next, whenever they were made, they were made inthesame proportions, say, 14percent r’sto86percent 0’s. Anyone inhisright mind realizes immediately that they must bethesame particle, thatwemerely produce anobject which hastwodifferent ways ofdis- integrating—not twodifferent particles. This object thatcandisintegrate intwo diflerent ways has, therefore, thesame lifetime andthesame production ratio (because thisissimply theratio oftheodds with which itdisintegrates intothese twokinds). However, itwaspossible toprove (and wecannot hereexplain atallhow), from theprinciple ofreflection symmetry inquantum mechanics, thatitwasimpossible tohave these both come from thesame particle-—the same particle could not disintegrate inboth ofthese ways. Theconservation lawcorresponding tothe principle ofreflection symmetry issomething which hasnoclassical analog, and sothiskind ofquantum-mechanical conservation wascalled theconservation of parity. So,itwasaresult oftheconservation ofparity or,more precisely, from the symmetry ofthequantum-mechanical equations oftheweak decays under reflec- tion, thatthesame particle could notgointoboth, soitmust besome kind of coincidence ofmasses, lifetimes, andsoon.Butthemore itwasstudied, themore remarkable thecoincidence, andthesuspicion gradually grew that possibly the deep lawofthereflection symmetry ofnature maybefalse. Asaresult ofthisapparent failure, thephysicists LeeandYang suggested that other experiments bedone inrelated decays totrytotestwhether thelawwas correct inother cases. Thefirstsuch experiment wascarried outbyMiss Wufrom Columbia, andwasdone asfollows. Using avery strong magnet atavery low temperature, itturns outthat acertain isotope ofcobalt, which disintegrates by emitting anelectron, ismagnetic, andifthetemperature islowenough thatthe thermal oscillations donotjiggle theatomic magnets about toomuch, they line upinthemagnetic field. Sothecobalt atoms willalllineupinthisstrong field. They then disintegrate, emitting anelectron, anditwasdiscovered thatwhen the atoms were lined upinafieldwhose Bvector points upward, most oftheelectrons were emitted inadownward direction. Ifoneisnotreally “hep” totheworld, such aremark does notsound like anything ofsignificance, butifoneappreciates theproblems andinteresting things intheworld, then heseesthatitisamost dramatic discovery: When weputcobalt atoms inanextremely strong magnetic field, more disintegration electrons godown than up.Therefore ifwewere toputitinacorresponding experiment ina“mirror,” inwhich thecobalt atoms would belined upintheopposite direction, theywould spittheir electrons up,notdown; theaction isunsymmetrical. Themagnet hasgrown hairs! Thesouth pole ofamagnet isofsuch akind thattheelectrons inaB-dis- integration tendtogoaway from it;thatdistinguishes, inaphysical way, thenorth pole from thesouth pole. After this, alotofother experiments were done: thedisintegration ofthe1r into].Land1/;uintoanelectron andtwoneutrinos; nowadays, theAintoproton and1r;disintegration ofE’s; andmany other disintegrations. Infact, inalmost allcases where itcould beexpected, allhave been found nottoobey reflection symmetry! Fundamentally, thelawofreflection symmetry, atthislevel inphysics, isincorrect. Inshort, wecantellaMartian where toputtheheart: wesay,“Listen, build yourself amagnet, andputthecoils in,andputthecurrent on,andthentakesome cobalt andlower thetemperature. Arrange theexperiment sotheelectrons gofrom thefoottothehead, then thedirection inwhich thecurrent goes through thecoils isthedirection thatgoes inonwhat wecalltheright andcomes outontheleft.” Soitispossible todefine right andleft,now, bydoing anexperiment ofthiskind. 52-9IO 1'0 " fi 7- \ \yo \ \ Fig.52-5. Aschematic diagram of thedisintegration ofa1+and a0+ particle. There arealotofother features thatwere predicted. Forexample, itturns out thatthespin, theangular momentum, ofthecobalt nucleus before disintegration is5units ofh,andafter disintegration itis4units. Theelectron carries spinangular momentum, andthere isalsoaneutrino involved. Itiseasy toseefrom thisthat theelectron must carry itsspinangular momentum aligned along itsdirection of motion, theneutrino likewise. Soitlooks asthough theelectron isspinning tothe left,andthatwasalsochecked. Infact, itwaschecked right here atCaltech by Boehm andWapstra, thattheelectrons spinmostly totheleft. (There were some other experiments thatgave theopposite answer, butthey were wrong!) Thenext problem, ofcourse, wastofindthelawofthefailure ofparity con- servation. What istherulethattellsushow strong thefailure isgoing tobe? Theruleisthis: itoccurs only inthese veryslow reactions,'called weak decays, and when itoccurs, theruleisthattheparticles which carry spin, liketheelectron, neutrino, andsoon,come outwith aspintending totheleft. That isalopsided rule; itconnects apolar vector velocity andanaxial vector angular momentum, andsaysthattheangular momentum ismore likely tobeopposite tothevelocity thanalong it. Now thatistherule, buttoday wedonotreally understand thewhys and wherefores ofit.Why isthistheright rule, what isthefundamental reason forit, andhowisitconnected toanything else? Atthemoment wehave been soshocked bythefactthatthisthing isunsymmetrical thatwehave notbeen abletorecover enough tounderstand what itmeans with regard toalltheother rules. However, thesubject isinteresting, modern, andstillunsolved, soitseems appropriate that wediscuss some ofthequestions associated with it. 52-8 Antimatter Thefirstthing todowhen oneofthesymmetries islostistoimmediately go back over thelistofknown orassumed symmetries andaskwhether anyofthe others arelost. Now wedidnotmention oneoperation onourlist,which must necessarily bequestioned, andthatistherelation between matter andantimatter. Dirac predicted thatinaddition toelectrons there must beanother particle, called thepositron (discovered atCaltech byAnderson), that isnecessarily related to theelectron. Alltheproperties ofthese twoparticles obey certain rules ofcorre- spondence: theenergies areequal; themasses areequal; thecharges arereversed; but,more important than anything, thetwoofthem, when they come together, canannihilate each other andliberate their entire mass intheform ofenergy, say 7-rays. Thepositron iscalled anantiparticle totheelectron, andthese arethe characteristics ofaparticle anditsantiparticle. Itwasclear from Dirac’s argument thatalltherestoftheparticles intheworld should alsohave corresponding anti- particles. Forinstance, fortheproton there should beanantiproton, which isnow symbolized bya17.Thepwould have anegative electrical charge andthesame mass asaproton, andsoon.Themost important feature, however, isthataproton andanantiproton coming together canannihilate each other. Thereason we emphasize thisisthatpeople donotunderstand itwhen wesaythere isaneutron andalsoanantineutron, because they say,“Aneutron isneutral, sohow canit have theopposite charge?” Theruleofthe“anti” isnotjustthatithastheopposite charge, ithasacertain setofproperties, thewhole lotofwhich areopposite. The antineutron isdistinguished from theneutron inthisway: ifwebring twoneutrons together, theyjuststayastwoneutrons, butifwebring aneutron andananti- neutron together, theyannihilate each other with agreat explosion ofenergy being liberated, with various -rr-mesons, 'Y-rays, andwhatnot. Now ifwehave antineutrons, antiprotons, andantielectrons, wecanmake antiatoms, inprinciple. They have notbeen made yet,butitispossible inprinciple. Forinstance, ahydrogen atom hasaproton inthecenter with anelectron going around outside. Now imagine thatsomewhere wecanmake anantiproton with a positron going around, would itgoaround? Well, firstofall,theantiproton is electrically negative andtheantielectron iselectrically positive, sotheyattract each other inacorresponding manner—the masses areallthesame; everything isthe 52-10 same. Itisoneoftheprinciples ofthesymmetry ofphysics, theequations seem to show, thatifaclock, say,were made ofmatter ononehand, andthen wemade the same clock ofantimatter, itwould runinthisway. (Ofcourse, ifweputtheclocks together, theywould annihilate each other, butthatisdifferent.) Animmediate question then arises. Wecanbuild, outofmatter, twoclocks, onewhich is“left-hand” andonewhich is“right-hand.” Forexample, wecould build aclock which isnotbuilt inasimple way, buthascobalt andmagnets and electron detectors which detect thepresence of5-decay electrons andcount them. Each time oneiscounted, thesecond hand moves over. Then themirror clock, receiving fewer electrons, willnotrunatthesame rate. Soevidently wecanmake twoclocks such thattheleft-hand clock does notagree with theright-hand one. Letusmake, outofmatter, aclock which wecallthestandard orright-hand clock. Now letusmake, alsooutofmatter, aclock which wecalltheleft-hand clock. Wehave justdiscovered that, ingeneral, these twowillnotrunthesame way; prior tothatfamous physical discovery, itwasthought thatthey would. Now it wasalsosupposed thatmatter andantimatter were equivalent. That is,ifwemade anantimatter clock, right-hand, thesame shape, then itwould runthesame asthe right-hand matter clock, andifwemade thesame clock totheleftitwould run thesame. Inother words, inthebeginning itwasbelieved thatallfour ofthese clocks were thesame; now ofcourse weknow thattheright-hand andleft-hand matter arenotthesame. Presumably, therefore, theright-handed antimatter and theleft-handed antimatter arenotthesame. Sotheobvious question is,which goes with which, ifeither? Inother words, does theright-handed matter behave thesame wayastheright-handed antimatter? Ordoes theright-handed matter behave thesame astheleft-handed antimatter? [3-decay experiments, using positron decay instead ofelectron decay, indicate that thisistheinterconnection: matter tothe“right” works thesame wayasantimatter tothe“left.” Therefore, atlong last, itisreally true that right andleftsymmetry isstill maintained! Ifwemade aleft-hand clock, butmade itoutoftheother kind of matter, antimatter instead ofmatter, itwould runinthesame way. Sowhat has happened isthatinstead ofhaving twoindependent rules inourlistofsymmetries, twoofthese rules gotogether tomake anewrule, which saysthatmatter tothe right issymmetrical with antimatter to_theleft. SoifourMartian ismade ofantimatter andwegivehiminstructions tomake this“right” handed model likeus,itwill,ofcourse, come outtheother wayaround. What would happen when, after much conversation back andforth, weeach have taught theother tomake space ships andwemeet halfway inempty space? We have instructed each other onourtraditions, andsoforth, andthetwoofuscome rushing outtoshake hands. Well, ifheputs outhislefthand, watch out! 52-9 Broken symmetries Thenext question is,what canwemake outoflaws which arenearly sym- metrical? Themarvelous thing about itallisthatforsuch awide range ofimpor- tant, strong phenomena—nuclear forces, electrical phenomena, andeven weak ones likegravitation—over atremendous range ofphysics, allthelaws forthese seem tobesymmetrical. Ontheother hand, thislittle extra piece says, “No, thelaws arenotsymmetrical!” How isitthatnature canbealmost symmetrical, butnot perfectly symmetrical? What shall wemake ofthis? First, dowehave anyother examples? Theanswer is,wedo,infact,have afewother examples. Forinstance, thenuclear part oftheforce between proton andproton, between neutron and neutron, andbetween neutron andproton, isallexactly thesame—there isa symmetry fornuclear forces, anew one, that wecaninterchange neutron and proton—but itevidently isnotageneral symmetry, fortheelectrical repulsion between twoprotons atadistance does notexist forneutrons. Soitisnotgenerally truethatwecanalways replace aproton with aneutron, butonly toagood ap- proximation. Why good ?Because thenuclear forces aremuch stronger than the 52-1 1 electrical forces. Sothisisan“almost” symmetry also. Sowedohave examples in other things. Wehave, inourminds, atendency toaccept symmetry assome kind ofper- fection. InfactitisliketheoldideaoftheGreeks thatcircles were perfect, andit wasrather horrible tobelieve thattheplanetary orbits were notcircles, butonly nearly circles. Thediflerence between being acircle andbeing nearly acircle is notasmall difference, itisafundamental change sofarasthemind isconcerned. There isasignofperfection andsymmetry inacircle thatisnotthere themoment thecircle isslightly off—that istheendofit—-it isnolonger symmetrical. Then thequestion iswhyitisonlynearly acircle—that isamuch more diflicult question. Theactual motion oftheplanets, ingeneral, should beellipses, butduring theages, because oftidal forces, andsoon,theyhave been made almost symmetrical. Now thequestion iswhether wehave asimilar problem here. Theproblem from the point ofview ofthecircles isiftheywere perfect circles there would benothing to explain, thatisclearly simple. Butsince they areonly nearly circles, there isalot toexplain, andtheresult turned outtobeabigdynamical problem, andnow our problem istoexplain whythey arenearly symmetrical bylooking attidal forces andsoon. Soourproblem istoexplain where symmetry comes from. Why isnature so nearly symmetrical? Noonehasanyideawhy. Theonly thing wemight suggest issomething likethis: There isagateinJapan, agateinNeiko, which issometimes called bytheJapanese themost beautiful gate inallJapan; itwasbuilt inatime when there wasgreat influence from Chinese art.This gateisveryelaborate, with lotsofgables andbeautiful carving andlotsofcolumns anddragon heads and princes carved intothepillars, andsoon.Butwhen onelooks closely heseesthat intheelaborate andcomplex design along oneofthepillars, oneofthesmall design elements iscarved upside down; otherwise thething iscompletely symmet- rical. Ifoneaskswhythisis,thestory isthatitwascarved upside down sothat thegods willnotbejealous oftheperfection ofman. Sothey purposely put anerror inthere, sothatthegods would notbejealous andgetangry with human beings. Wemight liketoturntheideaaround andthink thatthetrueexplanation of thenear symmetry ofnature isthis: thatGod made thelaws onlynearly symmet- ricalsothatweshould notbejealous ofHisperfection! 52-12 Index Aberration, 27-7, 34-10 Cavendish, H.,7-9 Dynamics, 7-2f,9-1ff Absolute zero, 1-5 Cavendish's experiment, 7-9 relativistic, 15-9 f Absorption, 31-8 ff Center ofmass, 18-1 f,19-1 ff Acceleration, 8-8ff Centrifugal force, 7-5, 12-ll Efficiency, ofideal engine, 44-7 f components of,9-3 Cerenkov, P.A.,51-2 Einstein, A.,2-6,7-11, 12-12, 15-1, 16-1, ofgravity, 9-4 Cerenkov radiation, 51-2 41-8, 42-8, 42-9 Activation energy, 42-7 Charge, conservation of,4-7 Elastic collision, 10-7 Adams, J.C.,7-5 onelectron, 12-7 Elastic energy, 4-2,4-6 Adiabatic compression, 39-5 Chemical energy, 4-2 Electrical energy, 4-2 Adiabatic expansion, 44-5 Chemical kinetics, 42-7 f Electric field, 2-4, 12-7 f Affective future, 17-4 Chemical reaction, 1-6ff Electromagnetic energy, 29-2 Airtrough, 10-5 Chromaticity, 35-6 f Electromagnetic field, 2-2,2-5, 10-9 Algebra, 22-1 ff Circular motion, 21-4 Electromagnetic radiation, 26-1, 28-1 ff Amplitude modulation, 48-3 Clausius, R.,44-2, 44-3 Electromagnetic waves, cosmic rays, 2-5 Amplitude, ofoscillation, 21-3 Clausius-Clapeyron equation, 45-6 ff gamma rays, 2-5 Analog computer, 25-8 Coefficient, offriction, 12-4 infrared, 2-5,23-8, 26-1 Anderson, C.D.,52-10 gravitational, 7-9 light, 2-5 Angle, ofincidence, 26-3 Collision, 16-6 ultraviolet, 2-5,26-1 ofreflection, 26-3 elastic, 10-7 x-rays, 2-5,26-1 Angstrom (unit), 1-3 Color vision, 35-1 ff Electron, 2-4, 37-1, 37-4 ff Angular frequency, 21-3, 29—2 physiochemistry of,35-9 f charge on,12-7 Angular momentum, 7-7, 18-5 f,20-1 Complex impedance, 23-7 radius of,classical, 32-4 conservation of,4-7, 18-6 ff,20-5 Complex numbers, 22-7 ff,23-1 ff Electron cloud, 6-11 ofrigid body, 20-8 Compound eye,36-6 ff Electron-ray tube, 12-9 Anomalous refraction, 33-9 f Compression, adiabatic, 39-5 Electron volt(unit), 34-4 Antimatter 52-10 f isothermal, 44-5 Ellipse, 7-1 Antiparticle, 2-8 Cones, 35-1 Energy, chemical, 4-2 Aristotle, 5-1 Conservation, ofangular momentum, conservation of,3-2,4-1ff Atom, 1-2 4-7, 18-6 ff,20-5 elastic, 4-2,4-6 metastable, 42-10 ofcharge, 4-7 electrical, 4-2 Atomic clock, 5-5 ofenergy, 3-2,4-1ff electromagnetic, 29—2 Atomic hypothesis, 1-2 oflinear momentum, 4-7, 10-1 lf gravitational, 4-2ff Atomic particles, 2-9f Contraction hypothesis, 15-3 heat, 4-2,4- 0-7, 10-8 Atomic processes, l-5f Copernicus, 7-1 kinetic, 1-7 —-5f,39-4 Attenuation, 31-8 Coriolis force, 19-8 f mass, 4-2 - Avogadro, A.,39-2 Cornea, 35-1 nuclear, 4- Avogadro‘s number, 41-10 Coulomb’s law,28-2 potential, 4-4, 13-1 ff,14-1 ff Axial vector, 52-6 f Cross section, forscattering, 32-7 radiant, 4-2 Crystal diffraction, 38-4 f relativistic, 16-1 ff Becquerel, A.H.,28-3 Energy levels, 38-7 f Birefringence, 33-3 ff Dedekind, R.,22-4 Energy theorem, 50-7 f Blackbody radiation, 41-5 f Degrees offreedom, 25-2, 39-12 Enthalpy, 45-5 Boehm, 52-10 Density, 1-4 Entropy, 44-10 ff,46-7 ff Bohr, N.,42-9 Derivative, 8-5ff Eotvos, 7-ll Bohr radius, 38-6 partial, 14-9 Equilibrium, 1-6 Boltzmann, 41-2 Dicke, R.H.,7-11 Euclid, 5-6 Boltzmann’s law,40-2 f Differential calculus, 8-4 Euclidean geometry, 12-3 Born, M.,37-1, 38-9 Diffraction, 30-1 ff Evaporation, 1-5f Boyle’s law,40-8 byscreen, 31-10 f ofaliquid, 40-3 f,42-1 ff Bremsstrahlung, 34-6 f Diffraction grating, 29-5, 30-3 ff Expansion, adiabatic, 44-5 Brewster’s angle, 33-6 Diffusion, 43-1 ff isothermal, 44-5 Briggs, H.,22-6 Dipole moment, 12-6 Exponential atmosphere, 40-1 f Brown, R.,41-1 Dipole radiator, 28-5 f,29-3 ff Eye, compound, 36-6 ff Brownian motion, 1-8,6-5,41-1 ff Dirac, P.,52-10 human, 35-1 f,36-3 ff Dirac equation, 20-6 Capacitance, 23-5 Dispersion, 31-6 ff Farad (unit), 25-7 Capacitor, 14-9, 23-5 Distance, 5-5ff Fermat, P.,26-3 Capillary action, 51-8 Distance measurement, color brightness, Fermi (unit), 5-10 Carnot, S.,4-2,44-2 ff 5-6 Fermi, E.,5-10 Carnot cycle, 44-5 f,45-2 triangulation, 5-6 Fields, 2-2,2-4,2-5,10-9, 12-7 ff,13-8 f, Carrier signal, 48-3 Doppler effect, 17-8, 23-9, 34-7 f,38-6 14-7 ff Catalyst, 42-8 Double stars, 7-6 superposition of,12-9f\).A.-1->9‘_I\)>-4-Q INDEX 1 Focal length, 27-1ff Focus, 26-5 Force, centrifugal, 7-5,12-11 components of,9-3 conservative, 14-3 ff Coriolis, 19-8 f electrical, 2-3ff gravitational, 2-3 molecular, 1-3,12-6 f moment of,18-5 nonconservative, 14-6f nuclear, 12-12 pseudo, 12-10 ff Fourier, J.,50-2 f Fourier analysis, 50-2 ff Fourier transform, 25-4 Four-vectors, 15-8 f,17-5 ff Fovea, 35-1 Frank, I.,51-2 Frequency, angular, 21-3, 29—2 ofoscillation, 2-5 Fresnel’s reflection formulas, 33-8 Friction, 10-5, 12-3 ff coefficient of,12-4 Galileo, 5-1,7-2, 9-1, 52-3 Galilean relativity, 10-3 Galilean transformation, 12-.11 Gauss (unit), 34-4 Gell-Mann, M.,2-9 Geometrical optics, 26-1, 27-1 f Gravitation, 2-3,7-1ff,12-2 Gravitational acceleration, 9-4 Gravitational coeflicient, 7-9 Gravitational energy, 4-2ff Gravitational field, 12-8 ff,13-8 f Gravity, 13-3 ff acceleration of,9-4 Green’s function, 25-4 Gyroscope, 20-5ff Harmonic motion, 21-4, 23-1 ff Harmonic oscillator, 10-1, 21-1 ff forced, 21-5f,23-3ff Harmonics, 50-1 ff Heat, 1-3,13-3 Heat energy, 4-2, 4-6, 10-7, 10-8 Heat engines, 44-1 ff Heisenberg, W.,6-10, 37-1, 37-9, 37-11, 37-12, 38-9 Helmholtz, H.,35-7 Henry (unit), 25-7 Hooke’s law, 12-6 Huygens, C.,15-2, 26-2 Hypocycloid, 34-3 Ideal gaslaw, 39-10 ff Impedance, 25-8f complex, 23-7 Incidence, angle of,26-3 Inclined plane, 4-4 Index, ofrefraction, 31-1 ff Inductance, 23-6 Inductor, 23-6 Inertia, 2-3,7-ll moment of,18-7, 19-5 ff principle of,9-1 Infrared radiation, 23-8, 26-1 Integral, 8-7f Interference, 28-6, 29-1 ff Interfering waves, 37-4 INDEX 2Interferometer, 15-5 Ion, 1-6 Ionic conductivity, 43-6f Ionization energy, 42-5 Isothermal atmosphere, 40-2 Isothermal compression, 44-5 Isothermal expansion, 44-5 Isotopes, 3-4ff Jeans, J.,40-9, 41-6 f Johnson noise, 41-2, 41-8 Joule (unit), 13-3 Joule heating, 24-2 Kepler, J.,7-1 Kepler’s laws, 7-1f,9-1, 18-6 Kerr cell,33-5 Kinetic energy, 1-7,4-2, 4-5f,39-4 rotational, 19-7 ff Kinetic theory, 42-1 ff ofgases, 39-1ff Kirchhoff’s laws, 25-9 Laplace, P.,47-7 Laser, 32-6, 42-10 Least time, principle of,26-3 ff,26-8 Leibnitz, G.W.,8-4 Lens formula, 27-6 Leverrier, U.,7-5 Light, momentum of,34-10 f polarized, 32-9 scattering of,32-5 ff speed of,15-1 Light waves, 48-1 Linear momentum, conservation of, 4-7, 10-1 ff Linear systems, 25-1 ff Logarithms, 22-4 Lorentz, H.A.,15-3 Lorentz contraction, 15-7Molecular motion, 41-1 Molecule, 1-3 Moment, dipole, 12-6 offorce, 18-5 ofinertia, 18-7, 19-5 ff Momentum, 9-1f,38-2 ff angular, 7-7, 18-5 ff,20-1, 20-5 oflight, 34-10 f linear, 4-7, 10-1 ff relativistic, 10-8 f,16-1 ff Monatomic gas,39-5 Motion, 5-1,8-1ff circular, 21-4 constrained, 14-3 harmonic, 21-4, 23-1 ff parabolic, 8-10 planetary, 7-1ff,9-6f,13-5 Music, 50-1 Nernst heat theorem, 44-11 Neutrons, 2-4 Newton, I.,8-4, 15-1, 37-1 Newton-meter (unit), 13-3 I Newton’s laws, 2-6, 7-3ff,7-ll - 10-1 ff,11-7 f,12-1, 39-2, 41- Nishijima, 2-9 Nodes, 49-2 Noise, 50-1 Nuclear cross section, 5-9 Nuclear energy, 4-2 Nuclear forces, 12-12 Nucleus, 2-4,2-8ff Numerical analysis, 9-6 Nutation, 20-7F10$o‘>—lF‘? Ohm (unit), 25-7 Ohm’s law,25-7, 43-7 Optic axis, 33-3 Optic nerve, 35-2 Optics, 26-1 ff Lorentz transformation, 15-3, 17-1, 34-8, geometrical, 26-1, 27-1 ff 52-2 Magnetic field, 12-9 f Magnetic induction, 12-10 Magnetism, 2-4 Magnification, 27-5 Maser, 42-10Oscillation, amplitude of,21-3 damped, 24-3 f frequency of,2-5 period of,21-3 periodic, 9-4 phase of,21-3 Oscillator, 5-2 Mass, 9-1, 15-1 harmonic, 10-1, 21-1, 21-5 f,23-3 ff center of,18-1 f,19-1 ff relativistic, 16-6 ff Mass energy, 4-2,4-7 Mass-energy equivalence, 15-10 fPappus, theorem of,19-4 Parabolic antenna, 30-6 f Parabolic motion, 8-10 Maxwell, J.C.,6-1,6-9,28-1, 40-8, 41-7, Parallel-axis theorem, 19-6 46-5 Parallel-plate capacitor, 14-9 Maxwell’s equations, 15-2, 25-3, 47-7 Paraxial rays, 27-2 Mayer, J.R.,3-2 Mean freepath, 43-3 f Mean square distance, 6-5,41-9 Mendeléev, 2-9 Metastable atom, 42-10 Meter (unit), 5-10 Mev (unit), 2-9Partial derivative, 14-9 Pascal’s triangle, 6-4 Pendulum, 49-6 f Pendulum clock, 5-2 Period, ofoscillation, 21-3 Periodic time, 5-1f Perpetual motion, 46-2 Michelson-Morley experiment, 15-3 ff Phase, ofoscillation, 21-3 Miller, W.C.,35-2 Minkowski, 17-8 Modes, 49-1 ffPhase shift, 21-3 Phase velocity, 48-6 Photon, 2-7,26-1, 37-8 Mossbauer, R.,23-9 Physiochemistry, ofcolor vision, 35-9 f Mole (unit), 39-10 Planck, M.,41-6, 42-8, 42-9|-I Molecular attraction, 1-3, 12-6 f Planck’s constant, 5-10, 6-10, 17-8, 37-11 Molecular diffusion, 43-7 ff Planetary motion, 7-1ff,9-6f,13-5 Poincaré, H.,15-3, 15-5, 16-1 Polarization, 33-1 ff Polarized light, 32-9 Potential energy, 4-4, 13-1 ff,14-1 ff Power, 13-2 Pressure, 1-3 Probability, 6-1ff Probability density, 6-8f Probability distribution, 6-7ff Proton, 2-4 Pseudo force, 12-10 ff Ptolemy, 26-2 Purkinje effect, 35-2 Pythagoras, 50-1 Quantum electrodynamics, 2-7,28-3 Quantum mechanics, 2-2, 2-6ff,6-10, 10-9, 37-1 ff,38-1 ff Radiant energy, 4-2 Radiation, infrared, — relativistic effects, — synchrotron, 34-3 - ultraviolet, 26-1 Radiation damping, 32-3 f Radiation resistance, 32-1 ff Radioactive clock, 5-3fl' Radius, ofelectron, 32-4 Ramsey, N.,5-5 Random walk, 6-5ff,41-8 ff Ratchet andpawl machine, 46-1 ff Rayleigh’s criterion, 30-6 Rayleigh’s law,41-6 Reciprocity principle, 30-7 Rectification, 50-9 Reflection, 26-2 f angle of,26-3 Refraction, 26-2f anomalous, 33-9f index of,31-1 ff Relativistic dynamics, 15-9 f Relativistic energy, 16-1 ff Relativistic mass, 16-1 ff Relativistic momentum, 10-8 f,16-1 ff Relativity, special theory of,15-1 ff Galilean, 10-3 theory of,7-11, 17-1 Resistance, 23-5 Resistor, 23-5 Resolving power, 27-7 f,30-5 f Resonance, 23-1 ff electrical, 23-5 ff innature, 23-7 ff Resonance interaction, 2-9 Retarded time, 28-2 Retina, 35-1 Rigid body, 18-1 angular momentum of,20-8 rotation of,18-2ff Ritzcombination principle, 38-8to F93-iwU34:-"'-°°ma?’iRods, 35-1, 36-6 Roemer, 0.,7-5 Root-mean-square distance, 6-6 Rotation, ofaxes, 11-3 f plane, 18-1 ofarigid body, 18-2 ff inspace, 20-1 ff intwodimensions, 18-1 ff Rushton, 35-9 Rydberg (unit), 38-6 Scalar, 11-5 Scattering, oflight, 32-5 ff Schrédinger, E.,35-6, 37-1, 38-9 Scientific method, 2-1f Screw jack, 4-5 Second (unit), 5-5 Seismograph, 51-5 Shannon, C.,44-2 Shear wave, 51-4 Side bands, 48-4 f Simultaneity, 15-7 f Sinusoidal waves, 29-2 f Smoluchowski, 41-8 Smooth muscle, 14-2 Snell, W.,26-3 Snell’s law,26-3, 31-2 Sound, 2-3,47-1 ff,50-1 speed of,47-7f Space, 8-2 Space-time, 2-6, 17-1 ff Special theory ofrelativity, 15-1 ff Specific heat, 40-7 f,45-2 Speed, 8-2ff,9-2 oflight, 15-1 ofsound, 47-7 f Spontaneous emission, 42-9 Standard deviation, 6-9 Statistical fluctuations, 6-3ff Statistical mechanics, 3-1,40-1ff Stevinus, S.,4-5 “Strangeness number, 2-9 Striated muscle, 14-2 Superposition, offields, 12-9 principle of,25-2 ff Symmetry, 1-4,11-1 ff ofphysical laws, 16-3, 52-1 ff Synchrotron, 2-5, 15-9, 34-3 ff,34-6 Tamm, I.,51-2 Temperature, 39-6 ff Thermal conductivity, ofagas,43-9 f Thermal equilibrium, 41-3 ff Thermal ionization, 42-5ff Thermodynamics, 39-2, 45-1 ff lawsof,44-1 ff Thompson scattering cross section, 32-8 Three-body problem, 10-1 Tides, 7-4f Time, 2-3,5-1ff,8-1,8-2 retarded, 28-2standard of,5-5 transformation of,15-5ff Torque, 18-4, 20-1 ff Transformation, Fourier, 25-4 Galilean, 12-ll linear, 11-6 Lorentz, 15-3, 17-1, 34-8, 52-2 oftime, 15-5 ff ofvelocity, 16-4ff Transient, 24-1ff electrical, 24-5f Transient response, 21-6 Translation, ofaxes, 11-1 ff Twin paradox, 16-3 f Tycho Brahe, 7-1 Ultraviolet radiation, 26-1 Uncertainty principle, 2-6, 6-10 f,37-9, 37-11, 38-8 f Unit cell,38-5 Unit vector, 11-10 Vector, ll-5 ff Vector algebra, 11-6 f Vector analysis, 11-5, 52-2 Vector product, 20-4 Velocity, 8-3,9-2f components of,9-3 transformation of16-4 ff Vinci, Leonardo da,36-2 Virtual work, principle of,4-5 Vision, 36-1 ff binocular, 36-4 color, 35-1ff Visual cortex, 36-4 Visual purple, 35-9 Wapstra, 52-10 Watt (unit), 13-3 Wave, shear, 51-4 sinusoidal, 29—2 f Wave equation, 47-1 ff Wavefront, 47-3 Wavelength, 19-3, 26-1 Wave number, 29-2 Waves, 51-1 ff light, 48-1 Weyl, H.,11-1 Work, 13-1 ff,14-1 ff X-rays, 2-5,26-1 Young, 35-7 Yukawa, H., - Yustova, 35-8Ix)O0 Zeno, 8-3 Zero, absolute, 1-5 Zero mass, 2-10 INDEX 3