[Arnold_Sommerfeld]_Lectures_on_Theoretical_Physic(BookFi.org)

Electrodynamics Lectures onTheoretical Physics, Vol. ILI BY ARNOLD SOMMERFELD University ofMunich ‘Translated by EDWARD G,RAMBERG NEW YORK, N.Y. ACADEMIC PRESS INC., PUBLISHERS 1952 Copyright 1952 By ACADEMIC PRESS INC. 125East 23rd Street, New York 10,N.Y. AllRights Reserved NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS Library ofCongress Catalog Card Number: 58-7481 SECOND PRINTING, 1956 PRINTED (N THE UNITED STATES OF AMERICA PREFACE Heinrich Hertz’s great paper onthe‘Fundamental Equations ofElectro- dynamics forBodies atRest” hasserved asmodel formylectures onelec- trodynamics ever since mystudent days (see §1). Following thisexample Iproceed inPart Ifrom Maxweli’s equations asanaxiomatic basis, ex- pressed notaswith Hertz, inthecoordinates and indifferential form, but invectorial integral form. InPart IItheseveral classes ofphenomena, in static, stationary, quasistationary, and rapidly variable fields, arederived from these equations, asinHertz’s paper. After Ihad heard Hermann Minkowski’s lecture on“Space andTime” in1909 inCologne, Icarefully developed thefour-dimensional form ofelectrodynamics asanapotheosis ofMaxwell’s theory andatthesame time asthesimplest introduction to thetheory ofrelativity; inreturn, this hasalways met with anenthusi- astic reception onthepart ofmyaudience. This four-dimensional electro- dynamics ispresented inPart III. Itstitle “Theory ofRelativity and Electron Theory” requires thecomment that itislimited tothespecial theory ofrelativity ontheonehand and tothetheory oftheindividual electronontheother.Thestatisticsofelectronsinmetalsandelectronsin +insulators belong inVols. IV*andV*oftheLectures. Schwarzschild’s prin- ciple ofaction, which establishes thefundamental relationship between Maxwell’s theory andthedynamics oftheindividual electron (orindividual electrons), ispresented attheend ofPart IIIwith certain modifications which appear necessary from ourpoint ofview. InPart IVisdeveloped theelectrodynamics ofmoving media, again following Minkowski rather closely. Asthemost important application thefields ofunipolar induction arediscussed and arecalculated asanexercise foraparticularly simple example. Part IIconstitutes themain portion oftheLectures. Idistinguish be- tween summatior. andboundary-value problems inelectrostatics andmag- netostatics. The computation oftheelectric potential forgiven charge distribution andthecalculation ofthemagnetic potential forgiven magneti- zation areexamples ofthefirst; thetheory ofthepermanent magnet, insofar asitfalls within thecompetence ofMaxwell’s theory rather than atomic theory, becomes simple andclear irom thisstandpoint. Ontheother hand, thesolution oftheelectric andmagnetic boundary-value problema properly belongs inVol. VI;only themost important cases aretreated in thepresent volume. The calculation ofstationary fields foragiven distri- bution ofthecurrent density, either bythemethod ofthevector potential in§15orthat ofthemagnetic shell in§16, isalsoasimple summation prob- *See p.xiiforlistofLectures onTheoretical Physics. iW Iv PREFACE lem. Among therapidly variable fields those ofthewire-wave type are treated with some completeness. The principal wave onasingle wire in§22 (symmetric electrical type) serves asprimary example; however, because oftheir recent practical applications (theory ofwave guides in§24) and their utilization inthetheory oftheLecher system, themagnetic type and theasymmetric secondary waves aswell aswire waves onnonconductors arealso dealt with in§23. Asconclusion ofPart IItheLecher system is treated fully forarbitrary separation and dimensions ofthetwo parallel wires, employing bipolar coordinates fortheexterior ofthewires and ordinary polar coordinates fortheir interior. Itisonly assumed that the two wires arerather good conductors. The dimensional character ofthefield entities istaken seriously through- out. Wedonotaccept Planck’s position, according towhich thequestion oftherealdimension ofaphysical entity ismeaningless; Planck states in §7ofhisLectures onElectrodynamics that thisquestion hasnomore mean- ingthat that ofthe“real” name ofanobject. Instead, wederive from the basic Maxwell equations thefundamental distinction between entities of intensity and entities ofquantity, which has heretofore been applied con- sistently intheexcellent textbooks ofG.Mie. The Faraday-Maxwell induc- tignequation shows that themagnetic induction Bisanentity ofintensity along with theelectric field strength E;B,rather than H,deserves the name magnetic field strength. H,like D,isbest designated as“excitation.” divHrepresents themagnetic density, just asdivDrepresents theelectric charge density. Hertz’s distinction between “true” and “free” electricity becomes pointless, since divEis,dimensionally, notacharge, but adiver- gence oflines offorce. The same applies tothedistinction between “true” and “free” magnetism, particularly since divBiseverywhere zero. The current density J,theelectric polarization P,and themagnetization M, areentities ofquantity likeHand D.Energy quantities always take the form ofproducts ofanentity ofquantity and anentity ofintensity, e.g. 4D-E, }H-B, J-E, EXH.The factthat Band E,andHandD,belong together follows unambiguously from thetheory ofrelativity, inwhich thequantities cBand —iE, and Hand —icD, respectively, arecoupled together inasix-vector (antisymmetric tensor). Wecallthefirst thefield tensor F,thesecond, theexcitation tensor f. The introduction ofafourth electric unit, independent ofthemechanical units, isdecisive forthefruitfulness ofthese dimensional considerations. Wechoose forthis theunit ofcharge Q,which, asamatter ofconvenience wemay identify with thecoulomb ifwewish. Inthismanner weavoid the “bed ofProcrustes” ofthecgs-units, inwhich theelectromagnetic quanti- ties are forced totake onthe well known unnatural dimensions. Since we must definitely give upthehope ofamechanical interpretation ofelectrical quantities, wemust regard thecharge asabasic, irreducible entity which can claim adimension ofitsown. We shall refer tothe “electrostatically” PREFACE v or“electromagnetically” measured charge only inpassing, and exclusively forhistorical reasons. With theparticular unit ofcharge Q=1coulomb theelectric current has thecustomary unit amperes =Q/sec. Asmechanical units, following thesuggestion ofG.Giorgi, weshall em- ploy the units meter M,kilogram (mass) K,and second S.The unit of energy then becomes 1joule (without apower oftenasfactor!) and that ofpower 1joule S~=1watt. Furthermore, thepowers oftendisappear also fortheelectric units volt, ohm, farad, and henry; wehave 1volt = 1joule/Q, 1ohm=1jouleS/Q’,1farad =1Q?/joule and1henry = 1joule S#/Q*. . Ontheother hand powers oftenmust appear asfactors when theunits ofthemagnetic field strength Bandthemagnetic excitation Hareexpressed intermsofthegaussandtheoersted, respectively, whichasweshallseein §8,have been adapted tothecgs-system. Asexpected, theunit Qauto- matically drops outoftheenergy densities }D-E,}H-B, and J-E referred toabove; their dimension becomes joule/M? directly, whereas that ofthe energyfluxEXHisjoule/(M’S). With our dimensional differentiation between entities ofintensity and entities ofquantity thedielectric constant and thepermeability evidently ,become dimensional quantities and therefore cannot besetequal to1in vacuum. Their choice, inwhich weaccept electrical engineering practice, happily permits ustomeet thedemand for“rational units” without diffi- culty. Itisonly necessary toset, as vo=4n-107 i inaccord with international conventions, and toderive éofrom therelation qo =1/c*, verified byHertz’s experiments. With this choice the 4x’s disappear wherever they donotbelong, asinPoisson’s equation and the energy expressions inPoynting’s theorem, and appear where they belong, asinCoulomb’s law and forthespherical condenser. We thus avoid the desperate expedient bywhich Lorentz achieves rationalization inhisarticles intheEnzyklopaedie, namely theintroduction ofthefactor ~/4x inthe definition ofthecharge and ofmagnetism. Atthesame time, with this choice of&and yo,thesquare root ofthe ratioofwoand9evidently becomes aresistance, namely theso-called ““wave resistance ofvacuum.” This quantity occurs inPart IIasafactor wherever the wave fields Eand Henter into formulas ofthe same dimen- sions. Jtoccurs again inthetheory ofrelativity intherelation between the excitation tensor fand thefield tensor F,which invacuum assumes the simple form forallsixcomponents f=VMF,& VI PREFACE ‘These questions ofunits, dimensions, andrationalization, often discussed toexcess inrecent years,aredisposed ofasbriefly aspossible inthelectures; however, thereader isrepeatedly urged inthem toconvince himself ofthe dimensional logic offormulas. Innumerical computations ourMKSQ sys- tem ofunits isfaund convenient throughout since itisadapted tothe practical andlegal units, volt, ampere, etc.Weleave thequestion open as towhether itisalso appropriate foratomic physics. Soastopermit an effortless transition totheGaussian system (€=40=1),which iscusto- mary inthis case, weexplain Cohn’s system in§9,which inourinterpreta- tion isbased onthefive units MKSQP (P.=magnetic unit pole). ‘The wonderful simplicity and beauty oftheMaxwell equations, which ismost striking intheir relativistic formulation forvacuum, lead tothe conviction that these equations, along with theequations ofgravitation (§38), arethedemonstration ofanall-inclusive world geometry. Approaches tothisbynomeans resolved problem aresummarily discussed in§37. The amazingly simple representation ofthegeneral theory ofrelativity in§38 isbased onaderivation ofSchwarzschild’s lineelement kindly made avail- able tomebyW.Lenz. Inthisnianner thethree tests ofthetheory open toastronomical observation may betreated without tensor calculus. This volume isbased onlecture notes prepared byH.Welker inthe winter semester of1933/34, atwhich timeIfirstabandoned thecgssystem and passed over tothemore general system ofthefour units. Inthefinal “formulation ofParts Iand IIIhave had the benefit ofthe constant advice ofProfessor J.Jaumann,. whose electrotechnical experience and point of view have been ofgreat advantage tothis volume. Iamgrateful toMessrs. P.Mann andE.Gora andtomycolleague F.Bopp forcritical remarks and suggestions forimprovements. Dr.W.Becker haskindly assisted mein reading theproof ofthis, asofpreceding volumes. Munich, April 1948 Arnold Sommerfeld Transtator’s Nore Aminimum number ofchanges has been made inthis translation of Sommerfeld’s ‘“Elektrodynamik” (the third volume ofthe Lectures on Theoretical Physics) toadapt itforuseinEnglish-speaking countries. As faraspossible, thesamé conventions regarding notation areemployed as inG.Kuerti’s translation ofVolume II,“Mechanics ofDeformable Bodies.” Thus vectors arerepresented bybold-face letters, vector components and scalars {aswell astensors and their components) byitalics; this inspite of thefact that theGothic letters employed intheoriginal text forboth vec- torsandvector components were used even inMaxwell’s Treatise. Toavoid confusion afewadditional changes ofsymbols were required inconsequence ofthis major change. E.G. R. CONTENTS Translator’s Note..... voceeccceeegeteeseseeeeseeseeeeesteneeeeres WE Parr I.FuNpamentats anp Basic Princrruzs orMaxweuu’s ELEcTropYNamics §1.Historical Review. Action ataDistance and Action byaField............ 1 Biographical Notes...........2..000000cccecseseeeeelessesdeeeegeesetereaee 8 Michael Faraday, 1791-1867.......... cettteeteseeeteseeeeeees @ JamesClerkMaxwell, 1831-1879..000.0...00000ccceecceeteeeeeeeeeeereee 8AndréMarieAmpére, 1775-1836 .......00.... cececececeeeeeeeetttess& HeinrichHertz,1857-1894..vecvetteteeseeteeetteeseeereeees cere8 §2.Introduction totheBasic Cortcepts oftheElectromagnetic Field....... 6 §3.Maxwell's Equations inIntegral Form...... tettetttteeseeeeesteee TH§4.TheMaxwell Equations inDifferential Form andtheMaterial Constants of 1.ConductivityandOhm’s Law....00...000000cceecccsceeeeeceeeeeeeeeees 20 2.Dielectric Constant. : beeeethetteteeeeceee eee Ql 3.Permeability........... ve tirereeereey BE §5.Law ofConservation ofEnergy and Poynting Vector ....................5 25 §6.The Role oftheVelocity ofLight inFlectrodynamics..................... 82 §7.The Coulomb Field andtheFundamental Constants ofVacuum. Rational andConventional Units. steeeeer cesseseeeeteteeeeeee BF A.Blectrostatics...0..... 0600000cceeccceeeceeeeeeeeeeeeeeeeeeesteeeereres 88 B.Magnetostaties........... veccetettestteeteetsesesses 4 C.Rational and Conventional Units. . sevens seseeees 42 D.Final Determination oftheFundamental Constants &%,uointheMKSQ §8.Four,Five,orThreeFundamental Units? -0.0..0.00.000csecceeeeeeeeee 45‘A.Supplementary Note onOur System ofFour Units................0.5. 45 B.TheFiveUnitsMKSQP eee ee secceecceeeeneeeerenes AT C.The Gaussian System ofOnly Three Units. . ceceteteeeeteeeeees 49 D.Supplement Regarding Other Systems ofUnits........0...:..0000000. 88 Part II.DERIVATION OFTHE PHENOMENA FROM THE MAXWELL Equations ~ §9.The Simplest Boundary-Value Problems ofElectrostatics.................. 55‘A.Charging Problems..............220cc:0cccecceeetecteeetertteeteeesees 55 B,Induction Problems and Method ofReciprocal Radii...............2.-. 56 C.Conducting Sphere inaUniform Field . cee severe 88 D.DielectricSphereinaUniform Field........... .....ceceeceereeeeees 6D E.Reflection and Refraction ofLines ofForce attheBoundary ofaSemi- infiniteDielectric. ...ceeeees seeeeeeeteeeeeeeeeneoes $10.Capacity andItsConnection with Field Energy... . testers Of A.ThePlateCondenser... beettereteset 65-ISpherical Condenser... ........ veecccttcceeees 6 C.Capacity ofanEllipsoid ofRevolution andofaStraight PieceofWire..-68 vu Vur CONTENTS, 1).Energetic Definftion ofCapacity. sees seeeeee 68 J:.The Capacities inanArbitrary System ofConductors...... seceee 70 $11. General Considerations onthe Electric Field sees sree TL A.The Law ofRefraction forthe Lines ofForce. seteereccesesssces TL B.Onthe Definition ofthe Vectors Eand D settee ceeeeeeeeeeeneees TB C.The Concept ofElectric Polarization; theClausius-Mossotti Formula.. 73 D.Supplement totheCalculation ofthePolarization..................... 76 E.Permanent Polarization deeeesececeeeecetsscceeees OE §!2. The Field ofthePermanent Bar Magnet... . wae secreecocsses 13 §13. General Considerations onMagnetostatics and Corresponding Boundary- ValueProblems.............. settee eeeeteteeeees88 A.The Law ofRefraction oftheLines ofMagnetic Excitation............ 89 B.Definition oftheVectors Hand B,Particularly inSolid Bodies........ 89 C.The Magnetization MinAny Non-Ferromagnetic Substance........... 89 D.Dia-andParamagnetism...... ....... oe .seeeeeceteseees 90 1.Soft Iron asAnalog totheElectric Conductor..............0.......... OL F.Specific Boundary-Value Problems. ... eee es seeeeeeeee OL G.The Uniform Field within anEllipsoid ofRevolution seeeeeee 92 H.The So-Called Demagnetization Factor sees OB §14. Some Remarks onFerromagnetism. . . sees 96 A.TheWeiss Domains sees : :seecesee B.The Electron Spin asElementary Magnet o ceteteeee eee 98 C.Hysteresis Loop and Reversible Magnetization settee teeters 9B D.Thermodynamics covet ceceeeeeeeeesees 100 §15. Stationary Currents and Their Magnetic Field. Method oftheVector Po- *tential. coe - cirettee teetaeteeeeeee 100 A.The Law ofBiot-Savart wees seveeeseees 108 B.The Magnetic Energy ofthe Field ofTwo Conductors................. 104 C.Neumann’s Potential asCoefficient ofMutual Induction............... 106 D.The Coefficient ofSelfinduction . sereeecseeececores 108 E.Selfinductance ofthe Two-Wire Line....... . veers 112 F.General Theorem Regarding Energy Transmission byStationary Cur- rents............. : . sevveeeeeeeees MB §16. Ampére’s Method oftheMagnetic Double Layer sevens U4 A.The Magnetic Shell forLinear Conductors . +16 B.Magnetic Energy and Magnetic Flux seeeceeeree 19 C.Application totheSelfinductance ofaTwo-Wire Line . ceceee 121 D.Application totheElectromagnetic Current Measurement ofWilhelm §17. Detailed Treatment oftheField ofaStraight Wire and ofaCoil......... 125 §18. Quasi-Stationary Currents... ceceeeveeececeeeeaeee 188 A.Energetic Interpretation oftheWave Equation.....................0++ 185 a.FreeVibrations. . . strescescocssesessees 186b.Forced Vibrations . rn ceceees 187 B.The Wheatstone Bridge wees vee, 140 C.Coupled Circuits . oe sevens M2 D.TheTelegraph Equation... . 43 §19. Rapidly Vgriable Fields. The Electrodynamic Potentials oo 145 A.The Retarded Potentials... . .M7 B.The Hertzian Dipole 148 C.Specialization forPeriodic Processes 152 1D.The Characteristic Vibrations ofaMetallic Spherical Oscillator 154 CONTENTS es E.Application totheTheory ofX-Rays....00.00000000000000 155 §20. General Considerations ontheStructure ofWave Fields ofCylindrical Sym- metry. Details onAlternating Current Impedance and Skin Effect 156 A.Longitudinal and Transverse Components. . . veces 157 B.The Wave Field ofSemiinfinite Space and ItsSkin Effect... 160 C.The Alternating Current Impedance ofaSemiinfinite Space. . 163, D.The Rayleigh Resistance ofaWire.. 166 E.The Alternating Current Inductance tees . . 167 F.Further Treatment oftheAlternating Current Field ofaCircularly Cy- lindrigal Wire eeeeeeecceeseeeeetencserstresccrtsnsceressesees168 §21. The Alternating-Current Conducting Coil... cocvevecceeeeeeeeeeveee 170 A.TheFieldofthe Coil................. Eee (') B.Resistance and Inner Inductive Reactance ofthe Coil.......... ..178 C.TheMultilayer Coil............. : veee ceeceeeees 175 $22. The Problem ofWaves onWires..... : veces seeeeeeeeeete IT A.The Field within and outside ofthe Wire........ see eeeeeeee eens 178 B.The Boundary Condition atInfinity.....0.. 0.020.000.2020. 0.e0eeeeeees 181 C.TheBoundary Condition attheSurface oftheWire................... 182 §23. General Solution ofthe Wire-Wave Problem. .......... setteeeeeceee185 A.Primary Wave and.Electrical Secondary Waves................00000.-. 186B.Magnetic Waves... veceeeeeees vecceeeeeeees 187 C.Asymmetric Waves oftheElectromagnetic Type.... seeeee erences 188 D.WireWaves onaNonconductor..........0.0.0..0.0 coceceeee usesens1901$24.OntheTheory ofWaveGuides. . .. serccscesceees 198 $25. The Lecher Two-Wire Line areeee weet eee 198 A.The Limiting Case ofInfinite Conductivity. . eee wee 200 B.The Exterior ofthe Wires. . cence ees202 C.The Interior ofthe Wires noses cece eee 2b D.The Boundary Condition H,=H, base sees. 206 E.The Boundary Condition forE,and theLaw ofPhase Propagation... 206 F.Supplement Regarding theRemaining Boundary Conditions cee, 28 G.Parallel and Push-Pull Operation Loe . coves 209 Parr III. Tuxory orRevarivity anv Exectron TuEorr §26. The Invariance oftheMaxwell Equations intheFour-Dimensional World... 212 A.The Four-Potential.... wee feaee 212 B.The Six-Vectors ofField and Excitution..... civteeeeeces 214 C.The Maxwell Equations inFour-Dimensional Form cannes 216 D,OntheGeometric Character oftheSix-Vector and ItsInvariants. 28, E.Relativistically Invariant Three-Vectors. .. : : ..220 27. The Group ofthe Lorentz Transformations and the Kinematics ofthe TheoryofRelativity wee eneeeeeeeeeeteeneens.222 A.TheGeneralandtheSpecialLorentzTransformation.................. 223 B.TheRelative Nature ofTime. ..............00.00 0000.ceceecveeeeeeees225 C.TheLorentzContraction... .ceceeeeeeeeeeeeeeeereer226 D.The Einstein Dilatation ofTime : seeeeeceees 227 E.TheAddition Theorem fortheVelocity ccceeeeeeee eesBDF.c'as Upper Limit forAllVelocities... . . .230 G.Light Cone; Space-Like Vectors andTime-Like Vectors; Intrinsic Time. .231 H.The Addition Theorem forVelocities ofDifferent Directions...... 233 J.The Principles oftheConstancy oftheVelocity ofLight and ofCharge 234 §28. Preparation fortheElectron Theory see eens BE x CONTENTS A.The Transformation ofthe Kleetrie Field. Preliminaries Regarding the Lorentz Foree.... bone : : seve BT B.The Magnetic Analog totheLorentz Force - 238 C.The Intrinsic Field ofanElectron inUniform Motion cece 230 D.AnInvariant Approach totheLorentz Foree; theFour-Vector oftheForeDensity...... veeveveveeeeeeeeevaceeses 241 E.The General Orthogonal Transformation ofaTensor oftheSecond Rank. .......eeeee cece cee cee3 §29. Integration oftheDifferential Equation oftheFour-Potential : m5 A.Four-Dimensional Form ofthe Potential Q............. 246 B,Retarded Potentials... 0.0.0.0... fecceeees 248 C.The Lienard-Wiechert Approximation. . bocce bocce 29 $30. The Field oftheAccelerated Electron. .................2.2085 251 A.Electron inUniform Motion ... cess 252B.The Accelerated Electron : : . .253 C.The Longitudinally Accelerated Electron... 254 §31. The Maxwell Stresses and theStress-Energy Tensor 255 §82. Relativistic Mechanics . . 262 A.The Equivalence ofHuergy and Mass......... . 264 B.Relationship between Momentum andEnergy 266 C.The Principles ofD’Alembert and Hamilton vecess266 D.The Lagrange Function and Lagrange Equations veces, 268 E.Schwarzschild’s Principle ofLeust Action... ........ . 269 .$33.Electrontagnetic Theory oftheElectron. ................0c0s0c0eeceeeees 1B Parr IV. Maxweit’s THEory rox Movixe Boues ano OTHER ADDENDA 434.Minkowski’s Equations forMoving Media......0.......0000.5e0gq000 grees 280 $35. The Ponderomotive Forces and theStress-Energy Tensor............ 290 $86. The Energy Loss oftheAccelerated Electron byRadiation and ItsReaction ontheMotion..veeeceeeees: cece eees208 $87. Approaches tothe Generalization ofMaxwell’s Equations and tothe Theory oftheElementary Particles. . : -..301 §38. General Theory ofRelativity; Unified Theory ofGravitation and Elec- trodynamies........ . .veceeeceese es807 A.Gravitational and Inertial Mass. we -.. B12 B.Observable Deductions from theGeneral Theory ofRelativity......... 315 C.Unified Theory ofGravitation and Electrodynamics. ..... ..B21 Syusors Emporen TarovaHour rae Text and THerm Dimensions.......... 823 Apprrionat Symsors tnParts III anv IV... _ coves BM Nomertcan Vatuzs, Resuuts orMEASUREMENTS, ANDDEFINITIONS ...826 PROBLEMSFORPART T..........000000000000 00ceeeeeeeeeeeeeceeee ves.327 1.1.The Boundary Conditions ofMaxwell’s Theory seceeeceeces B87 1.2,The Magnetic Excitation Inside andOutside ofanInfinitely Long Wire.. 327 1.3. The Magnetic Excitation within anInfinitely Long Solenoid... 327 1.4.The Cosine Law ofSpherical Trigonometry asSpecial Case ofaGen- eralVector Formula....... os aeteeeeseeeetes cece cecesBMTPropaemsrorParr II.......beteeeeeeeeee eeeeeeeeeces seeceeeeee B27 IL1. The Charging Potential ofaConducting Ellipsoid ofRevolution... 327 11.2. The Unilaterally Infinitely Long Rubbed Glass Rod and ItsCom- parison with the Conducting Paraboloid ofRevolution 328 11.3. Comparison oftheDielectric and theConducting Sphere . 328 CONTENTS: xI 11.4. Edge Correction forthePlate Condenser According toKirchhoff 328 II.5. The Capacitance ofaLeyden Jar(Cylindrical Condenser) : 328, IL.6. OntheDefinition oftheCapacitance ofTwo Conductors with Equa!andOpposite Charges . Leoeebeeeeeececeeteeeeeceeeeeeesees, 828 IL7. Characteristic Oscillations and Characteristic Frequencies ofaCom- pletely Conducting Cavity Bounded byaRectangular Parallelepiped... 330 H.8. Characteristic Oscillations and Characteristic Frequencies oftheIn- teriorofaCompletely Conducting Circular Cylinder ofFinite Length. .330IL9. Characteristic Ogcillations within aCavity Bounded byaMetal Sphere 330 IL.10. Determination ofthePropagation Constants ofWire Waves from Kelvin’s Telegraph Equation andfrom Rayleigh’s Alternating Current Resistance........... . cereeee ceeeeeees 830 PROBLEMS ForParts IIIanpIV. feteeteeeteteeeeeeeeeet encesBBO, IIL1. The Lorentz Transformation foraRelative Motion Deviating from thez-Axis....... seen seveeees ceeeeeeeees 830111.2. OntheAddition Theorem forTwo Differently Directed Velocities... 331 I1.3. The Field ofanElectron inUniform Motion sescccerecccrecccce OBL IIL4. OntheRelativistic Energy Theorem fortheElectron............... 331 IIL.5. The Electron inaUniform Electrostatic Field... ceeeeeeee BBL IIL6. The Electron inaUniform Magnetostatic Field cece eteeeeeeees BBL 1IL.7. The Electron inaUniform Electric Field and aUniform Magnetic Field which isParallel thereto eee veces BBL IIL8. The Electron in@Uniform Electric Field and aUniform Magnetic Field’Rerpendicular thereto. ....... . coveeeeeeeeees882 III.9. The Characteristic oftheThermionic Diode According toLangmuir andSchottky. ... .seaensenne ceeeeeesB82III.10. The Acceleration oftheElectron inthe Betatron.. -.333 1V.1. The Field ofUnipolar Induction. pecccescrassses B37 ANswers ANDCoMMENTS.. we . errs| AutuorInvEx sees reer rere seese365 Supsect Inpex... aseesenaees cette eeeeeects eeenesB67 Lectures onTheoretical Physics VouumeE I:Mechanics. 1952. Translated byMartin O.Stern Vouume II:Mechanics ofDeformable Bodies. 1950. Translated byG.Kuerti Votume IV: Optics. 1953. Translation inpreparation Votume V:Thermodynamics andStatistical Mechanics Votume: VI:Partial Differential Equations inPhysics. Translated byErnst G.Straus Part I FUNDAMENTALS AND BASIC PRINCIPLES OF MAXWELL’S ELECTRODYNAMICS §1.Historical Review. Action ataDistance andAction byaField Icanbestgiveyouanideaofthesweeping changes inviewpoint brought about bythetheory ofFaraday andMaxwell bytelling youofthetime I spent asastudent, 1887-1891. Mynative city, Kénigsberg, wastheearliest fountainhead ofmathe- matical physics inGermany, thanks totheactivity oftherevered Franz Neumann, 1798-1894. AttheUniversity ofKonigsberg hetaught, inaddi- tiontocrystallography, theoretical physics which wasnotatthetime given elsewhere inGermany. Hisstudents, ofwhom Gustav Kirchhoff ofKénigsberg wasthemost prominent, spread theteachings ofthemaster totheother German universities. Through theseminar inmathematical physics, founded byhimandC.G.J.Jacobi, healsosawtoitthatthe East Prussian secondary-school teachers received aparticularly thorough preparation, Thismaybearsome relation tothefactthattheGymnasium ‘intheAltstadt graduated themathematician Hermann Minkowski andthe physicists Max andWilly Wien shortly before myfinal examination, while atthesame time theonly slightly older David Hilbert andEmil Wiechert were attending other Kénigsberg schools. Neumann’s greatest successes inresearch were achieved intheelastic theory oflight andinthe physics ofcrystals; hismathematical formulation oftheinduction cur- rents discovered byFaraday willbediscussed in§15. Simultaneously with Neumann and Jacobi, and almost outshining them, F.W.Bessel taught inKonigsberg. Mytime ofstudy coincided with theperiod ofHertz’s experiments. Atfirst, however, electrodynamics wasstillpresented tousintheold manner—in addition toCoulomb and Biot-Savart, Ampere’s lawofthe mautual action oftwo elements ofcurrent anditscompetitors, thelaws of Grassmann, Gauss, Riemann, andClausius, andasaculmination thelaw ofWilhelm Weber, allofwhich were based ontheNewtonian concept of action atadistance. The total picture ofelectrodynamics thus presented touswasawkward, incoherent, andbynomeans self-contained. Teachers andstudents made agreat effort tofamiliarize themselves with Hertz’s 1 2 FUNDAMENTALS OFMAXWELL'S ELECTRODYNAMICS 1 experiments step bystep asthey became known and toexplain them with theaidofthedifficult original presentation’ inMaxwell’s Treatise. Itwas asthough scales fellfrom myeyes when Iread Hertz’s great paper: “Uber dieGrundgleichungen derElektrodynamik fiirruhende Ixérper.” Here Maxwell’s equations, purified byHeaviside and Hertz, were made theaxioms and thestarting point ofthetheory. The totality ofelectromagnetic phenomena isderived from them systematically by deduction. Coulomb’s law, which formerly provided thebasis, now appears asanecessary consequence oftheall-inclusive theory. Electric currents are always closed. Current elements arise only asmathematical incre- ments oflineintegrals. Alleffects aretransmitted bytheelectromagnetic field, which may berepresented byforce-line models. Action atadistance gives waytofieldaction,’ the“constructable representation” ofaspace- time propagation postulated already byGauss.‘ Thave held totheorder ofHertz’s paper inallmylectures onMaxwell’s theory. Inthis presentation, too, weshall notbegin with electrostatics, asisdone socommonly and also inMaxwell’s Treatise, buttreat itmerely asanextreme simplification ofthegeneral field theory. We shall deviate from Hertz only insofar asweshall start notfrom Maxwell’s equations indifferentiql form, butinintegral form. Itgoes without saying that we ‘shall replace therather extensive coordinate calculations ofHertz byvector algebra, which isperfectly suited tothe electromagnetic field. We shall seethat this algebra, extended tofour dimensions, leads directly tothe special theory ofrelativity. The latter will provide anapproach tothe electrodynamics ofmoving bodies, which Hertz unsuccessfully sought to master inthesecond paper cited. Inagreement with Hertz weseeinMax- well’s equations theessence ofhistheory. Weneed notdiscuss themechan- icalpictures, which guided Maxwell inthesetting upofhisequations. We have discussed one such picture inVol. II,§15ofthese lectures. 1The great student ofelectrolysis, Wilhelm Hittorf, who hadheard much ofthe new theory ofelectricity, inadvanced years attempted tostudy theTreatise, but was unable tofind hisway through the unfamiliar mass ofequations and concepts. Hewasthusledintoastate ofdeep depression. Hiscolleagues inMinster persuaded him totake avacation trip tothe Harz Mountains. However when just before hisdeparture they checked hisluggage they found init—the two volumes ofthe Treatise onElectricity and Magnetism byJames Clerk Maxwell. (As told byA. Heidweiller.) *Gdttinger Nachr. March 1890 and Ann. Physik, Vol. 40; continued inAnn. Physik, Vol. 41:“Uber dieGrundgleichungen derElektrodynamik firbewegte Kérper.”“Weavoidthealternative term‘“‘nearaction”whichsignifiesmerelyactionata small distance, and byournotation direct attention tothemedium transmittingthe effect, namely thefield. 7 4Inaletter toWilhelm Weber, of1845. See Collected Works, Vol. V,p.627. 1 HISTORICAL REVIEW 3 Biographical Notes Micuaex Farapar, 1791-1867 Hewas born asson ofablacksmith inimpecunious circumstances. The family belonged tothepious sect oftheSandemanians, towhich Faraday remained faithful tohisdeath. Hishigh ethical concept oflifeand human kindness derived from thereligious spirit ofhisfamily. Hewas first news- paper carrier, then bookbinder. Inscience and letters hewas entirely self-taught. The lectures ofSirHumphry Davy attheRoyal Institution were decisive forhiscareer; hewrote them upcarefully and found an opportunity topresent them tothegreat chemist. Hebecame hislabora- tory assistant intheRoyal Institution. His first important work was “the rotation ofacurrent about amagnet andtherotation ofamagnet about a curreut,” andalso theliquefaction ofchlorine. This work brought about hiselection asFellow oftheRoyal Society and later theindirect succes- sion toDavy attheRoyal Institution. In1832 hebegan thepublication ofthe “Experimental Researches.” His discoveries recorded inthese ex- tend tothemost diverse fields ofphysics, electrochemistry, and thestudy ofmaterials. Wemention asmost significant forus:The discoveries ofthe law ofelectromagnetic induction in1831, thedielectric constant, para- .and diamagnetic behavior, andthepicture ofelectric andmagnetic lines offorce. His' magneto-optical discoveries arediscussed inVol. IV. The failing ofhismemory forced many pauses inhiswork, aswellastherepeti- tion ofexperiments made atanearlier date. Itisuncertain whether this is tobeattributed tomental overexertion or,asiscommonly assumed today, tomercury poisoning inthepoorly ventilated basement rooms of theRoyal Institution. Certainly hispurely intuitive method ofworking, devoid ofany mathematical aid, required tremendous mental concen- tration. Inhislast years arestful summer retreat inthe royal palace, Hampton Court, wasmade available tohim atthesuggestion ofthePrince Consort, Albert. Athisdeath there were found ninety-five honorary diplomas oflearned societies, bound with hisown hand. James Crerk Maxwe.1, 1831-1879 Hecame from aprominent Scottish family (the father’s name wasClerk, theadded name Maxwell being derived from hismother) and was given the best ineducation that histime offered, both inthefield ofletters and that ofscience and mathematics. Thus, atanearly date, hecould translate Faraday’s pictures oflines offorce into amathematical form which could be.generally understood. See hispaper of1855 “On Faraday’s Lines of Force” (translated into German byBoltzmann inOstwald’s Klassiker Nr. 69). Inthepreface tohisTreatise hestates: “Faraday, inhismind’s eye, saw lines offorce traversing allspace where the mathematicians 4 FUNDAMENTALS OF MAXWELL’S ELECTRODYNAMICS 1 (from thepreceding discussion itisapparent that herefers particularly toGauss, Wilhelm Weber, Riemann, Franz and Carl Neumann) saw centres offorce attracting atadistance: Faraday saw amedium where they sawnothing butdistance: Faraday sought theseat ofthephenomena inreal actions going oninthemedium, they were satisfied that they had found itinapower ofaction atadistance impressed ontheelectric fluids. ‘When Ihad translated what Iconsidered tobeFaraday’s ideas into a mathematical form, Ifound that ingeneral theresults ofthetwomethods coincided, ...butthat... several ofthemost fertile methods ofresearch discovered bythemathematicians could beexpressed much better interms ofideas derived from Faraday than intheir original form.” The Treatise appeared in1873. Itsgreatest achievement istheunifica- tion ofoptics and electrodynamics. The simplified form ofthe Maxwell equations, later rediscovered byHeaviside and Hertz, istobefound al- ready inPart IIIofhispaper fortheRoyal Society of1864. Almost as important ashiselectromagnetic papers arethose onthekinetic theory of gases (Maxwellian velocity distribution) and ongeneral statistics, to which belongs also histheory oftherings ofSaturn. Heisalso theauthor ofpurely mathematical papers (oncycloidal surfaces, thetheory ofthe top, and the,determination ofmagnitudes inHelmholtz’s color triangle) &nd ofanimportant paper onlattice structures (seeVol. IIofthese Lec- tures, p.310). After abrief teaching engagement inAberdeen hebecame the first director ofthenewly founded Cavendish Laboratory inCambridge; he died there atanearly age. Anpr& Mantz Ampkre, 1775-1836 Weshall add abiographical note onAmpére notonaccount ofthefun- damental lawalready mentioned, norbecause oftheclassical experiments, which enabled him toderive itwith thesimplest possible means, but for hisdiscovery ofthegeneral relationship between themagnetic field and electric currents. Born inLyon, asaprecocious boy heoccupied himself with philological and mathematical studies. His father was avictim ofthe Revolution. Because ofhismathematical papers hewasnamed professor attheEcole Polytechnique inParis in1804. Here hesoon directed hisattention to chemistry, where hewas able tocompete with Avogadro inthefield of atomism. There follow fiveyears inwhich heisconcerned primarily with psychology and metaphysics, though accepted asamathematician into theAcademy ofSciences ofParis. Hisinterest inphysics isnotawakened until 1820, when hehears ofOersted’s discovery. Inafewweeks heverifies hisbelief that electricity inmotion, and notelectricity atrest, hasamag- 1 HISTORICAL REVIEW 5 netic effect. The years 1820-1826 hespent elaborating hisconcept ofthe connection between themagnetic field and theelectric current, which is equivalent tohalfofMaxwell’s equations provided that theconcept ofthe electric current isextended bytheaddition ofMaxwell’s displacement cur- rent. Weshall hence denote this portion oftheMaxwell equations (in integral form) in§8directly asAmpére’s law. From thispoint ofdeparture Ampere recognized theequivalence ofasolenoid traversed bycurrent toa permanent magnet. Thestrengthening ofthemagnetic field byasoft-iron core placed inthesolenoid isalsotobeattributed tohim. Ampére may thus beregarded asthefather ofthe“electromagnet.” Wemay mention in addition Ampére’s molecular currents and theelegant method ofthemag- netic sheet. When, in1826, however, Ampére obtained aprofessorship inphysics attheCollége deFrance hisinterests changed once more: hereturned to philosophy and logic and devoted himself finally tobiology and com- parative anatomy. Altogether ascientific career ofextraordinary breadth anddepth, ofintensity andversatility! (This material hasbeen taken from anessay byLouis deBroglie inhisbook Continu etDiscontinu, Paris, 1941.) Hernrich Hertz, 1857-1894 Hewasborn inHamburg thesonofarespected merchant family; his father was inlater years Senator oftheFree City. Initially hisgreat modesty prevented Heinrich Hertz from entering upon thecareer ofa scholar; instead, heturned toengineering attheTechnische Hochschule inMunich, Soon, however, hebegged hisfather topermit him totransfer topure physics. Hestudied first inMunich, then inBerlin, andbecame the favorite student and assistant ofHelmholtz. The relationship between teacher and student was the closest imaginable and finds touching ex- pression inthememorial addressed tohim byHelmholtz (reprinted in Vol. IofHerta’s Collected Works). Aprize problem setupbyHelmholtz directed him tothe testing ofMaxwell’s theory. After ashort term as Privatdozent inKiel hewas called tothe Technische Hochschule inKarls- ruhe. Even theearliest papers ofHertz show hismastery inrelating theory and experiment. Several ofthem received thewarm recognition ofhis colleagues, ashisquantitative determination ofhardness among engineers, and hisdescription ofthecondensation processes inrising aircurrents among meteorologists. Hisyears inKarlsruhe, from 1885 to1889, repre- sent thehigh point inhiscreative activity. Wemention inparticular his paper of1888: “Forces ofelectrical oscillations treated byMaxwell’s theory.” Itprovides thecharacteristic solution now generally designated astheHertzian vector and shows thefamiliar force-line pictures ofthe 6 FUNDAMENTALS OF MAXWELL’8 ELECTRODYNAMICS 1 Hertzian dipole. Itisamazing how much ofthe later development of radio telegraphy hasbeen anticipated inthis paper. Weshould also point out thegreat paper on“Rays ofelectric force.” The theoretical papers (basic equations ofelectrodynamics) have already been discussed. The discovery ofthephotoelectric effect also falls into this period. With hislastexperimental paper of1891 “On thepassage ofcathode rays through thin metal films” hereached beyond theproblems setbyMaxwell’s theory and without knowing it,blazed thepath totheelectron theory. The very thin metal films later designated as‘Lenard windows” aredescribed al- ready inthis paper. In1889 hewas called toBonn. Here heprepared hislast work, ‘“Prin- ciples ofMechanics,” which wehave discussed inVol. I,§39. The intro- duction ofnon-holonomic auxiliary conditions, thepolydimensional treat- ment ofmechanical systems ofmany degrees offreedom, theprinciple of thestraightest path attest thekeen logic and thegeometric intuition of their author. Increasing illness prevented experimental work. Hedied on January 1,1894, 37years ofage. §2.Introduction totheBasic Concepts oftheElectromagnetic Field Weregard theexistence ofelectric charges asanestablished fact, whether weproduce them byrubbing apiece ofamber, thegodparent ofelectricity, orrecognize them from thespark when connecting thepoles ofabattery. Weinterpret theobserved attraction, repulsion, and heat generation as theresult ofcharges which have been produced. Wetake care nottodefine thecharge verbally ortoascribe aderived dimension toitbysome ar- bitrary procedure. Instead weregard itashaving itsown dimension, as anentity beyond therange ofmechanics. We call this quantity Q.We could choose asunit ofcharge, whether negative orpositive, thefamiliar universal charge oftheelectron. Weprefer however toletQstand forthe coulomb, theaccepted unit inthepractical system, interms ofwhich the electron charge isexpressed bye=1.60-107* coulomb. Weassume that electrometer apparatus isavailable with which wecancompare different charges with each other andwith thecoulomb asunit ofcharge. The atomistic nature ofcharge isdisregarded intheMaxwell theory proper. The charge oftheatoms and elementary particles istoamuch higher degree anabsolute constant than themass (see §27J). Inaddition totheelectric unit Qwenormally employ asmechanical units oflength, mass, andtime theGiorgi units M(meter), K(kilogram mass), andS(second), which have been established internationally bythe decision, oftheappropriate commissions. Asalready pointed outinVol. I,p.8,there istheadvantage thatinthissystem theunits ofenergy and power correspond exactly (without multiplying powers often)tothejoule and watt introduced previously inthecgssystem. We designate them as 2.2 BASIC CONCEPTS OF ELECTROMAGNETIC FIELD 7 1joule =1M’KS~ =10?cm’-g-sec™ =10’erg 1joule/S =1M’KS~ =10’cm’-g-sec™* =10”erg/sec =1watt and define correspondingly 1newton =1MKS~* =10°cm-g-sec™? =10°dynes This unit offorce “newton” isseen tobeconveniently comparable insize with thepractical unitofforce, the“kilogram” =9.81-10° dynes. Wewill show presently that theannoying powers oftenvanish also for thepractical units ofthe volt and ohm when the MKSQ system isem- ployed. Wenow proceed toexamine insequence thebasic electromagnetic con- cepts. Inmost cases weshall beconcerned with adimensional description rather than with acomplete definition; thelatter will bederived from their interrelation through the basic equations ofthe theory, which can be tested byexperiment. Inthesucceeding section wewill follow directly the present day enumeration ofthebasic concepts. We begin with the electric fieldstrength, for which atrue definition is possible aud isgenerally conventional. Letthis quantity bedenoted by E.'We define itasthe mechanical force exerted inanelectric field onan (infinitesimally small) testbody, divided bythecharge ofthetesthody. Eistherefore 2vector with thedimension’ Force _newton E=Force _newton, 1Charge Q @ within thefield itvaries from point topoint indirection and magnitude. Infollowing everywhere thedirection ofEwedescribe anelectric line of force. Wenow consider the line integral B B [Bav= fBas (2) 4 ‘A between two points Aand B.E,istheperpendicular projection ofEonthe direction oftheline element dsand dsisthe line element regarded as vector; E-ds denotes, asusual, the scalar product. We call this line in- Maxwell employed gothic letters (rather than bold-face letters) forthevectors oftheelectromagnetic field (see Vol. IIoftheTreatise, art. 618): Except forthis distinction weusethesymbols here given. Jwill denote theelectric current density, Tthe total current inawire. 2Heré, and atmany other points, weusetheequality sign toindicate equality ofdimension. Where, asinEq. 2a,wewish todistinguish between actual numerical equality and mere dimensional equality, wewrite =..., i.e., “equal except fora numerical factor.” 8 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 2.28 tegral the“voltage” V: 5 v= [Beds =-..newtonM_..jo"dyne-om 4Q Q (2a) =...19788,Q Theconversion ofthedimension from theMKSQ- tothecgs-system shows that ourunit ofvoltage isidentical with 1volt=10°cgsunits (2b) if,asdecided above, wefixQat 1coulomb =Yscgsunit. (2c) Forthedefinition ofthevoltage itisnecessary that inaddition tothe terminal points A,B,thepath between them beprescribed. Only inlamellar fields (seeVol. I,Eq.6.16, andVol. II,p.137) istheindependence ofthe line integral with respect tothepath guaranteed byStokes’ law (see Vol. II,Eq.3.6). Inplace ofvoltage wemay then speak ofdifference of potential between thetwopoints AandB,designated byVs. We introduce ascompanion tothe fieldstrength Easecond electric vector D.Weshall call this preferably electric “excitation,” but shall also frequently employ, particularly inthefirst part ofthese Lectures, the customary term “dielectric displacement” (Maxwell’s designation). Wemake theintroduction ofDcomprehensible bythefollowing con- sideration: Charge, initshistorical origin, isaconcept based onthenotion ofaction atadistance. Toadapt ittotheviewpoint ofaction byafield itisnecessary toimagine anexcitation ofthesurrounding medium pro- ceeding from thecharge centers, which excitation willbedescribed bythe vector D.Forasingle point charge eweimagine “‘Dlines” leaving euni- formly inalldirections, with such density that the‘“D-flux” becomes fPate=« @) doisanelement ofanarbitrary surface surrounding e.If,inparticular, wechoose aspherical surface ofradius r,wefind 4n’D =e. (a) For.arbitrarily, including continuously, distributed charges, Eq.(3)is replaced by - fd.da=% =e (3b) 2.5 BASIC CONCEPTS OF ELECTROMAGNETIC FIELD 9 whereéindicates thetotalcharge within o,thealgebraic sumofpositive and negative charges. Wewillseein§4that this description ofD,foraa arbitrary choice ofc,isselfconsistent, butdoes notsuffice foraunique defi- nition ofD.Wewill also seethere that inthesimplest case (isotropic medium, linear relation between DandE)the‘“‘D-lines” areidentical with thelines offorce defined bytheEvector. Fromthepreceding equations thedimension ofDisseentobe ;charge QD=me 7 (4) This dimension isentirely different from thedimension ofthefieldstrength E,given byEq. (1).With regard toMaxwell’s designation “dielectric displacement,” wenotethatitfitsstrictly notthevector Ditself,butonly that fraction ofDwhich arises from thepresence ofponderable matter andwhich willlater (see §11C) bedesignated asthepolarization P.Thus thisportion Pvanishes forvacuum, themedium which isofgreatest im- portance tous.Nevertheless the“displacement” Dretains itsindividual] meaning, distinct from E,inthis case also. Wecompare Eq. (4)with thedimension oftheelectric current density J.One knows that this istobedefined asthequantity ofelectricity trav- ersing unit area inunit time inaconductor. Itsdimension istherefore —charge _Q J cea-time ~MS" (4a) Depending onwhether theunit area isplaced perpendicular tothedirec- tion ofthecurrent oratanangle thereto, theabsolute magnitude ofJ oracomponent ofitisobtained. Jisthus avector similar incharacter to D.Dimensionally, however, notD,butthetime rate ofchange ofD,the socalled displacement current D,corresponds toJ. Wehavehereassumed asharp distinction between conductor andnon- conductor (dielectric medium). Actually, noperfect insulator exists since even thebest nonconductor conducts tosome extent, e.g. under theinflu- ence ofcosmic radiation. Maxwell therefore supplements thedisplacement current toform the total current c=b+J; (5) thedesignation C(current) was introduced byMaxwell. This notion ofthe equivalence ofDandJisabasically newideaofMaxwell, which isapre- requisite forthe unified representation ofelectromagnetic phenomena. Similarly, hesupplements inthemetallic conductor theconduction cur- rent Jbytheaddition ofahypothetical displacement current D,although here thefirst term completely outweighs thesecond. We now pass tothe magnetic field. This quantity exerts amechanical 10 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS, 2.6 force onamagnetic pole P,which, tobegin with, may bethought ofas isolated. With thesame letter Pwedesignate alsothestrength ofthemag- netic poleandwith Ptheasyetundetermined dimension “pole strength.” Themechanical force divided byPweshould most properly callthemag- netic field strength. Wewillhowever, atleast inthebeginning, adhere to custom and call this quantity themagnetic induction B: _ force _newtonBepolestrength =P ® Weshall even goastep further inouradherence tocustomary notions and utilize therelation between current and magnetism elaborated by Ampére, whose systematic description must however bepostponed until §17.Thus, forexample, themagnetic field ofaplane circulating current J about thearea Fis,atagreat distance from J,equal tothefield ofabar magnet placed normal toFatJ,with themoment m=IF, (6a) This relation, which intheconventional cgssystem serves tomeasure the current I“magnetically,” weshall here employ todefine thepole strength Pinterms ofourelectric unitofcharge Q.Weset m=pole strength -pole separation =Pl (6b) andobtain from Eq.(6a) =-pFLQM_oMPelysgM 7s ” Ourdimensional equation (6)thusbecomes newton S B=“9M (8) Acomplete description ofthemagnetic field alsorequires inaddition to Basecond vector which weshall designate with H.Wecannot however adhere tothecustomary notation “magnetic field strength,” which, as wehave seen, rightfully belongs tothevector B,butwill callHthemag- netic excitation. Wefollow here thecarefully thought-out representation of electrodynamics ofMie.’ With thename magnetic excitation weplace Hinparallel withthe“electric excitation” D.Corresponding toEq.(4) wetherefore define Hdimensionally by =polestrength _P H—area OO? (9) ‘Gustav Mie, Lehrbuch derElektrizitét und desMagnetismus, 2nd Ed., Enke, Stuttgart, 1941, and Handbuch derExperimentalphysik Vol. XI,Part 1,Elektrody- namik. 2.98 BASIC CONCEPTS OFELECTROMAGNETIC FIELD i which, inview ofEq. (7),wemay write Q H=gs: (Ga) This representation alsojustifies adesignation which iscommonly employed inengineering and which, though rather awkward, ismore appropriate than theunfortunate name “magnetic field strength,” namely, thedesig- nation “ampere turns perunit length.” Forfurther details seetheend of§4. Thedirection ofthefield vector B,varying from point topoint, isrepre- sented bytheform ofthemagnetic lines offorce. Asiswell known, these aremade evident bytheautomatic alignment ofiron filings which are brought intotheneighborhood ofthemagnet andwere known long before thecorresponding electric lines offorce. Their expressive appearance still contributes greatly totheunderstanding ofthefield concept. Because of theequality ofdirection ofHandBinairoranyotherisotropic medium thelinepatterns corresponding totheHvector areidentical with thelines offorce ofthe Bvector. Wemay indicate finally asubdivision ofphysical entities intoentities of intensity and entities ofquantity. Eand Bbelong tothefirst class, D andH,tothesecond. Theentities ofthefirstclass areanswers totheques- tion“how strong,” those ofthesecond class, tothequestion “how much.” Inthetheory ofelasticity, forexample, thestress isanentity ofintensity, thecorresponding strain, oneofquantity; inthetheory ofgases pressure and volume form acorresponding pair ofentities. InDthequantity character isclearly evident asthequantity ofelectricity that haspassed through; inHthesituation isslightly obscured bythefactthat there are noisolated magnetic poles (see§3).Weareingeneral inclined toregard the entities ofintensity ascause, thecorresponding entities ofquantity astheir effect. §3.Mazwell’s Equations inIntegral Form After thisvery incomplete preparation wepass totheaxiomatic founda- tionofMaxwell’s theory. Theaxioms ofelectrodynamics, justastheNew- tonian axioms ofmechanics, rest onexperience—more exactly onthe ordering ofthetotality ofexperience intoasimplified andidealized form. Thus thelawofinertia ofmechanics appears very different from what is observed inaparticular case forterrestrial bodies. Similarly, ourelectro- magnetic axioms aremuch more abstract andmathematically generalized than what ismeasured with coils, wires, and pointer instruments. Never- theless likethemechanical axioms, they eresimply 2summary ofdivers: observations. Tobegin with wesetuptwoprincipal axioms which weshall then eupple- 12 FUNDAMENTALS OFMAXWELL'S ELECTRODYNAMICS 3.L ment bysecondary axioms. Oneofthese weshall callFaraday’s lawof induttion. Weshall state it,asfaraspracticable, inFaraday’s own line- f-force language. Theother axiom weshall name after Amptre, since he was thefirst toformulate therelationship between current and magnetic fields. The fact that Ampére’s lawalso rests onexperience hasbeen em- phasized byitsauthor.’ Weshall, however, state both axioms intheuni- versal form whose possibility wasfirst realized byMaxwell. Forthispurpose weconsider anarbitrary surface «with theboundary curve s.Weprovide thelatter with apointer indicating sense oftravel and Shall define that direction ofthenormal tothesurface aspositive which forms aright-handed screw with thes-pointer. Wecompute thesur- face integrals [Bede andfCxde ) extended over oand shall call them magnetic flux and electric curren! flux. Number oflines offorce andnumber oflines ofcurrent isanother common designation. This notation isofcourse audacious since these bundles of lines arenotcountable. Itisfirst necessary togroup them in“tubes,” justasinVol. II(p.136) thelines ofturbulence were grouped intubes ofturbulence. The tubes must beconstructed sothat their cross section becomes inversely proportional tothemagnitude ofBandC,respectively, atthepoint inquestion. The counting ofthe tubes offorce orcurrent traversing oursurface then amounts tothesame astheevaluation ofthe integrals (1). Next wecompute thefollowing lineintegrals extended over theboundary curve 8: fBedsandfWas. (2) Wecallthese theelectric andmagnetic loop tension. The first hasalsofora long time been called E.M.F. orelectromotive force; included inthis designation, itistrue, arealsoother “electromotive” causes, such asdiffer- ences intemperature andchemical effects. The word “force” ishere used initsantiquated meaning ofenergy. The remarkable thing inMaxwell’s point ofview isthat theE.M.F., which totheexperimenter hadhadmeaning only forclosed metallic cir- cuits, ishere defined forarbitrary loops, whether they pass through con- ductors, nonconductors, orthrough parts ofboth. The same geometric freedom then exists alsoforthemagnetic looptension ormagnetomotive force. ‘We now write down the two principal axioms which relate the 1Inthetitle ofhiscomprehensive paper: Iathéorie analytique desphénoménes électrodynamiques, uniquement déduite del'erpérience 3.4 MAXWELL’S EQUATIONS ININTEGRAL FORM 13 quantities defined inEqs. (1)and (2)inthis completely general sense. ‘They are: 4fBade=-fBeds, 3) ffCndo=fias « Inwords: Every change inthenumber ofmagnetic linesofforcewhichtraverse agiven surface oproduces initsboundary 8anelectric looptension which is numerically equal totherate ofchange, butopposite insign (Faraday’s law ofinduction) and Thenumber ofelectric current lines, which traverse anarbitrary surface o 1saccompanied byamagnetic looptension inthebounding curve of«which isequaltoitinbothmagnitude anddirection (Amptre’s lawrelating mag- netic field and electric current). Letusconvince ourselves first that this equating ofelectric andmag netic quantities isdimensionally proper. The twosurface integrals defined in’(1)(inspite oftheir dimensionally incorrect designation asnumbers offorce lines arcurrent lines) have, according to(2.8) and 2.4a), the dimensions sentenMS=taueand&,respectively. According to(2.9a) thelatter dimension agrees with thedimension ofthe lineintegral in(4).The time rate ofchange ofthefirst expression yields joule/Q, ie.thedimension ofanelectric tension (expressible involts), inagreement with theright side ofEq. (3).From this dimensional check ourfundamentally different conception ofBand Hbecomes apparent, anditisclear that ourspecial introduction ofthesymbol Qforthedimen- sion ofcharge isunavoidable. Next weconcern ourselves with thesigns inEqs. (3)and (4).They correspond totherules ofLenz and Ampére. Ampére’s rule issimply theright-handed screw rule, bywhich wecorrelated thepositive normal ofthesurface «with thesense oftravel along theboundary s.The various rules ofthumb commonly given intextbooks aremerely specializations of ourrighthanded screw convention. Tocheck Lenz’s rule weimagine in Eq.(3)theboundary curve stoberealized byawire loop, andthemagnetic fluxtraversing thesurface ointhedirection naslines offorce proceeding from thepositive polePofabarmagnet, thenegative polebeing assumed tobegufficiently faraway. Webring (seeFig.1)thebarmagnet near tothe wire loop andthus increase themagnetic flux, sothat theleftside of(8) becomes positive. Then, asshown bytheequation, thelineintegral onthe right sidemust become negative. The E.M.F. and thecorresponding cur- i FUNDAMENTALS OF MAXWELL’S ELECTRODYNAMICS 3.5 rentinduced inthewireloopthenformaleft-handed screwwiththedirec- tion ofmotion ofthebarmagnet. The magnetic field corresponding tothe induced current is,ontheother hand, represented, according toourright- hand screw rule, bythearrow P’inFig. 1.The positive pole ofthismag- netic field thus points inthedirection from which thepositive pole P ofthemagnet approaches theloop: The two poles repel each other orthe induced current inhibits the motion ofthe inducing magnet. This isthe meaning ofLenz’s rule: The appearance oftheinduced current opposes thedisturbance ofequilibrium produced bythemotion ofthebarmagnet. Weemphasized above that thebounding curve smay befixed quite arbitrarily; thesame remark applies also forfixed boundary tothesurface o.Iftwo different surfaces ,and o2arepassed through thesame curve s, theleftsides ofEqs. (3)and (4)computed foro;and oz,must turn outto 8. Pr P. <—_ EI Nut BMF Fra. 1.Lenz’s rule. beequal. This isequivalent tostating thatthey must vanish fortheclosed surface formed byo;and o¢ifthepositive normal (nalways pointing out- ward) isdefined inauniform manner. Werealize thisfact alsofrom the following: Weconsider aclosed surface owith aboundary curve which hascontracted toapoint. This does notcontribute totheline integrals inEqs. (3)and(4).Ifweindicate theintegration over thenow unbounded surfacewith},wethusobtain 4,fBade=oand$Cxde=0; 6) byEq.(2.5) thesecond equation mayalsobewritten fJade+5,$Dede=0. a)dt More particularly, ifthesurface oliesentirely innonconducting material and ishence traversed bynoconduction currents, ad4,$do=0. (5b) 3.6b MAXWELL’S EQUATIONS ININTEGRAL FORM 15 The first equation (5)and Eq. (5b) state, inintegrated form, $Bdo=const., fDdo=const. (6) while thesecond Eq. (5)and Eq. (5a) show that thetotal electric current isalways closed inMaxwell’s theory: thequantities entering and leaving just’ compensate each other; thecurrent lines traversing our surface ¢ form closed loops somewhere outside ofit.Furthermore, themagnetic lines offorce alsoarealways closed. Ifamagnet (orelectromagnet) issubdivided, north poles and south poles, which compensate each other asfarasthe total magnetic fluxisconcerned, areformed anew onevery part. Itfollows that theconstant inthefirst Eq. (6)must bezero, while inthesecond equation this constant isthealgebraic sum éofthecharges ¢enveloped bythesurface «.According totheabove this must beaconstant intime Joranonconductor: §Bede=0,fPade=é &=Qie=const. (6a) The first Eq. (6a) isasupplementary axiom, anaddition toourprincipal axiomsrequired byexperience. ThesecndEq.(6a)agreeswithourearlierEq.(2.3b) andstates theconstancy intime ofthecharge innonconductors. The D-lines and theE-lines coinciding with them geometricaily originate atpoints ofpositive charge and end atpoints ofnegative charge. Eq. (5a) generalizing thesecond Eq.(6a), may bedesignated inhydrodynamic terminology asthecontinuity equation ofelectricity. Ifthedefinition ofé inEq. (6a) isemployed ittakes onthe form eyfsde=0. (68) This expresses thefactthat theelectricity within asurface «may decreaze astheresult offlowing offthrough metallically conducting portions of¢. The fist Eq. (62) may beexpressed, with Heriz, intheform: There isnotruemagnetism. Inthisstatement oneproceeds from theassumption, formerly regarded asobvious, that Bisthemagnetic analogue ofD.From ourstandpoint, however, thisanalogue isH,and notB.Weshall hence have torelate the definition of“magnetism,” inparticular ofthe pole strength P(see§7),nottoBbuttoH. Wenow apply thefirst Eq.(6a) totheneighborhood oftheboundary surface between twobodies ofdifferent magnetic properties such asiron and air. Cet theclosed surface obethesurface ofavery flat prism (Fig. 2), whose height Ahisvery small compared tothebase Af,andletthisbase lieforexample iniron, while theparallel topside isinair.Eq. (6a) then 16 FUNDAMENTALS OFMAXWELL'S ELECTRODYNAMICS 37 demands, with arbitrary accuracy inview ofthearbitrary smallness of 4h, (B’y +By) Af=0. (7) Let B’refer, forexample, toiron, B,toair.The normal (n’iniron, nin air)points outward onboth surfaces Af.Then, inview ofEq. (7), By =—B, and hence also B’,=Ba, provided that now ndenotes thesame direction inboth media. Wehave thusobtained afirstboundary condition forthemagnetic field:Atthetran- sition between two magnetically different media thenormal component of the induction iscontinuous. ® 4 Fra.2.Derivation ofthecontinuity of Air 7B,atthe boundary between two media h “Wyfromtherelation$B,da=0. Zwon ‘af “We willshow that thesame applies tothetangential component ofthe excitation H.For this purpose weconsider avery small rectangular loop s (Fig. 3),with theheight 4hnormal totheboundary surface and the side length Asparallel toit.Here weassume that Ah<Assothat inthe limit 4h—0thearea Ac=AhAs vanishes. With theassumption that the current density parallel tothe boundary surface, referred toinEq. (4), does notbecome infinitely large’ weobtain from Eq.(4): 0=(H'y +H,)ds (8) sothat #A’y =—H, and hence also 4H, =H’, (8a) where again sdenotes thesame direction inthetwo media. From exactly thesame figure and thesame consideration fortwo elec- trically different media weareledfrom Faraday’s law ofinduction tothe conclusion that thetangential components oftheelectric field strength Eare continuous along the boundary ofthe two media: . E', =E,. (9) ‘Thislimiting caseisthenormal oneforgoodconductors athighfrequencies. - ‘Then H,becomes discontinuous and B,vanishingly small. 3.128 MAXWELL’S EQUATIONS ININTEGRAL FORM 7 Nothing has been said regarding thenormal component ofE.Further- more, thecontinuity ofthe normal component ofD(unlike that ofB) isnotrequired byEq. (6a). For, ifD,hasadiscontinuity attheboundary oftwo electrically different media (e.g. glass and air) oratany other sur- face, wesaythat asurface charge ispresent onthesurface. Ifwecallthis surface charge w(dimension Q/M’), thecharge present intheprism in Fig.2forthetransition tothelimitAh—0is, E=w df. (10) thus bytheconsideration leading toEqs. (7)and (7a), thesecond Eq. (6a) demands (Dv +Ds)Af=wAf, (10a) or,employing thesame direction ofthenormal n: D,—D's =w. (11) Discontinuous behavior ofthenormal component ofDsignifies that thebound- arysurface considered carries asurface charge; themagnitude ofthediscon- _tinuity indicates thesurface charge directly. .2 Fic. 3.Derivation ofthecontinuity of Air H,attheboundary between twomedia 4 oh fromtherelation§Heds=0 trontga % Finally, weobtain from Eq. (6b) fortheboundary surface between a conductor and anonconductor byutilizing Fig. 2and Eq. (10), dw atin =0 (12) that is,alossofsurface charge ifelectric current ispossible intheconduc- tor. Inelectrostatics, where theinterior ofconductors isfieldfree (D= 0,J=0),Eq.(12) isfulfilled identically andEq.(11) takes onthespecial form o=D,. (12a) Inthestatic field conductors bearasurface charge varying from point topoint andgivenbythenormal component ofD. 18 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS A §4.TheMaxwell Equations inDifferential Form andtheMaterial Constants oftheTheory Wepassfrom theintegral tothedifferential form byallowing theloops 8intheintegral form, andhence alsothesurfaces «passed through them, tobecome arbitrarily small. Ifwecallthelatter Acwecanwrite inthelimit: JBade=doB,—[Cade=AoC. @ Furthermore werecall thedefinition ofthevector operation “curl” bythe transition tothelimit ofaloop integral (Vol. II,Eq.2.21). Forourin- finitesimal loops this leads to fE,ds=Aocurl,E,$H,ds=Aocurl,H. (2) Wemust form thetime derivative ofthefirst Eq.(1).Wewillhere imagine thesurface Actoremain fixed, which obviously applies tomedia atrest, towhich weshall confine ourselves initially. Wethen obtain adJ 5,_0B =,|Bado =doBn, =>: a looB, Bea (3a) Atthesame time, using Eq.(2.5), wewrite Eq.(1)intheform fewde=ace+bo,B=B. (3b) With Eqs. (2)and(8a,b),cancelling thefactor Aowhich iscommon to allterms, aswellasomitting thecommon index x,theprincipal axioms (3.3)and(3.4)leadtothetwovectorial differential equations." B= —culE .(4) D+J= cul. Theuniversal importance andimpressive beauty ofthese equations le Boltzmann’ toquote: “Was itagodwho wrote these lines ...” 1Oursecond equation (4)isusually called thefirstsetofMaxwell’s equations, ou~ firstEq.(4),thesecond set.Weprefer thesequence ofthetextsince inourpresen- tation theintensity entities EandBwere introduced firstasbeing more readily interpreted. Wecanalsopoint to§7,where electrostatics willresult from thespe cialization ofthefirst, magnetostatics, from thespecialization ofthesecond Eq. (4),insupport ofourorder. Since itwould beimproper totreatmagnetostatics ahead ofthesimpler electrostatics thenumbering oftheMaxwell equations which differs from, ours appears unsuitable. 2Motto ofthesecond volume ofhis“Vorlesungen tiber Maxwells Theorie der Elektrizitat unddesLichtes,” Manchen 1893. Ourformulation, which deviates slightly fromBoltzmann's (vector inplace ofcoordinate notation), clearly only serves toenhance thebeauty andsimplicity oftheequations. 4.4e MAXWELL EQUATIONS iNDIFFERENTIAL FORM id Wecomplete them bythesupplementary axiom (3.6a) forB,and the relation between Dand the charge, contained inthesame equation. We shall now regard thelatter ascontiauously distributed inaccord with our differential point ofview. Thus, weshall not speak ofpoint, charges e, but offinite densities inspace p,sothat theinfinitesimal charge Ae=pAr iscontained inanelement ofvolume Avwhich approaches zero inmagni- tude. Atthe same time werecall the vector operation “divergence” and itsrepresentation (inVol. II,Eq. 2.20) bythelimit ofavolume integral.! For ourpresent purposes wewrite this representation lim+$Bydo=divB, lim$Dydo=divD Ar Ar andobtain forEqs.(3.6, b),omitting thefactor Ar,their differential form: divB=0, (4a) div D=», (4b) 2%+div=0. (40)ot Our Eqs. (4)and (4a, bsc)setuptheframework into which thephe- nomena ofelectrodynamics must befitted. But this framework isstill toowide. Five vectors E,D,J,B,and Hoccur inourequations, oralto- gether 15unknown functions oftime and space. (The scalar pisreferred back tothevectors Dand JbytheEqs. (4b) and (4c) respectively.) For their determination wehave twovector equations (4),i.e.,altogether only sixdifferential equations. Wemust narrow down theframework tobeable tofillitout with aunified electrodynamic model. The electromagnetic 1Wecontrast thevolume divergence here introduced with theterm surface diver- gence. Referring toFig. 2and theintegration there carried outover aprism with base 4fand vanishing height, weunderstand bythis the result oftheintegration divided by4f.According toEq. 3.7and with the meaning ofthe normals nand n’ there given, the surface divergence ofanarbitrary vector Ais: Ant +Aaj (4d) Eqs. 3.7a and 8.10a then state simply: The surface divergence ofBvanishes, that of Dequals thesurface charge. Similarly, wecan contrast thevolume curl with thesurface curl. Referring toFig. 3and theintegration over arectangle ofbase Asand vanishing height carried outin Eq.3.8, weunderstand bythesurface curl theresult oftheintegration divided by 4s.The surface curl ofanarbitrary vector Aishence, according toEq.3.8, _ Ay +Ag (4e) itrepresents thediscontinuity ofAatthssurface inquestion. 20 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 45 material constants serve this purpose. Weshall discuss them inthesequence conductivity, dielectric constant, permeability 1.Conductivity and Ohm's Law The electric current density Jdepends ontheelectrie field strength E within the conductor. We assume alinear dependence J=oE (6) and callthereal positive constant otheelectric conductivity. Eq. (5)ex” presses Ohm’s lawforunit length ofawire carrying astationary current. Torecognize this, wereplace Jbythetotal current 1=g/(g=cross section ofthewire) and multiply Eq. (5)with the length ofthewire. We obtain (pal Rizvi) (5a) 1 [v= =|Bas=voltage° The concept ofvoltage had been created already byVolta, while thecon cept ofresistance was first introduced byGeorg Simon Ohm in1827° *For usOhm’s law signifies theintroduction ofthematerial constant ¢. According toEq. (5)itsdimension is: - % ~_V_. °=MPSnewton MSjoule (6b) According toEq. (5a) omay also bedesignated asthereciprocal ofthe specific electric resistance, i.e.theresistance ofaprism ofthelength |= 1Mandofthecross section ¢=1M*.Thedimension oftheresistance is byEqs. (5a, b): jouleS R=ro. (Se) The unit ofresistance inthepractical system ofunits isthe@(pronounced “ohm”) =10°gsunits. Itisidentical with theunitinourMKSQ system provided that wechoose, according toourconvention, Qequal to1cou- lomb =yycgsunit Wethen obtain joule S 7ergsec " .1 =10’ 2 =10°ogsunits=19. (5d) ; e g * Ohm’s lawapplies onlytomacrophysical events, nottoAmptre’s moleeu- larcurrents, electron paths inatoms, Larmor precessions; cathode rays invacuum tubes are also resistance-free electric currents. ab MAXWELL EQUATIONS IN DIFFERENTIAL FORM 2 2.Dielectric Constant Thedisplacement Ddepends ontheelectric fieldstrength Eatthepoint inquestion. We assume the dependence tobelinear: D=e& (6) and callthereal positive constant ethedielectric constant. Itsdimension *is,byEqs. (2.4) and (2.1), -_@_.©=Moule a) Wedenote thedielectric constant ofvacuum by&.Italso isadefinite quantity ofthedimension (6a). The relation D= gE, (6b) valid forvacuum, waspointed outalready in§2.Invariably e>e. 8.Permeability Arelation alsoexistsbetween thetwomagnetic vectors HandB,which, asafirst approximation, weshall also assume tobelinear. Wewould like towrite itinthe form H=1B, since weregard Hasanalogue ofDandBasanalogue ofE.However, we areunfortunately obliged tofollow general usage and choose theform . B= uH. (7) The material constant 4iscalled permeability andhas, according toEqs. (2.8) and (2.9a), thedimension ajoules* onOre (7a) This introduction of4,which isillogical inview ofEq. (6),leads tothe obvious consequence that inlater formulas, such asCoulomb’s law, not», but itsreciprocal y’will take the place ofe.For vacuum wewrite B=wH; (7b) 4also obviously hasthedimension given inEq. (7a). Forparamagnetic bodies 1>so,fordiamagnetic bodies, u<uo,Ourformulas (5),(6),and (7)donothave thesame degree ofcertainty andgeneral validity asMax- well’s equations (4).This haslong been known fortheferromagnetic mate- rials, where ageneral functional relationship - B=BUH, T), T=absolute temperature 22 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 48 takes theplace ofthelinear relation (7).Rochelle salts' show adielectric behavior similar tothat ofthe ferromagnetic materials, exhibiting, like thelatter, both saturation and hysteresis phenomena. Forparamagnetic materials deviations from linearity occur only atex- tremely high field strengths orextremely low temperatures. Deviations from thelinearity ofOhm’s law have been expected atvery high field strengths; thefailure ofthis lawforsuperconductors isobvious. Further- more thesimple proportionality between corresponding vectors expressed byEqs. (5), (6), and (7)istrue only forisotropic bodies. Incrystals the dependence isexpressed instead quite generally byalinear vector function (see Vol. II,Eq. 1.10). The varied and interesting phenomena ofcrystal optics, which weshall treat inVol. IV,rest onthis fact. Ontheother hand, thegeneral field equations (4)apply also foraniso- tropic bodies. Beyond this, they appear tohold true even intheface ofall new proposals ofageneralized electrodynamics, proposals which arecon- cerned with extremely strong fields (such asmust occur, forexample, close toanelectron), but intrinsically amount merely toareplacement of thelinear relation (6)byageneralized variation ofDwith E(seethefinal section ofthis volume). The deeper reason fortheremarkable vitality of theform ofequation discovered byMaxwell will befound torest inits invariance’ properties, whichwillnot,however, betakenupuntilPartIII. We.can now undertake the required contraction ofour electtodynamie framework. If,inparticular, weemploy forthis purpose oursimple linear relations and treat c,©,and »asquantities independent of¢(restriction tomedia atrest), we obtain bysubstituting Eqs. (5), (6), and (7)in Eq. (4): Bon=—curlE,at (8) (e2+)k=curlH, ie., sixsimultaneous differential equations ofthefirst order forsixun- knowns, the2-3components ofEand H.Thus wefind ourselves pre- sented with awell-defined mathematical problem.* 1Also known asSeignette salts. Seignette was thename ofapharmacist inthe French fortress LaRochelle. Wearehere concerned with hydrated sodium potassium tartrate: NaOOC-CHOH-CHOH-COOK +24H:0. *Wecould ofcourse also have written Eqs. (8)asrelations between Eand B,or also between Dand H.However, the form inthe text isthe customary one and, in general, also the most convenient one. 49a MAXWELL EQUATIONS INDIFFERENTIAL FORM 23 Atthesame time theconditions (4a,b,c)take ontheform div(uH) =0, div(eE) =p, (8a,b) div{(ea4‘)x}=0. (80)ot Eq.(8a)istoberegarded asarestrictive supplementary condition onMax- well’s equations, Eq.(8b), asdefining equation forp.Eq.(8c)isobtained byforming thedivergence ofthesecond equation (8).Itssimplest solution results from setting theparenthesis {}equal tozero; itisrepresented by theexponential function E=Eexp(-<1),E,=arbitraryfunctionofspace. (9) We set . f=7, (0a) o andcallTtherelaxation timeoftheconductor. Itsdimension isthesecond byEq.(5a)and(6a), asmust bethecase, itsmagnitude forgood con- ductors avery small fraction ofasecond. The field decays within thecon- “ductor everywhere inaccord with this relaxation time and isknown, pro- vided that Eyisgiven. Wemight continue with thealready discussed conditions atthebound- arybetween twoelectromagnetically different media. Tousethedifferen- tial form ofthe Maxwell equations, however, itwould benecessary to regard thetransition between thetwomedia ascontinuous, i.e.,tospeak ofa“boundary layer” rather than a“boundary surface.” Wewillcarry outthisprocedure inproblem I.1,where weshall findthat thederivation becomes lessstraightforward than intheEqs. 3.7to3.12, which followed from theintegral form ofMaxwell’s equations. The same conclusion isreached inother problems distinguished bya particular symmetry: Thegeneral development ofMacwell’s theory must pro- ceedfrom itsdifferential form; forspecial problems theintegral form may, however, bemore advantageous. Thefollowing twofundamental problems, which willbetreated alsoby thedifferential method inproblem I.2and1.3,areexamples ofthis: 1.Aninfinitely long wire intheform ofacircular cylinder istraversed bycurrent distributed uniformly over itscross section. Thereturn ofthe current may take place through asimilarly traversed hollow cylinder which4scoaxialwiththewire.Themagneticexcitationistobedetermined within thewire, within thebollow cylinder, and intheregion between them. 24 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 4.10 2.Aninfinitely long, tightly wound coil istraversed similarly bysta- tionary current. The magnetic excitation istobedetermined atanypoint, within the coil. Regarding 1:Letabetheradius ofthewire, bandc,theinner andouter radius ofthecylindrical return conductor. Weintroduce aright-handed coordinate system about thecenter line ofthewire asz-axis. Letthecur- rent density have thedirection ofthepositive z-axis inthewire, that ofthe negative z-axis inthereturn conductor. Letthetotal current beJand —Irespectively: Lewd, -L=(0- 0)J+ The symmetry oftheproblem indicates that Hisindependent ofyand hasthedirection ofincreasing y.Wewrite H,=Hand carry outtheline Zz 4 AN YY, Fi.4.Astraight wirecarrying astationary cur-\ Y rentandahollowcylindersurrounding itasreturnWN HY conductor. Themagnetic excitation Hy=HwithinNS Y thewire,intheairspacebetween thetwoconduc- NN YY tors, andinthereturn conductor. eed | we integral ofHabout anycircle r=const inanycross-section plane ofthe wire. Since thedisplacement current vanishes everywhere inview ofthe assumed stationary condition, weobtain 2: ers=5 atiO<r<a: WwH=w'sl.=5h H=15 (10) a<r<b: QerH=I, Heb (1) . , 2er bercereHatt? Bs.=1(1-Sa r<e; 7 ms z=R) a) y-i¢2"treo e<r :2arH =1—I, H=0. (13) ‘Theboundary conditions forHatthesurface ofthewirer=aandatthe~ cylinder surfaces r=6,caresatisfied automatically byEqs. 10to13.The variation ofHisplotted inFig. 4. 5.2 MAXWELL EQUATIONS INDIFFERENTIAL FORM 25 Regarding 2:Weusearight-handed system r,,zwhich hasthecenter line ofthecoilasz-axis. Forsufficient length ofthecoiland sufficiently close winding nomagnetic lines offorce penetrate totheexterior ofthe coil; thecurrent Jhasthedirection ofincreasing ¢,theexcitation Hthat ofincreasing z.WeshallshowthatH,=Hisconstant within thecoil. Forthispurpose weconsider therectangular loop, oflength Jinthez- direction, shown inFig.5.Itsplane intersects thecoilinNlpoints, where N,isthenumber ofturns perunit length ofthecoil. Since H,=0both eie e: ( Fra. 5.The magnetic excitation Hwithin anin. @ | finitely longcoil.” 3: H e: i e{ e ae e;e ! within andoutside ofthecoilandH,=0outside ofthecoil,onlyoneside oftheloop contributes tothelineintegral. Wefind Hl=Nil, H=Ni. (14) Themagnetic excitation within thecoilisgiven bythe“number ofampere turns perunitlength” NJ. This explains thedesignation ofHcustomary inengineering practice which wasintroduced onp.12.The value ofH given byEq.14isindependent ofr,ie.thesame throughout theinterior ofthe coil. §6.Law ofConservation ofEnergy andPoynting Vector Starting from Eqs. (4.4) wecarry outascalar multiplication ofthefirst with H,ascalar multiplication ofthesecond with E.Weobtain asthe .sum ofthe two: H-B+E-D+E-J =E-curlH —H-curlE. () Ontheright-hand sideweapply thetransformation, valid forarbitrary vectors U,V: V-curl U—U-curl V=div(U XV). ,~ 4 Weprove thisrelation most readily byutilizing the ienabla- operator”see+, y-2,2,2 ak oz’ du’ dz 26 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 5.28 (seeVol.II,footnote 1onp.23)andinterpreting thedivergence asscalar multiplication, thecurlasvector multiplication with thisvector: div(U XV)=V-(U xV)=Wo-(U XV)+Vy-(U XV), (2a) curlU=VxU, crlV=V XV. (2b) InEq.(2a)thesubscripts U,Vindicate that theV-differentiations areto becarried outonly onthevectors Uand V,respectively. Since these- quence ofthevectors may becyclically interchanged in.thedouble prod- ucts,Eq.(2a)mayalsobewritten div(UXV)=V-(VXU)+U0-(V xV) (20)C) =VV XxU)—UV XxV). Here theright side,.in view ofEq.(2b), isthesame astheleftaideofEq. (2),sothat Eqs. (2c)and (2)become identical. This proof ofEq. (2)is only anabbreviated form forthedirect, butmuch more involved, calcula- tion with rectangular coordinates z,y,z. LetusnowsetV=EandU=HinEq.(2)andintroducetheabbre- viation S=EXH, (3) Then Eq. (1)becomes H-B+E-D+E-J +divS =0. (4) Eg.(4)isPoynting’s theorem, S,thePoynting vector. Weshall show that Sisthe energy flux vector. Weconsider first thedimension oftheindividual terms ofEq. (4} The first two terms have, according toEqs. (2.9a) and (2.8), and (2.1) and (2.4), respectively, the dimension Ea=‘owe=energyperunitvolumeandunittime.(4a) Thethird term has, asmust bethecase, thesame dimension (seeEqs. (2.1) and (2.4a)). The dimension ofEq. (3)is,byEqs. (2.1) and (2.9a), ‘joule_energy perunitareaandunittime. (4b)MS Theoperation div,which indicates adifferentiation with respect tothe space coordinates, yields forthedimension ofthefourth term inEq.(4) the same result. Weseethatourelectrical unitQdoesnotoccurin(4a,b).Jthasdis- creetly withdrawn from thecompany ofthemechanical units MKS. The same will benoted inmany later dimensional considerations inwhich 5.6b CONSERVATION OFENERGY AND POYNTING VECTOR 27 wearedealing with purely mechanical quantities, which areindependent ofthe choice ofthe electrical unit. Wepass tothephysical interpretation oftheindividual terms inEq. (4).Itissimplest forthethird term: thissignifies thework done bythe electric field onmoving electric charge perunit volume andperunit time. Itisgenerally converted into heat and isknown asJoule heat. Wedesig- nate itW,, transferring thesymbol W(work), which Maxwell generally employs fortotal energy, toenergy density. Thus weobtain W,=E-J. (5) Weshall seeright away that thetwo first terms of(4)arethetime rate ofchange ofthemagnetic andelectric energy densities; thelatter aredefined, inaccord with Maxwell, by “W. =4H-B, W,=3E-D. (6) Bythisdefinition theenergy islocalized inthefield; adefinite electric andmagnetic energy content W.drandW..drisascribed toeveryelement ofvolume dr.This constitutes afirst step intheadaptation oftheenergy concept totheideas offield theory. Thefactor1/2inthetwodefining equation (6)evidently pointstoacon-tinuous generation ofenergy, comparable with thestretching ofaspring. Inaccord with thepattern force Xincrease inpath length =intensity entity Xchange inquantity entity, weobtain w.=[E-aD, which, foralinear relationship between EandD,reverts, infact, to(6). ‘Thesituation isslightly different forthemagnetic energy. Here Poynting’s theorem (4)directs ustostart from Wa=[Hba= [Ha (6a) From thepoint ofview ofourgeneral system (B=intensity entity, H= quantity entity) itwould have seemed more reasonable torepresent the energy density notby(6a), butby / W,=[B-dH. (6b) Foralinearrelationship between HandBthisofcourse leadsagainto Eq.(6);foranonlinear variation, ontheother hand, itleads toresult whichdiffersfromfH-dB,andisthereforeincorrectbyPoynting’s theo- rem. From this welearn that work need notbeexpressible intheform intensity entity <change inquantity entity. 28 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 5.6c G.Mie, who takes thesame standpoint throughout inrespect tothe meaning ofBand Haswedo,onp.467ofhisexcellent textbook cited in§2,points tothefollowing mechanical analogue: amoving body carries with it,inunit volume, themomentum (intensity entity) p.For itsac- celeration theforce perunit volume dp/dt isrequired, and hence thework dp .un-ds=dp-v=v-dp; ‘Thisisaproduct ofthetypeH-dB, i.e.quantity entity Xchange inanin- tensity entity.’ Inthis representation themagnetic energy parallels thekinetic energy of mechanics. We shall meet the same correspondence inelectron theory. Also inHelmholtz’s analogy between vortices influids and electric cur- rents themagnetic energy corresponds tothekinetic energy ofthefluid. The same applies foroursemielastic ether model inVol. II,§15. Ifweshould refertoMaxwell inconnection withEqs.(6),wewould find thatinMaxwell’s work thefactor 1/2isreplaced by1/(8x), which from Maxwell has passed over into the major portion oftheliterature. Itevidently lacks thesimple logical basis ofthefactor 1/2and has only historical justification. . Wemust howbelatedly givetheproof thatthequantities H-B andE-D occurring in(4)areidentical with the time rates ofchange ofthe energy densities given by(6). For this purpose wededuce from (6) W,=4E-D+4E-D. (6c) The two terms ontheright areequal, tobegin with, inanisotropic me- dium, where D=eE.They arealso equal inananisotropic crystal, where a“linear vector function” replaces thesimple proportionality (see p.22): Di=Den Es. . ¥ From this wecalculate forthetwoexpressions ontheright side of(6) LER= LeLeah, LD =Diy wk 7 -LEDeak 1Fortheelementary relationship between pandv,i.e. p=Mv,and forconstant masswehaveagaindp-v=p-dv.However, foramassvarying withtime,inpartic- ular, the velocity-dependent mass ofrelativity theory, this isnot the case. Then theform dp-v ofthetextexpresses theenergy change uniquely. 57a CONSERVATION OF ENERGY AND POYNTING VECTOR 29 The twoexpressions areequal toeach other since, irrespective ofthecrys- talsymmetry" fa=xe (6d) Itfollows from (6c) that fortheanisotropic case, asfortheisotropic case, W.=E-D. (6e) The same applies forthemagnetic energy density both fortheisotropic medium (proportionality between Hand B)andforthemagnetic crystal (linear vector function with wa=usi). Here also W.=48-B +4H-B =H-B. (6f) Inview of(5)and (6e,f),(4)yields Wt+divS=-W, W=W.+ Wa @ Inthisform Poynting’s theorem expresses theenergy balance intheelec- tromagnetic field. The Joule heat isrecorded asalossontheright side of theequation. The leftside corresponds totheenergy exchange between the volume element dzinquestion and neighboring elements. This becomes even clearer ifEq.(7)isintegrated over agiven volume; then theapplica- ‘tion ofGauss’s theorem leads to . a]Wart[Sde=- SfWart [sido=-[Wear. (7a) The significance ofSasenergy fluxthrough thesurface ofthevolume con- sidered isnow evident. With theintroduction ofthisconcept Poynting passes beyond Maxwell’s localization oftheenergy. Wenow learn not merely how much energy exists atany place, butalso where itwill goor(fortheopposite sign ofS) from where ithas come. 'This restriction ontheotherwise arbitrary coefficients e.isnecessary inorder that thework done onanelement ofvolume, E-dD, may beacomplete differential. Otherwise theelectric energy density would notbeacharacteristic function ofthe state, aswepostulate forideal solid bodies. (Itistrue that forcertain known crystals hysteresis phenomena occur which make thenotion ofaquantity characteristic of this state illusory). Compare thequite analogous situation inthecase oftheelastic body, Vol. II,p.72and p.288. Intheerystal W,isageneral positive form ofthesecond order intheE;,nota simple sum ofsquares asfortheisotropic case. The notation inthetext asscalar Product isinanyeaseconceptually preferable, particularly sinceitbecomes necessaryforW,,where BandDneed nothave thesame direction even intheisotropic caee. Incontrast with W,and W., Wy,isnot astate function. The condition om=om should hence apply, inthecrystalline conductor, only foraparticular crystal sym- metry. 30 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 58 Intheideal nonconductor theright side of(7)vanishes, sothat (7) takes ontheform ofthehydrodynamic equation ofcontinuity (see Vol. Il,Eq.(5.4)): Wreplaces thehydrodynamic density p,Sreplaces pv. Continuing with thishydrodynamic analogy, wemay saythat even inthe insulator theenergy flows notlikeanincompressible butlikeacompressible fluid. Inaconductor itisabsorbed furthermore, inthemeasure inwhich heat isgenerated inany element ofvolume. Inoptics Splays adominant roleasrayvector; theemission andirradia- tionofagiven surface element doisdistinguished bythepositive andthe negative sign ofS. Weknow from mechanics that thelaw ofconservation ofenergy isnot only offundamental importance physically, butisalsohighly useful mathe- matically asafiretintegral oftheequations ofmotion. Something similar applies forourelectrodynamic lawofconservation ofenergy: From itmay bederived theuniqueness oftheintegration oftheMaxwell equations for tiven initial condition andsuitably prescribed boundary conditions onthe Boundaries oftheregion considered. Asusual, theproof isindirect: Weassume theexistence oftwosolutions, form their difference, anddeduce therefrom acontradiction. Letthetwosolutions beE,,H:andEs,H:(by§4thecorresponding vec- tors D,Bbrethen also known). Weput E=E£,-E, H=H8i- (8) Inview ofthelinearity ofMaxwell’s Eqs. (4.8), Eand Haresolutions as well asEi,H,and E,,H:.Hence Poynting’s theorem, e.g.intheform (7a), applies formally alsoforthem. However, thequantities W,8,W., because oftheir quadratic character, arecomposed notmerely ofthecor- responding quantities oftheindividual fields 1and2,butalsoofmized terms involving 1and2.Weshow thisforthequantity W,asexample, assuming isotropy forthesake ofbrevity. W,=4E-D=5B=5(i—By, O) orexpanded. W.=SEL+5Bi—cEEs. (9a) The lastexpression ontheright isthemixed term mentioned above, while,thefirsttwotermsdenote theelectric energy oftheindividual fields 1and2.However, weshall notneed thisexpanded form andshall refer below totherepresentation in(9).Now, including thequantity W,,and thecase ofanisotropic media inourconsideration, wecansay: The quan- 5.ila CONBERVATION OFENERGY AND POYNTING VECTOR 3h tityWin(7a)represents adefinitely positive quadratic form, formed with thecomponents ofthedifference fieldE,H.Thesame applies forthequan- tityW,.Finally thequantity Sis(irrespective ofthedifference terms arising initscalculation) thevector product EXHformed bythediffer- ence fields. Letthedomain over which (7a) isintegrated becomposed ofpartial domains a,b,... j,... with, ingeneral, different’ material constants e, u,0.Weindicate thisbyreplacing WandW,by));W and)),W,, which, according tothepreceding, can, just liketheindividual W,never become negative. Consider now theterm xf8,do; (10)7 which arises from (7) inthesame manner. Pairs ofterms which refer to thesame inner boundary surface cancel here because forthem theS, areequal andopposite—opposite because oftheopposite direction ofthe normal n,equal because oftheboundary conditions forthetangential components ofthefields E;,E,andH,,H»,from which follows theequality ofthetangential components ofthedifference fields E,Handofthecom- .Ponent ofS:normal totheboundary surface. Thesum(10)becomes, there- fore, simply equal tothesurface integral over theouter boundary ofthe region ofintegration f8.4e. (108) Lettheboundary condition tobeprescribed forthisouter boundary simply consist inthetangential component ofeither theelectric orthe magnetic field being given everywhere onit.Forthedifference field (8) thissignifies that thetangential components ofeither EorofHvanish. Ineither case thevector product Sformed with them and, hence, thein- tegral (10a) vanish also. Now (7a) applied toourcase takes ontheform ay[wan-- Efweds, (uu) or,integrated with respect tot: 'qi Efwal=-[aDf.Wear. (11a) 7 ° 7 Here theright sideislessthan oratmost equal tozero. Theleftsidevan- ishes atthelower limit t=0,since forprescribed initial values ofthefields land 2E=0andH=0in every oneofthedomains j,so.that W=0 also. Attheupper limit ¢,ontheother hand, theleftside of(11a) is,in 32 FUNDAMENTALS OFMAXWELL'S ELECTRODYNAMICS 6.1 view ofthemeaning ofW,certainly notnegative; itsleast value iszero. Only then theinconsistency with theright sideisresolved. Forthisvalue wemust have for all £>0 E=0, H=0, sothat, by(8), E.=E, Hi=F. This proof ofuniqueness satisfies any demand forrigor. Anunrigorous proof may bededuced directly from theform ofEqs. (4.8). Forthese equa- tionspermit thedetermination ofthechange withtimeofEandHif their distribution inspace isknown atany onemoment. This means ina sense: thevalues ofEandHatthetime¢+décanbecalculated from their values atthetime ¢.This calculation isunique since theMaxwell equations arelinear inEand H. Inthepreceding wehave confined ourselves toafinite closed domain. Physically theunlimited domain isofcourse ofgreater interest. Theunique- ness oftheintegration problem canbeproved here forthestatic case as in§10D. Wewillconsider thesignificance ofthePoynting vector forthe unique forntulation oftheproblem ofwaves along wires in§22. *§6.TheRole oftheVelocity ofLight inElectrodynamics Itappears reasonable toeliminate Hfrom Eqs. (4.8) andtoobtain in thismanner,a single vector equation forE.Forthispurpose theoperation curlisapplied tothefirstEq.(4.8), theoperation ud/dt, tothesecond. Adding thetwoequations yields “gE, a euae+opa7 curlcurlE, ()) i.e.,alinear differential equation ofthesecond order infour coordinates of space andtime. Wewillconvert this expression toaform which ismore familiar tothe mathematician. For this weutilize thegeneral transformation (3.10) of Vol. II: curlcurl E=grad divE—AE. (2) Asindicated there, thisequation istobeapplied with caution, since the Laplace operator Acan,byitsdefinition asdivgrad, onlybeapplied to scalar, quantities. Incidentally, (2)may also bederived from thewell- known vector formula ~ AxX(BX C)=B(A-C) —C(A-B) (2a) 66 ROLE OFVELOCITY OFLIGHT INELECTRODYNAMICS 33 bysymbolic calculation with thenabla operator (see thebeginning of §5), where ittakes theform VX (VX E)=WV-E) —(V-V)E. (2b) This isidentical with Eq. (2),term forterm. Weconsider similar vector formulas inProblem 1.4. Equation (1)isvalid inanycoordinates, curvilinear aswell asCartesian. Ontheother hand, Eq. (2),according totheabove, isrestricted tothe Cartesian coordinates x,y,zandthecomponents E.,Ey,E,,since only these may betreated asscalar quantities. With this restriction wefind from (1)and (2) oe oE . &3g+onay=SE—graddivE. (3) This canbefurther simplified ifwespecify that Erepresents asolution foramedium ofuniform dielectric constant andfreeofcharge. Then Eq. (4.4b), with e=const andp=0,becomes div D=ediv E=0.Thus the lastterm ontherighthand sideofEq.(3)vanishes andEq.(3)assumes theform ofthewave equation: oE oE cuoptoa =AE (4) Thesame equation evidently applies, under similar restricting conditions, alsoforH(aswell asforDand B). Thefirstcoefficient in(4)is,ascanberead directly outofEq.(4),the reciprocal square ofavelocity: Correspondingly, wefindfrom Eqs. (4.6a) and (4.7a): -_@_,joules’_gy "Mio gM ~\5) ®) What isthemeaning ofthisvelocity? Maxwell’s answer is:I¢isthevelocity ofpropagation ofelectromagnetic waves, which invacuum isidentical with thatoflight: (coms)?=¢=(29978+0.0002)10°F~310°.) Even atanearly date thevelocity oflight c,then denoted as“critical velocity,” maintained anelusive existence inelectrodynamics, asinthe theorem ofWilhelm Weber andthenumerous measurements oftheratio ofan“electromagnetically” and“electrostatically” determined charge ona condenser (§16D). However theroleof¢inelectrodynamics was first clarified byMaxwell’s theory oflight andHertz’s experiments. 34 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 6.7. Ifwepass from vacuum toanarbitrary electromagnetic medium, the velotity (ex)?appearing in(5)signifies, according toMaxwell, thevelocity oflight (more precisely, the‘(phase velocity ofthelight”) inaponderable body characterized by©and yu (eu)=»,‘=n=refractive index. (2) Itistrue that thestatement (7)hasbynomeans thesame certainty as statement (6).For itdoes notaccount fordispersion phenomena and hence cannot even explain theprismatic colors. Wewilllearn inVol. IV how these are tobefitted into electromagnetic optics. Eq.(6)isevidently asupplementation ofMaxwell’s theory derived from experiment, which establishes arelationship between thetwomaterial constants &,uoofvacuum. Inthefollowing section wewilldiscuss how theconstants aretobedetermined individually. Wenow turn totheintegration ofEq.(4)specialized forvacuum 10Eao7 4E (8) with theayxiliary condition already made useof : divE =0. (8a) Weseek, inparticular, solutions of(8)which areindependent ofyand2. Forpurely periodic time dependence these represent monochromatic plane waves which advance along thez-axis. Weshall show that they areneces- sarily transverse. Inview oftheassumed independence ofyandzofthe function Eq. (8a) reduces to OE. a 0. Equation (8)yields accordingly: oF. _oe7 (8b) E,would thus bealinear function oft,which isinconsistent with the periodic dependence on¢.Hence Z,=0.Thus wealready note adecided advantage ofelectromagnetic optics overtheoldelastic optics. Aswesaw inVol. II,§45, thelatter could never getridofthelongitudinal com- -ponent oftheplane wave: Even ifitwasoriginally absent, areflection or refraction would cause itsappearance along with thetransverse component. Ineontrast tothiswehave proved that theplane wave oftheelectromag- netic theory oflight isnecessarily transverse. Wecandesignate Eq.(8a) asthe condition oftransversality. 6.12 ROLE OFVELOCITY OFLIGHT INELECTRODYNAMICS 35 Ifthewave hasasingle electrical component or,intheusual terminol- ogy,isplane polarized, wecantakeitsdirection ofvibration’ asthey-axis, sothat, inaddition toEZ,=0,also HE,=0.Eq. (8)then becomes 10°F,_HE,So7ie ® The solution which ispurely periodic intime is E,=acos(kt—wt+a). (10) According to(9)thewave number &introduced here and theangular frequency warerelated by . i=6; (10a) interms ofthewave length and theperiod rwehave bat, ga2, (106)» T Omitting thesign Re,denoting “real part of,” weshall write (10) ina form which will prove more convenient forwhat follows: BE,=Ae“! A=ae". (1) This ispermissible aslong aswearedealing with linear relations, such as theMaxwell differential equations; indealing with energetic quantities which arequadratic inthefield components wemust obviously return to real expressions such as(10). Wenext investigate themagnetic component ofthe plane wave. It may bederived from thefirst vector equation (4.8), specialized forvacuum: oH a =curlE. ., aa . Since Z,=E,=0and—=—=0,thisleadsto oy az H.= Hy,=0 and furnishes thefollowing equation forH,: OH,__Ey_agiteietwae z ikAe™**", (12) 1Itshould benoted that wearehere dealing with thedirection ofvibration ofthe electric field, notwith thedirection ofanymaterial displacement. 36 FUNDAMENTALS OFMAXWELL’S8 ELECTRODYNAMICS 6.13 For purely periodic time dependence itsintegration with respect to¢ iscarried out simply bydividing the right side by—iw. Accordingly, oH,=Kygiteiet 1Agtet « c and, inview of(6), H=V2Ad, (13)Ho Thedimension ofthecoefficient (€0/uo)* isthatofareciprocal resistance, ie.2",For, by(4.6a), (4.7a), and(4.5c), eeQe/ses_(e Vi (4)»=jouleM/ QM jouleS, oe (uo/€0)' isdesignated as“wave resistance ofvacuum.” Weshall seein §18D that this quantity actually assumes therole ofaresistance (voltage/ current) inthetelegraph equation. ¥Sr H Ss z ' ‘He . E Fic. 6.The relative orientation ofE,H,and§foraplane wave progressing inthe x-direction. Fig. 6shows theorientation ofEandHrelative toeach other and rela- tive tothePoynting vector Satagiven instant. Inthis sequence they form aright-handed system. With increasing ¢thefigure isdisplaced with the velocity oflight inthedirection ofthepositive z-axis. Itmay notbesuper- fluous topoint outthat Eand Hbecome zero atthesame point and attain their maxima atthesame point. The situation differs from that ofpendu- lum vibrations inmechanics, where theenergy appears inturn initskinetic and initspotential form. For theexperiments ofHertz and many optical experiments airand our vacuum areequivalent. Adistinction between airandvacuum need only bemade inhigh-precision wave-length determinations. We have continually employed the term vacuum inpreference tothe term “ether” (“light ether”), which isfrequently used elsewhere. This negative term appears tohave more significance than thelatter scholastic 6.188 ROLEOFVELOCITY OFLIGHT INELECTRODYNAMICS 37 word, which gives risetofalse notions that cannot bereconciled with the theory ofrelativity. Wecanindicate thematerial constants ofponderable bodies bytheir relative values referred tovacuum instead ofbye,u,setting ©=Crethey = Hreltloe (15) €retaNdpre1then arepure numbers, which ingeneral donotdiffer greatly from 1.Inaponderable nonconductor Eq.(10a) must ofcourse bereplaced by o € RT’ Aa (16) and Eq. (13) by A=V<Actsiet (17) Plane transverse waves arepossible also inanabsorbing medium (¢#0). Thegeneral wave equation (4)issatisfied bytheform (11), forgiven w, bysubjecting &tothecondition generalizing Eq. (16): Bm ews!+iow,k=Vian ev=e+S. (18) e’isthe“complex dielectric constant” frequently employed intheoptics ofabsorbing media. Iftherelaxation time introduced in(4.9a) isemployed, weobtain e io tre7itorlter (18a) IfT,>>7theadded imaginary termofkismerely acorrection term;ifT,«+therealandimaginary partsofkbecome equal(because /7=(1+1)/+/2). Inboth cases thewave isdamped exponentially asitpro- gresses along thepositive z-axis. §7.TheCoulomb Field andtheFundamental Constants ofVacuum. Rational and Conventional Units Onthebasis oftheir time dependence weclassify fields asstatic, station- ary, quasistationary, and rapidly varying fields. Instatic fields notonly field anddensity variations, butalsocurrents of electricity andenergy aretobezero. Hence wedemand B=0, D=0, 4=0, J=0, S=0. According toEqs. (4.4) andthesucceeding equations these conditions are fulfilled ifweset: 38 FUNDAMENTALS OF MAXWELL’S ELECTRODYNAMICS 71 A. Electrostatics curlE=0, divD=pinnonconductors, D=E=0inconductors, qd) #H=0inallcases. B.Magnetostatics curlH =0, divB=0always, buteventually divH =pm,(2) E=0inallcases. Anexplanation ofthe“magnetic density” pqhere introduced willbegiven inconnection with Eq. (9a) below . | Instationary fieldsweretain tieconditions B=0,D=0,»=0,but prescribe current fields Jintheconductors, which according toEq.(4.4c) must befree ofsources. The electric field must still satisfy, both within and outside thecurrents, curl E=0;ontheother hand, curl H=0only outside the currents. Inquasistationary fields weshall determine thefields asinthestationary case, buttake account oftheir time dependence inthefirstapproximation. .The system oftheMaxwell equations isfully utilized only forrapidly varying fields. A. Electrostatics Wedefer allproblems requiring theuseofthetheory offunctions. These arethe boundary-value problems with conductors ornonconductors ofdif- ferent dielectric constant present inthefield. Weshall therefore deal first ofallonly with auniform dielectric, sothat w2may set¢=const. Inthis case wearefaced with asimple summation problem instead ofaboundary- value problem. Eqs. (1)then take thesimpler form curlE=0, @) divE=2. (Ba) Eq. (3)states that Emay betreated asgradient ofascalar potential E=—grad ¥, (4) which evidently brings about asubstantial simplification oftheproblem ofintegration. According to(3a) this potential must satisfy thePoisson equation =?Ayo (4a) 7.6b THE COULOMB FIELD 39 Lamellar field(curlE=0)andpotential field(E=—gradW)havethe same meaning; thesurfaces ¥=const. divide thefield into layers (lamel- jae), towhich thelines offorce areorthogonal. The line integral ofthe field strength 2ffBeds=vi- us (4b) 4 isindependent ofthepath; carried outover any closed path (B=4A)it vanishes. The voltage Vsisidentical with thepotential difference WW. —Vs. The summation problem mentioned above consists intheintegration of Eq. (4a) and canbecarried outdirectly with theaidofGreen’s theorem, forwhich werefer toVol. II,§20, Nr. la.We obtain arev=[Pdr r=ree. 6) Pisthepoint atwhich Wistobecalculated, Qisthepoint ofintegration. The leftside results from theintegration over asmall sphere surrounding ‘thepoint r=,0, Q=P;theintegral over thesphere bounding theregion ofintegration éxternally vanishes provided that thetotal charge enclosed bythis sphere isfinite. Ifthecharge isnotdistributed inspace, but concentrated onasurface oronaline, themathematical method employed in(5)leads to arey=[2de, (5a) » arev=[>as; (6b) wisthesurface density, \thelinedensity (charge perunit length). Afinal step inthis series leads ustothecharge econcéntrated inapoint: pa & adred==, ()E=E,= -F"ie (6a) This istheCoulomb field. Wecould also have read itoffdirectly from Eq. (8a), which, using Gauss’s theorem, wecould integrate over asphere of radius rdescribed about thecharge e.Weobtain then directly ee $Bade=[Barn’. (6b) _Inview ofthespherical symmetry wemust putE,=E,=const. onthe left,whereupon (6b) becomes infactidentical with (6a). 40 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS V7 The Coulomb force F,with which.two equal charges ¢atthedistance rrepel each other, follows from (6)according toourdefinition offield strength: é FeF,= ch,=7 ” Ithasbeen customary inthepasttowrite instead, forvacuum conditions, 2 F=h5 with f=. (8) Here Fissupposed tobemeasured indynes, rincentimeters. However, inthis manner thewhole structure ofoursystem ofdimensions iscast aside; wepassfrom ourformer system ofdimensions tothesocalled elec- trostatic' system ofcgsunits. Thecharge ¢would then, according to(8), take onthe unnatural and ungainly dimensions e=Vayaemt=om"gisec (8a) Furthermore thecharge ewould be1iftwoequal charges ¢atadistance of1cmwould repel each other inairwith aforce of1dyne. .Wemustteject, ondimensional grounds, Hertz’s distinction between “true charge density” expressed bydivDand“free charge density” ex pressed bydivE.Weshalldesignate thelatterquantity correctly as“diver- gence ofthelinesofforce”; inthepreceding wehaveavoided itbywriting p/e forit. B.Magnetostatics Although inmost treatments theanalogy between electrostatics and magnetostatics isemphasized, ourapproach compels ustopoint out clearly thedifferences aswell. Aswesawin(2),itisnottheintensity B,butthequantity Hthat is lamellar. Weagain denote thecorresponding scalar potential by¥,dis- tinguishing where necessary between ¥,and¥,,,andfind H=—grad ¥. (9) Weshall callthequantity pmdefined above inEq.(2)simply “magnetic density”; inview ofthecorrespondence ofHandDwecanregard itasthe direct analog oftheelectric density p. How canwereconcile itsexistence, i.e.theEq.divH»0with theuni- versally valid Eq.divB=0?Thisisonlypossible atpoints oflocalvaria- 1The term electrical system seems toustobepreferable inprinciple tothecus- tomary term electrostatic system, since iteapplication isnotlimited toequilibrium conditions, butmayalsobeextended toelectrodynamic processes. See§16D. 711 THE COULOMB FIELD 4l tionofpermeability, asisshown bythefollowing lines: divB=pdivH+H-grady=0 (0a) pm=divH=H-grad log(uo/n). Wewillgiveaphysical interpretation ofthisrather formal explanation oftheconcept ofmagnetic quantity anddensity byintroducing the“mag- netization” Min§12.Wehave noreason foradistinction between “true” and“free” magnetism, such aswasalsogiven byHertz. For,since we havealready interpreted divHasmagnetic density, thedifferently dimen- sioned quantity divBisnot«magnetic density. Furthermore itisevery- where equal tozero. Wenowreturn to(9)andform thedivergence ofthevectors ontheleft andtheright. Wethenobtain thePoisson equation ofmagnetostatics, i.e. AY=—pr (9b) Ifpmisgiven throughout thisisintegrated, inanalogy to(5),by Pn arv=|&ar, (10) or,inanalogy, to(5a), forgiven surface charge wm,by ; ary=[2ao. (108) Ifthedensity isconcentrated onapoint poleandifwecall p=fPmdr thepolestrength (anopfiosite poleisimagined tolieatinfinity), (10) leads to ary=?, (10b) :=-*. 2, H=H,or7ie, (100) ThisistheCoulomb fieldoftheisolated magnetic pole.TheCoulomb force, withwhich twopoles ofequal magnitude andthesame signrepeleachother, is,however, notpH,but,according toourdefinition oftheintensity B 2 : =F,=pB,= =Pe F=F,=pB,=pu, _ ay ‘Thefactthat4appears hereinthenumerator, although e,in(7)appears inthedenominator, results evidently from theinconsistency, pointed out 42 FUNDAMENTALS OF MAXWELL’S ELECTRODYNAMICS 7.12 inconnection with Eq. (4.7), intheintroduction of4ascompared with that ofe.(We would have liked todefine thereciprocal of4asthemagnetic constant atthat point.) In(11) »evidently signifies the permeability of thesurroundings ofthemagnetic pole p;forair(vacuum) weput»=po. Just asinconnection with (8)wetook cognizance ofanelectrostatic system ofunits and aunit ofcharge corresponding tothissystem, sowecan introduce, onthebasis of(11), amagnetic system ofunits and acorre- sponding unit ofpole strength. Tothis end (11) isreplaced, forvacuum inparticular, (wefollow thepattern ofEq.(7)andwhat follows literally) by:ry eFeIe (12) andfisputequalto1;Fissupposed tobemeasured indynes, rincenti- meters. Our former-system ofunits isonce more cast aside, and wepass over totheGaussian magnetic ogssystem.’ Inthissystem thepolestrength phas,according to(12), thesame unsatisfactory dimension asthecharge ¢ intheelectrical system (8a). Unity pole strength would correspond toa repulsion with aforce of1dyne oftwopoles ofequal sign andmagnitude separated by1cm(inairassurrounding medium). C.Rational and Conventional Units We must now deal with the factor 4inCoulomb’s law. Itistrue that thisismuch lessfundamental than thequestion ofdimensions and bears tothelatter only ahistorical relationship Historically theforms (8)and (12) ofCoulomb’s lawresult from aneffort toapproach asclosely aspos- sible thecustomary form ofNewton’s law. Weshall denote thesuppression ofthenumerical factor 4xinCoulomb’s law asconventional, ourretention ofitasrational. Itisinfact evident that inaproblem with spherical sym- metry, such asthe.Coulomb problem, thefactor 4xisappropriate (this follows inparticular from ourargument in(6b)). Ifwewish toavoid this factor, wemust rewrite Poisson’s equation (4a) aswell asthesecond of Egg. (1)asfollows: ay=42, divD=dap. (13) The factor 4xwould thus beimproperly introduced into thefundamental equations oftheMaxwell theory. Furthermore, thetransparent expression (5.6) fortheenergy density would bedistorted into W.=LED. 14) 1The fact that Gauss employed mm instead ofcmasunit oflength isasuperficial distinction. 7.16 THE COULOMB FIELD 43 Heaviside fought alife-long battle fortherational units. Inthiscon- nection hepointed alsototheexpression forthecapacity ofacondenser (fordetails see§10,where therelationship withtheexpression fortheenergy density isalsoindicated): Theplate condenser (area F,plate separation a) has,inrational andconventional units respectively, thecapacity Fe FeK-=7 and (15) thespherical condenser (radius ofsphere r,outer sphere imagined atin- finity), the capacity K=4ner and er. (15a) Weseethat, with rational units, thefactor 4xappears forthesphere, where itbelongs; withconventional units itismissing forthesphere and appears fortheplane condenser, where itdoes notbelong. Heaviside makes thefollowing striking comparison: Inpassing from the measurement ofdistance tothemeasurement ofarea onemight define as unit ofarea thearea ofacircle ofradius 1.This would belogically possible. Itwould however lead tothestrange result that asquare with theside 1 would havethearea1/x.Everyone would thensaythatxwasatthewrongplace. Wesaidthesame ofthefactor 4intheformulas totheright in (15) and (15a). D.Final Determination oftheFundamental Constants &,uointheMKSQ System Theviewpoint oftherational units together with therequirement of meaningful dimensions andadaptation tothelegal units leads toaquite definite choice ofthefundamental constant 0ofvacuum. Forwecanob- tain agreement between Eq.(11), which isdimensionally correct inour sense, and Eq. (12) byrequiring ‘to|jouleS*_[|SoSP=(7oe a) [f]isthenumerical value offinthecgs-system, which wewished toset equal to1.Thebrackets ontheleftarethenumerical value ofthequan- tityuo/(4m) inourMKSQ-system; itsdimension (seee.g.(4.7a)) isindi- cated. The conversion ofthese dimensions into the cgs-system follows from Q=1Coulomb =yycgs, M=10°cm, joule =10’erg. Accordingly joule8’_jg 1-3 =10 .eM a 44 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 7.168 With [f]=1wethus obtain, after cancelling thedimensional factor cgs onboth sides of(16) Ho) 197[2]-0 We obtain hence, entering the dimensions: = -1jouleS?_ 7OS bo=4r10"“Oa=4x10FF (16a) Regarding theunit ofresistance, 9=“ohm,” here employed, seeEq. (4.5d). We have thus determined one ofthe two fundamental constants of vacuum insuch amanner that the demands stated above are satisfied. Thenumerical value ofywsoobtained, which isaccurate toanarbitrary number ofdigits, shows clearly that ourdetermination isnot established bydirect measurements, but byourchoice oftheunit ofQand isequiv- alent tothelatter. Theother fundamental constant eqofvacuum follows then from therelation (6.6) which issupported bythesum total oftheHertzian experiments: 1 _10°M =—=— =. 1 f=[eetTeckOS an Ifwesubstitute forctheapproximate value c=3-10° M/S wefind 10° $ ©~36aM” (18) Wecan also write (17) intheform Arce=107Zz. (18a) Division of(16a) by(18) andtaking thesquare root leads tothefollowing value forthe “wave resistance ofvacuum,” introduced with (6.14), in terms ofthe unit Q: "2=12002&3770. (19)0 Inthepreceding wehave disregarded thesmall differences between the velocity oflight ¢anditsapproximate value 3-10° M/S, aswellasthe difference between the “international” and the “absolute,” i.e., the ideal, ohm. These differences, which concern only thehigher decimals, areof vourse ofgreat importance inprecision measurements and have called forth, inthedetermination ofthe@inrelation totheoldSiemens unit,’ Resistance ofamercury thread 1mlong and 1mmt incross section at0°C= 0.937 a. . 7 ‘THE COULOMB FIELD 45 thecompetitive efforts ofthebest experimenters (Kirchhoff, Lord Ray- leigh, F.Kohlrausch ete.). They play norole, however, inthegeneral theory. Insummary: Our form oftheMaxwell equations isadapted tothera- tional choice oftheunits MKSQ, with thevalue (16a) foruotaken over from theconventional Gaussian magnetic units. Below weshall usethenumerical values (16a), (18), (19) only inspecific numerical computations and not. introduce them, asoften happens inengineering literature, into thegeneral theory. Instead weshall always take account ofthedimensions ofallquan- tities, also those ofe,yo,and thus make ourselves independent ofthe particular choice ofQ=1Coulomb. §8.Four, Five, orThree Fundamental Units? A.Supplementary Note onOur System ofFour Units Our four units MKSQ aresimply intended totranslate Giorgi’s idea (introduction ofaseparate electrical unit) into aform which isparticularly convenient forthetheory. Itisbasically indifferent whether theunit of charge Qisemployed orastandard resistance R,asGiorgi hasoccasionally advocated, forreasons ofconvenience ofmeasurement. Inview ofthere- lationship Q=ampere-second wewould ofcourse alsobecontent with theampere asfourth unit. Wetakelesskindly tothedesignation ofGiorgi’s system bytheunits MKSVA. Inview of VA =watt =joule/sec these units arenot independent ofeach other. We can well understand that thelong-employed quantities VandAappear more convenient inuse than ourunit ofcharge Q.Nevertheless, ofthetwo dimensions E=aa andE=Yor thefirst appears tobethemore natural one. Kalantaroff’s system ofthe four units MSQO (magnetic flux) isselfconsistent, but seems, bythe elimination oftheunit ofmass, somewhat tooartificial forgeneral usein physics. Itistobewelcomed, from our point ofview, that, byinternational agreement, separate designations gauss and oersted have been introduced forthetwomagnetic vectors BandH.Historically, thename gauss also seems proper forB,since Gauss’ methods ofdétermining magnetic mo- Ment rest onmeasurements offorce and hence refer toBand nottoH. Theunhappy term “magnetic field” forHshould beavoided asfaraspos- sible. Itseems tousthat this term hasledinto error none less than Maxwell himself, who, inart.625oftheTreatise puts theforce exerted bythefield onamagnetic pole mequal tomH. . 46 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 8.1 Wehave repeatedly stressed asanadvantage ofoursystem that it avoids theannoying powers oftenofthecgs-system. Thisapplies tothe electrical aswell astothemechanical quantities. The converse istrue, however, forthemagnetic quantities. The unit gauss ofthemagnetic induction Binthecgs-system is,bydefinition, equaltotheunitinthecgs- system. Hence, transferred tooursystem ofunits, itacquires apower of ten. We determine the latter asfollows: Let[B]bethenumerical magnitude ofagivenfieldinoursystem ofunits, sothat newton S_ joule S B=[B)om ={B]OF qa) Wesubstitute again joule=10'erg,©M=10cm,©Q=Jogg-units If,inparticular, weset[B]=1,wefindfrom (1)ascorresponding value ofthis quantity Bincgs-units: 10° ; “ BeljgSunits =10*gauss, (2) and conversely =jotiouleS _19+V8 1gauss=107“Ore=10 (3) Itmay bementioned infavor ofthischoice ofunitthat thegauss isin- conveniently small forpractical purposes, sothatnotonlyinengineering, buteven inpure physics (except forterrestrial magnetism) thekilogauss must generally beemployed (e.g. intheZeeman effect). Hence our10,000 times greater unit istobepreferred inpractice. Inordertoexpress theoersted inoursystem ofunitsaswell,weproceed from therelationship between Hand B: w=8, (4) Mo WenowsetH=1oersted, B=1gauss,sothatby(3) =19-+joule S ;B= 104 and by(7.16a) =de.1977uleS* yo=4-10‘OM 8.6 FOUR, FIVE, ORTHREE FUNDAMENTAL UNITS? AT We then obtain from (4) =jotjoule8/“1977ioule8°. 1oersted =10OM 4n-10 GM (6) sothat =i &=1ig Loersted =710°ay=Ge10° (a) Haspna=44-10"Horses: (eb) B.The Five Units MKSQP Itmay bestated generally: Adimensional analysis willbemore suc- cessful’ inthedegree inwhich more independent units areatitsdisposal. Ourfour units arehence more informative than thethree units ofthe“‘ab- solute” system, inwhich thedimensional character ofthefundamental electromagnetic vectors isobscured. The fiveindependent units consid- ered below areofeven greater value from ageneral theoretical point of view. _Weintroduced themagnetic polestrength Pasadimension in§2,but expressed itimmediately in(2.7) interms ofthecharge Q,inaccord with Ampbre’s hypothesis. Isthishypothesis binding eventoday, after thedis- covery oftheneutron, anuclear particle asbasic anduniversal asthepro- ton? The neutron hasamagnetic moment which isnotassociated with anycharge, unlike theelectron andproton which, though endowed with equal charge ofopposite sign, have magnetic moments ofentirely diferent magnitude. Certainly anattempt toabandon Ampére’s hypothesis andto introduce Pasindependent fifthdimension isjustified andinstructive. We shall, forthepresent, refrain from fixing themagnitude ofP. Wewrite down thefollowing sets ofdimensional relations, which now show acomplete correspondence: newton newtonBo Boe . Q P D-ap H=ap . ©)e-P=_V_ LHL PTE joule M nu B jouleM Ep=newton _joule [HT newton _jouleED=—“e ~MF BH= yp "MF* 15.Fues, Z.Phys. 107,662,1937, indicates anupper limit totheuseful number of dimensions. 48 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 8.7 Theentries forE,D,B,andHareidentical with theoriginal formulas in (2.1) to(2.9.) Inaccord with thenote accompanying Eq.(4.7) wehave entered thereciprocal of4asanalog of¢inourtable.‘Thelastline,which isindependent ofQandP.hasthedimension of energy density. Ontheother hand the(scalar orvector) product ofEand Hhasadimension which depends onPand Q: Pnewton _P§joule TH-OM "OMMS’ ” Thelastfactor ofthelastexpression hasthedimension ofenergy flux (radiation vector). Letthefactor multiplying itbe1/TWethusput =9MT-=59 (8) and write (7)intheform _joule TEH=ps: (8a) This dimensional equation suggests that theenergy fluxSisnowtobe defined asTEXH.Wefurthermore compute theproduct exfrom (6)and +find -¢%_ (8y =m =ae): (9) Wesuspect fromthisthatthevelocity oflight¢isnolonger given by (cone)*,butbyT(caue)*.‘The same factor Ioccurs now also inMaxwell’s equations. Weassert that these should bewritten: B=-relE, D+J= Toul. (10) If,asin§5,weproceed toPoynting’s theorem (scalar multiplication ofthe firstequation withH,ofthesecond equation withE),weobtain HB+ED+EJ+IrdivE xXH=0; withtheformer definitions oftheehergy densities andoftheJouleheatin(5.6) and(5.5) andwith thedefinition oftheenergy fluxsuggested by (8a)thisexpresses thelawofconservation ofenergy: Wa+W.+ W,+divS=0. (11) If,ontheother hand, justasin§6,weintegrate Eq.(10)forthecaseof theplane wave invacuum propagated inthex-direction, weobtain the wave equation intheform ewe£=—F*curlcurlE=I*AE. (12) 8.138 FOUR, FIVE, ORTHREE FUNDAMENTAL UNITS? 49 Since thisissupposed torepresent aprocess with thevelocity otpropaga- tion c,our expectation suggested by(9)isconfirmed: y — Or Vau=° Veom =5: (12a) The general form (10) oftheMaxwell equations isnotnew. Itwas in- troduced byEmil Cohn, thefriend and fellow student ofHeinrich Hertz, andforms thebasis ofhisimportant book' “Das elektromagnetische Feld.” Wehave avoided Cohn’s notation V,taking theplace ofourI’,since we have otherwise disposed ofV.Itistrue that Cohn does notwork outthe relationship ofthis constant with ourunit Pofpole strength, nordoes he place dimensional considerations intheforeground ashasbeen done here. Students ofCohn, in,particular J.Zenneck, have used Cohn’s system by preference. : H.A.Lorentz clearly recognized theadvantages ofCohn’s standpoint when, in1902, hewrote histwo great articles onMaxwell’s theory and electron theory fortheEnzyklopadie dermathematischen Wissenschaften. Hewrote: “Cohn’s system hastheadvantage ofeasy transition toother systems, byspecific choice ofthevalues ofV,eo,and yo.Eventual later advances in‘the understanding ofthephenomena could beutilized forthe ultimate determination ofthe units. On the other hand we could not bring ourselves tointroduce indeterminate quantities intotheformulas which arecomplex tobegin with.” The “eventual later advances intheunderstanding ofthephenomena” contemplated byLorentz canonly beexpected when wehave atheory of theelementary particles which now constitutes thegreatest problem onthe program ofatomic physics; thiswould have toexplain notonly themag- netic moments, butalso thepossible masses and charges oftheelemen- tary particles. However, wecan even now benefit bytheflexibility of Cohn’s system. C.The Gaussian System ofOnly Three Units Weevidently return tooursystem with thefour units MKSQ and our former form (4.4) ofMaxwell’s equations ifweset -Tel. (13) Then Phas,according to(8),thedimension P=QXvelocity, (18a) 'First Edition, Leipzig 1900, Second Edition, 1927. *Vol. V,second part, p.87. 50 FUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 8.14 inagreement with Ampére’s hypothesis in(2.7). Furthermore, our spe- cific choice of4pand &in(7.16a) and (7.17) isevidently consistent with Eq. (12a) forthese values ofIand P. Weobtain another, also very simple, form ofMaxwell’s equations ifwe set Tec (14) Then, by(12a), theproduct gous:must beapure number. Itistempting tomake &andjp»separately pure numbers and toset w= 1, a=. (14a) Inthis manner wepass over totheGaussian system ofunits. Inview of (14)theMaxwell equations then become (weconfine ourselves firstto nonconductors) : le=-omz, d=owe (15) Inview of(8)Pand Qhave now thesame dimension. Hence, byTable (6),thedimensions ofEandB,aswellasthoseofDandH,alsobecome «mutually idéntical. (The same follows also from theform ofEqs. (15).) Furthermore thedimensions ofthetwo pairs become thesame, since now eand p,justase9andjz»in(14a), become purenumbers, equaltothepure numbers €re1and pre: introduced in(6.15). The Gaussian system obscures thedimensional character ofthefourfundamental vectors E,D,B,Hcom- pletely, while Cohn’s system expresses itmost clearly. The two Coulomb laws (7.8) and (7.12) were written intheconventional form (with thefactor 4xsuppressed). This hastheresult that the4do notoccur intheMaxwell Eqs. (15) fornonconductors butarise once more intheir integration. For, taking thedivergence and integrating Eqs. (15) with respect to¢leads to: divB=const., divD=const. The first constant is,ofcourse, equal tozero; thesecond must now not besetequal top,butequal to4xp: divD=4np, (15a) inorder that thefactors 4xcancel each other onthetwo sides oftheequa- tionifitisapplied toapointcharge¢=fodrandisintegratedovera sphere about ¢.Only inthismanner isthefieldstrength E,=e/(er"), obtained corresponding totheconventional form oftheCoulomb force F.From this follows also astheexpression forthecorresponding electro- atatic potential Y=¢/(er), unlike Eq.(7.6), where theappropriate factor 3.17¢ FOUR, FIVE, ORTHREE FUNDAMENTAL UNITS? 51 {appears ontheleft.Hence also(7.5) and(7.4a) must nowbereplaced aythelessappropriate expressions ev=[2dr av=—dap/e. (156) The same follows from theform (7.12) ofCoulomb’s lawforthemagnetic jensity p»and themagnetic potential V..: divH=4r~m, Yan[Pdr Ava=—44om. (150) Wenowextend Eq.(15)tothecaseofaconductor. Here wemust tem- yorarily multiply theconduction current J,which istobeadded toD, vith @numerical factor ywhich weshall determine inamoment. Hence wewrite inplace of(15) p=-one, 1}+o)=ne (16) Taking thedivergence ofthesecond ofthese equations, asin(15a), and itilizing thedefinition ofpgiven there leads to 4rE+divJ=0. (168) Wemust sety=4xinorder thatthisequation may express theconserva- tionofcharge, orinother words, theabsence ofsources ofthetotal current C;only then does (16a) become theanalogue (4.4c) or(3.6b) ofthehydro- dynamic equation ofcontinuity. Having entered thisvalue ofyin(16), weseek theexpression forthe Poynting theorem bytheprocedure followed atthebeginning of§5.Mul- tiplying thetwoequations (16)scalarly withHandErespectively, and utilizing thetransformation (5.2) weobtain twB+lED+2EJ+dvEXH=0. an Wecompare thiswith theearlier form (5.7) ofthesame theorem: Wat+W.+divS =—W. (17a) Since wecannot disturb Ohm’s lawW,isstill given bytheproduct E-J. Wemust hence divide (17) by4x/c inorder that (17)may correspond with (17a). Then acomparison oftheterms of(17)and(17a) leads to . 5 1 5 1We=Za8, We=ZF, (17b) S=£EXH,andW,=E-J=oB'asbefore. (17) 52 FUNDAMENTALS OF MAXWELL’S ELECTRODYNAMICS 8.17d Integration of(17b) with respect to¢,asonp.27,yields (for isotropic and anisotropic media): 1 1 Wn=wHB We= ED; (17d) They express thelocalization ofenergy inconventional units. Already in connection with Eq. (7.14) wepointed out the unsuitable form ofthe denominator 82, ascompared with the denominator 2inour rational notation (5.6). The same applies forthefactor c/(4m) inthepresent: ex- pression (17c) forthe energy flux. Even inthe Maxwell equations (16) thesuppression of41,carried outatthewrong place, avenges itself: These equations, intheform appropriate forboth conductors andnonconductors, become: tpe-omz, 1p+*y- ane (18) with the supplementary conditions, applying specifically for isotropic media ‘D=ecE, B=.H, J=o. (18a) Wehope thatbythissummary wehave facilitated forthereader the laborious transition between vurtwo systems ofunits MKSQ (rational) =cgs(Gauss, conventional) asfaraspossible. Wehave discussed thehistorical source ofthis annoy- ance attheend of§7.Itisunavoidable inview ofthepresent status of thequestion ofunits inelectrical engineering, experimental physics, and theoretical physics. The following remarks may serve toclarify thesitua- tion. H.A.Lorentz, when writing hisarticles fortheEnzyklopddie in1902, like Hertz, utilized the Gaussian system, postulating: Electrical quan- tities (including theelectric current) aremeasured electrically (electrostat- ically), magnetic quantities, magnetically. Contrary tohisoriginal inten- tion hedecided, inthecourse ofcomposing thearticles, toconvert the Gaussian system (unlike Gauss and Hertz) into rational units. Inthis manner the theoretical relationships became clearer and the 4x’s were eliminated from the Maxwell equations. Lorentz setforvacuum &%= wo=1,asinourEgg. (14a). Inorder toretain therational form ofthe Coulomb force law hethen had tointroduce the4x’s appearing initinto thedefinition oftheunit charge andtheunit polestrength, respectively. This somewhat artificial conversion ofunits' hasnotfound wide accept- ance, inspite oftheauthority ofLorentz. 1Seetable onp.87ofVol. 5,part 2,oftheEnzyklopadie. 8 FOUR, FIVE, ORTHREE FUNDAMENTAL UNITS? 52 Wehave here—also against ouroriginal intention—arrived atthede- cision towrite theGaussian system, insofar asweshall useit,inconven- tionol units. Thereason isthefollowing: Since theyear 1902 atomic physics hascome tobethemost important branch ofourscience. Itdeals always with conventional units, e.g.with theelectron charge e=4.80-10” (electrostatic egs-units) andwiththeelectric potential, e.g.inthehydro- genatom,¥=e/r(not¥=e/(4xr)). Weconsider itinadvisable toover-turnthiswhole formalism anew bypassing over totherational form ofthe Gaussian system oreven tooursystem offour units. Ontheother hand Giorgi’s system oftheunits MKSQ, freed of4x’s, ismost suitable forthemacrophysical problems ofthislecture. Weare hereinagreement with theinternational conventions, with thepractice ofengineering, and,inparticular, withthetextbooks ofMie(quoted on p.10)andPohl.’ Weregard thedogma ofthescientific superiority ofthe threepurely mechanical units em,g,sec,which forexample issupported * inKohlrausch, Praktische Physik, asoutmoded. D.Supplement Regarding Other Systems ofUnits Intherestriction tothese twosystems ofunits, theGaussian system in conventional form andtheMKSQ system, wefollow thepractice ofthe excellent textbook ofJoos.” TheGaussian system (whether inrational or conventional form) isamized system, consisting ofelectrical (electro- static) andmagnetic cgs-units. There arehowever, asiswell known, alsoapurely electrical andapurely magnetic system ofunits, ofwhich the latter isparticularly important, since thelegal units volt, ampere, ohm, ete. are based onit. Thereason forsetting upaseparate electrical system ofunits rests on& certain quantitative difference between electrostatics andelectrokinetics: Electrostatics deals with large voltages andsmall quantities ofelectricity, electrokinetics, with moderate voltages andlarge quantities ofelectricity. Togiveacomparison from hydrodynamics, theelectric spark ofacon- denser discharge corresponds toawaterfall (great height, small quantity), theelectric current, toariver (small grade, great flow), asisindicated in Fig.7.Thus, forélectrostatics, asmall unitofcharge andalarge unitof fieldstrength aresuitable. Theelectrostatic system based ontheelectrical Coulomb lawprovides suchunits. Interms ofthissmall unitofcharge thecharge oftheelectron (seeabove) hastherelatively large value 4.80- 10~”ogs-unit. Theunitofcharge intheelectromagnetic system (equal to 10coulombs), 6ntheother hand, islarger bythefactor c;interms ofit, thecharge oftheelectron appears smaller byafactor c,i.e.equal to1.60- 10cgs=1.60-10~" coulomb (seep.43).Ontheother hand, theunit offieldstrength intheelectromagnetic system, according tothedefinition ‘BR.W.Pohl, Elektrisitatslehre, Springer. 8thand9thEdition, 1943. 4G,Joos, Theoretical Physics, 2ndEd.,G.E.Stechert andCo.,New York, 1950. 54 YUNDAMENTALS OFMAXWELL’S ELECTRODYNAMICS 8 ofthevolt, is10~*volts/cm; thatintheelectrostatic system isctimes as large, or300volts/em. Weshall return tothisin§16D. Wehave frightened generations ofstudents with these two setsof values forcharge andfield strength (their number would beincreased to4if,inaddition totheusual conventional units, rational units would also beconsidered). Itis,inouropinion, aspecial advantage oftheintro- duction ofourfourth unit ofcharge, Q,which isindependent ofallother units, that weneed deal only with quite definite charges, expressed as multiples ofQ. Voltage Voltage |Largevoltage‘Small‘Largecurrent *mallcurrent CurrentRiver,stationary current Waterfall, condenserdischarge Fro.7.Thehydrodynamic representation ofastationary electric current andof ‘acondenser discharge. Wequote finally aninformative analog tothedouble (electrostatic and electromagnetic) measure ofcharge which, likesomany other clari- fications inthequestion ofunits, weowetoJ.Wallot:' Suppose thatsome- one had decided todescribe mechanical processes interms ofonly two independent units, cmandsec.Heeliminates thegram asunit bysetting either thedensity éorthemodulus ofelasticity Zofsome standard mate- rialsuch ascopper arbitrarily equal to1.Hecanthen express themass m ofsgiven copper rodintwo ways, either byameasurement ofvolume according totheformula 3=,which,becauseof5=1,leadsto:m=m=V orbyavibration experiment with longitudinal waves according tothe formula c=g=BY’,which,becauseofE=1,leadsto:m=m=V/c’. Ifhenow divides oneofthetwo values ofmsofound bytheother heob- tains—perhaps tohissurprise—the square ofthevelocity ofpropagation ¢ofelastic waves incopper. The analogy toelectrodynamics isstriking and requires nofurther explanation. 1J,Wallot, Elektrotechn. Z.,Vol. 43,Nr.44(1922), section 28ofthepaper “Phys- icalandEngineering Units.” The latest relevant publication isPhys, Z,44,p.17, 1943. Parr II DERIVATION OFTHE PHENOMENA FROM THE MAXWELL EQUATIONS $9.TheSimplest Boundary-Value Problems ofElectrostatics Wehavesetupthefundamental equations ofelectrostatics inthebegin- ningof§7andhave dealt with theresulting summation problem forauni- form médium inEq.(7.5). Wenowturn totheboundary-value problems arising from thepresence ofconductors ornonconductors ofdifferent dielectric constant. Wethink of-thesimplest electrostatic experiments: Letametallic conductor ofarbitrary shape, originally insulated, A,beconnected to asource ofpotential V(with respect toground) orB,begiven aknown charge (e.g. byapiezoquartz, seep.78). Wewish toknow thefield out- sideoftheconductor. Wedescribe thisfield bythepotential ¥associated withthefieldstrength E=—grad W.Let¥besetequal tozeroatinfinity inbothcases; AandB.Inboth cases AY=0outside oftheconductor; onthesurface, aswell asintheinterior oftheconductor, wehave Y=Vx =const. A.Charging Problems IncaseA,¥,=Visgiven,incaseB,¥,,mustbefound. According to (3.12) thesurface charge density atanypoint doofthesurface ofLis given by =—-(% o=Dy=eB,=~e(2%). a ¢isthedielectric constant outside oftheconductor, ntheoutward normal tothesurface ofL.According to(1)thetotalcharge onLis g=fode=-ef Xa, (2) IncaseA,gissought: incaseB,whereqisgiven, (2)determines ¥,. Forthecase ofasphere, ofradius a,theappropriate solution ofthe differential equation AY=0may bewritten down immediately, inthe form + vely, @) r 55 56 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 9.38 ThisyieldsforA,since¥,=V,~ avecv. (Ba) Incase B,(2)and (3)lead to =dratyy,= =! q=dra’ Y= are=2 (8b) The charge g,uniformly distributed over thespherical surface, thus acts atadistance likeapoint charge concentrated atthecenter. Itisalso possible toguess thefield ofaconductor oftheshape ofa prolate spheroid (ellipsoid ofrevolution with long axis asaxis ofsym- metry). Forthisitismerely necessary tostretch, sotospeak, thecenter of thesphere, which appears inthelastEq.(3b)aslocus ofthetotal charge, into theconnecting line ofthetwo focal points ofthegenerating ellipse and todistribute thecharge quniformly over this line. Ifwecallthe distance ofthetwo focal points from the center oftheellipsoid c,the linearly distributed charge density becomes g/(2c) and itspotential be- comes, byEq. (7.5b): +e a =i ——— feyaeVityt @—9) =Liogitet+veFPTEFY2° 2-et+Ve+¥ +&—oF Ini.volem II.1 wewill show that this expression assumes aconstant value ¥=W,oneach oftheconfocal ellipsoids pertaining tothegiven separation offociandthat therefore itsolves ourpotential problem for each oneofthese ellipsoids. Since only theseparation cofthefocal points occurs in(4), hisformula applies forallconfocal ellipsoids ofthefamily, inthesense that alloftheequipotential surfaces oftheconfocal ellipsoids with thesemiaxes a.>a,,b:>6,areincluded among. theequipotential surfaces oftheellipsoid with thesemiaxes a;,b;.The ellipso'd a=¢, b=0,which degenerates toastraight lineoflength 2c,also belongs to thisfamily. InII.2thelimiting case ofaparaboloid ofrevolution, andits degeneration, thefield ofasemi-infinite glass roduniformly charged by friction, arestudied from this point ofview. B.Induction Problems and Method ofReciprocal Radit The ‘induction problem,” which weshall specialize toaninducing point charge, ismore complex than the‘charging problem” treated thus far. Here also wecandistinguish between two cases: A.The (otherwise arbi- trarily shaped) conductor isgrounded andB.,itisinsulated. Thegenerally 98 SIMPLEST BOUNDARY-VALUE PROBLEMS 57 accepted meaning of“grounding” isaconducting connection with an infinitely distant surface atthepotential ¥=0.“Insulation” signifies, foranoriginally uncharged conductor, thateven after induction thetotal charge qcontinues tobezero. Problem Aissolved byGreen’s function G(P,Q)—more exactly, ““Green’s function ofthepotential equation fortheexterior oftheconductor L.” Qsthe“source point,” which willbeassumed torepresent a“unit source,” P,the“reference point.” @isdefined bythefollowing conditions: AG=0forallP#Qoutside ofL, G@—1/(4xrrq) forP—Q(definition ofunitsource), ©)G=0onthesurface ofL, G0 for P> @. Green’s function plays acentral rolenotonly inpotential theory, but generally inthetheory oflineardifferential equations, tobetreated inVol. VI.Here weshall merely point outitssignificance forourspecial problem. IfQrepresents theposition oftheinducing charge e,thesolution ofproblem Ais given by @ V(P,Q)=BoP, 0] ©) and that ofBby e H(P,Q) =GP, @)+a; (62) here¥isthesolution ofour“charging problem” Aforthesameconductor L,a,aparameter which, according toEq.(2),isdetermined bythecon- dition a¥(P,) 7[Ae aw=o. (6b) Asspecial case weconsider once more asphere ofradius a.ItsGreen’s function canbewritten down inclosed form bytheingenious method of theyoung William Thomson, later Lord Kelvin, which willbetreated in detail inVol. VI,§23. With Qassource point and Q’as“electrical image ofQwith reference tothesphere ofradius a”this solution is: 1al 4nG(P,Q) =—--—. (PQ)=ore (7) pe=-0Qistheseparation ofthesource point Qfrom thecenter ofthe sphere, p’=0Q’, that ofitsimage point Q’.They arerelated bythecon- tion of“reciprocal radii’: pe’=a, (8) 58 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 9.8a from which Thomson’s procedure hasbeen given thename “method of reciprocal radii.” Itisimmediately evident thatourformula (7)satisfies thefirst, second, andlastcondition (5);thefulfilment ofthethird condi- tion (5)canbedemonstrated byelementary geometry. According to(6),Eq.(7)yields forinduction onthegrounded sphere: , 4e¥(P,Q == -Sse Se (8a) req Tra’ P andfortheinsulated sphere, by(6a,b)and (3b): ,, 4ne¥(P,Q)= -=+5,¢=Se. (gb) Tro Tro’ ‘Tro Pe Thelastterm inthisformula corresponds totheadded term ain(62). Ithastheeffectof-raising thepotential ofourinsulated sphere tothe ele! ert =g ue)qTrg (zy) Fie. 8.Two charges -temoving toward infinity anc their electric images ata conducting sphere ofradius aproduce auniform electric field andanelectric dipole atthecenter ofthesphere. value V=e//(4rea) andofmaking thetotal charge onitssurface g=0, asitshould be. C.Conducting Sphere inaUniform Field Forthepractically unlimited possibilities ofapplication ofspherical images inpotential theory werefer totheportion ofVol.VIcited above. Here weshall treat only thesimple case ofthesphere inauniform field, whose lines offorce may, forexample, beparallel tothez-axis. Inthe absence oftheconducting sphere theuniform field isgiven by v=Fr,B=-=F,8,=8,=0. 4) oz Weimagine thisfield asresulting from thesuperposition oftwofields, originating intheinfinitely distant source points Q,Q(seeFig.8)onthe 9.1la SIMPLEST BOUNDARY-VALUE PROBLEMS 59 z-axis, withthecharges -te;theircoordinatesarez =-kp(p>©),y=z=0. ‘The superposition leads tothepotential ares ©Salat tet Tro Pa —etaityta4ytaeetet+y tz}? Qa) px px 2ex (1458 -14="—-)-2%(tre +x8o)- e Wehave thus infact auniform field ofthesame form as(9)provided that weletebecome infinite asp*.Toobtain quantitative agreement with (9)wemust put 2e= —4neF. (9b)ae ‘ InFig.8wehave alsoshown thesphere ofradius aandtheappropriate spherical images ofthesource points Q,Qconstructed forit: . @=',0,0andQ=—9',0,0. They approach eachother asthecharges +emove apart andform inthe limitanelectric dipolewiththemoment M=2%". (10) Here wesubstitute from Eq.(8)p’=a°/pandfrom Eq.(82,b)e’=ea/p. Inview of(9b), Eq.(10)then states thatthemoment Massumes inthe limit forp>©thefinite value 2 M=22% ~—4neFa. (10a) oP Weconclude therefore that theboundary value problem forthehomo- geneous fieldissolved byplacing avirtual electric dipole ofthefinite moment Matthecenter ofthesphere. Thehomogeneous field (9)isthen replaced bytheinhomogeneous field? Mal YaPetTop (1) which contains thedistortion created bythedipole. Since, now, rrepresents thedistance from the center ofthe sphere, re(@ityt 2},00that22=-3 Eq.(11)hence becomes M1 7 v=-Fe(1+ 263). (11a) The denominator 4reistobeadded here forthesame reason asthefactor 4ze onthe left side of(9a). 60 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 9.11b Ifwesubstitute herethevalue ofMfrom (10a), weobtain *) v=-Fe(1~a): (11b) Onthesurface ofthesphere, r=a,Yassumes theconstant value ¥,=0. The value ofMfound in(10a) bythemethod ofreciprocal radii isthus confirmed fortheconducting sphere. D.Dielectric Sphere inaUniform Field Weshall now show that theformula (11), with thevalue ofMundeter- mined, hasamuch greater range ofvalidity, i.e.willalsofulfill theboundary conditions foranon-conducting sphere ofarbitrary dielectric constant. Ifwedistinguish theexterior andtheinterior ofthesphere bytheindices Land 2(seeFig. 9),these conditions require that : i av:waa BB torr=a. (12) The firstofthese guarantees thecontinuity ofthetangential components ofE;thesecond indicates thecontinuity ofthenormal component ofD, which, foranoriginally uncharged, nonconducting sphere, isequivalent ‘totheabsence ofsurface charge, #=0(Eq. (3.11)). Weassert that both equations may besatisfied iffor¥,(exterior ofsphere, r>a)weuse formula (11a) andfor¥,(interior ofsphere, r<a)weassume ahomo- geneous field, inthesame direction as,butdiffering instrength from, the exterior primary field. With z=rcos6(@=geographic latitude onthesphere, measured from thefield direction) wewrite tentatively M1 w==H+ao})cos6,2=—Fircos8(13) and, according to(12), wemust demand that forr=@(cos@cancels in both Eqs. (12)): M1(1+mars)“Pe om 1(18a) & (1aa) ah From thiswefind, with ¢=é2/e, asrelative dielectric constant: FP; 3 M _er-l,Foe42' drake +2" a) Thus ourassumption ofahomogeneous fieldwithin thesphere hasproved adequate. This field F;isweaker than theprimary external fieldFif 9.148 SIMPLEST BOUNDARY-VALUE PROBLEMS 61 e>1,8, forexample, ifthesphereisinair.Thelinesofforcepenetrate the interior ofthesphere (seeFig.9);although they arecurved outside bythe action ofthe(virtual) dipole moment they arestraight andparallel tothe z-axis within thesphere. Tounderstand this figure properly itshould benoted that itdoes not represent thelines offorce E,butthelines ofinduction D.The twosystems oflines have thesame direction butdifferent density both inside and out- side ofthesphere (see theremarks onmagnetic lines offorce orbetter tubes offorce onp.11), and hence behave differently atthesurface. The D-lines aresource-free not only within and outside ofthesphere butalsoatitssurface, because ofthevanishing surface divergence (D,is continuous); this does not apply totheE-lines (£, isdiscontinuous). The fact that Fig. 9represents theD-lines isevident from thefact that just one line passes through every point ofthespherical surface. Inthe EO —_ See ———_- ——— Fig. 9.Adielectric sphere inauniform electric field. The excitation lines within and outside ofthe sphere. case oftheE-lines, more lines would arrive atthesurface ontheoutside than leave itonthe inside. Weconsider twolimiting cases, ¢+©and e—0.The first proves to beidentical with that oftheconducting sphere. Eqs. (14) yields then M=—4re,Fa’, F,=0, (14a) inagreement with Eq.(10a) andwith thefactthat theinterior ofthesphere isfield-free. Thisappears tocontradict Fig.9a,which showsafinitefield inside ofthesphere. Wemust note again, however, that this figure, as limiting case ofFig. 9,represents theD-field andthat “D=eE=finite” 38consistent with passing tothelimit e+©, E>0. Thelimiting case¢>0cannot berealized electrostatically,' butonly 1Oronly byassuming 2<¢&1,i.e.considering aspherical cavity inamedium-of very high dielectric constant. Then indeed the interior ofthesphere isrelatively free ofD-lines, asisshown inFig. 9b. 62 DERIVATION OFPHENOMENA FROM MAXWELL EQuavtons 9.14b byplacing anonconductor inthestationary current field ofaconductor. ‘Themagnetic analog would beasuperconductor; hydrodynamically it represents thecaseofarigid sphere immersed inaliquid whose flow is nonturbulent, incompressible andinparallel lines atinfinity. Incontrast with. (14a) wehave now M=2ne\Fo!, F,=SP. (14) Inspite ofthefinite value ofF;themagnetic induction linesinthecaseof thesuperconductor andthehydrodynamic flowlines donotpenetrate into —_—_—_—_—_—_—_ =‘ Fro. 9a.Aconducting sphere ina uniform electric field. Representation ofthe excitation lines. aS —_—_— Foua_fR]l_l™_MCSESRR Fie.9b.Streamlinesaboutarigid sphere. Atthe same time, magnetic linesofforeeaboutasuperconducting ———oe—sphere. —$WZ—=_ theinterior ofthesphere, asshown inFig.9b;they arepushed outofthe interior since they must runtangential tothesurface. Intheother limiting case, ¢—>©,thelines offorce areperpendicular tothesurface,! asforaconductor. InProblem II.3weshall indicate the close relationship between Figs. 9and 9a.Weshall return tothese im- portant formulas andfigures in§11. ++The force linesattheupper andlower poleofthesphere (inthree dimensions at the‘diametral plane passing through thepoles) form anexception. They make an angleof45°withthesurface ofthesphere (seefigure), anangleof90°witheachother.This may forexample beseen from thefact that theTaylor expansion of¥(Eq. (11b)) begins with aterm ofthesecond order. Maxwell calls such apoint a“‘point of equilibrium” (see art. 112oftheTreatise). 9.168 SIMPLEST BOUNDARY-VALUE PROBLEMS 63 E.Reflection andRefraction ofLines ofForce attheBoundary ofaSemi- For thesake ofcompleteness arather trivial problem will bedealt withhere,namely induction inadielectric bounded byaplane(seeFig. 10):Letaunit electric charge Qbeatthepoint +=aintheright half- space,z>0;itbrings aboutastateofinduction inthelefthalfspace, z<0; istherelative dielectric constant ofthelefthalfspace referred to theright halfspace. Asin(12), theboundary conditions are _wy, oe -He =eafor2=0. (15) Asolution may beobtained with theaidofthesimple reflection method familiar from optics: gvirtual charge Q’ofopposite signisimagined atthe point x=—a,which hasaneffectinmedium 1,buthasnoeffectinmedium Fra. 10.Inthe right-hand halfspace a_ = (air) atpoint Qispoint charge producing |inductionintheleft.halfspace(dielectric). =A Representation, oftheexcitation lines. In PSStheright,halfspace theyarecurved;their \e =<.continuations (dotted inthefigure)pass (SensesthroughtheimagepointQ’ofQ.IntheleftC= <A halfspace theyarestraightlineswhose — (dotted) continuations passthrough Q. <— 2,inview ofthefiniteness ofthefieldthroughout thismedium; hereall effects appear toproceed from theprimary charge. Introducing thetwo disposable parameters e’/eande”’/ewewrite tentatively: , ”greats =£+%, dro =©. (16)ror Tr Intheboundary plane z=0,1=rpgandr’=req’areidentical; atthe same time a1__%-a@ a1__z+ader ANd areequal andopposite, thatis,arerespectively equal tota/r*, Hence conditions (15) require ete =e” l-e 2pairs Ya 7, 1eneed! e=Tpe ipred (16a) 64 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 10.1 Employing optical terminology, e”might becalled the“refracted” charge; the“reflected” charge e’ofcourse becomes zero for¢=1(uniform dielec- tric, nodiscontinuity atz=0).Inthelimiting case e+©(conductor) e=—e,e”’=0,80that¥:=0;thiscorresponds totherequirement ofa constant potential forz<0.For e—>0thesecond condition (15) becomes o¥,/dx =0andleads toe’=+e.Seeinthisconnection Fig. 10(arbitrary €>1),Fig. 10a(c>©),andFig. 10b(¢ 0,ive.Exight >>Erett)> A))Fro.10a,Limitingcase©+©:theleft Past) SRF halfspace isaconductor.oy 7 HN , oid id\Hy { Fig.10b.Limiting case¢—0;dielec-~\/ i- tricconstantoftherighthalfepace veryMee 2 large compared with that oftheleft half- 4re oiPre|Space. §10. Capacity anditsConnection with Field Energy Weconsider two conductors ofarbitrary shape Z,and L,and give them charges +gand —qrespectively. Such asystem iscalled acondenser because thefield between them isconcentrated and limited totheir neigh- borhood. The lines offorce pass from L,toL,without diverging toinfinity. Letthepotential Yhave theconstant values ¥,and ¥,onJ,and L,. ‘Thepotential difference between them isthen bs Veu-v= |Bas. «) 1 Inthislineintegral wearepermitted toleavetheshape ofthepathin- definite; every pathfromL;toL,(itneedbynomeans bealineofforce) yields, asweknow, thesame value ofthepotential difference foralamellar electrostatic field. 10.3b CAPACITY ANDITSCONNECTION WITHFIELD ENERGY 65 Theratio g/Viscalled thecapacity, i.e.theability ofthesystem totake upcharge. Weemploy forittheletter K(instead oftheoften-employed. symbol C,which weusesooften with themeaning “constant”’): EK=7 (2) Theunit ofcapacity isthe“farad”: coulomb yg. lfarad=1sar} joule’ (3) Fig.11.ThepairofconductorsLy,L:ZX) withthecharges-iqandthepotentials¥;* Y.GY ¥:formanelectriccondenser. Y ty Gp es5 a 2 which isthusdefined inoursystem without theappearance ofinconvenient powers often.Ontheother hand, inelectromagnetic cgs-units, according to(3), Liarad=10oge=107ogs; (3a) jor8 88; thus thesocalled absolute cgs-unit ofcapacity would be=10° farad. Themicrofarad (=10~° farad) isemployed more often than thefarad. According to(3)and(4.6a)thedielectric constant ofvacuum hasthedimension farad/M and,according to(7.18), ithasthemagnitude 10~farad_107microfard«35M 73e (3b) However, ifonechooses togivethecapacity in“absolute electrostatic units” notonlyisthenumerical value ofthecapacity different butalso itsdimensions (here cm!)arechanged andaveryconfusing stateofaffairs results. Weshallconsider, assimplest example, A.The Plate Condenser Thiscondenser istoconsistoftwoconducting plateswith(large)area F,which areplaced parallel andfacing eachother with(small) separation 66 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 10.4 a,We,imagine theircharges +gtobedistributed overFwithuniform surface density w.Hence wehave forz=-a/2 o=FJ, andalow=D,(Eq.3.128). (4) Weregard thefield between theplates ashomogeneous, neglecting aswe already have in(4),theperturbation attheedge zones. Thelines offorce arethen normal totheplates throughout, andwefind, with thevalue ofD given by(4) =faldaeFe* (4s) Itfollows hence that tal?ve[) Bar=® (4b)alt Fe andthus, from (2) Ka, 6) @ This isthe“‘rational” value ofourcapacity, previously given in(7.15). Za? Fic.12.Theplatecondenserwith the field E,Dbetween plates consid- ofvs t eredasuniform. If,forexample, weconsider F=20-20 em’,a=1mm, and¢=2e (paraffin filling) wefind, with thevalue (3b)of&inoursystem of.units, 8010"farad=210°farad=2-10mi Kion10farad=iz10~farad=ie10”microfarad. (5a) Thus, toobtain acapacity comparable with amicrofarad, avery large number ofcondensers ofthetype considered have tobeadded together. The neglected edge correction and theresulting inhomogeneity ofthe field will betreated inProblem II.4. B.Spherical Condenser _This istoconsist ofaninner sphere ofradius r;and anouter sphere oftadius 7.The spheres need notbeconductors throughout; itsuffices iftheouter surface oftheinner sphere, and theinner surface ofthe-outer sphere are“coated with tinfoil.” (The same remark applies fortheplate condenser.) Lettheinner sphere have thecharge +, theouter sphere, 10.8b CAPACITY AND ITS CONNECTION WITH FIELD ENERGY 67 thecharge —g.Thefieldisspherically symmetrical—E andDdepend only onrand aredirected radially. Hence forevery rbetween r;andrz |Dido=4xr'D=g D(6) elie tf Beoo ier From £,weobtain 7= =-2(1_ )oece v[E,drare(;nm)4eenit” @ ‘This leads to K= gate", (8) v mon Wa y Yy ;nwNGG Fic.13.Thesphericalcondenserwithitsra- A dialfieldE,DandavoltageV=i—Ws.‘\\A\ Ifweallow rztobecome infinite weobtain. K=4rer,, (8a) ite.thevalue forthecapacity forthislimiting form ofspherical condenser which was designated as“rational” inEq. (7.15a). If,ontheother hand, both radii in(8)arepermitted toapproach infinity while yettheir difference 1:—1;=aremains finite, andifonly afinite segment Fofthetwospherical surfaces 4x7}and427iisconsidered, thecapacity ofthissegment becomes according to(8) . Fe x" K-=7 (8b) ie.,identical, asitshould be,with that oftheplate condenser inEq.(5). 68 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 10.9 The cylindrical condenser (Leyden jar’) willbetreated inProblem Is. C.Capacity ofanEllipsoid ofRevolution and ofaStraight Piece ofWire InEq. (9.4) wehave given thepotential ¥ofanellipsoid with the charge q.Ifwesubstitute forz,y,zthecoordiaates ofany point ofits surface, e.g.oneofthetwo endpoints ofitsmajor axisz =y=0,2 =a,¥ assumes thevalue V,where Visitspotential relative toaninfinitely distant ground (¥=0).If,atthe same time, weintroduce theminor axisbinplaceofhalfthefocalseparation c,weobtain ¢c=+/a?—b'.Eq.(9.4) thus becomes* Voll 1logttVe=e¢”Keene aVeo 1 a+Vane ===log—_-_—_.. 9) leva —b b ®) For b—a,inthelimit, thecapacity (8a) ofthesphere isobtained. Ina similar manner thecapacity oftheaxially symmetric ellipsoidal condenser, consisting ofaninnerandanouterellipsoid oftheconfocal family, maybe derived. Onthe other hand, ifweletb—0,our ellipsoid degenerates into a straight segment oflength 1=2c(focal separation). This may bethought ofasrepresenting astraight wire* ofradius b—0.Itscapacity becomes, according to(9) K=arci/log} (9a) Wewill obtain asimilar logarithmic formula fortheselfinduction ofa straight piece ofwire (see §15). D.Energetic Definition ofCapacity The nsual elementary definition (2)ofcapacity may appear rather arbitrary andformal. Weattain aphysically more significant understand- ingofitbyconsidering theenergy oftheelectrostatic field. 1Also known as“Kleist jar,” since itwasbuilt bythepastor ofKleist inCammin (Pomerania) and first demonstrated attheDanzig Scientific Society inDecember _1745, Atthebeginning ofthenext year itwasdemonstrated inLeyden andhashence become internationally famous. Inthis manner itacquired itspresent customary name, which isthus purely accidental inorigin. *Compare Kohlrausch, Praktische Physik, 12thEd.,p.631.Thefactor4mismissinghere (conventional system ofunits). Obviously thelast formula inEq. (9)results from thepreceding onebyrationalization ofthedenominator. +Not, itistrue, acylindrical wire, but onegetting thinner toward theends. 10.11 CAPACITY AND ITS CONNECTION WITH FIELD ENERGY 69 Weutilize Green’s theorem intheform (3.16) ofVol. II: _av [smdU-grdvar+fuavar= [ude (10) UandVaretwoarbitrary continuous functions. Theintegration onthe leftside isextended over anarbitrary region ofspace, theintegration ontheright, over itssurface; nisthenormal tothesurface ofthespace, pointing outward. Weput U=V=¥andobtain, because ofAY=0, E=.-grad ¥,from (10) Jaradv-gradvdr=[Bare fvMae (10a) Letthevolume integration ontheleftbecarried outthrough theregion exterior tothetwoconductors I,andL;inFig.11,thesurface integration ontheright, over thetwo conductor surfaces andasphere Kwith the very large radius r=R.Theintegral over Kvanishes.' (10a) then leads to 2 oy ow[Banu Zatul Bae (10b) After multiplication with ¢/2wecanwrite instead i --}(uf =1,-y) =} 5fFDdr=([Dade+¥% |D.de)4(m4—¥)=dav. Attheleftendofthismultiple equation wehave thetotal energy ofthe yolume considered, which weshall callW: we[Wear Wethen find, with thedefinition (2)ofK, 1, Kyp_¢W=5W =5V'= se an 1Wehave, byGauss’s theorem, foranarbitrary system ofnconductors L, In,+++Ly,sincedivD=0, owfretfDadetofDate=fDado~Ba. x ti Le x omIf¥onKisputequalto¥.,itfollowsthat ow Vo Vo[eae- WEffPaae=~"EBa Inthepresent case this vanishes because ofg=g,ge=—g.Ingeneral, withqsx0,itisonlynecessary toput¥,,=0,i.e.toreferthepotentials ¥;tothezerolevel atinfinity, tomake itpossible tocarry over thefollowing formulas alsoto unneutral systems. This istoberemembered forsection E. 79 DERIVATION OXPHENOMENA FROM MAXWELL EQUATIONS 10.110 ‘This fundamental relation isreminiscent oftheexpression forthekinetic energy ofaparticle inrectilinear motion intermsofitsvelocity vandits momentum p=mv: _i PomsW=50-2 -F0. (11a) Inboth cases theenergy isfactored (see p.11)into theproduct ofan entity ofquantity and oneofintensity orexpressed bythesquare ofone ofthese two quantities. The quantity isinonecase q,intheother, v, theintensity, Vand p,respectively. Comparison of(11) and (11a) shows that thecapacity Kcorresponds tothe reciprocal mass 1/m, which we might call “compliance,” incontrast with the“inertia” m.However, the analogy isnotveryprofound andwillhavetobemodified in§33. Wecancall(11) anenergetic definition ofcapacity, just as(Ila) may serve asenergetic definition ofinertia. E.The Capacities ofanArbitrary System ofConductors Ifwepass from twoconductors L,,L;with charges +gtoanarbitrary number of‘conductors L,,Lz,---L,with charges q:,g:,°-*gn,Where once more thetotal charge 2719; isassumed tobezero, Green’s theorem (10a, b)shows directly that thetotal energy Wofthesystem isthesum of nterms, according totheformula We5Sua. (12) it Here ¥;denotes theconstant value ofthepotential ontheconductor L;. Now, however, ¥;depends notonly ong;,butdepends linearly onallthe gs88well. This follows from thegeneral representation (7.5) ofthepo- tential. Inorder toperceive this, werewrite (7.5) interms ofthesurface charge w;,whose distribution onevery conductor L;wecan assume asknown, inplace ofthevolume charge p,andputw;=go;,where w;isthedis- tribution ofunit charge onL;(inthe presence ofthe remaining con- ductors!). Then (7.5) becomes Y= Dats. (13) i “The coefficients which appear here, Hy=2[tian -(13a) feel ry? arepurely geometrical quantities, which depend only onthe location oftheL;relative toeach other and relative toL;;they areindependent id GENERAL CONSIDERATIONS ONTHE ELECTRIC FIELD 7 ofthechoice oftheorigin r;;=0onL;,since ¥;hasthesame value for every choice ofthispoint. Eqs. (13)and(13a) thus confirm thelinear rela- tion between the¥;and q;. Thesolution ofthesystem (13)ofnequations forthencharges q;yields o=-DKu% with Ky=. rh)= A A;;isthesub-determinant ofthen-nrowdeterminant AoftheH,;as- sociated with theterm i,j.Maxwell calls theH;;“potential coefficients” ofthesystem inart.87ofhisTreatise andtheKi;,“capacity coefficients.” Substituting therelations (13) and (14) in(12) yields themultiple - W=420,q; =F22Hisgiq; =$2EK VY; - (15) which generalizes ourearlier Eq.(11). Since Wisaquantity determined by thestate ofthesystem (the work done incharging thesystem must bein- dependent ofthe“path,” i.e.thesequence oftheindividual processes), theKandHfulfillthereciprocity relations Kiy=Ky, Hi=Hy. (15a) Capacity andpotential coefficients play aroleincommunications, where complicated systems ofinteracting conductors areoffrequent occurrence. Their theoretical calculation isdifficult since itpresupposes thesolution of thepotential problem ofthemulticonductor system inquestion. Aswe have seen, even fortwoconductors thesolution ispossible only forpartic- ularly simple shapes oftheconductors (plane, sphere, ellipsoid). Ingeneral approximations arerequired. InProblem II.6 weshall discuss the (somewhat complicated) relation between these coefficients and the elementary definition ofcapacity in Eq. (2). §11.General Considerations ontheElectric Field The following statements andconcepts apply notonly toelectrostatic, butalso toarbitrarily varying fields. . A.TheLaw ofRefraction fortheLines ofForce Theboundary conditions applying attheinterface between twoinsulators ofdifferent dielectric constant, ” Etang continuous aiidDrom continuous qa) (the latter intheabsence ofsurface charge, Eq. (3.11)) show directly that the“angle ofincidence” a;andthe“angle ofrefraction” a;,both measured with respect tothenormal totheinterface andgiven by 72 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 11.2 =(Buse Erase tana=(fet), tener=(Fe), arerelated by tana;_tanar (2) a fa This “law ofrefraction ofthe electric lines offorce” deviates from the optical lawofrefraction notonly intheappearance ofthetangent instead ofthesine, butalso inthedirection ofrefraction: Inentering into theelec- trically denser medium alineofforceisrefracted awayfromthenormal tothe ‘interface. Wehere describe themedium with thehigher dielectric constant as“electrically denser.” Ifthis ismedium 2,then itfollows from Eq. (2) that tan ap>tan a. Examples ofthis phenomenon areshown inFigs. 10(refraction ataplane) and 9(refraction atasphere). The conductor (limiting case &/e: =~) satisfies this law ofrefraction inasmuch ashere generally =0(lines of force normal tosurface ofconductor). B.OntheDefinition oftheVectors EandD Now wedonot accept the “positivistic” standpoint, according to which only observables may beemployed intheoretical physics, butinstead areoftheopinion that theintroduction ofnotdirectly observable quantities isjustified whenever theresulting conclusions agree with experiment (as inthekinetic theory ofgases). Nevertheless wedemand that theconcepts introduced inahypothesis may bebased atleast onanimaginary experi- ment, i.e.anobservational method, even ifitcannot becarried outin practice. In§2wedefined theelectric field strength dynamically astheforce ona unit test charge. This force can, however, only bemeasured inair(more generally, inafluid) bythemotion produced byit.Thus thedefinition failsinthesolid body. Inorder todefine thecomponent ofEatagiven point inagiven direc- tionswithin thesolid body weproceed asfollows: Wedrill atube with the direction sinthebody atthepoint inquestion. Thetube issonarrow and soshort that itdoes notappreciably disturb thefield elsewhere; itremains empty orisfilled with air.According totheboundary condition (1)the field-strength component £,isthesame within itasinthesurrounding solid body andasitwasoriginally inthetube. Thus E,canbemeasured within thetube onatest body which hasbeen introduced, andbyvarying thedirection ofthetube allthree components ofthevector Ecanbeob- 11 GENERAL CONSIDERATIONS ON THE ELECTRIC FIELD 73 tained. Ifthefield isnotstationary, butvariable, wemust measure more rapidly than thefield changes. Ourdefinition ofDin§2requires supplementation inevengreater degree. Wedescribed, inEq.(4),Dasthequantity ofelectricity which, atagiven point, haspassed through anarea Fduring theexcitation ofthefield, divided bythemagnitude ofF.More precisely, weobtain inthismanner thecomponent D,ofDinthedirection ofthenormal ntoF.This explana- tionisunsatisfying since itdoes notcontain specific directions formeasure- ment. Wecan,however, obtain suchdirections bythefollowing imaginary experiment: Weplace atthepoint inquestion aplate condenser,’ whose surfaces F aremade normal tothen-direction; thespace between theplates istobe filled withthesurrounding dielectric (ifwearedealing withasolid body a slitmustbecutintoitintowhichthecondenser fitsexactly). Inviewof theboundary condition (1)thevalue ofD,inthecondenser isequal to that initssurroundings andhence also equal tothevalue ofD,which prevailed before theintroduction ofthecondenser atthepoint inquestion. Wecannowmeasure D,onourcondenser directly asthesurface charge onthat coating toward which theprescribed direction npoints. Inthismanner thedisplacement Dalsobecomes, inasense, an“observ- able quantity.” C.TheConcept ofElectric Polarization; theClausiue-Mossotti Formula Wegive uptemporarily thepurely phenomenological point ofview of Maxwell’s theory and attempt toconstruct amolecular model ofthe dielectric. Amolecule consists ofpositive andnegative charges (protons andelectrons), butacts asaneutral entity intheabsence ofafield. With theapplication ofthefieldthecharges areseparated andformadipole?Theinduced moment misproportional totheexternal fieldandisachar- acteristic ofthe molecule. Such amoment hasthedimension charge-lever arm =QM. Ifwepass from thesingle molecule tothesum ofthemolecular moments “per unit volume” the dimension QM _Q Mw Mm ‘IftheGeldisinhomogeneous thecondenser mustbemadeadequately small. Itsmetallic coatings distort thefield, butdonotdisturb themeasurement ofDx between theplates. 7Wethink here ofnonpolar molecules. Thepolar molecules, which have been studied with great success, both experimentally andtheoretically, byDebye, have adipole evenintheabsence ofafield.Theformulas ofthetextwouldhavetobealtered forpolar molecules andwould show adependence ontemperature. Fordetails seeDebye: Polar Molecules, Dover, New York, 1945. Inanalogy toparamagnetism, polar mole- culesmaybecalledparaelectric. 74 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 113 isobtained. Itcorresponds tothedimension (2.4) ofD.Weshall designate thissumEmdivided bythevolume withPandcallitthepolarization.* Weshall divide Dintoonepart which ispresent even intheabsence of themolecules, andanother part which isproduced bythemolecules. The first corresponds tothecase ofvacuum andisD)=&E,where Eisthe applied, macroscopically measurable field; thesecond isourpolarization P.Hence wewrite D=D+P=eaE+P. (3) Disthemacroscopically measurable excitation andishenceequaltocE. Accordingly (8)leads to .P= (e— aE. (4) Wealso wish todetermine Pfrom thebehavior ofthemolecules inthe electric field. Inagreement with thenotation ofEqs. (9.9)ff. wecallthis field Fandindicate bythisthat itdiffers from themacroscopic field E. ‘The difference between them results from theeffect ofthepolarized mole- cules according totheformula 1P FeE+3a (5) Inorder nottointerrupt ourtrain ofthought wedefer proof ofthisexpres- sion tosection D.The moment macquired bytheindividual molecule is proportional tothis F.Weput m=aoF, (6) where aisaconstant characteristic ofthemolecule. Here themolecule is assumed tobeisotropic; otherwise mandFwould nothave tohave the same direction. IfNisthenumber ofmolecules perunit volume, weobtain from (6) and (5): P=Dm=NauF=Na(oB+5P). @ ‘unit 3 volume Ifwesubstitute here expression (4)forPand cancel thecommon factor E,wefind e-a=Na(a+£52)=Mee+20) or,ifwepass totherelative dielectric constant ¢/e0: \Everyindividual moment hasthedirection ofitsleverarmasaxis.Hencethedirection ofPisobtained bythegeometric addition ofalltheminthevolumecon- sidered and passing tothelimit ofasufficiently small volume. 11.12 GENERAL CONSIDERATIONS ONTHEELECTRIC FIELD 75 &ret—1_Namt 3 (8) This istheClausius-Mossotti formula. Inoptics, where ér.1isthesquare of therefractive index, itisknown astheLorenz-Lorentz formula. Toclarify thephysical content ofEq.(8),wemultiply numerator and denominator with m,themass oftheindividual molecule. Thus weobtain, intheproduct Nm, themass ofunit volume ofthedielectric oritsdensity and, atthesame time, inthequotient «/m, anew constant characteristic ofthemolecule. Then inEq. (8)theleftside isproportional tothedensity This assertion canbetested directly oncompressed gases forwhich ére1 differs appreciably from 1.For highly diluted gases, where frei~1, ret+2~3,Eq.(8)leadsto ta—1=Na=pS. (8a) Historically itmay bementioned that Mossotti, inhispaper dating asfarback as1850, treated themolecules asconducting spheres, which were assumed tobedistributed insome fashion intheimponderable “ether.” Weknow frbm §9that anexternal field Finduces amoment Minsuch a sphere whose magnitude isgiven by(Eq. (9.10a)) M=4xea°F. Our molecular constant has then, according tothe definition (6), the value a=4a’, (8) Ifthisexpression issubstituted in(8)theright side becomes 4zen. (a) This issimply theratio ofthevolume ofthespheres contained inunit volume tounit volume (dimensionless, asitshould be). We summarize what wehave learned about the concept ofdielectric displacement which wasinadequately explained in§2.Discomposed of twoparts, 8vacuum portion D,=«cEandaportion arising frommatter P: D=D+P. (10) WecallPthepolarisation ofthematter; itisalsotheelectric moment per unit volume ofthedielectric. Similarly, thedielectric constant ismadexp oftwo parts, itsvacuum component ¢and itscomponent arising from matter fon: €=e(1 +7). qi) 76 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 11.12 ‘The material constant 7,which isdefined asapure number, iscalled the electrio susceptibility. Itispleasant tonote that inboth Eqs. (10)and(11) thefactor 4x,which otherwise occurs inPand7,isabsent astheresult oftherational character ofour MKSQ-system. Indetermining Pand » wehadtodifferentiate between thefield strength Facting onthemolecule and themacroscopically defined field strength E;thedifference between them arises from thefield oftheneighboring molecules. Asalready noted onp.9,the designation “dielectric displacement” forDreally fitsonly itspolarization component P.Invacuum there isno charge which canbedisplaced andDisnevertheless bynomeans equal tozero. This isthereason why wehave preferred (see p.8)theterm “excitation” for D. D.Supplement totheCaleulation ofthePolarization Weare here concerned with theproof ofEq.(5).Weconsider anarbitrary molecule, surround itwith asmall sphere whose radius bisnevertheless very great compared tothemolecular radius, andremove from thesphere allmatter except theonemolecule atthecenter; thismolecule istherefore invacuum. Itisthusacted uponbythefieldstrength E,corresponding to thefirst term ontheright side of(5).The removal ofthemolecules from theinterior ofthesphere does notresult inachange ofthefield resulting from thematter present, provided that themolecules were distributed randomly, i.e.that they were notoriented inany way bythemolecule under consideration. Ifwelimit ourselves toisotropic dielectrics wecan assume this.’ After exclusion ofthissphere wecantreat theremaining dielectric asa continuous medium, i.e.neglect itsmolecular structure and proceed according toMaxwell’s phenomenological theory. Hence weshall replace theaction oftheresidual dielectric bycharge densities wontheelements dooftheinner bounding sphere ofthecavity ofradius b;here weshall have todetermine wnotfrom thecomplete D,butonly from itsmolecular com- ponent PinEq.(3).SincePdiffers fromzeroonlyforr>6(invacuum, forr<b,P=0),wisnotgiven bythedifference oftwo P-values (asin (3.11) bythat oftwo D-values), butdirectly byw=P,. Inview ofthe fact that Phasthesame direction astheprimary field E,which shall have thex-direction, wefind o=P,=P,cos@with 6=angle ofnwith respect tox. (12) According toCoulomb’s law(7.7) thecontribution ofwdotothefield strength acting onourmolecule atr=0inthedirection oftheradius veetor is 1. A.Lorentz has proved this also forcrystals ofcubic structure; forother symmetries, aswell asforassociating liquids, theassumption inthetext isunproved. 11.14 GENERAL CONSIDERATIONS ONTHE ELECTRIC FIELD 7 odo_P.cosé oPGneght ~neal 8) anditsz-component, with which alone weareconcerned, Pzcos’6 dF,=ted do. (18a) Integration overthewholesphere, withdo=6”sin6dédyleadsto aFe=Pt.2e[cos!asinade=37. aa) 4xeo 3&0 This corresponds tothesecond.term oftheright side ofEq. (5),whieh is proved herewith. E.Permanent Polarization Wehave assumed sofarthat thepolarization iscaused byanexternal field andvanishes with it.That isnotthecase ingeneral. Wehave already noted (p.73,footnote 2)that permanent electric moments exist ona molecular scale. Itistrue that they compensate each other, particularly intheliquid prgaseous state, because ofthethermal disorder inanyfinite volume, sothat here also the resulting polarization vanishes with the external ‘fieldE.However, ifasubstance madeupofsuchpolarmolecules (awax orresin) isliquefied byheating andexposed toastrong electric field, thelatter forces themolecular moments largely into itsdirection. After solidification thesubstance retains itspolarization foratime even ifthefield issubsequently removed. Iftheenvironment could bemade completely insulating asubstance would beobtained with amacroscopically permanent electric field. Heaviside has christened asubstance treated inthis manner with the rather forced name “electret,” inview ofitsanalogy tothepermanent magnet. The assumption ofacompletely insulating environment is,however, never satisfied. Even pure, highly diluted airis,because ofradioactive emanation and, inparticular, because ofcosmic radiation, always some- what ionized andhence conducting. Anelectret hence tends tolose its effectiveness totheoutside inthecourse ofhours ordays. There arehowever also natural substances with similar properties. We find these among crystals which areasymmetric instructure (crystals with apolar axis). The most familiar example istourmalin. Acrystal is ingenéral made upofpositively andnegatively charged ions which, if there isanimperfect symmetry ofstructure, have anelectric moment in any elementary domain. Depending onthe lattice ofthe crystal, the elementary moments maycombine toform amacroscopic moment, which thenproduces anelectric fieldinitsneighborhood. 7 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 12 Such afield can infact bedetected onfresh fragments oftourmalin. Since theenvironment, asnoted above, never insulates perfectly, thefield decays inthecourse ofafew hours. Surface charges arebuilt upbythe conduction currents attheentrance and exit points ofthelines offorce which then compensate theexternal field oftheinterior electric moment. The difference between the electric and magnetic permanent moment consists merely inthefact that there arenosuch conduction currents in themagnetic field. Hence asteel magnet shows moappreciable change in itsfield inthe course ofdecades. This difference between electret and magnet isnotfundamental, butonly quantitative. The polar asymmetry existing intourmalin may beproduced artificially inother, lessasymmetric’ crystals bysubjecting them toadeformation. This distorts thecrystal lattice and impresses anelectric moment pro- portional tothe deformation. The crystal thus becomes piezoelectric. Quarts isthe typical representative ofthis class ofsubstances. Itwas used byPierre Curie, as“piezo-quartz,” toproduce well-defined quantities ofelectric charge. Itistrue that here also, because ofimperfect insulation, theelectric charge decays with acertain finite relaxation time. Ofeven greater inlportance inmorerecent times hasbeentheroleofthequartz crystal when vibrating with itscharacteristic frequency andproducing a corresponding oscillatory electric field. Inversely, byapplying analter- nating field ofthis frequency itispossible tomaintain thecharacteristic vibration ofthequarts ataconstant amplitude. Inthismanner oneobtains (Cady) anideal microscale oftime, which plays itswell known role in present-day radio engineering. Inallthese cases (electret, tourmalin, piezoquartz) theexternal electric field may becalculated from theinner moment which isassumed tobe known. Wewill omit this, however, since thecalculation isquite similar tothat oftheexternal field ofapermanent magnet, which iscarried out below. §12. TheField ofthePermanent Bar Magnet The forces which emanate from certain forms ofiron have excited the popular imagination since theearliest times. TheGreeks called thecarriers ofsuch effects magnets? The Chinese were thefirst toutilize their inter- action with thegreat magnet “Earth” forgeographic orientation onthe 1Thedegree oftherequired asymmetry may bepredicted exactly bymeans ofthe general rules ofVoigt. SeeVol. I,§40. *Apart from steel themetals cobalt andnickel, which arerelated toiron, show permanent magnetism, similarly theHeusler alloys, containing manganese, which adjoins iron intheperiodic system» The iron oreFe:0s-FeO which, crystallizes in acubiclattice, isknownasmagnetite, thehexagonal FeS(withadmixture ofFe:8), aspyrrhotin ormagnetic gravel; both arecharacterized bypermanent magnetism and magnetic anisotropy. 12.2 THE FIELD OFTHE PERMANENT BAR MAGNET 79 broad expanses oftheir country. Inthe18th Century itwas fashionable toattribute allmysterious processes inthehuman bodyto“animalmagnet- ism” (Mesmer). Today theimportance ofthenatural orpermanent magnets isoutdistanced bythat oftheelectromagnets. Inspite ofthisweshall begin with some consideration ofpermanent magnets; weshall then beina position tocover briefly thegeneral properties ofthemagnetostatic field, inanalogy tothose oftheelectrostatic field in§13. Itistrue that thenature ofthepermanent magnets liesoutside therange oftheMaxwell theory and canbeunderstood only with theaidofatomic physics. Itisbased onthespin oftheelectron and itsmagnetic moment, ofwhich the Maxwell theory is,ofcourse, ignorant. The same remark applies eventually totheelectromagnet: theelectric currents which produce theelectromagnet, unless they aregenerated electrodynamically byinduc- tion, have their origin inelectrochemical processes, which areforeign to theMaxwell theory; only themagnetic fields proceeding from theelectro- magnet aredescribed bythelatter. Insimilar manner, thefields proceeding from permanent magnets fitinto theframework oftheMaxwell theory. Wecommence with themagnetization, which weshall callMor,tobegin with, M*,ascounterpart totheelectric polarization P;theanalog ofour sequation ofdefinition (11.3) for P, D=HE+P. () would be,from our point ofview, H-iB+m* (1a) Ma (Here Hand M*are“quantities” like Dand P,Bisan“intensity” like E; 1/uo corresponds to€,asemphasized inEq. (4.7)). Solved forB,Eq. (1a) yields B= w(t —M*). (1b) The customary definition ofmagnetization is,ontheother hand, contained intheequation B=»(H +M), (2) which weshall utilize from here on.Weshall return toEq. (1b) in§13D, indiscussing diamagnetiam. Asdefined by(2),Msignifies thepart ofthe excitation derived from matter and isatthesame time thesum, referred to unitvolume, ofthemoments ofelementary magnets, justasPwasacorre- sponding sum ofelectrical elementary moments. Inthefollowing weshall imagine thedistribution ofMwithin themagnet tobegiven arbitrarily and shall calculate from theMaxwell equations thecorresponding fields of the vectors Band H. 80 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 12.3 Weknow from§7thatHisthroughout lamellar, Bthroughout solenoidal (free ofsources). Inview oftheabsence ofsources ofBEq.(2)leads to the condition divH=—div M. (8) Atasurface ofdiscontinuity ofMtheEq.divB=0,which presupposes thatBiscontinuous anddifferentiable, isreplaced bythecondition thatthe “surface divergence” ofBvanishes; i.e. B,+By=0, (3a) where nandn’,asin(3.7), denote thenormals ofthesurface ofdiscontinuity pointing toward opposite sides. Applied tothesurface ofamagnet, at which Mjumps from theexternal value M=0toavalue ofMwhich, in general, differs from zero, (3a)yields inview of(2) : Hy+Hy =~My. (3b) Tf,now, H=—grad Wissubstituted inEqs.(3)and(3b),thedifferential equation ofthe problem AY=divM, (4) and the. surface condition oy,ovint= M, (4a) areobtained. Weadd, asboundary condition atinfinity v=0. (4b) Since, inthetwo Eqs. (4)and (4a), theright sides canbeassumed to beknown, wearedealing here notwith aboundary value problem, but, in thesense of§7,Eqs. (10) and (10a), with asimple summation problem. The solution is divM M, ary=—[Ma —[Brae i) The first term sums allmagnetic volume densities p»intheinterior ofthe magnet, thesecond allsurface densities w»,onitsboundary. Thenegative signs result from thefactthat, according to(4)and(4a) - pm=—divM, wn=—M,. (ifwehadcontinued toemploy M*=—M inourcalculation, thesigns would have been positive, asinelectrostatics.) Weconsider twospécial cases: a.homogeneous magnetization parailel tothebaraxis, b.magnetiza- tion increasing from zero toward thecenter, alsoparallel tothebaraxis. 12.6a THEFIELD OFTHEPERMANENT BARMAGNET 81 Forathefirst integral ontheright inEq. (5)vanishes because of divM=0,for6,thesecond onebecause ofM,=0. a.Only thetwoendsurfaces contribute tothesurface integral in(5) since thenormal component ofMvanishes, byassumption, onthesides. Thepolestrengths ofthemagnet arethus, inasense, uniformly distributed over theendsurfaces; ifthecross-section area isFthetwo total pole strengths are +P =+FM. Approximate integration of(5)forlarge distance ofthereference point from themagnet (seeFig.14:p,r:,7:distance ofreference point from bar axis andfrom thecenters oftheendsurfaces, respectively, z,coordinate Ref. pt. 4 : “ee “7 Awt 5“ory 14; 1 ibe COA Yn Fro.14.Barmagnet, magnetized, longitudi- 1{ths /nally.Pointofreference ontheoutside. 1are4.ae-4#---~----Y_ M rly? tft 1 6 e, t H ofthereference point parallel tothebar axis, measured from itscenter, 21,length ofbar) leads to n=@-D+e4e¥=—MP(t-x)with"| Pr6) mM v= @+0 to Series expansion yields 1_1 12+?PaA(+4 ret-), 1_i 122+?neACae +-) withry="/2 +p?=distance ofreference point from center ofbar.Hence~ according to(6), ary=-2p4,=mp2} = ree ere (6a) 82 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 12.7 ‘Aswastobeexpected, theexternal action ofthemagnet fordistances >>21isthatofadipole with thelever arm2Iandpolestrength P=MF. Thesame representation bythesurface integral inEq.(5)applies for theinterior ofthebar,butthisrequires amore careful evaluation. For thesakeofbrevity welimit ourselyes tothecenter lineofthebarand || x| 1 Bin||- tall Fro, 15.Demagnetization ofauniformly magnetized bar. Inthedrawing itis assumed that d=1/4. assume thecross section tobecircular (2=radius, p=distance ofthe point ofintegration fromthecenter oftheendsurfaces). Wethenfind dev=vent{[ede_fet. oy oT (7) na=Gd—ate r= Ute +e The evaluation yields, since 0S$|z|S1: vaMya oteay—(d+ata +2 Ce) ov __M l-z +zaoe |ooo tHei - H=-y 7%lwareal t(satay 2](») SinceJisinanycasemanytimeslargerthana,weobtain H~O, aHLgforz=0 dz MoHM1 He=- >eta forz=+1; asshown inFig.15,there isasharp decrease of—Hatthetwoendsofthe barandavanishing ofahighorderatthecenterofthebar.ThesignofH 12.11 THE FIELD OFTHE PERMANENT BAR MAGNET 83 isopposite tothat ofM;Hhaswhat iscommonly expressed asa“de- magnetizing action.” Thisisseenalsointhepattern ofB.Though Battains almost thefullmagnitude yeM atthecenter ofthebar,itisonly halfas large attheends, thesame ofcourse, inside andoutside ofthebar. b.LetMbeconstant inevery cross section ofthebar,butbedependent onzinsuch fashion that Mvanishes attheends z=-t/and increases parabolically toward thecenter: Cc z . Cz m=(1-2), divM=—>. (8) Asalready mentioned, thesurface integral in(5)vanishes here, andonly thevolume integral remains tobecalculated. With £,p,¥forthecoordinates ofthepoint ofintegration, and2,p,g forthecoordinates of.thereference point Eq.(5)yields: ¢ ys teen5fffEatae, ©) P=(es +p+0—2pcos(e— y),da=pdpdy. Thefieldatadistance here, justasincasea,isthatofadipole. Ifthe average magnetization Miscomputed forthelength ofthebarandif asina,weputMF =P,themoment ofthedipole field becomes, asin (6a), 2UP. Within thebar, inparticular onthebaraxis p=0,weobtain from (9),bycarrying outtheintegrations with respect topandy +t v=Sfratte—pttales) 2PJs cr on (10)=f —pagit Ef fls_ ipml,bate )°+a}53 it) and wy _cst, ia ‘ota antag Lig HeRael adie ee+Se-0 2 -S[te-o teeictieet 42s") ay _ocLte taatt mL, I@-9? +a°|' ag. Thesymbol ||inthelastintegral indicates thatthesignofthissquare: Toot, just asthat ofthepreceding ones, istobepositive. Wewillshow that, justasinthecase ofuniform magnetization, His nearly zero (oftheorder a/l) everywhere except attheends ofthebar. Ex- 84 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 12.11a clusion ofthebarends signifies‘a «<1! —|z|.We may then neglect ain (11) and obtain Pa)woG[t-etrte fet24), (11a)2l U U ive.zero (more exactly, vanishing totheorder Ca/l). For z=+l, i.e.at thebar ends, Hissmall tothesame order ofmagnitude. This does not apply, however, todH/dz. This isgiven by’ dH _CEe7gite): (12) Thus Hhas positive values, appreciably differing from zero, only ina small region near thetwo ends ofthebar. This isillustrated byFig. 16. aN (| eee[t\.ot =! Fra.16.Demagnetization ofabarmagnetized accordingtotheformulaM= Ccia a2(-2),AsinFig.15,a=1/4. Also here Hhasa“demagnetizing action,” i.e.isopposite totheim- pressed moment M.B/yw approximates thefullvalue otMinthewhole middle portion ofthebaranddeviates from itslightly only attheends. The fact that Halways acts inthedirection ofdemagnetization (for arbitrary distribution ofthemagnetization andforanyshape ofthemagnet) mayberecognized from thefollowing: Thelines offorce (B-lines) areclosed 1Differentiation of(11) leads totheexact elementary formula aH£{we=Dta 22+D+a |2) on dz7Verte Verret} from which (12)isreadily derived. Thedots in(12)indicate terms oftheorder ¢/l. ‘Theupper positive signin(12)refers toz=+1andpositive dz;thegradient ofH toward theinterior ofthebar(negative dz)isthus negative, just asattheother end ofthe bar. 12.120 THE FIELD OFTHE PERMANENT BAR MAGNET 85 because ofdivB=0andpart oftheir path liesintheinterior, andpart liesintheregionoutside ofthemagnet. Wecarryoutalineintegral ofH over such aclosed lineofforce inthepositive B-direction. Then weobtain, because ofthelaracllar character ofH,asforanyclosed path, $H,ds=0. (12a) Thepart oftheintegral over thepath outside ofthemagnet, where the directions ofHandBcoincide, ispositive; hence thepart oftheintegral over thepath within themagnet must benegative: fiHeae<o (12b) Inaide Ontheotherhand, theintegral ofBoverthesamepartis,byassumption, positive. Eq.(2)shows that thisapplies even more totheintegral ofM: [iMede=2f Bas-f mds>0. (120)inside HoJinside inalde Thetwoinequalities (12b, c)forHandMshow together that Hhaswithin ‘the magnet,*along any lineofforce, ontheaverage theopposite di- rection ofM.° Wehave dealt with thepreceding rather arbitrarily selected problem insuch detail because most textbooks contain little ofaquantitative nature regarding thevectors B,H,andMintheinterior ofamagnet. We have seen that ifMisknown Hmay inprinciple beevaluated, inaccord with therules ofpotential theory, byasimple summation, whereupon B isalsoknown. Itistrue that theassumption ofaknown distribution ofM. isnotfulfilled inpractice. Barmagnets aretherefore unsuited forapractical study offerromagnetism: weshall return tothislater. Although ourcalculation waslimited tothecenter lineofthebarmagnet, Figs. 17and18givequantitative information’ regarding theshape ofthe Tinesofforce andlines ofexcitation throughout theinterior ofauniformly magnetized barmagnet: TheB-lines aredrawn intotheinterior, theH-lines pushed outofit.Ontheoutside thetwosetsoflines coincide, ofcourse, since B=oH. Theringmagnet, provided with anarrow gap, isboth simpler andof greater practical importance than thebarmagnet. Because oftheequiva- lence ofallcross sections themagnetization mayhereberegarded asuni- form,sothatdivM=0,andMiseverywhere parallel tothecenter line. 'These figures were kindly prepared byProf. J.Jaumann, byagraphical method which wasdeveloped byMaxwell forthenumerous line-of-force patterns attheend ofhisTreatise andwhich iswidely employed byelectrical engineers; seealsoart. 123oftheTreatise. 86 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 12 A“magnetic coating” existsonlyonthegapfaces;between themamag- netic field isformed which issimilar ingeometry totheelectric field ina plate condenser. This applies notonly toapermanent ring magnet, but almost identically tothering-shaped electromagnet with aniron core. Referring once more toEq.(12a), weconsider thelineintegral oftheex- citation Hcarried outover thecenter lineofthering. Itmay bedivided into twoparts, theshort section through theairgapofthickness ,which weshall traverse inthepositive direction ofthemagnetic condenser field Fie. 17.Lines offorce ofauniformly magnetized barmagnet; they aredrawn into the interior. Fro, 18.Lines ofexcitation of»uniformly magnetised bar maguet; they sre pushed outoftheinterior. whichexists there, andthelongpartthrough thering-shaped ironcoreof length J,onwhich theintegration istobecarried outinthesame sense. WeputH=HginthegapandH=H,intheironcore,respectively. Eq. (12a) yields * 1-H, =—a:He. Thus a“demagnetization” oftheiron core goes with themagnetization oftheairgap. Wehave here thesame state ofaffairs asinFigs. 15and16, only inamuch simpler and more obvious form. 12.15 THE FIELD OFTHE PERMANENT BAR MAGNET 87 Wereturnoncemoretothebarmagnetandtothedefinition ofitspolestrength P.Fromourpointofviewitisaquantity ofmagnitude P=$Hi,do, (13) where theclosed surface «envelops, starting from thecenter ofthebar, theoneortheother half ofthebarinarbitrary manner. This definition ofPcorresponds tothedefinition (7.2) ofthemagnetic volume density p»=divHandstatesthatPisequaltothesumofallmagneticquantities padrwhicharepresentinthehalfofthebarinquestion. Incontrast tothisthepolestrength isoftendefined asintensity inthe literature and described interms ofthemagnetic flux. Weshall callthe pole strength sodefined Pandwrite P=|Bade 4) Thesurface ¢cannot nowbeclosed sinceotherwise, inviewofdivB=0,P=0.Rather, thecrosssection qpassing through thecenter ofthebar must beexcluded from theintegration; or,asanalternative, theintegra- ,tioniscarried outonly over thiscross section with reversed signofthe normal 7: P=fB,dq. (14a) Itmay readily beshown with theaidof(13) and (14) that then, very nearly, P=wP (15) where poisthepermeability ofthesurroundings.’ Infact, weconvinced ourselves above that Hvery nearly vanishes inthecentral cross section, 80that theclosed integration in(13) may bereplaced bytheopen integra- tionin(14), where wemay putH,=B,/yo. The definitions (13) and (14) would thus bepractically equivalent inthepresence ofaplane ofsymmetry (which, incidentally, exists also forthehorseshoe magnet). However, for asymmetric shape orasymmetric magnetization relation (14) fails and onlydefinition (13)remains meaningful. Itisalsorecommended bythe factthat itexactly corresponds tothedefinition ofcharge: omfDede. _1While P,according to(13),mayberegarded asaninternal property ofthemag-net,Palsodepends, according to(15), onitssurroundings. This evidently arises from thefact that anenvironment differing from vacuum contains itself magnetic moments which, quite understandably, areincluded inP.Seeinthis respect Phys. .1985, p.424, 88 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 13.1 $13. General Considerations onMagnetostatics andItsBoundary-Value Problems While, in§12,wehave considered only theproper field ofpermanent magnets, weshall now approach thebehavior ofarbitrary bodies inan outside field, arising from either permanent magnets orelectromagnets. ‘Thelaws which apply hereclosely parallel those ofelectrostatics; however, contrary toourclassification ofthefieldvectors intoentities ofquantity andintensity, Hhere corresponds toE,BtoD.This follows from the familiar fundamental equations, e.g.(7.2), (7.9), and (7.98) curlH=0, H=-grad¥, a) divB =0, divH=pp @) andtheboundary conditions ataninterface: continuity ofthetang. comp. ofHandthenormal comp. ofB. (3) Thelineintegral ofthemagnetic excitation, which, by(1),isindependent ofthepath, i.e.depends onlyontheendpoints A,B,weshall callmagneto- motive force anddesignate byUs, inanalogy toVuzin§2,Us»bears following relation tothemagnetic potential: 2 Us=fBeds=VaUs. @ A The “loop magnetomotive force” isgiven by Use=fads=0, (5) irrespective ofthemanner inwhich theclosed path istraversed, whether itpasses through magnetized material orthrough air. ‘Wedesignate the“induction” orthe“flux” through anarbitrarily shaped surface obythe usual symbol : a=|Bade. () Inview of(2),&depends only ontheboundary curve ofthesurface a,ie. isidentical forallsurfaces passing through thesame boundary curve. For aclosed surface itvanishes, ofcourse: b=$B,do=0. (6a) Weshall now discuss briefly themagnetic analogs ofthetopics dealt with in§§11, 10,and 9. 13.9 MAGNETOSTATIOS AND ITSBOUNDARY-VALUE PROBLEMS 89 A.TheLawofRefraction oftheLinesofMagnetic Excitation ‘Thelawofrefraction (11.2) ofthemagnetic linesofforcemaybecarried overtothelines ofmagnetic excitation, butapplies alsotothemagnetic linesofforcebecause oftheidentical direction ofHandB.Iftheangles@anda;have thesame meaning asbefore, tana; _tana (2) mL Bs Weprefer tospeak hereofthelinesofforce because they, unlike thelines ofexcitation, areindividually continued intothesecond medium. Wemaytherefore say:Every individual B-line isrefracted away from thenormal inentering themore permeable medium (e.g.us>41). B.Definition oftheVectors HandB,Particularly inSolid Bodies Tomeasure thecomponent ofHinagiven direction atagiven point by means ofanimaginary experiment ashort narrow tube must bedrilled, tomeasure B,athinslitmust becut.Within thecavity (vacuum orfilled with air)soprepared deflection experiment canbecarried outandBor ‘H=B/w, respectively, bedetermined fromtheforce, someasured, acting énatestbody." C.TheMagnetization MinAnyNon-Ferromagnetic Substance AsinEq.(12.2) wedefine Mby B=4(H +M) (8) and set, M=cH. (8a) «isamaterial constant, themagnetic susceptibility ofthesubstance. Physi- cally more significant isthemolar susceptibility x=M/p, (8b) where Misthemass ofamole, theso-called molecular weight ofthesub- stance, andpisitsdensity. (Asweknow,theproportionality between M andHdoesnotingeneral applyforferromagnetic materials). B=m4andEqs. (8)and (8a) lead to 4 #=boca | —S. 9)poits «=ie (9) 1Amagnet needle, awiretraversed bycurrent, or,eventually, abismuth spiral from ourstandpoint such anexperiment yields theintensity Bdirectly; from itis determined, inthetubeexperiment, theproportional quantity H.Fordetails see aleo §11B, 90 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 13.10 ‘The conventional form ofEqs. (8)and(9)ismarred bytheappearance of444sfactor ofMand«(seeourcorresponding remark regarding P and 7inconnection with Eq. (11.3)). Wemust hence note that intheuse ofexperimental data employing conventional notation thefactor 4xmust beadded. That isindicated inthesmall table inthefollowing section D. D.Dia- and Paramagnetism Diamagnetism corresponds todielectricity; likethelatter itisageneral property ofponderable matter andindependent oftemperature. Both owe their origin totheelectronic (and nuclear) structure ofmatter. Paramag- netism occurs only formagnetically polar molecules, i.e.molecules which have amagnetic moment oftheir own. (Electrically polar molecules were discussed onp.73.)Paramagnetism istemperature-dependent andhence hasastatistical origin. Thisphenomenon, likeferromagnetism, liesoutside ofMaxwell’s theory. The paramagnetic susceptibility obeys thelawof Curie and Langevin: Cc C=Curieconstant x= ar (10)T T=absolute temperature Thisdependence onthetemperature indicates thatincressing thermal agitation interferes with thealignment ofthemagnetic moments inthe field direction, decreasing thermal agitation favors it. Inthediamagnetic case wehave B<m, «<0 (11) This apparent difference from thedielectric case e>e&, 7>0 (La) isexplained inthemanner already indicated following Eq.(4.7): thetrue analog ofeisnotu,but1/s. The magnetic parallel ofthestatement ©>e isthus }>i>B<m, «<0 uo po corresponding to(11). The negative sign ofM*inEq. (12.1b) isalso re- lated tothis. For, ifweputM*=«*H,thesusceptibility «*sodefined becomes, inviewofM*=—M=—xH, : = -«>0 inthediamagnetic case corresponding to7>0in(11a). Theintroduetion ofM*, which was suggested previously butimmediately given upabove, thus corresponds infact totheinner relationship ofdielectricity and dia- magnetism, 13.11b |MAGNETOSTATICS ANDITSBOUNDARY-VALUE PROBLEMS 91 Forparamagnetism, wehave, incontrast with (11), u>m, «>0. (11b) ‘Thenumerical value ofxisvery small forboth paramagnetic anddiamag- netic materials. The following represent extreme values: Paramagnetism Diamagnetismk=+4n-18-107 forOr k=—4x-0.007-10-* forNz «=+4x-782-10"* forPd k=—4x-160-10™ forBi The paramagnetic values refer here to18°C. TheClausius-Mossotti law, Eq.(11.8), with ereplaced byu,gives the dependence ondensity. E.SoftIron asAnalog totheElectric Conductor With certain restrictions soft iron may beclassified with theparamag- netic substances. The initial value ofthepermeability (referred tothe value forvacuum) isseveral thousandfold, according tothevariety of iron;forincreasing H,Bapproaches asaturation valueintheneighborhood .of21,000 gauss. Theequation B=»Hmusthence bereplaced bythe functional relationship B=B(H), aswasmentioned already onp.21. Justaswenotedonp.61,thattheelectric conductor corresponds, with respect totheelectrostatic boundary conditions andboundary-value problems, tothelimiting casee>ofadielectric, sowemayregard soft iron(u—©)asthemagnetostatic analog oftheelectric conductor. Eq. (7)shows, infact,thatthemagnetic linesofforce areperpendicular to thesurface ofsoftiron (a;—0follows from 4,—~).Iftwosuch pieces ofsoftironFe,andFe,areplaced atdifferent magnetic potentials (e.g.if theyareplaced onthepoles ofahorseshoe magnet), theH-lines be- tween them aresimilar totheE-lines inFig. 11.Thus, inasense, amag- netic condenser isobtained. Inengineering applications itisalsoconvenient tointroduce thecon- ceptof“magnetic resistance” andtoemploy a“magnetic Ohm’s law.” F.Specific Boundary-Value Problems Themethods ofsolution developed in§9may betaken over directly intomagnetostatics; anexample istheimaging at2plane, where itmakes nodifference whether theinduction inhalfspace 2inFig. 10isproduced byasingle poleoradipole inhalfspace 1.Thesame remark applies for themethod ofreciprocal radii asapplied toasphere (previously regarded asaconductor, hereasconsisting ofsoftiron). - Wereferinparticular tothesphere inauniform magnetic field. Within there isauniform field F:,while outside thefield becomes nonuniform, through thesuperposition ontheoriginal fieldFofthefieldofavirtual 92 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 13.12 magnetic moment Mfatthecenter ofthesphere, withitsaxisinthefield direction. (‘Field” here denotes “excitation field”.) Thevalues ofF;and Mare,by(9.14), 3= ola. PF,ron,3F M= Fa? AnusF. (12) Here a=radius ofsphere, »=2/4 =relative permeability ofthesphere referred toitssurroundings (air). F;isstronger than Ffordiamagnetic substances, weaker forparamagnetic materials. Intheinterior ofsoft iron H&0,justasintheelectric conductor. G.TheUniform Field within anEllipsoid ofRevolution Thesolution (9.13), forthesphere, translated intomagnetic terms, be- comes, after substitution ofthevalues (12) =—r(,-4a ie ---3YaP(>2535) coe,+2ayFreos6.(18) This solution satisfied theboundary conditions ou,_ON Mah,Fee ¥ (18a) because thefactor cos6,which varies over thesphere, factors outofthese equations. Wewill show that asolution ofthesame form isvalid forthe ellipsoid. Inpassing 1rom thesphere totheellipsoid wemust first replace the spherical polar coordinates r,@bycorresponding elliptical coordinates, which weshall callu,v.Weproceed here from thewell-known parametric representation oftheellipse (instructions forProblem II.1), inwhich we write fortheprincipal axes aand b @=ccoshu, b=csinhu, a4)c=Va—bt=independent ofu. Rotation about thelong axis(z-axis, angle ofrotation g)produces afamily ofelongated confocal ellipsoids ofrevolution, corresponding toEq.(19.178) inVol. II: 2 etyFeokra *Samba 7b (148) Lettheellipsoid considered byusbeoneofthese, namely thatwith‘the parameter u=uw.The relation between z,y,zandtheelliptical coordi- nates u,v,¢isthe following: 13.18 MAGNETOSTATICS ANDITSBOUNDARY-VALUE PROBLEMS 98 2=ccoshucos2, z=cainhusin»cosy, (15) y=csinhusinvsiny. The expression forthelineelement dsconsequently isgiven by ae=(cosh?u—cos’v)(du*+do*)+sinh*usin’vdg’.(15a) According totherule (3.9b) inVol. IIthepotential equation inthese coordinates, becomes: af. . av a . OW2(sinhusino2)+2(sinhwsinv4)=0(16) ifthepotential isindependent ofthecyclic coordinate ¢. One solution istheuniform field, parallel tothemajor axis, ¥=2=coshu008». (17) " Weseek aséoond solution ofsuch form that thefactor cosvwhich varies overthesurfaceoftheellipsoidcorresponding tothefactorcos6giving geographic latitude onthesphere, iscancelled outintheboundary condi- tions. We write for this second solution W=f(u) cose (17a) and obtain from (16) forthe differential equation &(inhuf"(w))~2sinkwflu)=0 (7b) which isevidently satisfied bytheuniform field f=cosh u.Following a general rule,’ weplace thedesired second solution equal totheproduct oftheknown first solution and anunknown function U(u): f(u) =cosh uU(u). (18) The resulting differential equation forU(u) wgSeinb?ut1py+ sinhucoshuu=o canbeintegrated directly andyields, withAandBasintegration constants, 1Itcorresponds tethemethod ofsolving analgebraic equation with oneknown root. 94 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 13.188 Ny)«Afa) <AfogCohu- ty4 UW)=Saveotty? UO=38cha bitcout _coshu,coshu—1 fu)=A(1+SSlogome). (18a) Inthelastformula wehave omitted theterm multiplied with B,since this corresponds toourknown solution representing auniform field. Wenowcomplete ourexpression (17)fortheexternal fieldbytheaddi- tionof(18a) andretain theuniform-field expression (17)fortheinternal field: coshu,coshu—1 wwPcoshwcos»+A(1+228ogSY=oso(19) V1=Fycoshwcosv. Weregard theconstant F’oftheexternal field, which hasbeen added, as known; thetwoconstants AandF;aretobedetermined from theboundary conditions. These areidentical with (13a), with drreplaced bydu(more exactly, byds., thelineelement inthedirection ofthenormal tothe ellipsoid, ds, =cr/cosh® u—cos?vdu, where, however, thesquare root cancels outinthesecond Eq.(18a)). The boundary conditions demand hence for u= 11,coshwm—1 r+a(che+poems) Py (198) coshuw,1,coshum—1 P+A(cot4HogHF) mis. (196) Subtraction yields thesomewhat simpler relation A=coshtsinh”o(u—1)F2 and substitution thereof in(19a) —4)ginh? 1 coshto—y)= m{1- 1)sinh’us(1+§coshtyloSOBMe=F.(20) ‘Thefieldstrength F;inside isthusexpressed interms oftheknown strength oftheoriginal external, homogeneous field. .Ifweutilize Eqs. (14)anddenote bya,b,ctheprincipal axesandfocal distance from thecenter ofourellipsoid u=uw,wemay write instead of(20) vla,a-a) rift~~8(14320g=} F(20a) 13.22 MAGNETOSTATICS AND ITSBOUNDARY-VALUE PROBLEMS 95 or,interms ofthenumerical eccentricity e=c/aandthemagnetic sus- ceptibility «=»—1: 1-e/1, l+e) wifitelS*(Gigitee)p=F. (21) We note here that: a.forx=0wehaveofcourse F,=F; (21a) b.for©—0series expansion of(21) leads to P=r/(1+*). (21b) The field inside isweaker than outside forparamagnetic bodies, stronger fordiamagnetic bodies. c.Thesame applies fore—1.Ifweput7=1—e,(21)yields her/{i+19(og2-2)}. (2t0) Inthelimiting case btheellipsoid becomes nearly spherical; infact, (21b) isidentical With Eq.(12)forthesphere. Inthelimiting casectheellipsoid degenerates toathin rod. Theproblem treated here isusually related toafamous formula ofDi- richlet forthegravitational potential ofatriaxial ellipsoid uniformly filled with matter. This procedure ismathematically more elegant than ours, butisrather indirect. Wehave preferred thedirect method ofthemag- netic boundary-value problem because itappears togive usmore profound insight into thephysical conditions. Oursolution ofthemagnetic problem isofcourse transferable without change tothecorresponding electrostatic problem. H.The So-Called Demagnetization Factor The ellipsoid anditslimiting forms (sphere, rod) isthestandard shape ofthemagnetic testbody because italone possesses auniform andeasily calculable internal field when introduced intoanoriginally uniform external field. For other shapes thedetermination oftheinternal field leads toa practically insoluble boundary-value problem; theinternal field isbyno means uniform, but varies from point topoint. Itisclear, however, that allquestions corfcerned with themagnetic properties ofthematerial depend ontheinternal fieldF;.This field inter- acts with themolecular components ofthematerial directly, while the external field Fhasnodirect effect onthem. Accordingly wemay express our Eq. (8a) more precisely by M=«F;. (22) 96 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 13.228 Suchquestions becomeparticularly important withferromagnetic mate- rials, where thedifferences between theexternal and theinternal field (Fand F;inourpresent notation) areextremely large; forpara- and dia- magnetic materials they arenegligible because ofthesmallness ofx.Since forferromagnetic, asforparamagnetic, materials F;<F,wewrite, with Pdenoting anumerical factor, Fi=F —PM (22a) or,inview of(22), F;=F—«PF;, (22b) Fil +«P) =F. (22c¢) ‘The numerical factor Pisameasure for the attenuation ofthe external field bythepresence ofthemagnetizable material and ishence called the demagnetization factor. Acomparison of(22¢) and(21) yields foritsvalue 1-e/1, L+e ) P15(Juejt?-e: (23) dtispurely geometric incharacter. Forthetwolimiting cases of(21b) and (21c) wehave‘ P=1/3fore—0,sphere, P=»(log?-2)-+0 for0,rod. Intermediate values arereadily calculated from (23) (log always signifies inthis book thenatural logarithm) and aretabulated forexample, by Kohlrausch.’ Itisclear that this factor Phasalegitimate meaning only fortheellip- soidanditsdegenerate forms, since only here wearedealing with theratio ofoneuniform field toanother. Clearly theboundary-value problem for other body-shapes cannot becircumvented bytheemployment ofanu- merical factor which isguessed insome manner. Inpractice theprocedure consists ofmeasuring thevalue ofBexperimentally with aninduction coil atsome characteristic points (e.g. thecenter ofthetest body). §14. Some Remarks onFerromagnetism This section does notpretend tobeanintroduction tothebroad field of ferromagnetism, butmerely intends tomention certain important features 1F,Kohlrausch, Praktische Physik, 12th Ed., p.540.Itshould benoted that the factor 4x,bywhich Kohlrausch’s Eq. (4)differs from ourEq. (23), is,with us,in- cluded inthedefinition ofx.Seetheremark atEq. (9)above. 4 SOME REMARKS ONFERROMAGNETISM 97 which, since they lieoutside oursubject, will beindicated rather than logically derived. Ashasalready been noted atthebeginning of§12,ferro- magnetism isnotbased onMaxwell’s phenomenological theory, buton themore profound laws ofatomic physics and onthestatistical behavior ofelectrons. A. The Weiss Domains The sign oftheferromagnetic susceptibility anditstemperature de- pendence indicate that it,liketheparamagnetic susceptibility, results from thealignment ofelementary magnets inamagnetic field. Thefact thatitdiffers from theparamagnetic susceptibility inorder ofmagnitude shows, however, thatweareheredealing notwithindividual, freely mobile, magnets, butwith whole groups ofthem which, perfectly aligned within thegroup, have diffetent preferential directions. Theindividual group is “saturated” internally even intheabsence ofanexternal field, while ina macroscopic block offerromagnetic material saturation occurs only ata field excitation of10to1000 oersted. This concept oftheferromagnetic state isillustrated byamodel con- structed byEwing. Magnet needles arearranged inalattice onaboard. ‘Withtheearth’s fieldcompensated byacurrent loop,theyarrange them-selves ingroups orrows inwhich they areparallel. This state isstable against external disturbances, such asshaking oftheboard. Aweak ex- ternal fieldproduces onlyaslight deflection from theequilibrium position since theinternal aligning field ismuch stronger than theexternal field. Complete alignment inthedirection oftheexternal field, i.e.saturation of theentire system, takes place onlyataverymuch higher fieldstrength. Pierre Weiss haselaborated thisinterpretation offerromagnetism inall directions, both experimentally andtheoretically, andhas,together with Langevin, given itathermodynamic basis. Theindividual groups are known asWeiss domains. Their sizeisestimated atabout 10~cminlinear dimension, corresponding to5-10° Fe-atoms. Thesmallest ferromagneti- cally active domains arehowever certainly very much smaller andcontain fewer than100Fe-atoms.’ Itistempting toidentify them withthesingle crystals ofwhichapolycrystal ofthematerial iscomposed onamicroscopicscale. However, itisnecessary toassume such asubdivision into Weiss domains evenforthemacroscopic singlecrystal, sinceitsbehavior isquali- tatively similar tothatofthepolycrystal. (Itistruethatquantitativelytheshape ofthehysteresis loop, discussed below, differs from thatforthe polycrystal; ithasarectangular shape.) ‘See H.Kénig, Naturwiss. 1946, p.1. 98 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 14 B.TheElectron SpinasElementary Magnet Allferromagnetic materials areconductors ofelectricity, i.e.contain free electrons.! Wehave very definite reasons forregarding these electrons as theelements whose alignment causes ferromagnetism. These aretheso- called “gyromagnetic effects”: magnetization byrotation ofabarofamate- rialoftheiron group (J.8.Barnett 1914) and rotation bymagnetization ofasmallferromagnetic rodsuspended onatorsion fiber(Hinstein andde Haas 1915). Inboth cases measurements gave asratio ofthemechanical tothemagnetic moment halfthevalue tobeexpected iftheeffects resulted from the electron orbits inthe atom. Itmust therefore beconcluded that itisnotthecharge oftherevolving electron andthemagnetic fieldproduced byitthat areresponsible, buttheinner structure oftheelectron itself. Theelectron possesses, apart from itscharge, aninner mechanical moment, a“spin,” andamagnetic moment whichistwiceaslargeasthemagnetic moment which would beassigned classically toitsspin. This magnetic anomaly oftheelectron isthegeneral result which may bededuced from thetotal observed data ontheanomalous Zeeman effects (Goudamit and Ublenbeck 1925). Itexplains directly theobserved results ofthetwogyro- magnetic effects andproves atthesame time that theelectron spin plays therole ofthemagnetic needles inEwing’s model. Onthebasis ofthisdiscovery Heisenberg, in1928, with theaidofmodern electron statistics, wasabletoproceed toward atruephysical understanding offerromagnetism and tocalculate qualitatively theextraordinary magni- tude oftheinner magnetic field inaWeiss domain. Weseefrom thishow long theroad isfrom theMaxwell theory totheactual theory offerro- magnetism anditbecomes evident thatwecannot travel thisroad. C.Hysteresis Loop andReversible Magnetization The figure which represents themagnetization Mofaferromagnetic material asfunction oftheexcitation Hwith increasing and decreasing H iswell known. Ifthematerial isoriginally unmagnetized, the“virginal curve” isfirst traversed, beginning intheorigin H=0,M=0andpassing over into thehorizontal asymptote M=Msofsaturation forsufficiently large H.If,from this point, Hispermitted todecrease, thecharacteristic liesabove thevirginal curve andcuts theordinate axis inapoint H=0, M=Map, which indicates theremanent magnetization. IfHispermitted todecrease still further, i.e.isreversed indirection, aregion isentered in which BandHhave opposite directions. The iron specimen hasthen be- -come a“permanent magnet”. With further decrease ofHthehysteresis loopcutstheaxisofabscissas inapointH=—He,M=0,wherethe 1Atomie theory has notdemonstrated fully why just theatoms oftheiron group areferromagnetically active. 14 SOME REMARKS ON FERROMAGNETISM 99 remanent magnetization isjust nullified. H¢iscalled the‘coercive force”. IfHisdecreased further, thenegative saturation M=—Msisapproached. If,now, Hisonce more increased, thegradually rising characteristic remains below thedescending branch and below thevirginal curve. Itdoes notpass through theorigin, but cuts theaxis ofabscissas inapoint H=+He, M=O.The ascending branch issymmetrical through theorigin tothe descending branch and approaches finally once more thepositive saturation M =+Msz. Asarule Mzisapproximately 3M; .Inorder that themagnet may retain itsremanence forallopposing fields which occur itisimportant that Ho may beaslarge aspossible. This isthecase forhard steel (tungsten steel has He ~70oersteds). The ascending anddescending branches form together thehysteresis loop;itsareaisthemagneticwork$H-4B,whichisperformed onthema- terial inacomplete cycle. Iftheincrease ofHisstopped, intraversing the virginal curve, before reaching saturation, e.g.forHi<Hs, and Histhen permitted todecrease toH=—H,, thentoincrease toH=+Hi,a smaller hysteresis loop isobtained, which liesinside oftheonepreviously «described. Porverysmall Hy,=65Htheloopdegenerates intoatwice tra- versed Jine; itsarea becomes zero and theprocess isreversible. The ratio 6M/6H defines theinitial susceptibility xo.Itispossible tocarry outsuch areversible process notonly attheorigin, butatany arbitrary point ofthe cycle and todefine forevery such point areversible susceptibility Krev. With reversible magnetization the elementary magnets are deflected only slightly outoftheir original position inthedirection oftheexternal field. With irreversible magnetization some reorientations take place as well. Inboth cases changes occur intheboundaries oftheWeiss domains, which aredescribed aswall displacements; they aresmall forreversible, large forirreversible processes. Inaninduction coilwith telephone connec- tion they become acoustically noticeable bynoises (clicks) andcanbemade visible onanoscilloscope asBarkhausen jumps. Infact, forsufficient oscillo- scopic magnification, theapparently continuous course ofthe hysteresis loop resolves itself, particularly inthedescending branch, into asequence ofsmall steps. ‘The individual processes which gotomake upthemagnetization curves arethus ofvaried nature. They depend onthecomposition oftheiron speci- men and onitsmicrocrystalline structure; even forthesingle crystal they depend ontheorientation relative tothemagnetic field.Itistheproblem ofthemetallurgist tofind thealloy (permalloy, perminvar, cobalt steel) suited foreach purpose (transformer laminations, communications engi- neering). 100 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 15.1 D.Thermodynamics Ferromagnetism iseven more dependent ontemperature than para- magnetism. Above acertain critical temperature ferromagnetism ceases andpasses over into ordinary paramagnetism. This critical temperature is designated with @andiscalled theCurie point. Onthecentigrade scale (0=‘ont +273) wehave: foriron cobalt nickel Goon= 770 1120 358 ForT>6wehave inplace ofCurie’s law(13.10) theCurie-Weiss law Cc x=Foy () This suggests that there may benodifference inprinciple between para- magnetism andferromagnetism, that, inother words, theCurie point, in theformer case, liesclose toabsolute zero. With this assumption aregion _ offerromagnetic behavior near7=0istobeexpected alsoforordinary paramagnetic materials. Itthenseemsreasonable totransfer Langevin’sstatistical and thermodynamic theory ofparamagnetic materials tothe conditions offerromagnetism. Infact Weiss inthis manner arrived atarepresentation ofthewhole complex offerromagnetic phenomena which, initsmain features, issatis- factory. However, this representation utilizes concepts with which wewill only beable todeal inVolume V.Also detailed questions ofatomic physics play here arole, such asthequestion oftheatomic unit ofmagnetic moment (Bohr’s magneton ascompared with theWeiss magneton, which issmaller byafactoroffive)andthequestion towhatextent, inaddition tothespin moment ofthefree electrons, the orbital moment ofthe electrons bound intheatom, which istwice aslarge, must beconsidered. The standard textbook’ ofR.Becker andW.Déring gives complete information onall relevant questions. §15. Stationary Currents and Their Magnetic Field. Method oftheVector Potential Since theassumption ofstationary fields demands 0/dt =0throughout, theMaxwell equations (4.4) reduce to curl E=0, J=culH. (1) 2Rerromagnetismus, Springer, Berlin 1999.Foralessdetailed treatment, seeF. Bitter, Introduction toFerromagnetism, McGraw-Hill, New York, 1937. 15.48 STATIONARY CURRENTS AND THEIR MAGNETIC FIELD 101 The second ofthese leads to divJ=0. (2) For thesurface ofaconductor carrying current wehave therefore Jn=0. (2a) (2)states that theelectricity within aconductor behaves like anincom- pressible fluid (increase and decrease ofcurrent density foranarrowing andbroadening oftheconductor, respectively). Also Kirchhoff’s branching laws forlinear conductors, which Kirchhoff worked out assolution toa seminar problem given byF.Neumann (seep.1),restinthefinal analysis on(2)and theexistence oftheelectrical potential (see below). Eqs. (1)state that Ehaseverywhere apotential, E=—grad ¥,,while Hhasascalar potential, H=—grad ¥,,,only outside ofthecurrent-carry- ingconductors. Wewilldeal with this scalar potential in§16. Here weshall give arepresentation ofB(and hence also one ofH)which isvalid both inside andoutside oftheconductors. Werecall here Helmholtz’s representa- tion ofthevelocity field vforgiven distribution ofturbulence winVol. I,Eq. (20.13). InHelmholtz’s analogy theelectric current density J corresponded’ ,tothesecond, themagnetic excitation H,tothefirst.Just agthere, weintroduce avector potential A,bysetting? B=curl A. (3) The second Eq. (1)then becomes 1eurl =curl A=J. (4) B For constant »wecan write instead curl curl A=yJ. (4a) 1Butforafactor #whose suppression wasjustified inVol. II,p.15bytherequire- ments ofelectrodynamics. ?Itiscustomary towrite instead H=curl A,which, however, assumes thecom- plete absence ofsources ofH,acondition whichisnotfulfilled inregionsofnon- vanishing magnetic density pm.Our formula (3)ismore satisfactory, since div B=0 throughout; inaddition, itwill generally simplify ourformulas, particularly in PartIII.Incidentally, both formulas amount tomuch thesame thing iftheassump- tion ismade thatuisconstanteverywhere, whichoccursalreadyinEq.(4a)ofthe text. Fornonconstant 4thesummation problem tobesolved in(7)would have to besupplemented byamagnetostatic boundary-value problem (determination ofthe discontinuity ofBuseattheboundary between media ofdifferent permeability for ourexpression forAanddetermination ofthemagnetic surface densities appearing there fortheusual expression forA,respectively). 102 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 15.4b Wehereemploy thegeneral transformation (6.2)withtherestriction toCartesian coordinates which isthere emphasized andobtain instead of (4a) theform which ismore convenient forintegration: AA~graddivA=—uJ. (4b) This may besimplified bythesupplementary condition divA=0, (6) which transforms (4b) into 4A=—al. (6) ‘Thecondition (5)may beadded since A,forgiven B,isdetermined by Eq.(8)withtheexception ofthegradient ofascalar function. Thelatter maybeutilized tosatisfy (5).For,ifA:isanysolution of(3), A= A.+gradf (6a) issimilarly asolution; ifwenow write divgradf=Af=—divAi, (6b) which, according tothewell-known integration procedure ofPoisson’s equation isalways possible, wefind divA=0. ‘Thismethod ofintegration yields atthesame timeasthesolution of(6): dp=fwJadre| (7)Tre Here thepoint ofintegration Q=&,n,¢traverses theentire interior ofthe conductors; P=z,y,zisthereference point forwhich theCartesian com- ponents A,,A,,A,aretobecalculated. Itwasshown inVol.I,§20,Nr. 2athatthisrepresentation satisfies (5)provided thatJissolenoidal, in accord with (2),andthat»isconstant. Theintegration inEq.7istobe extended over theclosed current field J(just asinVol.IIover theclosed vortex rings). Thecurrent density Jappearing in(7)maybeobtained assolution ofa potential problem. Since J=cEthepotential equation applies, forconstant ¢,just asmuch forJasforE: AJ=0. (8) -Forvarying «Eq.(8)takesonasomewhat morecomplicated form.The total current Jisobtained from Jbyintegration over anycross section ofthe conductor: T=|Jade. ® 15.11a STATIONARY CURRENTS AND THEIR MAGNETIC FIELD 103 ‘Thefamiliar factthatJhasafixed value independent ofposition and shape ofthecross section follows from integration ofEq. (2)over aseg- ment oftheconductor bounded bytwo arbitrary cross sections asforthe analogous spatial lawofconservation ofvortex theory (seeVol. II,p.136, Fig. 24). Weshall now give some applications ofour representation (7). A.TheLaw ofBiot-Savart We subdivide the three-dimensional conductor into current tubes with thecross section dg,normal tothetube axis, and theelement oflength ds; thecurrent J,dginsuch atube element, which hasthesame dimension asthetotal current J,weshall also callI,forthepresent. Wethen can set : 1 Fig. 19.The law ofBiot-Savart, derived from the vector potentialofanelementofcurrent. ‘s . e Jdr=Ids,where thedirection ofthecurrent flowJisindicated bythe vectorial character ofds.By(7)thecontribution ofourtube element toA then becomes 4xda=HLSr and,by(3),thecorresponding contribution toBis, 4raB=cuntLSS, 10) ‘Weshorten theremaining calculation byemploying thesymbolic vector V: curtMf9Svxae(eraa*)xalds. (a) For Idsisdependent onz,y,zindirection, but not inmagnitude; 1/r depends onz,y,zinmagnitude, butnotindirection (being ascalar). We seefurthermore from Fig. 19that . L r egad-=-5=-5, (1a) where rdenotes theradius vector from the current element tothe reference Point and ethecorresponding unit vector. Substitution of(11) and (11a) 104 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 15.12 in(10) yields: teB=Hexds,tejaB|=8sino, (12) #signifies theangle between thevectors eand ds;thedirection ofdB, inview ofthenegative sign in(12)andthemeaning ofthevector product, isthat ofaleft-hand screw for the direction ofrotation e—ds. Letus imagine amagnetic unit pole atthereference point and letthevector dB actonit;dBthen represents theBiot-Savart force exerted ontheunit pole bythecurrent element Ids.The corresponding lineofforce then surrounds thecurrent element, inaright-hand screw direction, asshown inthefigure and asexpected. Evidently thefirst Eq. (12) isthemore complete one, since itexpresses thedependence ondirection which ischaracteristic of themagnetic field inthesimplest and most appropriate manner; wehave added thesecond form merely because itisthehistorically more familiar one. B.TheMagnetic Energy oftheField ofTwo Conductors Ifwedenote theenergy integrated over space byWweobtain, forthe thagnetic energy density W,‘rom Eq. (5.6) aw=fH-Bar= fH-curlAdr. *(13) For theevaluation weutilize thevector formula (5.2), previously derived inconnection with the Poynting theorem, which werewrite interms of ourpresent symbols (A, Hinplace ofU,V)asfollows: H-curlA=A-curl H+divAXH. (14) We assert that thesecond term ontheright vanishes intheintegration over infinite space. According toGauss’s theorem thisterm yields faivax Har=[(AXBade, (142) where theintegration ontheright istobecarried outover asurface bound- ingtheregion atagreat distance, e.g.asphere ofradius R.Letthetwo conductors, whose total magnetic energy istobedetermined, beentirely confined toafinite region. The distance ofalltheir points from theinfinitely distant element ofarea doofthebounding sphere may then besetequal to theconstant value R.ByEq.(7)Aapproaches zero ondoas1/Rand, by thelawofBiot-Savart, Happroaches zeroas1/R*. Since do=R*da(da = solid angle intercepted bydo), theright sideof(14a) approaches zero as RR. 15.17¢ STATIONARY CURRENTS AND THEIR MAGNETIC FIELD 105 Inviewof(1),(14)and(13)leadto 2W=fAcoustwar=or (15) Theintegration isnowtobecarried outonlyovertheconductors 1and2, since Jiszeroeverywhere outside ofthem. Ifwedesignate thepoint of integration in(15)byP(drpinstead ofdr)andifwesubstitute A=Ap from (7),thesimple volume integral isreplaced byadouble volume in- tegral: 2w ff arpdre pL ee 0) Inevaluating thisintegral wehave todistinguish fourcases, depending ontheposition ofPand Qonconductors 1and 2: a.PandQonl, b.PandQon2,c.Pon1,Qon2, d.Pon2,Qon1; thecases canddarehowever alike inview ofthesymmetry of(16)with respect toPand Q.We can write theresult intheform W=Flu li+Lali +Wel 1), a7) ou ayandr _#ffsudndrinafff, tea2ffien@%, ar) #ffs.4dnidr tasEffiv are) Thefactor 2intheproduct term in(17)results from theequality ofcases cand d,which causes thecoefficient Lyalsotobegiven by(17b), i.e.Le= Ly.T,,Izarethetotal currents inconductors 1and2,which, asnoted at(9),areindependent oftheplace intheconductor. 1,1/aretwopoints ontheconductor 1;2,2’two points onconductor 2.Division ofthecur- rent densities Ji,Jzby1,J2leads tothepurely geometrically defined “current-line density vectors” a-2, yaka h-7- (17c) TheL’sarecalled induction coefficeents: Ly, Lmarethecoefficients of self-induction, Ly:isthecoefficient ofmutual induction. Maxwell uses the letter Minplace ofLy. Theunitofthecoefficients Z(orM)isthehenry.’ Inaccord with (177° 1Joseph Henry, 1792-1878, American physicist, discovered almost simultaneously withFaraday theappearance ofanelectromotive forceinacoilwhenthemagneticfield initsinterior ischanged. 106 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 15.18 this unit isfixed invalue and dimension bythestatement joulejouleS* Thenry =1° =1. (1g) Converted into electromagnetic cgs-units wefind, since Q=10cm'g', 1joule=10”cm’gsec, (18a)lhenry =10°cm=1quadrant oftheearth. From ourstandpoint wecan, however, attach nosignificance tothisappar- ently sosimple dimension andthisrelationship totheearth’s circumference, since itrests onthearbitrary assumptions oftheelectromagnetic system ofunits. Atthesame time weareglad topoint outthefollowing relation between thehenry andthepermeability invacuum given by(7.16a): =4-107joule-S? =dp-107?ReBEY bo=4n-10"Gag=4x10" (18b) Compare with this theanalogous relation between thedielectric constant ofvacuum and thefarad asgiven byEq. (10.3b). C.Neumann's Potential astheCoefficient ofMutual Induction In(17b) itispossible topass tothelimiting condition oflinear con- ductors, i.e.infinitely thinwires. In(17a) thisisnotpermitted since ri (orri)would vanish asthetwointegration points approach each other and theconvergence oftheintegrals would bedestroyed. Wewrite in(17b) dr,=dgids, dr,=dqsda (19) andcombine dg;andji,dg:andjy.The products j,dg:andjzdgthen have, by(17c), unit magnitude and their scalar product isequal tothecosine oftheangle :between thetwodirections offlow ds,andda,.Hence: an 008 613 dsy-ds.Flamfanfans=[Se a This amazingly simple andbeautiful representation wasdiscovered’ by Franz Neumann asearly as1845. Itisknown asNeumann’s potential; according to(17) itrepresents that portion ofthemagnetic energy which results from theinteraction ofthe two circuits. Arelative displacement “orrotation ofthetwo circuits with thecurrents J,and J;leftunaltered hence isaccompanied byachange inenergy 3Wintheamount — bW =Tl; bly. (208) 1Abhandl. Preuss. Akad., reprinted inOstwald’s Klassiker, Nr. 10. 15.22 STATIONARY CURRENTS ANDTHEIRMAGNETIC FIELD 107 Theworkwhich must bedone inadisplacement orrotation, aswellasthe forceortorque which onecircuit exerts ontheother, arerelated tothis. Ingpite ofthesimplicity ofexpression (20)theactual calculation ofthe mutual induction coefficient israther inconvenient. Tobegin withwework outaformal mathematical example, i.e.twostraight parallel wires oflength Iseparated bydistance a.Thecondition ofclosed circuits stated inEq. (7)istemporarily notfulfilled here. Weshalltakedueaccount ofitonly a u Fig. 20.The coefficient ofmutual induction oftwo SB dy:straight,parallelsegmentsofwireoflengthJ.Thefinite54 crosssectionofthewiresisindicated atthebottom of; a {Pa~ ' ‘ tl.be when wereach Eq. (24). Referring toFig. 20,wehave (dy,dy:inplace ofds,dey, cos A=1): ae :, dyspoemfon Ge @ The formula ofintegration already employed in(9.4) yields forthesecond integral [ dysjoVat +(ys—1)? (22) =log—n+Var+Owy)—log(—-m +VattW)- Afurther formula, which may readily bechecked bydifferentiation, flog(t+Vat+4)dt=2loge+Vat+2)—Vat+a+const yields fortheintegration ofthefirst term ontheright of(22) 2 [autoett—n+VET =Log+Ve+t)-—Ve+o+a 108 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 15.23 andfortheintegration ofthesecond term i =[aytog-+VED =<log( <4. VFR) —VFB + Asimple transformation under thelogarithm. sign yields forthesum of the two @+ — ng=logtVERE2VEE+20(23) Weassume 1>>a,and obtain asafirst approximation "Ly=21(log 1), (24) # a Itisworth noting inthisresult that wehave notobtained simple propor- tionality to/,sincetheparenthesis depends on/logarithmically. Accord- ingly wecannotspeak ofamutual induction coefficient perunit length ,ofourwires. '.Thereason forthisisthefollowing: Ourderivation assumes, ashasbeen stressed repeatedly, twoclosed circuits, while ourexample deals with two circuit segments. Ourresult (24) isnevertheless meaningful. For example, itispossible todetermine with itsaidthemutual inductance oftwo par- allel rectangles such asoccur inAmpere’s basic experiments. For such apair ofrectangles (the second rectangle issupposed tobeobtained from thefirst byaparallel displacement perpendicular toitsplane) only par- allel pairs ofsides contribute; formutually perpendicular sides theproduct. ds,-d8, occurring inEq.(20) isequal tozero. The mutual inductance of twosuch rectangles becomes equal tothesum offour terms oftheform of(24). D.The Coefficient ofSelfinduction Asalready noted wecannot inthis case pass tothelimit ofthelinear conductor, butmust return tothedouble volume integrals in(17a). We canreadily convince ourselves, however, that then any convergence diffi- culty isavoided. Forif,foranarbitrary position of1,weemploy polar coordinates r,#,ywith thispoint asorigin tolocate thepoint 1’,dr’ = 1°drsin6d8dgand thedenominator rn’=rcancels oneofthefactors rindry’. However thecarrying outoftheintegrations becomes now, in general, even more awkward than inC. a Wetherefore limit ourselves toasimple mathematical example, namely astraight wire ofthe(great) length Jandthe(small, butfinite) cross sec- tiong.Weconsider twocurrent filaments parallel totheaxisofthewire 15.27 STATIONARY CURRENTS ANDTHEIR MAGNETIC FIELD 109 (y-axis) and employ once again Fig. 20,where now thetwo linear currents areassumed torefertothesamewireofcrosssection g.Letdq:,dgzbethe cross sections ofthe two current filaments; their separation, formerly denoted bya,willnowbecalledpsinceitisvariable, depending onthe position ofthetwo current filaments within g.The volume elements are once again given by(19), with ds;=dy, ds=dy:, and thevectors defined in(17c) become A= =t q The defining equation (17a) fortheselfinductance then takes theform 4x dg:dgs f'f* ___durdys=L= SS. 25)ceara The second double integral hasexactly thesame form as(21). Wecan utilize the approximate evaluation (24) here also and obtain dn 21 21Fa=|]onan(v3) 21=AX[fanen(log21—1)—Ifatest6. Inthefirst term ontheright theintegration with respect todg,anddgz canbecarried outeasily; thesecond term requires more detailed discussion because ofthevariability ofp=pz=separation ofourtwo current fila- ments. Wenote thepreliminary result $1=atflog 1-1-toga}, logs=4fdorfdarlogpx.(26) # ¢ Maxwell calls thequantity ghereintroduced themean geometric separation oftheelements dq:,dg:within thecross section g.’Itcanbedetermined more elegantly byanelectrostatic consideration than bydirect calculation. Interms ofpolar coordinates thetwo-dimensional potential equation becomes vwob_1dde1ow Aboattatpde?dptae0. (27) 1Treatise, art.691f.Inexplanation ofthenotation weremark: The integral to beevaluated in(26)isthearithmetic mean ofallvalues oflogpoccurring onoursur- face g.Ipview ofthe relation Zhogaw=logOy this arithmetic mean ofthelogarithms isatthesame time thelogarithm ofthe geometric mean ofallthep;. 110 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 165.28 Apart from amultiplying and anadditive constant thesolution which is independent ofyisknown tobethe“logarithmic potential” &=logp. This signifies, intwo dimensions, anegative charge concentrated atthe pointp=0.Ifthecharge isdistributed overtheareaqwithapositive surface density f,andwith thesurface element ofgbeing designated with dq,,Green’s theorem yields foritspotential atthereference point 1. ae,=—|flogpudas. (28) " This isthetwo-dimensional analog tothefamiliar Eq. (7.5). If,in(28), weputf=—2x,weobtain theinnerintegral in(26), &=[108psdos (28a) G and ourdesired mean geometric separation may bewritten 1logs=5|dn. (28b) Theintegral (28b) canbereadily evaluated inthecasethat¢isacircle, e.g.ofradius b,and thepoint 1coincides with thecenter ofthecircle. We then have py:=9,i.e.equal tothepolar coordinate employed previously, anddg;=pdpdy.Wedenote by&pthecorresponding specific valueof&,. Eq. (28a) then takes the form 2% a=[de[108»»dp (29) The integral with respect topcanreadily beevaluated byintegration by parts; thus > 2 » % 2 =f -[2ar=% - ilogppdp=5tog0f340=5(log—9). Hence (29) leads to ;= b¥(log b—4). (29a) Furthermore #;,asthepotentialforaknownsurfacedistribution, ean becalculated forarbitrary location ofpoint 1directly from Poisson’s equation, which, inanalogy to(7.42), intwo dimensions takes theform Ab =—f, 15.38 STATIONARY CURRENTS ANDTHELR MAGNETIC FIELD iil where fisthesurface density. Inourspecial case (f=—2r, circle qof radius b,&,afunction ofponly)itbecomes ld d&3ha (30) Integrated twice this yields ds a°_77 +A, Hm>t+Aloge +B. Inorder that thisexpression for#,maypassover,forp=0,intotheexpres- sion (29a) for4), wemust set 1 A=0,Bud,hence=28(log6-1438). (1) Ifwesubstitute thisexpression for®,in(28b) weobtain . 1 _ 1 os tog3=+fdgilog—»)+5,fdn§ip (82) - r: = - =logb-3+H [ohde=logd—2. For#itselfweobtain fromthisthepeculiar value B=o/Ve. (32a) Here theproportionality with }israther obvious inview ofthedefinition ofthegeometric mean (seelastfootnote); however thenumerical factor is tobefound only bydetailed analysis, which following Maxwell, wehave herebased onpotential theory. Itmaybementioned thatMaxwell carried outthese considerations even forcross sections ofarbitrary shape. Topursue ourrealgoal, thecalculation oftheselfinductance L,wereturn to(26). We then find, utilizing (32) 4n 3 a3;L=at{og21—logb}21{tog7.(33) This formula confirms ouroriginal expectation that thetransition to thelinear conductor (6—>0)isnotpermissible fortheselfinductance. Regarding thedependence onIwemust make thesame remark asatthe endofsection C:Wecannot, bydivision of1,obtain theselfinductance per unitlength ofaninfinitely longwire; however, wecan, a8forthemutual inductance, piece together theselfinductance ofanyclosed circuit, made upofstraight wires, byadding upterms oftheform of(33). Weshall further answer rather obvious objection which may beraised against thesomewhat indirect derivation of(33). Themagnetic fieldofan 112 DERIVATION OF PHENOMENA FROM MAXWELL EQUATIONS 15.338 infinitely longwireisknownfromFig.4andthecorresponding equations (10) to(13) onp.24.Cannot theenergy and selfinduction ofthestraight wirebecalculated much more directly from them? » With Jascurrent wefound (writing now 6,cinplace oftheearlier a,b) rt I r<bH=5a;r>bH= a. Hence thecontribution oftheinterior ofthewire totheenergy perunit length is Bor[Hrdr= £Etar=HP (33a) 2 0 4xbt Jo 16x and the contribution ofthe exterior: a a# pdr He [Ho # Plog?Boefwrar=Ar[Fm#rg’. (336) From thiswefind fortheselfinductance perunit length, from theenergetic formula ofdefinition (17), xl coe¢+log‘). (33¢) This expression becomes logarithmically infinite ifweletc>©,i.e.pass over tothesingle wire without return conductor. Our intended simplified derivation hence fails—quite understandably—as the result ofthe un-- physical assumptions oftheproblem, which must lead toaninfinite energy content ofspace forany unit length ofthewire aswepass tothelimit cm. E.Selfinductance oftheTwo-Wire Line The system oftwo straight parallel wires traversed bycurrent inop- posite directions plays animportant role inelectric power transmission and isknown, inHertz’s experiments with high frequency waves, asa Lecher system. Weshall determine theselfinductance perunit length of such asystem (with appropriate restriction tothedirect-current case). Soastobeable toutilize Fig. 20,weshall call the(very great) length ofthewires J,their separation a,and their radius b.Weproceed from the energetic formula (17), where weput hel, he=-l Ly=ln=L. We then obtain W=34D!', Le=XL—Ly). (34) 15.36 STATIONARY CURRENTS ANDTHEIR MAGNETIC FIELD 113 Wecandesignate Ln,introduced here, asinductance oftheline,regarded asauniform system.’ Substitution from (33)and(24)yields: Kz 23 2t Lo#1(ioe? ilog>+1). Inevaluating thelogarithmic terms thetwoterms +log21drop outand thefollowing simple formula remainé Lou ().@4) T={eg ta) (35) L,/l isthedesired selfinductance perunit length ofthetwo-wire line, which evidently isindependent of!.Hence thetransition to1—©,which, according to(38)and(24), wasinappropriate intheexpressions forL/I andLy/l because oftheterm log2i,isnow feasible. This isobviously re- lated tothefact that thefield ofatwo-wire line traversed byoppositely directed currents corresponds tothat ofacircuit closed atinfinity. Onthe other hand thetransition tothelinear two-wire line (b>0),which was possible intheexpression forthemutual inductance, cannot becarried outeven now. \ F.General Theorem Regarding Energy Transmission byStationary Currents Weconsider thesection ofanarbitrarily shaped wire between two cross sections F;andF;.Letitcontain a“load” inwhich electrical energy is translated intowork orsome other form ofenergy, e.g.alight bulb. Weask what power issupplied totheload (theJoule heat generated inoursection ofwire tobecounted aspart oftheload), ‘Weextend thecross sections F;,F;toaclosed surface Fandcalculate thepower Nastheinward-directed energy fluxthrough thissurface. Ac- cording toPoynting’s theorem (5.7a) andthemeaning ofJoule heat (5.5) weobtain under stationary conditions N= dF=|E-Jav. [svar=fay (86) Visthevolume enclosed byF.Under stationary conditions wehaveevery- where within V curlE=0, E=-—grad¥, henceE-J =—grad ¥-J. Wetransform thiswith theaidoftheobvious anduniversally valid identity .div(¥J) =gradW-J+wdivJ - 1Asimilar definition isemployed inelectrical engineering formultiple conductor systems with thedesignation “operating selfinductance,” ifunder theconditions of operation allcircuit currents aredetermined byoneofthem. 114 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 15.37 where Jisanyvector, ¥,anyscalar. Forourmeaning ofJthelastterm vanishes, We hence deduce from (36) Ne=-fdiv(vJ) eV (37) and, byapplication ofGauss’s theorem v=[wnar. (38) p ‘This integral need only becarried outover thesurfaces ofentry and exit, F,and F;, ofthecurrent, since only here J,differs from zero. Itismost convenient tochoose F;and F;astheequipotential surfaces ¥=W,and W=W,. Wethen obtain from (38) Newflnartef Jadhs. (30) tay Pa With theconvention regarding thedirection ofnestablished above [tan =—faan=1 and henoe, by(39), N=(%—WI =VI. (40) Visthevoltage dropbetween F;andF;(involts) andJ,thecurrenten- tering and leaving (inamperes). This fundamental formula, which expresses thepower directly inwatts (joules/S), hashere been derived forstationary conditions; in§18itwill befound applicable also to“quasistationary” states. Toavoid misunder- standings itshould benoted that Eq.(40)makes nostatement regarding thebrightness ofthelight bulb ortheenergy radiated byit.This phe- nomenon liesoutside ofthedomain ofMaxwell’s theory andrests onatomic processes which aremade possible bytheJoule heat supplied tothefila- ment, butwhose energy balance hasnothing todowith Eq.(40). Our Eq. (40) follows theenergy conversion only uptothegeneration oftheJoule heat, notbeyond it. §16. Ampéere’s Method oftheMagnetic Double Layer Inthelastparagraph wehadtointroduce theconcept ofthevectorpo- tential'tn order toarrive atarepresentation ofthemagnetic field without and within thecurrent-carrying conductor. Ifwecontent ourselves witha representation which applies only outside oftheconductor wecangetalong with the ordinary scalar magnetic potential ¥. 16.38 AMPERE’S METHOD OFTHEMAGNETIC DOUBLE LAYER 15 Outside oftheconductors wehave, by(15.1), 0=curl H, H=~—gradv. () Ifweadd thecondition divB=0andassume uniform permeability, e.g. #=wo,outside, wehave furthermore divgrad¥=AY=0. (2) This potential ¥isnot, however, aunique function ofposition, asinthe magnetostatic case. For every closed loop about aconductor carrying Titchanges bytheamount J,independently oftheshape and length of the path: ‘Woke guds=+1. (3) Here ¥;and¥;arethevalues of¥atthestarting point 1and thecoinciding endpoint 2oftheloop. The upper orlower sign ofJapplies depending on whether theloopformsarightorleftscrew withthedirection ofthecurrent. ,Ontheother hand, forevery closed path which does not link such acon- ductor: . W-h=fa.ds=0. (8a) The proof ofboth Egs. (3)and (a) follows again from (15.1). Anarbitrary surface obounded bythepath ofintegration cuts theconductor inquestion inthefirst instance; inthesecond itcan always beplaced sothat itcuts noconductor. Ifthecomponent ofEq. (15.1) normal toevery surface ele- ment doisformed and integrated over alldc,weobtain [Judo =fcurl,Hde. By(15.9) theleft side isthe total current traversing thesurface o,i.e. JforEq.(3)and0for(3a).Therightsidemaybetransformed byStokes’s theorem into theline integral over s. Formultiple loops about theconductors inoneortheother direction achange inWisobtained which isequal tothesum ofthechanges corre- sponding toindividual circuits: . h- ve=Link. - Here n,denotes the number ofcircuits about the kth conductor. Thus ¥ isinfinitely multiplevalued. Any two “branches” of¥differ byaconstant which isasum ofthecurrents J,multiplied byinteger coefficients m,. 116 DERIVATION OF PHENOMENA FROM MAXWELL EQUATIONS 16.4 A.TheMagnetic ShellforLinear Conductors Togive aprescription forthecomputation of¥which isunique inspite ofthis multiplicity wemust confine ourselves tothelimiting case oflinear conductors (cross section —0);furthermore, itwill suffice forthepresent toconsider asingle conductor. Let itbeA.Through Aasboundary we place anotherwise arbitrarily shaped “branch cut” surface Sand forbid passage through S.Inthis manner weselect, from theinfinitely many- valued potential, a“function branch.” Carrying over thealready somewhat daring language oftheRiemannian surfaces into three dimensions, we could also say: Ofthe ‘Riemannian space,” whose infinite number of “leaves” have thebranch line Aincommon and arejoined inthebranch cutS,weseparate outoneleafasalone physically significant. This leafhas become “singly connected.” The calculation of¥may now becarried outbysimple application of Green’s theorem: J(wav—va)dr=[(u%-4)do. @) on én, Here weput, u=W, ve, r=Tq (4a) andcarry outthevolume integral ontheleftover allpoints Pofourphysical leaf, excluding asphere ofradius p—0about thesource point Qand a sphere ofradius R+©cutting offthe infinite; the latter sphere may have anarbitrary point Oascenter, which may forexample besituated on S.Correspondingly, thesurface integral ontheright istobecarried out over thetwo spherical surfaces K,and Kr, aswell asthetwo “sides” of thebranch cutS.Since onthesphere K,, av dl 1“n=©(3).7 are integration over dohere yields evidently 4rVq. (5) The sphere Kxcontributes 1 _i|ov . plvd-pfSa. Accoraing tothe law ofBiot-Savart Hdecreases with increasing Ras 1/R?. Hence thesecond oftheabove integrals isfinite and, with itsfactor, vanishes as1/R. The first integral becomes infinite only inproportion to 16.78 AMPERE’S METHOD OFTHE MAGNETIC DOUBLE LAYER 17 Randyields similarly avanishing contribution when ‘multiplied with 1/R’. Thus thecontribution ofKeiszero. Wecould confirm thisresult which ishere based onthelawofBiot-Savart, i.e.themethod ofthevector potential, also bythemethod ofthis paragraph. Finally, wemust consider thetwo sides 1and 2ofthebranch cut. In view oftheopposite direction ofnonthetwosides wehave ou ov(5), -G)- ®) Since du ov inan~Me isaphysical quantity which hasnothing todowith themathematical fic- tionofourbranch cutwehave, inaddition to(6), ou ou(@),_-(3), (6a) ‘The sum ofthecontributions ofthetwo sides ofthebranch cuttothe rightsideof(4)mayhencebewritten or 0 /{(-w(2)-@-%9 ()}de, (6) Here v,—2vanishes because ufthemeaning ofv=1/r,but,byEqs. (3) and (4a), u—uw=+J. Hence (6b) takes theform a1afr 22a. @ Tofixthesign, consider Fig. 21.Here 1istodenote that sideofSon which thenormal n,directed toward Sforms aright screw with thedirection ofcurrent flow.Theloopfrom1to2showninthefigurethenformsaleftsorew with thedirection ofthecurrent. Hence byprescription (3)wemust choose thenegative signinformula (7).Ifwenowwrite nformandplace thefactor Z,which isaconstant forallpairs ofpoints 1,2,ahead ofthe integral sign, thisbecomes al-1f hae. (7a) Together with(5)wethenobtain asthevalue oftherightsideof(4): a1 trvo—1fhae, 8 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 16.8 Since theleftside of(4)vanishes because Au=Av=0,weobtain asthe Sinal representation of¥: al fart=1[22as, @) The integral onthe right has avery simple geometrical significance: Itisthesolid angle &intercepted bythecircuit Aatthepoint Q.Infactthe integrand inEq. (8) ai, 1 donmr” "acos(n,7)do=z= isthesurface element dfoftheunit sphere about Qcutoutbytheradii directed toward théboundary ofdo;dc,isthecorresponding surface ele- ment ofasphere ofradius 7.Hence Q= 8a,faf (8a) isthetotal area cutoutontheunit sphere bythecone ofradii directed toward thtboundary ofS,i.e.theabove mentioned solid angle. ry Fra. 21.The magnetic line integral about the conductor A,ex f tended from side 1toside 2ofthe branch cut S. I a The potential jump atthebranch cut, ¥;—2,now also acquires acer- tain simple meaning. Forifweplace ourpoint Qonside 1ofSthecone ofradii degenerates into aflatfanandthesolid angle @to2x;ontheother hand, ifweplace itonside 2,thesolid angle becomes 2=—2m. Ifweform thedifference ofEq. (8)forthetwo cases wefind 4x(%i —Va)=(2x —[-2r]), i.e.thepotential jump demanded by(3). We will supplement this geometrical interpretation ofthe expression _(8)byamagnetic interpretation: Wespeak ofadouble layer onthebranch cutSwiththemagnetic surface densities +wm,which wethinkofasdis- tributed parallel toSatadistance dnfrom each other. Weregard thecur- rent Jasthemoment perunit area ofthis magnetic double layer: I=wadn. (9) 16.10 AMPERE’S METHOD OFTHE MAGNETIC DOUBLE LAYER go Eq.(8)may then bewritten al tee=fondn2hae. (9a) Ifwedesignate byr,thedistances ofthepoint Qfrom thepositive and thenegative layer respectively wehave al 11 onde fiademr ee[= foe. (9b) Accordingly Eq.(8)isifact thepotential ofamagnetic double layer whose moment hastheconstant valueJoveritsentire surface. Following Amp3re wecallthecarrier ofthis double layer amagnetic shell; thelinear con- ductor Aforms theboundary ofthis shell. Inthisconnection westate agenerally valid lawofpotential theory for simple anddouble layers: Asimple layer ofthetype (7.5a) eo : ary=[de ‘Jeaves thepolential¥continuousinpassingthroughthecarriersurfaceo, but leads toadiscontinuous normal component ofthepotential gradient, since ()-@)-»anji \an/s , ontbeother hand, adouble layer ofthetype (8) al are=[dha makes thepotential discontinuous, but leaves itsgradient continuous. We have here WU-wW= 1,butBH,—H,=0. B.Magnetic Energy andMagnetic Flux The calculation ofthemagnetic energy ofalinear conductor inamedium ofuniform permeability, ie.the carrying out ofthe integration inthe expression defining this energy: 1 ufWa5[apa=2faas, (10) becomes particularly simple bytheabove method. Weutilize here the so-called “second form ofGreen’s theorem” (Vol. II,Eq. 3.16) which 120 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 16.11. becomes, . ou frvsude+ fgradwgradudr=fuSde rh) ifthetwo functions uand voccurring there aresetequal toeach other. Weset_u =Wandextend theintegratiou ontheleftover theentire exterior ofour linear conductor, having made thepotential Yunique with the aidofthebranch cutS.The integral ontheright isthen tobecarried out over thetwo sides 1and 2ofS.Wecanomit theintegration over thesur- face bounding thespace considered atinfinity inview ofourknowledge regarding thebehavior of atinfinity. The first term ontheleftof(11) vanishes since AY=0;thesecond is identical with 2W/u since grad Y=—H. Summing over sides 1and2 with dueregard oftheobposite signs of2¥/dn, theright sideof(11)becomes [on wae =1fHade, (11a) on where theestablished rule regarding thecorrelation ofthesign ofthe normal nand thedirection ofthecurrent istobeobserved. Wethus obtain from (11) 1 = =-I¢: War|Hede=318; (12) Here®is‘hefluxofthemagnetic induction through ourconductor A. Aparticularly simple definition ofthecoefficient ofselfinduction Lmay bededuced from (12). For, ifwecompare (12)with theequation ofdefini- tion (15.17) specialized forasingle conductor, W=iur (13) wefind directly S=LI, L=2/I. (14) Wenow pass from thesingle linear conductor considered sofartotwo linear conductors A;and A;.Wemust then make space “singly connected” bymeans oftwobranch cuts S;andS;,which arebounded bythecur- rents J,and I,.Weshall cail thenormals correlated toJ;and Isbythe right screw rulemandn;.Bysuperposition themagnetic fields of7,and I;formthetotalfieldH=H,+Hz.Intheintegration overS;andS:,tobecarried outasin(11a), there appear theexpressions tfG+Bdedosondff(i+Hades.ay es 16.17 AMPERE’S METHOD OFTHE MAGNETIC DOUBLE LAYER 121 Thus (12) isreplaced byanenergy expression offour terms, which we shall write, asin(15.17), W=(Luli +(Li+Ln), +Lal), (15) Im=fHimdoyT=fHandoy (15a) I=FfHondoy In='ffHmdoe. (5b) IfHisexpressed by¥,and¥by(8),itwillberecognized that alsowith this definition . In=In. We then obtain v1 4b=wfder[dn 3, (18) where risisthedistance between anyonepoint onS;andanysecond point .onS;.Sincetherightsideof(16)issymmetrical withrespect totheindices1and2itrepresents alsoLy. . Bytherepresentation in(15a,b)themagnetic fluxes4,and4,through S;and 3;may bewritten hanfi+Bade=LaltLal, lay (17) =fG+ Wade =Lal+Lah. A Themagnetic fluxisthusnowexpressed bytwoelements (fornlinear con- ductors bynelements) interms ofthetwo(orn)currents. C.Application totheSelfinductance ofaTwo-Wire Line Asin§15E weregard conductor andreturn conductor asoneclosed circuit anddesignate, justasthere, theseparation ofthewires witha,the radius with b.Since wecannot proceed tothelimit b—0ourmethod ofthe scalarpotential, which isrestricted tolinearconductors, isstrictly inap-Plicable. However, wemayregard Eq.(14),quite apart fromitsorigin in thismethod, asthedefinition oftheselfinductance L,or,moreexactly, of thatpartofLwhich hasitsorigin intheexterior ofthewires (seeFig.22). Theportion ofthebranch cutSwhich isofimportance toushashere beenshaded. Itextends inthezy-plane (theplane containing thetwoaxes ofthewires) from theperiphery ofonewiretothatoftheother andisto havethelength1inthey-direction. Themagnetic fieldHresults, ingen- 122 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 16.18 eral, from thevectorial superposition ofthemagnetic fields H,and H;of thetwo wires. Ontheplane area S,Hiand H;have, however, thesame direction, since J;=—J;, and areperpendicular toS.With zand a—z, ’ |SNS Fig. 22.Selfinductance ofatwo-wire line, com- NsNN puted from themagnetic fluxthrough thebranch 1TNS cut8. Be hal respectively, indicating thedistance ofthereference point onSfrom the conductor and thereturn conductor weobtain Ifi 1 te=5(b+535): hence wefindfrom (14)forunity length ofthetwo-wire line eff 1wetf (t+, :)dz. (as) Thesubscript aofLindicates “external selfinductance”. Theevaluation of (18) yields: B a-b_ BN8HgtOANHogt Laoe(los=>log=) 7os>7083°(19) thelastsince bXa.Ifwecompare thiswith (15.35), weseethat thefirst partoftheearlier formula corresponds toour“‘ezternal” selfinductance and conclude from this that thesecond part willsignify the“inner” selfin- ductance L;. Weconfirm thisinthefollowing manner: Employing theenergetic defini- tion ofselfinductance in(15.17) weput W.=410. (20) W;isthemagnetic energy within aunitsegment ofourtwo-wire line. Within theindividual wire themagnetic field isgiven, asin(15.338); by riHm3556° 10.23 AMPERE’S METHOD OFTHEMAGNETIC DOUBLE LAYER 123 ifweneglect themagnetic fieldoftheother single wire, which isweak in comparison. Hence theenergy content ofunitlength oftheindividual wire becomes asin(15.38a) te : 2pb 2 “=ri =wt[Bp=HL ‘flBrdrde==[G5)nde=FahY=ier This isonehalfoftheenergy W,in(20). Ithence follows from (20)that atLeg (20a) Together with (19)thisconfirms ourearlier result (15.35). D.Application totheElectromagnetic Current Measurement ofWilhelm Weber Weproceed from Eq. (8)and assume tobegin with that thesurface S bounded bythecurrent Iisplane; thereference point forwhich ¥istobe computed issupposed tobeatagreat distance from S.Then thedirection ofdnandthemagnitude of1/rbecome thesame forallelements deofS. The integration over domay becarried outdirectly andyields al 4n¥=IS— (21) Atthesame time werecall Eq. (7.10b) forthepotential ofasingle mag- netic polep.From itweobtain forthepotential ofadipole ofthevery small separation 1between poles and thedirection noftheaxis: 4nv=M2h, M=pl=moment ofthedipole. (22) Comparison of(21)and(22)indicates: Themagnetic fieldofacurrent I bounding asurface Sis,atlarge distance, equal tothatofadipole, wemight also sayashort barmagnet, which isplaced perpendicular tothesurface S and has themoment M=IS. Anon-plane current path may beprojected onthree mutually perpen- dicular planes andtheequivalent barmagnets may bearranged perpendicu- larly tothe resulting plane current paths. Vectorial addition oftheir moments yields anobliquely oriented dipole, which atlarge distance again produces thesame magnetic field astheoriginal current path. This equivalence ofcurrent and magnetism isthebasis ofthefamous “electrodynamic determination ofunits” ofWilhelm Weber. Furthermore theelectromagnetic system ofunits, which hails back toWeber, isbased onit,i.e.onputting equal I-S=M. (23) 124 DERIVATION OFPHENOMENA FROM M\XWELL EQUATIONS 16.24 The electric quantity Ithus becomes amagnetic: quantity. Orrather: Fur theelectric quantity Jaquantity M/S, which appears different inchar- acter, issubstituted. This ispossible only ifadefinite relation isestablished between the dimension ofthemagnetic pole, which asbefore weshall designate asP,andthedimension Qofcharge. By(23) thisrelation is areaQime=P-length or P=gna=Q-velocity. Wethus arrive atAmpere’s hypothesis according towhich magnetism is merely electricity inmotion. However, wementioned already onp.47that this hypothesis istoday, after thediscovery oftheneutron asabasic ele- ment ofallnuclear matter, nolonger asbinding asahundred years ago; wealsosawin§8Bthat Cohn’s system ofunits isindependent ofthishy- pothesis and isrecommended particularly bythat fact. ,Oursystem ofthefour units MKSQ bears apeculiar relation totheelec- tromagnetic cgs-system introduced byWeber. Asweknow, ourunit Q=1 coulomb =1ampere-second isdefined as1/10 oftheelectromagnetic unit ofcharge. The fundamental constants ofvacuum, with dueregard forthe experimental fact eauo=1/c’, were therefore found tobe . 2 hebo=4x-107ont=4-107BY,Bg,(15.186) (24) 1 _10°QM_10”farad =a7iedjoule S36, Mo?Eqs.(7.17)and(10.3b)(25) .2 *=4ncl07nee=1200,Eqs.(7.19)and(4.5¢). (26) However, weknow that, apart from theelectromagnetic (more briefly magnetic) system, alsoanelectrostatic (more briefly eleciric) system isin use. Here thearbitrary, and only historically justifiable, convention (7.8) ismade: f=4re=1 (27) andfrom thisanelectrostatic unitofcharge ewisdefined. Weaskhow Qis tobeexpressed inthisunit, ie.what value Q/e. may have, having fixed thevalue ofQmeasured interms oftheelectromagnetic unit ofcharge magn bY co emer=10° (28) 17 THE FIELD OFASTRAIGHT WIRE AND ACOIL 125 From Eqs. (25) and (27): 10°(Q/en)*: 10°em dreaml= 107ergsec?* From this and (28) itfollows that =i ,Cm_¢3.19OO 1=5Cmsen/ea)’s ee0810M (29) Theunitofcharge inthemagnetic system is3-10" times larger thantheunit ofcharge intheelectric system. This corresponds toourmetaphor ofriver andwaterfall onp.53.The numerical values ofagiven physical charge behave ofcourse ininverse fashion. Thus thecoulomb has, intheelectric cgs-system, the numerical value 1Q=he 310%; seealso ourdata onthecharge oftheelectron in§8D: 4.80-10- electric and1.60-10~” magnetic units. The conversion rule forthenumerical value eofanarbitrary charge in andMKSQ system tothecorresponding numerical values @msgnaDdée1is é7 a= toe? 10(ioecnaCena’=Cal (30) Since thequotient D/e ontheonehand and theproduct Eeontheother areindependent ofourfourth unit, theconversion ofDandEmay readily bederived from Eq.(30); fortheconversion ofE(30)yields. 10(AresEYuxaa =5Bases!=Eat @y Rules (30)and(31)replace therulegiven byH.A.Lorentz onp.87ofVol. V:oftheMathematische Enzyklopadie. With thistheunpleasant business oftheelectrical units may beregarded asdefinitely disposed of. , §17. Detailed Treatment oftheField ofaStraight Wire andaCoil Weconsider theapparently trivial caseofaninfinitely longstraight wire carrying astationary current with areturn path inacoaxial hollow cylinder surrounding it.Lettheradius ofthewire bea,theinner radius ofthe hollow cylinder b,andtheouter radius ¢+>©.Forreasons ofsymmetry 126 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 17.1 the magnetic field isknown directly, and similarly thecurrent density. As in§4,Eqs. (10) to(13), wehave ri I r<a,H=adna’ Jz=mt a<r<bh H=a, J=0, q) 2ar bcrce, H= J, sno> ar’ ° The direction ofHiseverywhere azimuthal: H=H,. Intheinterior theelectric field iseverywhere axial indirection and has byOhm’s law thevalue : Jd . E=E, =>=73° rsa (2) Similarly byOhm’s law wehave within thehollow cylinder (conductivity 1) Eaten, bsrce. @) a Intheregion between wire and hollow cylinder thefield isyettobedeter- mined, from thedifferential equations E=-gad¥, Av=0 (4) and the boundary conditions I ®.-2,=xa forr=, (5) 20 for r=b Since these conditions areindependent ofzand ¢wecan write forthe solution ofAY =0 ¥=4(rz. (6) Wecanomit anadditive term ¥(r) independent ofzsince wecansatisfy alltheconditions oftheproblem with formula (6).Wethen obtain from (5) ¥@)=--; 40)<0. )ao The differential Eq. (4)demands d_ dy,Ba 7% VW,=Alogr +B, 1711 THE FI€LD OFASTRAIGHT WIRE AND ACOIL 127 which yields, with (7) =~ tog”/tog? nO=zare8i/log. ® Now thefield Eisknown alsointheintermediate space a<r<b.Accord- ingto(6)and (8)itisrepresented by _Tf r a B==SEtowt/toe, ) ~%,2/@ E, or?7xater.loga (9a) Thus thefield ishere bynomeans axial indirection, asintheinterior of thewire; rather, itaradial component isofthesame order ofmagnitude asitsaxial component. Inpassing from theinterior totheexterior ofthewire there occurs a jump in£,andhence alsointhecorresponding component oftheexcitation D,which indicates theexistence ofasurface charge: =D,=of,=2/10g wo=D,=ek,xalelogPa (10) "Thissurface charge decreases linearly along thewire,frompositive tonega- tivevalues, informal language from + to—©.Itdepends only slightly, i.e.logarithmically, ontheradius boftheouter return conductor. Thesero pointofthecharge remains undetermined sincethepointz=0canbe fixed arbitrarily. Wemay eventually identify itwith the“center” ofthe wire, which, forinfinite length, alsoremains indefinite. ‘Wecanobtain anidea astothemagnitude ofthischarge inthefollowing manner: The dielectric constant in(10), which refers totheexterior of thewire, does notdiffer materially from that intheinterior ofthewire (though formetals itisrather hypothetical). Hence thequotient e/¢does notdiffer materially from therelaxation time 7,defined in(4.9a), forthe material ofthewire; thisisoftheorder ofmicroseconds. The product eI/o occurring in(10), which represents #charge, isthusnotoftheorder ofanampere-second =Q,butoftheorderofamicroampere-second =10™* Q.Theperipheral charge ofthewire andtheradial field strength corre- sponding toitarehence very small. This isthereason whythey generally remain unnoticed bothintheory andinexperiment although, asweshall see,theyareessential foranunderstanding ofthecurrent transport. Implicit intheexistence ofaradial electric fieldistheappearance ofa potential difference between wire and return conductor. According-to (a) itisgiven by . __ iz va[Bar--75. ay 128 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 17.113 Wecompare itwith thecharge onunit length ofthewire, ¢=2xaw, or theequal and opposite charge onunit length ofthereturn conductor. We find & tre and (11a)ar This isthecapacity perunit length ofthecylind:ical condenser formed bythewire and thereturn conductor (see Problem II.5). Wespeak here and insimilar cases ofa“distributed capacity.” Fig. 23shows theshape oftheequtpotential lines ¥=const. inthez,r- plane, given according toEqs. (6)and (8)by ; slog?=¢. (12) ForC=0,z =0andr=b,corresponding tothebroken lines ABC and ABD ofthefigure. The equipotential lines forC>0accommodate them- selves within thearea bounded bythem. The angles under which they meet :the surface ofthewire deviate increasingly from aright angle with in- éreasing distance from A.The orthogonal trajectories totheequipotential lines represent thelines offorce; thearrows onthem indicate thedirection from positive tonegative surface charge w.Two bounding curves passing through the “equilibrium point” B(see footnote 2after Eq. (9.14b)) belong tothis family ofcurves. According to(12) theequipotential lines satisfy the differential equation bdr log=dz—z 7= ontheother hand, thelines offorce, which areorthogonal thereto, (re- placement ofdz/dr by—dr/dz) aregiven by ede+rlogar=0. Intheneighborhood ofBwefind, with p=b—r, z2dz—pdp=0. Thus twolines offorce, with tangent directions z=+p, pass through the point Bforming aright angle with each other. Atagreater distance from these Bounding curves, above andbelow them, thelinesofforce passmore orless radially from thewire surface totheouter conductor. The equipotential lines represent atthe same time the paths ofthe energyfluxS;thearrows marked onthemindicate thedirection ofS.From 17 THE FIELD OFASTRAIGHT WIRE AND ACOIL 129 the formulaS=EXH,Sisperpendicular toHandhenceliesinthe plane ofthedrawing, since Hiseverywhere perpendicular thereto; inaddi- tion Sisperpendicular toEandhence haseverywhere thedirection ofthe family ofcurves ¥=const. Anapplication oftheright-screw rule forthe three vectors E,H,Sshows that thearrows areproperly oriented. —S G Ht\eaIh | ECR|BRT = ‘ b f. a 7 H steals, KeZi| _ae id, = ~ 4 CT Hh RAANTHA |_EES Fro. 23.Energy flux about astraight wire carrying stationary current with co- axial return conductor. Equipotential lines =stream lines oftheenergy drawn out full, electrical lines offorce =excitation lines drawn dotted. They form tubes of constant charge starting and ending onthesurface ofthewire and thereturn con-ductor,respectively. Theirnumberperunitlengthofthewireindicatesthesurface charge onthelatter and itslinear increase with distance from A,positive forz<0, negative forz>0. Since E=J/othelinesofforce within thewire(notdrawn inthefigure) areaxial indirection; hence thevector Sisdirected inward, perpendicular tothesurface ofthewire; here also itliesintheequipotential surfaces WY=const. The energy fluxisdissipated intheinterior ofthewire since it becomes sero forr=0,inview ofH=0.Thisisindicated inthefigureby theterminated arrows atA.Such arrows should beimagined along the 130 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS =17.13 entire surface r=a.Theenergy entering from thesurface isconverted into Joule ‘heat intheinterior ofthewire. According to(9)and(1)themagnitude oftheenergy fluxforr=ais It |S]=BH, ==a theenergy supply tounit length ofthewire from allsides hence becomes 2ra|S|=PRs,Raz, q=rd. (13) R,istheohmic resistance ofunit length ofthewire; (13) thus yields, in fact, theJoule heat claimed bythis unit length. Accordingly weobtain thefollowing total picture ofthebehavior ofthe energy: Outside ofthewiretheenergy flowsfromtheelectrodes z=+0 from allsides toward thesurface ofthewire. After entering ititflowsradially toward theaxisofthewire,beingconverted atthesametimeintoheat.Thereisnoenergy fluxparallel tothewireaxiswithin thewire. This picture ismaterially different from thepopular concept ofthe energy transfer inawire carrying current. From theMaxwellian standpoint there isnodoubt, however, about theinner consistency andunique validity ofourpicture. Itindicates thefundamental change which Maxwell’s theory hasbrought intheconcepts conductor and nonconductor: Theconductors arenonconductors ofenergy. Electromagnetic energy istransported without lossonly innonconductors; inconductors itisdestroyed, orrather trans- formed. The notation “conductor” and “nonconductor” refers only tothe behavior with respect tocharge; itismisleading ifapplied tobehavior with regard toenergy. Wethus come totheconclusion that oursimple example isafter allnot astrivial asitappeared. ‘Thenext-simplest caseofthecircular conductor isalready beyond ele- mentary treatment. Even inthelimiting case ofthelinear conductor the geometrical formula forthesolid angle involves anelliptic integral. Infact, ifweusethepolar coordinates r,y,zforthereference point and p,«+9—x,0forthepoint ofintegration wefindfrom Eq.(16.8), denoting thedistance between point ofintegration andreference point byR, [eRe- [ode da26°+o+2rocosat 2 onR 0 0 oz f° pdp ” agol yeterel vores B= 0/2 pn 40 G+p+2 17.13 THE FIELD OFASTRAIGHT WIRE AND ACOIL 131 The preceding integral with respect to8isa“complete elliptical integral ofthefirstkindintheLegendre standard form,” andkisthe“modulus” thereof. We cannot ofcourse here delve further into the treatment ofthis formula. The exact treatment ofacoilofwire offinite thickness with finite pitch would beeven more complex. Wehence pass directly tothelimiting case ofvery small thickness and pitch, i.e.Ampére’s solenoid, whose magnetic field wasalready discussed inEq.(4.14), though only superficially. Wenow wish tocompare itwith thefield ofapermanent uniformly magnetized bar magnet ofthesame dimensions asthecoil. Weshall here assume theinterior ofthecoiland thecoilwire tobeunmagnetic (u=4). Weshall show that theH-field ofthecoilcorresponds totheB-field of such abarmagnet. Asproof wewrite down theboundary conditions and differential equations forthetwocases sidebyside: Coil BarMagnet (@.-H).= Nil, (Ba—Bi). =wo(H. —Hs—M). =—uoM, (H. —Hi). =0, (B.—Bi), =0, 4¥=0, divB=wdiv H=—pAW¥ =0. ‘Thefirstlinerelates tothemantel surface. Itslefthalfstates simply, inthe terminology introduced atEq. (4.4e), that thesurface curl onthemantel surface ofthecoilisequal toNi/, where N,isthenumber ofturns perunit length. The right half ofthe first line follows from our equation B=yo(H; +M)fortheinterior ofthebarmagnet, which fortheexterior becomes B=4H, .Itstates that thesurface curl ofBonthemantel sur- face ofthebarisequal to—u»M. Acomparison ofthetwo halves ofthe first line shows that, inthebarmagnet, thequantity —u»M/N, corre- sponds tothecurrent Jinthecoil. Weapply thesecond lineinparticular tothetwoendsurfaces. Here both BandHareofcourse continuous forthesolenoid, forthebarmagnet only B.Since wehave assumed »=jwthesame equations apply alsofor the mantel surface. The third line applies inthetwo cases both fortheinterior and forthe exterior; herewemust recall ourassumption regarding thebarmagnet that itsmagnetization was supposed tobeuniform, since otherwise theterm uodivMwould have tobeadded totheterm 4divHandthiswould spoil thecomparison ofthetwo cases. Weseetherefore that ourearlier Fig. 17,which represented theB-field ofthebarmagnet, reproduces simultaneously theH-field ofthesolenoid. Accordingly theearlier Eq.(4.14), which applies onlyforaninfinitely long coil,isrounded outgraphically byFig.17.This provides usnow with a complete picture ofthespreading ofthelines ofexcitation attheends of thecoiland oftheir exit through theconvex surface. 132 DERIVATION OFPHENOMENA FROM MAXWELI. EQUATIONS 17.14 Withrespect totherepresentation ofthecoilfieldbythepotential ¥we wish ‘tdpoint outexpressly itsfamiliar multivalued character; thebranch cutS,which makes itsingle valued, isahelical surface ofinfinitely small piteh which follows theturns ofthewire. Itfollows that theequation ¢H-ds=0 which isuniversally valid forthe bar magnet loses itsvalidity forthe coilifthepath ofintegration links oneorseveral turns. Hence also the conclusion with regard tothedemagnetizing character oftheH-field ofthe barmagnet isnotapplicable tothecoil. For toreach this conclusion, we employed (see p.85)apath which inside was along theaxis ofthebarand ontheoutside ledback tothebaraxis. This same path, forthecoil, inter- sects theabove-mentioned helical surface and therefore isnotapermissible path ofintegration. Wecanalso make thefollowing statement: The H-field ofthebarmagnet islamellar throughout, that ofthecoilisnot; instead ithasthecurlIcon- centrated attheturns ofwire. The H-field ofthecoilissolenoidal through- _out,since everywhere B=joH,thatofthebarmagnet isnot.Fortheuni- *formly magnbtized barthebarends have asurface distribution ofdivergence. Inviewoftheproportionality ofBandHforthecoil,Fig.17evidently also represents theB-field ofthecoil. The H-field ofthebarmagnet, Fig. 18,is materially different from Fig. 17. The near-uniformity oftheinternal field evident inFig. 17suggests the computation ofthemagnetic energy Wofthecoilbytheformula We5H, (14) where V=7a’l isthevolume oftheinterior ofthecoil. Inview ofthe relation Hl=NI(Eq. (4.14)), where Nisthenumber ofturns along the fulllength ofthecoil, (14) takes theform 2raal? W=uN’5. (14a) Acomparison with ourenergetic definition ofselfinductance in(15.17) leads tothefollowing value ofthelatter: 2 Le=7uN, (15) This formula applies, ofcourse, asmay also beseen from Fig. 17,only for avery long coiland hence canscarcely beused forforms encountered in practice. Ontheotherhand,itretains itsvalidity ifthestraight coilisbent into aring electromagnet, because oftheclose approach tofield uniformity 18.1 QUASI-STATIONARY CURRENTS 133 within thelatter. Here thefactor uin(15)alsobecomes significant ifthe ringisprovided with asoft-iron core. Eq.(15)shows thatthisarrangement, first suggested byAmpére, hasamuch greater selfinductance and hence realizes amuch greater concentration ofenergy than theelectromagnet without iron forequal coil current. This increase inenergy concentration applies qualitatively also tothe straight coilwith soft-iron core. Thequantitative computation offieldand energy would, however, bemuch more complex, since aboundary-value problem, corresponding tothepassage ofthemagnetic lines offorce from iron into air,would beadded tothesummation problem which wehave treated. §18. Quasi-Stationary Currents Most oftheproblems ofelectrical engineering andoflaboratory physics liewithin thedomain: oftheslowly variable fields. Itistrue that there isno one-word answer tothequestion “slow compared with what?”. Forvibra- tions which areperiodic intime orexhibit adamped periodicity theanswer maybeidentified withthedemand thatthelightpath corresponding tothe period +ofthevibration belarge compared tothedimensions Jofthe apparatus: er a) Itisthen permissible toneglect the“retardation ofthefields” tobeintro- duced in§19.Attheendofthisparagraph weshall, however, treat success- fully also very long lines, which donotobey thiscondition, onaquasi- stationary basis, bysubdivision intodifferential segments. Ingeneral terms thequasi-stationary approximation consists incalcu- lating allfields asforstationary processes. Inthismanner itbecomes pos- sible toestablish alinear relation between theexpressions occurring inthe integral form oftheMaxwell equations. Werefer tothemagnetic flux ®through aclosed curve, thecurrent Jthrough across section, and theelectromotive forces Vaalong segments ofthepath ofintegration, which adduptotheloop e.mf. Vforthispath: 2 a=(Bids, I=fond, Yam[Beds, VeDVa=¢E,da, Weknow that, under stationary conditions, thefollowing relations exist between them: &=Luh +Lals +Luls +-+-(magn. fluxthrough circuit 1) I=x(Ohm’slaw),I=WK(condenser charge). 134 DEXIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 18.2 The inductances L,resistances -R,and capacities Kdepend, apart from thematerial constants, only onthegeometry ofthefield andaretheresult ofanintegration overthespace coordinates. Hence onlyanintegration with respect tothetimeremains tobecarried out.Themathematical simplifica- tion achieved inthis manner isconsiderable: While the exact treatment ofrapidly variable fields demands theintegration oftheMaxwell partial differential equations, theintegration ofordinary differential equations with constant coefficients suffices forslowly varying fields; forperiodic processes these reduce even toalgebraic equations. ‘This method wasdeveloped byGustav Kirchhoff (seebeginning of§15) andwasapplied tometal wireloops andnetworks composed ofthem. Itis ingeneral appropriate tochoose theclosed path ofintegration along the metal wires; aninterruption ofthemetallic path bynonconducting gaps (condensers) does notinterfere with themethod, however. The case of thinwires isparticularly convenient, since here thefield within thewire is practically uniform. Thefirst Maxwell equation inintegral form then yields forthechosen closed path ofintegration —é=V=loopemf. (2) Thesecond Maxwell equation finds expression inthecomputation ofthe coefficients ofinduction insofar asitrepresents theproduction ofmagnetic fields bycurrents andinthecomputation oftheresistances andcapacities insofar asitrepresents theorigin ofthese currents from theelectric field. Forasimple current loopwithout branching thetwosides of(2)maybe expressed interms ofthecurrent Jwhich isthesame forallcross sections: IDL DR+[rao peee @) Itisseen that here theselfinductance coefficients ofallthemagnetic fields canbecombined in@single expression, which may berepresented byan imaginary coilatanarbitrary point ofthecircuit. Thesame applies forthe resistances andthecapacitances. #isthepresumably known e.m.f. between theterminals bywhich thecurrent enters andleaves, i.e.thesocalled ter- minal emf. Theintroduction ofthise.m.f. Hconveniently avoids carrying thepath ofintegration through apparatus whose action isnotcovered by theMaxwell theory (galvanic orthermoelements, photoelectric cells, elec- tron tubes) orthrough “machines” which, though they function fully within theframework ofthetheory, would unduly complicate theproblem. Ifseveral loops arejoined inanetwork, thelatter may beregarded as made upofelements connecting the“junction points”. Letthenthelement carry current J,from onejunction point toanother. Since wedonot asyetknow itssign, weplace amarker arrow ontheelement which isto 18.6b QUASI-STATIONARY CURRENTS 135 indicate inwhatdirection weshallreckonthecurrentaspositive. Inviewoftheabsence ofcurrent sources wehave forthejunction points XI, =0 i) and forevery circuit made upofarbitrary elements Eq. (2)applies again inthe form DR t+LDVn=-& (=emf). (5) (4)and (5)areknown asthe“first” and “second” Kirchhoff equations They date from thetime preceding Maxwell’s theory. Hence Kirchhoff places ontheright side of(5)not—4, buttheolder concept oftheeleciro- motive force (e.m.f.) ofall“current sources” which are inserted inthe closedcircuit. Ifwearedealing withcurrents whichresultfromFaradayinduction, e.g.incoils ofmachines, thise.m.f. becomes exactly identical with —anditsconcept issuperfluous. Ithas, however, theadvantage of covering also theeffect ofother current sources (batteries etc.) without re- quiring anexamination ofthephysical processes taking place therein. A.Energetic Interpretation oftheWave Equation Weconsider inparticular anunbranched circuit withselfinductance, capacitance, and ohmic resistance, which weimagine connected inseries and, asinEq. (3),concentrated atcertain points ofthecircuit. Thus we insert aresistance box inplace oftheresistance which isdistributed over thewire; acoil inplace ofthedistributed selfinductance with which we became familiar inconnection with thetwo-wire linein§15E; and weshall not consider distributed capacities inthecircuit but assume instead the presence ofanelectric plate condenser. The lawofconservation ofenergy offers aparticularly convenient ap- proach tothetreatment ofsuch asystem, just astothetreatment ofma- terial vibrations inmechanics. Wewrite itintheform ofEq. (5.7a): WatWet0=§Side. @) W,,isthemagnetic energy concentrated inthecoil, given according to Eq.(17.14a) by we=Zr. (6a)2 W.istheelectric energy within theplate condenser, according toEq. (10.11) given by 1a=e. b) Ww. ax? (6b) 136 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 18.6¢ Here -earethevarying charges onthetwocondenser plates and de T=3: (6c) QistheJoule heat generated within theresistance box: Q=kr. (6d) Theenergy fluxSrefers tothecurrent source. ByEq.(15.40) wewrite for theenergy supplied byitinunit time $S,do=EI(volt-ampere =watt). (6e) Eistheterminal e.m.f. ofthecurrent source mentioned above. From (6a, b)follows v= vypttWo=Lil, W.=xeRe (6f) Substitution of(6d,e,f)in(6)leads to,after canceling ofafactor I: Li+RI+fem8, @ ‘and, after aseoond differentiation with respect to¢,tothewave equation Lr+Rt+hr= 8 ; (7a) Intheterminology ofpoint mechanics there corresponds thus Etotheexciting force, Ltotheinertia, moreparticularly themassofthevibrating particle, Rtothedamping, and Ktothecoefficient oftherestoring force. Asinmechanics, wedistinguish between free and forced vibrations. a,Free Vibrations WesetE=0andaskforthesolution ofthehomogeneous equation LI+RI+Z7=0. () Wecanassume atrigonometric formforZ,butknowfrompointmechanics that itisdefinitely preferable touseinstead theexponential form stthe startandtopasstotherealpartofJonlyaftertheintegration. Wethere-fore’ set: I=Ine (8a} 1Temporarily weadhere tocustom inemploying theusual positive signoféia theexponent, although wegenerally prefer thenegative sign. Seee.g.$6,Eq.(11). 18.94 QUASI-STATIONARY CURRENTS 137 andobtain forthecircular frequency w)=2x/7» ofthefreevibration the quadratic equation —Lead+Ran+fe=0. (8b) Fornodamping wefind w= t=oeVEL 0) This istheKirchhoff-Thomson formula. Ifitisnecessary totake account ofdamping thesolution of(8b) yields ik 1 RoF -e (0s) Theprocess isaperiodic orperiodic, depending onwhether R 1 R 1 n,n (9)aL”WKL 2b*/KL @ Intheaperiodic case«ispurely imaginary andthecurrent (8a)decreases monotonically.,In theperiodic caseusually found with condenser discharges theangular frequency is 1RM4kL ae (Qc) Since theohmic damping occurs here only asacorrection ofthesecond order ascompared with thefirst term ontheright of(9a), (9c)canusually be replaced by(9)—in analogy with themathematical pendulum, where theformula +r=2x+/i/g isnotaffected appreciably byairresistance etc; hence itisgenerally permissible towrite inplace of(8a): T=[qePHO8),gtteitiro (9a) where 1»nowrepresents thevalue (9)withnodamping. Thedouble signin (9d)evidently becomes unimportant when finally passing totherealpart, butpermits taking account ofthephase oftheoscillation which, likethe amplitude, may beprescribed arbitrarily, through superposition ofthe two solutions. b.Forced Vibrations Here weproceed preferably from Eq.(7)andset E= Ee’; istheangular frequency ofthealternating current source: Thecircuit oscillates with thesame rhythm assoon asitscharacteristic vibtations, 138 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 18.10 which aredetermined byanarbitrary initial state, have decayed. Wehave therefore Tete, L=iol, e=ES(see(6c)). io We thus obtain from (7) . 1(iat+Rt+zx)I=8, forwhich weshall write more briefly RI=£. (10) (10) isOhm’s lawforalternating currents; therealohmic resistance Ris here replaced bythe,complex impedance .R=R+s(u- 3). (10a)oK, Ifweput R=([Rl{e*, (10b) ' aveevidently have _iy wL—1/(wK) in|=4/R+(ot-Ry,tana=>=. (10¢) We introduce the following designations: R=resistance wL—1/(wK) =reactance wL =inductive reactance 1/(wK) =capacitative reactance |R|=impedance. Wegiveaninterpretation ofEq.(10)inthecomplex Gaussian plane ofFig. 24,The two-dimensional “vector” Ilags bytheconstant angle abehind thetwo-dimensional “vector” E.Ofcourse only thereal parts ofFandI have physical meaning. The engineer calls thisrepresentation arotating vector diagram; The figure should infact bethought ofasrotating with theangular velocity w astime progresses. The projections ofthetwo-dimensional vectors EandI on.therealaxisgivetheinstantaneous values ofthesequantities.Weshall establish furthermore how theenergy flowing into thesystem 1Inelectrical engineering this term, incidentally, isused notonly for|R|,but alsoforourimpedance operator Riteelf. 18.12 QUASI-STATIONARY CURRENTS 139 isused up.Tothis end wemultiply Eq. (7)with I,where now Iand Eare torepresent their real parts. Wefind Ldn 2, Ldazal +R +5556 =IE. (11) Ifweaverage over aperiod r=2z/w ofthevibration, weobtain: 1[rra=! [ma 7 T sothat RP =TE. (11a) Thecontributions derived fromLandKvanish since theyaregiven by differentials. Theaverage power IHintroduced intothesystem isthusdis- Fra. 24.Representation ofthecomplex Eand Jin theGaussian plane. Lag ofJwith respect toEbythe angle a. £ T sipated entirely inthe ohmic resistance R.The imaginary part ofour impedance operator Rhasapurely wattless effect onthee.m-f,, i.e.itdoes notconsume energy onatime average. Wecancalculate directly Bea?[14cost(ot—a)dt=3105 (ub) 1 . 1 Tay=ViI,andcorrespondingly E.,,=WMEy; (Le) Iziscalled the“effective current”, E,,, the“effective voltage”. Forthe average power wefind EI=tTEolocoswtcos(wt—a)dt € a=:Bale[cos’wtcosadé+[cosutsinwtsinadt (12) , =FEo]ycosa=[yyyEu 008& Analog: Work =Path-Projection oftheforce onthepath, where thepro- jection must becarried outinthecomplex plane inthepresent example. This analogy applies however only forthetime average, inwhich thesecond 140 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 18.13 integral ofthemiddle lineof(12)vanishes. Thelatter signifies anoscillation oftheenergy between storage (K,L)and thecurrent source (E). B.TheWheatstone Bridge Wedistinguish between thefour bridge arms a,b,c,dandthearms eandf,containing thecurrent source and thegalvanometer, respectively. The disappearance ofthecurrent inthegalvanometer arm isattained by adjustment oftheslide-wire contact S(oreventually thetwosliding con- tacts Sand T).Thus anull method, viewed with such favor inphysical <_g a Fra.25.Wheatstone bridge: a,b,c,dbranches of bridgewithresistanceboxesandinductioncoilscon- oi ;|,nectedinseriesand,eventually, capacitances con- 4®nected inparallel; fande,branches containing galvan- . ‘Vy ometer and voltage source; $and T'slide contacts. 2 8 measurements, isrealized. The geometrical structure ofthebridge isbest represented byatetrahedron (Fig. 26),thesixarms being transformed into thesixsidés without altering their relationship, asinanalysis situs. Thetwo arms ¢andfthen become opposite sides, similarly thearms a,dand b, ¢,while thearms a,betc.become “adjoining sides”. 8 Fie. 26.Space representation ofWheatstone bridge as a Ul tetrahedron; adjoining and opposite branches. 4 T >dq B From thegreat range ofapplications oftheWheatstone bridge weselect two characteristic special cases; thetrivial case ofthecomparison oftwo ohmic d.c. resistances will beaby-product: a.Comparison oftwoselfinductances b.Comparison ofaselfinductance andacapacitance a.Letthetwoselfinductances Z,andLybeinserted intheadjoining arms aand b,inseries with theohmic resistances aandb,asshown inFig.25. Inosder that there may benocurrent inthearm f,weadjust thesliding contacts Sand7’sothatthere isnodifference ofpotential between ther. Sta:ting from Awe.have, from (10) and (10a), (a+tLe) =cle (13) 18.158 QUASI-STATIONARY CURRENTS 141 and starting from B: + twh)h =dh, (13a) sothat (a+twL,) d=(b+twha)e. (14) Ifthis equation istobesatisfied, theequality must exist individually for thereal andfortheimaginary parts. Thus: ad=b, Lad =Le. a_c_In (4a) 727k: The first half ofthisdouble equation applies alsofortheequilibrium ofan inductance-free bridge, irrespective ofwhether itistraversed bydirect orbyalternating current. b.Lettheselfinductance andthecapacitance lieintwoopposite arms of thebridge, e.g.aandd,insuch fashion that theselfinductance Land the ohmic resistance aare connected inseries, the capacitance Kand the ‘ohmic resistance d,inparallel. According toKirchhoff (seep.101)thepotential drop‘across twoparallel ohmic resistances R’andR”withthe currents J’and I”isgiven by . ,”111 RIwith I=I' +I", Rete The corresponding voltage drop foralternating current isevidently Rrwith I=r¢l, bates. 09) Forourbridge armdthetotalcurrent atitsendpoints isJ;(seeFig.25); wetherefore set 1 1_1,. I=h,R=d, R"=rme R7qt eK and obtain inplace ofEq. (13a) dy oh,= ———_ 1+tok (15a)d Incombination with theunchanged Eq. (13) this leads to: I,_a,.L_b,, Tre tem gtee 142 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 18.16 which yields, separating realand imaginary parts, L ad=be=x (16) Inthenotation ofp.138thelastterm ofthis double equation istheproduct ofinductive and capacitative reactance. Eq. (16) shows that thedeter- mination ofacapacity from bridge equilibrium rests, apart from ohmic resistances, onaknown selfinductance (orvice versa). Inourexamples wehave restricted ourselves tocases inwhich theequi- librium conditions, Eqs. (14a) and (16), donotdepend onw.Insuch cases thebridge equilibrium exists notonly forthepurely periodic alternating current here assumed, butforanarbitrary time variation, e.g.excitation by aninterrupter. There areother cases inwhich theequilibrium conditions depend onw;bridge equilibrium then exists only forsinusoidal alternating current. C.Coupled Circuits The lawofconservation ofenergy sufficed tosetuptheoscillatory equa- tion foronedegree offreedom. For asystem oftwo circuits (two degrees offreedom J,andJ;)theenergy theorem nolonger isadequate forsetting upthedifferential equations, justasinmechanics. Ontheother hand, Kirchhoff’s law(5)yields directly Inh+Inh+Rh+ 2=By a7) Lal: +Luh +Bile + =Bs. The generalization formore than two circuits isobvious. From (17) wecancalculate, ontheonehand, thefree vibrations ofour coupled system, ontheother, theforced vibrations produced byane.m.f. InthefirstcasewesetH,=E,=0andreduce theresulting homogeneoussystem ofequations, with theassumption T= Ae", Ih=Ae** toabiquadratic equation foruw»after elimination oftheratio A,/A,. We canbebrief indiscussing theconclusions derived therefrom since thesame problem hasbeen treated indetail inVol. I,§20andtheresults arerepre- sented there inFigs. 34and 36.The characteristic beat phenomena ofthe “coupled pendulums inthecaseofresonance occur, interms ofourpresent notation, when theperiods oftheuncoupled circuits areequal, which leads toKily =K2lm, and when inaddition thetwo “coupling coefficients” Lu/Ly andLn/Ly areequal, which leads, inview oftheuniversal equality 18.19 QUASI-STATIONARY CURRENTS 143 ofLyand Ly, toLy=Ly. These beat phenomena become particularly impressive ifweassume, asforthecoupled pendulums, that damping is slight, inourcase R,20, R:20. Also forforced vibrations thetreatment inVol.I,§19may serve asa model. Ifthefree and forced frequencies areidentical, #=«,damping becomes essential andtheamplitude maximum andphase lagrepresented inFig. 33ofVol. Ioccur. Atthebeginning ofthe§20mentioned above itwaspointed outthat in theearly stages ofwireless telegraphy coupled mechanical oscillations com- monly served asmodel forthecoupled electrical oscillations which occurred intheopen primary antenna circuit andthetuned secondary circuit intro- duced byFerdinand Braun. Itistrue that forthese rapid oscillations the quasistationary treatment isonly acrude approximation; only thecom- plete integration oftheMaxwell equations in§19canyield asatisfactory representation. D.The Telegraph Equation Quasistationary calculations may alsobeapplied toshort sections ofa longtwo-wire line,forwhich condition (1)isnotfulfilled. Ifthelength of these sectiohs ispermitted toapproach zerothetotaldifferential equations ofthesystem become apartial differential equation. This wassetupbyW. Thomson even before Maxwell inthetreatment ofthepropagation oftele- graph signals inmarine cables. Between twooppositely located andoppo- sitely charged points ofthedouble linethere isacharging current with a change involtage; Hence, inaddition tothevoltage V(z), alsothecurrent I(z)varies continuously along thelength oftheline.According toKirch- hoff’s second law these two variables arerelated by or avLat kits =5 (18) LandRrelatetounitlength ofthedouble line.Furthermore, itfollows from theabsence ofcurrent sources, i.e.Kirchhofi’s firstlaw(4),that al avgetkytev-o (18a) Forthesake ofcompleteness aconduction current GVthrough theeven- tually semiconducting dielectric hasbeen added tothecharging current KaV/at. Thistermmayalsoaccount forhysteresis losses inthedielectric. Gisknown asthe“leakage” perunitlength ofthedouble line;Kalso refers‘to this unit length. _ Elimination ofVfrom (18) and (18a) leads tothepartial differential equation {ox%+ex+103+a-Z}r=09) oe at oat , 144 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 18.20. which applies alsoforV.If,inparticular, thedissipative coefficients R andG@aresetequal tozero, itassumes thesimple form ofthedifferential equation ofthevibrating string: e_@{ux3B3I=0 andisintegrated, foraphenomenon advancing inthepositive z-direction, by =ofle— -,/1I=af@-¢), ¢WVte (20) According to(18)and(18a) thecorresponding value ofVisinthisspecial case V=Leaf(z—ct)=Et. (21) Theratioofvoltage Vandcurrent Jisthusthequantity (L/K)}, which isindependent ofzand¢andrepresents aresistance. Itisknown asthe ‘wave resistance. The following practical conclusion may bedrawn from (21): Ifafinite doublelineisterminated byanohmicresistance ofthemagnitude (L/K), nodiscontinuity occurs inthecurrent and voltage variation attheend, hence also noreflection. Eq.(20) states that thecurrent andhence also thevoltage propagate themselves along thelinewithout distortion and without damping. Wemay also askforthecondition forundistorted damped propagation along the line, i.e.that I= e“f(z—a) (22) represents asolution ofthedifferential equation (19). Bysubstituting (22) in(19) and setting theresulting factors off”,f’,andfequal tozero we find 1 1RK +LE _conte «=EE.vm 23) The velocity ofpropagation cisthesame asin(20). The double equation foraleads to . .RK LE .RK+LG=2VRKLG, ie.4/+4/8=2, and hence RK K_LVR=1oralsoG"E (23a) 19 RAPIDLY VARIABLE FIELDS 145 This signifies equal decay time’for thepure displacement current KV+ GV=0(Eq. (18a) with aJ/az =0)andofthepure conduction current LI+RI=0(Eq,(18)withaV/ax =0).By(23)and(23a) ourdamping coefficient athen becomes equal toR»/K/L. Inthegeneral casethecurrent variation changes withprogress along theline.Itisthen proper toanalyze theprocess intocomponent waves of theform expi(kz—wf)withcomplex, frequency-dependent kwhich are periodic intime anddamped spatially. Thetotal phenomenon isnowno longer distortion-free. Theideal caseofundamped plane waves isapproached ifthetwowires areimagined flattened intowidebands, theintermediate space being vac- uum andtheband material aperfect conductor. Then, apart from the marginal portions, theelectric fieldintheintermediate space isuniform, similarly themagnetic field.Eisperpendicular tothebands indirection,H,parallel thereto. Thecurrent becomes J=|H|b(6=band width), thevoltage V=|E|d(d=separation ofthebands). Thecharge perunit length is¢=e|E|bandthecapacity K=e/V=eb/d. Themagnetic fluxperunitlength, i.e.through arectangle withthesides 1andd,becomes ®=u|H |d,sothat theselfinductance L=@/I =yod/b. From Kand .Lwecompute, by(20) and(21), slocity¢=<x«—he wavevelocity ¢=7=Vem’ . v Lo ‘Hod waveresistance>=Vk=fei. Herewith wehave again come upon thequantity (yo/e)', which in§6we haddescribed asthewave resistance ofvacuum forthepropagation ofa plane wave. (Toobtain agreement wemust refer itnow toaquadratic section ofthewave surface, i.e.setb=d.) Wehave inserted thissketchy note onthetelegraph equation partly to refer theconcept ofthewave resistance (more generally wave impedance orsurge impedance) toitshistorical origin, partly toprepare thewayfor thetransition torapidly variable fields inthenext section. §19. Rapidly Variable Fields. TheElectrodynamic Potentials Only inthisparagraph dowemake fulluseoftheunabbreviated Max- wellequations. Weindicate ageneral method ofintegration, which how- ever islimited tothecase ofauniform medium, e.g.vacuum. Hence throughout spaceweput€=e,«=yoandinaddition imagine thecharge density pandthecurrent density Jtobegiven inallofspace andforall times ¢<&(&=instant ofobservation). Inthisformulation oftheprob- lemwealready take cognizance oftheelectron theory, which, however, we 146 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS. 19.1 shall take uponly inthethird part. Westart from theMaxwell equations intheform (4.4) with theauxiliary conditions (4.4a, b,c).Inview ofour assumptions €=&=const, u=4»=const wecan write instead B=-curl E, 16) 52+mJ=cu, @ i 2divE=a (3) divB=0, (4) ., opdivJ+go (5) Wesatisfy Eq.(4)byourearlier formulation (15.3) B=curlA. (6) Substituting this in(1)weobtain curl(E+A)=0. This hasthenecessary consequence that thevector after thecurlsign isa gradient. ‘Thus E=—grady -A. @ Wecall ©scalar potential, Avector potential. Wesubstitute expressions (6)and (7)in(2)and obtain -5G+grad¥)+woJ=curlcurlA =—AA+graddivA. (8) We have here utilized the transformation ofcurl curl which has been repeatedly employed before (e.g. inEq. (6.2)), butapplies only forCar- tesian components ofthevector A.Weshall simplify Eq. (8)bysplitting itupinto two vector equations, namely into 130A aA~Ssje=Ted (9) and grad(avA+4¥)=0. (9a) Ifwerefrain from theinappropriate addition ofafunction depending on¢ only,i.e.ofakindof“integration constant”, thesecond ofthesebecomes divA+54=0. (10) 39.13b RAPIDLY VARIABLE FIELDS 7 Ourinitial equations (1),(2),and(4)arethus satisfied; there remains, apart from Eq.(5),Eq. (3).Inview of(7)thistakes theform Av+divA =—% +divA & or,taking account of(10), 1oy 2 WY-35F a (l) Our twopotentials AandWthus satisfy two differential equations ofthe same form. Wecallthem “wave equations”. Asnoted above, their right sides aregiven functions ofx,y,zand of“‘past time,” ¢<t&.The desired solutions arerelated bycondition (10). Werecognize thatthiscondition isappropriate from thefollowing: If wecall itsleft side Xand ifweform div(9)+como3;(11), weobtain _1aX .2) AXae m(aiv5+2. (12) The right side ofthis equation vanishes however because ofourEq. (5), which here atlast isdrawn into the consideration. Thus Xalso satisfies thehomogeneous wave equation, which represents awave process without external excitation, i.e.notaforced, but afree vibration. Itcan before- seen from this that asuitable integration ofthedifferential equations for Aand ¥,which excludes theappearance offree vibrations, notonly leads toEq.(12) forXbeing satisfied, butalso toX=0,ie.thesatisfying of Eq. (10). Nevertheless this equation isneither superfluous nor obvious, since thesplitting upofEq. (8),i.e.thetransition from (8)to(9)and (11), rests expressly oncondition (10). A. The Retarded Potentials With respect totheintegration ofourwave equations (9)and (11) we shall bebrief, since thenext section will indicate therational procedure. Wewrite down directly theresult oftheintegration: Axel=|loldr (18a) r arta [Oe (13b)be r 148 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 19.13¢ WandArefertothereference pointz,y,zandthe“reference time”¢,for which wewish tocalculate thevalues of¥and A.&,7,¢isthepoint of integration anddrisequivalent todédydf.Theintegration iscarried out over allofinfinite space and wehave Pe@-t-Mt@-H% [p]and{J]are,however, notthevalues ofcharge andcurrent density at thetime ofobservation ¢,but attheearlier time t=t—r/e. (13¢) r/cisthetimerequired bythe“light” totravel from thepoint ofintegra- tiontothereference point. Hence theexpressions (13)arecalled retarded potentials. They arecalculated from charge andcurrent density atatime which issetback byr/crelative tothetime ofobservation. ‘Themethod ofintegration (13)ismathematically unique if,forphysical reasons, theaddition ofadvanced potentials, which correspond tothelater time M=t+r/c, (18d) isexcluded. Itshould benoted, however, that such advanced potentials have tentatively been introduced byDirac intothetheory oftheelectron andplayanimportant roleinmore recent investigations (see§37). Ifweapply thesame method ofcalculation (13)toourquantity Xwe find directly xX=0 since theright sideof(12)wasequal tozero. This result isalsomathe- matically unique withtheexclusion of“advanced” solutions, which alone, incombination with theretarded solutions, could give risetofreevibra- tions. This may betaken asconfirmation ofourearlier statement that condition (10)issatisfied automatically intheintegration ofthedifferential equations forAand¥.Wenotefinally that fullunderstanding ofthe structure oftheabove formalism including thesignificance ofourretarded _andadvanced potentials canbeobtained onlyonthebasisofthetheory ofrelativity. What uptonowmayhaveappeared arbitrary andasym-metric Willthere assume anastonishingly unique andsymmetrical form. B.TheHertzian Dipole -Wewillexplain ourmethod ofintegration (13)foraparticular case, thecaseoftheHertzian dipole. Thisisobtained ifwecombine moving charge +ewithaneighboring stationary charge —etoform amoment p(t)=elvarying withtime, where |signifies theseparation ofthetwo charges. 19.17 RAPIDLY VARIABLE ¥IELDS 149 Wesubstitute J=pvin(13b), denoting thespacedensity ofthemoving charge bypanditsvelocity byv,and obtain oncarrying outtheintegra- tion, where randvmay beregarded asconstant inspace, Uldr_Iv]aa2=aE berated ieeeEee de Wehence obtain from (13b), ifwetake dueaccount ofthemeaning of thebracket symbol asgiven by(13c) Ho8 _? tea=#29(1—2), aa) Itishistorically customary andconvenient tointroduce, inplace ofthe vector potential A,theHertzian vector 1bywriting "am 1 rAsm sen=}(1‘). (15) With thisnotation wefollow thegreat paper ofHertz, already discussed in§1,p.5:'“The Forces ofElectrical Oscillations, Treated byMaxwell’s Theory.” ‘Inallofspace except attheorigin ofthecoordinate system Isatisfies, by(9),thedifferential equation 1on an33a 7% (16) which can also readily beverified from theexplicit representation of11 given in(15). By(10) thecorresponding value of¥becomes ek=—divl. (16a) From (6)and (7)weobtain then astherepresentation oftheelectro- magnetic field ; . lon H=curlOy,&E=graddiv i—555- (17) Asanexample weassume thatthepathofthemobile charge eisrecti- linear andmake itsdirection, which isalsothat ofthevector M1,theaxis ofaspherical coordinate system r,3,y.Wethen have* TL,=cos0-0, Dy=—siné-0,0,=0, 1Ann.d.Physik $6,p.1,1888;Gesammelte Werke, Vol.II,p.147. *The positive r-direction forms theangle @with thedirection ofII,thepositive d-direction, theangle 9+/2; hence thefactors cos2atI,andcos(@+4/2) = —sindatHe. 150 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 19.18 where, according te(15), Mdepends only on¢andr,i.e.isindependent of 9and¢.Inthese coordinates weobtain, byProblem 1.3ofVol.II, —~sin8(a(rl)_)=—gngOU =_ curl,=a(“@BugsinoFcurl,=curlyI=0, . coed ys) 1 Ap. = ol div1rR emFand39in’?1)coos, . ou .—sin¢ol : grad,div1=cosese,gradsdiv1Fe grad,div 1=0. Hence, by(17): H,=Hy=E,=0 (18) and by(15) , ind/a1 4H,=225 -8), cos&/3°2a21 trad,~%2(2 >22y+2r-49), (19) sind /10 1 1 4reBy=—me(lo,_1y— 4). Weconclude from (18): The magnetic lines offorce arecircles about the direction ofp,while theelectric lines offorce lieinthemeridional planes through this direction. Because oftheargument t—r/cofpitispossible, inEqs. (19), to transform thedifferentiation with respect torintoonewith respect tot. We have ol ap 1or a) atabe (19a) Thenthetermwithacancelsthatwith§intheequationforZ,in(19). Atthesame time wewilllimit ourselves tothe“distant zone” (large dis- tances from theorigin, i.e.setr>«©.Wewillindicate themore precise meaning ofthisinamoment, indiscussing theperiodically oscillating dipole. Accordingly weneglect allterms in(19)which contain higherpowers of1/rthanthefirst: Wethenobtain 4rH,=Sa?(i:-‘), 4xeE, =0, =~(20) 4reEy=Bea(t-‘). 19.24b RAPIDLY VARIABLE FIELDS 151 Thevectors HandEareperpendicular toeachotherandtotheradius vector 1fromtheorigin. BothHandEvanish ontheaxis¢=0and3=x;the H-and E-fields have their mazima intheequatorial plane 9=x/2. From (20) wecalculate BtwyeHeec e° (2) This isthesame ratio asthat which was obtained from Eqs. (6.11) and (6.13) fortheratio E,/H,. The structure oftheradiated electromagnetic field isthus that ofaplane light wave. Itiscustomary tosayinstead, inboth cases, thatEandHareequal, which however isdimensionally meaning- less. The amount ofenergy radiated per unit area and per unit time be- comes . 1sind S=EXH=£EH,ieee P (22) The total energy radiated inunit time isobtained byintegration over the spherical surface ofradius r: 2, _ ot . _# 8=[Sde=2[Ssinods =2. (23) Since p=el(1=separation ofthemobile and thestationary charge) p=ev,p=ed,where, ofcourse, inaccord with themeaning ofp,5denotes the value ofthe acceleration attheearlier time ¢—r/c. We thus obtain from (23) ee8S=cae (24) Inatomic physics itiscustomary towrite, inelectric ormagnetic cgs- units, ace 8S=3— (24a) or 2gaze? (24b) 3¢ which, according to(16.30), corresponds to(24). J.J.Larmor’ firstgave this fundamental lawofradiation intheform (24b). Fig. 27shows theradiation density Sasfunction of#.Itissimply the 1Phil. Mag. 1897, p.512. Larmor points outtherelationship toHertz’s paper of 1888 inhisbook Aether and Matter, Cambridge, 1900, p.225. 152 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 19.25 Polatdiagram ofsin’3.Inthetheory ofwireless telegraphy (seeVol.VI, Chapter VI) itfinds extensive application inthe treatment ofalinear antenna radiating freely into space. Infact such anantenna does not radiate energy initsown direction; themaximum oftheradiation is directed transversally. Inplace ofasingle dipole pwemay, ofcourse, also consider adiscrete orcontinuous sequence ofdipoles. Inthelatter case wewrite inplace of (15): 4o1l=fdp(t—r/c) (25) e r Here theintegration istobeextended over agiven curve Candthedif- ference indirection ofthevectors dpmust beconsidered. a0 j tint? _-“Fia.27,Radiation ofanelectron acceler- Py (I=BcosG* ated longitudinally inthedirection d=0. y Lower pair ofcurves: Hertz’s formula (22), CYee,into vc.Upperpairofcurves:correctedrelativi- ---eS,SD,astically,vcomparablewithc. Acomparison ofthepreceding with Hertz’s calculation inCartesian coordinates, which isfound inmost textbooks; demonstrates thesuperiority ofourvector representation oftheproblem oreventually, ofourspherical polar coordinates, which fitthesymmetry oftheproblem. Itseems even more important that ourpresentation clearly indicates thedimensions of allfield quantities, while theGaussian system ofunits employed byHerts obscures them. C.Specialization forPeriodic Processes Weobtain thesimplest model foralight source byassuming that the electric moment poscillates monochromatically with acertain circular frequency w.For example weset p(t)=Acosut=ARee™', tan p(t—r/c)=AReexp{—ta(t—r/c)}. Tf,asin(6.10a, b),weintroduce thewave number k=w/candomit the signindicating therealpart, which ispermissible forallfieldquantities except thequadratic ones S,S,wefind Lae r/)=Ao, (27a) r r 19.29b RAPIDLY VARIABLE FIELDS 353 Wehave thus arrived attherepresentation ofthespherical wave inVol. I,Eq.(13.18) ifwealso suppress thetime factor in(27a). This may and willbedone inthefollowing. Wethen obtain from Eqs. (20)thefollowing representation oftheelectromagnetic field: AR ge” Akw.ye E=K,=ie, PF H=H,= a,sine. (28) This applies, asalready noted above (20), forthe“distant zone”; weare now however inaposition todefine thisterm exactly. For, ifweletthe wave-length A=oee o correspond totheangular frequency w,thedistant zone includes alldis- tances for which r>d, (29) ie.excludes only theimmediate neighborhood ofthelight source. For aperiodic processes (29) isreplaced bytheinequalities Ligiy lA! 1 lel, lelglel> i, glpl> tee. (20a) These statements justify precisely the approximations made inpassing from (19) to(20). Ascompared with anatural light source ourmodel isspecialized both with regard toitsmonochromatism and itsintensity distribution. Itradi- ates noenergy inthe directions 9=0and 3=-;foralso now Fig. 27 andEq.(22)apply totheradiation vector S.In(22)both thetimefactor andthephase factor e™”drop outinthetime average. Infact,by(27), 2BAe cos(hr—ut) r r and thetime average ofthesquare thereof is A’al/(2r*) =APtR/(21). (29b) Since k=2x/) this isinversely proportional tothefourth power ofthewave- length. If(29b) issubstituted in(22) or(23) weobtain thefamous lawofLord Rayleigh, explaining theblue sky. The sunrays falling ontheparticles of theair*generate inthem electric moments which vibrate inharmony and radiate light inturn. Their radiation ismuch stronger attheblue endof thespectrum thanattheredend.Since AreaS¥2viue their ratio isabout 2'. The same law explains also theredcolor ofthesun and moon when rising 154 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 19.30 andsetting. Initspath through theatmosphere, which ishere particularly long, theblue light isscattered much more strongly outofitsdirect path than theredlight; primarily redsun- ormoonlight reaches oureyes. We shall notinquire whether acertain selectivity ofwater vapor intheatmos- phere plays anadded role. D.The Characteristic Vibrations ofaMetallic Spherical Oscillator The problem ofeleciromagnetic characteristic vibrations became sig- nificant asaresult ofHerts’s experiments. Ametallic body consisting of two oppositely charged halves (Hertzian oscillator) discharges with the formation ofaspark andradiates exponentially damped vibrations toward infinity. How dowecalculate their frequency anddamping? Inthecase of thesphere, which weimagine assubdivided into two closely adjoining oppositely charged halves, thequestion isanswered directly byformulas which arealready familiar tous.Itistrue that wemust notstart here from Eq. (20) forthedistant zone, but must employ themore general formulas (19), since the wave length ofthecharacteristic vibration gen- erated will understandably beoftheorder ofmagnitude oftheradius of thesphere; thesurface ofthesphere thus belongs tothenear zone. Ifthe ‘sphere isasgumed tobeperfectly conducting wehave onthesurface, i.e. Yorr=a=radiusofephere, Ey=0forall3.Hence, by(19),wehaveas boundary condition 1a@_1_18G-a-dm)P oe (30) Weletptake thesame form asin(27), treating knot asareal number asuptonow, butasanunknown complex number. The same applies then also for «=ck.Todetermine k(30) yields ik 1 2aoatk =o (1) The solution ofthis equation, which isquadratic inka,is ta=ERSv3. (ia, ‘The imaginary part isnegative, aswemust demand, since wearedealing with avibration which decreases with time. Inthe real part the positive sign istobechosen inorder that wave-length and frequency bepositive Wethus obtain ‘ 2ra_ V8 =34 _ya TB i.e.infact Xoftheorder ofmagnitude oftheradius ofthesphere. Damp- 19 RAPIDLY VARIABLE FIELDS 155 ingisvery great; asfollows from (31a), the amplitude decreases bya factor ee) 8gta inthecourse ofasingle vibration. Herewith thenature ofthefundamental vibration ofour spherical oscil- lator isdescribed. There ishowever also aninfinite number ofharmonics forwhich thesphere isnot divided into two oppositely charged halves, butinto 4,6,---alternately charged zones. While thefundamental vibra- tion corresponds totheHertzian dipole, these harmonics cannot bederived from the Hertzian vector MI.For them wemust refer toVol. VIand more particularly toAppendix IIofChapter Vofthat volume. The case oftheprolate spheroid, which comes closer totheHertz oscil- lator than thespheri¢al shape, wastreated byMax Abraham, after the problem ofthe spherical oscillator had been solved generally byJ.J. Thomson asearly as1884. E.Application totheTheory ofX-Rays The primary x-rays areproduced bytheincidence ofcathode rays on ‘theanticathode. Classically theinitial velocity »oftheincident electrons isreduced to&lowvalue; theelectrons experience aretardation—v. From Fig. 27weexpect that noradiation occurs inthedirection ofthecathode rays, insofar asthis coincides with thedirection ofv.The proof ofthis is possible with extremely thin anticathodes (films afew microns inthick- ness), ifthe transmitted x-rays are observed; this has been shown by Kulenkampff and hisstudents. Forsolid metal anticathodes theretarda- tion takes place along azigzag path; hence thevariation with direction is smoothed out. We will show relativistically in§30, atEq. (11), that the maximum oftheradiation does notlie,asindicated bythepair ofcurves inFig. 27,at¢=+/2, butthat itadvances, instead, with increasing hard- ness ofthecathode rays (increasing magnitude ofv)more and more toward 0=0.The fact that thecontinuous or“brems”-spectrum discussed here hasashort-wave limit isaconsequence ofthequantum theory, with which weshall notdeal here. The same applies fortheregular increase inhard- ness and intensity ofthex-rays with thehardness ofthecathode rays. Here weshall only discuss theproof oftheéransversal nature ofx-rays, which was given byBarkla in1905, tenyears after Réntgen’s discovery. Inplanning hisexperiment Barkla assumed this transversality, drew the consequences ofthisassumption, andconfirmed them bytheexperiment. Weconsider, inFig. 28,abroken lineconsisting ofthree mutually per- pendicular segments, the“primary”, “‘secondary”, and “tertiary” segment. Theprimary z-rays, regarding whose polarization weshall make noassump- 1SeeEnzykl. d.mathem. Wiss., Vol. Vs,section 18,p.498. 156 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 20 tion, travel along thefirst segment (foravery thin anticathode even these would bepartly polarized). Weimagine their electric field strength tobe analyzed into itsccmponents along thedirections 2and3ofthetwoother segments. Theyfallonafirstscatterer Z,,whose electrons theysetinto vibrations. Those parallel to2have noeffect along thesecondary segment, while those parallel to3produce onitsecondary x-rays, which vibrate parallel to3and aretotally polarized. They fallonascatterer Z,and set itselectrons into vibrations indirection 3.Inthis fashion tertiary x-rays areproduced which, however, have theintensity zero along thetertiary segment. They have maximum intensity inthedirection oftheprimary segment. This behavior ofthetertiary x-rays proves both thetransversal nature oftheprimary andthetotal polarization ofthesecondary x-rays. 2 jor3a) : 1 Fig. 28.Barkla’s arrangement for demon- H strating thetransversal nature ofx-rays. Zi,oySecondary 7,scatterers (spheres ofparaffin). ‘ a 'KS.2%ACS-BY*>- ‘Tertiary Ory Noradiation The scatterers Z;and Z,were spheres ofparaffine; forheavier materials the“characteristic radiation” might have falsified theresult. §20. General Considerations ontheStructure ofWave Fields ofCylindrical Symmetry. Applications toAlternating Current Impedance andSkin Effect Inthefollowing section wewill concern ourselves almost exclusively with surface waves which areguided along bodies ofcylindrical shape. Let theexcitation besuch that theprocess isperiodic intime with thecircular frequency w.The calculation ofthepropagation and damping ofthewaves asthey progress inthedirection ofthecylinder axis, which weshall choose asthedirection ofthex-coordinate axis, then becomes ofprimary interest. We leave the cross section ofthe cylindrical conductor (oralso noncon- ductor) temporarily indeterminate. Weexpress propagation and damping byasingle complex wave number h,which differs from thereal wave num- berk=w/cinvacuum. Weconsider thusawavetypewiththedependence onzand ¢ exp{i(he —wt)}; 20.3 WAVE FIELDS OFCYLINDRICAL SYMMETRY 157 fortheassumed cylindrical structure ofthewavefield,4hasnecessarily thesame value outside ofand inside oftheguiding surfaces; thesame applies ofcourse tow. Wedefine, intheplane perpendicular tothez-axis, anorthogonal coordinate system u,v;dz,du,dv,inthisorder, aretoform aright-handed coordinate system. Forthelineelement inspace wewrite, inaccord with (2.22) ofVol. II: ds*=da?+gu’du?+9,do’. (¢)) guandg,arehere given functions ofuandv.The cross section (which is constant, i.e.independent ofx,forevery conductor) may differ fordifferent conductors. The coordinates u,varetobefitted, ineach particular case, totheshape ofthecross section: polar coordinates forthesingle wire, bipolar coordinates forthetwo-wire line, Cartesian coordinates forsemi- infinite space (limiting case ofthesingle wire ofinfinitely great radius). Wesetourselves theproblem ofcomputing thetransversal components E,, E,,H., H,from thelongitudinal components E,,H;.Thisispossible without making any special assumptions regarding thecross sections of thecylindrical conductors guiding thewave, without discussing thecorre- " sponding bolipdary conditions, andalsowithout assuming thatthewave equation isseparable inthecoordinates u,».The general structure ofthe wave field soobtained applies notonly fortheexterior oftheconductors, butalso, with altered choice ofthematerial constants, totheir interior. A.Longitudinal andTransversal Components The longitudinal components E,and H,, which weshall designate by thesingle symbol X,satisfy, asCartesian components, thesimple wave equation a a(1-5 -a3)x=0 (2) Ithasbeen written inthis form fortheinterior oftheconductors, but applies also totheexterior, where ¢=0,©=&,u=uo.Weput FoA=Prt+dw, where A,,isthe two-dimensional Laplace operator transformed tothe curvilinear coordinates u,v.Inview ofthedependence ofthephase factor on2:aiid¢wecanwriteinplaceof(2) ~ (du+P-M)X =0, B= ena’+inow. @) Intheexterior oftheconductors kisreal(=+/eouqw= w/c), intheinterior 158 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 20.38 itiscomplex. However, byintroducing thecomplex dielectric constant e’ from Eq.(6.18) wecanemploy thesame formula k=Vepw (Ba) inboth cases, ifweset &=&, B= Mo intheexterior, . . f (3b) e=e', g=xn__ intheinterior. Actually tointegrate Eq.(2)itwould ofcourse benecessary topassto specific coordinates u,v,whichareadapted totheshapeoftheconductor;weshallavoidthisforthepresent, however. Nevertheless wemayregard, forwhat follows, thelongitudinal components Z,andH,asknown func- tions ofspace. . Tocalculate from .them thetransversal components weutilize the definition ofthectirlofanarbitrary vector Ainanycurvilinear coordinates Pr,D2,Pscontained inEq.(2.26) ofVol.II: 1(ag.As)2tar42)) curlA=—(-—=—— (4) 1gga Ops op} ‘ ‘andsetinaccord with (1), m=2,Ps=U,B=U5a=1, G2=GuyGs=Jee Bycyclic interchange oftheindices 1,2,3andthecoordinates z,u,»we obtain from (4)allthecomponents ofthecurloccurring intheMaxwell equations. Wethus calculate from thefirst and second group ofthree Maxwell equations, taking account atthesame timeofthex,¢dependence ofthe phase factor, foul, =curl,E=she,—1%guOU ive, =cureH=ihe, +LUE,GeOv Onthebasis ofourconvention (3a,b)wesubstitute onthelefto=k/+/en and obtain - i(stt.-ne.)ee e@ GuJu (ig/tn,-ue.)=Lyfe. e ge eov 20.6 WAVE FIELDS OFCYLINDRICAL SYMMETRY 159 From these equations Z,andH,areobtained bysimple elimination: 2—hyp,=—2oBs |/uos selBNEgeueVev 6iat—1)g/m,=—EEe1g/ette ie’—'angedugyV&Ov" Thus ourobjective regarding thetwotransversal components E,andH, hasbeen attained, since ontheright there occur only thelongitudinal components which areassumed tobeknown. The calculation forE,and H,iscarried out similarly. Here the Max- well eanations . F 1OE, oul. thE,+nm —iweE, =thHy—1oH: guOU areemployed, and wsubstituted inthem once more inaccord with the convention (3a,b).After eliminating oneofthetwounknowns H,orE, weobtain i(k?—WE,=hae kp/m,GoOvguedu 6)iat—Wyg/t=*Eebgfate i(kw)4/tH. qooeth Wewillbeable tomake good useofthese rather brief and abstract considerations inthefollowing §21-25, where weshall replace ourgeneral coordinates u,vpartly bypolar, partly bybipolar coordinates. Thus, they show directly, e.g., that forpolar coordinates r,¢andnon-dependence of thefield on¢,thepairs ofequations (5)and (6),which ingeneral are coupled, separate intoonepairwhich contains onlyF.,E,,H,andanother which contains only H., H,,E,.This simplification corresponds tothe symmetry ofthesingle wire. Inthepresent paragraph weshalldealwith thestillsimpler caseofrectangular coordinates u=y,»=zfornon- dependence ofthefield onz.This corresponds tothetransition tothe limit: radius ofwire+ inFig.30.Inallthese special cases thepre- ceding general relations take onareadily understood form andmay, as weshall see, beverified directly. Independently ofthechoice ofuandvafurther conclusion may be drawn which applies forallcylindrical perfect conductors: Thevelocity of propagation onthem isalways equal tothevelocity oflight. Wenote first 160 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 20.7 that theelectric lines offorce must beperpendicular tothesurface ofthe conductor andconclude hence EZ,=0.Since, furthermore, theenergy flux canhave nocomponent inadirection toward theconductcr, wemust also have H,=0.Tobegin with this applies only tothesurface ofthecon- ductors. Weshall assume however that both longitudinal components vanish everywhere inthenonconductor without coming intoconflict with theMaxwell equations. Itthen follows from (5)and(6)(excluding the trivial solution E,=E,=H. =H, =0),that wemusthave h=k. The wave equation (3)which applies forevery Cartesian component of theelectric andmagnetic fieldthen passes over intothepotential equation. Weshall make useofthisconclusion inaspecific example ({25A). B.The Wave Field ofSemiinfinite Space and itsShinEffect Letthemetallic semiinfinite space bebounded bytheplane y=0.Let thepositive y-axis point upwards into theempty space (eventually filled with air)y>0and thewave progress toward theright (paitive z-axis). _Ofthetwo possible solutions (5)and (6)wechoosethefirst,inceitcorre- sponds tothewire traversed byalternating current. We hewe make the reasonable assumption E,=B)| E,EB,=F(y)exp[i(fer—o)}; Hsp=0 (7) H,=Gy)) A, : The differential equation (3)forZ,then takes theform a+(e-W)E=0, with#=ys+im, (8) Itssolution is Ey) =AéViiiy 4Beivitity, (8a) Fory<0&iscomplex, fory>0itisreal.However, wethall, forthe present, assume avery small «>0even fory>0and pasover tothe limit ¢—0atalater stage.’ Weshall choose thesign of~/k* —h?once and forallsothat ithasa positive imaginary part. Since thestete must remain finite for y>+ wemust#in(8a) for y>0:B=0, for y<0:A=0. ‘In this manner wecircumvent some basic questions regarding teexterior of thewire which will bedeferred until §22. 20.106 WAVE ¥IELD@ OFCYLINDRICAL SYMMETRY 161 Furthermore, sinceE,must becontinuous aty=0,Bmustbesetequal toAinthetworesulting expressions. Inthefollowing weshall reserve theletter kfortherealwave number fory>0inthelimiting case¢>0.Forthesakeofdistinction thevalueofkwithin theconductor willbedenoted bykr.Then (8a)assumes the final form: AeiVetaty y>0, Ey)= —— (9)Acivijity y<0 Withu=yv=2n=9 =1,and3=0weobtainfrom(5) hA |eR vo, FY)=hA (9a)Vira city <0. kA— /* -Fee a>, »—[Gy)=kiA (9b)TE oe ee Thefactors of@,ontheleft,denote “reciprocal wave impedances”, the upper onetherealreciprocal impedance ofvacuum asonp.36,thelower onethecorrespondingly defined complex quantity forourconductor. There welearned already thatthedimensions ofHandEmustdifferbyafac- torwith this dimension. Wenowtakeaccount ofthecontinuity ofH,aty=0.Inviewof(8a,b) this demands uot —ht aVki—a Oe. (10)Be ki From this wederive ne?keafR 1BEke Bo Tk ee (10a) Thisvaluedepends inasymmetrical fashion ontheconstants Ho,kandy,ky ofthetwomedia airandmetal. Wefind,inparticular, forx=jp 1 1 1 grata (10b) 162 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 20.11 Forawell conducting metal theconduction current (the o-term inkz) exceeds thedisplacement current (the e-term). Wemay thus assume ky&Vinow=vii =tia, f=. an Wehave then simultaneously «>k, ie by(0b) A&K, (1a) Inthis case oneobtains asphase velocity simply: PateaoETo ashasalready beenpointed outattheendofsectionA. From (9)and (9a) there follows with assumption (11a) intheexterior oftheconductor: |F|«|F|, intheinterior oftheconductor: |E|>|F|. 'The electric Jines offorce thus have essentially thedirection ofthey-axis outside ofthe'conductor, that ofthez-axis inside oftheconductor. Wewill now determine that value ofthey-coordinate within themetal forwhich E,and hence also. J,have decreased toafraction 1/e ofthe value atthesurface. Wecallthis value ofy,—d. According toourassump- tions (11) and (lla): rd=1, date. (12)ck Since k=2z/) this issmall compared tothewave-length \corresponding tothefrequency w.The current isconfined toathin skin attheperiphery of theconductor, while thewhole interior ispractically freeofcurrent. Wespeak ofaskin effect which isknown toplay animportant role inalternating cur- rent practice. The thickness oftheskin issmaller inthedegree that the frequency ishigher (by(11)«increases inproportion to~/«w). Thefollow- ingtable gives some values ofdforCu(=57.5-10°"M™, »=ws= 4n-10-'0M~'S). ‘AlternatingCurrent ‘Telephone WirelessTelegraphy|HertzianOscillations ——ePp zt 60/sec 1000/see 3-10*/s0€ 10¥%/rec~ A= 5:10 km 300km 1km 30cm c= 6 m= 4.7-108 m= 8.15-10' m™ 4.7-108 m=! d= 8.6 mm 2.1mm 0.13 mm 21-107? mm 20.138 WAVE FIELDS OFCYLINDRICAL SYMMETRY 163 Weillustrate this byFig. 29,which, however, does notindicate thecir- cumstances ofasemiinfinite space, but thepractically more interesting ones forawire ofcircular cross section. The straight line 00fordirect current orcommercial alternating current (60/sec) passes over into the slightly concave curve 11fortelephone frequencies (1000/sec); curve 22 applies forhigh frequencies proper (e.g. 1kmwave-length) andshows a on 2 Fro. 29.Variation ofalternating-current ampli- r tude incross section ofwire: 00,direct current; 11, -— te telephone current; 22high-frequency current. o1 2 rey, ~* Fra. 30.Transition tothelimit from thecircular cross ree sectionofthewire(coordinates z,r,¢)tosemi-infinite 2a space (coordinates 2,y,z). pronounced. ‘skineffect. Thethreecurves 00,11,22havebeendrawn for the same total current I. Wealso illustrate, byFig. 30,thetransition from thecoordinates z,r, ofthewire tothecoordinates z,y,zofthesemiinfinite space. C.TheAlternating-Current Impedance ofaSemiinfinite Space Wecutoutofthemetallic semiinfinite space arectangular parallelepiped which isinfinitely long inthey-direction andwhose upper endsurface lies intheplane y=0.Letthelength oftheside parallel tothez-direction be unity, that parallel tothez-direction beequal tothewave-length Aofthe wave propagating itself inthisdirection. If,inthefollowing, weneglect theimaginary part ofh(the slight damping ofthewave inthez-direction) wehave \=2x/h. The total current flowing through theparallelepiped is: 1 poe t=[[sededy, (13) Wedefine theresistance oftheparallelepiped energetically with theaidof theheat generated within it,which isgiven bytheJoule heat Qintegrated over theparallelepiped andaveraged over thetime. ByPoynting’s theorem wehave fortheparallelepiped: _ WetWatQ=-fSide. (13a) Here thefirst two terms ontheleftdrop outontaking thetime average because oftheperiodicity oftheprocess. The Poynting vector ontheright 164 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 20.13b isonlythatacross thezz-surface (shaded inFig.31);thecontributions ofthezy-surfaces arezerosince H.=H,=0andthecontributions ofthe ys-surfaces cancel because oftheperiodicity with respect tox.Hence 1pk » Jsao=—[ae [sae=[Bena. If,forZ,andH,,weusethevalues within themetal (which, asweknow, agreewiththoseinairaty=0andmaybesimplified because |h|<|kr), wefindfrom (9)and(9b), utilizing therepresentation (7) ‘F JS.do=4[Rele***#*] -Re{Vgconesdt.u If,a8asufficient approximation, wesubstitute herei/wfore’andwriteforViitsvalue (1-+'i)/+/2, wecanplace thefactor («/(2uw))' ahead of y i, ( es a7Fro.31.Ablockiscutoutofthemetallicsemi- TYE infinitespacey<0bythetwopairsofplanesz=0 FEBS\\\\\\ spaceyy paipl » oa\\ANAS ==X;2=0,2=1.Computationofitsresistancefor ’rn ‘asurfacewaveprogressing inthez-direction. ‘;=i H ' H theintegral sign,while thefactor 1+iremains intheargument ofRe.If wemake the further substitution du _ddu _ u=he—ot,daa ty=—ul, thepreceding equation passes over into Afofm .nA?V5 [Sedoel. cosu(cosu—sinu)du>Ja’13) Since thisvalue hasbecome independent of#,itrepresents atthesame time thetime average ofQ,which werequire forthedefinition ofthe resistance. Taking atimeaverage oftheenergetic definition ofFin(18.6d) andthus extending itforalternating current wewrite en. cRP=Q=>W/55° (14) Inorder tocompute thevalue of/*which occurs herewemake useof theloopintegral ofHabout theparallelepiped, e.g.intheplane +=0 20.15¢ WAVE FIELDS OFCYLINDRICAL SYMMETRY 165 (indicated inFig.31byheavy arrows). Inviewofthedirection ofH only theedge y=0,0<z<1oftheparallelepiped yields acontribution. We thus obtain 1t=[Hae=H,=Ge 0 ; (14a)=V2Ac=V=A(l+ae™ a 2uw Before taking themean square ofthiswemust passtotherealpart: I=VeA(coswt+sinwt). 2us We then obtain B= *A (14b)Qu Substitution in(14)yields B=r4/t=,roferringto(1). (15) Qo "Themeaning ‘ofthisformula becomes clear ifwesubstitute theskin thickness dfrom Eq. (12). Itthen becomes ay R=z. (15a) disthe“tength” ofourconductor segment measured inthedirection of propagation ofthewaves. Ifwecompare (15a) with theelementary formula fordirect current R=, (asb) qe weseethat thecross section qbecomes, inouralternating-current case, therectangle d-1(dinthey-direction, 1inthez-direction). (15¢) Inplace oftheinfinite cross section (theyz-surface ofourparallelepiped) available toitthealternating current utilizes, inasense, onlytherectangle (15¢); expressed differently: thealternating current, which drops offex- ponentially within theconductor, behaves, with respect toitsresistance, ~ just asadirectcurrent which isdistributed uniformly overtheskinthick- ness d. 166 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 20.16 D.The Rayleigh Resistance ofaWire Wenow pass from theresistance formula (15) foraconductor with plane boundary tothat foracircularly cylindrical wire ofradius a.We assume here a>d, sothat theskineffect may develop freely atthesurface ofthewire and itsinterior remains free ofcurrent. Wemust now, however, consider a width 2raoftheparallelepiped, rather than thewidth 1,andimagine its current-carrying layer tobebent into thecurrent-carrying surface layer 4 is Fro.32.Resistance and inner inductive rese- oo tance foralternating current asfunction offre- R by quency. Theabscissa isproportional tothesquareEz ory rootofthefrequency. OA=bisector oftheangle 7— between thetwo axes. The inductive reactance curve Tye” poate approaches itasymptotically, whereas theresistance“ +curverunsparalleltoitinthelimit.a ofthewire: Herewith wehave changed thecross section g=d-1defined in(150) tog=d-2na andtheresistance Rfound in(15a) toRe: R r Redna” Ieaod® (16) If,furthermore, weintroduce thedirect current resistance ofthewire R= (16a)mao which, likeRe,werefer tothesame length \andthesame conductivity o asourR,weobtain simply: Re _la_a@B73a73" (7) This isRayleigh’s resistance formula forhigh-frequency alternating current. Fig.32indicates itslimits ofvalidity. Forsmall w(stationary and quasistationary currents) Ra=R,,incontradiction with(17):Ourplotted curve atw=0istangent tothehorizontal atadistance 1from theaxis ofabsciseas (toahigher order oftangency). The approximate representa- tiom(17)applies onlyforsufficiently large w.Because ofourchoice ofthe scale ofabscissas thedirection ofthecurve forincreasing wfollows the asymptote OA,which isinclined by45°totheaxes. Theintermediate 20.19 WAVE FIELDS OFCYLINDRICAL, SYMMETRY 167 region between smallandlargewrequires amoredetailed analyticaltreatment (seeendofthissection). E.TheAlternating Current Inductance Themethod given sofarcould yield only theresistance. Toobtain in similar manner thereactance wewould have tocompute themagnetic energy Wasafunction ofthecurrent I: L Wa=zr andthisnotonlyfortheconductor (inner selfinductance L;,seep.122), butalsoforthesurrounding airspace (outer selfinductance L,).Onthe other hand ouranalysis oftheexternal field, ascarried outuptothis point, would notbeadequate forapplication tothewire, sothatinthe following weshall confine ourselves totheinterior field andtheinner selfinductance L;. Forthishowever, wehave amore general andalsosimpler method available, namely thatoftheimpedance operator ofEq.(18.10): RI=EF,R=R-— iol,.' “WeputEequaltothefieldstrength E,(voltage perunitlength ofour conductor) atitssurface y=0,sothat,suppressing thephase factor, we have byEq.(9)#=A,where now both RandLhave tobereferred to unitlength. Weobtain Jfrom (14a), where weagain suppress thetime factor: o o =zs =>A . rTVfganar z(L+4) Wefind astheratio ofthetwo E « zra-ae, sothat, by(18), E « KRey=(-a)-, R=uly=~. (19) With respect toRthisagrees with(15)if)isreplaced byourpresent unit oflength, and shows atthesame time that theinner inductive reactance 1Wehave changed thesign oftheimaginary unit ascompared with §18inorder tobeabletoemploy thepreceding formulas forJand£directly. Inthem thetime factor waswritten intheform exp(—iwt), while in$18exp(+iwt) occurred inthe corresponding formulas. 168 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 20.20 wL, isequal totheresistance R.‘This applies generally foraconductor bounded byaplane, but applies also tothecircularly cylindrical wire in Rayleigh’s limiting case ofsufficiently high frequency. F.Further Treatment oftheAlternating Current Field ofa Circularly Cylindrical Wire Inorder toclose thesubject ofthis section itisnecessary toutilize some formulas which will notbederived systematically until §22. Weare concerned first ofallwith thelongitudinal alternating current field ina wire ofradius a: E,=CJo(kr). (20) Jodenotes theBessel function oforder 0which iscontinuous atr=0." (Justaspreviously, thephasefactorshould beimagined tobeadded.) Inthefollowing kistosignify thecomplex wavenumber intheinterior ofthe wire, not, asbefore, thewave number inair.The coefficient Cin(20) is determined from thecurrent density J=oF, inthewire and thetotal current Jbythelater Eq. (22.34): Kl ~~deanFa} om) From (20)and’(20a) weobtain forthecurrent density J JL_boda)To~72Tila)’ (21) Jo=I/(xa’) isthedirect-current value ofJ Eq.(21)may serve tocheck ourFig.29.Forlowfrequencies theargu- ments krand kaaresmall inabsolute value. Then theexpansion (22.3c) and itsderivative may beemployed: -1- (+0)-A(he soo)=1~(8)+2)-E(B)t/t fet (21a) p)=—2(1--(2 =(2 see rin=~5(1-3(3) +0(8) *~) substituted in(21) this leads, with dueregard of(11), to kr1(kr\‘ toLe 7-()+1(¥)eeLaidLa Jo1(kay(is) teeLow .1-38)+5 >verLG(xa)?—5(xa) 1Weemploy here andinthefollowing (unlike Vol. VI)thesymbol J,inaccord with thepractice followed inVol. Ifofthese lectures andinAmerican physics and engineering literature generally. 20.23b WAVE FIELDS OF CYLINDRICAL SYMMETRY 169 From this follows . 1 4af 1+3(xr) RlTa (21c)a 1+B(xa)’ Ascompared tothedirect-current straight line 00inFig. 29there occurs thus adip,which ontheaxisofthewire (r=0)hasthedepth (xa)‘/48, and ariseattheperiphery ofthewire (r=a)toaheight which isfive times asgreat. Athigh frequencies weobtain asymptotically, according to(22.7) and (22.6a): Joller) =(wkr/2)"* cos(kr—x/4) (22) J _ka(a)!cos(kr—4/4)Fal=‘yeen) h7?()an@a=a7) |)~Var)? (22a) Inview ofthe sharp decrease for r<athe whole interior ofthe wire is practically freeofcurrent; themagnitude attheedge isxa/+/2 timesas (Great asinthedirect-current case. +Thesame formulas alsoyield aclosed expression fortheoperator Rin Eq. (18).By(20)and(20a)wehave E, _k_So(ka) kaJo(ka) R=~~SyarTuba)~~2Teka) =) where R,isonce more the.direct-current resistance perunit length, i.e. 1/(xa’s). Hence wehave forlowfrequencies (by(21a), expanding inpowers ofxa): R_iol tey2y 1‘Re Re™14)+Fy(ea ‘The separation oftherealandimaginary parts yields: R le ob ly asRB1+3(xa)*, R7i(xa)*. (23a) Ontheother hand, weobtain, from (22) and (22a), forvery high frequencies simply: R_wh ka xa, s- pt risgetl-IsRo Bo 2 2 - (23b) Rows Ro Rm 2° 170 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 21.1 This agrees with ourearlier results (17) and (19), obtained forthecon- ductor with theplane boundary. Our approximations (23a, b)permit ustocheck also Fig. 32and to interpolate fortheintermediate region between low and high frequencies. Atlowfrequencies, by(23a), theresistance curve approaches thestraight lineR=Ryasaparabola ofthefourth order and athigh frequencies it approaches thestraight lineOAinFig. 32from above. The curve forthe inner inductive reactance starts atlow frequencies asaparabola ofthe second order’ andapproaches’ athigh frequencies thesame straight line from below. §21. The Coil Carrying Alternating Current Inasmuch asin§17wehad todefer thetreatment ofthedirect-current field ofthecircular wire asmathematically toocomplicated, therigorous treatment ofthealternating-current field ofalong coilappears tobeout ofthequestion. Wehence make thesame approximation asonp.25, ie.teplace thecoil, which weassume toconsist ofasingle closely-wound layer, byaninfinitely long hollow cylinder ofuniform metal. Letitsaxis (x-axis) bevertical and letitbetraversed byhorizontal circular currents ‘whoseintensity distribution weshalldetermine. Lettheinnerradiusof thehollow cylinder bea,theouter radius a+d. A.The Field oftheCoil Asinthedirect-current case weassume that themagnetic excitation is zero outside ofthecoil, uniform? within itand parallel tothecoil axis, so that wemay write H=0Oforr2e+d, H=H,=He“'forr Sa. (ly Wemust then assume theH-field tobeparallel tothecylinder axis also inthe metallic conductor, i.e. H=H,= He“ fora<r<atd. 1This statement applies totheproduct wLplotted inthefigure; Litself hasfor #=0,inaccord with themeaning of2=pew/2, thenon-vanishing value Fepoo? a ra rs inagreement with §16C. +This approach isatrue tangency forw +;ontheother hand, thecurve for theresistance remains evenforw©afinitecmount Ro/4abovethestraight line OA, as-‘would beshown byamore precise formulation oftheapproximation (23b). *This customary and practically unavoidable agsumption forinfinite lengthof thecoilis,strictly speaking, notpermissible inMaxwell’s theory. Itcontradicts the equation, applying forthenonconducting interior space, D=curlHandis,inview ofcurlH=0equivalent totheneglect ofthedisplacement current D. 21.5 THE COIL CARRYING ALTERNATING CURRENT 171 H,must satisfy thegeneral wave equation (20.2), which yields, forH(r), the differential equation AH(r) +BH(r) =0, = eww? +ipow. (2) Transformed topolar coordinates x,r,¢itisintegrated interms ofBessel functions. Wedonotrequire here, however, theparticular solution Jo(kr) asin’(20.20), but thegeneral solution containing two constants, which wewrite preferably intheform CAHi(hr)+CaHChr). (20) H',H’arethetwoHankel cylinder functions, about which weshall give some information inthe next section. Inparticular weshall familiarize ourselves there with their asymptotic behavior forlarge values ofthe argument p=kr—@: HG)>V2HBG) V2ein, (2b)™p ™p Since forhigh-frequency alternating current invariably |k|a>>1,wecan limit ourselves intheintegration of(2)tothese asymptotic values and ,can write uoVz(Cael4.Gyertter, )ES From Eqs. (1)wehave theboundary conditions Haa+d)=0 and H(a)=H.. They aresatisfied if,byspecial choice ofC,, C2, expression (3)istrans- formed into @7,sin[k(a+d—1)] Hy)=4/28,2hee—, @ Having found inthismanner H.asfunction ofr(wemay also sayasa function ofthepolar coordinates z,r,y)wecannow utilize §20A.Itistrue that now wearenotdealing, asthere, with awave advancing along the z-axis, butwith ordinary (stationary) alternating current, forwhich the wave number hgiven there vanishes. Infactourpresent Eq.(2)passes over into theearlier (20.3) for X=H,,h=0.Furthermore wemust notemploy, asinthepreceding paragraph, Eq. (20.5) (electric type), butEq. (20.6) (magnetic type). This yields thetransversal components H,,Z,expressed interms ofthelongitudinal component H..Weareparticularly interested inE,“With g.=1,9.=r,H=0,(20.6) yields: 8 faa) pu7 E,kVEa°aa (5) 172 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 21.6 Ifwecarry outthedifferentiation with respect toronly inthefactor sin{k(a +d—r)}(the square-root factor is“slowly variable”) and if wedenote thevalue ofJontheinner surface ofthecoilbyJ.,weob- tain facos{k(a+d—r)} =ies/tcos(kd)int J1.4/9leteanh,Je=ta,oein(ka)©~(6) Passing tothediscussion ofthefield, wedistinguish twocases: a|k|d>1 and bjkl|d«1; weofcourse continue toadhere totheoriginal assumption |k|a>1. a.High-Frequency Alternating Current with nottooSmall Coil Thickness. Since wetake theimaginary part ofktobepositive wehave then Je |>>|eit# and, more particularly near theinner coilboundary also: |EBONY [5|gitlotenn | Hence (4)yields, forr=a, Hr) =He, @ which signifies asteep exponential falling offattheinner boundary ofthe coil. By(6)thecurrerit density shows asimilar steep exponential decline: Jeter, (7a) Je=ioViHee*. (7b) Wethus have apronounced skin effect attheinner surface ofthecoil. b.Small Coil Thickness andRelatively Low-Frequency Alternating Current. Wemay then expand Eqs. (4)and (6)inpowers ofkdand, particularly intheneighborhood ofr=a,alsoinpowers ofk(a+d—r).Weindicate only thefirst term ofthese expansions: woyen tte", ses, ® Fig. 33illustrates this graphically: Atthe left isshown thebehavior of H(r), attheright that ofJ.The curves 0correspond tothelimiting case b, thecui'ves 2tocase a,and thecurves |toanintermediate case. Allthree pairs ofcurves refer tothesame Hwithin thecoiland hence also tothe same total current Jinthe coil. 21.10 THE COIL CARRYING ALTERNATING CURRENT 173 B.Resistance andInnerInductive Reactance oftheCoil Wewish tocompute these quantities perunitlength ofthecoilandmust firstknow thetotal current passing through thisunitlength. Itisobtained byintegration of(5)with respect tor: +rfJdr=C{H(a+d)—H(a)je™' =—CHe'; le inviewofthemeaning ofkande’wehave -¢ = 4/ha~CHENE21 (9) Se, 4 AifRd G _-1_ DR ne 272red a atd a atd +£16,33.Malneticfieldandcurrentdistribution inasingle-layer coil:Atthe left,magnetic fieldH(r) =H,,attheright current density J(r)=J,.Thecurves 0and2correspond tothelimiting cases ofdirect current andhigh-frequency alter- nating current, curves 1tointermediate frequencies with thesame total-current as inthetwo limiting cases. Torefer thistotal current tothecurrent Jflowing inasingle wireofthe coil wesetitequaltoNJ,whereNisthenumber ofturnsperunitlength: 1oytet Iyee (9a) With thisvalue ofJwewrite theequation (R—wl) =E, wherewemustsetthevoltage Fequaltothefieldstrength ontheinner surface ofthecoil,i.e.equal toJ./c. Wethusobtain from (6)and(9) iNkcos(kd)_.Nke“ +6™ Rtoh = 7ae (10) Foradiscussion ofthisexpression wewrite, asin(20.11), k=(1+ae. The desominator of(10) then becomes _ Qitind _0-oed 174 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 21.11 Wemultiply numerator anddenominator withthecomplex conjugate of this quantity, whereupon they can beexpressed interms ofthetrigono- metric and hyperbolic functions of2xd. We obtain ~gol=(1—4)NAsinh(2nd)+ésin(2nd) R~tol=(14)“cosh(xd)—cos(xd) (4) sothat, separating real and imaginary parts, Nesinh(2nd)+sin(xd) Bmcosh@xd)=cosed)’ (ia) Nesinh (2d)—sin(2xd) wh=cosh(sd)—cos(Od)* (1b) Fordirect current (kd—>0)(11a) yields N R=ca’ (12) corresponding tothedirect-current resistance perunit length ofawire of rectangular cross section with thewidth dofthecoil (not tobeconfused with thelayer thickness dofp.162)andtheheight 1/N,Wehavetherefore R sinh (2xd) +sin(2xd) Roacosh(2xd)—cos(2xd)’ (12a) ob sinh (2xd) —sin(2xd)Re~"4Cosi(oud)—cos(xd) (2b) .Moreparticularly weconsider thetwolimiting casesaandbofp.172: a.xd>>1.Thensinh(2xd) =cosh(2xd) —©,sothat R _oL~~ = 1Re”Bexd, (18a) b.xd<1.Byexpansion inpowers of2«d, and retaining only thefirst nonvanishing term wefind from (12a, b) R 4 one ob 27 nsKRL+z(ay, BR3(ed). (13b) These results (13a, b)arerepresented qualitatively once more byFig. 32.Also now thedirect-current straight line R/Ro =1isapproximated at -lowfrequencies tothefourth order andtheaxisofabscissas isparabolically tangent tothecurve wL/R, attheorigin. Athigh frequencies both curves again approach the 45°line asymptotically. Only the scale ofabscissas, now given by«d,differs from that given before since theskin effect occurs here unilaterally, onthe inner surface ofthe coil. 21.15 THE COIL CARRYING ALTERNATING CURRENT 175 However much more important than theinner selfinductance, towhich wehave limited ourselves, isofcourse theexternal selfinductance ofthecoil Inthediscussion ofthecoil traversed bydirect current (§17, Eq. (15)) wewere exclusively interested inthelatter. With thelimitation mentioned infootnote 3,p.170(neglect ofthedisplacement current) wecantransfer the value found there directly tothe alternating-current case. With ouridealization oftheproblem (closed ring currents inhorizontal planes) wecould circumvent thedetermination oftheelectric field within thecoilandleave theboundary conditions forthisfield outofconsideration. Actually thecurrents are,inview ofthefinite pitch ofthecoil,notexactly horizontal andthere exists, from turn toturn, anexternal electric field of complex character which ispredominantly axial indirection. Our elemen- tary treatment evidently does notsuffice todetermine thisfield. C.TheMultilayer Coil Weimagine theseveral layers, ninnumber, tobeplaced oneover the other without intervening space, idealized ashollow cylinders ofthickness d/n,andtraversed inseries byalternating current Jofuniform magnitude. Lettheinsulation ofthesuccessive layers beperfect, though byassumption infinitely thin;\the sameistoapply fortheinsulation ofsuccessive turnsin any single layer. Inview ofthemagnitude oftheloop integral ofHabout asingle layer, i.e.over arectangle ofheight 1andwidth d/n, themagnetic field decreases byNIforeachlayer (V=number ofturns perunitlength along theaxis ofthecoil). Hence wefindattheboundaries ofsuccessive layers, from the inside outward: Ho=H, Hi=H-NI, H:=H-2NI,---,Ha= (14) H—-nNI =0. Thefactor exp(—twt) must bethought ofasadded here, andinthefollow- ing.Inthe»thlayer themagnetic fieldsatisfies thedifferential equation and the boundary conditions: Awr=at— vf, 4H+RH=0, H= a(15) H, r=at+y a Thedifferential equation isagain integrated bythesuperposition ofthetwo Hankel functions, forwhich wecansubstitute their asymptotic values from Eq.(2b)since krisverylarge; furthermore wecantreat thedenominator kr asslowly variable incomparison withtheexponential functions and i76 DERIVATION OFPHENOMENA FRCM MAXWELL EQUATIONS 21.16 include italong withtheremaining constants (exp{—ix/4} and+/2/m) intheamplitudes C;, C;. Eq. (8)isthen simplified to H=Ce“ +Ce, The following expression, which atthesame time satisfies theboundary conditions (15) and hence represents themagnetic field inthevthlayer, isofthis form: eos{t(+-a-»2} H= Hea = 2n, (16) sine(+-a-b- oH) -WNI& sin(é‘)n, Inparticular weobtain forthefirstlayer,a <r<a+d/n,y=1 wo{t(r—2-5) yanthe0 H=H, —)\— 94 —ypSENS) (168)) (8) cos|k— sin|k—2n, Nn, Weneed consider only thisformula ifnow wewish todetermine theelec- tricfield Z,attheinner surface ofthecoiland from this theimpedance operator R=R—iol,ofthemultilayer coil. Tobegin with weobtain forthecurrent density J,,forwhich weshall consider right away thevalue forr=a,from (5)(seealso (9)): . dsin(«a) s.=0(#)=—Hk +wk7... dr Ja d F d. cos(:z) sin(«‘) 2n, n, Wehere setH=nNI (seelastofEqs. (14)) andobtain after simple trigo- nometric transformation, neos(x2)-(n—1) Jo=NIk——*7___. . dsin(«“n, 22 THEPROBLEM OFWAVES ONWIRES 177 Hence we obtain d1G3)—(n—1) B=8,<14, andReZ NkON o I o . adsin(«‘)n which, forn=1,isidentical with Eq. (10). Wedivide Rbythedirect-current value RyofR,obtained bypassing tothelimitw—0,ie.k+0: NnRo=cd’ and find Rcos(i“)-(1-2)Sek AM NM )rm n(e9) an sin{k~n, .Theseparation of(17)intorealandcomplex parts israther complicated. If+as before we\set k=(1+i)x, ityields sinh(2‘)asin(2i)—2(1-) n, n n, .(sinh(«2)+008(«2)cosh(+2).sin(«*)) R/ie=wd n, n n. n L/Ro ~ °cosh(x‘)—cos(2‘) n, n, Thus, thanks toourextensive (possibly excessive) idealization oftheprob- lem, wehave obtained aquite simple final formula. The frequency band forwhich ourformula, isvalid hasanupper limit determined bythechar- acteristic frequency ofthecoil; asweapproach thelatter ournotion of equal current inallturns obviously becomes invalid. Itshould also be emphasized that ourformula presumes the’regular superposition ofthe layers and does notcover thespiral interweaving oftheturns (Dolezalek, litzwire) which ispreferred forpractical reasons (suppression oftheskin effect). a §22. The Problem ofWaves onWires Asiswellknown, theexperiments ofHeinrich Hertz dealtwith“surface waves” progressing along wires aswellaswith “space waves” propagated freely through theair.Hertz expected their velocity also tobeequal toc, i78 DERIVATION OF PHENOMENA FROM MAXWELL EQUATIONS 22.1 butcould confirm thisresult neither experimentally northeoretically. The reason forhisexperimental failure was theinfluence ofthewalls ofthe laboratory; thereason forhistheoretical failure, anexcessive idealization oftheproblem. Hetreated thewire asinfinitely thin andhence could not setupelectromagnetic boundary conditions. This wasfirst accomplished in.apaper bytheauthor’ which yielded forthevelocity ofpropagation a value nearly equal to—more precisely, slightly lessthan—c. Itwashere essential that aphase velocity exceeding ¢could beexcluded byacondition atinfinity. The experimental difficulties were overcome byE.Lecher (see§25) byusing @two-wire line. Inthepresent section weshall confine ourselves toHerts’s problem ofthesingle wire. A.TheField within andoutside oftheWire While inthepreceding section theMaxwell equations were utilized only inpart, ‘inasmuch asnotonly wasthedisplacement current within theconductor neglected, buttosome extent alsothat infreespace, wemust now adhere strictly tothese equations. The problem issymmetrical about theaxisofthewire. Wemake itthez-axis ofacylindrical coordinate system z,7,g.Then forallcomponents 3/dy =0andonly thecomponents E,,E,, H, q@ differ from zero. Asin§20wesetthem equal toproducts ofthecommon factor eriette (1a) with afunction ofronly. Letthetime variation bepurely periodic, i.e.wbe real, and thephase propagation take place along thepositive z-axis; h must then have apositive realpart. Wedeal first with theCartesian longitudinal component Z,.Itsatisfies thewave equation (20.3), inwhich weputu=r,v=y.If,inplace ofr, we introduce the dimensionless variable p=VBhr (1b) and set EB,=F(p)e**'*™, (2) weobtain forFthedifferential equation 1d(a)Pe (ela ra0aap? de+ (3) ‘Ann. d.Physik, Vol. 67,pp.233-200, 1899. 22.5 THEPROBLEM OFWAVES ONWIRES 178 or,withthedifferentiation carried out, @F .1dF4+ :S4 Ppnd.aw+adp+ (3a) This, aswell asthemore general equation aP,1dr(*) a5t+-5- 1-5)F=0 aw+pap+ 2 (3b) isknown asBessel’s differential equation. Wehave dealt with italready inVol. II,§27. The solution which iscontinuous forp=0was represented there, in(27.7), bytheseries al(ey__td“yn 1ay 1.)=3.(6)mori +eas “+Ge) Forallthatfollows nmaybeassumed tobeaninteger. Forn=0weobtain therepresentation (21a) forJo(p) employed in§20, from which wesee directly d - 9:0)=—2sl), [/ndolo) do=plo (3d) lp Within thewire, r<a(a=radius ofwire), where EZ,must nowhere become infinite, ourfunction Fisthus determined but foraconstant by itsdifferential equation (3): F=CJlp) for O<r<a. (4) Outside ofthewire(inair)¢=0,=&,»=wo,80thatk=w/cis real. Forthesake ofdifferentiation weshall denote thecomplex value ofk, which applies within thewire, bykzasin§20.Since thecondition ofcon- tinuity atr=0plays noroleoutside ofthewire, Eq.(3)hastobeintegrated generally. This isdone bythetwo“Hankel functions oforder 0”,already mentioned onp.171: Hilo) and Hale). Weshall deal indetail with these and thegeneral Hankel functions of order ninVol. VI,§19. Itmust here suffice toenumerate some oftheir principal properties: a.Thefunctions HjandHgbecome logarithmically infinite forp=0. sincé wehave, forsmall p: Bp)=Lelogepeeemetelog28... Ho"(p) 14Flogy+ &log (5) 180 DERIVATION OFPHENUMENA FKUMMAAWSLL BYUAIUND =26.00 1isrelated,totheBuler-Mascheroni constant * 1,1 1Lim(1+3tatooto-log»)=0.5772+++; For logy=0.5772 ---,y=1.781--- (5a) (loghere denotes, asalways, thenatural logarithm.) b.The Hankel functions arebranched inthecomplex p-plane, just as thelogarithm. Tomake them single-valued wemust provide abranch cut inthep-plane, e.g.along thenegative imaginary axis.Ifwesetp=|ple”, wethuslimittheangle#tothevalues —x/2 <8<31/2.Inthissensewe speak oftheprincipal branch oftheHankel functions, justaswespeak of theprincipal branch-of thelogarithm.c.TheHankel functions H'\(p) andH’,(p) oforder naredefined assolu- tions ofEq.(3b) insuch fashion that alsotheir representation contains logarithmic term forp—>0,i.e.thatthebranching mentioned inbapplies also tothem. However, thedetermining factor fortheir singularity isnot thislogarithmic term,butthetermwhich becomes most,strongly infinite: 1)ooniay 27)(naDay Hy)=SSMZY, a-2=F(Z). Thelogarithmic singularity andthebranching disappear forthesumof thetwofunctions H,. The regular solution ofthedifferential equation (8b) isobtained intheform Talo) =AHe) +HX(6))- (6a) d.Forp— ©wehave asasymptotic representation ofthetwoprincipal branches: HYG)-V2gio) gx.(o)-fiero, 7) alo) aod ,al ‘Thus A’,vanishes forlarge pinthepositively-imaginary p-halfplane, H’., inthenegatively imaginary p-halfplane. The twotogether vanish for large ponly ontherealaxis. J,becomes, by(6a), infinite everywhere at infinity except ontherealaxis.Inviewof(6a)and(7)wehaveatinfinity inthepositively imaginary p-halfplane Fao) - Ti)7t* @) After these insertions, which unfortunately were necessary forwhat follows, wereturn toouractual problem. Wemake theconvention thatthe signofthesquare rootin(1b)isalways tobechosen sothatitsimaginary 22.128 THEPROBLEM OFWAVES ONWIRES 181 partispositive. Thiscoversthecasethattherootisitselfcomplex. Wemust however alsotakeintoaccount thePossibility thattherootisreal and that h<k. Inthefirstcasetheonlypossible formula fortheexterior ofthewireis F(p)=AHie), a<r<o, A=const, (8) since, by(7),H(p) becomes infinitely great forr+©. Inthesecond case F(p)=AHi(e) +BH), 9a<r< oe. (9) isapossible formula sincenow,forrealp,bothHvanish, by(7),a8p74.AandBareforthepresent arbitrarily disposable constants. What isthemeaning ofthiscase? Since according to(2)thephase velocity ofthewaveisequaltow/handsincew/kisequalto¢,itimplies phase velocity >velocity oflight. Wesupplement ourrepresentation ofH,bythatofthetransversal com-ponents Z,andH,,which isobtained mostreadily withtheaidofthegen- eralrule(20.5) (where however theterms withH,areofcourse omitted). Afforthepresent wedesignate thefunctions ofappearing inE,andH,with G(p) and\K(p) thisyields : thaF() th , GO)=BR eT Veaero: (10) tkaF(o) tk fi=~, Fb)—_* lp 280)=pe VeamPO aw B.TheBoundary Condition atInjinity Asforthesurface wave in§20wearealsonowdealing withaprocess whichdrawsitsenergyfromtheendofthewireats=—0.Wehenceshalldemand thatthetotalenergy fluxthrough acylindrical surface r=Rcoaxialwith the wire vanishes: S=2r{rS,} un=0. (12) Ifasanabbreviation wedesignate thephasehr—wtby®weobtain —tk—dF(o)} i®,& P)ie S,=E,H,=Re{F(p)e*} re{4/®Ve=B do°fs Weconsider thesecond case,inwhich h,Vi?—fi,andpwerereal.Eq.(9) thenappliesforF(p)andweobtain,indicating allimmaterial constant fac-_torsby... andutilizing theasymptotic expressions (7)withn=0: Sm+2[A¥cos?(+9—2/4)—BYcoe!(—p+/4)). (128) 182 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 22.13 Itissignificant herethatthefactorr/premains finiteforarbitrarily large r=Rand thatthephase may take onanyrealvalues. Ourrequirement (12) canthen besatisfied only by A=B=0. ‘Thissignifies: Inthesecond casewirewaves cannot occur; theywould have tobefedbyanartificial arrangement ofenergy sources atinfinity, which contradicts thephysical meaning oftheprocess. Conditions aredifferent inthefirstcase, where phasapositive imaginary part. According toEqs. (8)and(7)thefieldoutside ofthewiredecreases hereexponentially asr—>©;theenergy fluxSvanishes insimilar manner. Only thiscaseisrelevant forus.ByEqs.(2),(4),(8),(10),and(11)we calculate thecorresponding fieldwithin andoutside ofthewire, anddis- tinguish thecomplex %,withinfromthereal&outside, aswellasthecom-plexe’/uwithin from thereale0/yo outside asin(20.9a, b).Furthermore itisconvenient toredefine theconstants CandAsothat allcomponents aremultiplied withVEL insideandwithYF— outsideofthe wire. Wethus obtain thefollowing tabulation: O<r<a p=Viewr |a<r<~, p=Ve—hr 7 Ve—g,~VERE oni) B=VERE an) 4 ’ (18) E,=CJole) E,=AHoo) k, k 5H one |ft=Bastin Hyisidentical with H};theprime atJoandHoindicates differentiation with respect totheargument p. C.TheBoundary Condition attheSurface oftheWire E,andH,must becontinuous for7=a.Hence wemust require fe!’ mCiod=AW), 4/%buCIiled =4/BKAHO 9 p=Veh, p=VPHa. Byeliminating theamplitude factors AandCweobtain thetranscendental equation . Hole)_fae (14a) Hoe) eyekz,Solo.) © = Weregard thisasthedetermining equation fortheasyetunknown wave number h.However, Eq.(14a) canbegreatly simplified bytaking account 22.16 THEPROBLEM OFWAVES ONWIRES 183, ofthefactthat ifthematerial ofthewire isagood conductor p;isalarge complex number with positive imaginary partsothatEq.(7a)isapplicable. With itsaidtheright side of(14a) may betransformed into g/t #9 wig Jem Since |e’|>>&itsabsolute value issmall compared with 1.Furthermore theleftsidemay alsobesimplified. Since itmust besmall wemay use Eq.(5).Wethen obtain fortheleftside of(14a) 2top 2 r_(wy Pplog55=ytloguwithu=(2y. (14e) Comparison with (14b) then yields asthefinal form ofourtranscendental equation a2 wlogu=vwithv=Fy he (15) 2 uo Foritssolution itispossible toemploy apeculiar method reminiscent of ‘thecontinued fraction. This rests onthefactthatloguvaries slowly in comparison with u.Hence ifannthapproximation wu,hasbeen found, an n+1*approximation may beobtained from Ung logUn=v. (15a) Wemay begin, forexample with up=vandput, inaccord with (15a), hs (15)logv theexact initial value isoflittle importance since itiscorrected step by step inthe subsequent approximations. Furthermore, by(15a): epee, = ete, (150)OB tog log—“— logv log8logv Consider, forexample, acopper wire with radius a=1mm and thefre- quency given farthest totheright onthetable onp.162, which corresponds toawave-length of30cmandavalue ka=2.1-10"’, Forthecorresponding value ofxtaken from thesame table, (15)yields v= —(L+8)-7.2-107. Webegin with um=(1+1)-3.6-10, (16) 184 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 22.168 wherelogv,occurring in(15b),hasbeenapproximated by~—20.Wethen find from (15a) uu,=(4.1+4.51)-107, ta=(4.2+4.6i)-107; (16a) Wethus have already arrived atthelimit ofconvergence ofour“continued fraction.” From this value ofuwefind by(14c) p=-Su=-63+58i)-10%; (16b) and by(1b) KR=aR+Fs(53+58i)-10%, hh=k{1+(6.0+6.6i)-107}. (17) We form i=i{1—(6.0+6.6:)-107} (17a) ,andconclude from this, since w/k=¢,that thephase propagation lagebehind “ebyonly6-10;*c.Ontheother hand weseefrom ourfieldfactor exp(thr) that theamplitude isreduced byafactor 1/eonly after traversal ofadis- tance zgiven by k-66-10%2=1, +=720m. (17b) This corresponds completely towhat isexpected: Nearly undamped propaga- tion with casphase velocity. There arehowever also conditions forwhich this expectation proves to beerroneous. Consider, forexample, aWollaston wire ofplatinum with a radius a@=2-10‘ cm;theconductivity ofplatinum is8timeslessthan that ofcopper. Letthewave-length inairbe1meter. Then x=9.2.10"cm™,f/%=(1—)-034-:10%, pp=(1+1)-02. The argument ofJoin(14) isthen nolonger large, sothat wemust use Eq. (8c) inplace of(7a). Ityields To(ox) Fe) =—(1—1-50 andfottherightsideof(15),withthemeaning ofuunchanged, thevalue v=—i-7.0-10". Ourtranscendental equation thus becomes - ulogu=v, v= —i-7.0-107 23 GENERAL SOLUTION OFTHEWIRE-WAVE PROBLEM 185 If,onceagain,weputu,=—v/20, weobtainby(5c), ‘ty=(—0.29 +3.54)-10°" &uy and hmSu=(036-440-10", @value ofthesame order ofmagnitude asthat found before in(16b). However thefurther calculation becomes quite different because ofthe smallness ofa=2-10~* em.Inplace of(17)wefindinom 1=i—0.0009 +0.011 « Itisnownolonger adequate toretain afirstterm inabinomial expansion, since here k*=(2x/d)* =0.0039. Instead weobtain h=0.085 +0.0651. Fromthisfollowsforthelengthofwirealongwhichthewaveamplitude has been reduced byafactor 1/e: 1 ones ~ em andforthephase velocity «/0.085. Division bythevelocity oflight w/k yields k v0 =0.74 astheratio ofthevelocity ofwave propagation andthevelocity oflight. Theformer lagsbehind thelatterby26percent. ‘Thereason forthisabnormal behavior evidently liesintheextreme thin- ness ofthewire, which increases thealternating current impedance and prevents thedevelopment ofanormal skin effect. The interior ofthewire isthen nolonger freeofcurrent; thecurrent distribution nolonger hasthe character ofcurve 22ofFig.29,butthat ofcurve 11.The field isthen no longer “immunized” against Joule heat loss. Damping andpropagation become anomalous. §23. General Solution oftheWire-Wave Problem Inthepreceding section wehave derived thatparticular solution which isrelated toHertz’s original problem ofwirewaves. Thequestion arises astowhether there isamore general solution. Thisquestion wasproposed~ toD.Hondros assubject forhisMunich thesis.’ Atthetime itseemed of ‘Ann. d.Phys. $0,p.905, 1909. 186 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 23.1 purelytheoretical interest, remote fromanypractical application. It hasbeen found since that itpossesses close analogies with thetheory of cavity conductors (§24), atpresent afavored fieldofcommunications en- gineering. Furthermore, itmaybeutilized advantageously forthetheory oftheLecher two-wire linewhich firstconverted Hertz’s single wire intoa functioning system. Also the“wire waves innonconductors” which fit intoHondros’s formulation oftheproblem have found practical application A.Primary Wave andElectrical Secondary Waves Wecallthesolution in§22theprincipal wave; forit|p|wassmall and hence |p:|large. Intheconverse casethat|p|islarge wespeak ofa secondary wave. Then, by(22.7), theleftsideof(22.14a) becomes equal to—ip, i.e.also large,inabsolute value.Thedenominator ontherightside of(22.142) must then become very small. p,must hence approximate one oftheinfinitely many andfrequently tabulated roots of : Tule)=0,p=wi,Wyre (a) generally w,,where weshall noteinparticular w;=3.83. (Ia) ByEq..(22.3) thesew,areidentical withtherootsofJi(p) =0.(Wewant toreserve thesymbol w,fortherootsofJo(p) =0.)Inviewofthemeaning ofpzgiven byEq.(22.14) wehave then approximately WN 2 went-(2). @a Forawell-conducting wireh,isthusapproximately equal tok,andwecan write,forallmoderate valuesofy(see(20.11)): h(t oe. (2a) From theformula forphase anddamping exp(—iut +thz),which istobe interpreted asbefore, wefind:Allsecondary wavesareexceedingly stronglydamped intheirprogress alongthewire;theiramplitude decreases bythe factor 1/eintheshort distance 1/x.Their phase velocity w/xissmall com-paredtothevelocity oflightc=w/k,theratioofthetwobeingequaltok/k. Thesecondary waves alsobehave oppositely totheprincipal wave with respect tothecharacter ofthefield.By(2a)wehaveoutside ofthewire : p=VP=r (-1+der and by(22.7) - Hp) ~oe" for a<<. By(22,8) thissignifies askineffectoutside ofthewire.Initsinterior, on 23.5 GENERAL SOLUTION OFTHE WIRE-WAVE PROBLEM 187 theotherhand,theargument ofJa(p)isrealsince p=Vibhr=w=(wyreal) andhence Je(p) isoftheorder ofmagnitude 1for0 SrSa. Theentire interior isfilled bycurrent. Considerable Joule heatisgenerated here, which explains therapid damping ofthewave initsprogress along thewireandmakes anyobservation ofthesecondary waves illusory. B.Magnetic Waves While wederived theelectrical principal andsecondary waves from thegeneral Eq.(20.5), weobtain themagnetic waves from (20.6). We heresetE,=0,H.equal butforaconstant totheBessel function Jo inside, andequal tothefirst Hankel function Hpoutside ofthewire. If wetakethecorrésponding transversal components from (20.6) weobtain, withu=r,0=9,9.=1g,=7: O<r<a p=Vki- hr a<r<o, p=VR hr 14/5He=ME=Fyj) en.=VE-Ppy) e \th £0 th 5He=DI) of/28.=BH) ® e & k,~8,=on -E,=FBG). From therequirement ofcontinuity ofH.andE,wenowobtain thebound- ary conditions exDidlo.) =4/SEpBHole), —f.DIs(ou) =KBE). (4) Here wehave putp=»/i? —ha,p,=+/k2 —Wa,asin(22.14). The teader may prove tohisown satisfaction that thesecond ofthese condi- tions assures atthesame time thecontinuity ofB,,taking account ofthe relation k*=eougo’, whichweshallalsouseinthefollowing. Elimination ofBandDin(4)leadstothetranscendental equation pHole)_,[emkpxJalox) (8) Hoe) coukySole) ’ whose right sidediffers materially, even inorder ofmagnitude, from that of(22.14a). Weaskwhether (5)hasasolution ofthetypeoftheprincipat wave h=k,i.e.pK1,Then theleftsidewould, by(22.5), beoftheorder ofp’logp,i.e.inabsolute value <1,while therightside,by(22.72) and since p,&ka,would beapproximately equal to{euo/(eu)'ka, i.e.inab- 188 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 23.6 solute value >>1since |e’|>>&.This applies even forsoftiron where #ismuch larger than yo.The assumption h=kthus leads toacontradic- tion. There isnomagnetic principal wave; themagnetic waves allhave the character ofsecondary waves: ,2 [o]>>1, Solon)0,ERE-(%), asin(1). The earlier comments ontheelectric secondary waves may be transferred without change tothe magnetic secondary waves. Their field also shows askin effect outside ofthewire and israpidly damped inthe interior byJoule heat. C.Asymmetric Waves oftheElectromagnetic Type Wenow consider processes without rotational symmetry about theaxis ofthewire andemploy fortherepresentation ofEZ,themore general solu- tions oftheBessel differential equation Jn(p)cos(np) or—Hy(p)cos(ng) (6) inplace ofthefunctions Jo(p) and Ho(p). Wereadily convince ourselves then that theformer three-component solutions E,,E,,H,and H.,H,, E, arenolongergufficient, butthatallsixcomponents ofEandHmustoccur inthesolution. We must now combine with theformula (6)forE,the formula rolsin(ng) (6a) Halo) forHz, soastogive allterms in(20.5) thecommon factor cos(ng), all terms in(20.6) thecommon factor sin(ng). We thus obtain from (20.5, 6) fortheinterior ofthewire, with p=»/k? —h'r ViiB,=VEX 01,6)cos(ng) ’ kun Ey=SCIalo)+ooDJ(9)cos(ng) n kuyy. —E,=;CIa(o)+zDJ'(p)?sin(ng)(7) of =VEEP Ds.)sinne e th 4H,=4E705) +Ds'lp)}sin(ng) geHe=47aCFale) alo) ing) obtte=(Ero) +®DIAG)008(ne). e A ° 23.11 GENERAL SOLUTION OFTHEWIRE-WAVE PROBLEM 189 Thephasefactorexp(—tw +‘the)isagaintobethought asincluded. The constant coefficients inZ,and H,have been sochosen that (7)passes overinto(22.13) forD=0andn=0and(after thepermissible inter- change ofcosandsin)into(3)forC=0andn=0. Proceeding likewise for the exterior ofthe wire we obtain, with p=VE =Hr VEF--RB,=YER Ans)008(ng) ’ kn BE,=5ANa(9)+ipBH,(0)}cos(ne) ~B,={8aio+f3H}inne : (8) =H,=YE="Bu,sin(ne) & th of,={4atta+Biro)sin(nw) & ph ie ko nWate {hantun)+"BriG)\cs(ve). Wenow turn totheboundary conditions between interior andexterior atr =a.Withp=/—Waand p,=ki—iaweobtain fromthe continuity ofEZ,and H, prCIn(or) =pAHa(o), (9) exDJa(ox) =9eBHA(e), 9=4/ (92) and from thecontinuity ofZ,and H,thetwo conditions CEdahon)+DHSelo)=ABHale)+BEHG), (10) PL h p A C8Son)+D™Jules)=ASHAGo)+BEHal). (108) % Theconstants A,B,C,andDaretobeeliminated fromthesefourequa- tions (9), (9a), (10), and (10a), most simply intheform ofafour-row determinant. Wedivide their columns immediately byH,and J,and find: ° 0 Pu 0 0 ~ 0 PL 2 bH, keds) =o au) r hoe poo hda RH, gm keJy omTRH. op ide 190 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 23.118 Thistranscendental equation istoberegarded astheequation determining thewave number h,which occurs notonly explicitly, butalso implicitly in p,px,H’/H, and J'/J. The electric andmagnetic components ofthe wave arecoupled byit.Anuncoupling occurs only inthesymmetrical case n=0,forwhich (11) may beseparated: Jo Ho Ho Jo{ot} pigkHepukapgsxf0.(11a) When setindividually equal tozero thetwoparentheses correspond ex- actly withthetranscendental equations forthesymmetric magnetic andthe symmetric electrical case, i.e.with theearlier Eqs. (5)and (22.14a). Weutilize (11)below only toanswer thequestion whether intheunsym- metrical case astate ofthecharacter oftheprincipal wave ispossible. Wethus assume ” Js .Ay h=k=p#0, |m|>1, 5ana |ae thelasttwo statements follow from Eqs. (22.6) and (22.7a). Wethen can neglect, inthefirsttworows of(11), notonlygp,butalsopincomparison withp:,.Thenthedeterminant (11)breaks upintotheproduct ofthetwo subdeterminants k pp0. ‘n\?1-5 qnt )=and = =-s- 12 0 psa4()wk ah(i The product ofthetwo yields a wm) Setequal tozerothisleads, forn =0,toh=+k, which contradicts our requirement h&k.There ishence noasymmetric principal wave. Wemust refer toHondros’s thesis fortherather complicated solution ofthetran- scendental equation forthesecondary waves. D.Wire Waves onaNonconductor ‘Thedissipation ofthesecondary waves onthemetallic wirebyJoule heat raisesthequestion astowhether secondary waves onadielectric wiremight beobservable. According toHondros andDebye thisquestion isto beanswered intheaffirmative.’ 1D. Hondros and P.Debye, Ann. d.Phys. $8,p.465, 1910. 23.15b GENERAL SOLUTION OFTHEWIRE-WAVE PROBLEM 191 Weshall consider a“water wire” (which may beimagined surrounded byaninfinitely thin-walled glass cylinder). Inview oftheabsence ofab- sorption Aisreal,sothat~/k*—h?iseither realorpurely imaginary. The first possibility (h<k,propagation with avelocity exceeding that oflight) isexcluded bytheprohibition ofradiation, inaccord with §22B. Hence Vi? —h?and ourformer p=+/k? —h?abecome purely imaginary. On theother hand, pz,=~/k? —h'aisreal. Forwehave now, since «=0 Ki=ena?=“H eouou* =n°k’,£0Mo where nisnowtodenote therefractive index, inaccord withMaxwell’s law inEq. (6.7) (not, asuptonow, the order oftheBessel functions!). For water wehave inthehigh-frequency range (decimeter waves) n=9. We introduce the two real quantities t=VR—-Ba, 1=Vit? —Ha, (13) which will serve asrectangular coordinates foragraphical representation. All our earlier formulas, inparticular those forsymmetrical waves, remain valid forourpresent case ofreal £,7insofar asthey donotcontain approximations. Eq. (22.14a) now takes the form -,Holt) _9Jo(n) ee ie 14) *are)~aJG) oa) For§>0and§—©itsleftsidevaries, according to(22.5) and(22.7) as ilog . = Blog2andas &=&,respectively.W/E or i Itthus becomes equal tozero and infinity along with ¢Hence theright side of(14) must also vanish for =0;this isnot the case when 7=0 (since Jo(0) =0),butonlywhen Jo(n) =0. (15) Ontheother hand, theright side of(14) becomes infinite for Jon) =—A(n) =0. (18a) We shall represent the variation given by(14) graphically inthe &- plane. Ontheordinate axis wemark theroots of(15) and (15a), which al- ternate with each other. Asonp.186wecall thesequence ofpoints Wi,Wr,Ws,ss? and=Wh,Wa,Wey and note inparticular w,=2.40. (15b) 192 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 23.16 Wedraw linesparallel totheaxisofabscissas through thepoints 7=wi, &=0.Thepoints »=w,,&=0areinitial points ofcurvebranches of thedesired representation, which must approach thehorizontal straight lines »=wiasymptotically for£==. There exists however, according to(13), also therelation 2 Etat=ot—nee=ot-»(7) (16) between ¢and 7.This means that thedesired solutions of(14) must also lieoncircles about theorigin oftheéy-plane with radius n= v= 18, (162) where )isthe“wave-length inair” corresponding toourstate ofvibration. Hence wemust letthe curve branches intersect with the circles (16). ln wif----55 Fie.34.Wirewavesonanonconductor. Planeof wy therealcoordinates ¢=\/i?—kta,n=Vn=Tta.Construction oftheroots ofEq. (14) with theaid tw Se oftheroots wm,ws,-+-ofEq.(15),Jon) =0,and “ey theroots1;,w;,«++ofEq.(15a),Jala)=0. Eva) | Depending onthemagnitude ofAthere arezero, one, two, oreven more intersections. Our figure shows directly: For »<w,there isnointersection. Forw<m<1;there isoneintersection. Forw;<7,<wsthere aretwointersections etc. According to(15b) thefirstrootofJo(n) =0isw:=2.40. Themaximum permissible value of\which corresponds tothisleast value of7,forwhich awire wave isjuststillpossible is,according to(16a): CLons Dmx 0Vit—Ta For our “water wire” with radius a=1cmthis is 2aAmax340 23.4cm. ‘Longer wave-lengths than thiscannot bepropagated along it.Thus wefind 24.1 ON THE THEORY OF WAVE GUIDES 193 ourselves intherange ofthe“decimeter waves” which isofsuch great interest atpresent.° As2isreduced thereisonepossibility ofpropagation, represented by thefirst intersection S,inthefigure; as\isreduced still further, corre- sponding to1>5.52 being increased, there aretwo possibilities, given by thetwo intersections S,,S;inthefigure. The first, atsmall £,yields h=k (velocity along thewire nearly equal toc),thesecond atlarger ¢(hmate- rially larger than k,velocity ofpropagation appreciably lessthan c)corre- sponds toan»which isnearly equal tothefirst root ofJi(n) =0,which according toEq.(23.1a) isw;=3.83. Inthefirstofthese twocases the electric lines offorce arenearly perpendicular tothesurface ofthewire and the decrease ofthe field outward isslow (the asymptotic decrease ¢ isattained onlyforlarger).Inthesecond case,where£isquitelarge,we have askin effect outwards, asforourauziliary waves forthemetallic wire. The first case hasthécharacter oftheprincipal wave ontheoutside, with thedifference that theinterior ofthedielectric isfilled bycurrent (7real andofmoderate magnitude). Similarly, forstillsmaller A,thelarge number ofvibration states then possible arearranged between thelimiting cases of principal andauxiliary waves. Atthesame time thephase velocity ofthe wave varies between thevelocity cinvacuum and that “inwater.” The above results predicted byHondros and Debye were verified most successfully byG.Southworth intheBell Laboratories. Also inGermany such dielectric wire waves have been profitably applied incommunications. §24. OntheTheory ofWave Guides Inthepreceding section wehave seen that electromagnetic fields may be held together andguided bythesurface ofanon-conducting rodand that they protect themselves against outward radiation byaskin effect. This pro- tection willbecomplete ifweembed thenon-conductor inametallic tube, whereupon thecondition ofasufficiently high dielectric constant may be omitted and thedielectric within thetube may also beair.Wethus arrive attheconfiguration ofthewave guides, which have become important in high frequency practice. Weconsider inparticular thecylindrical wave guide, since itstreatment may bededuced directly from thepreceding formulas. Letabetheradius ofthemetallic envelope, which forthepresent will beassumed tobea perfect conductor, and hthewave number ofthepropagation. Itisreal since thewave isdamped neither byJoule heat norbyradiation. There are electric andmagnetic waves ofsymmetric type andalsoofasymmetric type. Wewrite forthesymmetric electric waves, asin(22.13), omitting theampli- tude coefficient Cand thephase factor exp(—twt +thx): —#=VERB, B=so, 4/28 EK. 194 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 24.2 ForknownE,theseformulas alsofollowdirectly fromtherelationship oftransversal andlongitudinal components in(20.5). Theboundary conditions reduce tothesingle equation E,=0forr=a,since forHytherequired condition ofcontinuity issatisfied byasurface current induced inthe envelope. Thus, with p=+/i* —h?awehave Joo)=0,p=WW,t2,°°'w,-++ withwm=2.40. (2) From thedefinition ofpitfollows that a new-(2), h<k fogre (2a) Thephase velocity «w/h,along thetube thus exceeds thevelocity oflight. Asforourdielectric wire there isalower limit forthewave number k, i.e.anupper limitforthecorresponding “primary* wave-length” A=2x/k. Itcorresponds toh=f;=0(phase velocity infinite) andyields by(2a) w Qe oe On te=Gyme=ay? 240 &) Sincéaisoftheorderofmagnitude ofcentimeters allthefollowing con- siderations apply tothecentimeter-wave region. Thenumber ofpossible states (or“modes”) oftype (1)depends onthefrequency wor,what is thesame, ontheprimary wave number k=w/c.According to(2a)this number isequal tothenumber ofroots w,which arelessthan ka. Themagnetic symmetrical waves arerepresented, according to(23.3), _—ht ky o/Bte=PEP 19,4/Bi=HO,~Be=ZT, corresponding tothegeneral scheme ofEq.(20.6). Thesingle boundary con- dition which mustherebefulfilled is#,=0forp=V/ié—Wa.Itdemands Jol) =0, p=wh,wh,=wy,+++withwi=3.83 (5) andyields, asin(2a),values ofh,which are<k.Theupper limit forthe primary wave-length liessomewhat lower than fortheelectric type. Itis 2x Qa max=wt=383% (5a) Topass overtotheasymmetric types westart from Eqs. (23.7). Inview ofthereduced number ofboundary conditions wemay now however set 1Bythe“primary” wave-length weunderstand that oftheexciting oscillation, which ofcourse hasthesame frequency wasthewave guide oscillation excited byit. ‘Thisprimary wave-length isactually simplyameasure ofthefrequency wwhichisfamiliar totheengineer andconvenient indimension. Thewave-length inthewave- guidecanbedetermined uniquely onlyintheaxialdirection andis\=2x/h,whereastheprimary wave-length is2x/k=2rc/w,ForXprim Aeax,Nex=©sinceh=0. 24.7 ON THE THEORY OF WAVE GUIDES 195, oneofthetwo amplitudes CandDequal tozero, theother equal to1.This simplifies the formulas considerably and leads toanasymmetric electric (D=0)and anasymmetric magnetic case (C=0). For theasymmetric electric type weobtain: B,=VEX 10)cos(ng), He=0 E,=Jp)c0s(ng), 2H.=#2jG)sin(ng)(6) & ho Ey=~-"Ja(o)sin(ng), fet.=©1,06)cos(no) p £0 h and fortheasymmetric magnetic case, if,forconvenience, ngisexchanged fornp+x/2, = ey, 2VBE.=0, of.=LEE10)conne) Bp=FEsulp)sining),4//BHy=Jp)008(ne) fo) k Ho : +BemFao)oomng), 4/BHy=—2Jap)inne) For n=0(6)and (7)pass over into (1)and (4). The boundary conditions forr=a,p=~/* —fiarequire for(6): E,=E,=0, ie.J,(o) =0, for(7) E,=0, ie.Jn(o) =0. ‘Asin(2)and(5)wecalltheroots ofthese twoequations again w,andw, and distinguish them when necessary from theformer bytheaddition of theargument n,writing thusw,(n) inplace ofw,(0), w,(n) inplace ofw;(0). The following table indicates therelative position ofthesmallest roots inthedoubly-indexed twofold system w,w’: Jy=0 Jo=0 at =0 wy(1) =1.84 w,(0) =2.40 wi(0) =3.83=w,(1), wy(1) =5.33 w,(0) =5.52 w,(0) =7.02 =ws(1). Itshows, contrary toexpectation, that themagnetic asymmetric wave with n=1,andnottheelectric symmetric wave with n=0,possesses thesmallest root. AJso forthe second root v=2,inthe second row ofthe table, the sequence ofthese two waves isthesame asfortheroot y=1.The third column shows finally that theelectric symmetric wave issucceeded bythe 196 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 24.8 magnetic symmetric wavewithn=0,whichyields thesamerootastheelectric asymmetric waveforn=1sinceJo=—Ji. Inview ofthis thewave-length \msx given by(3)isnottheabsolute upper limit forallwavee that canbepropagated intheguide, butrather thewave-length Qn Dees=5g (8) Nr 341 an2.61 2.05 Tato 1.64 P22SS YY ee thy Rees] ee S er KespHa—~— 5e7/ y. 1.1.18FR14ONKa\(BOxexHONun PSSEZeS Fra. 35.Transversal fields ofcylindrieal-guide waves, ordered according tothe limiting wave-length \max/a. Full lines, electric; broken lines, magnetic lines offorce, both represented inthecross section oftheir antinodes. Open beginnings orends of thelines offorce indicate their being bent outoforinto thedirection oftheaxis. Customary engineering notation: TM magnetic, TEelectric type; first subscript, azimuthal number ofnodal lines; second subscript, radial number ofnodal lines within theguide. Thus, ifthefrequency isincreased continuously (continuous reduction of theprimary wave-length) andiftheexciting elements aresuitably disposed inspace, theelectric symmetric wave isnotthefirst toappear, butinstead themagnetic asymmetric wave with n=1.The electric symmetric wave follows and, after it,themagnetic symmetric wave simultaneously with thefirstasymmetric electric waven=1,asillustrated intheseriesofpic- tures inFig. 35. - Sofarwehave discussed only exactly circularly cylindrical tubes. Every deviation from circular symmetry occurring along thetube isequivalent 24.10 ONTHETHEORY OFWAVE GUIDES 197 toadisturbance inthesymmetry oftheexcitation and hence occasions theappearance ofnew secondary waves ofdifferent symmetry, which leads todifficulties inthepractical application ofthedesired modes ofoscillation. Alsoatubeofelliptical crosssection maybetreated directly bythegeneral method of§20A, since thewave equation isseparable intheelliptical coordinates u,v(see Vol. II,Problem IV3).Itisonly necessary toput E,, H,equal toaMathieu function (function oftheelliptical cylinder) F(éu) orequal totheproduct oftwosuch functions F(iu)-F(v); thetrans- versal components E,,H,;H.,Z,may then bewritten down immediately with theaidofEqs. (20.5) and (20.6). Weshall still glance briefly attubes with rectangular cross section. Since there canbenoquestion ofasymmetrical wave inview oftheshape ofthe rectangle (sidesbandcinthey-andz-directions) wegiverightawaythe general formulas (napdmarbitrary integers) corresponding toEqs. (6) and(7);theamplitudes ofE,in(9)andH,in(10)have, justasin(6)and (7),been chosen in8manner convenient forwhat follows. Asbefore, the phase factor expi(he —of)must beimagined asincluded. 2Ee=#(z+%)sin(ne)sin(me), H,=0, ‘EB~05%co(ne¥)sin(me) Hn,=—*E,,©) . b bye)” a” kh” =th™sin(neY z eH,=+* E,=th;sin(ne¥)oon(me), fet +;By nom’ y 2f/ane=o{e +}om(ne#)cos(me2),E,=0, Hy =—ihcin(neYz=ky/m Vet atsin(ne¥)coo(me?) E, y/en., (10) MH, =—in ¥)sin (ma? =—*,/™2m. th;oo(ne#)sin(me?), E,i4/ety. ‘The wave number hisdetermined inboth cases bythedifferential equation AX+KX =0,which must besatisfied forevery oneoftheCartesian components ofEandH.Substitution ofeither (9)or(10)readily leads to . wee(heS) w=(2), >=primarywave-length. Themaximum valueofdforgivennandm,belowwhich thetubeiscapable ofoscillations, occursforh=0andis2 ate 198 DERIVATION OFPHENOMENA FROM iuAXWELL EQUATIONS 25 If,aswemay assume, 6>cthe-absolute maximum isattained forn=1, m=0and is Amex =2b. AsLeon Brillouin hasnoted, these and similar oscillations inguides can beconstructed elegantly andinstructively bythesuperposition ofordinary plane space waves which interfere atthetube walls. The same idea leads also directly from theprogressive waves derived above tocharacteristic standing waves ine.g.arectangular parallelepiped oracircular cylinder offinite length. Essentially, thewave number hmust simply bereplaced byaninteger multiple ofx/a where aisthelength ofthe third side oftheparallelepiped orthelength ofthecylinder, respectively. Wewilldiscuss this ingreater detail inProblems II7and II8and treat, in Problem II9,theradially symmetric characteristic vibrations ofthesphere aswell. The characteristic vibrations oftherectangular parallelepiped, in particular, find auseful application inmicrowave practice forthedeter- mination ofthe frequency ofthe primary excitation bythe resonance principle. More difficult questions arise inthepractical application ofwave guides, ,Where, instead ofperfectly conducting walls, thefinite conductivity ofreal ‘metals andtheheatlossinthem must beconsidered; thelatter has,upto the present, prevented the propagation ofwaves inguides over great distances. Also theshaping oftheends oftheguides into conical orhorn- shaped openings raises questions upon which weshall notenter here.’ §25. TheLecher Two-Wire Line Mathematically this isthegeneralization forhigh-frequency alternating currents ofthequasistationary two-wire linetreated in§18. Itsadvantage, compared with thesingle wire traversed byalternating current, rests inthe fact that thefield outside ofthewires decreases more rapidly than forthe single wire, sothat thedisturbances bythesurroundings discussed onp.178 are avoided. The phase ofthealternating current inthetwo wires advances inthe same direction, sayinthepositive x-direction; thedirection ofthecurrent itself, ontheother hand, asin§15E, isopposite inthetwo wires. Wemight say: For thesame zpositive charge flows inonewire through agiven cross section, negative charge intheother. Also thecharge accumulated onthe surface hasatany moment, forequal z,theopposite sign inthetwo wires. !Werefer tothecomprehensive textbook of8.A.Schelkunoff, “Electromagnetic Waves®?VanNostrand,NewYork,1943,whichwaspublishedasaBellMonograph and iswidely employed intheUnited States, aswell astothelectures ofL.deBroglie, “Problémes depropagation guidée desondes électromagnetiaues.” Paris. Gauthier- Villars, 1941. 25 ‘THELECHER TWO-WIRE LINE 199 Wecallthis apush-pull excitation. However, themode inwhich charge of equal sign flows inboth wires (and isaccumulated attheir surface) may also berealized. Wethen speak ofparallel excitation. Theconditions ofexcitation determine which ofthetwo states occurs. Any asymmetry ofexcitation results intheappearance ofboth wave types. However, weonly callthe push-pull arrangement a‘“‘Lecher system.” Forparallel excitation thesitua- tion isquite similar tothat forthesingle wire (§22) andisfraught with the same experimental drawbacks. G.Mie’ succeeded ingiving acomplete theoretical treatment ofthe Lecher problem asearly as1900. Thefollowing representation,” which is bothsimplified androunded outtosomeextent, deviates fromthatofMie more inform than insubstance. Like Mie, weintroduce asystem ofbi- polar coordinates, towhich thecircumferences ofthetwo cross sections, assumed circular, belong. These coordinates would betheideal mathemati- calmedium ifthewave equation were separable inthem. Unfortunately this isnotthecase (see Vol. II,Problem IV.1). Wehence must employ methods ofapproximation which restonthereplacement ofthewave equa- tion bythepotential equation intheyz-plane. However, thisapproximation isvalid only forsufficiently good conductivity ofthematerial ofthewire ,andintheexterior ofthewires. Inside wemust caleulate with ordinary eylindrical polar coordinates. Thecomparison ofthetwoformulas atthe surface ofthewires leads toaclear-cut equation forthedetermination ofh, thewave number, which inthepush-pull case becomes even simpler than forthesingle wire, being algebraic inplace oftranscendental. Intheparallel case itispractically identical with that forthesingle wire. G.Gentile Jr.hasproposed aprocedure which differs from ours andfrom Mie’s.’* Inaccord withthegeneral methods ofperturbation theory hesuper- poses onthesymmetric wave propagated along thefirst wire thetotality ofasymmetric Hondros waves from §23, each multiplied byadisposable coefficient. Heseeks tofitthese coefficients totheboundary conditions onthefirst andsecond wires, inwhich process hehastoutilize thegen- eralized addition theorems ofthe Bessel and Hankel functions. This leads him toaninfinite system ofsimultaneous linear equations fortheco- efficients. However heand hiscollaborator T.Magri failed toobtain an approximate solution ofit.Ontheother hand, ourprocedure leads toa direct and explicit determination ofthe infinite number ofcoefficients which must be introduced. ‘Ann. d.Phys. 2,201, 1900, 1Itrests onadetailed study oftheproblem byMr.J.Jaumann; healso hasmade available tometheelegant treatment ofthelimiting case ¢+©given inthesus- ceeding section A,which weowe hislate father, the wellknown physicist G.Jau- mann of Brinn. +Nuovo Cimento, Vol. I,pp. 161and 190, 1943. 200 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 25.2 . A,TheLimiting CaseofInfinite Conductivity For¢>©thewaves propagate themselves with thevelocity oflight c, sothat h=k,ashasalready been pointed outattheendof§20A and follows directly forthesingle wire from Eq.(22.15). Then thethree-dimen- sional wave equation foreach Cartesian field component becomes the two-dimensional potential equation, inaccord with Eq. (20.3). This may besolved bythemethod ofconformal mapping forarbitrary cross-section peripheries (which need notbecircular, noreven thesame forthetwo wires). The method canalso beapplied when theexcitation isnotpurely periodic andmonochromatic, i.e.,when ourphase factor exp{i(hz —wf)} isreplaced byanarbitrary function f(z—ct). Itistrue that thelongitudinal components E,, H., which were con- sidered firstinEq.(20.3), vanish inthelimit h—k,since forinfinite con- ductivity theelectric lines offorce areperpendicular tothesurface ofthe twoconductors, andthemagnetic lines offorce also lieintheplanes z= const. Hence wehave forthe longitudinal components, inafirst ap- proximation E,=0, H.=0. (2) Ontheotherhand,theCartesian transversal components Z,,£,,Hy,Hsmay bedetermined almost directly from thefactthat assolutions ofthe two-dimensional potential equation they form anelectrostatic andacorre- sponding magnetostatic field. They aremost simply combined inthevector formula’ B+i4/2H=gad,wa=utiv (2a) w=w(t) isafunction ofthecomplex variable =y+tz,which may be constructed byconformal mapping; thetransversal E-andH-components areobtained asgradients oftherealandimaginary parts, uand»,ofthis complex function. The conformal mapping forourtwo identical circular cross sections is known tousfrom §19ofVol.II.Fig.26given there isreproduced inthe following figure with thenotation tobeemployed here. Both systems of lines offorce arecircles. The electrical lines offorce v=const. proceed from thefixed points Q,,Q:ofthefamily ofcircles; themagnetic lines u=const. have their centers ontherealaxisofthe¢-plane, onwhich Q,andQ:alsolie. Letthecenter ofthesystem Mbetheorigin ¢=0.Ourfunction wisgiven byEq.(19.10) ofVol.IIwhich inthepresent notation (u,»,f,foinplace ‘ofp,¢,2,¢)andwith aconvenient choice oftheconstant Atakes theform =loofT ~ w=logteh. (3) 1Asbefore, thefactor (ue/es)!/# must beapplied toHfordimensional reasons. 25.3b THELECHER TWO-WIRE LINE 201 +tarethe(real) values of¢corresponding toQ:and Q,,respectively. uandv,asrealandimaginary partsofw,havethesamemeaning asthe parameters pand¢ofthebipolar coordinate system defined inEq.(19.10b) ofVol. II.Ofthemagnetic lines offorce, those have been drawn heavy in Fig. 36which aresupposed tocorrespond tothecross sections ofthewires (radius a).Their centers 0,O;donotcoincide with thepoints Q:,Q:. Wecall thelatter, assources oftheelectric lines offorce, source points (three-dimensionally they aresource lines parallel totheaxes ofthewires). { \ ote;.wepoaeORerr etaumer Sta ARSE EE eS-=»—| ipa arer} | ; \SBS SL]OESREEFSOLED’SPREE: cae \ / Fre. 36.The families ofcircles ofthe bipolar coordinates u=const, »=const with thefixed points (source points) Q,,Q:.The peripheries ofthetwo wire cross sections u=ue areindicated byheavier lines; 0,,O:are their centers, atheir radius. The centerMofthefigureistheoriginofthecomplex variable {=z+iy. Inthesense ofEq. (9.8) Q:and Q;aretheir mutual electrical images with reference tothefwocircular cross sections, i.e.aretransformed into each other bythe“transformation ofreciprocal radii.” With thedesignations 00-00: =f, O%2=-O0%=F, OM=0M=b wehave hence .Pad, f+F=%, F-f= 2%. (Bay Solution ofasimple quadratic equation then leads to Febt+vVe—a, f=b-Vei=—a, f= ViF—a (Bb) 202 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 25.4 B.TheExterior oftheWires According toProblem IV.1 ofVol. IIthelineelement ofthebipolar co- ordinates may bewritten 1_coshu—. dst=gi(du’+do’),9-=SRT 4 adu+ae), oR (4) Ascompared with thegeneral orthogonal line element inEq. (20.1) we thus have here thespecial (isometric) case g.=go=g. Theelectric lines offorce, represented inFig.36bythefamily ofcircles »=const, have thedirection ofincreasing u,themagnetic lines offorce, represented bytheorthogonal family ofcircles u=const, have thedirection ofincreasing v.Fortheparticular lineelements ds,,ds,ofthetwosystems oflines offorce weobtain from (4) ds,_ds,m7 (4a) Wewrite Eq.(2a) separately fortheu-andv-directions: ig/MH,=2ae Eats/n. duds,9’ p/n, =085 B+ig/eH, avde,9" Separation ofrealand imaginary parts leads to 0Hr,=2_,coshu—cosy =*=0.6 B=4/#H, a7ape? qe=0 6) The lastisobvious since theelectric lines offorce have thew-direction, the magnetic, thev-direction. Furthermore, inspite ofthevanishing ofthe longitudinal components EZ,andH.noted inEq.(2),wewish toobtain a somewhat closer approximation forthem assolutions ofthetwo-dimen- sional potential equation. This is,interms ofuandv, Cn du=sat 5p=O Itisintegrated byparticular solutions oftheform sinh (nu) cosh (nu)\cosh(mhcos(nt),Sih(nfSi(me). (5a) Hereniisapositiveinteger,Forn=0thesefunctions arereplaced bythelinear function au, (5b) 25.8 THE LECHER TWO-WIRE LINE 203 Theaddition ofaterm bvisnotpermissible since E,andH.,must beunique functions ofspace, whereas thecoordinate vchanges by+27 after revolv- ingonce about oneofthewires, i.e.ismultivalent. Furthermore, the addition ofaconstant cto(5b) isexcluded since FE,and H,must vanish atinfinity (u=»=0).Inview ofthesymmetry ofourproblem wewrite E,as.an odd function ofuand asaneven function of»,H,,vice versa, as eneven function ofuand anodd function ofv: E,=Ey+E,sinhucosv+Hzsinh(2u)cos(2v)+--+ io (6)fai=Hi,coshusinv+Hzcosh(2u)sin(2v)+--- Injustification weconsider two symmetrically placed points u,»and —u, »totheright and totheleft inthe figure. Inthe push-pull case, which alone interests ‘ustobegin with, thecurrents flow inopposite direc- tions inthetwo wires; thesame applies tothez-components ofthedis- placement currents outside ofthewires. Hence £,is,inour two points, equal and opposite. Ontheother hand H,hasthesame sign inthetwo points inview oftheir position totheleftand totheright ofthetwo wires. ,Consider now twopoints u,vandu,—»inthefigure, above and below thestraight line»={2}.Inthem Z,hasthesame signandH,opposite signs. The formulation (6)ishence justified. We already know ofthecoefficients 2), £,,-*-, Hi,-++, from (2), that they vanish inthefirst order for h—k.Toobtain more detailed information webest turn back tothegeneral relations (20.5) and (20.6) which before served forthe calculation ofthe transversal components from thelongitudinal ones, and which weshall now employ todetermine thelongitudinal components tothefirst order from thetransversal com- ponents known tothezero order ofapproximation. Since g,=g»=g,sub- stitution of(5)and (6)in(20.5) and (20.6) leads to i(k?—WY)=—hEy —(hE, +kHj) coshucosv++ iQ—Rt)=hE—(4B,+hE)coshweosv»--0m20-5) 0= (hE,+kH;)sinhusinv---O= =(BE,+AA,)sinhwinv«70m (20.6) We conclude therefore: sat) oeByeAE) AE) won —wy, a)h k h kMhe-,h2-_;ns-h,. (3) The sign &signifies here “equal but forhigher terms inh—k”.Itis readily seen that thesame applies forH., H;,-+-- asforH,. We have 204 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 25.9 thusdetermined EyandH,,H:,Hs,--:H,.TheE,,Ey,--»E,remain indefinite from this point onand aredisposable forwhat follows. To conclude the consideration ofthe exterior ofthe wires wewrite down theexpression forH,ontheperiphery ofthefirst wire asweapproach thelatter from theoutside: Since this periphery isamagnetic line of force, wehave here u=const, say =+%. Weutilize theabbreviation p=e“ (9) and obtain from (5) imtm1(pph— - (et ra(e+} 2cos»)foru=+u. (9a) C.The Interior oftheWires Since, asinthepreveding sections, wehave within thewire ky=Vena? +tnow =Velo inplace ofkandsince|k,|>>h,wedonotattain ourgoalwithsolutions ofthepotential equation inbipolar coordinates, butmust employ actual solutions ofthewave equation inordinary cylindrical coordinates. Hence {weintroduce atthecenter e.g.ofthefirstwireanewcomplex variable “n=re”with'the origin O,andwemust deal with themutual transforma- tion ofourtwo systems, polar and bipolar coordinates, particularly atthe periphery ofthewires. This isfurnished byEq. (8)ifthere weexpress ¢ interms ofthenew variable .Referring toFig. 36andEqs. (8a,b)weset bHben PEpen btviRaa= (POP and obtain from (3) ee te fo ho LaF= = =i; 10)ONS Fh af 0) inversion leads to wtie_pe fev -Fgare = (10a) The coordinates r,yarethus expressed interms ofthecoordinates u,v and vice versa. Thus byforming theabsolute value of(10)andsquaring itwefindfor theperiphery ofthefirstwire, where weshould have u=-+uandr =a, eM atat—fac*—-f_a+f' —2afcoy Pe—Fae*—F a+P—2aFcosy 25.15 THE LECHER TWO-WIRE LINE 205 Since pisindependent of¢thisequation issatisfied onlyif,after multiply- ingthrough with thedenominator, thefactors ofcos¢onthetwo sides areequal, i.e. —2ap'F=—2of,p=Vi. or,ingreater detail, inview ofthereciprocity relation inEq. (3a), ficif PV4Fra (11) From thesame Eq.(10) wefind forr=asince p<1: iee*—F/a_pe* ~1_ _oe ieyttip OsPaap (p—e-*)(1 +pe”+pre?”+..-) =pt (pi—Ne? +(p—pe +--- (11a) and hence cos»=p+(p’—1)cos¢+p(p*—1)cos(2¢) 2) +pip =1)008(Bp)++++ Ontheother hand wefindfrom (10a) forr=a,making useof(11): \ is ef=Poiemo ef=(p—(1—pe, (13) The real part ofthis equation is,forn=1,2,3, cose=p—(1—p*)cos.»—(p—p’)cos(20) ~GF=pen B) Fo cos(2¢)=p’—2(p—p*)cos+(1—4p*+3p*)cos(2v)+«++ cos(3g)=p*—3(p'—p')cos»+(3p—9p*+6p’)cos(20)+++ Asthegeneral expression forthefield inside ofthefirst wire weusethe superposition ofthesystem ofpartial waves in(23.7), where however we need write down only theexpressions forE,andH,: VERSB,=VEX! ¥0,526)cos(ne), =fk (15) 5H=E{8Oodle)+Dusalo)}008(ne) e ano lh ° Bysupérposing here allpossible asymmetric Hondros waves weimplicitly andinthemost general fashion take account oftheunilateral effect ofthe “second wire” onthe interior ofthe “first”. 206 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 25.16 D.TheBoundary Condition H,=H, Previously theamplitudes C,Dcould bechosen arbitrarily forthe individual partial wave. Herewedonotconsider theindividual partial wave, butthesuperposition ofallofthem, andmust fixtheamplitudes C,,Dzofeach partial wave bytherequirement ofcontinuity inpassing over tothe exterior field. Tothisendwecompare theexpression (9a)forH,,after having replaced initcos»with theseries (12), withtherepresentation ofH,in(15). Tak- ingaccount alsoof(11)weobtain asfactor ofcos(ng) in(9a) e2p" V22p"2 _— = -—. 16)V2Pav =/2% 6) Wewrite forthecorresponding factor in(15): ; zo.ke hn DadaVSeehTio{+Rete Beh, cea HereJ,/J’, hastheorder ofmagnitude unity, aswasnoted at(22.7a). Of themultiplying factors thefirst two arevery small quantities; thesame applies forD,/C,,a8weshallconfirmlateron.Acomparison of(16)and +(16a) thus yiglds directly h2p” Cafo ty 17 ©pokiaSa(p) an The case n=Orequires special consideration because oftheconstant term appearing intheexpression (12)forcosv.Wehere obtain inplace of (16) and (16a) ®t(pat)281wna—4/eHest , V2( pt)=2andV27,CoFo(e),respectively and hence hod-of8ttay oo eokt,aSo(p) (ava) With these expressions forC,,and C,thecontinuity condition H,=Hy issatisfied. E.TheBoundary Condition forE,andtheLaw ofPhase Propagation Substitution of(17) and (17a) in(15) leads totheexpression forE, fou VE G2 sxJoho)0)1 emthaToo)+22Tip)P08(re)>CBF 25.20 THELECHER TWO-WIRE LINE 207 which, inview of|kz,>>|h|andJ,/J, &z,maybesimplified to -4/20) ep Betta decos(ng). (18a) Here weimagine theFourier series (14)substituted forcos(ny) andcom- pare then (18a) with therepresentation (6)fortheexterior ofthewires, where weput u=+t: Eyuo +E,sinh upcos»+Ezsinh (2ue) cos(20) ++++ (18b} Equating ofthefactors ofcosv,cos(2v), cos(3v)onthetwosides yields E,sinh tu =4/88 1tap +BPA=A+ 2s!few (19) Bysinh(Qua)=7B4/%(1—pt—(1—4p"+Bp)+--+), aew Eysinh(31)=2/3%{1-pte} a e!po . ‘Thus thecopfficients EZ,oftheexterior field, left indefinite uptothis point, aredetermined forn>0;their values may befurther simplified by utilizing . 1 2 A ~lg_ao»y...sinhwsapt—p’),sinh(2ue)=ap P), What, however, isthestatus ofthecoefficient Ey,which wasfixedalready byEq.(7)andhence isnotdisposable forsatisfying theboundary condi- tion? Fortunately itcontains theasyetundetermined quantity h.The remaining condition, containing E,thus serves forthefinal determination ofthepropagation constant h,which hereasforthesingle wireisofprimary jnterest. Theequation inquestion isobtained bycomparison oftheterms jn(18a, b)which areindependent ofvandis BoweVsti +220): Ifhere wesubstitute values from (7)and (9)forZoand uwand carry out thesummation ontheright, weobtain: =jfseries /1 Q heakt+s maine log5- (20) Weseethathisgivendirectly andbyanelementary expression, not,asfor thesinglewire,byatranscendental equation. Furthermore foraperfectly 208 DERIVATION OFPHENOMENA FROMMAXWELL EQUATIONS 25.208 conducting material, e0/e’ -»0,h=k,asshould bethecase. Forfinite conductivity e’isessentially positive imaginary, ~/e’hence ofthetype exp(ix/4) and(e0/e’)* ofthetype oto 1a-tv2° Hence thecorrection term in(20)isofthetype 1+4.Therealpartofh thusbecomes greater thank;thissignifies apropagation velocity lessthane. ‘Atthesame time therealpart ofihbecomes negative. Forourform exp{i(he —wt)}thismeans damping forpropagation along thepositive z-axis, The correction term thus indicates areasonable physical behavior inboth itsreal and itsimaginary parts. Finally, totestthedependence ofthecorrection termonthegeometrical data oftheLecher system, i.e.thewireradius aandthewireseparation 2b,wemay, for2b>>a,replace Fby2b.Wethenfindfromformula (11) _2 1+¢/1 2 p=5%4 ieJog5&I/log| (20a) ‘The correction term then hasonly alogarithmic dependence onthesep- ,aration ofthewiresandisinversely proportional tothewireradius.“+If,conversely, bisonlyslightly greater thana,i.e.b=a(1+a)with a 1,weobtain from (3b) and(11) 1 Fe=a(l+V2a), p=l— V2a, log=V2a, and hence lite es /gw ew1 tt|i,a(Fa1)20=Daa=2b—a)"(200) ‘Wehave carried outthisshort calculation toshow that ourfinal formula (20)alsocovers thecaseofslightly separated wires, where themutual influence ofthewires isverygreat andtheir skineffect must beenhanced unilaterally. Wethusmake itclearthatourtreatment isquitegeneral withrespect tothegeometric circumstances. Ontheother hand, thelimi- tation towires ofhighconductivity already introduced insection Aapplies throughout. F.Supplement Regarding theRemaining Boundary Conditions Todemonstrate thecompleteness ofoursolution anditsfreedom from contradiction weshall survey briefly theremaining boundary conditions. ‘These arethecontinuity conditions forH.ontheonehand, for7,=2, onthe other. ‘Therepresentation oftheinterior fieldtobeobtained from (23.7) by 25.21 THELECHER TWO-WIRE LINE 209 summation over ncontained, inH;,thecoefficients D,which tillnowwe have suppressed. The boundary condition relates them totheconstants H,oftheexternal fieldwhich occur in(6)andwhich, according to(8),are equalandopposite tothealready known constants H,.Theboundarycondition forH.thusserves thedetermination oftheconstants D,ofthein- ternal field. Without entering intothecalculation, wefind [Dil «|Cal, which wehavealready usedin(17).Finally, nodisposable coefficients are leftoverforthefulfilment oftheboundary condition E,(inside) =E,(outside). Ontheother handwe,know fromourapproximate calculation thatevery- where outside, andin-particular atthesurfaces ofthewires, E,=0.Itis thusnecessary toshow thataccording toourrepresentation (23.7) also thesumofalltheE,(inside) ismuch smaller than that oftheother field components. Thisisinfactso,byseveral orders ofmagnitude. Theproof must however beomitted here. G.Parallel andPush-Pull Operation Theapproach already described isentirely adapted topush-pull excita- tion. Intheparallel casethetransversal components must bederived not from thepotential (3),butfrom w=log(f—fo)+log(¢+f), (21) andthisonlyforsufficiently thinwires (a«b;otherwise theequipotential lines of(21)arenotapproximated bycircles!). Thebipolar coordinates then losetheir usefulness, since they nolonger coincide with thesystem ofequipotentials of(21). Inparticular theelectric linesofforce nolonger pass, asinFig.36,from Q:toQ:,butrepel each other andpassfrom Q; andQ;separately toinfinity. Wehence arenowobliged touseordinary cylindrical coordinates atthecenters ofthefirstandsecond wires, r,9 and#,,respectively, fortheexterior ofthewires aswell. The field isnow tobeconstructed bythesuperposition ofthecontributions ofthetwo wires with equal sign. Furthermore, itisnownecessary’ toformulate these representations directly assolutions ofthewave equation, which formerly, because ofthelimited range ofthefield, could beavoided. Wethus return alsofortheexterior toHondros’ formulation (23.8)where, restricting our- selves tothin wires, wecanlimit ourattention tothezero-order terms. In 1Fortheradiation condition forr>»,which nowbecomes essential, seethe discussion forthe single wire in§22. 210 DERIVATION OFPHENOMENA FROM MAXWELL EQUATIONS 25.22 therepresentation of#.wethen obtain forthesuperposition ofthetwo wires the sum: VEER a eeMO HolVB=Br)+HolVie=FF)P(22) Ontheperiphery ofthefirstwire r=a,#&2bweobtain by(22.5), be- cause ofthesmallness of+//? —At(weofcourse continue toassume high conductivity), 2—2 Je R BRYEaRw (Ce{ve=t a+log{1ve=#ail).(22a) This much about theexterior ofthewires. Inside itisnecessary tousethe earlier perfectly general formulation (15)forE,.Thecoefficient Cyappear- inghereisdetermined with theaidofthecontinuity condition forHy. Herewith thelongitudinal fieldZ.inside ofthewire, more particularly its zero-order partial wave, isalsodetermined. This must agree with the term (22)oftheexternal fieldforr=a.Wethusobtain anequation of the form ap)(og(wea a+log{ve=e 2})=const, whose right sideisknown. Itmust befulfilled byproper choice ofAandis transcendental incharacter, aswith thesingle wire. (Itevidently canbe made tocorrespond tothesingle-wire equation (22.15) bycombining the twologarithms.) Wenowseethereason whythecorresponding equation forhbecomes elementary instead oftranscendental intheLecher case: ‘Thetwologarithms areheresuperposed with the—signinplace ofthe +sign; incombining them thefactor 7~/k? —7#/(2i) under thelog-sign cancels andonlylog{a/(2b)} remains, asinEq.(20a). Wealsoseethat thecalculation here outlined, without bipolar coordinates, would have beensuccessful alsointheLecher case,butthatitwould havebeenmuch more involved than theearlier method, particularly without restriction to extremely thin wires. Notonlyinmathematical formulation butalsoinphysical structure the parallel wave resembles thesingle-wire wave. Itdecreases much more slowly outwards than thepush-pull wave andishence much more dis- turbed bythesurroundings. Itisobvious thatatlarge distance thetwo similarly directed currents oftheparallel wave must produce thesame fieldasthealternating current ofthesingle wire. *Experimentally apureexcitation ofthepush-pull wave isalways desir- abletoavoid disturbances from thesurroundings. However ifthearrange- ment isnotquite symmetrical parallel waves arealsooccasionally excited, 25 THELECHER TWO-WIRE LINE 211 which maketheposition ofthenodalpoints, onwhich thewave-lengthdetermination rests, unsharp. Hence evenfrom apurely experimental standpoint itisimportant to keep inmind thepossibility oftheparallel processes andtotake account ofthetheory ofthesingle wire, which wehave treated before therestin §22,although ascompared withthetheory oftheLecher system itisof secondary practical importance. Parr III THEORY OF RELATIVITY AND ELECTRON THEORY §26. TheInvariance oftheMaxwell Equations inthe Four-Dimensional World ThepathtakenbyEinstein in1905inthediscovery ofthespecial theory ofrelativity wassteep anddifficult. Itledthrough theanalysis of the concepts oftime and space and some ingenious imaginary experi- ments. The path which weshall take iswide and effortless. Itproceeds from theuniversal validity oftheMaxwell equations and thetremendous accumulation ofexperimental material onwhich they arebased. Itends almost inadvertently atthe Lorentz transformation and allitsrelativistic consequences. A. The Four-Potential _We refer tptheelectrodynamic potentials in§19, which atthat point &till remained ‘inthe fogofanunsatisfactory formalism. Iwish tocreate the impression inmy readers that the true mathematical structure of these entities will appear only now, asinamountain landscape when the foglifts. Inthetwo differential equations (19.9) and (19.11), satisfied byAand Y,wehad onthelefttheoperator 1é4Bae w Wenow introduce inplace ofz,y,z,¢thenew coordinates A=% mY M2 B=tet, (2) where theproper remarks regarding theimaginary unit in2willbemade later. We call these z;world coordinates since allevents inthe world are determined inspace-time. The operator (1)then becomes thefour-dimen- sional generalization oftheLaplace operator andmaybedesignated by* a =z 3 5p>axt @) 'Ceftain more advanced theories ofEinstein and Kaluza employ also thefive- al - dimensional symbol ()==."et azt 212 26.7 INVARIANCE OFMAXWELL EQUATIONS 213 Like theindependent coordinates z;wecombine thepotentials A,¥in afour-dimensional entity, thefour-potentialQ. Letitsfour components be =A, =A, =A, Gale ) Thefactor ¢inisreasonable inviewofthedefinition of2,;thefactor 1/c gives thefour components ofQthesame dimension and will bejustified below (7).The differential equations (19.9) and (19.11) then take theform Oo =—wr. (6) The quantity Frhere introduced may becalled thefour-current density. It follows from (4),(19.9), and (19.11) that itsfour components are Tr=Js,° Tr= Jy, Ta= Js, Te=tcp; (6) they allhave thedimension Q/M’S ofacurrent density. Wenow turn totheEq. (19.10) relating thepotentials Aand ¥.The second term onitsleftis,inview of(4), Ay=18 _om é teat ox Eq.(19.10) thusbecomes 8%,WM,A,Ay_oe,tontantam=0. Wewrite this more briefly ‘. . aDiva =0,Div=Sin: (7) Weshall calltheoperator Div thefour-dimensional divergence. Itsfour- dimensional symmetry indicates teofrepy inspace and time ofworld events. Our operator ()shows thesame isotropy. Thus weseethereason that wemay regard ourfour-potentialQandour four-current density Iasvectors infour-dimensional space, more briefly, asfour-vectors. This notation contains astatement regarding thebehavior ofthequantities@and©whenthecoordinates x;arechanged. Aswas shown inVol. II,§2,anordinary ‘‘three-vector”’ isaquantity which, for orthogonal transformation ofthez,y,z,behaves just astheradius vector r=(z,y,2).Thus thefour-vector attains ameaning inthefour-dimen- sional world which isindependent ofthechoice ofthecoordinate system. Atthesame point inVol. IIwedefined ascalar asaquantity which is invariant with respect toorthogonal transformations; inparticular, we showed byasimple calculation that thedivergence ofathree-vector Podsesses this property. This calculation may betransferred directly to 214 THEORY OFRELATIVITY AND ELECTRON THEORY 26.8 four dimensions andshows that thedivergence defined in(7),applied to any four-vector, yields ascalar, ie.afour-dimensional invariant. Byreducing theMaxivell equations tothefour-vector Qandtheinvariant- operators 1,Div wehave demonstrated atthesame time their general validity, independent ofthe coordinate system. The isotropy ofthree- dimensional space found adequate expression inthe vector calculus of parts Iand II.Itisnow replaced, inview oftheworld isotropy, bythe four-dimensional vector calculus. This states that, for atransition toa “primed” coordinate system x;,theMaxwell equations remain invariant, i.e.have thesame form intheprimed field components andcoordinates as intheoriginal “unprimed” ones. This invariance issimply theprinciple of relativity initselectrodynamic formulation. The Maxwell equations satisfy therelativity postulate from thevery beginning. They need notbesub- sequently, adapted toit,liketheequations ofmechanics (see§32). B.TheSiz-Vectors ofField andExcitation Wenowturn totliérepresentation ofthefield component E.ByEq. (19.7) wehave e.g. oy 0A, E,Or According toEqs: (2)and (4)thisisequivalent to =ig{9%_aM E, = i¢{—— —- —}."@2) ®) This relation suggests theintroduction ofthefour-dimensional curl Curl,@=We—22, @On Om Asatwo-indices quantity ithassizcomponents (according toVol. II, Eq. (2.17) itwas preferable togive thethree-dimensional curl aswell two indices instead ofone index). Evidently Curls, =0, Curlan =—Curlan. (9a) Curl iscalled asiz-veclor or,preferably, inview ofthesymmetry proper- ties indicated by(9a), anantisymmetric six-tenso#. The term “surface tensor” isalso applied toit.The sixcomponents which differ from zero and from each other may bédivided into three space-time and three space- space forms, corresponding tothearrangement oftheindices 14,24,34 and 23,31,12. 9b) According to(8)thefirst three belong totheelectric vector, thelastthree tothemagnetic vector. However, wemustnotcouple theentityofquan- 26.13 INVARIANCE OFMAXWELL EQUATIONS 215 tityHwiththeentity ofintensity Etoformafour-dimensional unit,but must employ forthis theentity ofintensity Borrather thequantity cB, which hasthesame dimension asE.Wetherefore write forexample, 8%_a cB,=ccurlyA=c(22), (10) Combining (8)and (10) and extending them cyclically totheremaining field components weobtain the following representation ofthesiz-com- ponent field vector, where (_)istoindicate merely thecombination ofthe two three-dimensional vectors into onefour-dimensional quantity: F=(cB, —iE) =cCura. (11) This field vector Fistobegiven twoindices inaccord with thesequence (9b), like thesix-vector Curl Q. Wenext askabout theexcitation vector, which weshall denote byf.We form itfrom thetwo equally dimensioned quantities Hand cD.Topass from Hand DtoBand Eweemploy theconstants forvacuum and &, since our part IIIwill belimited throughout tospace free from matter (e.g. avacuum tube). Since : , n=2.4/aa andDxcoe-4/@E Ho Ho Ho weobtain from (11) simply j=(,-id)=y/26Cua. (12) Hence f=V2PF (13)Mo Tooursatisfaction thegeometric mean ofthetwo three-dimensional con- stants ofvacuum €and yu’=1/yo appears here. Already attheintroduc- tion ofthepermeability in§4,p.21weemphasized that thetrue magnetic analog ofthedielectric constant €isnot»,butitsreciprocal y’,Inany case, Eq. (13) combines the three-dimensional relations between excita- tions and fields symmetrically, with asingle constant ofvacuum. Atthe same time our formulation (11) and (12) translates the earlier, highly heterogeneous representations (19.7) and(19.6) ofEandHintoancntirely symmetric andharmonious form.Tocreate aclear visual image ofthestructure oftheantisymmetric tensor wewrite down thearray ofallthecomponents offinmatrix form; thearrangement ofthecomponents ofFisobtained herefrom bymultipli- cation with ~/y/e, and simultaneous exchange ofHwith cBand ofcD 216 THEORY OF RELATIVITY AND ELECTRON THEORY 26.14 withE.Wedistinguish thecomponents ofHandDinthecustomary three-dimensional fashion bytheindices x,y,z;from ourpresent stand- point itwould bepreferable todesignate them bythedouble indices (9b), asinthematrix attheleft. Wepoint outspecifically theorder 12,13,14 inthefirst row, 21,23,24inthesecond row, etc.aswell asthechange in sign fortheconverse order: 0fa fuSu 0 4H, —H, —icD,' feJu0fufu) |-H. 0H,—%D, (14)fafa 0fu H, -H. 0 —iD, fafa fa 0 tcD, teD, icD, 0 C.TheMaxwell Equations inFour-Dimensional Form We also want towrite the original Maxwell equations with double indices. Proceeding from theequation ioH,,HyDewy+z Je weobtain from thefirst row ofthearray (14), making useofthedefinition ofFin(6), _fe_ay_au _pOz, Ot, Ome , and corresponding equations forthe second and third ofthis triplet of Maxwell equations. Wehence have ingeneral form form=1,2,3 +AanGee Ty.Uae (15) Ifweextend this form tom=4,weobtain by(14) and (6) io(2D:4.2Deud)in= io(2B4Pr4Be=tep=Ty. Our original definition ofthecharge density pin(4.4b) thus proves tobethe four-dimensional completion ofthesecond Mazwell triplet (4.4). Theoperation xzcarriedoutin(15)bearsthename“reduction” or “divergence” ingeneral tensor analysis (see Vol. II,p.60); itreduces a four-dimensional tensor toafour-vector. Wesymbolize itbyDiv, placing itparallel totheoperation Div defined in(7),which reduces afour-vector toascalar. Wethus write inplace of(15) SSfmn Divaf=D5=Te (16) 26.17f INVARIANCE OFMAXWELL EQUATIONS az Generally forany antisymmetric tensor T DivDivTam =0, (16a) which follows directly from Tan =—Tan .Weconclude therefore that the divergence ofthefour-current vanishes: Divr=0; (16b) thisissimply thecontinuity equation (4.4c) inarefined, four-dimensional form. What ofthefirst triplet oftheMaxwell equations (4.4)? Itsz-com- ponent is p,+2B_ae B+we 0. Inview ofthemeaning ofourfieldtensor F,which isanalogous to(14): 0 Fs Fu Fy 0 cB, —cB, —iE, pa[% 0FaFu)_[-cB, 0eB,~i8) 4)Fu Fn 0 Fu cB, —cB, 0 -iE, Fa Fa Fo0 iE, iE, iE, 0 it.may berewritten intheform ,(Fa,oFufe)_ (FesFeeBe=0. 7a) Thissomewhat confusing distribution ofsubscripts becomes quite plain if weintroduce the“dual” six-vector ofF, F*=(—1E, cB), (17b) which isobtained fromFbyanexchange oftherealandtheimaginaryconstituents. Thedetermination oftheindividual components ofF*is fixed bytherule Fran =Far (17e) with theprescription thatthesequence ofsubscripts klmn arisesfrom 1234 (17d) byanevennumber ofexchanges. Bythisrequirement wehaveuniquely Fu= Fu, Fu=F*, Fo=F*s, (17e) sothat Eq. (17a) becomes _ OF",,Ft,oF;tntn ta 7% (17) 218 THEORY OFRELATIVITY AND ELECTRON THEORY 26.18 Ithasthus become thefirst component of Div F*=0. (18) The other two components oftheMaxwell triplet inquestion take on similar forms. But what isthemeaning ofthefourth component of(18)? Itis OF,|OF%.|OFa,+am+, 7° and, by(17b) and (17) may betransformed into _,(Be4aBya) c(z=+wy+or0. Itisthusidentical withthefamiliar absence ofsources ofthemagnetic field intensity B.This appears now, from thefour-dimensional relativistic stand- point, asaformally necessary completion ofourfirst Maxwell triplet, while originally, inEq.(4.4a), ithadtobepostulated separately asanempirical fact. Wehave thus acomplete representation ofMaxwell’s theory forvacuum inthe statements Divf=r, DivF*=0, F=ee (19) which parallel and areequivalent tothepotential relations Oa=-pr, Diva=0, F=cCulg, sf=Cura. (20) Allquantities and operations appearing inboth formulations have proper citizenship inthefour-dimensional world and hence satisfy theprinciple ofrelativity. Itshould beemphasized onthis occasion that thetheory ofrelativity leaves nodoubt that thevectors Eand Bontheone hand, and Dand H ontheother, belong together, asparts ofthehigher entities Fandf.This seemed clear tousfrom the beginning, partly fordimensional reasons, partly because oftheir different significance asentities ofintensity and quantity. Inparticular, thetheory ofrelativity leaves nodoubt that this distinction isasnecessary invacuum asinany ponderable medium, i.e. that here also both six-vectors Fandf(the four three-vectors E,B,D,H) have tobeemployed side byside. " *D.OntheGeometric Character oftheSix-Vector anditsInvariants Thefour-vector isrepresented, asamatterofcourse,byastraightlinesegment infour-dimensional space (byanR,with sense ofdirection). It might appear appropriate torepresent thesiz-vecior byatwo-dimensional 26.21 INVARIANCE OFMAXWELL EQUATIONS 219 segment ofaplane, i.e.byitsmagnitude andposition infour-dimensional space (anRy,oneofwhose sides isdesignated aspositive). However, this representation istoospecialized. Such asegment ofarea hasonly 5inde- pendent parameters, not6,asasix-vector, ie.oneparameter indicating itssize(shape istobeindifferent) andfour’ indicating itsorientation (parallel displacements donotcount). Toobtain ageometrical interpreta- tion ofthegeneral six-vector wenote that every R,infour-dimensional space hasuniquely correlated with itasecond Rzperpendicular’ toit.Ifa segment ofareaisalsoprescribed inthesecond, asingle further parameter isobtained (since theorientation inspace isalready determined bythe orientation ofthefirstsegment ofarea), leading tothedesired total num- berofsixindependent parameters. Thegeometric picture ofthesix-vector isthusnotonesegment ofarea, buttwomutually perpendicular segments of areaofarbitrary size."The components ofthesix-vector areequal tothe sums oftheprojections ofthetwosegments ofarea onthesixcoordinate planes (z_,Zn).Ifthesizes ofthetwosegments areinterchanged, the original six-vector Fpasses over intothedual six-vector F*;thisconfirms therelation (17c) between thecomponents ofFand F*. Thefour-vector hasonlyoneinvariant, thesquare ofitslength, equal to ‘thesum ofthesquares ofitsfour components (thefourth ofthem taken, 6fcourge, withnegative sign,inviewofitsimaginary character). Onthe other hand, every six-vector Fhastwoinvariants F-F and F-F* both given, inaccord with theruleforthescalar product, bysummation over thesixcomponents with equal indices. Wecarry thisoutforthe electrodynamic case. By(17) and (17a) POF =Fy)+Fal+Fut+Fad+Fad+Fu=CB—EP en P-F* =FoFy +Fok +FuFu ++++ =—2icB-E. Here ---signifies repetition ofthethree preceding products with reversal ofthesequence ofthefactors, i.e.simply thedoubling ofthesum. Forthe vacuum light wave both invariants (21) arezero. Infact, by§6,B1E andc|B| =|E|. Inview ofourstatement regarding invariance the 1Ifitsorientation isthought ofasdefined bytwofour-vectors proceeding from thesame point andlying inRs,these aregiven by3quantities,e.g.theratiosoftheir fourcomponents. Both may however berotated arbitrarily within R:,sothat the number 2-3 isreduced to2-3 —2=4. *OneRimay bedesignated asthe“‘axis”’ oftheother, since onecanberotated about theother arbitrarily within itself. The relation ofthetwo isofcourse mutual or“‘dual’’. TheaxisofanR;inR,isthus atwo-dimensional, notasinR;aone-di- mensional manifold. 220 THEORY OFRELATIVITY ANDELECTRON THEORY 26.218 light wave retains thisproperty inallreference systems ofthefour-dimen- sional world. : ‘The excitation vector hasofcourse thecorresponding invariants ff=H-¢D’, (21a) f-f* =—2icH-D. Thefollqwing mixed invariants differ from (21)and(21a) onlybyafactor: f-F=ful’+fala+faa+ful+fuFu+flu (22)=cH-B —cD-E, f-F*=fil+falta+falta+fuPhic+ful+fuPa (23) =—-iH-E —icD-B =—2E-H. We define 1 1 1 AwgfF=3H-B-5DE (24) astheLagrange density (Lagrange function ofthemoving electron perunit volume ofthefield). Ontheotherhandtheenergy density W,given by }H-B +4D-E, willprove tobeacomponent ofaworld tensor; byitself jthasnomeaning independent oftheframe ofreference. Ourinvariants maybeexpressed asfollows interms oftheareas aand bwhich are correlated inthe six-vector: PP=@+, F-Pt=2b, A=B+). (25) Itfollows fromthisthataparticular six-vector (b=0)isdistinguished from thegeneral onebythecondition F-F* =C. E.Relativistically Invariant Three-Vectors Wenowaskwhat properties athree-vector must have inorder thatit mayexistlegitimately inthefour-dimensional world. Forthispurpose we consider asiz-vector which isdualtoitself. Wemaywrite itintheform P=Prin+F¥mn=Fan+Fir (26) This six-vector hasinfactonly three independent components. Wecan define them by a,=Py=Py=Fu+Fu, a,=Py=Py=Fu+Fa, (26a)- a,=Py=Py=Fut+Fo, andobtain asspecific tensor arrangement forthethree-vector @by(17): 26.288 INVARIANCE OFMAXWELL EQUATIONS 221 0. a a, a -a 0a,a, Pe . (26b) a, -a. 0 @ —a,—a,—a,0} Inthespecial caseoftheelectrodynamic tensor Fweobtain ascorrespond- ingthree-vector a,=—i(E,+ 1cB.), a,=—i(E,+ iB), a,=—i(E, +icB,). Thus thecomplex three-vector E+ iB=ia (27) may beregarded asafour-dimensional tensor oftheform (26), which is dual toitself. Ontheother hand, wemay alsowrite, inplace of(26), P=Pan—Fan=Fan—Fut This six-vector isoppositely dual toitself and leads tothethree-vector E- B= —ib. (27a) Even before thetheory ofrelativity itwasoften noted that thecomplex combinations E+icBandtheanalogous H+icDhave certain character- istic advantages fortheintegration oftheMaxwell equations. Wecannow also give tleHertzian vector IIitsproper place inthe four-dimensional world. Itappeared in§19asathree-vector, but itisin fact adisguised siz-vector, with thestructure ofanelectrostatic field vector Fat.ForitBast=0,Estatequals athree-vector which, temporarily, we willdenote byP., P,, P,.By(17) wethen have 0 0 0 -iP, F, 000—tPy (28)“lo 00-4P,/” uP, iP, +P, 0 Reduction ofthistensor leads toafour-vector which forthepresent will bedenoted byQ,likeourfour-potential: Q=Div Fux. (28a) Ithas thecomponents~ .OPs, -ae thay=~é2Eeet 1B, ay=divP, 222 THEORY OFRELATIVITY ANDELECTRON THEORY 26.28b 1f,byEq. (4),wepass from this%.1.24 totheelectrodynamic potential A and W,wefind A=-1B, weedivP. (28) Tf,finally, weset P=—yocII, ourEqs. (28b) pass over exactly into Eqs. (19-15) and (19.16a), bywhich wehad otiginally defined the Hertzian vector. Thus, like P,thethree-vector Mfhasbeen reduced tothesix-vector (28). This six-vector ishere referred toacoordinate system inwhich the Hertzian dipole rests. The transition from this “system atrest” toanarbi- trary reference system canbecarried outbytherules ofthenext section. §27. The Group oftheLorentz Transformations and theKinematics ofthe .Theory ofRelativity Inhis“Erlangen Program” Felix Klein’ hasclassified theseveral geo- metric disciplines onthebasis ofthegroup oftransformations permitted inthem. Projective geometry regards allfigures asthe same, which pass over into each other bycentral projections inthree-dimensional space. Affine geometry keeps theinfinitely distant plane fixed, andhence permits onlyparallel projections. Forelementary geometry alsotheshapes offigures, their angles and ratios oflinear dimensions, areofimportance. Itsgroup isthat ofthe orthogonal transformations inthree-dimensional space, ex- tended bythesimilarity transformations. Here theimaginary sphere circle contained intheinfinitely distant plane iskept fixed inaddition tothis plane itself. The geometry ofthegeneral point transformations cantrans- form any surface into any other, butsubjects any small region ofspace only tolinear (projective) changes. The group ofthecontact transforma- tionsdissolves eventhecontent ofsurfaces inspace andleaves onlythe combined position ofsurface point andtangential plane untouched. The fact that wehave spoken here notofindividual transformations, butonly oftransformation groups evidently derives from thenecessity of regarding transformations resulting from asequence orcombination of transformations asequally valid. Ifattention isfocused ontheunaltered properties, rather than thechanges ofgeometric structures, wespeak of thetheory ofinvariants belonging tothetransformation group inquestion. The transformation group ofclassical mechanics isthat oftheGalilei transformations (see Vol. I,p.10).Itmay bedivided into thegroup of theorthogonal transformations ofspace and thedisplacements ofthe time scale, corresponding toNewton’s idea ofabsolute space andabsolute times Itsinvariants arethesquare oftheseparation inspace andthetime difference. ThegroupofMazwell’s electrodynamics is,aswesawinthélast 1Comparative Study ofMore Recent Researches inGeometry, Erlangen, 1872. 27.5 LORENTZ TRANSFORMATIONS AND RELATIVITY THEORY 223 section, that oftheorthogonal iransformations inspace-time. Inhonor of thegreat Dutch physicist Hendrick Antoon Lorentz, Poincaré hascalled them theLorentz transformations. Just asthenature oftheseveral geome- tries ischaracterized bytheir particular group, theessence ofMaxwell’s theory rests initsinvariance within theLorentz group. Itsfundamental invariant istheseparation oftwoworld points, inparticular thefour-dimen- sional lineelement, i.e.theseparation oftwo neighboring points inspace- time. A.The General and theSpecial Lorentz Transformation Wehave recorded thegeneral pattern oftheLorentz transformation already in(2.10) ofVol. I.Itiscontained intheformulas t=Daam =Laer, i.142,34, (1) ‘ OkziDajons=Doaoe=ba= _Gt (2) ft ct lk=i ,The orthogonal transformations ofthree-dimensional space, oritsrotations svithin itself} form asubgroup. Correspondingly thegeneral Lorentz trans- formation may bedesignated asarotation inspace-time. Wehave also already derived, inVol. I,Eq.(2.14), thespecial Lorentz transformation in which twocoordinates remain unchanged. Ifwechoose forthelatter the y-and z-coordinates, thetransformation matrix reduces to a a as % a a 0 0 au cA 0 1 0 0 (2a) EA 0 0 1 0 PA Ce 0 0 cy According toconditions (2)wemust have ahtak=ohtak=ah+ah= aktak=1, @) sothat ah=ah, oe=ah. (4) weput ay=ay=a (5) and find from (2) 224 THEORY OFRELATIVITY AND ELECTRON THEORY 27.6 anna +ana =a(ay +an)=0. (6) Wethus may write, introducing anew constant 8: ou=—on =tap. 7) Theadded factor iisnecessary bceause oftheimaginary character ofx and2,ifrealvaluesaretobeobtained forz;andz,intheequations of our system manta and =am+ant. Substitution of(5)and(7)in(3)finally leads to 1 ASS aaB)=1, «Vick (8) ‘Thus oursystem (2a)-becomes 1 m=Vian&+i800),mom, t=ty(9) 1 Hopalm on+) ‘or,inreal terme, 1 Y=pape —bel, yun ¢341, (10) t=Vi-#al:-52), Tf,in(10), wecarry outthetransition tothelimit c+, 6-0, but fe=v =finite, weobtain : ver-t yoy vy Ht (10a) Wefollow Ph.Frank incalling these equations theGalilei transformation. From thestandpoint ofthistransformation group time andspace have become “absolute”. Ittakes theplace oftheLorentz group only when v<e, 1e. B<1. (10b) ‘Theuniversally accepted notation 8=v/cmayrecall the6-rays, which have avelocity comparable withc,sothattheGalilei group isnotappli- cable to,them. Eqs. (10)signify thatthetwosystems (z,?)and(z’,2’)move withrespect toeachotherwiththevelocity v=fc.Ifweconsider aparticular point 2’=const, wefindforitfrom (10) a—Pet=x—vt=const. 27.1la |LORENTZ TRANSFORMATIONS ANDRELATIVITY THEORY 225 The primed system thus progresses along thepositive z-axis, which coin- cides with thez’-axis, with thevelocity ».The two other axes y’and 2’ displace themselves, inspace, with thesame velocity v,remaining parallel totheaxes yand z.For¢=0the“movingsystem”andthe“systemat rest” coincide. Problem III.1 will treat thesomewhat more general case that therela- tivemotion ofthetwosystems willnotbealongthez-axis, butforexam- pleinsome other direction lying inthezy-plane. B.The Relative Nature ofTime From Egg. (10) and (10a) weseethat thecourse oftime isabsolute only inthelimitec+©,whereas forfinite¢itdepends ontheframe ofreference oftheobserver: The “primed” observer measures adifferent time than the “unprimed” observer.” This becomes obvious ifwereturn from the real representation (10) to thecomplex representation (9),butnevertheless plot theentities occurring af . .Fro. 37.The, system ,%istransformed into the ‘system z;,z,bytheimaginary angleofrotation y.The Rl @twoevents R,Qwhich aresimultaneous in¢then re- NT er ceivedifferent coordinates z,,justasthetwoevents “*\b--" iSea’ P,Qwith thesame z-coordinatehave different coor- ct dinates 2; } es,ee there asrealquantities (Fig. 37).Wemay then write, asinplane analytic geometry, n= nesytausny, 2= —msiny+meosy (11) with cosVi-#’ sinyViz tany=if.(lla) The first Eq. (11), asiswell known, signifies theprojection ofthebroken lineOPQ onthez}-axis, thesecond thatonthez-axis, However, therela- tiveangle ofrotation ofthesystems ishere imaginary andsimilarly also sinyandtan7;cosyisactually ahyperbolic cosine andhence >1,since B<h _ Fig. 37alsoshows directly that two“events” (points inspace-time) Q +Itisofcourse equally permissible toregard theprimed system as“system at rest,” relative towhich theunprimed system moves with thevelocity vinthedirec- tion ofthenegative z-axis. This will occur occasionally in§§28 and 33. 226 ‘THEORY OFRELATIVITY ANDELECTRON THEORY 27.12 and R,which aresimultaneous intheunprimed system, arenolonger simultaneous intheprimed system. This removal ofthesameness intime (simultaneity) now surprises usnomore than theremoval ofthesamenesa ofthez-value oftheevents QandP. Our pseudo-real representation inFig. 37willalso prove useful inthe future andcanscarcely lead tomisunderstandings. Itistrue that we deviate herein from thegreat example ofHermann Minkowski. Inhis classic lecture “Space andTime” before theKélner Naturforscher-Gesell- schaft in1908 hetalculates throughout with real quantities. What we wouldcalltheunitcirclez?+2?=1isforhimthehyperbola z*—c’'t’=1; twostraight lines which, tous,areperpendicular toeach other, then be- come conjugate diameters ofthishyperbola. Itneed scarcely beemphasized that, inspite ofthis(only superficial) difference, wehave stood onMin- kowski’s shoulders even inthepreceding paragraph and will continue to follow hisconception ofthetheory ofrelativity. C.The Lorentz Contraction ThiswasproposedbyH.A.Lorentzevenbeforethetheoryofrelativity asanadhochypothesis toexplain thenegative resultoftheMichelson*experiment. Deferring discussion ofthisexperiment toVol. IV,westate Lorentz’s hypothesis inthefollowing manner: Toanobserver atrestarod of“intrinsic length” lyappears, ifmoving with uniform velocity vinthe direction ofitslength, shortened tol=ly»/i—#. Lettherodrestinthemoving system 7’,¢’and letitsendpoints inthis system havethecoordinates z,,zs;theirdifferenceistheintrinsiclength l=24—2.Weareherenotconcerned withthetimes&,t4ofthemeas-urements. The situation isdifferent forthe observer atrest. Hemust ar- range themeasurements ofthetwoends oftherodsothat, from hisstand- point, they occur simultaneously, i.e.atthesame instant 4,=4.Hethus findsthepoints x,and2,andregards theirdifference, x—z.,a8the length oftherodJ.From thefirstequation (10)itfollows that , 1 , 1 =app(Me—Betas =pap —Be), andhence,sincet=&,%—2=1,25—2=he, L ——. b=Fae lthvin#. (12) Thehypothetical Lorents contraction isthus adirect consequence ofthe Lorents transformation. Itisnotsuperfluous tointerpret thisresultgraphically. Thelocation of therodinthemoving system isrepresented inFig.38bythestripwhich 27 LORENTZ TRANSFORMATIONS AND RELATIVITY THEORY 227 isshaded parallel tothez-axis. Theobserver atrestmakes acutofthis strip parallel tothez-axis. From thefigure itslength is i=, cos y Inour pseudo-real representation itappears longer than the intrinsic length by,although infactitisshorter since cosy=(1—8)! Asaresult ofthiscontraction amoving sphere ofradius aisflattened intoanoblate spheroid with thesmall axis b=a+/1 —#andwith the large axis a.Lorentz based onthis hishypothesis ofthedeformable electron, firststated in1903 asfinal result ofhisgreat paper onelectron theory.’ The frequently raised question whether the Lorentz contraction is“real” or“apparent” isofcourse justasidleasthequestion whether abody “actually” moves. The distinction between amoving system and asystem Ezy a! Fra.38.Arodatrestintheprimed ZZsystem, moving withrespecttotheun- aaprimed system,’ isrepresented bythe 2Bshadedstrip.Its‘length1intheun- 2B) 2!primed system appears longerthanite @Aintrinsic lengthJ,inthefigure,butisin CANfactshorter, inview ofthefactthat y ER Dyisimaginary: Lorentzcontraction. A EAA atrest, which wepermitted inthepreceding forthesake ofsimplicity of expression, isequally meaningless andarbitrary. D.TheEinstein Dilatation ofTime Letaclock’ restintheprimed system andmark, byitspendulum swings, the times and successive time differences COS Oeeeeeee They project themselves intheunprimed system into Goh, b-h=h-ha= sat. InFig. 39+seems shortened ascompared with 1’,butisactually ex- panded inview of - 1Enzyklopaedie derMath. Wiss., Vol. V:.Pages 277-279 ofthis article arefore- runners ofthe theory ofrelativity. 2Instead ofspeaking, with Einstein, ofaclock wemay adhere toelectromagnetic patterns bythinking ofatuned circuit and itsnatural period. 228 ‘THEORY OFRELATIVITY ANDELECTRON THEORY 27.13 : ; racer=ae. (13) This isrealized most simply analytically ifEq.(10) 1sinverted, inwhich »simply changes sign, asmay beverified bycalculation. Thus 1 1 &) =o / =es (7+BE). Pepa te, t=reg (+H). an Furthermore, since x’=const, thesecond ofthese equations yields for thesuccessive differences h—1,4—,-*+ 34-4, + 7 Vie Suchaclockisréalized inarapidly moving atomwhichemitsamono- chromatic spectral line, e.g.ahydrogen canal ray.Einstein regarded the expected redshift ofthespectral line asthecrucial experiment ofthe % + Fra. 89.The period ofoscillation ofthe \ moving clock +’appears shortened inthe an‘cen Oo figureforanobserver atrestinthem1,z- ator 1’ system tor,ishowever actually lengthenedalain inviewoftheimaginary character of7: Einstein's time dilatation. ay theory ofrelativity andspoke ofa“transversal Doppler effect,” consider- ingobservation at90°relative tothecanal rays. Ives hasshown, however, that theobservation may aswell becarried outatanarbitrary angle, preferably asmall angle, where theprimary light ofthecanal rays canhe compared with light reflected byamirror from the(oppositely directed) canal rays. Wethen observe, e.g.fortheH,line ofthehydrogen canal rays, inaddition totheprimary light, which isshifted toward theblue, thereflected light, which isshifted toward thered. The arithmetic mean ofthetwowave-lengths does not,however, coincide with thespectral line H,oftheatom atrest, butisdisplaced from itbytherelativistic redshift independently ofthedirection ofviewing. Theexperiment’ fully confirms Einstein’s expectation. _ 1H. E.Ives and G.R.Stillwell, J.Optical Soc. Am. £8,215, 1938; H.E.Ives, J. Optical Soc. Am. £9,188and 294, 1939. G.Otting, Munich thesis, Phys. Z.40,681, 1939. There is#difference inthetheoretical interpretation oftheAmerican papers and thesimultaneous German thesia which isnotable inview ofthetimes (19891): 27.15 LORENTZ TRANSFORMATIONS AND RELATIVITY THEORY 229 Aradioactive sample also hasanintrinsic time 7’intheform ofitsmean life. Hence, observed inthecanal ray, itshould have alonger lifethan at rest. This experiment isrealized under themost favorable circumstance: (8nearly equal to1)inthemeson disintegration ofcosmic rays. Rasetti found forthelifeofmesons which had become trapped inanabsorber andhence were practically atrestthevalue 7’&1.5-10~* sec,determining thetime difference between theincidence ofthemeson andtheappearance ofthesecondary electron produced inthedisintegration. Ontheother hand, absorption measurements onthemesons ofcosmic rays lead toa most probable range oftheorder of20km. Inthe(unprimed) time measure oftheterrestrial observer thiscorresponds toamean lifer=20km/c = 7-10-* sec.The time expansion thus hashere theenormous value’ tg TOF 50.a 15-104 ~ Wehence find forthevelocity ofthemesons, by(13), 1 1) Vi-# =50,v=c(1000)" This consideration isconfirmed bytheexperimental determination ofthe energy ofthemesons; forthemost commonly occurring mesons approxi- mately 50times therest energy isfound, which fully agrees with the dependence ofthekinetic energy onthevelocity (see §32). E.TheAddition Theorem fortheVelocity Two velocities v,and v;having thesame direction donotcombine rela- tivistically according totherule vent. We have instead p=2te 102" (13) 1+% Here v;isthevelocity with which apoint 2moves relative toabody 1, which itself moves inthesame direction with thevelocity v;.When Ein- stein in1905 proposed this formula, itnaturally aroused surprise. Itbe- comes entirely reasonable, however, when wenote that wearehere dealing with the composition oftwo Lorentz transformations, and that each of them, according toFig.37,denotes arotation.’ Let72betheangle ofrota- theAmerican papers seek toretain theconcept oftheabsolute ether, while the German paper assumes therelativistic standpoint from thevery beginning. 1W.Heisenberg, Vortrige tber kosmische Strahlen, Springer, 1943, pp.78ff. ?The tworotations 7:and +:a8well astheir resultant 7take place about theaame “axis”, i.e.theRsperpendicular tothez,2,-plane (seep.219, footnote 2). -280 THEORY OFRELATIVITY ANDELECTRON THEORY 27.158 tion which, byEq.(11a), pertains to»,andy;that pertaining to»,.The result ofthe composition ofthetwo rotations isarotation through the angle yentn (15a) Thus theangles ofrotation areadded, nottheir tangents. Forthelatter we have instead =any+tan tany 1—tanytany:" By(11a) this leads to Bi+Best 15b) . PTT Ra (a5) which agrees with (15). The addition theorem forthevelocities ishence inessence merely theaddition formula forthetangent function. The same formula may beobtained quite readily, though more in- directly, bysuperposing thetwo Lorentz transformations. They may be written, e.g., intheform ofEqs. (14): Vir Beat Ach, VIBin=%+Bich, VinHientin, Vinsnanthn, |) Elimination ofx,,t,then yields forthedirect transition from z,¢to22,ts: Vi- ivi ~ Bit BsMITA VIA Pty TP op,T+hm 7~*titan (16.) Vi-BVI-#, ygthm 1+Bibs U+BiBc° ‘This isagain aLorentz transformation oftheform (14) ifweput Bi+be pope Vie Biv 8 =Th — f=YAY 15d) o-Tram VWF 1+Bibs (5a) The first ofthese formulas agrees with (15a); thesecond follows from it, asmay beverified byasimple calculation. For small velocities (v;«¢and tz c)(15) ofcourse passes over into theelementary superposition formula. F.cas Upper Limit forAll Velocities *Ifbyrepeated superposition ofvelocities theresultant approaches the velocity oflight, thefurther addition ofany arbitrary velocity iswithout efféct. Infact wehave, by(15b), for6;&1: 1+ b= SLCST yee 27.16 LORENTZ TRANSFORMATIONS ANDRELATIVITY THEORY 231 Thevelocity oflight ccanonly beapproached, never exceeded. Even acyclo- tron orbetatron, which operates with continuous increases invelocity, cannot yield velocities greater than that oflight. Weshall define ourstatement more precisely. Tobegin with, weobviously mean by“velocity” “relative velocity”. But that does notsuffice. Consider asample ofradium. Itemits electrons with almost thevelocity oflight. Two electrons which flyoffsimultaneously inopposite directions have very nearly therelative velocity 2c,viewed from thelaboratory inwhich the sample ofradium isatrest. However, inorder toproperly define relative velocity asused inour statement wemust view one electron from the other. Then andthen only theseemingly paradoxical equation c+¢=¢ .applies. Wearethus concerned, inourstatement, with therelative velocity ofamoving point withrespect toareference system which istransformed toa state ofrest. ‘The moving point need notbeamaterial point; itmay also beaprocess resulting inmaterial changes. Such aprocess iscalled asignal and wethen speak ofthesignal velocity. Ifthis should ever exceed c,thewhole time sequence would bedisturbed (seebelow). Inwireless telegraphy andradar &bundle ofelectromagnetic waves serves assignal; amonochromatic wave, ontheother hahd, constitutes nosignal, since apurely periodic wave has neither beginning norend. Itsvelocity ofpropagation ishence notgoverned byourstatement. Infact, wefound forwave guides, in§24, phase velocities w/hwhich were greater than c.Similarly, wewillseeinVol. IVthat phase velocities greater than cmay occur intheanomalous dispersion oflight waves. There arealso quite trivial processes with velocities exceeding that oflight, which, then, obviously cannot serve assignals. Anexample isthe intersection oftheedge ofaruler with astraight line with avery acute angle. Ifwedisplace theruler atright angles toitself even with only moder- atevelocity, theintersection willmove along thestraight linewith avelocity exceeding that oflight, provided only that theangle hasbeen chosen small enough. Aformal indication oftheprohibition ofv>cisevidently givenalreadybytheLorentz transformation intheform(10),sincehere~/1—f?and consequently also x’and ¢’would become imaginary. @.Light Cone; Space-Like Vectors and Time-Like Vectors; Intrinsic Time The four-dimensional form : “ “Beat =0,inrealtermsr*—c''=0, (16) ischaracteristic forthemetric oftheLorentz transformations. Itrepre- sents, inthree dimensions, asphere expanding with thevelocity oflight, infour dimensions, aconic R;with rotational symmetry about thet-axis. 232 THEORY OFRELATIVITY AND ELECTRON THEORY 27.17 Wecallit,with Minkowski, thelight cone. The interior iscalled theforecone and theaftercone, depending onwhether ¢<0ort>0. Allfour-vectors leaving theorigin which lieoutside ofthelight cone arecalled space-like, those which lieinside ofit,time-like. Thus thevector r intheequatorial plane ofthelight cone isspace-like, whereas allpermitted velocities leaving theorigin aretime-like. The sequence ofallfour-dimensional positions assumed byamoving material point iscalled itsworld line. The world lineofapoint atrestis parallel tothe¢-axis. Allworld lines passing through theorigin lieinthe aftercone for¢>0,intheforecone fort<0. Weconsider theelement ofaworld line z ds=VYDazi. =f Asthedistance between two neighboring world points, itisLorentz-in- variant (seep.213). The same applies toMinkowskt’s intrinsic time dr=BKfe—Jade?+ay!+de’) tc/ ec : (17) =atg/tYadtVink.e Wewillnow define thefour-vector ofthevelocity along aworld line. The form dz dy dz | -Pea ©7Me would notbeapermissible definition, since dthasnoinvariant meaning. This does notapply, however, to dzdydz.dt_d). \ali el (18) The square ofitslength is 2 a 2 tat vy tata ed__ (18a)dr? ie.infactaninvariant which, furthermore, hasthesame value forall yelocityvectors V.Thefour-vector oftheacceleration should bedefined corre-_ spondingly by _W_ dz dy dz .dtWom @a“a (18h) 27.198 |LORENTZ TRANSFORMATIONS ANDRELATIVITY THEORY 233 Itis,inthefour-dimensional sense, perpendicular tothefour-vector ofthe velocity; differentiation of(18a) with respect to7leads to: vV-W=0. (18¢) H.TheAddition Theorem forVelocities ofDifferent Directions Velocities ofthe same direction were tobecombined insuch fashion that their angles were added. Since intheir elementary meaning angles denote ares ontheunit circle, their addition isequivalent tothejoining ofarcs onacirclewhichinourcase,itistrue,hastheradius7instead of1.The formulas ofplane trigonometry could beapplied tothejoining ofthese ares. Inorder tocombine velocities with different directions itisnecessary topassfromthecircle.to thesphere, i.e.fromtheformulas ofplane tothose ofspherical trigonometry, and forasphere ofradius i,notofradius 1. Combination ofthe velocities »,and v;toform the resultant »istherefore equivalent tothecombination oftheangles 7;and2totheresultant angle 7,i.e.theconstruction ofaspherical triangle with thesides 71,y:,and y. Tfaistheangle ofinclination of»;relative to»,,then aappears asthe ‘external anglebetween thesides7,andy;inthespherical triangle. Wethen have bythecosine law(see Problem 1.4): C08Y=C081CO8‘Y2—8iN71SiN72COsa. (19) This isthedesired generalized addition theorem. For a=0weobtain cosy =cos(y: +¥2);¥ =71+72, ie.theearlier Eq. (15a). Inview oftherela- tioncosy=(1—6")ete.(19)isequivalent withtherather untransparent formula _Bi+Bi+281A:cosa—BiBsin’a id(+BiB:cosa}? ; (10a) which was given already byEinstein. InProblem III.2 itwill beproved analytically byapplication ofthe Lorentz transformation. The introduction ofour sphere ofradius 7may seem anarbitrary trick; actually, itismerely anexpression ofthefact that thearcs 7,y2which wemust combine areimaginary, according to(11a). Weshall mention one more interesting result, which may beread off directly onFig. 40:Inthetheory ofrelativity thesequence ofdifferently oriented velocities isnotexchangeable; theresult ofthecombination of vand wdiffers from that ofthecombination ofv,and v,.Though the magnitude oftheresultant isthesame, thedirection differs. The difference inthe two directions increases asthe velocities increase; infact, aswe shall know, itisequal tothespherical excess ofthespherical triangle formed inour construction. InFig. 40theangle «between v,and hasbeen chosen equal to4/2 234 THEORY OFRELATIVITY ANDELECTRON THEORY 27.20 tosimplify thedrawing andtheare7,corresponding tov,hasbeen placed ontheequator ofthesphere. Theextension ofthearcy:then passes through thenorthpole N.If,ontheother hand, starting from thesame point A, wefirstrecord yp(denoted by72onthefigure), perpendiculartotheequator andwith itsextension alsopassing through N,wemust draw through the endpoint B’of7;agreat circle perpendicular tothemeridian AB’and measure offonityi=11=AB.Thepoint A’located inthismanner does notcoincide withC;instead, theconnecting arcsACandC’A’enclose a certain angle e.Inview oftheequality ofthetwotriangles ABC andA’B’C’ wehave here 4.BAC =x.B’A’C’ and XACB =x.A’C’B’. Ifwecall N. Fra.40.Combination oftwodifferently wo‘. directed velocities andvztoform there-BL SC sultant »,corresponding tothecircular arcsan ‘1,72,andonasphereofradiusi.For ,convenience inrepresentation theanglebe- aid[2 4!tweenmandvzhasbeensetequalto+/2. |Thefigureshowsthenon-commutative Veh 7)_---] character ofthecomponente: 901=ABC A=eC’. YE B xv2,0=C’B’A; theangle ebetween AC and C’A’ isequal tothe spherical excess ofthe triangle ABC (and that ofthe tri- angle A’B'C’ which iscongruent toit). these two angles 7and #,weseethat theright angle atAisformed by n,8,and ¢inthefollowing manner: 7 priten-e Hence e=ntd—x/2=qnt+d9+n/2—7. (20) ethus isinfact thespherical excess ofourright spherical triangle ABC andthecongruent triangle A’B’C’. (The same applies forageneral spherical triangle.) The limiting case y,=y2=x/2, where thetwo triangles ABC and A'B'C’ become equal tothesame spherical octant, isparticularly simple. Here theresultants areevidently perpendicular toeach other and, inview of7=8=2/2, thespherical excess isalso 1/2. J.ThePrinciples oftheConstancy oftheVelocity ofLight andofCharge Einstein in1905 expressly added thefirst ofthese principles tothe principle ofrelativity asanempirical postulate. Itstates that thevelocity 27 LORENTZ TRANSFORMATIONS ANDRELATIVITY THEORY 235 ofpropagation oflight isindependent ofthestate ofrest ormotion ofthe emitting body. This principle isalready included intheoriginal formulation ofourworld geometry insofar aswehave demanded theuniversal validity oftheMaxwell equations. Like thevelocity, thespherical propagation of thelight isinvariant inthetransition from x+++xto7-++24.The Loreritz transformation does not change thelight sphere into alight el- lipsoid, butleaves italight sphere. (This does notapply tothewave-length ofthelight, which isnotinvariant butisknown todepend ontheframe of reference oftheobserver: Doppler effect.) Intheearlier but long since discarded theory oftheuniversal ether, the independence ofthelight wave from thestate ofmotion oftheemitting body was readily understood: once transferred totheether, itpropagates itselfinaccord with,the(elastic orelectromagnetic) properties ofthis medium. Constancy ofthevelocity oflight was here equivalent with field action. The same does notapply foramechanical emission theory such as that surmised byNewton. Here atransfer ofthevelocity oftheemitting body totheemitted light particles seems almost unavoidable.’ Wemaysay: The constancy ofthevelocity oflight istoday theonly valid remnant of theether concept. Ifatpresent weshould speak ofanether, wewould have toassign aseparate ether toevery frame ofreference, i.e.speak e.g.ofa primed and anunprimed ether. We now regard Lenard’s “absolute ether (Urather)” merely asafreak and theAristotelian and scholastic “quintes- sence” (the fifth element, added tofire, water, air,andearth) asanhistorical curiosity. Thus inparts Iand II,wehave almost never spoken ofthe “ether”, butused instead thenotreadily misinterpreted word “vacuum”. The principle ofconstant charge isasimportant asthat oftheconstancy ofthevelocity oflight. The charge isthesame forevery frame ofreference. This isnotobvious, butfollows from theMaxwell equations ifwecanclaim their universal validity forallframes ofreference. Ontheother hand the principle oftheconstancy ofmass with change ofthesystem ofreference, formerly regarded asobvious, cannot beupheld, asweshall seepresently. The charge isanabsolute invariant with respect toLorentz transformations; mass and, asweshall also see, energy, arenot. Summarizing thecontent ofthisandthepreceding section wemay say: From thestandpoint oftheMaxwell equations thetheory ofrelativity is obvious. Amathematician whose eyes had been trained byKlein’s Erlangen program could have read from theform oftheMaxwell equations itstrans- formation group along with allitskinematic and optical consequences. 1Théfactthat Newton’s emission theory could inasense, experience aresurrse- tion inthepresent theory ofthelight quanta rests solely ontheaddition theorem of thetheory ofrelativity according towhich effectively c+o=c(c=velocity oflight quanta, »=velocity oftheemitting body). 236 THEORY OFRELATIVITY ANDELECTRON THEORY 28 §28. Preparation fortheElectron Theory Maxwell had directed attention away from thecharges and toward the lines offorce. Since thediscovery oftheelectron and Helmholtz’s earlier remarks' ontheatomism ofelectricity, interest hasonce more returned tothesources ofthelines offorce, theelectrons andions. H.A.Lorentz’ hascreated thesecure mathematical basis forthis new electrodynamics (which might becalled electron dynamics). The judgment exhibited byhim here isremarkable; heintroduced only concepts which retained their substance inthelater theory ofrelativity. We will abbreviate our treat- ment byinverting the historical development and basing the electron theory onthetheory ofrelativity. Unlike Maxwell, Lorentz does notrecognize ahost ofmedia differing electrically and magnetically; allevents take place inasingle uniform medium, thevacuum. The different properties ofmatter arise simply from thevaried binding and state ofmotion oftheelectrons and ions. Indi- electrics theelectrons arebound toions, inconductors they aremore or lessfreely mobile, and inmagnetic materials wearedealing with electrons which, asthe result oftheir spin, arealigned inthe magnetic field. Inthisexplanation oftheelectromagnetic properties ofmatter wehave thesimultaneous action ofgreat numbers ofelectrons, i.e.astatistics of electrons. We shall treat this subject ingreater detail inVol. V.Inthe present volume wemust limit ourselves tothetheory oftheindividual electron. Itistrue that thebasic question regarding thenature oftheelectron will remain unclarified. The electron isastranger inelectrodynamics, as Einstein hassaid onoccasion. Wecannot comprehend, from theelectro- dynamic standpoint, how thefinite electron charge e,concentrated ina point orinavery small volume, cancohere stably inspite oftheCoulomb forces between parts ofthecharge. Forasolution ofthisproblem wemust look toageneral theory oftheelementary particles, the electron, proton, neutron, neutrino, positron, meson (and other elementary particles which areyettobediscovered). Itisclear, however, that such atheory isatthe moment still remote. 1InhisFaraday Lecture in1881: ‘Ifweaccept atoms forthechemical elements wecannot avoid concluding that aleo both positive and negative electricity issub- divided into certain elementary quanta which behave like atoms ofelectricity.” 2Inhisbook “Versuch einer Theorie derelektrischen und optischen Erschein- ungen,in bewegten Kérpern,” Leyden, 1805; unaltered reprinting, Teubner, 1906.Seealsothelater‘‘Theory ofElectrons,” Teubner, 1909.EmilWiechert reached the same conclusions andformulas independently ofLorentz atalmost thesame time in “The Theory ofElectrodynamics andRéntgen’s Discovery,” Abh. derPhysikaliach- okonomischen Gesellschaft zuKénigsberg. 28.6b PREPARATION FOR THE ELECTRON THEORY 237 A.The Transformation oftheElectric Field. Introduction toThe Lorentz Force Inorder tocalculate inthemost elementary fashion, i.e.only with four- vectors, wereturn tothefour-potential Q,which transforms itself like the coordinate vector. Weemploy Eqs. (27.11) which, applied toQ,take the form %=cosyQ+siny&®, W=%, BW=%, %=—siny% +cosy. (69) The same equations (27.11), solved forz,yield m=cosy —sinyt%, w=%21, m=24, ., , (2) ‘am =siny2+cosy%. Weconcern ourselves firstwith E,andcorresponding to(26.9), form q i Ms —9% _aM CurlyQ=aahai” (3) From (1)and (2)weobtain am_3% inyO aXaa7oa siny55,+008¥5 (3a) M5_AMx,OO,Ory inyOM aiaanaa+a,Srl siny55+cosy5(3b) and asthedifference oftheright sides of(3a, b) 20 8) gin(9%_aM ony(2m4sn(2) (4) =cosyCurly +sinyCurly Q. This isatthesame time theright side of(3).Wehence have Curl,Q!=cosyCurly@+sinyCurlsQ. (6) ByEg. (26.11) weconclude therefore —iky, =cosy (—42,) +siny(cB,). (6) Inview of(27.1la) wemay write instead »_Ev—BeB,Ey=vVi-# (6a)7 Acorresponding calculation yields 1BetfcBy naw FoR (6b) 238 THEORY OFRELATIVITY ANDELECTRON THEORY 28.7 Thecalculation forthez-compénent issomewhat more complicated insofar asitleads firstto8terms, ofwhich 4aremultiplied with sinycos7,and twoeach with sin*yandcos*y,respectively. Thefirstcancel each other, whereas theremaining ones may bereduced to 30_amy Ox Oxy andyield simply EL=E,. (7) Soastoremove the distinctive treatment ofthe z-axis weindicate bythesubscripts ||and|thedirection parallel andperpendicular tothe relative motion ofthetwosystems. TheEqs. (7)and(6a,b)then become , ,_(B+vx?) Ei,=Ey,a-®5L ys (8) Since (vXB),, =0wemay write instead ! / E+vxBBieB+eXBin z=(252). (8a) The quantity E+vXB,which appears here automatically, when multi- plied with e,hasthedimension “newton” and iscalled theLorentz force .K=eE+vXB). (9) Through itsformulation (more precisely, the formulation oftheforce density kKtobeintroduced presently) Lorentz putanend tothefruitless discussions oftheolder theory with regard-to theponderomotive forces on moving charges. Inspite ofitsamazing simplicity Eq. (9)represents the sum total oftheforces acting inarbitrary electromagnetic fields. Anex- periment ofW.Wien onhydrogen canal raysconfirms thisdirectly.’ After J.Stark had demonstrated thesplitting oftheBalmer lines inanelectric field, Wien could produce qualitatively thesame effect byletting amag- netic field corresponding totheelectric field actontherays. Hethus replaced Ebythe vXBwhich isequivalent toit. Itmay incidentally benoted that vXHiscommonly written inplace ofvXBin(9);fromourdimensional standpoint thisisanabsurdity. B.TheMagnetic Analog totheLorentz Force -We,mustnowcalculate B’,i.e.thespace-space components ofthecurl ofQ’,instead ofthepreceding space-time components. Thisbecomesvery 1Preuss. Akad., January 1914. 28.12 PREPARATION FORTHEELECTRON THEORY 239 simple forthecomponent inthedirection ofmotion. Inview of(1)and(2) itbecomes 80;3M,_a,a yoo =SP SRLS _ Curl’s:Q!Ozh~On,~Ou,OtCurhsQ. ByEq. (26.11) this leads directly to Bi=Bs. (10) ForByweproceed asbefore forE,,noting that a0;os_ - % Curl’,Q’= -—S = -— I’Qag7aahae,208¥%+sin7%}ax’ aM, _a,. ain{cosYontsin3}%,sothat QQ «.OQ,—Qs 1’, = —_-— sS-S Curl'nQ=008@*)+siny(@=) =cos7Curly Q+sinyCurly Q. From thisweobtain byEqs. (26.11) and(27.11a): , _By+BE,/e B= re oe (10a) Similarly wefind B,—BE,/cB=. 10b)vi- B (ob) These formulas (10) may begeneralized vectorially to B-vXE/¢c By=(B-vXE/e)y, BL=Cre ).:wy Thequantity B—vXE/c’ appearing in(11)isatthesame time the ponderomotive forceonamagnetic poleofstrength 1,i.e.themagnetic analog oftheLorentz force exerted bythefield onthecharge 1. C.TheIntrinsic Field ofanElectron inUniform Motion Inaframe ofreference z,y,zwhich moves with theelectron the infrinsic field ofthe electron iselectrostatic incharacter. Thus, forr =V2 +y+ 2: e 1_ E=--i grad=andB=0. (12) 240 THEORY OFRELATIVITY ANDELECTRON THEORY 28.128 Foranobserver atrest, with respect towhom theelectron moves inthe direction ofthenegative z-axis (seefootnote attheendof§27A), wethen have inview of(6a, b),(7),(10), and (10a, b) ! ’ 1 ! 1E,=E, E,=Viza E,Vian (12a) Be=0, B=5p =e, B-- ap=Gh 2=0, =aT me aaia Weexpress these primed fields interms oftheprimed coordinates 2’,y’,2’ ofthepoint ofthefieldconsidered, which, likethez,y,z,weshall measure from themomentary position oftheelectron and consider theLorentz contraction along thez-coordinate: g=Vi-f, yy geez (13) Simultaneously wesets(x’, y’,2’)=r(z,y,2)or e=4/ +y+ 2 (18a)1-f . Wethenobtain from (12)and(12a) 1 op gt é yaE.,Ey,E.=avi-® °° (14) “ . ev 0,2’,~y"Bi,By,Be=heatvVizf 8 (14a) Thus ourprimed observer, unlike onemoving with theelectron, isaware ofamagnetic field inaddition totheelectric field. By(14a) itslines of force are circles about the direction ofmotion; itsintensity is,ifwere- place eoc’inthedenominator of(14a) bywointhenumerator andpass over from thefield strength Btotheexcitation H=B/w, -eosind ngaVite |H|Wisk sing=YY. (14b) This expression should becompared with theexpression (15.12) forthe Biot-Savart force, from which (14b) differs only byrelativistic corrections ofthesecond order in8.Thus inasense amoving electron inacathode rayrealizes thecommonly mentioned, butunreal, current element ofthe earlier theory; evhere takes theplace ofJds. The electric lines offorce, ontheother hand, areaccording toG4) straight lines diverging from theinstantaneous position oftheelectron in theprimed system (inview oftheproportionality ofthecomponents ofE’ in(14) with 2’,y’,2’)aswell asintheunprimed system; however, they do 28.16 PREPARATION FORTHEELECTRON THEORY 241 nothave thesame density inalldirections intheformer asinthelatter case. Rather, they aresqueezed together intheequatorial plane z’=0. Because ofthemeaning ofsin(13a),s>©andE’—0for6>1unless x’=0.Inthislimiting casetheelectrit fieldwould beconcentrated entirely intheequatorial plane. Thus theelectron isflattened inthelimit v+¢not only inrespect toitsshape (with which wearenotconcerned here), but also inrespect toitsfield. Inthepreceding wehave convinced ourselves that thedetermination of thefieldforuniform motion ismerely amatter ofalgebraic transformation, while intheolder electrodynamics itinvolved atleast some integration.’ §30will deal with thefield ofaccelerated motion. Weemphasize ingeneral: Theelectric andmagnetic fields form asingle unitandcanbedistinguished only with reference totheparticular refer- ence system employed. Together they form asix-vector. Inchanging the frame ofreference itselectric components contribute tothemagnetic ones andvice-versa. Wearehere dealing with aneffect ofperspective infour dimensions. Theaspect ofacube furnishes thethree-dimensional analog: Foraparticular choiceoftheviewing direction weseeonlythe(“electric”) ‘front face, forother, oblique, directions the(“magnetic”) lateral faces aswell. ‘ D.AnInvariant Approach totheLorentz Force; theFour-Vector ofthe Force Density From thefour-vector 2;--+24weobtain asthedifference inposition of twoneighboring world points thefour-vector dr;,dary,drs,icdt. (15) Furthermore thefour-dimensional volume element dx,dx,dzy-icdt (15a) isalsoindependent ofthechoice ofcoordinates, justlikethevolume ele- ment dz,dzzd, inthree dimensions. Since thecorresponding charge Ae (thenumber ofelectrons contained intheelement ofvolume), just like¢itself,isalsoinvariant, division ofAeby(15a)leadstoanother in- variant scalar andmultiplication ofthisscalar by(15)toanother four-vector. Asin(26.6) wecallitthefour-current density T°: _ __4e dx, dx, dz; .\ _. PmSe(BS,Bei)=lei. a6) eistheusual three-dimensional charge density. Acomparison ofthepre- ceding definition ofFwiththatin(26.6) shows thatthecurrent density J 1SeeOliver Heaviside, Phil. Mag. 1889. Thesurface s=const (Eq.(13a)) isknown. astheHeaviside ellipsoid; seeProblem III.3. 242 ‘THEORY OFRELATIVITY ANDELECTRON THEORY 28.168 ofelectrodynamics passes over intotheconvection current density pvinthe electron theory and, furthermore, that thefour-vector ©hasthesame direction asthevelocity four-vector Vdefined in(27.18), inview ofthe relation v=pty=pVvi-BV. (16a) Wenow multiply thefour-vector Fwith thesix-vector Fofthefield.. This results, bytheprocess of“reduction”, again inafour-vector, justas fortheoperation Divin(26.16). After having divided itbyc,fordimen- sional reasons, wecallitforce density anddenote itbyk,itsnthcomponent bykn: . ‘ kalrk, cheDleFe 2=1234 (7) ot Written term byterm this becomes ck,= PFe+Fu+Thu, ck,=Fn +Tifa+Tu, (17a) cky=Fan +TsFe +Tu, cky =Fa +TiFe +TiFa or,inthree-dimensional coordinates, ky=ke=p(vyB, —1-By +Ez) ky=ky=p(B, —v2B, +E,)} =o(E+vXB). (17b) ky=ky=p(vsBy —vyB:+E,), Inhisoriginal theory Lorentz operates primarily with thethree-dimen- sional vector ontheright. We are ofcourse also interested inthe fourth component. Itis,by (17a), ck,=ip(v.B. +v,E, +vB) =ipv-E =ipL. (17) Lisherethepowerexpended bytheelectric fieldstrength onaunitcharge moving with velocity v. Itmay beshown readily from therepresentation (17a) that thefour- vector*k isperpendicular totheworld lineofthecharge. Inview ofthe proportionality ofFandVin(16a) andtheantisymmetric character ofF weobtain forthescalar product ofkand V: V-k=0. (17d) 28.19¢ PREPARATION FORTHEELECTRON THEORY 243 Wepass from theforce density totheforce itself. Itisnot permissible here, however, tosimply change pinto theelectronic charge ebyintegra- tion over space, since the three-dimensional volume element isnot a relativistic invariant, but anarbitrary section through the“world tube” described bytheelectron (perpendicular tothealso arbitrarily chozen time axis). Itismuch more appropriate toplace thesection perpendicular to theworld line oftheelectron, which isindependent oftheorientation of the¢-axis, or,what isthesame thing, perpendicular tothegeneratrices of themantel surface ofthe world tube. Ifwedenote theangle between the worldlineand¢-axisby7,thethree-dimensional dzdydzprojects itselfinto the “world tube cross section” dzdydz dxdydzcosy=Vi-#8 (18) with thegeneral meaning ofygiven byEq. (27.11a). Byintegration over this cross section we obtain e fpaxdydecosy=wo (18a) +From therepresentation (17b) fortheforce density wethus finddirectly . kdedydz é ___[GS -FE txD-wviR-B ul) KistheLorentz forceofEq.(9).Itisnotdirectly apartofafour-vector, but becomes oneafter division by~/1 —B. The corresponding fourth energetic component ofthis four-vector is according to(17) [ited tops th (ida) Vi-B VIB VI * We call thefour-vector oftheforce, completed inthis manner, F;Itsfour components may beexpressed collectively by K etL F=( >= [=F 19b)wee view Inview ofitsderivation from the Lorentz force density kand ofthe relation (17d) itisperpendicular totheworld lineoftheelectron: V-F=0. (19¢) E.*The General Orthogonal Transformation ofaTensor ofthe Second Rank Asgeneralization oftheantisymmetric field tensor Fwenow consider anarbitrary (symmetric orasymmetric) tensor ofthesecond rank Tam, 244 THEORY OFRELATIVITY ANDELECTRON THEORY 28.20 whose components T’,,neednotvanish andforwhich wedonotneces- sarily have Tam=T'nn. Here wedefine astensor aquantity whose com- ponents T'nnandTambehave likethesquares andproducts zhandZntmof thefour-dimensional coordinates intheorthogonal transformation (27.1). Theformula which, by(27.1), applies fortheproduct zatm : 44 Inlm=zDeOnsOm25Le St tt may betransferred toTinthefollowing manner: a4 Tam=Do2enicnnTa (20) Weshall encounter 9symmetric tensor in§31.Itremains symmetric in the transformation. Foranantisymmetric tensor (20)may bewritten ,Gini Onk Pin=LD (erick —Omitins)Pa=DY Ta.(20a) adOR|cmtOme Here thecomponents T;,=—T's arealready accounted forwith the “components 7.Hence thedouble sumin(20a) must becarried outin suchfashion that, whereas itraverses allvalues from 1to4,konlyassumes thevalues k<i.Thevalue k=ievidently need notbeconsidered since T;=0and,inaddition, since allthedeterminants in(20a) vanish inthis case. Anantisymmetric tensor remains antisymmetric inthetransforma- tonsince theinterchange ofnandmreverses thesignofallthedeter- minants. After these general considerations wereturn oncemore tothebehavior ofthesix-vector inthespecial Lorentz transformation (27.2a). Wecon- vince ourselves that allthesubdeterminants ofthismatrix vanish with theexception of (=|oyanai|=|1 OyOn anan) jaeou |easou vi- |"=||="=|“1 os, cas] ay tse, E=|_|"“|_|=|_["=|__-#8 oean as, ou,ae anaul Vim inviewofthevalues oftheaugiven byEqs.(27.5, 7,8).Onthisbasis the sums ofsixterms in(20a) arereduced tooneortwoterms. Wefindspe- cifically 29.3 INTEGRATION OFDIFFERENTIAL EQUATION OFFOUR-POTENTIAL 245 Ty=Alla oT. Te=?apap (Fu—Tw). ‘2s=Tn, 1 5 Tu=Viza Ta+BTx), 1 p TumTu, Tu=Vip lu+Bw), 1 p Tu=pap (Tu—Pn) Ifhere wesubstitute fortheT'stheelectromagnetic equivalent oftheFu given bythematrix arrangement (26.17) weseereadily that thepreceding transformation formulas correspond toEqs. (8) and (11). The present procedure forthederivation ofthese equations may beslower than the earlier one, but it-is fundamentally more elementary and certainly more general, since itcovers any tensors ofthesecond rank. §29. Integration oftheDifferential Equation oftheFour-Potential We now turn tothedifferential equation (26.5), . OQ =~wI, (1) where wearedealing with aproblem offour-dimensional potential theory. The three-dimensional potential theory, Eqs. (7.4a) and (75), may serve usasexample. Tobegin with, werequire thefour-dimensional analog to Newton’s potential 1/r. Itisgiven by 1 Usp B= -a+ G-at+G-a't+G- @ Asproof wecalculate a1_ 48#-% Fl _2,8, osa Re aa Rtme85 From this wededuce fF1 88k OU=Dade —etRo &) which isvalid forallpoints except the“source point” &=2,,¢ =1,---4. Itmay beproved similarly that inaspace ofp+2dimensions thecen- trally symmetric potential isrepresented byU=R™” ifthemeaning of Risgeneralized correspondingly. - Furthermore, asapreliminary, weshall determine the“surface” ofthe sphere R=const, i.e.athree-dimensional structure infour-dimensional space. Ifwsignifies thesurface area oftheunit sphere, itis 246 THEORY OFRELATIVITY ANDELECTRON THEORY 29.4 oR*® with w=2x. (4) Inproof weconsider theintegral ° 4 J=fff0{Zeatsdds (4a) Bycarrying outtheintegral foreach coordinate separately andutilizing thefamiliar value oftheLaplace integral weobtain thefourth power of ~/z orx*.Ontheother hand, ifweintroduce polar coordinates, with f= De, wefind aof cTrda® Jaw[trans (4b) Acomparison of(4a)and(4b)proves Eq.(4).Similarly, forspace with p+2dimensions wo=2eP*/P(/2 +1). (40) (The reader may check thevalidity ofthisformula inthethree-dimen- sional and twodimensional cases, p=1andp=0). +A.Four-Dimensional FormofthePotential Wenowapply Green’s theorem toourtwopotentialsQ andU: aU3a foov-voo) a--a=[(a%d-022)ae6) Theintegration attheleftistobeextended overinfinite four-dimensional space, withtheexclusion ofthesource point x;=&bymeans ofasphere Kofradius R—0.Theintegration ontherightistobecarried outover thissphere Kandasphere R—©,which however, asinthethree-dimen- sional case(seep.39),does notcontribute totheintegral. Substitution from thedifferential equations (1)and(3)then yields for(5)(Inowde- notes thefour-current, nottheI’-function): di++dh_91, _fide w[rats [oxme(Te (se) SincebyEq.(4)fdo=2x°R*,thesecondintegralontherightvanishes forR—>0,Since furthermore dn=—dR (thenormal istobetaken posi- tiveifitpoints outward fromthespace ofintegration, i.e.inward intothe sphere K), 2[apa =4a. 20.68 INTEGRATION OFDIFFERENTIAL EQUATION OFFOUR-POTENTIAL 247 Qhere denotes thevalue ofourpotential atR=0,i.e.for£;=2;.Hence wefind from (5a) POG, 2,24,2)/m=frm, ©) inperfect analogy to(7.5). (6)represents thefour-potential inanarbitrary world point x+++x4by@four-dimensional integration overthefour-current density 1,which isassumed tobeknown. However, ©isknown tousonly fortherealtimes +<t,which precede thetimeofobservation t;wemight alsosay,forthetimes r<0,if,without lossofgenerality, weputthetime ofobservation ¢temporarily equal to0. Imaginary axis VV Ecplane xr. tr. Real axis L. Fie. 41.Integration ofthefour-dimensional potential equation 02 =wl". Defor- mation oftheoriginal path ofintegration along the real &-axis into aloop about the“light point” Lonthenegative imaginary axis. Accordingly T°isnotknown tousalong thereal &axis, aswehave im- plicitly assumed tillnow, butonly forthenegative imaginary values &=ter=—tc|r]. (6a) Accordingly weshall distort thepath ofintegration for&along thereal axis, —© <&<©into aloop-about thenegative imaginary half-axis, which leads from —ico byway oftheneighborhood oftheorigin ofthe complex &-plane back to—i«, asshown inFig. 41.This does notalter ourrepresentation (6)orthefact that (6)satisfies ourdifferential equa- tion (1). Westillwant toconvince ourselves that (6)satisfies also theauxiliary condition (26.7), DivQ =0.This follows from thefact that I”satisfies thecontinuity equation (26.16b) Div r=0.Infact, ifweindicate the differentiation with respect toz;and &,bysubscripts, and carry itoutunder thefourfold integral sign wefind 248 THEORY OFRELATIVITY ANDELECTRON THEORY 29.7 4a"DivO/y=f(1Grad.5)dia++ate "Re 1(7) =—|(FGradyfe)as+dk, ie.,after carrying outanintegration byparts, 4a"DivO/ny=[DiverSH —0, (7a) which was tobeproved. For thefurther treatment oftherepresentation (6)wecancarry out first either theintegration with respect to&orthat with respect to&, &,and &.For thepresent weshall follow thefirst course. B.Retarded Potentials With reference toFig. 41,welook forthose points ofthecomplex &- plane atwhich thedenominator R’vanishes. We write Ratt w-e), 8) where rsignifies the three-dimensional distance between the point of ‘iptegtation &,&,&andthereference point 2{,21,t3,andwhere we have dropped ourtemporary convention x,=0,which merely served the moreconvenient description ofFig.41. ‘There aretwopointa atwhich R’=0,ie. u&—& =tir (8a) and %—&=tr. (8b) Wecallthefirst, with Minkowski, the“light point” L;thesecond’ is designated with L’inthefigure. Intheneighborhood ofLwehave according to(8)and (8a) RY=(4—&—tt)(ee—Be+Hr)&Dire —&—ir). (Be) ByCauchy’s theorem wecan now distort thepath ofintegration inFig. 41into acircuit about L,yielding bythemethod ofresidues ad Tt dt ai we)=TEDfr& Keoaot kw Ew Tzisthevalue ofIatthelight point. The factor (+277) results from the fact that, ontheone hand, —& occurs inthedenominator, ofttheother; 1The digtortion oftheoriginal real path ofintegration into #loop about L’would lead tothe“advanced” instead oftheretarded potentials (see p.148). 29.12 INTEGRATION OFDIFFERENTIAL EQUATION OFFOUR-POTENTIAL 249 theintegration path istraversed clockwise about L,i.e.inthenegative direction from afunction-theoretical standpoint. Substituting (9)in(6)wefind 4rQ/mo=[Tabi disdt. 0) Resolved into components thisyields, by(26.4) and (26.6) 4rA/uy=[¥anaeae, tre=|%dt,diadts.(100) These are,however, exactly therepresentations oftherelarded potentials inEq.(19.13). Infactourpreseht J,andp,have thesame meaning as ourearlier [J]and[p],in §19.Forifwedesignate thetime ofthelight point, which precedes that oftheobservation, by7,asin(6a), wefind from (8a) ict=tertir, =r=t—r/e. (10b) This ishowever exactly thetime defined in(19.13c), forwhich [J]and{p] were tobecalculated. Inthis manner theformerly suppressed proof of '(9.13) hasBeen brought inamathematically particularly appropriate fashion. Itshould benoted thatG.Herglotz haddevised themethod given here even before thetheory ofrelativity, just onthebasis ofmathematical symmetry andelegance.' C.TheLienard-Wiechert Approximation Wenow take thesecond course mentioned above and carry outthe integration over &,&,&.Wehereimagine current andcharge tobecon- centrated inasingle point, theelectron, rather than spacially distributed asuptonow.WeuseforIitselectron-theory value(28.16), bywhichthe conduction current Jwasinterpreted asconvection current, andwith eas electron charge, obtain from it Jvdesdesdts=ev,ic)=—eR. (uy Ristheradiusvector fromtheelectron tothereference point,Ritsderiv- ative with respect to¢forfixed reference point: a.(4dedeswe)=-5 R(&.Ra (y,tc). (11a) Wetherf obtain from (6),carrying outthefirst three integrations, ~ te'0/u=~e$dis a2) ‘See Gdttinger Nachr., 1904. 250 ‘THEORY OFRELATIVITY ANDELECTRON THEORY 29.18 Asin(9),theintegration istobecarriedoutaboutthelightpointinFig. 41.However, thelocus ofthe electron &,&, isnot anindependent point ofintegration, asuptonow, but itself depends ontheintegration variable &.Wemust therefore consider theworld line oftheelectron inthe neighborhood ofthelightpoint Landexpand R’asfollows: Ra=R+@-wEt, aa) soastobeable toapply themethod ofresidues. Here wehave Le GR1dR-R)2 Ri=0 and hz a %RR ‘Theexpansion ofR*becomes hence Rm —a)ERRE (13a) and Eq. (12) passes over into :420/=—22Bfe. (13b) We" RRS omee Inview ofthesign of&inthedenominator, which isopposite tothat in (9),theintegral isnow equal to—2xi. Wethus obtain eR 4nQ/m =—S=. 14) /woRE (14) According to(8a) thevector Rhasthetime component ir,whereas its space component (light point toreference point) isr;by(11a) thetime andspace components ofRare—icand—v.Hence Ri=rovr=re(1-T4) =re(1-4),(14a) re e where v,denotes theprojection ofvonthedirection ofr.Substitution in (14) and separation into real and imaginary parts yields theremarkably simple formulas ofLienard (1898) andWiechert (1900): eov eol 4rA/us=2, je OM (15) Ourderivation shows that, liker,vandv,must betaken fortheearlier time ofthelight point. Itisinteresting tonote that thedenominator 1—»,/cwillrecur inVol. IVinconnection with theDoppler effect. Actually theoriginal integral form (6)ofthefour-potential will prove 30.2 FIELD OFTHEACCELERATED ELECTRON 251 more useful forwhat follows than theformulas (15) or(10a), where the integration has been carried out. §90. TheField oftheAccelerated Electron Theadvantage ofEq.(29.6) rests inthefactthat thevariables 2---1 ofthereference point occur here only inthedenominator R*.Wehave to differentiate only thelatter ifwewish tocalculate thefield ofanelectron inanystate ofmotion. Inthismanner weobtain from (29.6) first: 2 al a1 4x?Curln o[(meh rs2h)aea) Since 31 ym— 2mOt, Rt RS Re theparenthesis in(1)becomes ~FinRe—TaRe)=+Z(0XRan. Here wehave'transferred theusual symbol (X) ofthethree-dimensional vector product totheproduct ofourtwofour-vectors, which evidently is quantity with sixcomponents. The same applies fortheleftside of Eq.(1),where by(26.12) Curlnn @isthenm-component ofthesix-vector Hof.Wethus obtain from (1) Pefom=fOEXRoands++te. (1a) Wenowcarry outtheintegration with respect toé,&,and,inwhich process, byEq.(29.11), Ftransforms itself into—eR, and&refers, from thispoint on,tothepoint electron. Wefind af=—e$BRRae, (2) Theintegration isheretobecarried outoveracircuit aboutthelightpoint, asinFig.41.Thedifference from theprevious calculations consists onlyin thefactthatthedenominator nowvanishes tothesecond order, sothat wehave tocarry theexpansion indenominator andnumerator oneterm further. Ifweabbreviate &—ka_ ee wewrite inplace of(29.13a) 252 THEORY OF RELATIVITY AND ELECTRON THEORY 30.3 R= UR-R+uv{R-R+R-R) +: R=40@-R)(1+wRRARR -). and, since RXR=0, RXR=(RXR)L+uURXR+-- Thus, ifatthispoint wetransfer thedenominator inpart tothenumerator and suppress thesubscript L(2)becomes: Ydu Ps R-R+RR af=fun R+uRX® (1-AR+E4). afae 2(RXR+u(R XR)) uRR Here weneed write outonly theterm multiplied with u™’,since only this isinvolved indetermining theresidue, andcanomit theterms with u™, w,u'-.Wethus obtain idu a ,R-R+R-R Ont=gyfU(RxRRxREBT), l= eS ER Bince theintegration indicated inFig. 41amounts simply totheaddition ofthefactor —2zt, wefind finally 4xf RXR gRR+R-RSam -RXRO. (3) ec (R-R)* (R-R)* According to(29.8a) and (29.1!a) theexpressions ontheright must be formulated specifically for R=(r,i7), R=-—(v,ic), R= (-¥,0). (8a) Weexamine this general representation first forthespecial case ofthe A.Electron inUniform Motion Here, since R=0, Anf 5_R-R , ov—=-R XR =RXR=,-—_1- 4ec (R-R)? eF(1—v,/e)* ) According to(3a)thesix-vector RXRisgiven, inmatrix notation, by . TeTyTstr RXR=- _de (5) Ye wy % 1. We calculate itsspace-space and space-time components assubdeter- minants ofthe matrix. Inthe notation ofordinary three-dimensional vector calculus weobtain 30.7 FIELD OFTHEACCELERATED ELECTRON 253 . vxXr forthespace-space components, RXR=. (5a) i(rv —cr)forthespace-time components. Ifthis issubstituted ontheright side of(4)and fisseparated into its space-space portion Hand itsspace-time portion —7cD onthe left as well, wefind 4sH 1-v/¢ “eX RT ole . (6)~DLry-r)1—w/e e ¢ RI —»,/e)* These expressions appear basically different from theexpressions (28.14) and (28.14a), with which werepresented previously thefield oftheelec- tron inuniform motion (there designated byH’,E’}, butcanactually be y Fia. 42.The field ofanelectron in P uniform motion. The electron moves ‘alongthez-axiswiththevelocity v;0isthe location ofthe electron which issi- multaneous with theobservation atP, y Lthelightpoint, sothatLO=vr,where (a ristheretarded time ofthelight signal emitted from LtoP. LO . as transformed into each other byelementary geometrical considerations. We willshow this inProblem III.3. Wewillthen make useofFig. 42,which pictures atthesame time thedifferent viewpoint ofthepresent and the earlier formulas: Inthepresent formulas randrrefer tothelight point L, inwhich theelectron was atthetime ¢—r/c,tbeing thetime coordinate ofthereference point P.The earlier formulas, ontheother hand, con- cerned theposition oftheelectron simultaneous with ¢,which isdesignated inthefigure byO;thecoordinates ofthereference point with respect toO aregiven by2’,y’,2’asin(28.14). B.The Accelerated Electron Ifin(3)weomit thepart (4)weobtain thepure “acceleration field” 4xfRXR_ RXRRR (7)ec|(RR) @R 254 THEORY OFRELATIVITY ANDELECTRON THEORY 30.75. Toanalyze this,wecalculate from (3a) RxR= —rX¥,thespace-space portion, (7a) ary, thespace-time portion aswell as R-R= -rv. (7b) Wethen obtain from (7)with due regard to(5a) and (29.14a) 45H_XY_&Xv)(r-¥) ecer(1—v,/e)?er(1—v,/c)”” ®) 4xD re 4=WMEE-W)€ ,&r(l—v,/e)? ©A(t —v,/c)* From this weconclude directly . rH/e=rXD and rD=0. (8a) H,D,andror,aswemight alsosay,H,E,andraremutually perpendicu- lartoeach other. Furthermore, taking theabsolute value inthefirst Eq. !(Ga) inview ofthesecond Eq.(8a), leads to taj=|DI (sb) or,expressed inother terms, to Hl=VfE|. [HI=4/218 Wethus have atypical transversal field, asfortheplane light wave inEqs. (6.11) and (6.13). Itsstrength decreases with increasing ras1/r; forthe denominators in(8a) have each onefactor rmore than thenumerators, not éwo asfortheelectron inuniform motion (Eq. (6)). Hence atgreat distances (6)may beneglected ascompared to(7),and (7)represents the entire field ofthe accelerated electron. C.TheLongitudinally Accelerated Electron Letusassume specifically that vand ¥have thesame direction (recti- linear motion, longitudinally accelerated electron); wethen seereadily that ¥(r-v) =v(r-¥) and hence also (tXW(t-v) =(FX v)(E-¥). 31 MAXWELL STRESSES AND STRESS-ENERGY TENSOR 255 If,now, thedenominators oftheright sides ofEqs. (8)aremade the same, two terms cancel each other ineach case. These Eqs. (8)then re- duce to 4H Xe 4D rv+ti, () ec er(1 —v,/c)®’ € ert(1 —v,/c)** Jwemakethecommon direction ofvand¥theaxis¢=0ofasphericalpolar coordinate system r,J,y,wehave v,=vcos8, v%=—vsin d, v»=0, o=dcosd, i=—dsind, t,=0, rXv=(rXWy, H=H,, D=Ds andweobtain from (9) e@ sin3 eb sin& AteOapeop OOReap 0) These arethesame expressions as(19.20), with theaddition oftherela- tivistic denominator (1—8cos9)’,which ofcourse waslacking inthe honrelativisti¢ calculation (6—>0).Infactourearlier factor p(¢—r/c)is thesame asourpresent factor ¢,computed forthelight point. Corre- spondingly wefind inplace oftheradiation Sin(19.22) eet sin’3 S=eeteaar (=Boos {) Accordingly themaximum oftheradiation nolonger liesatd=4/2, but advances, as6approaches 1,from 9=x/2toward 3=0.'Wealready referred onp.155tothis phenomenon, which ischaracteristic forx-ray theory. §31. TheMaxwell Stresses andtheStress-Energy Tensor Sofarwehave only dealt with thekinematics oftheelectron, prescribing itsmotion andinquiring regarding theaccompanying field. Wenow tum tothestatics andthen tothedynamics oftheelectron. With respect tothe *Bydifferentiation of(11)with respect to0weobtain ascondition forSmax @quad- ratic equation forcos8,which, forsmall 3,yields con=38,8=5~38, and for6nearly equal to1, 1-#-1-6 code ~—R, 8Vz: 256 THEORY OFRELATIVITY ANDELECTRON THEORY 311 statics oftheelectron wehave familiarized ourselves tillnowonly with theLorentz force, acting atthelocus oftheelectron. Afield concept cannot besatisfied herewith, however, butmust follow upthetransfer of force actions invacuum, where there arenocharges. This wasFaraday’s intimation when hespoke oflinesofforce asofelastic bands which trans- mittension andcompression. Maxwell wasalsohereabletoplace Faraday’s notions intoclear mathematical focus. This wastheorigin ofMaxwell’s stress tensor, which may beexpanded relativistically into astress-energy tensor. ‘Weproceed fromtheLorentz force density inEq.(28.17), ‘ kalrk,ck=UTPer, i) wrt andreplace I’,inaccord withMaxwell’s equations (26.16), bythesix- vector oftheexcitation f.Wethen obtain from (1) ‘ 4haf, ke=Div, fF =DL Pw (2)rt Falmat OLm Wewillshow thatkmaybeexpressed asthefour-dimensional divergence ‘ofatensor T,;i,e. that “é= DLT m= 3) b=Dat ) and ié Tam=—=2,Forde+Samy (4) where Adenotes theLagrange density in(26.24). Since wehereenter thedomain oftensor quantities thefollowing rather abstract computations withdouble indices cannot beavoided. Totransform theright sideof(2)weutilize theidentity Om _9 og OFaedan” =dam(SomFar)—fem55+ (5) Change oftheorderofsummation yields forthefirsttermontherightof (5),summed asindicated in(2), a aLfUbePe=-ZLgeLiar (3) Thesecond term ontheright sideof(5)becomes, after carrying outthe summation over randm(including thenegative sign’ OFnr 318 MAXWELL STRESSES AND STRESS-ENERGY TENSOR 257 Wewrite this expression once more, reversing both thesymbols forthe summation subscripts r,mand thesequence ofthesubscripts offand F: OFwn ELlea and form half thesum ofthese equai expressions: 1 OFmn1OFnr aDie(Ge+FE) ® Now wemake use ofMaxwell’s Eq. (26.18). According toit(the three terms areformed bythecyclic interchange ofthesubscripts m,n,r) OFan,OFnr)OFpm tm Taz, +a, 7% Taking care ofthenegative sign bychanging thesequence ofsubscripts ofF,wethen canwrite instead of(7) 1 OFne>xFortn: (7a) !This istheresult ofthesummation ofthesecond term ontheright sideof (5),whereas thatforthefirstterm wasgiven by(6).Hence weobtain, finally, from (2), (6),and (7a), a L OFnr thy=—Dg farPar+90LDherGe 8) Here thefirst term isalready identical with thefirst half oftherepre- sentation ofthetensor 7in(3)and (4).Toprove fully thecorrectness of therepresentation wemust still demonstrate that thesecond term onthe right of(8)isequal to SOA aAcD>Samincan" By(26.24) this isactually thecase. The statement (3)isthus proved. Ourexpression (4)for7represents asymmetric tensor ofthesecond rank, Itssymmetry follows directly from the proportionality offand Fand fromthemeaning of5,»;thetensor character inthesenseofp.244follows from thebehavior ofthesix-vectors fand FinaLorentz transformation. This calculation, overloaded with indices and formal asitmay seem, leads tofar-reaching physical consequences. From (4)wecompute thecomponents ofTindividually, beginning with” thediagonal terms ofthematrix, allofwhjch have the term with Ain common. We find 258 ‘THEORY OFRELATIVITY ANDELECTRON THEORY 319 Pus—2tfaFat faFa+fuFu)+A =—H,B, —HB, +DE. +4H-B- 4D-E =—H-B+ H.B.+ DE. +}8-B —4D-E =H.B.+ DE. —W, where Wdenotes theenergy density. Similarly, Tn=H,B, +D,E, —W, Tu=HB, +D,E, —W. On the other hand Tu=—2fautfaFatfaPu) +A =DE. +DyEy +DE, +¥H-B —4D-E =4D-E+4H-B=W. Wenow turn tothenondiagonal elements, beginning with those having thesubscript 4,such as Tu=Ta=—1aFu+faFu) =i(D,B, ~D.B,)=—iceow(E XH).=—£8. Tu=Ta= -5S, TuaTa=—2S, The remaining nondiagonal elements are H,B,+DE,=H,B,+D,E, Ta=Tu=—}(uF +ful) = oe ¢ H.B, +DsE, =H.B, +D,E., where the equality ofthe last four expressions again follows from the proportionality ofDand E,and similarly HB,+DE, B,B,+D,E, tu=te[et.Eru-ta{1B.+DyEs, HB, +DE, (HB, +D,E,. Thus thecomplete 7’matrix becomes, inabbreviated notation «|-is 7 T=|—— 9)--S| W c 31.12 MAXWELL STRESSES AND STRESS-ENERGY TENSOR 259 Hereoisthethree-dimensional matrixofthesocalledMaxwell stresses: HB, +D.E.—W, H,B. +D,E., HB, +D,E,, H.B, +D,E,, H,B, +D,E,— W, H,B, +D,E,, (10) ALB, +D,E,, H,B, +D,E,, HB, +DE, —W. The electrical portion oforepresents atension ofthemagnitude Winthe direction ofthelines offorce andacompression ofthesame magnitude in thedirections perpendicular thereto. This isseen immediately ifthez-axis isplaced inthedirection ofEandBisputequal to0.Then oes=4D-E, ow=on=—4D-E, cu=0. Thesame may beshown forthemagnetic lines offorce ifthez-axis is placed intheir direction and Eisputequal to0.Wehave thus returned tothemodel which Faraday had constructed purely onthebasis ofin- tuition. This stress tensor ¢,byitself, ishowever not alegitimate physical quantity inthesense ofthetheory ofrelativity. Itbecomes oneonly by itsextension with theenergy quantities Sand W,forming the“stress- epergy tensor” 7.This hasthecharacteristic property that its“trace” (sum ofallthefour terms ontheprincipal diagonal) vanishes. Wehave in fact: Tu+Ta+Ta+Ty =H-B+D-E -3W+W =0. Wenow return totherelationship between 7andtheLorentz force density kasgiven by(3),and consider first thefourth lineof(3).Inview of(9)itis i. ow k=—(divS +57. Ifwereplace kybyitsvalue ipL/c from (28.17c), wefind, since «=ict, Wdst pb=0, Lave, an This isPoynting’s theorem, Eq.(5.7), where theformer energy loss by Joule heat isreplaced bythework done onthemoving charge p. Consider now oneofthespace components ofEq.(3),e.g.thefirstline: ky=hy=div,o—1SscOx which, written outindetail, becomes doen Boys, Bone 1OSs_aetytaea (02) 260 THEORY OFRELATIVITY ANDELECTRON THEORY 31.13 Ifweomit thelast term ontheleft, i.e.confine ourselves toastationary state, weobtain thecharacteristic equation (8.11) ofelastic equilibrium in Vél.'II. Just asthere thevolume force F;isabsorbed andbalanced bythe stresses oj, insofar asthey point inthex-direction. The Lorentz force density may becompletely replaced bythese stresses inour case. They aredefined throughout thefield bythetensor array (10), even where, in view oftheabsence ofcharge density, theLorentz force isnonexistent. We have thus attained thegoal setatthebeginning ofthis section, of following upthetransmission oftheforce through vacuum (without the useofatest body). However, what doweknow ofthenonstationary state and the term with dS/dt which isthen added in(12)? The answer isgiven byEq. 14.1 inVol. II,where thecorresponding term, there designated by—pd’s/al’, represented theinertial resistance ofunit volume oftheelastic body or, with positive sign, itschange inmomentum. We learn from this that there exists amomentum perunit volume Galso intheelectromagnetic field, and that itistobedefined, indirection and magnitude, by 1 Ge=2Ss. (13) Wealready know thattheelectromagnetic fieldpossesses energy andhow this istobelocalized inspace. Wenow seethat wemust also attribute to the field momentum, continuously distributed through space wherever there isanenergy flux Sand ofthesame direction with thelatter. Correspondingly alight wave carries momentum and exerts apressure on& nonreflecting (black) body onwhich itisincident—the light pressure dis- covered byMaxwell. Similarly, ifalight wave isemitted byabody, it imparts toitarecoil which isequal andopposite tothemomentum carried byit.Wecallthelatter body the“transmitter,” theformer the“receiver,” and assume that both were atrest fort<0.At¢=0,when thewave is emitted bythe transmitter, the latter receives arecoil. The center of gravity ofthetransmitter andreceiver isthen setinto motion andremains inmotion fortheduration 0<¢<T.Atthe time ¢=Tthelight wave isabsorbed bythereceiver (without reflection, asweshall assume forthe sake ofbrevity). The wave then imparts tothereceiver anequal impulse forward. From this point onthecenter ofgravity ofthetwo bodies is once more atrest, though ithasbeen displaced acertain distance during theinterim 7,corresponding tothebackward motion ofthetransmitter. This contradiction with thelawgoverning thecenter ofgravity vanishes only ifweassign amomentum tothelight wave itself during itslifetime Z. ‘Then momentum isneither created nordestroyed, both inemission and in absorption, and thecenter ofgravity remains permanenily atrest. Asiswell known, itisdifficult todemonstiate thepressure oflight in 31.158 MAXWELL STRESSES ANDSTRESS-ENERGY TENSOR 261 thelaboratory. The radiometers constructed forthis purpose indicate generally convection currents ofresidual gases, caused bythethermal effect oftheradiation. Theproof ofthepressure oflight intheheavens is much grander. Thetails ofthecomets, pointed away from thesun,show it(Lebedew), alsothesolar corona, where luminous particles arebalanced bythepressure oflight orradiation ataheight equal toasmuch asthe radius ofthesun. The inner constitution ofthesunand thebright fixed stars generally isalsocontrolled bythecommon action ofthepressure of radiation andthethermodynamic gaspressure (Karl Schwarzschild, +1916, forthesurface ofthesun; quite generally, A.S.Eddington, 1944). We must here content ourselves with pointing outsome general relationships between momentum, energy, and light pressure. From thedefinition ofSweobtain foratransversal plane wave (eaial=4/218): vo V2EP=ecK* iS| =|EXB| =|E||Hi = =W. eye=BH=yocH? From thisfollows, inview ofthedefinition (13) ofG ig}=¥ (14) The momentum incident onascreen ishence equal inabsolute magnitude, butforthefactor 1/c,totheenergy density infront ofthescreen (this applies notonly forvacuum, butforanynon-absorbing medium). ‘Weconsider abundle ofparallel rays, a“wave packet”, oflength Jand cross section g.LetWbetheenergy contained init,Gthemomentum contained init: W-a", Gaqe=9%=%, (1s) thelastfollows from (14).Wespeak ofa“photon” ora“light quantum” iftheenergy Wofthebundle isequal tohy(h=Planck’s constant, y»=number ofvibrations persecond). By(15) themomentum ofthis bundle is a=. (15a) Inthetheory oflight quanta thelight pressure isthus identified with a “hail ofphotons”, towhich every photon contributes thequantity hy/c. 262 THEORY OFRELATIVITY AND ELECTRON THEORY 32.1 Wetestthisstatement once again with theaidoftherepresentation (10) ofthe,stress tensor o.Letthelight wave, assumed plane, beincident per- pendicularly, inthepositive z-direction, onaplate. Because ofthetrans- yersal nature oflight FE,,D,,Bz,andH,arezeroandthefirstrowof(10) reduces to Ox=—W, oy=Ou=0. Tfwearedealing with thelight bundle described by(15), theforce ong =Wa actsontheplateduring thetime7=J/c;itstimeintegral yieldsthe impulse imparted totheplate. Wecalculate: 7 '= ai Wd _W Ga[wean >|wea "B=, which agrees with (15). Ifthelight wave isnotincident perpendicularly ontheplate, butatan angle awith respect tothenormal, thequadratic character ofthecoeffi- cients inthetensor transformation formula (28.20) leads to ozs=Wcos’a. This dependence onangle isreasonable since thearea bombarded bythe light pencil isnowg/cos aandonly thecomponent inthez-direction of themomentum ofthelight rays iseffective aslight pressure. ' §82. Relativistic Mechanics Unlike electrodynamics, which fitstherequirements ofthetheory of relativity from thevery start sothatwecould actually base thistheory onit,classical mechanics must undergo fundamental revision toharmonize itwith thetheory ofrelativity. This revision even affects, fortheindividual particle, thedefinition ofitsmomentum (its“quantitas motus”) asafour- vector. Inagreement with Vol.I,§2weassume ittobeproportional tothe four-vector Vofthevelocity inEq.(27.18) and callitagain G: _ _oh day dz; da,G=mV, Vt, @& F F&F () Thecoefficient msistherestmass oftheparticle, dr=+/1 —'dtisthe differential oftheintrinsic time. Wecanalso write inplace of(1) mo . _a dy dz GFram t Ya a at (le) The quantity ‘™o naois (2) 32.5 RELATIVISTIC MECHANICS 263 iscalled themass inmotion. Itisnotconstant, asinclassical mechanics, butincreases for8—1,»—ctoward infinity; accordingly itdepends on theframe ofreference andishence notalegitimate world entity. This applies notonly fortheelectron, butforevery mass—though alarge mass cannot beaccelerated tovelocities close tocinthe same manner asan electron. (However, incosmic rays with their tremendous energies the variation ofmass finds expression also fortheheavy and semi-heavy particles, theprotons and mesons.) The law ofinertia, Newton’s first law, now becomes inrelativistic formulation G=const. (3) Correspondingly thesetond lawmay bewritten asfour-dimensional vector equation asfollows: dG an F. (4) HereFistheexternal force,extended toafour-vector. Weknowthatfor theindividual, electron theLorentz force density kissuch afour-vector, bitnottheLorentz force Kitself. The latter becomes oneonly after it hasbeen divided by~/1 —#?andhasthus been placed intotheinvariant relationship with kwhich isexpressed byEqs. (28.19) and (28.19a). This leads tothe following definition ofthe four-vector Fforthe individual electron interms ofK: K e(E+vXB) et Fiaa=Vice@- vi-8? Fy"Wink v-E.(4a) When thisissubstituted in(4)thefactor »/1 —6ontheright cancels thefactor +/1—#*contained indr.Wethusobtaininsteadofthefirst three components of(4) d dmv a) =arom =& (6) ‘This istheequation ofmotion (4.6) inVol. I.Aswasfirstpointed outby Planck,’ theLorentz force Khere takes theplace oftheclassical New- tonian force. Forthesake ofdistinction thefour-force Fin(4)isdesignated asMinkowski force. Aswasalready shown inVol. I,§4,(5)becomes, forlongitudinal and transverstl direction oftheforce (K||vand K1v), ™ dv_ ™o dv_ A awa =Kand iPta K,respectively. 1Verhandl, d.deutach. phys. Ges. 4,p.136, 1906. 264 THEORY OFRELATIVITY ANDELECTRON THEORY 32.6 Thedesignations longitudinal andtransversal massforma(1—6°)**and me(1 —6*)~"* were discussed andcriticized atthesame place inVol.I. Wesupplement Eq.(5)bythefourth energetic component, which fol- lows from (4)and (4a): a _meavi-# =e-E=v-K. (6) Theequality ofev-E andv:Kpostulated herefollows intheelectrodynamic casesimply from thefactthatv-(v XB)=0.Applied toanarbitrary force lawitsignifies that thefour-force Fmust beperpendicular tothe world lineoftheparticle (seep.243). _v-K isthework done onthemoving particle bytheforce Kinunit time, ie.itisequal tqdA/dt. Accordingly theleftsideofEq.(6)issimply thechange inthekinetic energy Teffected bytheforce K.Wehence have moet T=Vi-# +const. (6a) InExercise III.4 weshall convince ourselves ofthefact that Eq. (6) maybederived fromtheequation ofmotion (5)alsobytheformalism customary ih,thederivation oftheenergy theorem inelementary me- chanica, namely scalar multiplication with v.Since, bydefinition, Tmust vanish forv—»0,theconstant in(6a)mustbeputequalto—mec’. Henee, inview of(2), =me 14). : T=me(FR 1)=(m—mc’. () Theclassical expression T=mv’/2 follows from thisbypassing tothe limitc>©,aswasalready notedattheendof§4inVol.I. A.TheEquivalence ofEnergy andMass Just asweconsidered therestmass moapart from themass inmotion m, weintroduce apart from theenergy inmotion Etherestenergy Eo,where “then7’=E—Ey.Wecanhencewriteinplaceof(7) E—Ey=(m—m)c. (7a) Werender thisequation more specific bythestatement E=me (8) andtheconsequent relation Ey=me’. (6a) This isthetheorem oftheinertia ofenergy, which according toEinstein isthemost important result ofthe(special) theory ofrelativity. We 32 RELATIVISTIC MECHANICS 265 quote Einstein literally: “The mass ofabody isameasure ofitsenergy content; iftheenergy changes byAZ,themasschanges inthesamedirec- tionbyAE/c*. Itisnotoutofquestion thatforbodies whose energy con- tentisvariable inahighdegree (e.g.forradium salts)atestofthetheorymay besuccessful.”” This testhassince been carried outonahuge scale: The atomic trans- formations which have been discovered inthemeantime and been studied indetail formost ofthelight elements have led,bytheuseoftheequiv- alence theorem, toanundreamed ofincrease intheprecision ofthechemical atomic weights’ andthefission oftheheaviest element, uranium—more precisely, theuranium isotope ofatomic weight 235—which wasdiscovered byOtto Hahn only toward theendof1938 has, inaccord with thelossof mass occurring init,hadaterrifying effect inthedestruction caused bythe uranium bomb. Weshall concern ourselves here only with thesecond example andthisonly briefly andsuperficially. The uranium 235atom, after capture ofaneutron (atomic weight 1) hasassumed theatomic weight M=236, but retained theatomic number Z=92oftheoriginal uranium atom.Itmay,e.g.,splitintokrypton, Z=36,andbarium, Z=56,orinto xenon, Z=54,andstrontium, %=38.Both:fission possibilities areobserved. Theconservation ofthe nuélear charge eZ-is here assured, since 92=36+56=54+38. However, themass isnotconserved. Instead, themass excess oftheatomic weight over theinteger 235(the so-called “packing fraction”) issetfree, ie.transformed into energy. Ifweassume that itamounts tooneunit in thefirst decimal (fortheheavier isotopes theatomic weights arenotyet precisely known), weobtain fortheenergy available from onegram-atom 0.1. =9-10" g-cm’-sec* =9-10" joules. Computed forakilogram offissioned uranium 235 itis1000/235 times as much, or38-10 joules. Wetransform thisintoheat units (onelarge calorie &4.2-10* joules) andobtain 38mie 10”cal©10”cal. +A.Einstein, “Does theinertia ofabody depend onitsenergy content?”, Aun. Physik, Vol. 17,1905. Einstein here explains thespecialization ofEq. (7a) tothe equivalence theorem (8)byanimaginary experiment:amovingbodyemitsradiation and isobserved from asystem atrest. 1H. Bethe, Phys. Rev. 47,683, 1985; Oliphant, Kempton, and Rutherford, Proc. Roy. Soc. London 14, 406, 1985. The almost simultaneous publication ofthese two papers onthe two sides ofthe Atlantic shows once more the inevitable course of development ofthe understanding ofphysics asprescribed bythe experimental material available atthe time. 266 THEORY OF RELATIVITY AND ELECTRON THEORY 32.9 Ifwenote that theenergy transfers ofordinary molecular processes liein therange from 100to1000 calories, weseethat oururanium process sup- plies many million times asmuch energy. Onthisbasis wemay understand both the terrible effect ofthe uranium bomb and the beneficial effect ofthe uranium engine, i.e.acontrollable, continuously operating uranium process, which could remove alleconomic illsofthetimes. The fact that theprac- tical realization oftheuranium process differs from that here considered, i.e.that itiscarried outbyway ofatransuranium element (plutonium), does not require mention. The validity oftheproof indicated byour simplified process isnot affected thereby. B.Relationship between Momentum andEnergy Inclassical mechanics thecomponents ofmomentum arederivatives of thekinetic energy with respect tothevelocity components, e.g.foran individual particle in’Cartesian coordinates: orGame, T=St+ ata) withm =const. (9)Oa 2 This nolonger applies inrelativistic mechanics. Itmay readily beverified however that therelativistic momentum components (1a) arederivatives of thefollowing quantity: , K==me'V/1 —#+const. (9) with respect tothe.Following Helmholtz,’ afunction which accom- plishes thisiscalled a“kinetic potential”. If,again, Kisnormalized so that itvanishes for6=0,theconstant must bechosen equal tomac’, yielding K=me(1 —VI=#). (9b) Hence thedefinition ofthemomentum ofanindividual point mass replac- ing(9)becomes ok mot, Oeoi VTBF (00) inagreement withthedefinition (1a).Forc> Kevidently passes over into Tand (10) into (9). C.ThePrinciples ofD’Alembert andHamilton What aretheconsequences ofthischanged meaning (10) ofthemomen- tum coordinates forthegeneral principles ofmechanics? Weshall first discuss D’Alembert’s principle. Theinertial reaction forcesintroduced byD’Alembert (seeVol.I,Eq.(10.1)) arealsonowgiven by—G,. (The 1Inhisgeneral studies ontheprinciple ofleast action. 32.128, RELATIVISTIC MECHANICS 267 usual definition asmass Xacceleration isofcourse now invalid). The statements ofD’Alembert’s principle inVol. I,§10then continue toapply literally: “The inertial reaction forces balance themselves against the physically impressed forces” (Vol. I,p.57). “The sumtotal ofthelost forces isinequilibrium onthesystem.” (Vol. I,p.58). The condition on p.49ofVol. Iserves asdefinition ofthe word “mechanical system:” “The virtual work ofthereactions within thesystem isequal tozero.” Hamilton’s principle isderived from D’Alembert’s principle inthe manner of§33 ofVol. I.Here 7istobereplaced byKand, forforces possessing apotential, (33.12) ofVol. Iisreplaced by 4sfK-vaqo. (11)te Here thevariation istobecarried outasintheearlier example: The space coordinates arevaried, whereas theendpoints ofthepath andthetime for itatransversal remain fixed. Application toasingle point mass yields 4 in 4 afKar=sfmo(t—Vi=B) dt=-8fmochVT=Fal.(118) to te te .Here wehave already taken account ofthefactthatthetimes &and ‘are nottobe'varied, i.e.that 5(t,—&)=0.Hence (11) becomes 4af(mcs/T =B+)dt=0. (12) . We can readily convince ourselves that this variational prescription agrees with ourequation ofmotion (5),and this notonly fortheLorentz force K,towhich (5)was limited, but forany given potential energy and anarbitrary force K=—grad Vderived from it. Inthevariation wemust replace x,y,zbyx+62,y+dy,z+82,ob- taining dz_dbx ev _ weeSeta: |Wa5peet =—Kebe , where the«--represent corresponding expressions inyandz.Similarly we must form 1ffdzdy][% bV1-#=5-=<(|/ = a a ine ane(a+[#]tla 1 2) fardie“avi-wl ella at" and hence fh ts mo dxdix2/1— = aoe eea 8ffme1—#dt|Apa at}ata2 268 THEORY OFRELATIVITY ANDELECTRON THEORY 32.12b Anintegration byparts, inwhich theterms without integral sign vanish (because dx=0fort=tandé=4,)according toouroriginalassumption, transforms this into fd m dx[alvetpa]= +} ) Thus (12) yields altogether aC d mo dx’[iP w]e+Sane (120) Since 52,éy,and 6zareindependent ofeach other thefactor of6xmust vanish, aswell asthose of5yand éz.Wethus obtain infact our earlier equation ofmotion (5)forarbitrary K;itisvalid, incidentally, even ifK cannot bederived from apotential energy. Ifthere arenoexternal forces (V=const) (12) may beabbreviated to ‘ a sfVirRaas["arno. (13) te ro This isFermat’s principle ofleast time, which now however does not felate totheconventional time¢,buttotheLorentz-invariant intrinsic time. Since drcorresponds tothefour-dimensional line element dsbut for thefactor ic,wecanalso write inplace of(13) xsfao (13a) This istheprinciple oftheshortest path forgiven starting point Aandend point E,or,aswecalled itinVol. I,Eq.(37.14), theprinciple ofthegeo- detic line, extended tofour dimensions and made Lorentsz-invariant. We shall therefore callitmore precisely theprinciple oftheshortest world line. D.Lagrange Function and Lagrange Equations Inourformulation (11) ofHamilton’s principle therelativistic Lagrange function La=K-V (14) replaces theclassical Lagrange function Ly;=T—VinVol. I,Eq.(33.13). The general Lagrange equations forarbitrary position and velocity coor- dinates arederived from Ly. bycarrying outthevariation prescribed in (11), just.as they arederived from Ly:in§34ofVol. I: CL —Les’ aoe~an7° (14a) 32.17b RELATIVISTIC MECHANICS 269. Inspite oftheir similarity with those ofclassical mechanics, these equa- tions, when applied tospecific cases, yield results which differ decidedly from those ofthelatter. Forexample, intheKepler problem ofthehydro- genatom they lead toanellipse with precessing perihelion instead ofto4 closed ellipse asaconsequence oftherelativistic variation ofmass; see also §38regarding theperihelion ofmercury. E.Schwarzschild’s Principle ofLeast Action Inhisfundamental papers “(OnElectrodynamics” Schwarzschild’ intro- duced with thedesignation “electrokinetic potential” thequantity L=Wv—-vwaA. (15) Weshall show that when multiplied with thecharge density pthis isa relativistic invariant. Tothis end weform thescalar product ofthefour- vector Iofcurrent density (Eq. (28.16)) and thefour-potential Q(Eq. (26.4)). Weobtain T-Q=p(v-A—W). Wecall—I°-QtheSchwarzechild invariant. Inviewof(15) r-Q =—pL. (16) Schwarzschild adds tothis invariant the Lagrange density Afrom (26.24) which, weknow, isalso Lorentz-invariant, and forms, with T= kinetic energy, T—A-— pb. (17) Weshall replace (17) by K’=7’—2A—pL=T’—-2A4+1T-aQ. (17a) Here T”represents therelativistic value ofthekinetic energy from (7), where, however, therest mass myistobereplaced bytherest-mass density iw.(Also theremaining terms in(17a) aredensities, referring tounit volume.) Hence T= we((l —#4 —1). (17) The factor 2ofAin(17a), ontheother hand, derives from ourbasic dis- tinction between theentities ofquantity fand ofintensity F;Schwars- schild, whoputsD=EandB=Handhence writes inourEq.(26.24) 1K,Schwarzschild, Géttinger Nachr. 1903.Seeinparticular thefirstofthethree. papers. The notation Listhesame asSchwarzschild’s; Schwarzschild uses ¢inplace ofour¥.Note thedate ofpublication 1903! Thus Schwarzschild arrived intuitively atthecorrect postulate ofthetheory ofinvariants sixyears ahead ofMinkowski. 270 THEORY OFRELATIVITY AND ELECTRON THEORY 32.18 H!—E*instead ofH-B—D-E,gainsafactor 2inthevariation, which wetinast supply in(17a). Indetail ourformula (17a) becomes K’=we'((1 —#4—1)-244+ 7-Q. (18) Fromthispointonwefollow Schwarzschild’s procedure. Heintegrates(17)over anarbitrary region ofspace-time andconstructs inthismanner anaction function W,which hesubjects totherequirement 5W=0.We form correspondingly W=fffwexayaca (19) and also set -W=0. (19a) According toSchwarzschild thisvariation istobecarried outinfollowing fashion: a.Thecomponents 9;,%,2,and%ofthepotential and ; b,thecoordinates z,,22,x,and2oftheelectrons aresubjected to arbitrary small variations; these variations aretovanish ontheboundaries oftheregion. Thevariations aandbareindependent ofeach other’ and «may becarried outindividually, e.g.also foreach component ofQ.If ‘several electrons arepresent wecanlimit ourselves tooneofthem since theeffects oftherestonitarecontained inthepotential Q.Thefactthat weusethefour-potential Q,originally introduced forconvenience ofcal- culation, rather than thesix-vector Fasfundamental fieldquantity repre- sents anew departure, towhich weshall return in§37. a.Since thefirstterm ontheright ofEq.(18)isindependent of@we areonlyconcerned with AandI°-Q. Ifthevariation islimited to6Q,we find a(r-Q) =Tem. (20) Intheexpression (26.24) forAwemust imagine Fasexpressed (by(26.11)) by¢CurlQ,whereasfistoberegardedasanunknown. Hence,forthe specific variation mentioned above, (—2A) reduces tothefollowing three terms (theremaining terms oftheCurl, tobeformed with%,0%,and %,drop out): 2A)=fq28M4p80,pBIESoffe] . 8(—2A) faz,thage,thus dntan+anB+ ++. 'Schwarsschild does nottake secount oftheauxiliary condition DivQ=0which wassatisfied automatically inourmethodofintegration in§29.WeadheretoSchwars-schild’s prescription also inthis respect. 32.22 RELATIVISTIC MECHANICS 271 The dots refer topartial derivatives with respect tothecoordinates which vanish inthelaterintegration overourworld region (since 6Q=0onits boundary). Together with (20) wethus find forthefactor of49,inthe integrand of(19) _[afe4as,au[i++ Be]+n. Itmust vanish since SW =0.Ifwesubstitute forfxthevalues inthe array (26.14) and forI"itsvalue from (28.16) wefind ~The4a4+one=0. 1) This isexactly the first component ofthe three Maxwell equations D+J=cwlH, where here,forvacuum, theconvection current density pvrepresents J.Thesecond and third component areevidently obtained similarly bythevariation of%and 9and thecondition div D=pfrom that of&.Wecannot, ofcourse, expect toderive theother setofMaxwell equations and thecondition div B=0inthesame manner, since these arealready implicit intheexistence ofthepotential. This clarifies also thebasis forourearlier name “Lagrange density” for ‘A(seep.220): Inourelectrodynamical variation principle Atakes the place oftheearlier Lagrange function LorL,.1 (Eq. (32.14)). b.The potential Qisnotvaried; hence A=0. Ontheother hand, theworld line oftheelectron istobecompared with neighboring world lines, sothat thefirst term ontheright of(18) and the Schwarzschild invariant '-Q are tobevaried. We shall first deal with the term I'-Q. Itishere convenient toreplace the world volume element dzdydzdt=dVdtin(19)bydV,dr,wheredV,represents itsthree- dimensional cross section perpendicular totheworld line. Since thecharge density poccurring inIisconcentrated ontheworld line oftheelectron weobtain inthe integration over dV, (not inthat over dV!) theelectron charge e. Atthe same time, inthe expressions forTand Q,wepass from the coordinates 2;---<4used sofartothecoordinates §---&oftheworld line element considered atthemoment (d& =icdr,dr=element ofthe intrinsic time). Then dé;/dr replaces dz;/dt and weobtain [fffv-aav.ar =efDBaa. (22) Wemustnotethatinthevariation notonlyé;ischanged by4£;,butalso Q;ischanged by 30; 80,=Oi(s$fBE,«+EeBE)—(Er,+BD=aeoe 272 THEORY OFRELATIVITY ANDELECTRON THEORY 32.228 (thecharge¢isofcourse conserved). Hence (22)leadsto . = ty od;09;:) afffr-aaviar =ef(z=BE0,+3EFts)dr.(220) Thefirstofthetwoterms ontheright istransformed byintegration by parts andyields (since 8;=0ontheboundary oftheworld region) -ySe, =- 205dee5ee Thus (22a), after interchange ofthesubscripts 7,jinthedouble sum, becomes a[30_20),g) «{(BEES -x])« @)Herewehaveinthe.‘parenthesis Curl,,Q,ie.exceptforthefactorcthe component Fj;ofthefield(see(26.11)). Ontheother hand, wehave, by (28.17) Y-F =ck —(k=force density) (24) and by(28.19) B¢ av. =Fay K=Lorents force). Hence ourexpression (23),which wasobtained bycarrying outtheinte- gration over V,,signifies simply dr [ka Zeer [Pwaea (25) Wemust addtothisfrom thefirst term oftheright side of(18), ifwe again putdVdt=dV,drandintegrate overtheworld linecross section avy: me's[(1—BY=1)dr=mes[U-VI=PamafKae Wehave already carried outthevariation ofthisintegral overKinEq. (Ila) andthesucceeding equations. Wefound there, translated intothe present notation =;oftheworld linecoordinates (see(12b)): _fr?(—™#) [Save asat 28) ‘Together with(25)weobtain asvariation oftheaction integral =- d[__mo _dé)_ ; w=[=‘late S|xh8g;dt.(27) 33 ELECTROMAGNETIC THEORY OFTHEELECTRON 273 Werequire that this integral should vanish forarbitrary displacements. This isonly ‘possible ifthe{}vanishes forj=1,2,3,4.Inthismanner wehave derived ourearlier Eq. (5)including thecorresponding fourth component, and this inmore explicit form: Our present derivation yields notonly this equation ofmotion, butalso theLorentz forceimpressedon. theelectron bythefield. Schwarzschild’s principle ofleast action thus com- bines Maxwell’s electrodynamics and theLorentz electron theory inasingle Sour-dimensionally invariant formulation. Fromahistorical pointofviewitmaybenoted thatSchwarzschild, starting with thekinetic potential (17), alsoobtains theMaxwell equations andtheequation ofmotion oftheelectron including theexpression forthe Lorentz force. Only thevariation ofmass oftheelectron escapes him, since, in(17), heemploys theclassical value ofthekinetic energy. Itistrue that hisderivation ofthe"Maxwell equations isnotquitecorrect fromour point ofview because ofthemissing factor 2inA,which iscompensated inSchwarzschild’s treatment byputting fand F(D=E,H=B)equal. Inourrepresentation theproportionality offand Fisalso contained in theSchwarzschild principle. Itisonly necessary toeliminate Ifrom the Eqs. (21) and (26.5), which wecanwrite ‘. Divf=T and DivF=cwF. Schwarzschild’s action principle isvery suggestive. Itcould bemade thestarting point ofthetheory andtheMaxwell equations beregarded as itsconsequences. There would beatthesame time theinviting possibility ofrefining theMaxwell equations byextending thekinetic potential (18) (addition ofother field invariants, taking account ofinteractions between theelectrons, their magnetic moment and spin). Wewillenter upon such questions in§37. §33. Electromagnetic Theory oftheElectron Attheturn ofthecentury interest was focused onthevariable mass of theelectron. The assumption oftherigid electron, which appeared appro- priate inthetheory oftheabsolute ether, ledtoadifferent, much more complicated lawoftransformation (Max Abraham) than Lorentz’s assump- tion ofthedeformable electron, which soon afterwards attained anassured basis inthetheory ofrelativity. Theexperiments ofKaufmann, Bucherer, Neumann andmany others were concerned with thislawoftransformation. The theoretical treatment oftheproblem (also fortherigid electron) Tested onthedefinition (31.13) oftheelectromagnetic momentum. Without detaining ourselves with therigid electron weshall show that thesame~ Starting point, with therelativistic treatment ofthemomentum, leads to thesame law (32.2) ofthevariation ofthemass asthetheory ofrela- tivity, which however extends itimmediately toanyarbitrary mass m. 274 THEORY OFRELATIVITY ANDELECTRON THEORY 33.1 Weshallobtainasaby-product aninteresting formula fortherestmass moofthe electron. Below weunderstand byGthetotal momentum ofthefield ininfinite space; weshall callthemomentum perunit volume, designated byGin (31.13), g.With dVasthree-dimensional volume element wethen have 1 6=fea=4|sav. (1) Tobeable tocarry outtheindicated integration weutilize theideas and symbols ofEqs. (28.12). Let2,y,zbetheframe ofreference moving with theelectron, 2’,y’,2’acoordinate system atrest, with respect towhich theelectron hastheinstantaneous velocity vinthepositive z’-direction. Inthez,y,zsystem wethen have ofcourse G=0;thefieldiselectrostatic sothat H=0and S=0.We areinterested inthe momentum G’and more particularly initsz-component: . YLfay 2)fet —eeap Gahfsav=3faim-eimav. @) Weexpress theprimed quantities interms oftheunprimed ones inaccord with Eq.(28.12a), inwhich however, inview oftheopposite direction of motion, thesign’of»must bechanged: ’ ’ 1 , 1 E.=E, E= 7Bey E, aRragca) a a aHi=0 WeDap eeHywatpevia ee av’=dVV/1 =# (Lorentz contraction). (2b) Thus weobtain from (2) ! 9 2 0=aay |+BDar. @) Inthexyz-system theE-field isspherically symmetrical, sothat 27 o207=2aolf usav=fwav=[etav=}fwav. (3a) ‘Thesameapplies forthecharge distribution. Itseemsmostnatural to spread thecharge euniformly over asphere ofradius a(“radius ofthe electron”). Then, asfollows e.g.from (7.6a), 0 for r<a E=E,=) ¢fi > (3b)iron joroor2a 33.7 ELECTROMAGNETIC THEORY OFTHEELECTRON 275 and hence 2 0 2= ttre 2fee [Bana faremalS-sky (3e) (3)and (3a) then lead to eeee peee OeCLOVIS RB4ecla~GradaViae @ Wemayalsoreadily convince ourselves that, G,=G=0 (4a) asmust beexpected forthespherical symmetry inthexyz-system. For ifweform . rl fern, 1 ha mtay oy@=5fsav=4fwin:-Ba)av inanalogy to(2)andagain make useofEqs. (2a,b), . vGi=4[Bz,av=0 since foraspherically symmetrical field E,andE,areproportional toz andy,andzyintegrated over thesphere vanishes. Eqs. (4)and (4a) canbecombined to m . re = G=m, mWoe (5) Themass factor mhereintroduced hasthus thedependence onvelocity familiar tousfrom §32.Fortherestmass mowefindfrom (4): 2é ™=rae’ (6) ‘The reader may check thedimensional correctness ofthis formula, i.e. theindependence ofthechoice oftheunit ofchargeQandtheunitof length M.Thefactor eo,which intheGaussian system issetequal to1, isfrom ourpoint ofview indispensable. Ifitissuppressed theformula becomes dimensionally meaningless. With thevalue oftherestmass computed by(6)Eq.(5)states: The mechanical momentum oftheelectron isequal tothemomentum contained in theelectromagnetic field asdefined byEq.(2): ~ Getectron =Grieta- (7) We read directly inEq. (6)that thetransition tothelimit a>0is unfortunately impossible; itwould lead tomp—©and ¢/m» —0.To 276 THEORY OF RELATIVITY AND ELECTRON THEORY 33.8 determine thenumerical value ofa,wemust know theexperimental values of¢and e/mo. InMKSQ units, with Q=1coulomb, these are: e=1.60:10°"Q, —e/m =1.76-10"Q/K. (8) From this wecompute ‘my=0.9-10-°K, (8a) Eq. (6)then yields, with (7.18a), 2 @_2&MK_2(1.60-10)?,) 4anu °=3iradm ~31mG3ogee MSP %em©) This isasubatomic dimension, ofthesame order ofmagnitude asnuclear dimensions. . Itisofcourse quite arbitrary that wehave here assumed asurface charge. Wemight equally well have distributed theelectron charge euniformly over the electron volume. Ifwethen callitsradius once more a,wefind’ instead of(3b) (_e=; for rsa4rea*pez, ~("" (10) {_¢Fer for r2a and instead of(3c) _@fftrtdr ©dréfi [ew-2hetl Sheele tyom The factor 6/5isthus tobeadded totheformula forG’in(4)sothat we obtain inplace of(6) 2G mo=ae (10b) The order ofmagnitude ofthevalue ofafound in(9)isnotaffected. The following remark isofgreater importance: Who canguarantee that theMaxwell equations canbeextrapolated right uptothesurface orinto theinterior oftheelectron? May nottheir simplicity and linearity bea consequence ofthefact that they areexactly valid only forweak fields andthat they must becorrected, intheimmediate neighborhood ofcon- centrated charges, byhigher terms, insome such manner asthetheory of 1The first lineof(10)evidently follows from thefact that elements ofcharge whose distance from thecenter isless than rmay bethought ofasconcentrated at thecenter ofthesphere, while those distant bymore. than rfrom thecenter donot contribute tothe field strength. 33.128 ELECTROMAGNETIC THEORY OFTHEELECTRON 27 dilute solutions inthermochemistry? Weshall return tothisquestion in §37. Itwillhere merely beemphasized that thederivation ofthelaw governing thevariation ofmass with velocity isnotsubject tothiscriti- cism, since, in§32,itcould bederived fromthegeneral principles ofrela- tivistic mechanics, whereas ourpresent computation ofmoisaffected; the latter isanyhow beyond experimental verification, inview ofthehypo- thetical character oftheelectron radius. Thederivation ofthemass-veloc- itylawin§32is,likeallconsiderations ofthespecial theory ofrelativity, only tiedtothecondition thattheoccurring relative motions should be nearly uniform. Weexpress this here bythedemand that theelectron motion bequasistationary. Wemean hereby thatitsvelocity change in thetimetakenbyalightwavetosweep overtheelectron (i.e.thetime 2a/c) besmall compared tov.Wethugdemand only; + Ke, an) Allprocesses invacuum tubes satisfy thisrequirement. With respect toformula (6)fortherestmass wenote furthermore that itmaybederived inthefollowing very elementary manner: Weconsider slowly moving electron. Itsmass isequal totherestmass moandits Kinetic energy ‘ m™ pTez. (12) Ifthisisofelectromagnetic origin wemust setitequal tothemagnetic energy ofthefieldsince theelectric energy isconstant forsmall fields, i.e. notproportional tov*.Wehence put wo fwT=2[wav (12a) Here wecansubstitute forHthevalue (15.12) from thelawofBiot-Savart evsin& H=Rr Wethen obtain forsurface charge * * te =m(2)fe[sw [ T#(2),ehSmeds|de. The three integrals are, insequence, 14 aR2x, 278 THEORY OFRELATIVITY AND ELECTRON THEORY 33.13 Hence . 22 =wevlTSa (13) Comparison with (12) yields 2 =mel ™Ora which isidentical with (6)since equc’ =1. Since thekinetic energy oftheelectron computed with (6)proved to beequal tothemagnetic energy ofthesurrounding field wemay suspect that itsrestenergy willcorrespond totheelectrostatic energy oftheCoulomb field. Inthesimple case ofsurface charge wefind that this isequal to &fpe & ae Eu=SfBay=Bae[Etter and obtain inview of(8c) 2é Eva=fea’ (14) Ancontrast tothisEq. (6)yields fortherestenergy ofourelectron by ‘Einstein’s lawofequivalence ofmass and energy: 2é Ey=moc’=Sraa’ (15) Thus only %ofthis restenergy isexplained electromagnetically byourpre- ceding (admittedly primitive) considerations. The program indicated by thetitle ofthis section isasyetincapable ofrealization. Aswas already said onp.236, theelectron isastranger inelectro- dynamics. The forces which, opposing the Coulomb forces, prevent its explosion areunknown tous,justlikethetheory oftheelementary particles ingeneral. Poincaré introduced (asearly as1906, intheRendiconti di Palermo) acohesion pressure ofunknown origin which wassupposed to envelop theelectron atrest like »membrane under uniform tension; the missing quarter oftherestenergy wassupposed tobehidden herein. The hypothesis ofrigidity oftheabsolute theory could transfer this cohesion pressure totheelectron inmotion. Itdidnotsuffice, however, forapurely electromagnetic description oftheelectron. Even theassumption ofrigidity contradicts thegroup-theoretical nature ofMaxwell’s electrodynamics which, asweknow, demands thedeformable electron ofLorents. Altogether, weshould facethefactthat ourelectrodynamic theory of theelectron isasyetvery incomplete. Wehave known for20years that theelectron possesses inaddition toitscharge aquite definite spin and8 33 ELECTROMAGNETIC THEORY OFTHEELECTRON 279 quite definite magnetic moment. Both can only bedefined onthebasis ofthequantum theory and areinaccessible toMaxwell’s electrodynamics. The secret ofthespin was first discovered inthemore precise analysis of theZeeman effect; thesecret ofthemagnetic moment wasactually, aswe know now, clearly and tangibly demonstrated inferromagnetism. Itis strange that practical electronics remained untouched bythese fundamental facts andcould getalong with thenotion ofthecharged point mass orthe minute charged sphere. OurProblems III.5 toIII.10 deal with thisapplication ofelectron theory. The varied electron trajectories which occur invacuum tubes andwhich, inthee/m experiments, first served toclarify thenature oftheelectron areatthesame time inaway thesimplest and best defined examples of themechanics ofanisolated point mass. Part IV MAXWELL’S THEORY FOR MOVING BODIES AND” OTHER ADDENDA §34. Minkowski’s Equations forMoving Media The extension ofMaxwell’s theory from media atrest tothose inmotion wasafavorite problem oftheolder electrodynamics. Heinrich Herta had failed inthis effort (see hispaper cited infootnote 2onp.2)because he adhered consistently toclassical theory (the ‘“Galiler transformation”). Hisfriend Emil Cohn’ came closer tothegoal, butwasnotyet(in1902!) inpossession ofthenecessary tools, the Lorentz transformation. Even H.A.Lorentz didnot quite attain the final form inhispapers inthe Enzyklopidie (1903), particularly notformagnetizable bodies. Einstein called hispaper of1905 “On theelectrodynamics ofmoving bodies” and indicated inthismanner aprincipal goal ofhistheory ofrelativity; how- ever hedoes not enter upon thegeneral structure oftheequations for ponderable bodies butconfines himself instead tothequestions arising for theisolated electron. Minkowski, in1908, atlong last infullpossession of theprinciple: ofrelativity, wasthefirsttosolvetheproblem completely.” Minkowski’s logic was simple: The Maxwell equations forastate of rest apply within thelaboratory. Consider apoint ofspace-time Pofa body moving’ withrespect tothelaboratory atthelaboratory time ¢;let ithave thevelocity v.LetPbetransformed torestbytheintroduction of thecoordinates 2’;y’,2’,¢’forthedescription oftheprocesses inthe neighborhood ofP,¢.Inthissystem Maxwell’s equations forastate of restapply tothequantities E’,B’,D’,H’,J’,p’: a Be-cune, M4y=uk, © divD'=, divB’=0, with material constants differing from those forvacuum: D=ecE, Bo=,H', Ji=oF'. (2) 1Géttinger Nachr. 1901, p.74;Ann. Physik 7,29,1902. 2Géttinger Nachr. 1908, p.53;Geeammelte Werke II,p.352. +The motion may bevariable inspace and time and must merely becapable of quasistationary treatment inthesense ofoq. (33.11). Thus vneed notbeapure trangJation andthebody need notberigid. Only thefixed value ofvinthespace-time point P,tenters inthefollowing Lorentz transformations. 280 34.5 MINKOWSKI’S EQUATIONS FORMOVING MEDIA 281 These constants have thesame values asifthebody were atrest with respect tothelaboratory, since itknows nothing ofitsmotion. Theopera- tions curlanddivin(1)refer ofcourse, justlikethetime¢’,totheprimed system. Now theinverse Lorentz transformation istobecarried out,which transforms theprimed system back intotheoriginal oneofthelaboratory. Inthelatter Eqs. (1)apply once more ifallprimes areomitted, inview of thebasic property ofcovariance oftheMaxwell equations withrespect to theLorentz transformations. However, Eqs. (2),transformed totheun- primed system, takeonanewform. Weknow therelationship oftheE’,B’andtheE,Bfrom Eqs. (28.8a) and (28.11): E+vxB B=(@+vXBy,B=lesll . 1 @) Bi,=(B-4vxz) Bi.={P7avX® ; ¢ Wy Vi-B ja ||and«.signify asbefore “parallel” a.d“perpendicular tothevelocity v”. Weshall supplement thisbythecorresponding relations between D’,H’ and D,H.Inview ofthedefinition ofthesix-vectors ‘f=, -tD), F=@B,-i®) they areobtained from (3)byreplacing EbycDandBbyH/c. Wethus obtain 1 1D+5 H Di,=(D+vx) =|tarxL Tf Vie (4) ’ ’ H-vxDHy=(H-vxXD)i, Higeseale Substitution of(3)and(4)in(2)yields, forboth theparallel andthe perpendicular components forwhich thedenominator cancels onthetwo sides, D+4vxH =cE+7XxB) 1 () B-3v XE=a-vxD). Here Bmay, forexample, beeliminated inthefirstequation bymeans of thesecond, sothatDisexpressed onlyintermsofEandH;similarly ‘Here wemake useofthetransformation AX(BXC)=B(A-C) —C(A-B) andoftherelation eqsc* =1.Itshould benoted thataccording to(5a,b)theidentity indirection ofDandEaswellasthatofBandHbasceased toapply even forthe isotropic medium. 282 MAXWELL’S THEORY FOR MOVING BODIES AND OTHER ADDENDA 34.58 elimination ofDleadstoanexpression ofBintermsofEandH,The resulting equations become simpler ifthey arewritten separately forthe components ||and1.They then become Dy=eEy, By=wy (Ga) Da=e(1—6)Es+(eu—cm)vXH, p-e] * aTSalty ”(6b) fone”J(Ba=wll—B)Hs+(Come—envXEB. Having taken care ofthefirst two Eqs. (2)wenow turn tothethird Eq. (2),“Ohm’s lawformoving conductors”. What istherelationship of J’and J?Weknow from Eq. (26.6) that Jisthespace component ofa four-vector I,whose time component isicp.We also know that every four-vector transforms itself like thecoordinate vector 2:,22,1s,%4.We therefore have forthespecialized Lorentz transformation (v||x): Js— p-Sde Va Fa I I =e.vi-# ji—# Foranarbitrary direction ofvthisbecomes 1 J= ev ’ e- avyn-S »thats, gah8" &@Vi- Bi Vi-# where wemay also write themiddle equation intheform Ju= J vs, (6a) since bydefinition v.=0. With this meaning ofJ’and themeaning (8)ofE’ourOhm’s lawbe- comes j=.)=~ =) =coE xBy,(SAS), =ob+exDe © (0—pon=o(EBova=0ae These twoequations canalso becombined into asingle one, though only inasomewhat artificial manner. Wehere make useofthefollowing nota- tion, which iscustomary also elsewhere intheliterature and will beuseful latern:* E*=E+vXB, H*=H-vxD. 8) 'The *here employed ofcourse bears norelation totheearlier *ofthedual six- vector 34.98 MINKOWSKI’S EQUATIONS FORMOVING MEDIA 283 Wecanthenwriteinplaceof(7) «V(¥ge J-weo™ £(;#) (9) vi-F Since ».=0theperpendicular component ofthis isidentical with the second Eq. (7).Furthermore, since v-E* =vE,, theparallel component of(9)is G vv o O=won=ecglahF-bat]=pare -O, which agrees with thefirst Eq. (7).Wecall i=J-pv (9a) the “conduction current”. Eq. (9)expresses thefact that theconvection current pyandtheconduc- tioncurrent J;aresuperposed andthat their differentiation depends onthe frame ofreference oftheobserver. The reason forthis evidently rests inthe four-dimensional combination ofJandpvinthefour-vector I’.Just asfor ,thesix-vector Fthedistinction between itselectric andmagnetic aspect depended onthereference frame oftheobserver (seep.241), achange in thereference frame now adds thetime component icpoftheIvector and thecorresponding convection current pvtotheconduction current Jy. The former, likethelatter, produces amagnetic field. This conclusion was contained already intheRowland effect discovered in1878. Since wearehere dealing exclusively with charge inmotion and since therefore theconduction term in(9)islacking, theconvection cur- rent pvalone ismagnetically active and takes theplace ofJintheap- propriate Maxwell equation. Thequestion naturally arises whether alsotheso-called “free charge”,! which occurs atthesurface ofahomogeneous dielectric inanelectric field, ismagnetically active when thedielectric issetinto motion. This led Roentgen tohisfundamental experiment:’ Adielectric plate isplaced ina ‘We have avoided this notation elsewhere (like Rémtgen, who expressly desig- nated hisdielectric plate asuncharged) since the“free charge” isnotacharge dimen- sionally, but #divergence offield strength (see p.40), inour case asurface di- vergence oftheelectric field strength. 2W.C,Roentgen, Ann. Physik Vol. 35,p.264, 1888. Inasupplement tothispaper Roentgen reports thenegative result ofanexperiment with arotatably suspended condenser soorionted with respect tothemotion oftheearth that the“ether wind” passed through thecondenser plates. Does thisether wind generate amagnetic field and, asaresult, adeflection ofthecondenser? From ourpresent relativistic point ofview thenegative result oftheexperiment isaforegone conclusion. Asimilar, refined, arrangement became famous atslater date inthe Trouton-Noble experiment, 284 MAXWELL’S THEORY FOR MOVING BODIES AND OTHER ADDENDA 34.10 plate condenser, parallel iotheplate electrodes, andismoved perpendicu- larly tothelines offorce inthecondenser (itwas rotated about anaxis normal tothecondenser plates intheexperiment). Does thismotion pro- duce amagnetic field? Roentgen could answer this affirmatively and Lorentz, asaresult, named thecurrent equivalent tothemotion the Roentgen current. Inagreement with later experiments and considerations ofEichenwald' themagnitude ofthiscurrent fortheexperimental arrange- ment inquestion is: R=v(e—&)|Eo]=v|Pol. (10) Theplate ishereassumed tobeunmagnetic (4=40),itsmotion aparallel displacement v;Eyisthefield strength inthecharged condenser andPp thecorresponding polarization oftheplate, both referring totheplate at rest, asindicated bythesubscript 0.Since the“free charge” isconcen- Condenser electrode Fia. 43.Explanation oftheRéntgen cur- rent. Section perpendicular tothe direction ofmotion ofthe dielectric plate and the Ey gam: © condenser electrode. Thelocation ofthe Réntgen current isthesurface ofthe plate. ‘The portion linked byarectangular loop a,Z—— 5,a,bisindicatedbyaheavyline. Dielectric plate trated onthe surface ofthedielectric plate, the Roentgen current isalso @pure surface current: Itisabsent both from theairgap and from the interior oftheplate and occurs only attheir irterface; itsdirection isthat ofv,just asfortheRowland current. Weshall show that (10)follows from (5a,b)if6°=(|v|/c)’ isneg- lected (which isofcourse fully justified under theconditions oftheexperi- ment) and iffurthermore »=jo,and ontheright thevalues ofEand H fortheplateatrestaresubstituted, namely Ei=Ey,Ey,=0,H=Hy)=0. We then find Dy=By=0, D=Di=e&, B=Bs=wl—e)vX EB. Fig. 43represents asection normal tov(¥visdirected into theplane of the paper) inwhich the shaded portion below indicates the dielectric plate, theupper portion, theairgap ofthecondenser. We compute the line integral ofH=B/yo about therectangular loop which has been drawn, thedirection oftheintegration beingrelated tothedirection ofy byaright-handed screw motion. Inview ofthedirection ofvXEy,H has thedirection ofthearrow ontheupper side oftherectangle a,but 14. Fiebenwald, Ann. Physik, Vol. 11,pp. }and 241, 1903. 34.10a MINKOWSKI'S EQUATIONS FORMOVING MEDIA 285 vanishes onitsince¢=&;thesameapplies forthesidesboftherectangle.Thus there remains onlythelower sideaoftherectangle, which istrav- ersed inadirection opposite toa.Ityields fHas=—a(e&—evXEy. Thismagnetic circuit isequal tothesurface current flowing through its interior, which inthefigure isindicated bytheheavy linethrough the middle oftherectangle. Itisa-Rifwecallthesurface current perunit length R.Wethus obtain fods=~0R, R=(e—e)v|Eo| =v{Pol, (10a) where bythevectorial symbol vwealsoindicate thepositive direction of R(Pointing intotheplaneofthepaperlikev).ThusEq.(10)isverified: The experiments ofEichenwald inwhich thedielectric plate andthe twocondenser plates were rotated about their common normal asaunit, sothat theconvection currents v|Dj| ofthecondenser plates (surface tdensity w=.|D/|) areadded totheRoentgen currents v|Po| onthedielectric plate,areofspecial interest. SinceDandPdiffer,aresidual magnetic field arises here also, contrary toHertz’s earlier theory andin spite oftheopposite signs oftheRowland andRoentgen currents. (The signofthecondenser charge isopposite tothat ofthecharge ontheplate induced byit.)Eichenwald (onp.331) states expressly regarding this residual field: “The magnetic effect isindependent ofthematerial ofthe dielectric.” Infact, D—P=eEisthevacuum component ofD,for which theterm “dielectric displacement” isnotparticularly appropriate, butwhich isvery characteristic forMaxwell’s theory and itsoptical application. Wefinally want tomention 2kind ofinversion ofRoentgen’s experi- ment, theexperiment ofH.A.Wilson’: Ahollow dielectric cylinder is placed between theelectrodes ofanuncharged cylindrical condenser ina uniform magnetic field parallel tothecylinder axis. Ifthecylinder is rotated thecondenser ischarged. Wehave followed Minkowski closely sofarand believe tohave thus even improved ontheclarity oftheotherwise insurpassable representation inW.Pauli’s article intheEnzyklopidie. Weshall now establish contact with H.A.Lorentz’s article intheEnzyklopadie which, initsmathematical formulation, follows thepaper ofH.Hertz (1801) andolder papers of Helmholtz. Tothis end weintroduce forthequantities referred tothe 1Phil, Trans. Vol. 204, p.121, 1904; seealso H.A.Wilson and M.Wilson, Proc. Roy. 80e., Vol. 89,p.99,1913. 286§MAXWELL’S THEORY FORMOVING BODIES ANDOTHER ADDENDA 34.11 laboratory (i.e. theunprimed quantities) ontheright ofEq. (1)inplace ofEandHthequantities E*andH*from(8): B=—curlB*+cutl(7xB), qu) D+J=culHt+oul(vXD). Weshift thelastterms ontheright over onto theleftandtake account of theauxiliary conditions in(1): divB=0, divD=p. (11a) We then can write instead of(11) &+vdivB—curl(vXB)=—curl E*, > (11b)2+vdivD—curl(XD)+J—ov=curlHY. We have encountered the aggregates onthe left already inVol. II (18.7c). There wecomputed foranarbitrary vector Aand asurface ele- “ment dowhich moves with thevelocity v,varying from point topoint, and intheprocess changes size and shape itself, the“A-flux through do” $(Ando)=[244vdivA—curt(wxA)|do.dt at In Here weemploy theabbreviation introduced byLorentz’ A=Asvaiva—cul(xA), (12) The preceding equation then passes into 4(A,do)=A,do (12a)dt = or,forafinite surface c, d .SfAndo=fdade. (12) Wethen obtain inplace of(11b) thebasic form oftheMazwell-Minkowski equations inmoving bodies (viewed fromourlaboratory) given byLorents and Pauli: B=—curl E*, @3) D+J-pv =culBt 'SeeEnzyklopaedie, Vol. V,part 2,p.75,Eq. (5). 34.14 MINKOWSKI'S EQUATIONS FORMOVING MEDIA 287 Their advantage rests inthefactthat they leaddirectly totheintegral form: [Bade=-$E*-ds, (14) Jcade~ftas,C=D+J-m. Attheleftweintegrate overasurface moving with thevelocity v,onthe right, over itsboundary s,thedirection oftraversal ofsandthenormal n ofobeing correlated bytheright-screw rule. Wehere recall footnote 3on Pp.280, according towhich vcanbearbitrary, i.e.oandsbeattached to anarbitrarily moving anddeformed body. Inthismanner wehave arrived ataformulation which isclosely related toouroriginal axioms, Eqs.(3) and(4)of§3,andgeneralizes them greatly. Only inthepresent generalized form dothey dojustice tothefacts, already mentioned in§3,ofthein- duction formoving conductors andmoving magnets. Theboundary conditions formovingbodiesalsofollowfromEqs.(14), inthesame manner asthose formedia atrestin§3.They require thecon- tinuity ofthetangential components ofE*andH*aswellasofthenormal component ofB:Herewemustnotethatthevelocity voccurring inE* andH*istoberegarded asaconstant oftheLorentz transformation, by which thepoint Pofthemoving body istransformed torest. Thus vhas thesame value onthetwosides oftheboundary surface, namely thatin thepoint P,orisatleast continuous inpassing through theboundary. Thesituation isdifferent ifvjumps discontinuously from thevalue 0 (laboratory) tothevaluev(moving solidbody). Weconsider inparticular thecase, important forunipolar induction, that thefield isstationary (a/at =0)andthesurface ofthebody isdisplaced with thevelocity V(v=Uang). Weshall show that then notthetangential components of E*,H*,butthetangential components (asseenfrom thelaboratory) ofE, Hmust becontinuous along theboundary layer. Wenote inpreparation that thetwo conditions which arehere com- pared signify thesame foratangential direction ofEorHparallel tov (because ofthemeaning ofthevector products in(8)), butthatthey are actually contradictory forevery other tangential direction, inparticular that atright angles tov. Letusconsider now, just asinFig. 3,arectangular loop AsAh,which isinitially placed normal totheboundary surface; thisisnowdistorted,_. since thesideAsparallel totheboundary within thebody isdisplaced, whereas theopposite side, invacuum, remains fixed. Then, since 8B/dt =0 anddivB=0,Bis,by(12),equal to—curl (vXB);theintegral onthe 288©MAXWELL’S THEORY FORMOVING BODIES \NDOTHERADDENDA 34.158 leftofthefirstEq.(14)becomes, making useofStokes’ theorem, —fcurl,(vxB)do==f(7xB)-ds, i.e.ingeneral notequal to0asforconstant orcontinuously varying Vv. Ontheother hand, theintegral ontheright ofthesame equation, car- ried outover thesame distorted loop, becomes inview ofthemeaning ofE* ~ftasf(7xB)-ds Equating.the twoexpressions leads totherequirement feds=0,i.e.continuity ofBruns, inaccord with ourearlier conclusion (3.9). The same consideration applied tothesecond Eq. (14) yields (since oD/ét =0anddivD=p) C=we-curl(vXD)+J—pw=—curl(VvXD)+J andfortheintegrals ontheleftandright sides ofthesecond Eq.(14) -$(xD)-ds+fIndoandfHs-$(@XD)-ds,respectively. Equating ofthetwo leads to [eae=fads Ifinthelimit Ah—0thesurface integral overJisputequal tozeroasin Eq.(3.8) (seealsofootnote 1atthat point) weobtain fHas=0,ie.continuity ofHuns. (15a) ‘This closes ourconsideration oftheboundary conditions inthespecial case ofamoving interface between twodifferent media. Theexistence oftheRowland andRoentgen currents attests thefactthat thepreceding theory isnotonlyofimportance forthelarge velocities of thetheory ofrelativity. Thesame follows from theproblem ofunipolar induction which hasbeen famous since thedays ofArago andFaraday— Theliterature onthissubject isvoluminous endbynomeans freeof contradictions, since thisproblem isconcerned withtheexact lawsofthe electrodynamics ofmoving bodies. Weshall discuss thisproblem only 34 MINKOWSEI'S EQUATIONS FORMOVING MEDIA 289 qualitatively here anddefer quantitative considerations toProblem IV.1. Furthermore, weareprimarily interested inthefields which occur here; hence wepass over thephenomena ofmotion, which arerealized inappa- ratus ofmany typos and have claimed most attention inexperimental work, Tf,forexample abarmagnet, suitably supported, isrotated about its axis, induction currents arise inawire ofwhich oneendglides, forexam- ple,onthemiddle ofthemagnet while theother isconnected tothebear- ingatoneoftheends oftheaxisofthemagnet. Since here only themagnet, pole adjoining thisendiseffective, wespeak of“unipolar induction”. This arrangement hasbeen employed notonly inlaboratory experiments, but attimes also onalarge scale inelectric generators. Wesimplify thestatement oftheproblem ifweseparate theconductor from thebody generating themagnetic field. Letusconsider forexample acopper diskbetween thepolepieces ofanelectromagnet. Itisknown that such a“Faraday disk” israised toincandescence ifmaintained inrotation andthat alternatively aninitial rotational momentum otthedisk israpidly damped bythemagnetic field. This occurs, however, only foraninhomogeneous field, such asisnor- mally realized experimentally, where thediskextends beyond theinner-most homogeneous portion ofthefieldofthepolepieces. Inorder todeal with awell-defined and easily solvable problem weassume that themag- netic field isuniform throughout and introduce into itametal rodwith itsaxis perpendicular tothemagnetic field B,which wesetinto uniform translatory motion along itsaxis. Itssurface ischarged hereby. NoJoule heat arises intheinterior since theconduction current iseverywhere zero; thetotal charge isofcourse also zero. Interior and exterior field join con- tinuously, butwith discontinuous normal gradient, corresponding tothe presence ofsurface charge. The inéerior field isperpendicular both tothe axis oftherodand toBand canbegiven immediately forany form ofthe cross section. Incontrast with the interior field, the exterior field cannot begiven immediately but requires the solution ofaboundary-value problem: the continuous fitting ofthepotential intheexterior tothesurface values of thepotential known from theinterior field. Inthespecial case ofthecir- cular cross section this boundary-value problem isreadily solved; see Problem IV.1. . Along with theinterior field thepotential difference between twosurface points isdetermined. If,bymeans ofsliding contacts onaconnecting wire, itistobeused forthegeneration ofcurrent, theinterior isnolonger- free ofcurrent. The present description oftheinterior field then becomes invalid. Astatic magnetic field, arising from theRowland currents atthesur- 290 MAKWELL’s THEORY FORMOVING BODIES AND OTHER ADDENDA 35.) faceoftherod,occurs along with theelectric field. Itis,however, evi- dently very small odmpared totheoriginal inducing fieldandcanhence beneglected, Ourdescription applies throughout foranobserver atrestinthelab- oratory; foranobserver moving with therodtheelectric fieldwithin the rod iszero. Intheactual realization oftheexperiment therodisofcourse replaced byametallic body ofrevolution andthetranslation byarotation about theaxisofsymmetry ofthelatter. Amathematical difficulty which arises here isalso indicated inProblem IV.1. §98. ThePonderomotive Forces andtheStress-Energy Tensor ‘Wereturn to§31andgeneralize theconcepts introduced there from vacuum toabody ofarbitrary e,u,which, however, weshall assume to beboth homogeneous andisotropic, although theanisotropic body would be ofspecial interest inconnection with electro- andmagnetostriction. We shall, furthermore, regard thebody asatrest, since, forthequestions at issue, wecanplace ourframe ofreference onthisbody. Our earlier definitions of§26 F=(8,-i®), f= (,-iiD), r= (i) aswellastheMaxwell equations inthedifferential formgiventhere DivF*=0, Divf=T (la) retain their validity; itcanbereadily demonstrated that thefactors c arising in(1)arenotderived fromthevacuum constants e,yo,butfrom thetime measurement 2,=ict,which, inthespecial theory ofrelativity, applies quite generally forallponderable media. However, therelation € = oF 2:1-3 ® between excitation and field istobechanged to Mo/wf={2", (2s) 2/e Bo where theupper linerefers tothespace-space, thelower tothespace-time components offandF. . Acomparison of(1)and(2a)shows thatthischange leads infacttothe required relations between excitation andfield: ua"4/®cB,ieH=B/p,aV we -iD=2,/%(-#),ie.D=cE.eV mo 35.5 PONDEROMOTIVE FORCES ANDSTRESS-ENERGY TENSOR 291 Weseefrom (2a) that thesimple proportionality (2)between fand F which wascharacteristic forvacuum passes over, fortheponderable body, intoatype oflinear vector function with twodifferent proportionality con- stants forthespace-space and thespace-time components. Intheaniso- tropic body this isreplaced byamuch more general vector function (see p.28)with ingeneral 12different material constants. Starting from theuniversally valid representation oftheLorentz force density in(31.1) weconvince ourselves, byacritical consideration ofthe individual steps, that thetransformations uptoEq. (31.8) remain un- altered, and arenotinfluenced bythedifferent proportionality factors for theelectric andmagnetic quantities in(2a). The same statement applies also forthediagonal elements ofthetensor 7’,sothat thediagonal sum of the latter retains itsearlier value ‘ 2,Tm=0. (3) The. same holds forthe nondiagonal elements 7.,provided onlythat nandmdiffer from 4.Ontheother hand wecompute from (31.4) e.g. Ty=-1Fafa+Fafa) Ta=-1Fafa+Pafy) =—ie(B,D, —B,D) =2G,H.-EH) () =~icen(B XH) =-245,, =-'@xm,=-!5.. €£0po c c Ifthesubscript 1ishere replaced by2or3theresult remains thesame except that thesubscript xofSisreplaced byyandz,respectively. This different behavior ofthetwo groups oftheTin (nand m#4as against n.or m=4)has the result that whereas thethree-dimensional stress tensor canstill bewritten intheform (31.10), thecomplete four- dimensional tensor 7takes ontheasymmetric form o |-i#s5 ©Foo T=|—__,____ |. (5) -‘s| Ww ¢ This asymmetry hasquestionable consequences. Weknow from hydro-.. dynamics and thetheory ofelasticity that anasymmetric stress tensor leads totorques which donotcorrespond toobservation (seee.g.Vol. II, §10and §8).Also inelectrodynamics torques may bededuced from the 292=MAXWELL’S THEORY FORMOVING BODIES ANDOTHER ADDENDA =35.6 asymmetric character ofourtensor 7’with respect toitsprincipal diagonal; these' torques arevery small andscarcely observable, butareeven soim- probable. M.Abraham hashence proposed asymmetric form ofthetensor T,differing from Minkowski’s, andM.vonLaue hasfollowed Abraham’s suggestion.’ The twopoints ofview arecompared with respect totheir physical consequences byW.Pauli inhisoft-quoted article intheEnzyklo- piidie, p.665. Following once again Minkowski, wededuce from thematrix (5)that thefourth component ofourearlier Eq. (31.3) remains unchanged and corresponds toPoynting’s theorem also inaponderable body. The first three components ofthesame equation, which areaffected bythechanged upper portion ofourmatrix (5)ontheother hand, leadtoadefinition of theelectromagnetic momentum density differing from (31.13). Whereas wefound forvacuum 1 1 G=as=3EXH (6) wenow obtain the different (though, ofcourse, dimensionally equal) expression 1ew Gmane DXB (6a) This conclusion ofMinkowski’s theory isalso notuniversally accepted. Asnoted initially, wehave been able toconfine ourselves tobodies at rest inthis section. Inview ofthe behavior ofthe world tensor Tina Lorentz transformation, known tousfrom (28.20), ourformulas can be transferred directly tobodies inmotion. The problem ofthepunderomotive forces would besolved forthem also assoon astheultimate form ofour tensor 7for bodies atrest had been determined. The fact that this has notbeen accomplished inaunique fashion signifies physically really only anesthetic defect and iscertainly noserious objection tothe theory of relativity. Infact from ourpresent electron-theoretical standpoint all processes take place invacuum, forwhich thequestion ofthestress-energy tensor hasreceived asatisfying and generally recognized solution in§31. From thispoint ofview theponderable bodies with their continuous mate- rialconstants €,»aresimply convenient abstractions andarenotphysical realities. “1Inhisexcellent textbook “Die Relativitatetheorie,” Vol. 1:“Das Relativitats- prinzip derLorentztransformation,” and Vol. II:‘Die allgemeine Relativitiits- theorie und Einsteins Lehre von der Schwerkraft.’’ which have been published as Nrs. 38and 58ofthe series “Wissenschaft” byVieweg. 36.4 ENERGY LOSS OFACCELERATED ELECTRON BY RADIATION 293 §96.TheEnergy LoseofanAccelerated Electron byRadiation andTis Reaction ontheMotion Weknow that, unlike theelectron inuniform motion, theaccelerated electron radiates. According to(19.24) theenergy radiated perunit time is,foravelocity small compared with c, ee8Grae ® ‘This energy lossmust ofcourse findexpression intheequation ofmotion oftheelectron. Totake account ofitwewillreplace itbyanequivalent force. Consider theeffect ofashort acceleration interval from t,to4. Before andafter theinterval, aswell asatitslimits, themotion istobe regarded asuniform, i.e.¢asequal tozero. Inview ofthebriefness ofthe interval thevelocity ischanged butlittle, sothat wemay put6,&6:&8. Wecallthedesired force the“reaction force oftheradiation” anddenote itbyR.(Please excuse theuseofthesame symbol RasfortheRoentgen current and, before that, fortheimpedance operator!) Itmust satisfy the condition that thework done byitontheelectron intheinterval from 4 {0t2beequal tothenegative radiation lossoftheelectron, i.e. ts [ran -f"sa (2) From theidentity w=2@)—vb andourassumption ¢(4) =¢(t) =0wefind ts eaqewl — =_f3fea wlfia fvas, sothat, by(1)and (2), 2é= [Rra- 5,fra @) Wethus obtain asthesimplest formulation eo IRl=soa (4) furthermore, itmay beshown that other expressions consistent with (3) deviate from (4)only byterms ofasmaller order ofmagnitude (see the 294=MAXWELL’S THEORY FORMOVING BODIES ANDOTHER ADDENDA 36.48, discussion after Eq. (27)). Inthemagnetic cgs-system this becomes, by (16.30), 2b, |Rl= 3° (4a) thismay becompared with Larmor’s formula (19.24b). Weshall study theeffect ofthereaction force forthevery simple case of anelectron vibrating about itsposition ofrest, which, asin§19c, may serve asanidealized model ofalight source. Letthevibration berecti- linear; inview ofthereaction force itisdamped. Wecallthedistance of theelectron from itsposition ofrest ¢and set g=he™, w= w(l+a). (6) wy=2n/ristheangular frequency intheabsence ofdamping, rthecorre- sponding period, aacomplex number; itisvery small inabsolute value and, forourexpression for£,must have anegative imaginary part. We shall demonstrate both facts. Theequation ofmotion oftheelectron is mk+fe=R. (6) The restoring force, which may arise insome fashion from theatomic binding, hasbeen setequal to—f¢andbeen transferred totheleft. We divide (6)bymyand put Z=o, (7) Ra; mec & (7a) (7)follows from Eq.(5),according towhich a»isthecharacteristic fre- quency oftheoscillation forR=0;in(7a)asignifies, by(4),alength of thesame order ofmagnitude astheelectron radius in(33.6). Eq.(6)then becomes Etude =38 ®) Substitution from (5)yields, after cancellation ofa*t, —C+a)$1=Fant +a)=2S(+a) @) )isthewave-length oftheemitted light; even inthex-ray region itisvery large compared totheradius aoftheelectron. Wemayhence neglect aas 36.12b ENERGY LOSS OFACCELERATED ELECTRON BYKADIATION 295 compared with1ontherightsideoftheequation anda”ascompared with 2aontheleftside. We thus find from (9) a=nrit. (10) The sign ofourresult agrees with theexpectation expressed at(5).The fact that, inourapproximate calculation, ahasbecome purely imaginary indicates that theperiod oftheoscillation isnotchanged appreciably by thereaction force, just asin(18.9d), where theperiod ofthequasista- tionary current oscillation didnotdepend materially ontheresistance. Substitution of(10) in(5)yields |.(-ont)=exp(—20°8! li=expryatexpony=. (11) The amplitude ishence reduced byafactor 1/einatime, measured in periods oftheoscillation, t x 7ea (aa) The correspanding light path measured inwave-lengths, z/A, hasthesame value. For\'=4-10 cm,a=2-107" cm(12)yields 5210, x=400em =4M. (128) The “distance ofcoherence” oflight waves, measured forparticularly sharp (ie. particularly monochromatic) spectral lines, isofthesame order ofmagnitude. There arenoabsolutely sharp spectral lines. Every broken ordamped wave train, when subjected toFourier analysis (see Vol. VI, Exercise 1.4), yields afinite spectral width (more precisely, half-value width). The Doppler effect, which istheresult ofthethermal motion of theemitting particles and ishence temperature-dependent, hasthesame consequence. The reciprocal ofthedamping time ¢given by(12) isdefined asthe natural classical line width. With the notation D=1/tand with 7=Neweobtain ac D=2x"x or,with thevalue ofagiven by(33.6), nrél D3mc” (126) This classical linewidth isthelower limit oftheobservable linewidth, at thelowest possible temperatures (elimination oftheDoppler effect) and 298©MAXWELL’s THEORY FORMOVING BODIES ANDOTHERADDENDA 36.12c¢ thelowest possible pressures (elimination ofso-called collision damping). Wespecialize theexpression forthecgs-units customary inspectroscopy (€/eo =4nc'emaen’ byEq.(16.30) andfind : 2 emus =1.6010, ¢=3.10", D=SfSenerfre, (12e)Cangn/™o =1.76-10". Ifinparticular weset\=4-10~* cmasbefore, weobtain D=7-10" sec”. (12d) This isvery small, even compared with theminute frequency separation ofthehydrogen doublet ofthefirst Balmer line: : Re=Rydberg frequency vn=fe=10"seca=fine-structure constant The reaction force Rplays animportant role inPlanck’s theory of black-body radiation (seeVol. V).Itdetermines theamplitude towhich alinear oscillator israised inequilibrium with thermal radiation, from which itthen accomplishes itsemission ofquanta. Sofarwehave considered only theslowly moving electron insofar aswe have determined the force ofreaction only from the standpoint ofan observer moving with theelectron. The theory ofrelativity makes it possible, however, tochange theframe ofreference andtodetermine then thereaction force foranelectron moving with arbitrary velocity. The factthat aresult differing materially from (4)willbeobtained follows from thefact that Eq.(1)then ceases tobevalid and, instead, (30.11) yields fortheenergy radiated perunit area andperunittime atanangle 3with respect tothedirection ofmotion ee sin’3 S=Teetegdh (=pcos5)" {18) The acceleration ishere assumed tobelongitudinal, i.e.inthedirection ofmotion. Itshould furthermore benoted that (13) refers tothe time scale oftheobserver atrest, whereas wemust know forthedetermination ofRtheradiation perunit time inthesystem ofthemoving electron. The retardation relation (29.10b) exists between thetwotime scales (¢,time of the-observer, 7,time oftheelectron aswell as“intrinsic time” ofthe latter): =~ r DY, at Dy _rat—, dr=dt+2dr, Fa1-<=1-8cos¥. 36.184 ENERGY LOBS OFACCELERATED ELECTRON BYRADIATION 207 Thus torefer Stothetime scale ofthemoving electron wemust multiply (13) by H1Bcos9. (14)ar Furthermore, topass from Stothe total radiation Softhe electron we moust integrate over thesphere ofradius rand with theelement ofangle do=2xsinédé.With theabbreviation u=1—8cos&wethus obtain *gin'dsinddd2xft?(:-*)du arfGres ehla) fe This integral may beevaluated inanelementary manner andyields Qr 4 3 -ey Hence ee 1 8*Greed=BF a8) «Thevalue ofRwhich isnowsought—we shall callitR’/—must (forlongi- tudinal acceleration) beimplicitly related to(15) by(2).However, itis notdetermined uniquely hereby; forthis itwould benecessary, following Abraham, toadd conservation ofmomentum toconservation ofenergy. Wehence prefer toproceed from therelativistic equation ofmotion ofthe electron, which will yield anexplicit value forR’;atthesame time we need nolonger limit onrselves tothe longitudinal case. Inaccord with Eq. (32.4) etc.andadding thedesired reaction force R’ wewrite theequation ofmotion intheform: mW =F +R’. (16) Wisthefour-vector oftheacceleration introduced in(27.18b), F,the four-force acting atany moment, which inthe electrodynamic case is related totheLorentz force KbyEq. (28.19b): Fark, 2=(1-6y™. a7 WandFareperpendicular totheworldlineoftheelectron inthefour- dimensional meaning oftheterm, i.e.(see (27.18c) and (28.19¢)) - V-W=0. and V-F=0. (18). Accordingly (16) leads totherequirement that R’also beperpendicular to the world line: V-R’ =0. (18a) 298©MAXWELL’S THEORY FORMOVING BODIES ANDOTHERADDENDA 36.19 Itwould seem most reasonable, following (4),todefine R’by 2 y= W =~, we Rw,6eae’ WwWadr (19) (dots willindicate differentiation with respect totheintrinsic time alsoin what follows). This would, infact, beaLorentz-invariant definition ofR’ which, specialized totheframe ofreference oftheelectron, would agree directly initsspace components with (4).However, thisdefinition would contradict therequirement (18a). Wehence modify itto R’=b(W +aV) (19a) anddetermine theconstant ahereintroduced from thecondition 5 . vw VW+WV=0,a=oy. (20) Like (19)thisdefinition ofR’satisfies therequirement ofbeing identical with (4)intheframe ofreference oftheelectron, since thefirst three components ofVvanish here; furthermore if,asin(4),correction terms of asmaller order ofmagnitude areneglected thedefinition isalso unique. The value (20) of@can befurther simplified: First, by(27.188), V-V=—c’;second, wemaydeduce fromV-W=0bydifferentiation with respect tor: VW+V-W=0, V-W=-V-W= -W-W. Hence wemay write a=iwew _(20a) and =o(w- Wy). 1) Thisistheveryconcise formulation ofthereaction force, validinevery frame ofreference. The conciseness islostwhen wepass over tothree dimen- sions, topermit comparison with theformulations known from thelitera- ture. Weproceed from Eqs. (27.18) and (27.18b): V=(qv,ten),=W=V=(av+4,ict) (22) and compute with theaidofthedefinition (17) of» tt ~--1 weortys, taty."TIF "=Toe wera |ave (22a) 36.258 ENERGY LOSS OFACCELERATED ELECTRON BYRADIATION 299 Wethen obtain from (22) 2 - w=(t(v-¥)v+tinir), Int on ‘ (2b) “W=atv”(eg)? +2(e.g)? -(wg)? wwTyee +SwwtyLovey Thefirstandlastterm ofthe{}canbecombined toyield “ 2 2 7oe on =—7 (y.¥)?od ee ee which combines with the second term. We thus obtain : 2WW={ft(vee)?+“ (23) WWy tit yay.3va2{e ate? Wore. (23a) Inthelastequation wehave written down onlythethree space components tofthefour-dimensional vector. Wedothesame inthecomputation ofW, ive.thedifferentiation of(22b) with respect to7: 2 : W=2ray+Et(it (OH)+1ta Ifwetake account of(22a) wefind 5 3 Wh yytS(Wt(wie+AH) Hav (24) Weobtain forthedifference of(24) and (23a) , 5 2 x=a(w-d)'v+E{(-¥)v+200-¥)¥) +av. (25) Finally wepass from theintrinsic time oftheelectron, towhich ¥and ¥ arereferred, tothetime scale ¢oftheobserver, inwhich weshall denote thecorresponding quantities byv’andv”.Weset dv_ dv togwer 1g= tegie 4 2=Ze)=atat=Bove tt. Ir Thus weobtain from (25) ’ 7 5 Fev +Evy +(yytat”, (250) 300=MAXWELL’S THEORY FORMOVING BODIES ANDOTHER ADDENDA =36.26 ThisvalueofR’istobeemployed intheequation ofmotion (16).Ifwe transform thelatter into thecustomary form ofthemomentum equation (32.5), i.e.replace moW bydG/dt =m)W/n andFbyK=F/n, R’must also bereplaced byR*=R’/y. Wethus obtain forR*from (25a) after substituting thevalues ofband 7: 2 + 1 Oo bree= (26)3v-v' v[”3(v-v')?}) v+v5-—— +5-—, |v-v"+—|). (trate mtartal tan Forsmall velocities (8—0)weobtain, ofcourse, R*=R,i.e.thevalue from Eq.(4).Eq.(26) wasderived first byAbraham from electrodynamics and byv.Laue from thetheory ofrelativity. Our procedure follows the suggestions ofPauli.’ Wemust still establish thelimits ofvalidity oftheformulas derived here. Byusing therelativity transformation foruniform motion wehave assumed theacceleration tobe“small” without otherwise restricting the magnitude ofthevelocity. Wearehence dealing with aprocess ofapproxi- mation oraseries expansion which isbroken off.H.A.Lorenta’ carries outthisprocess, representing theretarded potentials aspower series of therelaxation time ¢—r/cand computing themechanical force ofthe intrinsic field. The first term ofthis expansion istheinertial reaction of theelectron, which, forsmall velocities,’ isgiven by . ev~mit =— (27) The reaction force RofEq.(4)appears assecond term: ew R=irae (27a) Lorentz emphasizes that Ristheonly term oftheexpansion which does notdepend ona(‘on theshape oftheelectron”). The higher terms, which arenotcomputed, areoftheform 2 3 G WAG Here a/cisthetime required bythelight totraverse the“radius ofthe electron”. Theterm (27a) canbebrought intothesame form; inview of ‘Enaykl. d.Math. Wiss. Vol. Vs,p.654. +The Theory ofElectrons,” Teubner, 1909, Note 18,p.251. +y', vw--»arethen identical with, v#---. 37 MAXWELL’S EQUATIONS AND ELEMENTARY PARTICLES THEORY 301 the meaning ofmitcan bewritten. oy,morag ‘The terms oftheseries must decrease inorder that theseries may con- verge, -i.e.bepractically useful. Hence wemust have ‘ ¢ ¢ I< flv, WWE<E1eL (28) Itisclear however thatinprocesses involving very large energies, such as theflight ofanelectron close toanatomic nucleus, notonly very great accelerations, butalso very great changes inacceleration canoccur. The termination oftheseries with theterm Rwould then benolonger per- missible. Thesame applies fortheacceleration process inthebetatron (see Problem III.10) andsynchrotron. The formulation oftheradiation resist- ance insuch extreme cases constitutes anasyetunsolved problem which hasledtomany discussions (Wessel, Dirac, Bopp, Stiickelberg).’ §37.Approaches totheGeneralization ofMaxwell’s Equations andtothe ae Theory oftheElementary Particles Gustav Mie took thefirst step inthis direction in1912 inhisfamous papers’ “Foundations ofaTheory ofMatter.” Their goalisno*less than thegeneralization oftheMaxwell equations sothat they include the existence oftheelectron. Inorder that thegeneralization may notloseitself inlimitless possibilities itissubjected from thestart totheprinciple of relativity andderived from a“world function” which may depend only on Lorentz-invariant quantities. Here adistinction ismade—possibly forthe firsttime inaconsistent fashion—between entities ofintensity andentities ofquantity, i.e.written inournotation andunits, between F=(B,-), a=(a£¥) onthe one hand and f=(H,-iD), 1=(py,tpc) ‘The most recent contributions tothisquestion aregiven bythepapers ofW. Heitler andH.W.Peng, Proc. Cambridge Phil. Soc. $8,296(1942) and, from the standpoint ofEinstein’s latest methods, N.Hu,Proc. R.Irish Academy, 67,87 (i947). + - ?Ann. d.Phys: First communication, Vol. 37,p.511; second communication, Vol.39,p.1.Thethird communication (Vol. 40,p.1,1913) deals with thetheory of gravitation andisofcourse outduted, having beenoriginated before thegeneraltheory ofrelativity. 302 MAXWELL’S THEORY FOR MOVING BODIES AND OTHER ADDENDA 37.1 ontheother. Mie tests theinvariants which may besetupwith theentities ofintensity andtheentities ofquantity respectively. Werecord three’ of theformer (see (26.24), (26.21), and (26.4)), suppressing constant factors insofar asthey aredimensionally superfluous: A=3(CB-E), (1) M=E-B, (2) jafi=a’— w/e. (3) Hefinds that only theinvariants (1)and (3)need beconsidered forthe description ofquasistationary processes and constructs aworld function? Wsuchthatatlargedistance fromtheelectron theordinary Maxwell equations apply, wheréas theequations aremodified attheelectron andin itsimmediate neighborhood. Like Schwarzschild’s function, the world function istobeintegrated over anarbitrary region ofthefour-dimen- sional world and tobevaried insuitable manner. What must betheform oftheworld function ifitistoyield theordinary Maxwell equations atanadequate distance from theelectron? According !toourexperience with Schwarzschild’s principle ofaction wemust then have W=A.'In fact, inpure vacuum (F=0andnokinetic energy of matter) thekinetic potential K’inEq.(32.18) reduces tothemiddle term, proportional toAand when subjected tothevariation, yields theMaxwell equations ofvacuum. Atthesame time thechange intheworld function intheneighborhood oftheelectron istobesuch that from itadefinite value¢oftheelectron charge (or,atleast,ofthespecific charge e/mof theelectron) may becomputed. This iscertainly notsoifthetwoinvari- ants Aand|Q|?atesuperposed linearly, since then alsotheresulting differential equations would belinear inthefield andpotential components respectively and their integrals would consequently involve coefficients which could bechosen arbitrarily. Ontheother hand, both requirements might besatisfied bytheformula W=A+alQ|* (4) where nisasufficiently large number. Actually, thesecond term may then beneglected atsufficiently large distance from theelectron since it 1Weyl, in§28ofhisbook tobequoted onp.321,points outafourth field invariant, constructed from Fand 2. 1Mie himself calls the function constructed with entities ofintensity ‘“Hamil- tonian function H”and designates as“world function” that constructed with en- tities ofquantity. Wehave taken theliberty ofreversing thenomenclature soasto establish correspondence with Schwarzschild’s action function. The entities ofin- tensity arethen obtained from Mie’s world function bydifferentiation with respect totheentities ofquantity. 37.68 MAXWELL’S EQUATIONS AND ELEMENTARY PARTICLES THEORY 303 vanishes asr~", whereas asingularity ofhigh order occurs atthelocation oftheelectron. For mathematical reasons Mie puts specifically n=6;in thismanner heobtains aspatially highly concentrated charge distribution, which, however, isnotstable inthefield ofanother electron. Itwould, after all,have been indeed surprising ifthefundamental problem ofthe elementary particles could have been solved byclever guessing. Today weareconvinced that much experimental preparation willberequired instead. Nevertheless, blazing thepath totheproblem was anactofgreat merit, asisevident from thefact that alllater workers in the field have followed inMie’s tracks. Pauli, inNr.64ofhispaper intheEnzyklopiidie, hadalready emphasized that dependence ontheabsolute values oftheelectrodynamic potentials ledtoserious difficulties inMie’s theory. Hence weshall avoid use of invariant (8)intheformulations tobediscussed below. Born andInfeld’ inparticular utilize intheir theory aworld function which depends only oninvariants (1)and (2). The nonlinearity oftheelectromagnetic field, which isrequired here also, follows from thechoice ofWasnonlinear function ofAand M.The conjugate four-current Idrops outalong with thepotential Qand, just asinLorentz’s electron theory, must bebrought irtasaforeign element. Wischosensothataninfinityofthefieldatthelocationoftheelectron isavoided. Inthismanner afinitevaluefortheself-energy oftheelectron isobtained andadifficulty ofclassical theory, which yields infinite energy forthepoint electron (a=0),iscircumvented. Forquasistationary prob- lems, inwhich theabove invariant (2)does notenter, theformulation of Born and Infeld is . . 2 2A wmatly/i4Heih. (5) The universal constant bhere introduced has the dimension ofanelectric fieldstrength since, by(1),Ahasthedimension ecE*. This formulation follows thepattern oftheaction function ofclassical and relativistic mechanics. Inthe classical mechanics ofthe point mass notacted upon byforces wehave astheintegrand oftheHamiltonian principle thekinetic energy m2 T=” ) Inrelativistic mechanics this isreplaced bythe’ “kinetic potential” in Eq.(32:9b) Keme{i— 4/1-%. (6a) 1M. Born, Proc. Roy. Soc. London (A)143, 410, 1938/34; M.Born and L,Infeld, loc. cit. 144, 425, 1934. M.Born, Ann. de|’Inst. Henri Poincaré, Tome VII. 304 MAXWELL’S THEORY FORMOVING BODIES AND OTHER ADDENDA 37.7 which passes over into(6)forv<candthen becomes independent ofc. Similarly (5)passes forA«cob”over intothevalue W=A,which corre- sponds toMaxwell’s theory, andbecomes independent ofb.Whereas (6a) setsanupper limit ctov,(6)imposes norestriction onv.Similarly (5)sets anupper limit btothefield strength Eintheelectrostatic case (B=0, A=—e0E*/2), whereas theformula W=Apermits anunlimited increase inthefield strength. The Maxwell equations invacuum forE,BandD,Hareretained in thetheory ofBorn andInfeld. Intheelectrostatic case, towhich weshall limit ourselves inthefollowing, weobtain foracentrally symmetric field andapoint charge eatr=0,just asfortheconventional theory el DeTae 7) D;may alsobedetermined from theworld function bythegeneral ruleof Mie’s theory: ow D,=-3E, (7a) Aécording to(6)thisyields, withB=0andW=eb*{./1 —E*/b—1} tok, D,=VizEye (8) Itfollows that — &E= DeiDyan (9) If(7)issubstituted in(9)weobtain ——1 /@ E,=avi 7 Fred" (10) The quantity romay beregarded astheelectron radius. H,isnow every- where finite, since forr=0wehave H,=b=e/(4mear). D,ontheother hand, becomes infinitely large atthesame place. For r>1,E,differs little from theCoulomb fielde/(4xecr*). Theelectrostatic potential is - e r wo)=[Bdrmpes(:) (ut) with - =dy 10= Tee 37.15 MAXWELL’S EQUATIONS AND ELEMENTARY PARTICLES THEORY 305 Attheorigin wehave - 6 . ¥O)=FaIO 10)=1.854. The Hamiltonian function Hisrelated toourworld function Wbythe general formula H=W+E-D —B-H. Thus wefind inthe electrostatic case H=eb'(/1 —Bi—1)+E,D,. (12) IfZ,iseliminated with theaidof(9)this contracts to H=eb°(V1 +D3/(e38%) —1), (13) Weobtain therefore forthetotalenergy an) Womaef°Hear, Substitution ofD,from (7)yields finally é iWap [wie day a4) The numerical value oftheintegral is1.236. If(14) isputequal tothe selfenergy oftheelectron mec*weobtain 2é ro=1.236Doe (15) i.e.very nearly theclassical radius aoftheelectron from (33.9). Then (10) yields forbthevalue baie 2Aregri —4xeqa?” very nearly equal totheclassical field strength EHatthe“edge ofthe electron”. Inthismanner anelectron radius oftheproper order ofmagnitude and avery high critical field strength bareobtained. The field ofseveral point charges, also, can bedetermined inunique fashion. Within these limits thetheory ofBorn and Infeld thus leads tosensible results, although its fundamental formula (5)canclaim only heuristic validity. Weshall finally discuss theproblem ofthe“scattering oflight bylight”. This problem arose from Dirac’s theoretical discovery ofthepositron and pair production. (Pair production, i.e.the simultaneous generation ofan eiectron and apositron from hard gamma radiation, was realized experi- 306 MAXWELL’S THEORY FOR MOVING BODIES AND OTHER ADDENDA 37.17 mentally soon afterwards byIrene andFrederic Joliot-Curie, whereas the positron wasobserved incosmic radiation byAnderson andBlackett and Occhialini.) Itisclear that this problem also involves achange inthe Maxwell equations forvacuum which isequivalent toa“non-linear theory oftheelectromagnet field”.’ The linear Maxwell equations could not account forsuch scattering, butwould imply that thefields oftwo inter- penetrating light waves aresimply tobesuperposed. _The problem has been treated quantum-mechanically byEuler and Kockel under thedirection ofHeisenberg’ andhence liesentirely outside oftheframework ofourpresentation. Wecanmerely indicate thepro- cedure foritssolution. The world function ishere chosen sothat forweak fields itreduces, as before, totheLagrange density A.Inthesecond approximation itis written asafunction ofthesecond degree inAand M.Only even powers ofMmay occur here, however, since foratransition from aright-handed toaleft-handed coordinate system Band hence Mchange their sign (in theterminology ofEuler Misnot“mirror-invariant”). The next term of theexpansion must hence have theform aa’+Bes’. az Inorder that aand 8may bepure numbers and this second term have thesame dimension asthefirst term A,wedivide (17) by&Xthesquare ofacritical fieldstrength, which, asbefore, weshall callb.Itishere defined asthefield strength “attheedge oftheelectron”, i.e.atadistance r=a from its“center” (a=classical electron radius, multiplied bythefine- structure constant 1/137). We obtain thus the formula W=a+(an+pelt’)+-- (1s) AsinMie’s theory thecomponents oftheentities ofquantity DandH areobtained from this world function aspartial derivatives with respect tothecorresponding components ofEandB: ow owD=- H=5g- (19) 1This isthetitle ofthepaper ofBorn mentioned inthelastfootnote. +H, Euler andB.Kockel, Naturwiss, 28,1935; Euler, Leipzig thesis, Ann. d.Phys. 26,1936; Heisenberg andEuler, Z.Physik 98,1936. Seealso thesimplified representa- tioninthepaper ofM.Born cited attheendoffootnote 1onp.303.Theproblem of. pairproduction wasapproached simultaneously from adifferent angle byR.Serber and E.A.Ubling, Phys. Rev. 48,1933. 38 GENERAL THEORY OFRELATIVITY 307 Since, by(1)and (2), oA aM oA= okBs op7CB=Bim, , aM op=E> weobtain from (19) 2 D=eofE+Fy(ad—BeoMcB)+\, 1 2 (20) H=L{s+=,(eAB+BeoME/c)+oo} Ho ob? The most difficult part isthedetermination ofthenumerical coefficients «and§from Dirac’s theory ofpairproduction, forwhich werefer tothe original papers. The final result isthefollowing: Maxwell’s equations forvacuum, relat- ingB,Eand D,Hretain their form. However, just asinthetheory of Born and Infeld, Disnolonger proportional toE;acorrection term occurs intheexpressions forDand Hwhich depends onE,B,A,and Mand is negligible compared with theprincipal term forweak fields. Noarbitrary assumptions ofany kind aremade here; itismerely presumed that an expansion inascending powers ofthefield strength ispossible, this being indicated inEqs. (18) and (20) by---.Inthepaper ofHeisenberg and Euler the expansion has been extended byanadditional term and has even been expressed inclosed form. Inany case, this work demonstrates thenecessity ofmodifying Maxwell’s equations forextremely strong fields even invacuum. §38. General Theory ofRelativity; Unified Theory ofGravitation and Electrodynamics Inthis paragraph also wemust limit ourselves toamere outline. Afull presentation ofthesubject would require aseparate textbook; itwould be premature towrite such aoneatthis time since many pertinent questions areasyetundecided. For thepresent weshall follow theoriginal presentation ofEinstein as recorded particularly effectively inhisPrinceton lectures.’ Heproceeds entirely inthespirit ofKlein’s Erlangen program: Classical physics belongs tothegroup ofelementary geometry (isotropy ofspace within itself), ‘Thefourlectures onthetheoryofrelativity, heldatPrinceton inMay1921,~ have been reprinted inA.Einstein, The Meaning ofRelativity, 3rd Ed., Princeton University Press, 1950. 308©MAXWELL’S THEORY FORMOVING BODIES ANDOTHERADDENDA 38.1 extended bythedisplacement ofthetimeaxisalong itself. Thespecial theory ofrelativity isfounded onthegroup ofthelinear orthogonal transformations ofthefour world coordinates x,%2,a2,and x,=ict(isotropy ofthe four-dimensional world, Lorentz transformations). We areledtothe general theory ofrelativity ifwestart from thebroader group ofpoint transformations which, inVol. I,p.16wehave characterized bythe formulas wy=feltr, t,t, %), k= 1,2,3,4 qa) Inthismanner allpossible frames ofreference become legitimate, notonly those which move with aconstant velocity v<crelative toeach other. “Space andtimelosethelastvestige oftheirabsolute character postulated byNewton andbecome merely means forthedescription ofphysical phenomena.” Such_‘a program hadalready been setupbyErnst Mach.However, hegaveupwithanegative point ofview(which, strangely enough, hedenoted aspositivism) andremained anopponent ofEinstein’s theory ofrelativity totheendofhisdays. Thelatter, ontheother hand, assumed apositive attitude byinquiring intothose space time relations which areconserved inallpoint transformations. Thegeneral theory of relativity signifies theinvariant orcovariant theory ofthisgroup oftrans. formations. ‘Thebasis forthishadbeen created inpartbyGauss’ inhistheory of surfaces andbyRiemann’ inhisinitiation lecture. Gauss studied theinner properties ofasurface, apart from itsexternal shape andposition inthree-<dimensional space. Forthispurpose herepre- sents thelineelement ds,i.e.theseparation oftwoneighboring points of thesurface, bytheformula ds*=Edp*+2Fdp dq+Gdg’. (2) pandqareparameters oftwofamilies of(ingeneral notorthogonal) curvesonthesurface, andE,F,andGparticular functions ofpandq.Forapure bending ofthesurface (without dilatation orshearing) thetotality ofline elements andhence alsothesystem ofthecoefficients E,F,andGiscon- served. Gauss shows that themeasure ofcurvature 1 K=RR (3) "1Pisquisitiones generales ciressuperficies curvas 1827,Ges.Werke, Vol.IV,trans- tedinto German inOstwalds Kiassiker Nr.5. ~ 2Uber dieHypothesen, welche derGeometrie zugrundc liegen, 1854, Ges.Werke 2nd Edition, p.272. 38.5 GENERAL THEORY OFRELATIVITY 309 introduced byhim(R,andRarethetwo“principal radiiofcurvature” ofthesurface) may beexpressed bytheZ,F,Gand their first and second derivatives with respect topand q.Inthis manner hearrives athis“The- orema egregium”’: Ifacurved surface isbent into another shape (without dilatation!) themeasure ofcurvature remains invariant inallpoints. The measure ofcurvature hence expresses aninner property ofthesurface, whereas thedefinition 1/(R,R:) (just like that bythe“spherical image”) appears todepend ontheexternal shape ofthesurface and does not in- dicate itsinvariance. Gauss commends hismethod offixing attention ontheinner properties ofsurfaces as“most worthy ofbeing diligently exploited bygeometers”. Weshall seethat this challenge was heeded byRiemann and Einstein. The character ofthegeodetic, orshortest, lines is,ofcourse, also conserved inthebending since itrestsolely ontheextremal property oftheinte- grated lineelement. Wealsomention theapproximation ofthesurface by oneofitstangential planes, although itdoes notbelong totheinner rela- tions ofthesurface; locations will here beindicated not inthecurvilinear coordinates p,q,butinordinary Cartesian coordinates. Wenow consider, with Riemann, ann-dimensional manifold ofvery general structure. Asthe generalization of(2)and already written in Einstein’s notation, itslineelement is dst=3)Goedendtr, —Gr=Only=1,2+++). (4) The g,,are given functions ofthe quite arbitrarily chosen parameters %,%--+-2,. Riemann studies the inner, invariant (inmore general terms, covariant orcontravariant) properties ofsuch amanifold. Tobegin with, however, weshall answer thesimple question: What must bethedimension NofanEuclidean space inorder that then-fold manifold may beembedded init?Weshall employ Cartesian coordinates X,, +++Xwinthis Euclidean space. Onthen-fold manifold they may be represented asfunctions ofthenparameters 2;,-++2m: Xi=Flt, +++an)3 +Xw=Fr(ay, +++tn). (5) Ifweform the Euclidean line element aXi+dXi+++.+dXh this contains thefirst derivatives ofthefunctions F;, ---Fy. Inorder that itmay assume theform (4)ontheembedded n-fold manifold with arbitrarily prescribed g,,,thenumber Nofthearbitrarily prescribable F must suffice forthedetermination ofthealso arbitrarily prescribable g,, 310—MAXWELL’S THEORY FORMOVING BODIES ANDOTHER ADDENDA 38.6 andhencebeequaltothenumberoftheg,,,whichisn(n+1)/2.Wethushave’ n(n+1) N=—: (6) Inthe case ofEinstein’s four-dimensional world wehave . Nea=10; (6a) intheGaussian case ofthetwo-parameter surface wehave ofcourse nya=*3a3, (6b) For every point ofthen-fold manifold a“plane” (Euclidean) manifold may beconstructed which, departing from then-fold manifold, plays the same roleasthetangential plane inthree-dimensional Euclidean space. One oftheinner properties ofthen-fold manifold is,inparticular, the minimal property ofthegeodetic (shortest orstraightest) lines. Ason the two-parameter surface, they areatthesame time thepaths ofapoint mass not subjected toforces. We demonstrate this with the aidofan example: Letaplane table topbecovered with acloth under which there imay lieastone. The geodetic paths areingeneral straight, but curved close tothe stone. The mass point, which isassumed not tobeacted upon byforces, including gravity, willhere bedeflected outofitsstraight path, inaccord with theprevailing surface curvature. This isthesimplest example ofEinstein’s theory ofgravitation (stone =sun, point mass = planet). Riemann investigates the generalization ofthe Gaussian concept of curvature tothen-dimensional manifold. Following Riemann, Christoffel hasdefined histhree-indices symbols andEinstein hisI';,whichareiden- tical with them (seeAppendix I,Eq. (3)inVol. IIofthese Lectures). They depend only onthecomponents g,,ofthe“fundamental tensor” and itsderivatives with respect tothecoordinates and arehence inner properties ofthemanifold. The“Riemannian curvature tensor” isformed from theg,I’,andtheir derivatives; itsvanishing isthecondition forthe manifold being “plane” (Euclidean). The Riemannian “symmetric curva- ture tensor” R,,isderived from itby“reduction” (summation with respect tooneofthepairs ofindices). The“Riemannian scalar” Risderived from thecurvature tensor R,,insimilar fashion; itisthegeneralization ofthe Gaussian measure ofcurvature K. [email protected],Ann.Physik, Vol.61,1919(Munich thesis). Iaminformedthat theorem (6)was stated bySchlafii asearly as1871 (Ann, Mat. pura appl. 5,f: 190), and hasbeen proved byE.Cartan and M.Janet (E.Cartan, Lagéométrie riemannienne etsesgénéralizations. Encycl. Frangaise, t.1,1937). 38.7 GENERAL THEORY OFRELATIVITY 311 Aswesawfrom ourprimitive example ofthetable top,thecurvature properties findexpression inthepaths ofmass points subjected tonoother forces; theyactonthem likeforces ofphysical origin. Einstein recognized herein theorigin ofgravitation, giving quantitative content toanidea of Mach Asthemost general force action, superseding allother physical agencies, itisattributed byEinstein solely tothecurvature conditions of thespace-time continuum. However, howarethese curvature conditions determined? They are determined bytheenergies distributed inspace andtime. Space andtime exist only byvirtue ofthephysical processes which occur inthem. Their structure isderived from thelatter. Weareinclined torecall Goethe’s grand vision ofthe“Mothers” inFaust II(corresponding, inasense, to thePlatonic ideas which existed before thecreation oftheworld): Géttinen thronen hehr inEinsamkeit, UmsiekeinRaum, noch wen’ger eineZeit. Von ihnen sprechen istVerlegenheit. Nichts wirst dusehn inewig leerer Ferne, Den Schritt nicht héren, dendutust, Nichts Festes finden, woduruhst. ‘The approach tothecurvature conditions ofEinstein’s world which will nowbedescribed mayseemasdisconcerting tothereader asthevoyage to theMothers seemed toFaust; weshall guide thereader along alessfor- bidding path presently. Thecurvature tensorRy»istoberelatedtothestress-energy tensorTyr ofmaterialandelectromagnetic phenomena bythesystemof10equations (u,»=1,2,3,4)Re—tgwR=-1Tp, @ asisshown byEinstein. Thefactor ofproportionality xhereintroduced is inessence theconstant @ofNewton’s lawofgravitation. Since theRy andRmaybeexpressed bytheI’sandg’s,andtheT’s,inturn, bythe g’sandtheir derivatives, Eqs. (7)areineffect asystem ofdifferential equations fortheg,,.Ageneral solution ofthissystem wouldofcourse be extremely involved. Einstein couldshowhowever thattheyleadinafirst approximation tothestatements ofNewton’s theory ofgravitation for weak fields org,,which differ only little from theEuclidean ones ofthe special theory ofrelativity (or,moreexactly, thepseudo-Euclidean ones,inview ofthenegative sign ofdz/).! 1Einstein wrote theauthor November 28,1915: “Last month Ipassed through oneofthemost exciting, absorbing and, atthesame time, most productive periods ofmylife. Icould notthink of writing. Irecognized that myformer fieldequations ofgravitation were quite without basis. This isindicated bythefollowing factors ... Having lostallconfidence intheearlier theory, Isawclearly thatasatis- 312 MAXWELL’S THEORY FOR MOVING BODIES AND OTHER ADDENDA 38 Even thefirst approximation, i.e.themere fact ofNewtonian attraction, reveals the non-Euclidean structure ofthe scale determination. In the second approximation there occur deviations from the Newtonian law, which areofcourse greatest intheneighborhood oflarge concentrations ofenergy. Hence theanomaly inthepath ofMercury, theplanet closest tothesun, thedeflections (observable only during solar eclipses) oflight rays passing very close totheedge ofthesun, and theredshift ofthe spectral lines ofthewhite dwarfs resulting from their extraordinarily high densities. Gravitational and Inertial Mass ‘These aretheabstract mathematical foundations ofEinstein’s theory of gravitation. Atamuch earlier date,’ almost immediately after thedis- covery ofthespecial theory ofrelativity, Einstein recognized aconcrete physical basis intheequivalence ofgravitation and acceleration. The phe- nomena observed inanelevator which isimagined tobefreed from the influence ofgravitation andismoving upward with theconstant accelera- tion gareexactly thesame asinthesame system atrestorinuniform mo- tion when itissubject totheinfluence ofgravity. Inboth cases athrown body describes @parabola, abody atrestonthefloor presses against it swith theforee mg,and apendulum ofequal length hasthesame period of oscillation. Conversely, anelevator falling freely inagravitational field is notsubject totheinfluence ofgravitation: Thepressure onthefloor ceases, theperiod ofoscillation ofapendulum becomes infinite, andathrown body describes astraight line. Such afreely falling system realizes aworld free from gravitation and curvature, inwhich pseudo-Euclidean measure is valid, and hence corresponds tothetangential plane totheRiemannian space which wasdiscussed before. The prerequisite forthis istheidentical character ofgravitational and factory solution could beattained only onthe basis ofgeneral covariant theory, i.e.ofRiemann’s covariant R,,.Unfortunately Ihave immortalized thelast errors ofthis conflict intheAcademy papers which Ishall send you soon.Thefinalresultisthefollowing: ...TheChristoffel symbols(*)are toberegarded asthe natural representation ofthe “components” ofthe gravitational field ... ‘The splendid thing which Iexperienced was notonly that now Newton's theory was obtained asfirst approximation, butthat inaddition, thepreces- sion oftheperihelion ofMercury (43” percentury) followed assecond ap- proximation. The magnitude ofthe deflection oflight atthesun became twice aslarge asbefore.” Andon February 8heremarks onapostcard: - “You will beconvinced bythegeneral theory ofrelativity when you have studied it.Hence Idonotlose aword todefend ittoyou.” 1Jahrbuch f.Radioakt. und Elektronik, Vol. 4,1907, further elaborated inAnn, d.Phys., Vol. 35,1911. 38.88 GENERAL THEORY OFRELATIVITY 313 inertial mass, which wasexpressed inVol. I,§3by:fheequation M,=m- (8) Only ifthisissatisfied isthe“weight” mzrar gequal tothe“inertial reac- tion” minert gandonly then istheperiod ofoscillation thesame forall pendulums ofequal length. Indetail theformula forthisperiod is vateg/Ree, (8a)Mervg Already Newton sawthataprofound physical problem washidden herein andBessel pursued theproblem bymaking extremely careful measure- ments onpendulums ofdifferent materials.’ R.Edtvés increased thepre- cision ofsuch measurements bypowers oftenwith historsion balance. However, Einstein wasthefirsttointerpret Eq.(8)inthefinal form gravitation =inertia (=world curvature). Weshall show that thisequivalence principle suffices fortheelementary calculation ofthegyinaspecific case,’ i.e.tosolve aproblem which was formulated generally inEq.(7),butwaspostponed asbeing toodifficult. Consider acentrally symmetric gravitational field, e.g.that ofthesun, ofmass AM,which mayberegarded asatrest.LetaboxK,,fallinaradial direction toward M.Since itfalls freely, K,,isnotaware ofgravitation and therefore carries continuously with itself theEuclidean metric valid atinfinity. Letthecoordinates measured within itbez,,(longitudinal, i.e. inthedirection ofmotion), y,,2.(transversal), and ¢,.K,arrives at thedistance rfrom thesunwith thevelocity v.»andraretobemeasured inthesystem Kofthesun, which issubject togravitation. Initweuse ascoordinates r,8,g,and¢.Between K,,andKthereexisttherelations ofthespecial Lorentz transformation, where K,,plays theroleofthe system “moving” with thevelocity v=8c,Kthatofthesystem “atrest’. The relations are dz,=dr//t —# (Lorentz contraction), dt,=dt-/i —B (Einstein dilatation), dy, =1d9 . (Lnvariance ofthetransversal lengths) dz, =rsin 3ddde \F,W.Bessel, “Experiments ontheForce, with which theRarth Attracts Differ- entKinds ofBodies.’ Abhandlgen d.Preuss. Akad! 1830; “Studies ontheLength oftheSecond Pendulum”, loc.cit.1826—reprinted inOstwald’s Klassiker Nr.7. *Onthehasis ofanunpublished paper ofW.Lenz. which bekindly communicated totheauthor in1944. Intheplanned publication hewillrender theargument given inthetext more rigorous. Healso intends, fo!!uwing Schwarzschild (see below), toextend theconsideration totheinterior ofasphere filled with anincompressible fluid. 314 MAXWELL’S THEORY FOR MOVING BODIES AND OTHER ADDENDA 38.9 Heyce theEuclidean world lineelement dat=dax+dy®,+ded,—c'dit (9) paases over into ad=2+Hao"+sin!ode)—ca—6.(e) Thefactor 1—6°,which occurs here twice, ismeaningful sofaronly in sconnection with our specific box experiment. Inorder todetermine its meaning inthesystem ofthesunwewrite down theenergy equation for K,,,a8interpreted byanobserver onK.LetmbethemassofK,,,moits rest mass. The equation then is: (m—mye?—GM 0, (10) Attheleftwehave thesum ofthekinetic energy inaccord with Eq.(32.7) andofthe(negative) potential energy ofgravitation. Theenergy constant ontherightwastobeputequaltozerosinceatinfinity m=moandr =~. Wehave computed thepotential energy from theNewtonian law, which weshall consider asafirstapproximation. Wedivide (10)byme*and obtain then, sincem=mo//1 —BF, 1Via =8,a=MM Goe0Ha.(70)).(100) WithM=3.3-10° Marnnandg=GMou/R’,R=radiusofearth= (2/) 10’ meter we obtain a=33-10%9(?i)=146-10"meter&1km.° r3-108, ~ Itfollows from (10a) that vi-Be1-%, 1-e21-%, (1) and hence, from (9a), 2 a? 209974gin? 2 as=aap +70"+sin’ode!)—oP—2a/r)dt.(12) This isthelineelement derived byK.Schwarzschild’ from Einstein’s Eqs. (7).InEddington’s presentation” the40components I,ofthegravi- 1Preuss. Akad., Sitzungsber. 1916,p.189. 3Seehisexcellent book: “The Mathematical Theory ofRelativity,” Cambridge, 1923. 38.15 GENERAL THEORY OFRELATIVITY 315 tational field arecomputed and (12) isshown tobetheexact solution of thetenequations contained in(7).Ourderivation claims only toyield an approximation, since itutiliges theNewtonian lawasfirst approximation and‘neglects, inthesecond Eq.(11), theterm (a/r)*; nevertheless, our result is,asshown bySchwaraschild andEddington, exact inthesense of Einstein’s theory. Itmight beasked atthispoint: What istherelativistically exact formu- lation oftheNewtonian law? The question iswrongly putifavector law ismeant hereby. The gravitational field isnotavector field, buthasa much more complex tensor character. Forthesingle point mass itiscom- pletely described bythefour coefficients gofthelineelement (12) and thevanishing oftheremaining gy. B.Observable Deductions from theGeneral Theory ofRelativity Weshall first deduce theanomaly oftheperihelion ofMercury from the line element (12); the general formalism oftensor calculus will not be required here. The law ofthe geodetic paths demands afds=o. (13) Ofthefour coordinates r,#,y,tin(12)wechoose yas“independent vari- able” and hence write inplace of(13) 3fudp=0, (18a) Fa2 F 205?4gin?o)—271— o=iat P(8*+sin’8)—2PL—2a/r), (14) .@ ;_@ ;atre -% i-e (14a) Wedesignate the“dependent variables” r,#,¢collectively byg.Themethod ofthecalculus ofvariation, which weutilized inVol. I,§34fortheproof oftheLagrange equations, leads tothe“Euler equation” (see thefirst footnote inthe section referred to) ddv av eae =0 15)dpaqag (9) For.g =#(14) yields dv_roav_rin8cos9 an » , 316 MAXWELL'S THEORY FORMOVING BODIBS ANDOTHER ADDENDA 38.16 and hence, by(15), - dd _sin8cos3 dgv » . The lastequation isfulfilled for¢=const =x/2; theother possibility 8=const =0represents noplanetary orbit, butameteor falling centrally straight into thesun. Ourdenoting theplane oftheplanetary orbit by§= 1/2isobviously simply aconvenient choice ofoursystem ofpolar co- ordinates _ Forq=¢(14)yields ov Ci(l—2a/r) av di(1—2a/r)aztO 0,sotht =0. From this weconclude ret=Bait)=e (16) Since iisrealandv(aswell asds)ispurely imaginary, theconstant also ispurely imaginary. Weputitequal toikandfindfrom (16) ; tke 7i= (17) kisafirst integration constant ofthepath oftheplanet. Forg=r(15)would yield adifferential equation for#,whose integration would provide asecond integration constant oftheproblem. Itissimpler, however, toemploy ageneral theorem, which wehave established in Vol. Iforanarbitrary variation problem and which corresponds inme- chanics tothelawofconservation ofenergy, i.e.Eq. (41.188). Wereplace Linthisequation byvinourpresent problem andtheconstant onthe right sidebyzh(numerical value oftheHamiltonian function HinEq. (41.18) ofVol. Iand, atthesame time, second integration constant ofour problem). We thus find ov P=—g-v= 1. Et ee (18) or,after multiplication with », av .<q —v= the (188)thee o ) Weevalnate theleftsidewiththeaidof(14),whereby alltermswith7,oa and ¢cancel. Itthen reduces to —1sin’9=1"since#=1/2. 38.248 GENERAL THEORY OFRELATIVITY 317 Hence (18a) yields 2r ye5. (19) (17) thus becomes 5 kor '*RTWar (20) and (14) yields, with¥=1/2, rt # 2.ck rt ipTaGap+TaBape @) Thisisadifferential equation forrwhich takes theplace ofthe(once integrated Euler) equation forthedependent variable g=rinEq. (15). Itissimplified ifu=1/risintroduced asanew variable andifitismulti- plied with 1—2a/r =1—2aw: 242 a+ul.=Qau)+LEH Pam0, forwhich wemay also write 22 Bpa=oat—LEE (22) We differentiate with respect tothe independent variable and cancel out t=du/dp. We thus obtain atua itBan. (23) Forcomparison wetreat thesame problem byNewton’s theory. Westart with theenergy equation (W=sum ofkinetic and potential energy): 1ffary +(x)_GM_W H(#)+°G@)}- 2-F. eo According tothelawofequal areas wehave, with thearea constant de- noted byhe: 209es he. (24a) Ontheleftsideof(24)wefactor out(dp/dt)* =h’c'/r', putonce more u= 1/r,andObtain after division byh'c* lose. in GM, _W gt) —retmae 318=MAXWELL’S THEORY FORMOVING BODIES ANDOTHER ADDENDA 38.25 Also here wedifferentiate again with respect toy,whereupon %cancels out, and find inview of(10a) a GM _aabun FH pe (25) The relativistic equation (23) differs from (25) only inthe correction term 3au’.This hasnoappreciable influence onthesizeorshape ofthe urbit, but affects merely theposition oftheperihelion. Torecognize this weplace thedirection ¢=0attheperihelion, which may bedefined by U=Umax and hence t=0.Then ubecomes aneven function of9.We may then, beginning with the solution of(25), expand this inaFourier cosine series: u=A+Beospt-:-; (26) itisthen found that thehigher terms, indicated by---,vanish. In(23) wesubstitute’ u=A+Bos (yg) +Ccos(2ry) --- (26a) and find forthedetermination oftheconstant here introduced from (23) theequation A+ (1—7)Bcos(ye)+(1—4')Ccos(2re)++++ =at3a(A+Bcos(yg))* =5+Bad?+6aABcos(vy)+3aB%(t+cos(279). Here wehave already dropped thehigher terms oftheseries inthecorrec- tion term. Acomparison ofthecoefficients yields @ 2 8a peAagt BaA*+rzB (1-7)B =6aAB (27) a-arto=3apt 1Ing manner similar asforthefine structure ofthehydrogen atom; thefollowing calculation maybeclearer than thecustomary astonomical one(Eddington). Itshould benoted that every deviation from Newton’s orCoulomb’s laweffects amotion of ttheperihelion oftheKepler ellipse. The motion brought about bythevariation of mass is,however, much smaller (byafactor ¢)than that arising from thegravita- tional correction. 38.30 GENERAL THEORY OFRELATIVITY 319 SinceBdropsoutofthemiddleequation, itservestodetermine 7: 1-7=60A, yX1—3ed4, 1-7=3a. (28) Amay bedetermined geometrically interms oftheperihelion andaphelion distances (aandedenote themajor axisandthenumerical eccentricity oftheellipse): Umer = tA TBH =,mxTain@(l—e) orwe= tion=2=—1_ apt... fk =nie= aPe) lorae= sothat 1 A- aa) Hence wefind from (28) 3a erte “Theprecession 6oftheperihelion inthecourseofonerevolution isalso determined geometrically, namely bytheformula - - l-7., 64a Qe+63)=2x,bj=Be—Te (29) ifaterm witha’isneglected. ForMercury thesecular displacement ofthe perihelion ishence found tobe43”, inagreement with observation. With theaidofthepreceding calculations thesecond testofthegeneral theory ofrelativity, thelight deflection attheedge ofthesun, can also be readily treated. Light paths aregeodetic lines, forwhich ds=0.Inthe special theory ofrelativity they were thegeneratrices ofthelight cone 2dz;* =0;nowthey aregiven byZgudz;dz,=0,i.e.inourcase, accord- ingtoEq.(14),by»=0.Hence wemustseth=©inEq.(19).Eq.(23) then becomes a+u= 3an’. Intheintegration itispermissible, asanapproximation, toletyapproach 1. Then(27)leadstoaA—0,C>—baBY,A>$aB’(thelastinviewof h=©), Hence, by(26a) u=82B+Bcose—SBcos(24). (80) 320 MAXWELL’S THEORY FORMOVING BODIES AND OTHER ADDENDA 38.32 For¢=0thelightistobetangent totheedgeofthesun(r=2).We must hence have 1_3a aRFR+B-SB =Btob, petit owt 1ylRivtoaB~Rita/R~ R’ since «&1kmisvery small incomparison with R.With 2=rcos9, +y=rsin g(80)then yields, after multiplication with rR: =3* aarp -letoyRmoRVe tht oRVere Thelight path comes toresemble ahyperbola, justasthepath ofthe planet resembled anellipse. Withtheassumption |y|>>|x|weobtain 3a le 2aReappytrts Ryne Ry (31) Theangle between thetwoasymptotes, which isequal tothedeflection ofthelight from itsoriginal path, is4a/R =1.75” andagrees wellwith theresults ofthesolareclipse expeditions. Itistwiceaslargeasthevalue obtained byanéarlier more primitive calculation (Soldner aswellasEin- stein before 1915; seefootnote onp.311). Wenote furthermore inthisconnection that inaddition tothedirection, thevelocity ofthelight ischanged bythegravitational field. Inaradial direction, e.g.along theradius 9=0,taking account ofds=0,itis drqi7(t~2a/ryeby(12). (81a) Finally, thelastoftheenumerated tests ofthetheory, theredshiftofthe spectral linesinthegravitational field, canbeunderstood without anycaleu- lation. Consider apoint ofthecurved world andconstruct there the(grav- ity-free, Euclidean) tangential plane. Letthecoordinate changes inthe latter, dX,,--- ,dX, =tedT coincide indirection with thecoordinate changes dr,---,dz=icdtinthegravitational field. Inview oftheequal- ityofthetwolineelements wethen have foraparticle atrestwhich is radiating light —dT=—3—2a/r)dt. (32) The measures oftime dtand dTarehence different; thesame applies to_ thefrequencies »and (inabsence ofgravity), which areinversely propor- tional tothese times. According to(32) wehave 38.34 GENERAL THEORY OFRELATIVITY 321 v=VI—Qajrm =(1—a/r)m ~_e (33) ym—=H. . Thefrequency isreduced bythegravitational field.Inviewofthemeaningofa,given byEq.(10a), themagnitude oftheredshift is 1GM _|P|ay @? (4) where V(Eq.(10)) isthegravitational potential. Thespectrum ofSirius Bandofother white dwarfs provides theexperimental confirmation. C.Unified Theory ofGravitation andElectrodynamics Following Gauss andRiemann, Einstein putmetric first,ie.,required theinvariance ofds*andthetensor character oftheg,,.“Balance, rod, andclock” werethebasicelements which hemanipulated inthegeneral, justasinthespecial theory ofrelativity. Withthemhewasabletogeo-metrize gravitation. However, Maxwell’s electrodynamics ofvacuum alsoconstitutes acom- plexofphenomena overshadowing material Processes. Theamazingly sjmple formwhich itassumes inthespecial theory ofrelativity andwhich thaybetransferred without appreciable changes totherealm ofthegeneral theory ofrelativity, covering arbitrary frames ofreference, suggesta simi- larly geometrization. However, themetric Proves toorestricted forthis. Einstein attempted tobroaden itbydemanding, instead oftheinvarianceofds*,merelythatofde”=0(i.e.thatofthelineelements ofthelightcone).*Wearethenconcerned onlywiththeratiosoftheQs,ratherthanwiththega,themselves. Hermann Weyl hadrecognized even ashort time before thisthatitwas simpler andmorenatural todropthemetric departure andtobegindirectly withEinstein’s I,.Theresulting system isknown asaffineworld geometry. Itprovides arulefor“parallelism atadistance”, i.e.aprescription for proceeding along aworld linewithout departing from theinitial direction. Bothgravitation andelectrodynamics fitintothissystem quite naturally. TheTy,were hereassumed symmetric inthe»and».Theresult ofthis theory isrecorded inhisclassic book “Raum-Zeit-Materie”, Springer, Berlin, 1918.7 However thesystem canbegeneralized even further: I,andI,can ‘“On @reasonable extension ofthebasis ofthegeneral theory ofrelativity”, Preuss. Akad.1921,p.261,aswellasthefollowing notesontheunified fieldtheory:loc,cit.1995, p.414;1928, p.3;and1929, p.3. - *English edition: H.Weyl, “Space-Time-Matter,” Methuen, London, 1922, 322 MAXWELL’S THEORY FOR MOVING BODIES AND OTHER ADDENDA 38 bechosen tobedifferent. There results anasymmetric affine theory, which gives risetoanew antisymmetric tensor. Ithasbeen sketched byErwin Schrédinger! and isbeing developed byhim infriendly competition with ‘Theincentive wasprovided bythefollowing: Nuclear physics hasjoined atomic physics asayounger sister science. Whereas atomic physics from the corpuscular standpoint rests ontheelectrodynamic interactions be- tween electrons and protons, theforces ofnuclear physics must beascribed tothemesons, which, inthemeantime, have come to.be demanded by theory andhave been discovered experimentally. (The name meson derives from thefact that these elementary particles have amass intermediate between those oftheelectron and theproton). Nuclear physics hence is meson theory. Itdemands anantisymmetric tensor differing from that of electrodynamics. Such atensor isfurnished bytheasymmetric affine world geometry, which thus would create atriple bond between gravitation, electrodynamics, and nuclear theory. Itsdetailed structure has notyet been determined, however. When atlast ithasbeen fully elaborated Max- well’s theory, too,willberevealed initsfullbeauty andsymmetry. 1Beehisnote inNature, May 13,1944, and, following it,several papers inthe Proceedings oftheIrish Academy fortheyears 1944-46, thelast, with thetitle “The general affine field laws”, inVol. 61,p.41. . SYMBOLS EMPLOYED THROUGHOUT THE TEXT AND THEIR DIMENSIONS Note: Asamatter ofcourse allequations inthis volume arewritten ina dimensionally consistent manner and hence nottied toanyparticular choice ofunits (e.g. M=meter, Q=coulomb). They are,touseapre- >ferred current expression, “equations ofquantities”. The “numerical equations”, which arecorrect only foraspecific choic ofunits, arefortu- nately falling more and more into disuse even inengineering. e,q charge Q,practical unit: 1coulomb pcharge density Qu? «surface charge density Qmu* p,Pmagnetic polestrength QMS" (in§8Pserves asfifthinde- pendent unit) pmmagnetic density Qu’s* @m magnetic surface density Qm™'s* E.,electric fieldstrength newton/Q =MKS“Q™, 1newton =10°dynes Delectric excitation (displace- QM™, divD=p,D,—D',=u, ment) Eq.(3.11) Jconduction current density QMS” =AM™, A=ampere Itotalconduction current QSst=A l=|J,do DBdisplacement current density QMS" =AM Ctotalcurrentdensity =J+D QMS" =AM*Velectric potential difference joule Q™=volt =V; =[Bas Bmagnetic field strength (induc- newton/P =KS’Q" =VSM”, tion) divB=0;1gauss=10vsM* Hmagnetic excitation (amperee PM =QMS =AM“ divH turns permeter) =pm, Hn —H'n =wm, 1oersted =10°/(4x) AM™ U_mag.tiePotential difference =ilH-ds Qs?=A &magnetic flux=fB,de M'ks"Q"! =VS 328, 324 SYMBOLS AND THEIR DIMENSIONS (electricpotential - woOE=—grad¥, joule/Q =V WY, magnetic potential H=—grad¥,, Qst=A . Avectorpotential, B=curlA MKS“Q™=VSM7*Sradiation vector =Poynting joule M“S* =watt M~ vector =energy flux density Wenergy density, joule M* W=W.+Wa, W.=4D-E, Wa=}H-B W,Joule heat perunit volume joule M“S* =EJ W energy ingiven volume joule edielectric constant Q/Goule M) =SMa" = farad-M™* permeability MKQ™ =98M" =henry-M~ £0,#oVacuum constants (eau) =velocity oflight,MS(uo/e0)' =wave resistance, 2 ¢conductivity M“K"'SQ’ =Ma” , e’=+ic/w =complex dielectric M“K™'S'Q’ =SMa" constant Rtresistance ofawire volt/ampere =M’KS’Q™ =@ L_selfinductance 98=henry R_impedance =R+toL Q Kcapacitance Q'/joule =9S =farad Pdielectric polarization QM™, D=&E+P,seep.74 M_magnetization QMS”, B=wo(H+M),seep.91 1electric susceptibility pure number, P=1eQE, €= eo(1 +2) xmagnetic susceptibility pure number, M=«H,»=yo(1+x) angular frequency S7, wo=2x/7, 7=period of vibration kwave number invacuum M7, k=2x/d,\ =wave-length k,wave number inconductor M7, ki?=ewe+inow=e'ya" hwave number ofsurface waves M™* oncylindrical guide xin§20to25=VWyow/2 foral-M™', 1/x=d=layer thickness in ternating currents skin effect Additional Symbols inPartsITIandIV 2%imaginary timecoordinate n=it deworldlineelement ae=2de} SYMBOLS AND THEIR DIMENSIONS 325 drelement ofintrinsic time =~dr=/1 —fdi,6=v/c ds/(ic) yimaginary angle ofrotation intany=if the Lorentz transformation R_four-dimensional radius vector R= 2,22,2, 2% V,W four-vectors ofvelocity and V=dR/dr, W=dV/dr acceleration Qfour-potential =A,i¥/c vsM" .Ifour-current =J,icp QM~S™, invacuum =p(y,ic) Fsix-vector ofthefield VM", F=cB,—iE=cCurla fsixvostor oftheexcitation AM,=H,~ieD=2Curla F*,f*dual six-vectors F*=—iE, cB,f*=—icD, H 5 1 1 1 ALagrange density A=gfF=5H-B-5D-E Msecondinvariant ofthefield M=grr =cB-E kforcedensity=1r-F yaa=QE+¥XB)b=ve K_Lorentz force, three-dimensional K=e(E+vXB) F Ybur-foree, F-V'=0 Pua=og =OEE four-force,F-V= a8=iF Vi B Tstress-energy tensor Tan==2ParSur +BumA = Gmomentum ofpoint mass G=m(y, ic),m=m/V/1 —B Eyrestenergy Ey=mo? K_kinetic potential K=me? (1-VI —#) —Y-Q Schwarzschild invariant r-Q =pvA—¥) £*\ Abbreviations intheequations E* = E+v XB Ht formoving media H*=H-vxD ALorentzsymbol a+vdivA—curl(vxA) J.conduction current density h=J-ov R_in§36: reaction force ofradia- depending ontheframe ofreference tion also denoted byR’orR* < sealar product oftwofour-vectors ramDro, yields&scalar f APow Pp scalarproduct ofafour-andasix-mr,p>TaFam vectoryieldsafour-vector 1 , sealar product oftwosix-vectors SP=5LLSomPan yieldsascalar ‘ =BDfonPoma<m mot 326 SYMBOLS ANDTHEIR DIMENSIONS (RX V)am =RaVin —ReanVn “vector product oftwofour-vectors yields asix-vector ..8, divergence ofafour-vector yieldsa Diva=x,ar, scalarCurke =a2,_32, curlofafour-vector yieldsasix-orOm vector ( . OTmn vector divergence ofa(general or Div? x,“Orn; alsoantisymmetric) tensor .+ .iwiF=Div,F*=OFmn yieldsafour-vectorDiva=DivaFx,Orn dualvectordivergence ofasix- vector Div*F=0 results inF=CurlQ DivDivF=0 applies forevery six-vector : 43DivCula=GrdDiva-Oa o%= >5% : nat Or, Numerical Values, Results ofMeasurements, andDefinitions ¢=velocity oflightinvacuum =3.00-10* MS (measurement) uo=‘permeability ofvacuum =4x-10-"9SM™ (definition)e0'=dielectric constant ofvacuum =10'/(4xc*) M“'so* +107*/(36.00%)M~'SQ* (consequence) (uo/)! =waveresistance ofvacuum =120.002 (consequence) e=electronic charge =1.60-10"" Q(measurement) e/m=specific charge oftheelectron =1.76-10" Q/K(measurement) my=restmass oftheelectron =0.90-10 K(consequence) eV=electron volt =1.60-10~" joule (consequence) mec=restenergy oftheelectron =}million eV=0.81:107"joule(con- sequence) PROBLEMS FOR PART I I.1. The Boundary Conditions ofMaxwell’s Theory. Derive Eqs. (3.7a) to(3.12) forE,B,H,and Dbythedifferential method. The transition from medium 1to2must then beassumed tobecontinuous (“boundary layer” instead of“boundary surface”). Use arectangular coordinate sys- tem z,y,zand letzbeperpendicular totheboundary surface which, in thelimit ofinfinite smallness, may beregarded asplane. Intheboundary layer (h—0)thederivatives with respect tozoccurring inthedifferential equations (4.8) must becontinuous inorder that these equations may be meaningful. . I.2. The Magnetic Excitation Inside and Outside ofanInfinitely Long Wire. Proof ofEqs. (4.10) to(4.13) from thedifferential equations. 1.8. The Magnetic Excitation within anInfinitely Long Solenoid. Proof ofEq. (4.14) from Maxwell’s equations. ¢ bo? Fig.44, / D=A By (fe f / y y oO 1.4. TheCosine Law ofSpherical Trigonometry asSpecial Case ofaGeneral Vector Formula. Prove the vector formula: (AXB)-(C XD)=(A-C)B-D) —(A-D)B-C) anddeduce from it,forthespecial case D=Aandwith reference toFig. 44, the cosine law cosa=cosbcosc+sinbsinccosa. PROBLEMS FOR PART II II.1. The Charging Potential ofaConducting Ellipsoid ofRevolution. Let abethemajor axis, btheminor axis, and c=+/a? —&thelinear eccen- tricity. For-fixed ¢andvariable a tty 7a-atan! 327, 328 PROBLEMS, ANSWERS AND COMMENTS represents thefamily ofconfocal ellipsoids with theseparation offocal points 2c.Show thatoneachofthem theexpression for¥given iaEq. (9.4) isconstant (independent ofz,y,z). 11.2. TheInfinitely Long Rubbed Glass RodanditsComparison withthe Conducting Paraboloid ofRevolution. Calculate thepotential ofaninfinitely long uniformly charged straight lineterminated atoneend, andshow that itsequipotential surfaces arethesame asthose fortheconducting parab- oloid ofrevolution which isobtained from (9.4) bytransition tothelimit C4 ea w, 11.8. Comparison oftheDielectric andtheConducting Sphere. Foradielec- tricsphere r=aplaced inanoriginally uniform electric fieldthere always exists aconcentric conducting sphere r=b<a,whose exterior fieldagrees, forr>a,with theexterior fieldofthedielectric sphere. Fig.45shows, for r>a,thefield ofthedielectric sphere, fora>r>b,notthe(uniform) field within thissphere, buttheanalytical continuation oftheexterior field, which isidentical with thefield oftheconducting sphere ofradius b. Prove that :e-—l baag/Sat. ‘Thefigure shows howthesingularity oftheequilibrium pointofthecon- ductor (seefootnote concerning Fig. 9a)develops continuously from the regular behavior ofthe force lines forthe nonconductor. 11.4. Edge Correction forthePlate Condenser According toKirchhoff. Con- vince yourself that the relation _2 _ Qeit Qxit’ z=a+tyz=5SO),S)=142—exp(24) ery represents thefringe field oftheunilaterally terminated condenser in Fig.46.¥=const aretheequipotential lines inthez,y-plane,@ =const, thelines offorce. Show that thetwofamilies ofcurves correspond qualita- tively tothedotted lines inthefigure, and that theline offorce 6=0 (drawn asfulllineinthefigure) isanareofacycloid which joins thetwo edgepoints z=0,y=Oandz=0,y=a. II. TheCapacitance ofaLeyden Flask (Cylindrical Condenser). Letthe dimensions be:height h=20cm,inner radius r;=5cm,wall thickness d=1mm. Letthedielectric constant oftheglass be6&.Boundary cor- rections aretobeneglected. The capacity istobeexpressed inmicrofarads. II.8. OntheDefinition oftheCapacitance ofTwo Conductors with Equal andOpposite Charges. Ifin(10.15) weputE,=—E, =Eandv=¥,—Ws weobtain 2W=VE=(Wu+Ha—2Hy)E* a)=Kui tKev: +2Kwhis. PROBLEMS, ANSWERS AND COMMENTS 329 rid oy t 1 og ‘Point ofequilibrium LoS 1 !aymoX< \[Prefth}— aryi[ssa rayOe 1SEIS<oy i \1otKIS pot\ !1 toy +t \\ Vt Uot _rt Fra. 45.The field ofthedielectric sphere ofradius r=a,which isproduced by& urfform field ontheoutside, yields, when continued analytically intotheinterior, atthesame time thefield intheexterior ofaconducting sphere ofradiusr =b<a which isproduced bythesame uniform field. a Pa tiNES wav i ATK veneer ererearers es=gnalae Seyee2 SEESEEPRY —T— Bethenagintnne\fo ~. on ‘ 1 Fie. 46.Shape oftheequipotentials ¥=const and lines offorce #=const at theedge ofaplate condenser. Show bycomparison with (10.11) thatthefollowing relation exists between theelementary definition ofcapacitance Kandthecoefficients H,;and Ki: 1 at »|KuKu| =o eS —“‘é‘ “= K=4Hu—He~Ky+Ku+Ku [KuKu) 330 PROBLEMS, ANSWERS AND COMMENTS 11.7. Characteristic Oscillations and Characteristic Frequencies ofaCom- pletely Conducting Cavity Bounded byaRectangular Parallelepiped. Using Eqs. (24.9) and (24.10), represent thecompletely continuous field within arectangular parallelepiped with thesides a,b,c,with thecondition Htang = 0onthethree pairs ofbounding surfaces ral? _{0 70a? Y= Br c* 11.8. Characteristic Oscillations and Characteristic Frequencies ofthe Interior ofaPerfectly Conducting Circular Cylinder ofFinite Length. Using Eqs. (24.6) and (24.7), represent the continuous field within acircular cylinder ofradius aandlength J,with thecondition Frag =0both onthe mantelsurfacer=@dndonthetwoendsurfacesz={. 11.9. Characteristic Oscillations within aCavity Bounded byaMetal Sphere. Asin§19, start with aHertzian vector «which isdirected along a diameterofthesphere(0={),isperiodicin¢,andotherwisedepends only onr.Incontrast with §19+must now becontinuous also atthe center ofthesphere. The state then corresponds nottoaspherical wave emitted from this.point, but toasuperposition ofaspherical wave radiated outward and aspherical wave (reflected bythespherical surface) radiated inward. Determine the characteristic wave numbers kand the corre- sponding characteristic frequencies w=kefrom theboundary condition Evang =0atthesurface ofthesphere r=a. II.10. Determination ofthePropagation Constants ofWire Waves from Kelvin’s Telegraph Equation and from Rayleigh’s Alternating Current Re- sistance a.foraLecher two-wiré line, b.forreturn conduction through the ground (let conduction intheforward direction beperfect). PROBLEMS FOR PARTS III AND IV III.1. TheLorentz Transformation foraRelative Motion Deviating from the x-Azis. Letabetheangle between therelative motion vand thez-axis ofthe “system atrest”. Letthezy-plane ofthelatter coincide with theplane through zand v.We consider an“intermediate system” 11, 1,4%,4 whos 2;axis istocoincide with thev-direction and whose z,y:-plane coin- cides with thezv-plane. The trahsformation /.BAStIH Wah, eZ kat (1) isthenanordinary rotation through ainthezy-plang. Letamoving sys~ temzi,yi,#1,tibesoplaced thatitsz;-andyi-axes agree with the2- andy;-axes oftheintermediate system fort=0,’=0.Thetransformation PROBLEMS, ANSWERS AND COMMENTS 331 TMA hah, MM, AA (2) isthen aspecial Lorentz transformation and ishence represented byEq. (27.10). pffinally, 21,yi,21,¢iisrotated againthrough theangle—aintheziyi-plane, corresponding tothetransformation ya, 472, ,2,0, mer,hat (8) thesystem ofcoefficients oftheresulting total transition Zy2ztoz yi2e (4) issimplified. Convince yourself oftheobvious fact that thistransformation isorthogonal infour dimensions and ofthenotobvious fact that itmay berepresented byathree-dimensional vector formula. IIT. OntheAddition Theorem forTwo Differently Directed Velocities. Prove Einstein’s formula (27.192) bythe method oftheLorentz trans- formation. IIIS, The Field ofanElectron inUniform Motion. Transform the representation (30.6) bymeans ofconsiderations ofelementary geometry applied toFig. 42into therepresentation (28.14), (28.14a). "HIT4. Onthe‘Relativistic Energy Theorem fortheElectron. Derive the expression (32.7) forthekinetic energy and theenergy theorem (32.6) from theequation ofmotion (32.5) oftheelectron bytheusual method (scalar multiplication with thevelocity). III.5. The Electron intheUniform Electrostatic Field. Anelectrou enters a(vacuum) condenser with atransparent upper plate with thevelocity v atanangle «.Lettheplate separation bed,thepotential difference ofthe upper with respect tothelower plate, Vvolts. What curve does the electron describe inanon-relativistic treatment? How closely does itapproach the lower plate? For what velocity does itreach thelower plate? (Example: »=5-10° meter/sec; d=107meter; V=110volts.) What potential field must anelectron which isinitially atrest traverse toattain thevelocity »=5-10" M/S? How dothe conditions change forarelativistic treatment? ITI.6. TheElectron inaUniform Magnetostatic Field. Iftheinitial veloc- ityoftheelectron isperpendicular tothelines offorce acircular path is described. Determine itsradius. Ifacomponent parallel tothelines of forceispresent, thepathbecomes «helixwithcircular projection. This applies intherelativistic justasinthenon-relativistic case. - III.7. The Electron inaUniform Electric Field and aUniform Magnetic Field which isParallel thereto. InKaufmann’s arrangement forthemeasure- ment of¢/m thef-rays emitted byaradium sample pass first through a 332 PROBLEMS, ANSWEKS ANDCOMMENTS narrow aperture Dandthen cross auniform electric field +#andmagnetic field Bparallel thereto (+2 signifies reversal ofpolarity ofthecondenser). Assume that both fields begin attheaperture Dand reach tothephoto- graphic plate, mounted perpendicular tothe beam direction atadistance afrom DWhat curve isrecorded onthe plate ifthe B-rays areemitted with allpossible velocities v?Neglect thechange inthetotal velocity as compared with thelarge v,and represent thecoordinates ofthepoints of incidence asfunction ofthe parameter v. III8.The Electron inaUniform Electric Field and aUniform Magnetic Jjield Perpendicular thereto. The path isatrochoid. Under theinfluence of r 110 volts a +Ovolts Fac. 47.The electron describes aballistic parabola inthe uniform condenser field.« Phot. PI. > AA 1 v v Fic. 48.Kaufmann’s arrangement forthemeasurement ofe/m. Attheleft: Lat- eral view, Ontheright: The pattern observed ontheplate for8-rays with acon tinuous velocity spectrum emitted bytheradium sample. theelectric field thecircular motion preduced bythemagnetic field iscon- verted intothemotion ofapointonarolling circular disk.Forwhatinitial conditions isasimple cycloid obtained? Motions ofthis type occur inthe“magnetron” electron tube. IIT.9. The Characteristic oftheThermionic Diode According toLangmuir andSchottky. Inpractice acylindrical configuration isgenerally employed: The cathode isawire along theaxis ofcylinder whose mantel surface coin- _ cides with theanode. The plane configuration ismathematically simpler: Here thecathode at«=0and theanode atz=/areplane circular disks separated attheedge byaninsulating cylindrical tube. LetV(z) bethe PROBLEMS, ANSWERS AND COMMENTS 333 potential atthepoint zbetween thecathode V(0) =0and theanode V(I) =V.Letthenumber oftheelectrons leaving thecathode persecond besolarge that itispermissible toassume acontinuous space charge —p oftheelectrons. Between cathode andanode thePoisson equation AV(z) = @V(z)/dz* =p/e isvalid. The current density J=pvtransported through: thetube isindependent ofx.visdetermined from mv"/2 =eV(z). Integrate thePoisson equation byassuming apower lawanddeduce here- from thesocalled characteristic ofthetube (Iasfunction oftheapplied ~voltage V). Convince yourself that thesame method ofassuming apower law is applicable also tothecylindrical configuration. III.10. TheAcceleration ofanElectron intheBetatron. Inthebetatron’ electrons areinjected intheplane ofsymmetry between theaxially sym- metric pole pieces ofanalternating-current electromagnet. The magnetic fieldforces themintoacircular orbit. Thepulsing ofthemagnetic fieldis accompanied byavortex-like electric field which accelerates theelectrons intheir orbit. There isaradius r=7»ofthepath which remains unaltered with thepulsation ofthemagnetic field and with increasing electron veloc- ity.After countless revolutions theelectrons reach avelocity approaching thht oflight; they then resemble thebeta-rays ofradioactive materials, whence the name “‘betatron”. Let the axially syrometric magnetic field distribution B(r, ¢)(its axial component) between thepole pieces, which decreases monotonically out- wards, and the tangential initial velocity v,ofthe electrons begiven. Werequire 1.the attainable momentum mvoftheelectrons, their velocity, mass, and energy ineV(electron volts), 2.theradius 7»oftheequilibrium orbit, 3.thefrequency ofrevolution attheend oftheacceleration period and the total number ofrevolutions, and 4.the reaction force oftheradiation atthis point. IV.1. TheField ofUnipolar Induction. Letabarbeinserted inauniform magnetic field Bperpendicular tothelines offorce and bedisplaced with uniform velocity along itsaxis. Compute a.theelectric field intheinterior and itspotential, b.the voltage between itstwo sides, c.theexternal field, specifically foracircular cross section, and d.thesurface charge. e.What thange results ifwepass from uniform translation ofthebar 'This hasalso been called arheotron, beam transformer, orelectron centrifuge. The original idea ofthedevice was given inthe Aachen thesis ofR.Widerde inthe year 1928. 334 PROBLEMS, ANSWERS ANDCOMMENTS along itsaxistouniform rotation ofabody ofrevolution about itsaxisof symmetry? ANSWERS AND COMMENTS 1.1. The derivatives . ok, oH,ede occur inthex-components ofEqs. (4.8). These must remain finite inthe transition tothelimit h—0sothattheleftsides oftheequations inques- tion donotbecome infinite, since this would make B,and D,infinite. In view ofthey-components ofEqs. (4.8) thisapplies alsotothederivatives OE, “oH, Thecontinuity ofthetangential components E.,Ey,Hs,Hy(see(3.9) and (3.8a)) follows herefrom. IndivB=0(Eq. (4.4a)) there occurs thederivative 0B,/az, which must alsobecontinuous intheboundary layer: hence thecondition forthecon- ;tinuity ofthenormal component B,,Eq.(3.72).Thez-component ofthe “Maxwell equation B=—curlEdoes notsuffice forthisconclusion. Itis truethat ontheright there occur only thetangential components EF,and E,anddifferentiations with respect tozand y,sothat theright side is continuous. However, thecontinuity oftheleftside, which may bededuced herefrom, would beconsistent with atime-independent discontinuity of B,.Hence theauxiliary condition div B=0becomes necessary. The same considerations, applied toEq.(4.4b) fornonconductors, divD =p,leadinthelimith+0top—>«.Thisconclusion isbynomeans toberejected, butindicates that attheboundary oftwononconductors there may exist asurface charge ,which corresponds toaninfinitely great charge density and isequal tothejump inthenormal component ofDonthetwosides. Bythez-component oftheMaxwell equation D= curlH,thisjump must betime-independent. Inthegeneral case ofacon- ductor andanonconductor surface charge may alsooccur, inaccord with Eq.(4.4c) orthez-component oftheMaxwell equation D+J=curlH; however, thissurface charge need notbeconstant, butmay decay asindi- cated bythecurrent density J,intheconductor. 1.2.Weconcern ourselves only with themagnetic field. Oftheelectric field, which weshall investigate ingreater detail in§17,weneed only know that it-gives risetoauniform current field J,within thewire, 0<r<a, andtoanalsouniform current fieldJ_,,correspondingtoanequalbut oppositely directed total current J,inthereturn conductor: I=rad, =—2(¢ -BJ4. (1) PROBLEMS, ANSWERS AND COMMENTS 335 Weemploy polar coordinates r,y,2with r=0astheaxisofthewire. In view ofthe symmetry ofthe problem CJ aRe 0 forallthree components H,, H,, H,. The g-and z-components ofthe Maxwell equation curl H=Jthen, with reference tothetable inProblem 1.3 ofVol. II,reduce to J, ,0<r<a, dH,_ ld _10sa<r<db,OD md EHO Ve <rcc, ® 0,c<r<oe, whereas ther-component takes ontheform 0=0. From (2)weobtain A,=const =0, thelatter since H,certainly must vanish forr =©.The fact that H,must also vanish follows from the condition div H=0.The magnetic lines of excitation H=H,arehence coaxial circles about r=0.Wewrite H,= ‘Handtabulate theintegration of(3)below: Differential Equation Solution Determination of Constants o<r<a Smad Had 44A=OsinceH(0)is finite dome 2B B=J,a'/2=I/(2x)by a<r<dgH) =0 Hes continuity ofHat r=aand byEq.(1) d _ r,¢ ,_I e ber<egh) =JorHeJagts C=-5 Jas -21¢. Pe a bycontinuity ofHat r=bandbyEq.(1) d D D=0bycontinuity ofe<rc@gGhn=0 Has Hatr=candby Eq. (1) This solution agrees with Eqs. (4.10) to(4.13). 1.3.Coordinates r,g,zandfieldeymmetry 8/ap =9/dz =9asin13; aand binner and outer radius ofthe solenoid. Now adie fora<r<h, Fate=G% Jfforr<aandr>b. 336 PROBLEMS, ANSWERS AND COMMENTS Thedifferential equation curl,H=0isfulfilled throughout bysymmetry, whereas thedifferential equation curl ,H=0demands d(rH,)/dr =0, H,=A/r, A=0because ofthecontinuity ofHatr=0.Similarly H,= Osince div H=0.The table shows how H,istobeevaluated: Differential Equation Solution Determination of Constants aH, const=H=inte- O<r<a379 H,=const riorfield OH, _ -_f A=Hbecause of e<r<bor JeHe[Jedr+Acontinuous joining with interior field b<r<oMig .a8 B-H-[ Jar because. ofconti- nuity atr =b » SinceH,=Oforr=2wemusthaveB=Oandtherefore H=[Jdr, :le witighisequivalent withH=NilinEq.(4.14):[Jydristhetotaleur- rentwhich passes through thecross section ofthesolenoid ofwidth b—a andlength 1inthez-direction. This current isNJ, where N,isthenumber ofturns perunit length. J.4. The proof ofthevector formula isobtained directly iftheab- breviation P=AXBisintroduced, andthecyclic permutation ruleP.(C xD)=C-(XP),andtheformula (6.2a), areemployed. Toprove thecosine lawset D=A.With A,B,Casradii oftheunit sphere wehave A-B=cosc,---,|AXB| =sinC,--- ‘Theanglebetween thedirections ofAXBandAXCisequaltothe angle ainthespherical trang mapped outbyA,B,andC. IL.1. From theequation oftheellipsoid given intheProblem wecalculate ‘ 2 2 2 (gt— ~% t+ty=@ A(t‘), é cz\" fgt+yY4+eted +58+re=(042). Inorder coobtain thenumerator anddenominator occurring intheloga- - rithm in(9.4) therootmust beextracted sothatitispositive forall|z|= a.This leads to PROBLEMS, ANSWERS AND COMMENTS 337 tet ViftPF ES=ztotatS= a+o(1+2), toot VEERFEW =ete Fa@-H(i42). Thequotient ofthetwoisequal to(a+c)/(a —c),i.e.aconstant foreach one ofthe confocal ellipsoids. Thus Eq. (9.4) isproved. Itistrue that thisconstant changes ifweproceed from anellipsoid with principal axes a,>a,b;>6,cinstead offrom that with axes a,b,c.This does not, however, affect theidentity ofthefields intheexterior ofthe former ellipsoid, since nophysical significance attaches tothenumbering oftheequipotential surfaces. 11.2.Lettherubbed glassrodhavetheconstant chargeAperunitlength and beinfinitely thin. According to(7.5b) itspotential isgiven by at aev=|fVety+e- =doge-f+VFFRFEDS = etvetyte \ »ee +¥ +const. The constant here becomes infinite, namely inasense equal to—)log0. Eq. (9.4), with z’=z+c, E=2cleads tothesame expression (with 2!replacing z)inthelimit c+©.Theequipotential surfaces oftheparab- oloidal conductor agree with those oftheglass rodofcourse only outside oftheformer, since within it¥=const. IL.3. According toEqs. (9.13) and (9.14) thepotential ofthedielectric sphere, forr>a(e=relative dielectric constant ofthesphere with refer- ence toitssurroundings), is: _ e-1a w=F(+£48)cos This isatthesame time theanalytical continuation ofthepotential into theinterior ofthesphere. Forr=b<aityields __ e-1d w=roaSef) ose. Ifhere we set s/e—1 braV5+2? W,becomes independent of©,namely, aswemust demand forthegrounded ~ conducting sphere, ¥,=0. 338 PROBLEMS, ANSWERS ANDCOMMENTS 11.4. This problem ismathematically related totheconformal mapping problems inVol. II,§29, 30,31,and makes useoftheidentity oftwo- dimensional potential theory and thetheory offunctions ofacomplex variable z=2+iyelucidated inVol. II,§19.The combination ofthe potential ¥and thestream function toform thecomplex variable ¢of thepresent problem wasdiscussed atthat point. Wegive zasfunction of {,rather than ¢asfunction ofz,because zisasingle-valued function of¢. The proof ofthemapping function f(¢) rests onthefollowing: 0 0 - Qed IfwesetY={0wefinds)-(o,}+L-e-—ee=; Hencey={2,2£1—9-6)S$0for—w <<+m. The{lonercondenserplatey={°,2&0isthusatthepotential upper a 0v-{f. Also thebisecting plane ofthecondenser isanequipotential surface. IfwesetY=V/2,wefindf(t)=xt+1—¢+e;hencey=a/2,2= Pati eo+€*).However, nowz20as¢varies between —©and +. Wearetherefore dealing notwith asemi-infinite straight line or plane, butwith abilaterally infinite line orplane. The boundary points =0,y=0and z=0,y=a,corresponding tov =0, =Oand¥ =V,@ =0,respectively, arebranching points of theconformal mapping. Thelineofforce#=0,whichjoinsthetwobound- arypoints, isgiven inparametric form (with y=2x¥/V) by a a .z=7(1—cosy), y=9,—siny). This istheequation ofthesimple cycloid (seee.g.thequite similar repre- sentation inVol. I,Eq. (17.2)). The lines offorce intheinterior ofthecondenser and atalarge distance from theboundary points belong totheparameter values eri 0<¥<2r. Since here exp(iy—¢)vanishes with increasing ,weobtain simply ‘ iy=Zt), i =f£a- -%rts t+w—9), ie s=F 0)vray Since @=const onthese lines offorce, wehave also g=const and hence a=const. Ontheother hand,¥varies onthembetween 0andV,andhence PROBLEMS, ANSWERS AND COMMENTS 339 ¥between 0and2xandybetween0anda.Thelinesofforcehenceapproxi- matecloserandclosertostraightlinesperpendicular tothecondenser plates, asistobeexpected. IL5. Thedifferential equation ofthepotential inthecylindrical co- ordinates r,y,zis,when independent ofyandz, ld dv rarar = >Ityields dv A —eAoT >D,=——. Hence thesurface density ontheinner andouter electrode is =—24 =t4 =SS andthecharge perunit length ofthez-coordinate €=2nrw, =—2red, &=Qnryo, =+2reA =—G. Atsecond integration leads to -Ww=eVe- Tr1joglt V=Alogr+B, W-WwW=aV Alog|inelogPa sothat Ki=+=Ine/log?=capacityperunitlength. a For d«7wefind TL 4)a4logs=log(:+2)25 and, neglecting end corrections, K=Kh=27he _surface aisectric constant,dad separation InMKSQ-units: nh=100cm’ =10°M’, d=1mm =10~M, Lio? _@_x1.10fared=!-10-*microfarad ©=be&10jouleM’K-35 10™farad 310“microfarad. _ 11.6. Thefirstoftherelations given inEq.(2)oftheProblem isobvious inview ofEq. (1).The second isobtained asfollows: 340 PROBLEMS, ANSWERS AND COMMENTS By(10.14): E=Kw +Kis, (1) —E =Ko, +Kath. Hence 0=(Ku +Kn) +(Kir +Kno. (2) Inaddition, thefollowing linear relation exists between Y,and ¥:: V=y—W. (3) From (2)and (3)wecompute Ku+Kn Ku+Ky SeeeA ee ee Ku +Ku +Km *” Ru 2Ku +Kn Substitution in(1)yields” KnKn —Ku! Es SV: Ku +2Kn +Kn The factor ofVisthecapacity intheelementary sense. Hence also the second relation (2)oftheProblem hasbeen proved. IL.7. Ifthephasé factor which must bethought ofasadded to(24.9) namely exp(iha),isreplaced by{°xl=, whereJisaninteger (standing instead oftravelling wave)andif°°ischosen withdueregard ofthe boundary conditions prescribed fortheindividual E-components, apar- ticular characteristic electromagnetic oscillation oftheinterior oftheparal- lelepiped isobtained forwhich Hz=0;similarly, proceeding from (24.10), one for which Zz=0isobtained. However these arenotyetthegeneral characteristic oscillations ofthe parallelepiped, asisevident from thespecific values H,=0and EF.=0, respectively. The general system, which has complete symmetry with respect tothethree axes, is E.=Acos(x2)sin(=¥)sin(xm‘). a, b. Ve, =Bsi z Y)gi 2 E,=Bsin(=2)cos(=A)sin(=m2), (1) EB,=Csin(x5)sin(=»)cos(em‘), a, b ¢, PROBLEMS, ANSWERS AND COMMENTS 341 2H.=A’sin(=®)ood(mn?)cos(m2), &o a b ¢, f/*H,=B’cos(x®)sin(m¥)cos(rm‘), (2) fo Q, 6. ¢, f#H,=C’cos(+12)cos(rn)sin(+m2) & a, 6. ¢, Since divE=0theA,B,Cmust satisfy thecondition Aleptsct=o, @) a 6 c whereas theA’,B’,C’aredetermined from theA,B,Cbytheequations +thAl=Fe-=B, iBa™4FG, 4) c a ine = Ba. a 6 Thissysteipwilhplayanimportant roleintheproblem ofblack-body radiation inVol. V,just asthegeneral system ofelastic characteristic vibrations in§44ofVol. IIwas ofimportance fortheproblem ofspecific heats. iL8. Wefind from (24.6) with h=xm/l (m=integer, 1=length of theeylinder), if,again,exp(ih2)isreplaved by“"Azandcareistaken, bytheproper choice ofthecosine orsine, tofulfill theboundary conditions Ere =0, £,=VEAPs.(9)cos(ng)ostis),4/BH=0 E,= —J'n(e) cos(ng)sin(hz), kn . fa=aofHe=B®Jn)sia(ng)08), E,= ©Ja(9)sin(rg)sin(ha), MH, =*sopV2A,Recos(ny)cos(hz). 342 PROBLEMS, ANSWERS ANDCOMMENTS Here, asin(24.6), p=Wk?—rand /@—a=w,isoneofthe infinitely many roots ofJ,(w) =0.The characteristic wave number k andthecharacteristic frequency wareaccordingly given by 2 ®R= += w=ke(c=velocity oflight). (2) Thesystem ofcharacteristic functions represented by(1)istriply infinite andisordered bythenumbers n,»,andtheinteger mcontained inh.Con- vince yourself that (1)satisfies notonly theboundary conditions, butalso “therelations between EandHdemanded bytheMaxwell equations. Similarly weobtain from (24.7), with thesame meaning ofh, fst.=weaJa(0)cos(ny)sin(hz),Ex=0, fH=S49)00g)cos(2), E,=—E®7G)sin(ng)sin(he), (8)thp VsH,=~"Jule)sin(ng)o08(2), Ey=—JG)cos(ng)sin(ha). Thecharacteristic wave number andthecharacteristic frequency arenow given by ’ Bae +, omke @) where w,isoneoftheinfinitely many roots ofJ‘,(w’) =0.Theseries of characteristic oscillations (3)isagain triply infinite. Do(1)and(3)supply thecomplete system ofcharacteristic vibrations oftheinterior ofthecylinder? IL.9, Except foramultiplying constant theappropriate solution, con- tinuous atr=0,ofthedifferential equation (19.16) is =Bin(kr)emt— : Theseperation intotwoparts =1GB_-wirten BigSe(o' é ) PROBLEMS, ANSWERS AND COMMENTS 343, indicates thesuperposition, mentioned intheProblem, ofaspherical wave radiated outwards andoneradiated inwards. Eqs. (19.17) andthefollowing equations yield, without the time factor & 2\sin(kr) .ld 2\sin(kr) cE,=coso(+H) ks=-sino(24+ae), 7tw ad. in(kr) By=0,y=Ho=0,Hy=“sino(sin(oy—2). The boundary condition Zy=0demands for r=a conten) —2.8) ty—(hay=0.ka Therefore the transcendental equation tanz=i= z=ka Itsgraphical solution yields afirst root x,which issomewhat smaller than xandaninfinite series ofadditional roots which asymptotically approach thevalue x,=vx. *dnaddition tethissingly infinite system ofcharacteristic functions, for which theelectric lines offorce lieinthemeridional plane =const and themagnetic lines areperpendicular thereto, there are©”lesssymmetric characteristic functions, with aLegendre function dependence in#and y. TI.10. Thetelegraph equation (18.19), with @=0andtheassumption I=Ipexp i(hx —wt),yields W?=aK(oL +iR). (0) Forlarge w,more exactly, forwl>>|R|, this leads to . iR h-k=27° (2) Here k=oV/KL (2a) isthewave number oftheperfectly conducting line and L i.VE (2b) isitswave impedance. Ristheimpedance operator from Eq. (20.19), composed: ofthereal resistance Rand theinner inductive reactance wl;— (numerically equal totheresistance inthepresence oftheskin effect) to form acomplex quantity: R=R- iol,=(1-OR. (20) 344 PROBLEMS, ANSWERS ANDCOMMENTS Itmay benoted, incidentally, that Rmay bereplaced directly byRin (1)ifweinterpret LasthesumL,+L;oftheouter andinner selfinductance and not, aswas done inthetelegraph equation, astheouter selfinductance L,alone. We assume that wislarge enough that afully developed skin effect occurs. According to(20.12) thedepth ofpenetration isthen given by d=1/Vuow/2 (3) We shall furthermore assume that the wave amplitude may vary slowly ~along thecircumference ofthewire sothat thevalidity ofthesolution originally obtained fortheplane problem in§20Bisnotimpaired. The alternating current resistance ofasurface strip oflength 1and width 1, measured inthedirection ofthecircumference, isthen by(20.15a) 1 Ryza (4) The current through thismetal strip (lying directly underneath) isequal tothelineintegral ofHabout thisstrip, which inourcasereduces tothe value ofHatthesurface. Hence weobtain fortheJoule heat developed in thestrip, utilizing (4), 2oig RiadH and fortheJoule heat developed inunit length oftheconductor asawhole ‘ds=lineelementofthecircumference, $=integration overthecir- cumference): ifwa=erwithT=pHas 6) Rand/areresistance andtotalcurrent ofthisunitlength. From (5)we compute 2 n-Lfwa/(f Has). 6) od His tobeobtained from thequasistationary field inthedielectric. This yields also theexternal selfinductance L,ofunit length oftheconductor and itscapacity K,aswell asitswave impedance zinKq.(25). a.The Lecher Two-wire Line. Wire radius a,separation ofwire axes 2b, separation ofthetwosource lines 2¢o(seeFig.36);u,»bipolar coordinates forrepresenting thefield inthedielectric, u=const magnetic, »=const electric lineg offorce; = =——o _ =Ve 7 ds=gdy, gcoshw—0088” fo(J2G Onthecircumference ofthewire u=tu,cosh u=b/a. PROBLEMS, ANSVERS AND COMMENTS 345 Since thebipolar coordinate vsignifies themagnetic potential directly, thedesired magnetic fieldcomponent attheperiphery ofthewireis d@_t_1 HelSesle2 - . 8)degh(coshua—cos»). @) From thistheintegrals occurring in(6)may becalculated: ae fwas=[50a=2, am 1/* Qe 2xb 2==== -dy== =a? fwasfnomif(eoshwo—008»)do=7coshuy=7F? andhence (6)leads to(thefactor 2tobeadded to(6)arises from thetwo- wire line): 22rd/21b a Rmcino/OPeda © ® Since furthermore thebipolar coordinate udetermines theelectric po- tential, thedifference inpotential between thetwo wires is Que=2arccosh =2log2+£0,a a From thiswefindforthecapacity andselfinductance perunitlength &L tm 1 bthKo dy78G (10) and forthewave impedance =JBtogBEL FA/*;logaa (11) Taking account of(2c), (9)and(11)yields forthepropagation constant inEq. (2) _pelti eb /b+he ho-keVedat log——. (12) This expression agrees with thevalue calculated in(25.20). Torealize thisitismerely necessary tosubstitute thevalue ofdfrom Eq.(3)in (12), andtoexpress theconductivity «bythecomplex dielectric constant e’=e+io/w &io/w, andtonote themeaning oftheabbreviation P(l/p_=(b +$0)/a) in(25.20). . b.Returnthroughground.Theearth’ssurfacenowtakestheplaceof theplaneofsymmetry u=0ofthebipolarcoordinates. Informula(6) forRthefactor 2isnowtobeomitted, since theforward conduction through 346 PROBLEMS, ANSWERS ANDCOMMENTS thewireistobeassumed toberesistance-free, sothatFrefersonlytothereturn conduction through ground. OntheotherhandKistobedoubled,Landztobehalved. bnowsignifies theheightofthewireaboveground, $otheheight ofthesource lineabove ground, which does notdiffer appre- ciably from 6.Wethen obtain from (9),(11), and (12) i ‘Ho1 2b Rmsh ongfe, a3) -kaltt fel/2b h-k 32d log=. (14) Numerical example:frequencyintherangeofradiotelephony 10°S~’, a(earth) =10°—10-“2"M™',b =10M,a =1mm.Wecompute from(3) d=50.4Mto5.04M and obtain from (13) and (14) R=3.2t00.320/M |h—k|=38t03.8-10* M7. Atlowfrequencies thepenetration issogreat compared with theusual height ofthewirebthat(incontrastwiththatin§25)ourpresentmethod ofcalculation fails. For thesingle wire without return conductor theexternal field can no longer becalculated inaquasistationary manner, sothat theabove ap- proximate method nolonger constitutes asimplification ascompared with §22. Our numerical example indicates, bytheway, how greatly thefield of thesingle wire isdisturbed even byanon-metallic return conductor; see inthisconnection thenote atthebeginning of§22regarding thefailure of Herts’s original experiments with wire waves and theinfluence ofthe laboratory walls. III.1. The transformation (4)oftheProblem istobebuilt upoutof thetransformations (1), (2),and (3)inthefollowing manner: L'=DED™ (a) (L'andL=Lorentz transformations, D=rotation, D™=inverse rota- tion). Here D: mM=resatysna, 4=2, w= —zsina +ycosa, =e; L: wee 4Be to2 1 jt—fBe nmNs f= 2mble roa.a= Vi-B# B =45 PROBLEMS, ANSWERS AND COMMENTS 347 D>: w=zicosa—yisina, g=a, y=risnatyicsa ft=h. Bytheir successive combination wefind, with theabbreviation a=(1-eyt thefollowing system ofcoefficients forL’: ® x . ie z 1+@—1)costa (@—1)cosasina |0ifycosay (y—1)cosasinge 1+@—1)sinta 0 1sina2 0 0 1 0 tet! |—18mC08a+ —16nsin 0 7 Itisfour-dimensionally orthogonal (sum ofthesquares equal to1,sum of theproducts equal to0,both inthehorizontal rows andthevertical col- umns) andhence canbereadjustaswellfrom toptobottom asfrom left toright. !,Thevelocity voftransformation (2)forming anangle «withthez-axis has,inthez,y,z-system, thecomponents »cosa,vsin«,0.With r=2, y,2,0 =2’,y’,2wetherefore obtain Larasrcosatysina, Tv=acosaty sina With these abbreviations oursystem, read from lefttoright, yields Y=r+i{o-yir- ont} (b) =nit—8F.vaftop*} and, read from toptobottom, rert‘a-olrt brat} () t=fe+at}.cv Whereas (b)follows fromsymbol (a),(c)corresponds totheinversion of(a): L=D"UD. @® Bytaking thederivative of(b)with respect to¢’orof(c)with respect tot,respectively, weobtain forthethree-dimensional velocity vectors q’=dr’/dt’ and q=dr/dt S48 PROBLEMS, ANSWERS AND COMMENTS: . 1247 Vin-@—Dra} q7d=va) > (e) qedtvlat =Dra} 0a(l+v-q/e) . III.2. The Problem canbeformulated inthefollowing manner: Leta system 2;,4move with respect toasystem =,¢with thevelocity 6,calong thex-axis. Insystem 2,t,letapoint Pmove with thevelocity #:¢atan angle awith respect tothez-axis (and atthesame time, thez-axis). What istheresultant velocity ofpoint Pasobserved from system z,t? Apart from additive constants themotion ofPinthe7,t-system is *described by a=Bach,cosa, 1=Arch, sina. (1) Atthe same time the Lorentz transformation _2=fret _ t_t=fizfe (2)Bima BOB BOE applies forevery point ofthe x, é4-system. Substitution ofx;from (1)and 4from (2)leads to &—Byct=yccosa(t—Ayx/c) or,after collecting terms with xand ¢, 2(1+BiBe cosa)=(6;+B:cosa)et. Hence dz_fitfcosa di1+Bipcosa” (3) inaddition, (2)and (1)lead to bre sinx u-Fine (t—biz/c), dy_Bcsina(:~a2) a Vi- Bs cedi)’ and, inview of(3), dy_Sean(1__Bi+BiB.cos3)~esina Vi-Bwy a” Vi-B 1+BiB:cos@ 1+BiB:cosa* ‘Theresultaht inthez,t-system is - /(@y (ay a=(G)+H): PROBLEMS, ANSWERS AND COMMENTS 349 Ifweput g/c =8,weobtain: ge(Bi+6:cosa)”+63(1—B})sin’a (1+BiB:cos«)* 6) —Bi+2618:cosa+Bi—Bibzsin’aQED (1+81: cos«)* . III.3. In(80.6) rdenotes thevector L—P,and 7,hence, oneside ofthe triangle LOP inFig. 42.LetObetheposition oftheelectron atthetime ofthefield observation atP.Letthevelocity voftheelectron bedirected, unlike §28, along thepositive z-axis, leading toaspecialization ofthefor- mulas in§30. The length LOthen becomes equal tov7with 7=r/c, so that LO =or=Br. The length OPistheseparation between electron and point ofreference atthetime oftheobservation and will bedesignated byr’, raVEEP EH, where 2’,y’,2’arethecoordinates ofPrelative to0.8and0’aretheangles at~L and Oshown inthefigure and wehave eos’ =2', reosd=2' +Br. q@ Furthermore, bythePythagorean theorem, t= 1?+(Br)? +2r/Br cos9, @) sothat (1—6)—2Bre’ =1”. The solution ofthis quadratic equation forryields rrot Vino ety tt+ae (@) Asin(28.13a) wedenote thesquare root ontheright bys: 8=w(i+Patrteny/ eaten TB) 7" 1-# and obtain by(3) $B arVizeti-# @) Now, »,=vcos8and, inview of(1), =Brcos8=Ba!+Ar; 350 PROBLEMS, ANSWERS ANDCOMMENTS Accordingly, by(4),. r(1-%) =10-6)~pe=Vimo C) Thus thequotient occurring in(30.6) becomes 1-v/i |1-6 1 ©A o/c =BR “Atthesame time thevectorial factors multiplied herewith inview ofthe assumed direction ofv,areresolved into their components: rXv= 0,2, —yr. (7) rtr=1B.1cos8=1B—2—B=-*,(7a) a Mg mearent yy ofran @. Tfallofthisissubstituted inEq.(30.6) taking due account ofthechanged sign ofv,Eqs. (28.14) and (28.14a) areobtained. ‘Theequation s=const defines thefamily ofmutually similar “Heavi- sideéllipsoids”’, flattened inthedirection ofmotion (see§28C); theelectric lines offorce in2’,y’,2’space aretheorthogonal trajectories ofthefamily. IIL4. Scalar multiplication with voftheleft side of(32.5) results in d__mov vv @__iVavi-g” ™Viet ayo The segond term ontheright isequal to 266 vv modge ™Oae Together with thefirst term this yields vi¥ e vv d_ mcma(tres) -maleate © Ontheright side of(32.5) scalar multiplication with vyields vK=v-E+v-(v XB)=v-E+B-(v Xv)=v-E. (2) Hence, setting (1)and (2)equal toeach other, weobtain dmc avi-e 77% ® asinEq. (82.6). PROBLEMS, ANSWERS ANL COMMENTS 351 IIL.5. Thepath isofcourse a.ballistic parabola with theacceleration ueYoond’ Thedistance ofitsvertex from theupper andlower plate, respectively. is .sina)’ _mdvtsin’aae(_mo'sin’a h%te dohea-Fh). The lower plate isreached with vsina=VvSov.m Foravelocity v=5-10* M/S thisisnotsatisfied even fora=*/2With thevalue ¢/m from Eq. (33.8) wehave then instead d-h 25-10" a7}>xro-T7e-10 7085 ‘Thevoltage required toproduce »isfound tobe,fromeV=mv*/2, 125-10" V=5reign =70volts The“electron volt”evisaunitofenergy much usedinatomic phys-ies,particularly. intheform “million electron volts”, Mev. Since e= 1,60-10-" Qwehave inourunits 1Mev =10°-1.60-10- joule =1.60-10~* joule. Onehalfofthisisalmost exactly equal totherestenergy oftheelectron, ie. mec=0.80-10-™ joule, my=0.90-10"K. Tf,inthecondenser field, thevelocity suffers changes which arecompa- rable with c,theconstancy ofthez-momentum (zparallel totheplates) results inthefact that v.cannot beconstant, and hence xcannot bepro- portional to¢.Correspondingly, theequation ofmotion forthey-direction shows thatyisnotproportional to¢*.Hence thepath isnotaparabola, butatranscendental curve (catenary). Similarly in§32d theKepler orbit, which inthelimiting case ofaninfinitely distant center ofattraction be- comes theballistic parabola, was notanellipse, butatranscendental curve (ellipse with precessing perihelion). Tocompute extremely highvelocities fromthenumber zofthecorre- sponding Mev wemay usetheenergy equation ij +1=Mevi-# me” 352 PROBLEMS, ANSWERS ANDCOMMENTS: Foranenergy of200Mev, such asoccurs incosmic radiation, wehave iL. 24-10 =1-2.10+Vick 1+40024-10, B=1 3310”. II1.6. weconsider immediately thecase ofhigh velocities (the familiar case v<c iscontained therein). Let thedirection ofBbethez-direction. Intueplane perpendicular “thereto letsbetheprojection ofthedirection ofmotion, nthedirection perpendicular tos,8,n,andzforming aright-handed system. Wethen have always v,=0,(vXB),=0, (vXB),=»,B=0, (vXB), =—0B. Hence the momenta inthe s-and z-directions are constant: Y% Y% ae SS =".Vi-#7o% Vine Since»,=0,squaringandaddingleadstote=const,sothatalso 8,%, and v,areconstant. Theequation ofmotion forthen-direction is(thecharge oftheelectron isnegative!) d Un LA -e € oS ee Se B), =— vB;VIB VTSAryXBOB Hence i,=MizFy=~0B.‘m m d,isthecentrifugal acceleration, andassuchequaltov3/p,wherepisthe tadius ofcurvature ofthepath projected onthes,n-plane. Therefore 128 Pp my The same formula applies also forthenon-relativistic calculation, where however m=mp)=const. The curvature 1/pvanishes nonrelativistically onlyforv,=©,whereas relativistically itbecomes zerofor8=1,i.e.02+ v}=c’,Theproduct pB(commonly written pH)istheexperimental meas- ureofthe“stiffness” ofthecathode ray. IIL.7.If%isthecommon direction oftheelectric andthemagnetic fieldandzisthedirection ofthe6-rayleaving D,theequations ofmotion of theA-particle are,with Lorentz’s expression F=—e(E+vXB)forthe force (negative sign because ofthenegative charge oftheelectron) PROBLEMS, ANSWERS AND COMMENTS 353 ad Us cE©He avi-# ™" qd y evXB) eB ds eryB dtV1— mg tig? dt1=Bttg* Since vzandv,carbeneglected incomparison with v,&v,wehave 6°+ v*/c’, andfrom thethird equation ofmotion tothesame approximation v=const. Hence thefirst two equations ofmotion canbeintegrated directly andweobtain, ifthetime¢ismeasured from themoment ofpassage through Dsothat theinstant ofincidence onthephotographic plate may beset equal tot=a/v, : ee / ea z=ad 1-2a --2,f\-§ a : Ue 2d This isthe parametric representation ofthe two branches ofthe curve which result when thepolarity oftheelectric fieldisreversed. *«Ifv’isneglected incomparison with c’elimination oftheparameter » leads tothetwo branches ofaparabola 2 ~eBa y*FCz, C=mE2° They touch atthepoint s=y=0with avertical tangent. This point corresponds tothevalue oftheparameter y=«. Iftherelativity factor \/1—1?/c isretained, elimination ofvleads to the curve ofthe fourth order 4 ht =cee _Ba y+Diy=Cx’,D=T5(Casabove), which takes theplace ofboth branches oftheabove parabola. Atthepoint x=y=0,which now corresponds totheparameter v=c,ithasacusp; thetwo tangents atthis point have thetwo distinct directions dy C_i.Bedz *p> *3 andformaccordingly afiniteangle2a(seeFig.48,ontheright) witheach other. This isclearly evident from Kaufmann’s photographic records. However itwas not possible toarrive atadefinite decision between Lo- rentz’s andAbraham’s variation ofmass (see thebeginning of§33), as intended byKaufmann, although thisshould bepossible inprinciple from 354 PROBLEMS, ANSWERS ANDCOMMENTS thecomplete shape ofthecurve; thefields employed werenotuniform and their distribution hadtobeestablished bylaborious probe measurements. TIL8. The twomutually perpendicular fields Z,=EandB,=Bgive rise tothe Lorents force qj dzFe=-e9B, Fo-(a- 8),F,=0. Fornottoogreat velocities (m=mo)theequations ofmotion are | dt dyae+eaB=0, dy dz ;mon aB eB. Ifthesecond equation ismultiplied byi,andifweset$=z+iy,we obtain byaddition f-te ~-if, a=. The general integral is reads Erte. The time¢=0canbesochosenthatforitdy/dt=0,80thatfo= becomes real.Wethenhavei=iaA+E/B,f{=A+Candhence ~p=t(a—-®)a- cy +8 r-boL(%FaeD+Rt Separating realandimaginary partsweobtain withy=at,a=E/(aB), b=(4—E/B)/a: Z—-HM=ap+bsing, y— y= b(1—cosy). This istheequation ofthegeneral cycloid ortrochoid (overlapping or stretched, depending ona§b).Fora=0(i.e.b=—a)weobtainarepre- sentation oftheordinary ¢ycloid, such asoccurred, with thesame notation, inVol. I,Eq. (17.1) inconnection with thecycloidal pendulum (where howeverwehadput2=yo=0). III.9. Since wearedealing with astationary state, Jisconstant inboth time and space as . aydivJ=an”0 qa) PROBLEMS. ANSWERS AND COMMENTS 355, Atthesametimevandpareconstant intimebutnotconstant inspace since v=W2ive,p=Jf, ‘@) Poisson’s equation becomes VIGO 6,0=sean @) Itmay beintegrated byputting V(z) =Ax*. (4) (8)then leads to A™ola ~1)2%** =C, ie. a_ a4. or 9\*" gte@2=0,a=33A’67%A(ic)".(5) For x=|weobtain from formula (4)and themeaning ofCin(3) 9FP Cn vV=Ar=¢aviaa) w, (6) oefe 2e/m yanJey (a Thetotal current J=xa’J (a=radius ofcathode andanode) becomes 2 T=ZaVvEm ev. ®) This isthedesired equation ofthecharacteristic. The reader may con- vincehimself thatowing toourfactor eitiscorrect dimensionally, namely hasthedimension Q/S also ontheright side. Wenote expressly that, according to(3)and (4), Vdoes notincrease linearly with z,and that 0V/dzx isequal tozero atthecathode. Here, according to(1), p=©and v=0.Thelastcorresponds tothefactthat, inthestatement oftheproblem, wehave neglected the(small) velocity ofemission ofthethermionic elec- trons ascompared with thevelocity impressed onthem bythefield. For thécylindrical arrangement (radius ofthehot-filament cathode:~ r=0,radius ofthecylinder-mantel anode: r=a,length ofthecylinder mantel: 1)thepreceding equations change asfollows: 356 PROBLEMS, ANSWERS ANDCOMMENTS Lary) _ I rdivJ=to 0,rJ=const=Onl ay) p=J/v=I /(oniMe25vo) @) VIG E(rMO)6,Cm1/eratvEm). — V(r) =Ar® (4) 2 _(9,.\" ;a=3, Aa(c) (5) 9 a 28= =Ag = (9—2_ 8 .forr=a, Va’@al"saaP(6) 1=Fev3em! ve. @) Atthecathode dVdrnowbecomes infinitely large according to(4’) because ofr=0,incontrast withdV/dr fortheplane configuration. Never- theless thetotal charge onthefilament approaches zerowithvanishing r; thjsisthereason fortheabsence ofthelogarithmic singularity ofthe potential occurring otherwise foracharged wire, whereas by(4’)Vvanishes forr=0.Forthisreason (8’)applies notonly forr=0,butalsoforwires ofsmall finite thickness with sufficient accuracy. TII.10. Themagnetic fluxthrough theelectron pathofradius r &=ae[BerDrdr 0 yields ae ab vs riFem2ePBlr, SP=BefBlnDrdr i) According tothelaw ofinduction wehave ro 2x[Blr,Drdr=—2arvB(ra, 1. @) 0 Bymultiplication with theabsolute value eofthecharge oftheelectron weobtain herefrom asaccelerating force intheorbit r=rp: _ef™, e3b ~Bl=[Berd = @) The equation forthechange inmomentum oftheelectron canthen be integrated with respect to¢and yields PROBLEMS, ANSWERS AND COMMENTS 357 mo—(mo),=5°(—4), @ where ®,denotes themagnetic fluxfortheinitial state v=u,m=m, Accordingly, forgiven initial momentum, mvistoberegarded asknown. Wehence calculate fortheanswer toquestion (1): a =1 (my PrVitmya ™mog/'+(2)*6) eV=(m— m)c. Numerical example: Forapathdiameter 27)=107'M afluxamplitude Basx=107VSisreadily attainable inpractice. Lettheinitial fluxbevery small, 6,=0.Wecanalsoassume v,&0since theinitial velocity isin- significant incomparison withthegreatfinalvelocity. Wethenfindfrom(4) (mv)mex_ _¢Bmax 16-10" 10 =188,“imge Baremec ® 0.9-10-*-3-108 4 i\" 1 ® fou~(1425)1719?Max=188mo, Marx —Mo=17.81m, Vnax =17.8moc’. Since, fromthediscussion ofProblem III.5, mec’isequal to$-10* electron volts, wehave eVimax&9-10°ev. 7) Onquestion 2.:Theorbit r=7wasassumed tobeknown tillnow; it willnowbecomputed. Oneverycircular orbittheremustbeequilibrium.between thecentrifugal force andtheforce ofBiot-Savart: 2 Me=eB). 8) This signifies according toEq. (1) €Ob m= 5 (9) Substitution fromEq.(4)with6,=0,v4=0yields & a 7"ke (10) Plotordinate B,asafunction oftheabscissa r,asamonotonically dc- creasing curvewhicheventually mayhavetobedetermined experimentally.Multiplication with2xrthenyields, byEq.(1),thecurve for00/dr, and mtegration withrespect torthatfor®.Itsordinates aretobedivided by 358, PROBLEMS, ANSWERS ANDCOMMENTS rand theresulting curve must be-pursued toitsintersection with the curve ford4/dr. The abscissa ofthepoint ofintersection isthedesired value r=1. Inorder thatthisorbit may bestable thefield distribution must satisfy certain conditions, which,forexample, havebeenclearlysetforthbyGans. Onquestion 3.:From Bmax weobtain forthefrequency ofrevolution Bmaxl 199gt tox= 10°S™. (11) Inorder tobeable tocompute thenumber ofrevolutions with ease, we assume that theflux&does notincrease sinusoidally, butlinearly from the initial state 4,=0tothefinal state Saax .For500-cycle alternating cur- rent inthewindings oftheelectromagnet thetime ofrise(=aquarter period) isthen 1/2000 S."We hence obtain =2000dauz8=20V. 12) Since by(3)theaccelerating force isthen also constant intime, this isat thesame time thegain inenergy inonerevolution, measured inelectron voltg. Since by(7)themaximum kinetic energy measured inthis manner was’9-10°, thenumber ofrevolutions becomes 9-10°7 450000. (13) Onquestion 4.:The state ofthe maximum number ofrevolutions is reached for =Pmax,i.e.06/dt=0.According toEq.(3)wethenhave E(ro) =0,ie.trang =0.From theequation ofthecircle r= rei wefind forthemagnitude and direction ofthederivatives ofr: v=ture, v=—ale = —tele, (ay Weconclude herefrom: ¥isopposite indirection tov;#=—wv,further- more, v-¥=0,v-¥=—o0’, Forthereaction force ofradiation wefind hence by(36.26) (the symbols v’,v’given there have thesame meaning asour¥,¥): ee an *Gracti—B\~* ~ea—8), 2wt a)éwa?= 4) =-—49 Oo )8savin (itiee fredi-py5 ey ~2 +]= - [Rt]=goog=8 'R. Gans, Zeits. f.Naturforschung, Vol. 1,p.485, 1946. PROBLEMS, ANSWERS AND COMMENTS 359 Inournumerical example wefind with v=¢,8=1—1/710: (710)? a)-AON [R*}=Qwears” With2=10Mand36re=10<2(Bq,(718)): ° joule M StSS """ 6(710)*eV. 2-.eV eV *|=So =6-7.17-1.6- ay&0.048—. [|R*|10"MQ6-7.17-1.6-10 wr&004835 (16) This may becompared with theforce oftheelectric circulating field, which by(8)and (12) is 20.eV_200eV 2xro M «M’ The reaction force increases with thefourth power oftheparticle energy. Asthereaction force becomes comparable with theaccelerating force of thecirculating field thebalance between increase inparticle mass andcen- tripetal force isupset. Thus thereaction force setsalimit tothemaximum energy which thebetatron canimpart toanelectron. .Theprincipal purpose ofthebetatron istheproduction ofz-rays ofvery great hardness. Since their limiting energy hvisgiven bythemaximum energy ofthebetatron electrons itdepends, inaccord with Eq.(5),onthe momentum mvwhich can beattained. ByEq. (4)this isdetermined by theratio 6/ry. Foraproportional increase ofallofthedimensions ofthe magnet (Bmax andhence alsoB,,arefixed bythesaturation ofthemagnet) ®increases quadratically, &/rp hence linearly. We found forour path diameter 2r9=107'M in(7) eVmex =9-10" eV=9-1.602-10"* erg=1.45-10~ erg. Hence, for2-rays 145-107erg=hy=we=2-10"=, d=14-10em =14X. The X-unit =10 cmhere introduced istheunit oflength customary inx-ray spectroscopy; thus theK-radiations oftheheaviest elements have wave-lengths ofabout 100X-units. Wethus find ourselves with ourbeta- tron ofrelatively modest dimensions inadomain farbeyond theshort- wave-length limit ofordinary x-ray spectra and even beyond that ofthe natural y-rays which isreached, forThC, at\=4.7X.Byincreasing the betatron dimensions thelimit can belowered still further and theenergy of9Mev, found above, beincreased. IV.1. We proceed from thefact that foranobserver moving with it, thefield within therodisboth free ofcurrent and free ofcharge: J’=0 360 PROBLEMS, ANSWERS ANDCOMMENTS and p’=0.Itthen follows from Eqs. (34.6) with thedefinition ofthecon- duction current in(34.9a), except forcorrection terms oftheorder f” =J-ppv=0, p=0. (1) The rodhence hasneither conduction current norvolume charge foran observer atrest inthelaboratory aswell. However, itpossesses surface charge wand aRowland current (which, infact, isdemanded byJ;=0) J=ov. Viewed from thelaboratory, thefield thus consists ofthesuperposition “ofastationary electric andmagnetic field, thelatter added totheoriginal uniform field and derived from the Rowland current J=wv. We there- fore have cnE=0, E=-—grad¥, curd =J. (2) ShowthatthisagreeswiththegeneralEqs.(34.13), ifinthemthemeaning ofE*,H*issubstituted from (34.8) andthatofB,Dfrom (34.12). For d/at =Oand theauxiliary conditions from (34.1la) they then become ‘—curl(v XB)=—curl(E +vXB),sothatcurlE=0, pv—curl(v XD)+J—pv=curl(H —vXD),sothat curlH=J. By‘(34.7)weinférfromJ,=0,E*=0,ie.tosufficientaccuracy(B= By=original field): E=-vXB, E=-B, W=vBrt+C. (3) Wehave here assumed that Byhasthez-direction, v,they-direction, and that 2,y,2constitute aright-handed system. Cisaconstant ofintegration which isindependent ofxand, inview ofthesymmetry oftheproblem, also ofyand z.Question a.isthus answered. Onquestion b.:Consider two points z;,z2ontheperiphery oftherod, e.g.21=point ofentrance, z2point ofexit ofthez-axis. The difference of Potential isthen by (3): : V= —%=vBo(n —2). (4) Asadifference ofpotential thisisindependent ofthepath (connecting wire ofinfinitely high resistance) bywhich weimagine points 1and 2tobe joined; this path may beimagined either intheexterior ortheinterior ofthebody. c.Weconsider theexternal field. Here theboundary conditions (34.15) take effect. They demand continuity ofthetangential component ofE (not ofE*)ahdareequivalent tocontinuity ofthepotential atthesurface oftherod,whereas nothing isstated regarding thenormal derivative. Since Wisknown intheinterior by(3),thesurface values of¥arealso PROBLEMS, ANSWERS AND COMMENTS 361 known. Wehence must solve’a boundary-value problem fortheexterior, withthenormalization condition ¥=0atinfinity. Asolution canbeat- tained foranyshape oftherod. Itbecomes elementary forthecircular cross section, towhich, byconformal mapping, every other cross section may bereduced, transferring theboundary values prescribed forthe latter. Ithence suffices todeal with thecircular cross section. Ifr,¢areordinary polar coordinates, r=0isthecenter ofthecircle, r=aitsperiphery, and¢ismeasured fromthez-axis, (3)yieldsfortheinterior andtheperiphery ofthecircle E=E,=—vB, E,=E:cosy=—vBcosy, ® E,=—E,sing=vBsing. , Thepotential intheexterior cangenerally beexpressed asFourier series v=(2)(Aqcos(ng)+B,sin(ng)); Inview oftheboundary condition only theterm with A,differs from zero Hence a low a. == =s--— = = ) ¥=Arcosy, Ey rae Asing, (@ and, inview ofthecontinuity ofE,forr=a 2 2 Ar=aB, Y=2Bycosy, 2%=-(?)»Bycose.(7)r or r d.Todetermine thesurface charge wemustpassfromEtoD.Wemayutilize forthispurpose intheinterior oftherodEq.(34.5), whose right side vanishes inview ofE*=0.Thus within the rod wedonot have D=eK, but D=-4VXH= -amvXH2-gvX B=aE(8) Since thiseoisderived from thegeneral relation eouo=1/c’,e&represents thedielectric constant ofvacuum andisnot, ingeneral, identical with thedielectric constant ofthesurroundings. Itischaracteristic andsatis- fyingthatinanexactapplication ofMinkowski’s theory in(8)thereappearsthewell-defined dielectric constant ofvacuum rather than thesomewhat problematical andscarcely measurable dielectric constant ofthemetal. Weconclude from (8)tobegin with forthewhole interior oftheréd (sinee £=const): divD =p =0, 362 PROBLEMS, ANSWERS AND COMMENTS which agrees with theinitial equation (1).Theinterior oftherodisfree from space charge, even asobserved from thelaboratory. Atthesurface oftherod, asjudged from itsinterior, wehave by(5) and (8), Dz=8,=—&E, =ewBycose ifndenotes the normal directed toward the interior. Forthesake ofsimplicity wesetthedielectric constant oftheexterior (air) alsoequal toe.Then, by(7),wehave forr=a,asseen from the outside (ndenoting thenormal directed outwards), D, =-a =evBocos¢. Thesum ofthese twoD,yields thesurface divergence ofDatthesurface, ie.thesurface charge w=2ewBy cos¢. (9) Itvaries from place toplace andhasitsmaximum values for¢=0and yg=7,ie, +2ewBo. Theelectric linesofforce, which intheinterior arestraight linesand perpendicular totheaxisoftherod,arebent intheexterior from thepoints ofpositive surface charge tothose ofnegative surface charge along the shortest possible paths, particularly intheneighborhood ofthetwopoints ¢=r/2and y=3x/2. Only fory=0andxarethelinesofforee perpendic- ular tothesurface and flow offtoinfinity. e.Ifthestraight rodisbentintoacircular ring, andthisisrotated about theaxis ofsymmetry perpendicular toitsmidplane, every section ofthe ringissubject toapproximately thesame conditions asthecorresponding section ofthestraight rod,provided only that theradius ofcurvature of theringislarge compared totheradius ofitscross section. Thesame ap- pliesforacircular diskring,provided thattheradius ofitsinnerbounding cylinder isnottoosmall. Since however thevelocity issmall intheex- cluded section ofthedisk, andthephenomenon ofunipolar induction be- comes insignificant atsmall velocities, thisrestriction maybeoverlooked and our results beextended tothewhole disk and eventually also toan arbitrary body ofrevolution. Wecanthen apply ourEqs. (3)and(8)also tothe field initsinterior: E=-vXB, D=&E (10) anddedtice therefrom thecorresponding values ofthevoltage Vandthe interior potential ¥,wherexs thepotential ontheoutside must beobtained bythesolution ofacomplex three-dimensional boundary-value problem. PROBLEMS, ANSWERS AND COMMENTS 363 However, thefollowing interesting difficulty arises: Ifthegeneral (or, rather, toospecialized) rule p=divDisemployed tocompute thespace charge within therotor, weobtain by(10), since nowvandhence alsoD vary inspace, . p=—&odiv(vXBy)=—ecBy-curl v=~2eeBo, (@=angular velocity oftherotation). This isnotzero, aswasthecase forthetranslation andaswemight have expected from thestandpoint of theobserver rotating with thebody. This contradiction is,however, no objection toMinkowski’s theory ofmoving media, which (seefootnote 3 atthebeginning of§34)isbased ontheLorentz transformation ofuniform translation, butmerely anindication that itisnotdirectly applicable to problems involving rotation. Author Index A F Abraham, M.,165,278,292,300,353, Faraday, M.,8,108,256,250,288 Ampere, A.M.,4,119 Frank, Ph., 224 Anderson, C.D.,306 Fues, E.,47 Arago, D.F.,288 6 B Gans, R.,358 Barkla, C.G.,155 Gauss, K.F.,2,42,52,308,309,321 Barnett, J.8.,98 Gentile, G.,Jr., 190 Becker, R.,100 Giorgi, G.,45 Bessel, F.W.,1,313 Goethe, J.W.von, 311 Bethe, H.,285 Goudsmit, 8.,98 Blackett, P.M. 8.,308 Boltzmann, L.,3,18 H Bitter, F.,100 Haas, W.J.de,08 Bopp, F.,301 Habn, O.,265 Born, M.,303, 306 Heaviside, O.,2,43,77,241 Braun, F.,143 Heisenberg, W.,98,229, 306 Brillouin, L.,198 Heitler, W.,301 Broglie, L,de,5,198 Helmholtz, H.von, 5,101,286,266, 285 Bueherer, A.H.,273 Henry, J.,105 Herglotz, G.,249 cHertz, H.,2,5,36,49,52,112, 149, 151, Cady, W.G.,78 152,164,177,185,280,285 Cartan, E.,310 Hilbert, D.,1 Christoffel, E.B.,310 Hittorf, W., 2 Cohn, E.,49,280 Hondros, D.,185,190, 193, 200 Curie, P.,78,90 Hu, N.,301 D I Davy, H.,3 Infeld, L.,308 Debye, P.,73,190,193 Ives, H.E.,228 Dirae, P.A.M., 148, 301, 305 Déring, W.,100 J Dolezalek, F.,177 Jacobi, C.G. J.,1 z Janet, M.,310 Jaumann, G.,199 Eddington, A.8.,261,314,315,318 Jaumana, J.,85,199 Eichenwald, A.,284,285 Joliot-Curie, F.,306 Einstein, A.,98,212,227,228,229,283, Joliot-Curie, I.,306 234, 236, 264, 265, 280, 301, 307, 309, Joos, G.,53 310, 311, 320, 321, 322 Eatvis, R.,313 bed Euler, H.,306 Kalantaroff, P.L.,45 Ewing, J.A.,97 Kaluza, T.,212 285, 366 AUTHOR INDEX Kaufmann, W., 278, 381, 353 Poincaré, H.,223, 278 Kempton, A.E.,265 Pohl, R.W., 53 Kirchhoff, G.,1,45,134, 185, 141, 328 Klein, F.,222,235,307 R Kockel, B.,306 Rasetti, F.,220 Konig, H.,97 Rayleigh, Lord, 45,188 Koblrausch, F.,45,68,96 Riemann, G.F.B.,4,308, 309, 310, 321 Kulenkampf, H.,155 Roentgen, W.K.,155, 283, 285 L Rutherford, E.,265 Lang, H.,310 s Langevin, P.,90,97,100 Schelkunoff, 8.A.,198 Langmuir, I.,332 Schladi, L.,310 Larmor, J.J.,145 Schottky, W., 382 Laue, M.von, 292, 300 Schrodinger, E.,322 Lebedew, P.N.,281 Schwarsechild, K.,261, 269, 270, 273 Lecher, E.,178 318, 314, 315 Lenard, Ph.235 Serber, R..,306 Lens, W., 818 Soldner, J.von, 320 Liénard, A.,250 Somnmerfeld, A.,178 Lorentz, H.A.,49,52,76,125, 223, 226, Southworth, G.,193 227, 236, 288, 242, 278, 278, 280, 285, Stark, J.,238 286, 300, 258 Stillwell, G.R.,228 u Stackelberg, E.C.G.,301 Mach, E.,308,311 T Magri, T.,199 Thomson, J.J.,155 Maxwell,J.Clerk,3,7,9,12,22,28,105,‘Thomson,W.,148 100, 111, 148, 286, 256, 273, 278, 280 Mesmer, F.,79 u Mie, G.,10,28,53,199,301,302,303 Uhlenbeck, G.E.,98 Minkowski, H.,1,228,232,248,260,280, Ubling, B.A.,306 285, 202 Mossotti, O.F.,75 v N Voigt, W.,78 Neumann, C.,4 Ww Neumann, F.,1,4,101,106 Wallot, J.,54 Neumann, G.,273 Weber, W.,2,4,123,124 Newton, I.,285, 308,313 Weise, P.,97,100 Wessel, W.,301 °Weyl, H., 302, 321 Ocehialini, G.P.8.,306 Widerve, R.,333 Ocrsted, H.C., 4 Wiechert, E.,1,236, 250 Oliphant, M.L.,265 Wien, M.,1 Otting, G.,228 Wien, W., 1,238 Wileon, H.A.,285 PWilson, M., 285 Pauli, W.,285,286,292,300,303 Z Peng, H.W., 801 Planck, M.,268 Zonneck, J.,49 Subject Index A Centimeter waves, 194 , . Charge, constancy of,234eeerintor,239 lansingMomotiformu,75,911 Coerciveforce, AdvancedaadPou8 Coil,alternatingcurrentfield,170 Aiongroneeisy, 2, magnetic field,5,131,827,335 Aeron +impedance, 167 maultilayer, 175‘Ampare’s hypothesis, 124 resistance, 178ampare's laws13 selfinductance, 192,178Ampire's method, 114 Condenser, oylindrical, 328,330Ampére’s rule,13 plate,65 Anisotropic medium, 28 spherical, 66Antenna,linear,152.” cnpieiey 20,338Asymmese waves,waveguide,195 Constants, physical,328* Continuity equation, 15,30,217 B Cosine law, 327, 336 Coulombfield,39,41 Deaneiarape,~ Coupledcircuits,142 Bessel funotion, 268,179 Critical fieldstrength, 305Betarays,224 Curieconstant, 90Betetren ‘533,356 Curie-Langevin law,90Biot-Savart, lawof,108 Curiepoint, 100relativistic modification, 240 Curie-Weiss law,100Bipolar coordinates, 201 Curl,general formula, 158Boundary conditions, ‘moving bodies, C¥Fent, magnetic measurement of,123 287, 361*Current density, 9 normal component, 17,827,384 Curvature, measure of,308tangential component, 16,327,334 Cyoloid, 354Branch, principal 180 Cylindrical cavity, 930,241Brench'eat, 118,180 Cylindrical condenser, 328,3392 Cylindrical fields, 156 c Cylindrical wave guide, 198 Capacity, 65 D arbitrary system, 70 distributed, 128 D’Alembert’s principle, 266 ellipsoid, 68 Decimeter waves, 193 energetic definition, 68 Demagnetigation factor, 83,95 measurement, 140 Diamagnetiam, 21,90 wire, 68 Dielectric, wave on,190 Capacity coefficients, 71 Dielectric constant, 21 Cavity, eylindrical,'390, $41 Dielectric constant, complex, 37 reetemgular, 380, 340 Dielectric constant, gases, 75 spherical, 390, 342 Dielectric constant, vacuum, 44 367 368 SUBJECT INDEX Dielectric displacement, 8 Excitation, electric, 8 Dimensions, 6,45,323 Excitation, magnetic, 10 Diode, 332, 354 Excitation vector (tensor), 215, 216 Displacement, dielectric, 8 PDivergence, four-dimensional, 213, 216 Doppler effect, 235 Faraday disk, 289 Doppler effect, transversal, 228 Faraday’s law, 13 Dual six-veetor, 217 Fermat’s principle, 268 z Forromagnetism, 21,96Field vector (tensor), 215, 217 EMF,, 12 Field strength, electric, 7 Einstein dilatation, 227 Field strength, magnetic, 10 Electret, 77 Fission, 265 Electric field, ellipsoid, 56,327, 336 Flux, magnetic, 88 energy, 27 Force, four-veetor, 243 sphere, 55 Forced vibrations, 137 thin rod, 56,328, 337 Forecone, 232 transformation, 237 Four-current density, 213, 241 Electrical image, 57,63 Four-potential, 213, 247 Electrodynamic potentials, 145 Four-veetor, 213 Electrokinetic potential, 260 Free charge, 40 Electromotive force, 12 Free vibrations, 137 Electron, accelerated, 252, 253 éritical fieldstrength, 305 6 deformable, 227,273 Galilei transformation, 222,224 inuniform motion, 252, 331, 349 Geodetic line, 309 intrinsie field, 239 Giorgi units, 6 motion inparallel fields, 331, 352 Gravitation, Einstein’s theory, 310 motion inperpendicular fields, 382, 354 Green’s function, 57 motion inuniform electric field, 331, Green’s theorem, 69,116, 246 351 secondform,119motion inuniform magnetic field, 331, Ground return, 345 352 Gyromagnetic effect,98radiation, 255 radius, 276 H rest mass, 275 Hamilton’s principle, 267 spin, 278 Hamiltonian funetion, 305 Electron field, flattening, 241 Hankel functions, 171, 179, 180 Electron theory, 236 Heaviside ellipsoid, 241, 350 Electrostatics, 38 Henry, the, 105 Ellipsoid, capacity, 68 Herteian dipole, 148 electric field, 56,327, 336 Hertzian vector, 149, 221 magnetic field, 92 Heusler alloys, 78 Elliptical wave guide, 197 Hysteresis loop, 98 Energy, conservation, 20 1 Energy density, 27 Energy, relativistic formula, 331, 350 ‘Image, electrical, dielectric halfapace. 63 _ Equivalence, gravitation-inertia, 312 conducting aphere, 57 mass-energy, 264 Impedanee, 138 Ether, 36,235 Induction, law of,13 Euler-Mascheroni constant, 180 Intensity, entity of,11 SUBJECT INDEX 369 Intrinsic time, 232 ellipsoid, 92 Invariants, six-vector, 219 pole strength, 87 ring, 85 Joule, 7 J Magnetic density, 40 Joule heat, 27 Magnetic double layer, 118 " Magnetic field, energy, 27 K ring conductor, 130 Kinetic potential, 266 solenoid, 25,131,327,335 Kircbhoff’s laws, 101,134 wire, 24,125,327,334 Kirebhoff-Thomson formula, 137 Magnetic flux,88 Magnetic shell; 116 L Magnetic susceptibility, 89 Lagrange density, 220, 271 Magnetic waves, 187 Lagrange equations, 268 Magnetite, 78 Lagrange function, 268 Magnetization, 79 Lamellar field, 39 : Magnetomotive force, 12,88 Laplace operator, four-dimensional, 212 Magnetostatics, 38,40 Larmor’s formula, 161 Mass, gravitational and inertial, 313 Teast action, Schwarzschild’s principle Mathieu function, 197of,269 Maxwell equations, differential form,18 Lecher system, 112,198,330,443 forvacuum, 146,218 Propagation, constant, 345 integral form, 13 Least time,principle of,288 invariance, 214 ‘eft screw rule, 13 ponderable bodies, 200 Lenz's rule, 13,14 Maxwell-Minkowski equations, moving Leyden jar,68,328, 339 bodies, 286 Light, deflection bysun, 319 Maxwell stresses, 255, 250 velocity, 32 Mechanics, relativistic, 262 Light cone, 232 Mereury, precession ofperihelion, 315, Light point, 248 319 Light pressure, 260 Meson, 322 Light quantum, 261 Mesons, life, 229 Light source, 152 Minkowski equations, moving media,280 Line width, 205 Minkowski force, 263 Lines offorce, 61 Minkowski theory, moving media, 280, refraction, 68 363 Litz wire, 177 Momentum, electromagnetic field, 260 Logarithmic potential, 110 Momentum four-vector, 262 Loop tension, 12 Moving media, field transformations, 281 Lorentz contraction, 226 Mutual induction, coefficient of,105,106 Lorentz force, 238 -Mutual induction, parallel wires, 107 magnetic analog, 239 " Lorentz transformation, general, 330,347 Neumann’s potential, 106special, 223 Neutron, 47 Lorenz-Lorentz formula, 75 Newton, 7 M Newton’s laws,relativistic form,263 MKSQ units, 6,45 ° Magnet, 78 Ohm’s law, 20 bar, 80 Ohbm’s law, alternating currents, 138 370 SUBJECT INDEX Ohm’s law, moving conductors, 282 R Oscillator, spherical, 154 Radiation, accelerated charge, 255,206 P dipole, 151 reaction foree, 203, 298, 300, 358 Packing fraction, 265 Ray veetor, 30 Pair production, 305 Rayleigh’s law, 153 mn Parallel excitation, 199, 200 Rayleigh resistance formula, 166 Paramagnetism, 21,90 Reactance, 138 Perihelion ofMercury, precession, 315, Reaction force, radiation, 203, 208, 300, a9 358 Permanent magnet, 80 Reciprocal radii, 57,201 Permeability, 21 Rectangular cavity, 330, 340 Permeability, vacuum, 43,108 Rectangular wave guide, 197 Photon, 261 Red shift, 320 Physical constants, 326 Refraction, lines offorce, 63,71 Piesoelectric crystal, 78 lines ofmagnetic excitation, 89 Plane wave, 34,145 Refractive index, 34 Planetary orbit, 316 Relativity theory, 212 Plate condenser, 65 Relativity, general theory, 307 edge correction, 328, 338 Relaxation time, 23 Poisson’s equation, 38,110, 333 Remanent magnetization, 98 Polar molecules, 73 Resistance, specific, 20 Polarization, 74 Retarded potential, 147, 248 permanent, 77 Riemannian surface, 116 Pole strength, 10,87 Right screw rule, 13,104 Positron, 305 Ring magnet, 85 Potential, advanced, 148 Rochelle salts, 22 electrokinetic, 269 Roentgen current, 284 kinetic, 266 Rotation, inspace-time, 225 logarithmic, 110 Rowland effect, 283 retarded, 147 scalar, 115,148 s vector, 101, 148 Scaler, 213 Potential coefficients, 71 Scattering, light bylight, 305 Potential field, 39 x-rays, 156 Potential theory, four-dimensional, 245 Schwaraschild invariant, 269 Poynting theorem, 26,250 Secondary waves, 186 Poynting vector, 28 Seignette salts, 22 Propagation, damped, 144 Selfinductanee, coil, 132 Propagation constant, 156 energetic definition, 120 forground return, 346 external, 122 Lecher system, 207, 345 measurement, 140 Push-pull excitation, 199, 200 two-wire line, 112, 121 Pyrrhotin, 78 wire, 111 Selfinduction, coefficient of,105, 108 Q Signal velocity, 231 Bix-vector, 214 Quantity, éntity of,11 dual, 217 Quartz, 78 invariants, 219 Quasistationary field, 38,133 representation, 218 SUBJECT INDEX 371 Skin effect, 162 magnetic ¢.g.s., 42,124 Sky, color, 153 rational, 43Solenoid, magnetic field,25,131,327,385Uranium, bomb,265 Solenoidal field, 80 v Sphere, conducting, inuniform field, 58, 62,328,337 Vector diagram, 138 dielectric, inuniform field, 60,328,387 Vector potential, 101 electrical image, 57 Velocities, addition, 229, 233,331, 348 superconducting, inmagnetic field, 62 Velocity, four-vector, 232 Spherical cavity, 330, 342 light, 32 Spherical oscillator, 154 signal, 231 Spherical wave, 153 upper limit, 230 Stationary fields, 38,100 Velocity oflight, constancy, 234 Stress-energy tensor, 256, 291 Vibrations, forced, 137 Surface charge, 17 free, 138 Surface curl, 19 Virginal curve, 98 Surface divergence, 19 Surgeimpedance, 145 w Susceptibility, electric, 76 Wave equation, 138 initial, 99 Wave guide, cylindrical, 193 magnetic, 80 elliptical, 197 molat, 89 rectangular, 197 xeversible, 99 transversal fields, 196 “Symbols, 323 Wave resistance, 144 vacuum, 36,44,145 TWaves, asymmetric, 188 Telegraph equation, 143 magnetic, 187 ‘Tensor, transformation, 244 onnonconduetor, 190 Thermionie diode, 332, 354 secondary, electric, 186 Three-vectors, invariant, 220 wire, 177 Time; intrinsic, 232 Weiss domains, 97 relative nature of,225 Wheatstone bridge, 140 Tourmalin, 77 Wire, capacity, 68 ‘Transvoreality, light, 34 alternating current field, 168, 178 x-rays, 155 electric field, 125 Trochoid, 354 energy transfer, 130 ‘True charge, 40 magnetic field, 24,125, 327,334 Wirewaves, 177 udielectric, 190 Unipolar induction, 287, 288, 333, 359 Wollaston wire, 184 Units, conventional, 42 World funetion, 301 conversion, 125 World line, 232 electrostatic ¢.g.s., 40,124five(MKSQP), 47 x Gaussian, 50 X-raya, continuous spectrum, 155 Kalantaroff (M8QO), 45 scattering, 156 MKSQ/6, 45 X-unit, 359