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Published book by Andre Koch Torres Assis and Julio Akashi Hernandes (Apeiron, 2007), kept in a folder of downloaded physics books. It presents experiments and calculations of the potential, electric field and surface charges in and around resistive conductors carrying steady currents, including straight wires, coaxial cables, transmission lines, plates, strips, and cylindrical, spherical and toroidal shells. Appendices cover the work of Wilhelm Weber and Gustav Kirchhoff, and a large bibliography is included.

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Andre Koch Torres Assis and Julio Akashi Hernandes The Electric Force of a Current The Electric Force of a Current Assis/Hernandes Apeiron About the Authors Andre Koch Torres Assis was born in Br azil (1962) and educated at the State University of Campinas – UNICAMP, BS (1983), PhD (1987). He spent the academic year of 1988 in England with a post-docto ral position at the Culham Laboratory (United Kingdom Atomic Energy Authorit y). He spent one year in 1991-92 as a Visiting Scholar at the Center for Elec tromagnetics Research of Northeastern University (Boston, USA). From August 2001 to November 2002 he worked at the Institute for the History of Natural Sc iences, Hamburg University (Hamburg, Germany) with a research fellowship awarded by the Alexander von Humboldt Foundation of Germany. He is the author of Weber’s Electrodynamics (1994), Relational Mechanics (1999); and (with M. A. Bueno) Inductance and Force Calculations in Electrical Circuits (2001). He has been Professor of physics at UNICAMP since 1989, working on the foundatio ns of electromagnetism, gravitation, and cosmology. Julio Akashi Hernandes was born in Brazil (1977) and educated at the State University of Campinas – UNICAMP, BS (1998), MS (2001), PhD (2005). He has always been keenly interested in basic physics, especially electromagnetism. He has published many papers on the electric field outside resistive wires carrying steady currents in ma jor international journals of physics. He is Professor of physics at Universidade Bandeirante de São Paulo, Brazil. The Electric Force of a Current analyzes the elec tric force between a charge and a circuit carr ying a steady current when they are at rest relative to one another. It presen ts experiments and analytical calcu- lations showing the existence of this force, contrary to the statements of many scientists. The force is pr oportional to the voltage of the bat- tery connected to the resistive circuit. It also includes calculations of the potential and electric field inside and outside resistive conductors carrying steady currents, and the di stribution of charges along the sur- face of the conductors that generate this field. It contains two appen- dices that discuss the pioneering and revolutionary works of Wilhelm Weber and Gustav Kirchhoff, and a substantial bibliography of mod- ern literature on the topic. 0-9732911-5-X Weber and the surface charges of resistive conductors carrying steady currents ,!7IA9H3-cjbbff! The Electric Force of a Current Weber and the surface charges of resistive conductors carrying steady currents Andre Koch Torres Assis Julio Akashi Hernandes Apeiron Montreal Published by C. Roy Keys Inc. 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com © Andre Koch Torres Assis and Julio Akashi Hernandes. 2007 First Published 2007 Library and Archives Canada Cataloguing in Publication Assis, André Koch Torres, 1962- The electric force of a current : Weber and the surface charges of resistive conductors carrying steady currents / Andre Koch Torres Assis, Julio Akashi Hernandes. ISBN 978-0-9732911-5-5 Includes bibliographical references and index. 1. Electric circuits. 2. El ectric conductors. 3. Electrostatics. 4. Electromagnetism. I. Hernandes, Julio Akashi, 1977- II. Title. QC610.4.A47 2007 537'.2 C2007-901366-X Front cover: Portrait of Wilhelm Eduard Weber (1804-1891) around 1865. He was one of the pioneers of the study of surface charges in re- sistive conductors carrying steady currents. Back cover : Figure of the 2 circuits: A constant current I flows along a resistive wire connected to a battery V. At the left side there is a qualita- tive representation of the charges along the surface of the wire. At the right side there is a representation of the internal and external electric fields generated by this distribution of surface charges. Contents Acknowledgments iii Foreword v Vorwort vii I Introduction 1 1 Main Questions and False Answers 7 1.1 Simple Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Charge Neutrality of the Resistive Wire . . . . . . . . . . . . . . 9 1.3 Magnetism as a Relativistic Eﬀect . . . . . . . . . . . . . . . . . 13 1.4 Weber’s Electrodynamics . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Electric ﬁeld of Zeroth Order; Proportional to the Volta ge of the Battery; and of Second Order . . . . . . . . . . . . . . . . . . . . 20 2 Reasons for the Existence of the External Electric Field 23 2.1 Bending a Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Continuity of the Tangential Component of the Electric F ield . . 26 3 Experiments 29 3.1 Zeroth Order Electric Field . . . . . . . . . . . . . . . . . . . . . 29 3.2 Electric Field Proportional to the Voltage of the Batter y . . . . . 30 3.3 Second Order Electric Field . . . . . . . . . . . . . . . . . . . . . 42 4 Force Due to Electrostatic Induction 45 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1.1 Point Charge and Inﬁnite Plane . . . . . . . . . . . . . . 45 4.1.2 Point Charge and Spherical Shell . . . . . . . . . . . . . . 46 4.2 Point Charge and Cylindrical Shell . . . . . . . . . . . . . . . . . 4 6 4.3 Finite Conducting Cylindrical Shell with Internal Poin t Charge: Solution of Poisson’s Equation . . . . . . . . . . . . . . . . . . . 47 4.3.1 Cylindrical Shell Held at Zero Potential . . . . . . . . . . 4 9 4.4 Inﬁnite Conducting Cylindrical Shell with Internal Poi nt Charge 50 3 4.4.1 Cylindrical Shell Held at Zero Potential . . . . . . . . . . 5 0 4.5 Inﬁnite Conducting Cylindrical Shell with External Poi nt Charge 52 4.5.1 Cylindrical Shell Held at Zero Potential . . . . . . . . . . 5 3 4.5.2 Thin Cylindrical Shell Held at Zero Potential . . . . . . . 56 4.5.3 Inﬁnite Cylindrical Shell Held at Constant Potential . . . 58 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5 Relevant Topics 65 5.1 Properties of the Electrostatic Field . . . . . . . . . . . . . . . . 65 5.2 The Electric Field in Diﬀerent Points of the Cross-secti on of the Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Electromotive Force Versus Potential Diﬀerence . . . . . . . . . . 67 5.4 Russell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 68 II Straight Conductors 71 6 A Long Straight Wire of Circular Cross-section 75 6.1 Conﬁguration of the Problem . . . . . . . . . . . . . . . . . . . . 75 6.2 Force Proportional to the Potential Diﬀerence Acting up on the Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.3 Force Proportional to the Square of the Current . . . . . . . . . 82 6.4 Radial Hall Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7 Coaxial Cable 93 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2 Potentials and Fields . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.3 The Symmetrical Case . . . . . . . . . . . . . . . . . . . . . . . . 97 7.4 The Asymmetrical Case . . . . . . . . . . . . . . . . . . . . . . . 98 7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8 Transmission Line 103 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 8.2 Two-Wire Transmission Line . . . . . . . . . . . . . . . . . . . . 103 8.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 9 Resistive Plates 113 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.2 Single Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.3 Two Parallel Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 116 9.4 Four Parallel Plates . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.4.1 Opposite Potentials . . . . . . . . . . . . . . . . . . . . . 118 9.4.2 Perfect Conductor Plate . . . . . . . . . . . . . . . . . . . 120 4 10 Resistive Strip 123 10.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 10.2 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.4 Comparison with the Experimental Results . . . . . . . . . . . . 128 III Curved Conductors 133 11 Resistive Cylindrical Shell with Azimuthal Current 137 11.1 Conﬁguration of the Problem . . . . . . . . . . . . . . . . . . . . 137 11.2 Potential and Electric Field . . . . . . . . . . . . . . . . . . . . . 1 38 11.3 Surface Charge Densities . . . . . . . . . . . . . . . . . . . . . . . 14 1 11.4 Representation in Fourier Series . . . . . . . . . . . . . . . . . . . 143 11.5 Lumped Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 12 Resistive Spherical Shell with Azimuthal Current 151 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 12.2 Description of the Problem . . . . . . . . . . . . . . . . . . . . . 151 12.3 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 12.4 Electric Field and Surface Charges . . . . . . . . . . . . . . . . . 156 12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 13 Resistive Toroidal Conductor with Azimuthal Current 163 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 13.2 Description of the Problem . . . . . . . . . . . . . . . . . . . . . 163 13.3 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 13.4 Particular Solution for a Steady Azimuthal Current . . . . . . . . 167 13.5 Potential in Particular Cases . . . . . . . . . . . . . . . . . . . . 1 70 13.6 Electric Field and Surface Charges . . . . . . . . . . . . . . . . . 173 13.7 Thin Toroid Approximation . . . . . . . . . . . . . . . . . . . . . 174 13.8 Comparison of the Thin Toroid Carrying a Steady Current with the Case of a Straight Cylindrical Wire Carrying a Steady Cur rent180 13.9 Charged Toroid without Current . . . . . . . . . . . . . . . . . . 18 1 13.10Comparison with Experimental Results . . . . . . . . . . . . . . 184 IV Open Questions 189 14 Future Prospects 191 Appendices 195 A Wilhelm Weber and Surface Charges 195 B Gustav Kirchhoﬀ and Surface Charges 213 5 Bibliography 217 Index 236 6 This book is dedicated to the memory of Wilhelm Eduard Weber ( 1804- 1891). He was one of the main pioneers in the subject develope d here, the study of surface charges in resistive conductors carrying s teady currents. We hope this book will help to make his fundamental work better k nown. i ii Acknowledgments The authors wish to thank many people who collaborated with t hem in pre- vious works related to the topic of this book, and also severa l others for their support, advice, suggestions, references, etc.In particular they thank Waldyr A. Rodrigues Jr., A. Jamil Mania, Jorge I. Cisneros, Hector T . Silva, Jo˜ ao E. Lamesa, Roberto A. Clemente, Ildefonso Harnisch V., Robert o d. A. Martins, A. M. Mansanares, Edmundo Capelas de Oliveira, ´Alvaro Vannucci, Iberˆ e L. Caldas, Daniel Gardelli, Regina F. Avila, Guilherme F. Leal Ferreira, Marcelo de A. Bueno, Humberto de M. Fran¸ ca, Roberto J. M. Covolan, S´ ergio Gama, Haroldo F. de Campos Velho, Marcio A. d. F. Rosa, Jos´ e Em´ ıli o Maiorino, Mark A. Heald, G. Galeczki, P. Graneau, N. Graneau, John D. Jackso n, Oleg D. Jeﬁ- menko, Steve Hutcheon, Thomas E. Phipps Jr., J. Paul Wesley, Junichiro Fukai, J. Guala-Valverde, Howard Hayden, Hartwig Thim, D. F. Bartl ett, F. Doran, C. Dulaney, Gudrun Wolfschmidt, Karin Reich, Karl H. Wiederke hr, Bruce Sher- wood, Johann Marinsek, Eduardo Greaves, Samuel Doughty, H. H¨ artel and C. Roy Keys. AKTA wishes to thank Hamburg University and the Alexander vo n Hum- boldt Foundation of Germany for a research fellowship on “We ber’s law applied to electromagnetism and gravitation.” This research was de veloped at the In- stitut f¨ ur Geschichte der Naturwissenschaften (IGN) of Ha mburg University, Germany, in the period from August 2001 to November 2002, dur ing which he ﬁrst had the idea to write this book. He was extremely well rec eived in a friendly atmosphere and had full scientiﬁc and institutional suppor t from Prof. Karin Reich and Dr. K. H. Wiederkehr. JAH wishes to thank CNPq, Braz il, for ﬁnan- cial support. The authors thank also FAEP-UNICAMP for ﬁnanc ial support to this project, and the Institute of Physics of the State Unive rsity of Campinas - UNICAMP which provided them the necessary conditions to un dertake the project. A. K. T. Assis∗and J. A. Hernandes† ∗Institute of Physics, State University of Campinas, 13083- 970 Campinas - SP, Brazil, E-mail: [email protected], Homepage: http://www.ifi .unicamp.br/˜assis †Universidade Bandeirante de S˜ ao Paulo - UNIBAN, S˜ ao Paulo - SP, Brazil, E-mail: [email protected] iii iv Foreword Is there an interaction - some reciprocal force - between a cu rrent-carrying conductor and a stationary charge nearby? Beneath this simp le question lie some remarkable misunderstandings, which are well illustr ated by the fact that the answers to it commonly found in the scientiﬁc literature and also in many text books are incorrect. In case there is any uncertainty about the answer, all doubt w ill be eliminated by this book. It tackles the question in a brilliant and compr ehensive manner, with numerous hints for relevant experiments and with impre ssive mathematical thoroughness. It is astonishing to learn that, as early as the middle of the 1 9th century, the German physicists Weber and Kirchhoﬀ had derived and publis hed the answer to this problem; however, their work was poorly received by t he scientiﬁc com- munity, and many rejected it as incorrect. The reasons behin d this scientiﬁc setback, which are presented in detail in this book and suppo rted with numer- ous quotations from the literature, represent a real treasu re trove for readers interested in the history of science. It becomes clear that even in the exact science of physics peo ple at times violate basic scientiﬁc principles, for instance by referr ing to the results of ex- periments which have never been carried out for the purpose u nder discussion. This book helps readers not only to develop a detailed knowle dge of a seriously neglected aspect of the so-called simple electric circuit, but reminds us also that even eminent physicists can be mistaken, that mistakes may b e transferred from one textbook generation to the next and that therefore persi stent, watchful and critical reﬂection is required. A didactic comment is appropriate here. The traditional app roach to teach- ing electric circuits based on current and potential diﬀere nce is called into ques- tion by this book. When dealing with electric current one usually pictures dri fting electrons, while for the terms “voltage” or “potential diﬀerence” one d irectly refers to the abstract notion of energy, with no opportunity for visualiz ation. Experience shows that only few school students really understand what “ voltage” and “po- tential diﬀerence” mean. The inevitable result of failure t o understand such basic terms is that many students lose interest in physics. T hose whose conﬁ- dence in their understanding of science is still fragile, ma y attribute failure to grasp these basic concepts as due to their own lack of talent. v Physics remains a popular and crucial subject, so the large n umbers of stu- dents who each year study the subject implies that the search for less abstract and therefore more readily understood alternatives to trad itional approaches is urgent. This book oﬀers such an alternative. It shows that in respect to surface charges there is no fundamental diﬀerence between an electr ostatic system and the ﬂow of an electric current. It refers to recent curriculu m developments concerning “voltage” and “potential diﬀerence” and presen ts a comprehensive survey of related scientiﬁc publications, that have appear ed since the early pa- pers by Weber and Kirchhoﬀ. Why should we refer to drifting electrons when we teach elect ric current and yet not refer to drifting surface charges when teaching v oltage or potential diﬀerence? The ﬁnal objective of the curriculum when voltage is covered will certainly be to deﬁne it quantitatively in terms of energy. For didacti c reasons, how- ever, it does not seem to be justiﬁable to omit a qualitative a nd more concrete preliminary stage, unless there is a lack of knowledge about the existence of surface charges. In the present market there are newly devel oped curriculum materials that cover basic electricity, to which the conten t of this book relates strongly. Comparison of the approach that this book propose s with more tra- ditional approaches should dispel any doubts about the need for the methods that it describes. This book provides a crucial step along the path to a better un derstanding of electrical phenomenon especially the movement of electr ons in electic circuits. Hermann H¨ artel Guest scientist at Institut f¨ ur Theoretische Physik und As trophysik Universit¨ at Kiel Leibnizstrasse 15 D-24098 Kiel, Germany E-mail: [email protected] vi Vorwort Gibt es eine Wechselwirkung zwischen einem stromf¨ uhrende n Leiter und einem station¨ aren Ladungstr¨ ager? Diese lapidare Frage enth¨ a lt eine erstaunliche Bri- sanz, zumal die Antworten, die man bis zu diesem Tag in der Fac hliteratur und auch in weit verbreitenden Lehrb¨ uchern ﬁndet, h¨ auﬁg unzu treﬀend sind. Das vorliegende Buch beantwortet die Eingangsfrage in brillan ter Weise: umfassend, mit zahlreichen Verweisen auf entsprechende Versuche und m it rigoroser, ma- thematischer Gr¨ undlichkeit. Sofern Zweifel an einer positiven Antwort vorhanden waren, sind diese nach dem Studium des Buches ausger¨ aumt. Erstaunlicherweise wurde bereits Mitte des 19 Jahrhundert s von den deut- schen Physikern Weber und Kirchhoﬀ eine zutreﬀende Antwort ver¨ oﬀentlicht, die jedoch von der wissenschaftlichen Gemeinde kaum rezipi ert, teilweise sogar als unzutreﬀend zur¨ uckgewiesen wurde. Die Gr¨ unde f¨ ur di esen wissenschaft- lichen R¨ uckschritt, die in dem Buch ausf¨ uhrlich dargeste llt und mit zahlreichen Literaturzitaten belegt werden, stellen eine wahre Fundgr ube f¨ ur wissenschafts- historisch interessierte Leser dar. Sie machen deutlich, daß auch in der Physik als exakte Wissen schaft manch- mal gegen methodische Grundprinzipien verstoßen wird, in d em zum Beispiel ein Verweis auf Experimente erfolgt, die nie gezielt durchgef¨ uhrt wurden. So verhilft dies Buch seinen Lesern nicht nur zu einer fundierten Kenntn is ¨ uber einen stark vernachl¨ assigten Bereich des sogenannten einfachen elek trischen Stromkreises, sondern bringt in Erinnerung, daß auch die f¨ uhrenden Vertr eter unserer Diszi- plin irren k¨ onnen, daß unter Umst¨ anden solche Irrt¨ umer v on einer Lehrbuch- generation auf die n¨ achste ¨ ubertragen werden und somit be st¨ andige, wachsame und kritische Reﬂexion geboten ist. Eine didaktische Anmerkung erscheint angebracht. Die im Ph ysikunterricht ¨ ubliche Vermittlung des elektrischen Stromkreis mit den G rundbegriﬀen Strom und Spannung, wird durch den Inhalt des vorliegenden Buches grundlegend in Frage gestellt. W¨ ahrend zum Begriﬀ des elektrischen Stromes noch Bilder vo n driftenden Elektronen angeboten werden, ﬁndet die Einf¨ uhrung der Spa nnung bzw. des Potentials auf der abstrakteren Ebene der Energie statt und l¨ aßt daher keinerlei Veranschaulichung zu. Wie die Erfahrung zeigt gelangen nur wenige Sch¨ uler zu ein tieferes Verst¨ andnis des Spannungsbegriﬀs. Dagege n f¨ uhrt bei vielen Sch¨ ulern ein solches Scheitern gerade an einem so grundleg enden Begriﬀ wie vii dem der Spannung zur Aufgabe des Interesses an physikalisch en Inhalten. Vor allem j¨ ungere Sch¨ uler mit noch schwach entwickeltem Selb stvertrauen m¨ ogen ein solches Scheitern sich selbst und dem eigenen Unverm¨ og en zuschreiben? Physik ist ein allgemein bildendendes und wichtiges Fach un d da hiervon gr¨ oßere Sch¨ ulerpopulationen betroﬀen sind, stellt die S uche nach weniger ab- strakten und damit verst¨ andlicheren Alternativen eine dr ingende Aufgabe dar. Das vorliegende Buch verweist auf eine solche Alternative. Es zeigt auf, daß es im Hinblick auf Oberﬂ¨ achenladungen keinen entscheiden den Unterschied gibt zwischen einer elektrostatischen Anordnung und einem stat ion¨ aren Stromﬂuß. Es verweist auf curriculare Neuentwicklungen zum Spannung sbegriﬀ und gibt einen umfassenden ¨Uberblick ¨ uber die wissenschaftlichen Ver¨ oﬀentlichung en, die seit den Arbeiten von Weber und Kirchhoﬀ erschienen sind. Warum sollte man also bei der Behandlung des Begriﬀs “elektr ischer Strom” auf das Driften von Elektronen verweisen, beim Begriﬀ “elek trische Spannung” aber nicht auf die Existenz driftender Oberﬂ¨ achenladunge n? Sicherlich wird es das Ziel des Unterrichts sein, den Spannu ngs- und Poten- tialbegriﬀ auf der Ebene der Ernergie quantitativ zu behand eln. Eine qualitative und anschauliche Vorstufe auszulassen ist jedoch didaktis ch nicht vertretbar, es sei denn, man hat von der Existenz von driftender Oberﬂ¨ ache nladungen keine Kenntnis. Es gibt curriculare Neuentwicklungen zur Elektrizit¨ atsl ehre, in denen die Inhalte dieses Buches ausf¨ uhrlich zur Sprache kommen. Ver gleiche mit tradi- tionellen Kursen hinsichtlich Lernerfolg und Lernmotivat ion sollten durchgef¨ uhrt werden, um letzte Zweifel an der Notwendigkeit einer eigene n curricularen Neuen- twicklung zu beheben. Auf dem Weg zu einem tieferen Verst¨ andnis elektrischer Ph¨ anomene, ins- besondere der Bewegung von Elektronen in Stromkreisen lief ert dieses Buch einen entscheidenden Beitrag. viii Part I Introduction 1 The goal of this book is to analyze the force between a point ch arge and a resistive wire carrying a steady current, when they are at re st relative to one another and the charge is external to the circuit. Analogous ly, we consider the potential and electric ﬁeld inside and outside resistiv e conductors carrying steady currents. We also want to discuss the distribution of charges along the surface of the conductors which generate this ﬁeld. This is a n important subject for understanding the ﬂow of currents along conductors. Unf ortunately, it has been neglected by most authors writing about electromagnet ism. Our aim is to present the solutions to the main simple cases which can be so lved analytically in order to show the most important properties of this phenom enon. It is written for undergraduate and graduate students in the following courses: physics, electrical engineering, mathematics, history an d philosophy of science. We hope that it will be utilized as a complementary text in cou rses on elec- tromagnetism, electrical circuits, mathematical methods of physics, and history and philosophy of science. Our intention is to help in the tra ining of critical thinking in students and to deepen their knowledge of this fu ndamental area of science. We begin by showing that many important authors held incorre ct points of view regarding steady currents, not only in the past but also in recent years. We then discuss many experiments proving the existence of a f orce between a resistive conductor carrying a steady current and an exter nal charge at rest relative to the conductor. This ﬁrst topic shows that classi cal electrodynamics is a lively subject in which there is still much to be discover ed. The readers can also enhance their critical reasoning in respect to the subj ect matter. Another goal is to show that electrostatics and steady curre nts are intrin- sically connected. The electric ﬁelds inside and outside re sistive conductors carrying steady currents are due to distributions of charge s along their surfaces, maintained by the batteries. This uniﬁes the textbook treat ments of the sub- jects of electrostatics and steady currents, contrary to wh at we ﬁnd nowadays in most works on these topics. We begin dealing with pure electrostatics, namely, the forc e between a con- ductor and an external point charge at rest relative to it. Th at is, we deal with electrostatic induction, image charges and related su bjects. In particular we calculate in detail the force between a long cylindrical c onductor and an external point charge at rest relative to the conductor. We then move to the main subject of the book. We consider the fo rce between a resistive wire carrying a steady current and a poin t charge at rest relative to the wire, outside the wire. In particular, we dea l with the component of this force which is proportional to the voltage of the batt ery connected to the wire (we discuss the voltage or electromotive force of a b attery, together with its distinction from the concept of potential diﬀerenc e, in Section 5.3). We embark on this analysis by ﬁrst considering straight cond uctors of arbitrary cross-section in general and a general theorem on their surf ace charges. Next we deal with a long straight conductor of circular cross-sec tion. Then we treat a coaxial cable and a transmission line (twin lead). We subse quently deal with conducting planes and a straight strip of ﬁnite width. 3 In the third part we consider cases in which the closed curren t follows curved trajectories through resistive conductors. Once more we ar e interested in the force between this conductor and an external point charge at rest relative to it. Initially we deal with a long cylindrical shell with azimuth al current. Then we consider the current ﬂowing in the azimuthal direction alon g a resistive spherical shell. And ﬁnally we treat the case of a toroidal conductor wi th steady azimuthal current. Although much more complicated than the previous c ases, this last situation is extremely important, as it can model a circuit b ounded in a ﬁnite volume of space carrying a closed steady current, like a resi stive ring. Our intention in including analytical solutions of all thes e basic cases in a single work is to make it possible to utilize this material in the undergraduate and graduate courses mentioned earlier. Although the mathe matical treatments and procedures are more or less the same in all cases, they are presented in detail for conductors of diﬀerent shapes, so that the chapte rs can be studied independently from one another. It can then easily be incorp orated in standard textbooks dealing with electromagnetism and mathematical methods for scien- tists. Part of the material presented here was previously di scussed in textbooks and research papers. We feel that the reason why it has not yet been incorpo- rated into most textbooks, which actually present false sta tements related to this topic, is that all these simple cases have never been ass embled in a coherent fashion. We hope to overcome this limitation with this book. At the end of this work we present open questions and future pr ospects. In an Appendix we discuss an important work by Wilhelm Weber where he presented a calculation of surface charges in resistive conductors car rying a steady current, a remarkable piece of work which has unfortunately been forgo tten during all these years. We also discuss Kirchhoﬀ’s work on surface charges an d the derivation by Weber and Kirchhoﬀ of the telegraphy equation. A full bibliography is included at the end of the book. In this work we utilize the International System of Units SI. When we deﬁne a concept, we utilize the ≡symbol to denote a deﬁnition. We represent the force exerted by bodyjoniby/vectorFji. When we say that a body is stationary or moving with velocity/vector v, we consider the laboratory as the frame of reference, unles s stated otherwise. The laboratory is treated here as an approximate ly inertial frame of reference, for the purpose of experiment. When we say that a “ charge” exerts a force, creates an electric ﬁeld, or is acted upon by an exter nal force, we mean a “charged body,” or a “body with the property of being electr ically charged.” That is, we consider charge as a property of a body, not as a phy sical entity. We consider the concepts of electric and magnetic ﬁelds to be ma thematical devices embodying the physical forces between charged bodies, betw een magnets or between current carrying conductors. That is, it is possibl e to say that a current- carrying wire generates electric and magnetic ﬁelds, as usu ally expressed by most authors. In this sense an alternative title of this book migh t be “The electric ﬁeld outside resistive wires carrying steady currents.” But the primary reality for us is the force or interaction between material bodies (gene rating their relative accelerations relative to inertial frames), and not the abs tract ﬁeld concepts existing in space independent of the presence of a charged te st particle which 4 can detect the existence of these ﬁelds. 5 6 Chapter 1 Main Questions and False Answers 1.1 Simple Questions Consider a resistive circuit as represented in Figure 1.1. Figure 1.1: A battery supplying a constant voltage Vbetween its terminals generates a steady current Iin a uniformly resistive wire. Is there a force between the circuit and an external point charge qat rest relative to the wire? Is any component of this force proportional to the voltage of the battery? A stationary, homogeneous and isotropic wire of uniform res istivity con- nected to a battery (which generates a voltage Vbetween its terminals) carries a steady current I. The main questions addressed in this work are the following : a) Will the resistive wire carrying a steady current exert a f orce on a station- ary chargeqlocated nearby? Will any component of this force depend upon the voltage generated by the battery? This is the most important question discussed 7 in this work. b) A related question is the following: Will this wire exert a ny action upon a conductor, or upon neutral dielectrics placed nearby? In p articular, will the resistive wire carrying a steady current electrically pola rize a neutral conductor placed nearby, attracting the conductor? We can also rephrase these questions utilizing the concepts of electric and magnetic ﬁelds. In this case we can say that the current-carr ying wire creates a magnetic ﬁeld outside itself. This magnetic ﬁeld will act u pon mobile test charges. We can then rephrase our question in terms of an elec tric ﬁeld: c) Does a resistive wire connected to a battery and carrying a steady current produce an external electric ﬁeld? If so, is this electric ﬁe ld dependent upon the voltage V generated by the battery? Other related questions: d) Is the resistive wire carrying a steady current electrica lly neutral along its surface? If not, how does the density of surface charges vary along the length of the wire? That is, how does it change as a function of the dis tance along the wire from one of the terminals of the battery? Is this density of surface charges a function of the voltage of the battery? e) Does the wire carrying a steady current have a net distribu tion of charges inside it? That is, is it electrically neutral at all interna l points? If it is not neutral, does this volume density of charges depend upon the voltage of the battery? Will this volume density of charges vary along the l ength of the wire, i.e., as a function of the distance along the wire from one of the te rminals of the battery? f) Where are the charges which produce the internal electric ﬁeld in a current- carrying wire located? This electric ﬁeld is essentially pa rallel to the wire at each point, following the shape and curvature of the wire, ac cording to Ohm’s law. But where are the charges that create it? Are they all ins ide the battery (or along the surface of the battery)? These are the main questions discussed in this work1[1]. One force which will be present regardless of the value of the current is that due to the electrostatically induced charges in the wire. Th at is, the external point particle qinduces a distribution of charges along the surface of the co n- ducting wire, and the net result will be an electrostatic att raction between the wire andq. Most authors know about this fact, although the majority fo rget to mention it. Moreover, they neither consider it in detail n or give the order of magnitude of this force of attraction. Is there another force between the wire and the stationary ch arge? In par- ticular, is there a force between the stationary charge and t he resistive current carrying wire that depends upon the voltage of the battery co nnected to the wire? Many physicists believe the answer to this question is “no,” and this opinion has been held for a long time. There are three main rea sons for this belief. We analyze each one of them here. The quotations pres ented herein are not intended to be complete, nor as criticism of any speciﬁc a uthor, but only to 1All papers by Assis can be found in PDF format at: http://www. ifi.unicamp.br/˜assis/ 8 indicate how widespread false beliefs about basic electrom agnetism really are. 1.2 Charge Neutrality of the Resistive Wire The ﬁrst idea relates to the supposition that a stationary re sistive wire carrying a steady current is essentially neutral in all its interior p oints and along its entire surface. This leads to the conclusion that a resistiv e wire carrying a steady current generates only a magnetic ﬁeld outside it. Many scie ntists have held this belief, for more than a century. Clausius (1822-1888), for i nstance, based all his electrodynamics on this supposition. In 1877 he wrote ([2] a nd [3, page 589]): “We accept as criterion the experimental result that a close d constant current in a stationary conductor exerts no force on stationary elec tricity.” Although he stated that this is an experimental result, he did not cite any experiments that sought to ﬁnd this force. As we will see, he based his elec trodynamics on an untenable principle, as a force between a stationary wire carrying a steady current and an external stationary charge does exist. This f orce has been shown to exist experimentally, as we discuss below. We conﬁrm the e xistence of this force with calculations. Recently the name “Clausius postulate” has been attached by some authors to the following statements: “Any current element of a close d current in a stationary conductor is electrically neutral” [4]; “For a s tationary circuit the charge density ρis zero” [5]; “Φ = 0,” namely, that the potential generated by a closed circuit carrying a steady current is null at all exte rnal points [6, 7]. We even ﬁnd statements like this in fairly recent electromag netic textbooks. As we will see, the electric ﬁeld inside and outside a resisti ve wire carrying a steady current is due to surface charges distributed along t he wire. On the other hand, Reitz, Milford and Christy, for instance, seem t o say that no steady surface charges can exist in resistive wires [8, pp. 168-169 ]: “Consider a con- ducting specimen obeying Ohm’s law, in the shape of a straigh t wire of uniform cross-section with a constant potential diﬀerence, △ϕ, maintained between its ends. The wire is assumed to be homogeneous and characterize d by the constant conductivity g. Under these conditions an electric ﬁeld will exist in the wi re, the ﬁeld being related to △ϕby the relation △ϕ=/integraltext/vectorE·d/vectorℓ. It is evident that there can be no steady-state component of electric ﬁeld at ri ght angles to the axis of the wire, since by Eq. /vectorJ=g/vectorEthis would produce a continual charging of the wire’s surface. Thus, the electric ﬁeld is purely long itudinal.” Although Russell criticized this statement as it appeared in second e dition of the book (1967) [9], the third and fourth editions were not changed si gniﬁcantly on this point. Here we show that there is a steady surface charge in th is conductor, and that there is a steady-state component of electric ﬁeld a t right angles to the axis of the wire, contrary to their statement. In Jackson’s book we ﬁnd the following statement ([10, exerc ise 14.12, page 503] [11, exercise 14.13, page 697]): “As an idealization of steady-state currents ﬂowing in a circuit, consider a system of Nidentical charges qmoving with constant speedv(but subject to accelerations) in an arbitrary closed path. Suc- 9 cessive charges are separated by a constant small interval △. Starting with the Li´ enard-Wiechert ﬁelds for each particle, and making no as sumptions concern- ing the speed vrelative to the velocity of light show that, in the limit N→ ∞, q→0, and △ → 0, butNq= constant and q/△= constant, no radiation is emitted by the system and the electric and magnetic ﬁelds of t he system are the usual static values. (Note that for a real circuit the statio nary positive ions in the conductors will produce an electric ﬁeld which just canc els that due to the moving charges.)” Here Jackson refers to the second order electric ﬁeld and the lack of radiation produced by all the electrons in a current carrying resistiv e wire, even though the electrons are accelerated. However, a casual reader of this statement, specially the sentence in parenthesis, will conclude that Clausius wa s right. However, we will see here that there is a net nonzero electric ﬁeld outs ide a stationary resistive wire carrying a steady current. Despite the wordi ng of this exercise, it must be stressed that Jackson is one of the few modern authors who is aware of the electric ﬁeld outside wires carrying steady currents , as can be seen in his important work of 1996 [12]. In the third edition of this b ook the sentence between parenthesis has been changed to [13, exercise 14.24 , pages 705-706]: “(Note that for a real circuit the stationary positive ions i n the conductors neutralize the bulk charge density of the moving charges.)” In this form the sentence does not explicitly mention whether or not an exter nal electric ﬁeld exists. But even the statement of charge neutrality inside a wire carrying a steady current is subject to debate. See Section 6.4. Edwards said the following in the ﬁrst paragraph of his 1974 p aper on the measurement of a second order electric ﬁeld [14]: “For over a century it has been almost axiomatic in electromagnetism that the electri c ﬁeld produced by a current in a stationary conductor forming a closed circuit i s exactly zero. To be sure the ﬁrst order ﬁeld, dependent upon the current I, is experimentally and theoretically zero, but several early electromagnetic the ories, including Weber’s, Riemann’s and Ritz’, predict a second order eﬀect dependent uponI2orv2/c2 wherevis the charge drifting velocity.” We will see here that there is an electric ﬁeld outside a resistive wire carrying a steady current. Thi s external electric ﬁeld is proportional to the voltage of the battery, or to the p otential diﬀerence acting along the wire. Edwards, Kenyon and Lemon had the following to say about ﬁrst order terms,i.e., to forces proportional to the current due to a resistive wir e carrying a steady current, or forces proportional to vd/c, wherevdis the drifting velocity of the moving charges in the wire and cis the light velocity [15]: “It has long been known that the zero- and ﬁrst- order forces on a charged o bject near a charge- neutral, current-carrying conductor at rest in the laboratory are zero in magnitude.” The experiments discussed below and the calcul ations presented in this book show that a normal resistive wire carrying a stea dy current cannot be electrically neutral at all points. Moreover, it will gen erate a zeroth order force upon a charged body placed in proximity. It will also ge nerate a force proportional to the voltage or electromotive force of the ba ttery connected to the wire. This force will act upon any charged body brought ne ar the wire. It 10 will also polarize any neutral conductor that is brought nea r the wire. A similar statement can be found in Griﬃths’s book [16, p. 273 ]: “Within a material of uniform conductivity, ∇ ·E= (∇ ·J)/σ= 0 for steady currents (equation ∇ ·J= 0), and therefore the charge density is zero. Any unbalance d charge resides on the surface .” We will show that there is also an unbalanced charge in the interior of a resistive wire carrying a steady c urrent. And similarly [16, p. 196] (our emphasis in boldface): “Two w ires hang from the ceiling, a few inches apart. When I turn on a current, so th at it passes up one wire and back down the other, the wires jump apart – they pl ainly repel one another. How do you explain this? Well, you might suppose tha t the battery (or whatever drives the current) is actually charging up the wire, so naturally the diﬀerent sections repel. But this “explanation” is inco rrect.I could hold up a test charge near these wires and there would be no force on it, indicating that the wires are in fact electrically neutral. (It’s true that electrons are ﬂowing down the line - that’s what a curren tis– but there are still just as many plus as minus charges on any gi ven segment.) Moreover, I could hook up my demonstration so as to make the current ﬂow up bothwires; in this case the wires are found to attract! ” Here we show that the statement in boldface is wrong. Despite these statements it should be mentioned that Griﬃth s is aware of the surface charges in resistive conductors with steady cur rents and the related electric ﬁeld outside the wires [16, pp. 279 and 336-337]. A similar statement is made by Coombes and Laue [17]: “For a st eady current in a homogeneous conductor, the charge density ρis zero inside the conductor.” Lorrain, Corson and Lorrain, meanwhile, state that [18, p. 2 87]: “A wire that is stationary in reference frame Scarries a current density J. The net volume charge density in Sis zero:ρ=ρp+ρn= 0.” Here ρpandρnrefer to the positive and negative volume charge densities, respect ively. Although aware of the distribution of charges along the surf ace of resis- tive conductors carrying steady currents and the correspon ding external elec- tric ﬁeld, Seely also believed that the internal density of c harges is zero [19, p. 149]: “Note that the net charge in any element of volume insid e a conductor must be zero in either the static case or the electron-ﬂow cas e. That is, the net charge per unit volume when the electrons and the ions of the m etal lattice are considered just balance. Otherwise, an unstable component of an electric ﬁeld will be developed. Hence, all net electric charge in a conduc tor resides on the surface of the conducting material. It is the function of the generator to pile up electrons on one end of the conductor and to remove them from t he other end. The internal ﬁeld is thus produced by a density gradient of th e surface charges.” Like Seely, Popovic was also aware of the surface charges and external elec- tric ﬁeld of resistive wires with steady currents, and he eve n presents a qual- itative drawing depicting them [20, pp. 201-202]. But in the next section he “proves” that the volume density of free charges ρgoes to zero at all internal points of a homogeneous conductor [20, p. 206]: “ ρ= 0 (at all points of a homogeneous conductor). This is a very important conclusio n. Accumulations 11 of electric charges creating the electric ﬁeld that maintai ns a steady current in homogeneous conductors cannot be inside the conductors. Ch arges can reside only on the boundary surfaces of two diﬀerent conductors, or of a conductor and an insulator.” The ﬂaw in all these statements is that the authors have forgo tten or ne- glected the azimuthal magnetic ﬁeld inside conductors with steady currents which is created by the longitudinal current. The magnetic f orce due to this ﬁeld acting upon the conduction electrons will lead to an acc umulation of neg- ative charges along the axis of the conductors, until an elec tric ﬁeld is created orthogonal to the conductor axis. This will exert an electri c force on the mobile electrons, balancing the magnetic force. As a consequence, a steady current conductor must have a net negative volume density of charges in its interior, as we will discuss quantitatively in Section 6.4. Despite this shortcoming, Popovic’s important work is one o f the few text- books that calls attention to the external electric ﬁeld of c urrent carrying resis- tive wires, and that even presents a qualitative drawing of t his ﬁeld in a generic circuit. One of us (AKTA) also assumed, in previous publications, tha t a resistive wire carrying a steady current was essentially neutral at al l points. On the topic of positive qi+and negative qi−charges of a current element i, we wrote [21]: “In these expressions we assumed qi−=−qi+because we are considering only neutral current elements.” The same assumption was made a ye ar later [22]: “We suppose this current distribution to have a zero net char geq2−=−q2+.” In 1994 we wrote [23, p. 85]: “To perform this summation we sup pose that the current elements are electrically neutral, namely dqj−=−dqj+,dqi−=−dqi+. This was the situation in Amp` ere’s experiments (neutral cu rrents in metallic conductors), and happens in most practical situations (cur rents in wires, in gaseous plasmas, in conducting liquid solutions, etc.)” And similarly, in a Section entitled “Electric Field Due to a Stationary, Neutral and Co nstant Current” we wrote [23, p. 161]: “In this wire we have a stationary current I2which is constant in time and electrically neutral.” Here we show in d etail that these statements are not valid for normal resistive wires carryin g steady currents. When we wrote these statements we were not completely aware o f the external electric ﬁeld proportional to the voltage of the battery, wh ich is the main subject of this book, nor of its related surface charges. We were foll owing most other textbook authors in assuming resistive wires carrying stea dy currents to be essentially neutral at all points. We were concerned only wi th the second order electric ﬁeld, a subject which we also discuss in this work. I t was around 1992 that we began to be aware of the surface charges in resistive w ires and the corresponding external electric ﬁeld proportional to the v oltage of the battery, due to a study of Kirchhoﬀ’s works from 1849 to 1857 [24, 25, 26 ]. All three of these important papers by Kirchhoﬀ exist in English transla tion [27, 28, 29]. We tried to understand, repeat and extend Kirchhoﬀ’s derivati on of the telegraphy equation based on Weber’s electrodynamics. We suceeded in 1 996, and the result of our labours was published in 2000 [30] and 2005 [31]. Simul taneously for several years we sought a solution for the potential outside a straight cylindrical 12 resistive wire carrying a steady current, until we found the solution in 1997, presented here in Chapter 6. In the same years we discovered J eﬁmenko’s book and papers with his experiments, and also many papers by othe r authors cited earlier. In the following Chapters we show how much have we le arned from important recent authors who studied surface charges and re lated topics in speciﬁc conﬁgurations. We quote them in the appropriate sec tions. Our ﬁrst paper on this subject was published in 1999 [1]. Since then we have published other works dealing with several other spatial conﬁguratio ns. We only became aware of Weber’s 1852 work [32] dealing with related subject s in 2001-2002 during our research in Germany quoted in the Acknowledgment s. In the period from 2004 to 2006 we had the opportunity to study Weber’s work in greater detail, and we present a discussion of it in the ﬁrst Appendix of this book. Our hope in publishing this book is that others will not need t o follow this tortuous path of discovery. In the bibliography at the end of the book, we have collected many important references by recent authors who h ave dealt with the subject of this book. We hope that others can draw on these wor ks to achieve new results in a more eﬃcient manner. 1.3 Magnetism as a Relativistic Eﬀect The second idea leading to the conclusion that a normal resis tive current- carrying wire generates no electric ﬁeld outside it arises f rom the supposition that magnetism is a relativistic eﬀect. A typical statement of this position can be found in Feynman’s Lectures on Physics , speciﬁcally in Section 13-6 (The rel- ativity of magnetic and electric ﬁelds [33, p. 13-7]) (our em phasis in boldface): “We return to our atomic description of a wire carrying a curr ent.In a normal conductor, like copper , the electric currents come from the motion of some of the negative electrons - called the conduction electrons - while the positive nuclear charges and the remainder of the electrons stay ﬁxed in the body of the material. We let the density of the conduction electrons be ρ−and their velocity inSbev. The density of the charges at rest in Sisρ+, which must be equal to the negative of ρ−, since we are considering an uncharged wire. There is thus no electric ﬁeld outside the wire , and the force on the moving particle is just F=qvo×B.” The statement that there is no electric ﬁeld outside a resistive wire (like copper) carrying a constant current is certainly false. One of the main goals of this book is to calculate this electric ﬁe ld and compare the theoretical calculations with the experimental results pr esented below. In Purcell’s Electricity and Magnetism we ﬁnd the same ideas [34]. In Section 5.9 of this book, which treats magnetism as a relativistic ph enomenon, he models a current-carrying wire by two strings of charges, positive and negative, moving relative to one another. He then considers two current carry ing metallic wires at rest in the frame of the laboratory, writing (p. 178): “In a metal, however, only the positive charges remain ﬁxed in the crystal lattice . Two such wires carrying currents in opposite directions are seen in the lab frame in Fig. 5.23a. The wires being neutral, there is no electric force from the o pposite wire on the 13 positive ions which are stationary in the lab frame.” That is , he believes that a resistive wire carrying a steady current generates no exter nal electric ﬁeld. For this reason he believes that this wire will not act upon an ext ernal test charge at rest relative to the wire. This is simply false. A normal re sistive stationary metallic wire carrying a steady current cannot be neutral at all points. It must have a distribution of surface charges which will produce th e electric ﬁeld driving the current inside it, and which will also exert net forces up on the stationary charges of the other wire. Other books dealing with relativity present similar statem ents connected with Lorentz’s transformations between electric and magne tic ﬁelds, about mag- netism as a relativistic eﬀect, about a normal resistive wir e carrying a steady current being electrically neutral, etc.For this reason we will not quote them here. The examples of Feynman, Leighton, Sands and Purcell i llustrate the problems of these points of view. It is important to recall here that Jackson [11, Section 12.2 , pp. 578-581] and Jeﬁmenko [35] have shown that it is impossible to derive m agnetic ﬁelds from Coulomb’s law and the kinematics of special relativity without additional assumptions. 1.4 Weber’s Electrodynamics The third kind of idea related to this widespread belief is co nnected with the electrodynamics developed by Wilhelm Eduard Weber (1804-1 891), in particular his force law of 1846. Weber’s complete works were published in 6 volumes between 1 892 and 1894 [36, 37, 38, 39, 40, 41]. Only a few of his papers and letters ha ve been translated into English [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. The best biographies of Weber are those of Wiederkehr [56, 57 , 58]. Some other important biographies and/or discussions of his work s can be found in several important publications and in the references quote d in these works [3, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. Modern applications, discussions and developments of Webe r’s law applied to electrodynamics and gravitation can be found in several r ecent publications [23, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93 , 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 1 11, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127 , 128, 129, 130, 131, 132, 133]. Several other works and authors are quoted in these books and papers. Weber’s force is a generalization of Coulomb’s law, includi ng terms which depend on the relative velocity and relative acceleration b etween the interacting charges. Charges q1andq2located at/vector r1and/vector r2move with velocities /vector v1and /vector v2and accelerations /vector a1and/vector a2, respectively, relative to a frame of reference O. According to Weber’s law of 1846 the force exerted by q2onq1,/vectorF21, is given by (in the international system of units and with vectorial not ation): 14 /vectorF21=q1q2 4πε0ˆr12 r2 12/parenleftbigg 1−˙r2 12 2c2+r12¨r12 c2/parenrightbigg =−/vectorF12. (1.1) Hereε0= 8.85×10−12C2N−1m−2is called the permittivity of free space, r12≡ |/vector r1−/vector r2|is the distance between the charges, ˆ r12≡(/vector r1−/vector r2)/r12is the unit vector pointing from q2toq1, ˙r12≡dr12/dt= ˆr12·(/vector v1−/vector v2) is the relative radial velocity between the charges, ¨ r12≡d˙r12/dt=d2r12/dt2= [(/vector v1−/vector v2)·(/vector v1− /vector v2)−(ˆr12·(/vector v1−/vector v2))2+(/vector r1−/vector r2)·(/vector a1−/vector a2)]/r12is the relative radial acceleration between the charges and c= 3×108m/s is the ratio between electromagnetic and electrostatic units of charge. This constant was introd uced by Weber in 1846 and its value was ﬁrst determined experimentally by Web er and Kohlrasch in 1854-55 [62, 134, 135, 136]. One of their papers has been tr anslated into English [55]. In the works quoted above there are detailed di scussions of this fundamental experiment and its meaning. The ﬁrst point to be mentioned here is that the main subject of this book can be derived from Weber’s law. As a matter of fact, we are mainly concerned with the force between a resistive wire carrying a steady current and an external point charge at rest relative to the wire. To this end we employ esse ntially Coulomb’s force between point charges or, analogously, Gauss’s law an d Poisson’s law. And these three expressions (the laws of Coulomb, Gauss and Pois son) are a special case of Weber’s law when there is no motion between the intera cting charges (or when we can disregard the small second order components o f Weber’s force, which are proportional to the square of the drifting velocit y of the charges, in comparison with the coulombian component of Weber’s force) . With this force Weber succeeded in deriving from a single exp ression the whole of electrostatics, magnetostatics, Amp` ere’s force between current elements and Faraday’s law of induction. When he presented his fundamental force law in 1846, Weber su pposed that electric currents in normal resistive wires are composed of an equal amount of positive and negative charges moving relative to the wire with equal and opposite velocities, the so-called Fechner hypothesis [13 7, pp. 135 and 145 of theWerke ]. This model of the electric current had been presented by Fe chner in 1845 [138]. Ideas of a double current of positive and negativ e charges somewhat similar to these had been presented before by Oersted [139, 1 40] and by Amp` ere [141, 142]. At that time no one knew about electrons, they had no idea of the value of the drifting velocities of the mobile charges in cur rent-carrying wires, etc.Later on it was found that only the negative electrons move in metallic wires carrying steady currents, while the positive ions rem ain ﬁxed relative to the lattice. Despite this fact, in his own paper of 1846 Weber already considered that Fechner’s hypothesis might be generalized considerin g positive and negative charges moving with diﬀerent velocities. Speciﬁcally, lar ger particles might ﬂow slower, while smaller particles might move faster [137, p. 2 04 of the Werke ]. That is, the particle with a larger inertial mass would move s lower inside a current carrying wire than the particle with a smaller inert ial mass. In his paper of 1852 which we analyse in the Appendix, Weber even con siders the 15 situation in which the positive charges were ﬁxed in the cond uctor while only the negative charges did move relative to it! He was certainl y one of the ﬁrst to explore this possibility, being much ahead of his time. Two main criticisms were made against Weber’s electrodynam ics after it was discovered that Fechner’s hypothesis is wrong. The ﬁrst is r elated to Amp` ere’s force between current elements and the second is related to t he force between a current-carrying wire and an external point charge at rest r elative to the wire. We will discuss each one of these criticisms separately. Amp` ere’s force d2/vectorF21exerted by the current element I2d/vectorℓ2upon the current elementI1d/vectorℓ1(located at /vector r2and/vector r1relative to a frame of reference O) is given by (in the international system of units and with vectorial n otation): d2/vectorF21=−µ0 4πˆr12 r2 12/bracketleftBig 2(d/vectorℓ1·d/vectorℓ2)−3(ˆr12·d/vectorℓ1)(ˆr12·d/vectorℓ2)/bracketrightBig =−d2/vectorF12.(1.2) Many recent publications deal with Amp` ere’s work and his fo rce law [103, 114, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154 , 155, 156, 157]. Further papers are quoted in these books and works. Weber knew Amp` ere’s force and derived it from his force law, assuming Fechner’s hypothesis. However, many people believed wrong ly that without Fechner’s hypothesis it would be impossible to derive Amp` e re’s law from Weber’s force. For this reason they criticized Weber’s law as experi mentally invalidated. But it has been shown recently that even without Fechner’s hy pothesis it is possible to derive Amp` ere’s law from Weber’s force [78, 21, 23, 92, 103, 114]. That is, even supposing that the positive ions are ﬁxed in the lattice and that only the electrons move in current-carrying wires, we deriv e Amp` ere’s force between current elements beginning with Weber’s force betw een point charges. This overcame the ﬁrst criticism of Weber’s law discussed he re. The second criticism is connected with the main subject of th is book, the force between a stationary charge and a current carrying res istive wire. Sup- posing Fechner’s hypothesis, we conclude that there would b e no force between a stationary current-carrying wire and an external charge a t rest relative to the wire (apart from the force of electrostatic induction), if t he wire were neutral in its interior and along its surface. This has been known sin ce Weber’s time. Later on people began to doubt the validity of Fechner’s hypo thesis. It was with the utilization of the Hall eﬀect in the 1880’s and with t he discovery of the electron in 1897 that the order of magnitude of the drifting v elocity of the con- duction charges inside metals was determined. The sign of mo bile charges was also discovered [63, pp. 289-290] [3, Chapter XI: Weber-Rit z, Section 2: The electronic theory of conduction, pp. 512-518]. As a result, it was found that Fechner’s hypothesis was wrong and that only negative charg es move relative to the lattice in normal resistive metallic conductors carr ying steady currents. Supposing (a) that only one kind of charge (positive or negat ive) moves in a current-carrying wire, (b) Weber’s force, and (c) that the w ire is neutral in its interior and along its surface, people concluded that there would be a net force between this stationary current-carrying wire and a statio nary charge nearby. 16 This force is proportional to v2 d/c2, wherevdis the drifting velocity of the con- duction charges and c= 3×108m/s. Based on the erroneous belief that a current carrying resistive wire exerts no force on a station ary charge nearby, unaware even of the larger force between the wire and the char ge, which is proportional to the voltage of the battery connected to the w ire, many authors condemned Weber’s law as experimentally invalidated. This trend goes back at least to Maxwell’s Treatise on Electricity and Mag- netism (1873). He considered the force between a conducting wire ca rrying a constant current and another wire which carries no current, both of them at rest in the laboratory. He then wrote [158, Volume 2, Article 848, page 482], our words between square brackets: “Now we know that by charg ing the sec- ond conducting wire as a whole, we can make e′+e′ 1[net charge on the wire without current] either positive or negative. Such a charge d wire, even without a current, according to this formula [based on Weber’s elect rodynamics], would act on the ﬁrst wire carrying a current in which v2e+v2 1e1[sum of the pos- itive and negative charges of the current-carrying wire by t he square of their drifting velocities] has a value diﬀerent from zero. Such an action has never been observed.” As with Clausius’s comment mentioned earli er, Maxwell did not quote any experiments which tried to observe this force a nd which failed to ﬁnd the eﬀect. Nor did he calculate the order of magnitude of t his eﬀect. This calculation would determine whether it was feasible to try t o detect the eﬀect in the laboratory. Maxwell does not seem to have been aware of th e surface charge distribution in wires carrying steady currents, a subject w hich had already been extensively discussed by Weber twenty years before, as we di scuss in the ﬁrst Appendix of this book. Clausius’s work of 1877 (On a deduction of a new fundamental l aw of electro- dynamics) was directed against Weber’s electrodynamics [2 ]. To the best of our knowledge this paper has never been translated into English . What we quote here is our translation. Clausius supposes only one type of m obile charge in a closed stationary current-carrying wire. He integrates We ber’s force exerted by this wire on an external stationary charge and shows that it i s diﬀerent from zero (in his integration he does not take into account the surface charges generating the electric ﬁeld inside the resistive current-carrying wi re). He then writes, our words between square brackets: “Then the galvanic current m ust, like a body charged with an excess of positive or negative charges, caus e a modiﬁed distribu- tion of electricity in conducting bodies placed in its neigh bourhood. We would obtain a similar eﬀect in conducting bodies around a magnet, when we explain the magnetism through molecular electric currents. Howeve r these eﬀects have never been observed, despite the various opportunities we h ave had to observe it. We then accept the previous proposition, which states th at these eﬀects do not occur, as an acknowledged certain experimental proposi tion. Then, as the result of Equation (4) [Weber’s force diﬀerent from zero act ing upon a station- ary external charge, exerted by a stationary closed current -carrying wire with only one kind of mobile charges] is against this proposition . It follows that We- ber’s fundamental law is incompatible with the point of view that in a stationary conductor with galvanic current only the positive electric ity is in motion. ” He 17 also mentions that this conclusion was reached independent ly by Riecke in 1873, which he became aware of only in 1876. It seems that he was also unaware of Maxwell’s previous analysis. In the sixth Section of his pap er he once again emphasizes his fundamental theorem, namely, “that there is no force upon a stationary charge exerted by a stationary closed conductor carrying a constant galvanic current.” The ﬁrst two paragraphs of the seventh Section of this paper a re also relevant. We quote them here with our words between square brackets: “To deal with the quantity X1we can utilize a similar experimental propo- sition, namely: a stationary quantity of electricity exerts no force upon a s ta- tionary closed conductor carrying a constant galvanic curr ent.” “This proposition needs clariﬁcation. When there is accumu lation of one kind of electricity at any place, for example positive elect ricity, then this elec- tricity exerts the eﬀect of electrostatic inﬂuence [or elec trostatic induction] upon conducting bodies in its neighbourhood, and this eﬀect will also aﬀect the con- ductor in which there is a galvanic current. The previous pro position says only that beyond this eﬀect there is no other special eﬀect depend ing upon the cur- rent, and therefore dependent upon the current intensity. I t should also be remarked that if a closed galvanic current did suﬀer such an e ﬀect, then a mag- net would also suﬀer the eﬀect. However it has always been obs erved, that stationary electricity acts upon a stationary magnet only i n the same way as it acts upon a nonmagnetic piece of metal of the same form and s ize. For this reason we will accept the previous proposition without furt her consideration as a ﬁrm experimental proposition.” Clausius shows here that he is completely unaware of the elec tric ﬁeld outside resistive wires carrying steady currents, which is proport ional to the voltage of the battery. This electric ﬁeld originates from surface c harges which are maintained by the electromotive force exerted by the batter y. For this reason it is probable that no analogous electric ﬁeld should exist o utside a magnet. Therefore, Clausius’s conclusion that if an electric ﬁeld e xisted (as we know nowadays it really exists) outside resistive wires carryin g steady currents, then, necessarily, it would also exist outside a permanent magnet , also seems incorrect. In this paper Clausius obtains a new fundamental law of elect rodynamics which does not lead to this force exerted by a closed stationa ry resistive con- ductor carrying a steady current upon an external charge at r est relative to the conductor, even if only one kind of electricity is in motion i n current-carrying wires. His electrodynamics led to this prediction: “The fun damental law formu- lated by me leads to the result, without the necessity to make the supposition of double current, that a constant stationary closed galvan ic current exerts no force on, nor suﬀers any force from, a stationary charge” [15 9] [3, page 589]. Clausius’s work was not the ﬁrst to criticize Weber’s electr odynamics in this regard; after all, Maxwell had done so before. Despite this f act his work was very inﬂuential and is quoted on this point by many authors. Writing in 1951 Whittaker criticized Weber’s electrodynam ics along the same lines [63, page 205] (our emphasis in boldface): “The assump tion that positive and negative charges move with equal and opposite velocitie s relative to the 18 matter of the conductor is one to which, for various reasons w hich will appear later, objection may be taken; but it is an integral part of We ber’s theory, and cannot be excised from it. In fact, if this condition were not satisﬁed, and if the law of force were Weber’s, electric currents would exert forces on electrostatic charges at rest (...)”. Obviously he is here expressing the v iew that there are no such forces. As a consequence, Weber’s electrodynamics m ust be wrong in Whittaker’s view, because we now know that only the negative electrons move in metallic wires. And applying Weber’s electrodynamics to this situation (in which a current in a metallic conductor is due to the motion of conduction electrons, while the positive charges of the lattice remain stationary) implies that a conducting wire should exert force on a stationary ele ctric charge nearby. Whittaker was not aware of the experimental fact that electric currents exert forces on electrostatic charges at rest . See the experiments discussed below. To give an example of how this misconception regarding Weber ’s electrody- namics has survived we present here the only paragraph from R ohrlich’s book (1965) where he mentions Weber’s theory [160, p. 9]: “Most of the ideas at that time revolved around electricity as some kind of ﬂuid or at le ast continuous medium. In 1845, however, Gustav T. Fechner suggested that e lectric currents might be due to particles of opposite charge which move with equal speeds in opposite directions in a wire. From this idea Wilhelm Weber ( 1804 - 1891) de- veloped the ﬁrst particle electrodynamics (1846). It was based on a force law between two particles of charges e1ande2at a distance rapart, F=e1e2 r2/bracketleftBigg 1 +r c2d2r dt2−1 2c2/parenleftbiggdr dt/parenrightbigg2/bracketrightBigg . This force seemed to ﬁt the experiments (Amp` ere’s law, Biot -Savart’s law), but ran into theoretical diﬃculties and eventually had to be dis carded when, among other things, the basic assumption of equal speeds in opposi te directions was found untenable.” Other examples of this widespread belief: In 1969 Skinner sa id, relative to Figure 1.2 in which the stationary closed circuit carries a c onstant current and there is a stationary charge at P[161, page 163]: “According to Weber’s force law, the current of Figure 2.39 [our Figure 1.2] would exert a force on an electric charge at rest at the point P. (...) And yet a charge at Pdoes not experience any force.” As with Clausius’s and Maxwell’s generic statem ents, Skinner did not quote any speciﬁc experiment which tried to ﬁnd this forc e. Amazingly the caption of his Figure 2.39 states: “A crucial test of Weber’s force law.” To most readers, sentences like this convey the impression that the experiment had been performed and Weber’s law refuted. But the truth is just the o pposite! In fact, several experiments discussed in this book show the existen ce of a force between a stationary charge and a resistive wire carrying a steady cu rrent. Pearson and Kilambi, in a paper discussing the analogies bet ween Weber’s electrodynamics and nuclear forces, made the same kind of cr iticisms in a Section called “Invalidity of Weber’s electrodynamics” [162]. The y consider a straight wire carrying a constant current. They calculate the force o n a stationary 19 Figure 1.2: There is a stationary charge at Pand a steady current ﬂows in the closed circuit. charge nearby due to this wire with classical electromagnet ism and with Weber’s law, supposing the wire to be electrically neutral at all poi nts. According to their calculations, classical electromagnetism does not y ield any force on the test charge and they interpret this as follows (our emphasis underlined): “The vanishing of the force on the stationary charge qcorresponds simply to the factthat a steady current does not give rise to any induced electr ic ﬁeld.” With Weber’s law they ﬁnd a second order force and interpret t his as meaning (our emphasis): “that Weber’s electrodynamics give rise to spurious induction eﬀects. This is probably the most obvious defect of the theor y, and the only way of avoiding it is to suppose that the positive charges in t he wire move with an equal velocity in the opposite direction, which of course they do not.” As we will see, the fact is that a steady current gives rise to an external electric ﬁe ld, as shown by the experiments discussed below. In this work we argue that all of these statements are mislead ing. That is, we show theoretically the existence of a force upon the st ationary external charge exerted by a resistive wire connected to a battery and carrying a steady current when there is no motion between the test charge and th e wire. We also compare the theoretical calculations with the experimenta l results which proved the existence of this force. For this reason these false crit icisms of Weber’s electrodynamics must be disregarded. 1.5 Electric ﬁeld of Zeroth Order; Proportional to the Voltage of the Battery; and of Second Order In this work we discuss the force between a resistive wire car rying a steady current and an external point charge at rest relative to the w ire. Both of them are assumed to be at rest relative to the laboratory, which fo r our purposes can be considered a good inertial frame. We consider three compo nents of this force or electric ﬁeld. The wire is a conductor. Let us suppose that it is initially ne utral in its interior and along its surface, carrying no current. When we put a charge near it, the free charges in the conductor will rearrange themsel ves along the surface 20 of the conductor until it acquires a new constant potential a t all points. As a result of this redistribution of charges, there will be a ne t force between the external point charge and the conductor. We will call it a zer oth order force, /vectorF0. We can describe this situation by saying that the conductor h as now produced an induced electrostatic ﬁeld which will act upon the extern al charge. This electric ﬁeld is independent of the current in the conductor , depending only upon the external charge, its distance to the conductor, and the shape of the conductor. That is, this electric ﬁeld will continue to exis t even when a current begins to ﬂow in the conductor, provided the shape of the cond uctor does not change. We will call it a zeroth-order electric ﬁeld, /vectorE0. We now consider this resistive conductor connected to a batt ery. In the steady state there is a constant current ﬂowing along the wir e. Will there be a force between this wire and the external point charge at res t relative to the wire, with a component of this force depending upon the volta ge of the battery? This is the main subject of this book, and the answer is positi ve. That is, there is a component of this force proportional to the voltage of th e battery. We will represent this component of the force by /vectorF1. It is also possible to say that this wire will generate an external electric ﬁeld proportio nal to the voltage of the battery and depending upon the shape of the wire. We will r epresent this electric ﬁeld by /vectorE1. If there is no test charge outside the wire, the force /vectorF0goes to zero, the same happening with /vectorF1. If it is placed a small conductor, with no net charge, at rest relative to a wire without current, no force /vectorF0is observed between them. On the other hand, when this resistive wire is connected to a c hemical battery and a constant current is ﬂowing though it, there will be an at tractive force between this wire and the small conductor placed at rest near by. That is, there will be a force /vectorF1even when the integrated charge of the small conductor goes to zero. The reason for this force is that the battery will cre ate a redistribution of charges upon the surface of the resistive wire. There will be a gradient of the surface charge density along the wire, with positive charge s close to the positive terminal of the battery and negative charges close to the neg ative terminal of the battery (and with a null charge density at an intermediat e point along the wire). This gradient of surface charges will generate no t only the internal electric ﬁeld (which follows the shape of the wire and is esse ntially parallel to it at each internal point), but also an external electric ﬁel d. And this external electric ﬁeld will polarize the small conductor placed in th e neighbourhood of the wire. This polarization of the conductor will generate a n attractive force between the polarized conductor and the current-carrying w ire. With this eﬀect we can distinguish the forces /vectorF0and/vectorF1. This eﬀect has already been observed experimentally, as will be seen in several experiments desc ribed in Chapter 3. Many papers have also appeared in the literature discussing a second order force or a second order electric ﬁeld, /vectorF2or/vectorE2. As we saw before, usually the people who consider this second order eﬀect are not aware of t he zeroth order eﬀects nor of those proportional to the voltage of the batter y. This second order force is proportional to the square of the current, or propor tional to the square of the drifting velocity of the mobile electrons. Analogously , we can talk of a second 21 order electric ﬁeld generated by the wire. Sometimes this se cond order electric ﬁeld is called motional electric ﬁeld. It has long been known that the force laws of Clausius and of Lorentz (the ones adopted in classica l electromagnetism and presented in almost all textbooks nowadays) do not yield any second order electric ﬁeld [11, p. 697] [15] [23, Section 6.6]. On the othe r hand, some theories like those of Gauss, Weber, Riemann and Ritz predict a force o f this order of magnitude by taking into account the force of the stationary lattice and mobile conduction electrons acting upon the external stationary t est charge [163] [3, Vol. 2, pp. 588-590] [63, pp. 205-206 and 234-236] [162] [15] [23, Section 6.6]. For typical laboratory experiments, as we will show later, t his second order force or electric ﬁeld is much smaller than the force or elect ric ﬁeld proportional to the voltage of the battery, which in turn is much smaller th an the zeroth order force or electric ﬁeld. That is, usually we have |/vectorF0| ≫ |/vectorF1| ≫ |/vectorF2|or |/vectorE0| ≫ |/vectorE1| ≫ |/vectorE2|. In this book we will be concerned essentially with /vectorE0and /vectorE1, discussing only brieﬂy /vectorE2, due to its extremely small order of magnitude. 22 Chapter 2 Reasons for the Existence of the External Electric Field In this Chapter we discuss essentially the electric ﬁeld pro portional to the volt- age or to the electromotive force of the battery connected to the circuit. That is, the electric ﬁeld proportional to the potential diﬀeren ce which is acting along the resistive wire carrying a steady current. 2.1 Bending a Wire Consider a resistive wire of ﬁnite conductivity gconnected to a battery and carrying a steady current I, as in the left side of Figure 2.1. The ideal bat- tery generates a constant voltage or electromotive force (e mf)Vbetween its terminals. Figure 2.1: The electric ﬁeld at Cpoints along the xdirection in the Figure at left. When the wire is bent as in the right side, the electri c ﬁeld atCnow points along the ydirection. However, the directions and intensities of the electric ﬁelds at A,B,DandEhave not changed. 23 The current density /vectorJis given by /vectorJ= (I/A)ˆu, whereAis the area of the cross section of the wire and the unit vector ˆ upoints along the direction of the current at every point in the interior of the wire. Ohm’s law i n diﬀerential form states that /vectorJ=g/vectorE, where/vectorEis the electric ﬁeld driving the current. By the previous relation we see that /vectorEwill also point along the direction of the wire at each point. Where are the charges which generate the electric ﬁeld at eac h point along the wire located? It might be thought that this electric ﬁeld is due to the battery (or to the charges located on the battery), but this is not the complete answer. To see that the battery does not generate the electric ﬁeld at all points along the wire, we can consider Figure 2.1. We know that the electri c ﬁeld driving the constant current will in general follow the shape of the w ire. At a speciﬁc pointCinside the wire the electric ﬁeld in Figure 2.1 (left circuit ) points along the positive xdirection. When we bend a portion of the wire, the electric ﬁe ld will follow this bending. In the circuit at the right side of F igure 2.1 it can be seen that at the same point Cthe electric ﬁeld now points toward the positive ydirection. If something changes inside the battery when we bend the wire , the electric ﬁeld at points closer to the battery would also change. Howev er, the electric ﬁeld changes its path or direction only in the portion which w as bent and in the regions close to it, maintaining the previous values and directions in the other points (like the points A,B,DorEin Figure 2.1). As the electric ﬁeld inside the wire has changed only in the bent portion, somethi ng local must have created this change in the direction of the electric ﬁel d. The shape of the wire has obviously changed. But as the shape or spatial co nﬁguration does not create an electric ﬁeld, the reason must be sought elsewh ere. What creates electric ﬁelds or the electric forces exerted upon the condu ction electrons must be other charges, called here source charges. So there shoul d exist a change in the location of the source charges when we compare the conﬁgu rations of the left and right sides of Figure 2.1. And this change in the loca tion of the source charges should happen mainly around the bent portion of the w ire, but not at the battery. We then arrive at Weber’s and Kirchhoﬀ’s idea th at the electric ﬁeld inside a wire carrying a constant current is due to free c harges spread along the surface of the wire [32, 24, 25, 26]. Kirchhoﬀ’s three pap ers have English translations [27, 28, 29]. In the Appendices we discuss thes e works in more detail. The role of the battery is to maintain this distribut ion of free charges along the surface of the wire (constant in time for steady cur rents, but variable along the length of the wire). There will be a continuous grad ient of density of surface charges along the length of the wire, more positive t oward the positive terminal of the battery, decreasing in magnitude until it re aches a zero value at an intermediary point, and increasingly negative toward th e negative terminal, Figure 2.2. If there were no battery, the charge density would be zero at a ll points along the surface of the wire. It is the distribution of these surfa ce charges in space which creates the electric ﬁeld inside the wire driving the c urrent. When we bend a portion of the wire, the free charges redistribute the mselves in space 24 Figure 2.2: Qualitative distribution of charges along the s urface of a resistive wire carrying a steady current in two diﬀerent conﬁguration s. along the surface of the wire, creating the electric ﬁeld whi ch will follow the new trajectory of the wire. This can be seen qualitatively in the right circuit of Figure 2.2. Supposing the wire to be globally neutral, the integration of the surface charges σalong the whole surface of the wire must always go to zero, althoughσis not zero at all points along the surface. A qualitative representation of the surface charges in a res istive ring carrying a steady current Iwhen connected to a battery generating a voltage Vbetween its terminals is shown on the left side of Figure 2.3. A qualit ative representation of the internal and external electric ﬁelds due to these surf ace charges is shown on the right side of Figure 2.3. Figure 2.3: Qualitative representation of the surface char ges (left) and of the internal and external electric ﬁeld (right) of a resistive r ing carrying a steady current. A qualitative discussion of this redistribution of surface charges when a wire is bent was given by Parker [164], and by Chabay and Sherwood [ 165, 166]. An order of magnitude calculation of the charges necessary t o bend the electric 25 currentIaround a corner has been given by Rosser [167]. However, most authors are not aware of these surface charges and the related electric ﬁeld outside the wire, as we can see from the quotati ons presented above. Fortunately this subject has been revisited by other author s in some important works discussed in this book. 2.2 Continuity of the Tangential Component of the Electric Field A second reason for the existence of an electric ﬁeld outside resistive conductors carrying steady currents is related to the boundary conditi ons for the electric ﬁeld/vectorE. As is well known, at an interface between two media 1 and 2 (wi th ˆnbeing the unit vector normal to the interface at every point) we have that the tangential component of /vectorEis continuous, Et1=Et2or ˆn×(/vectorE2−/vectorE1) = 0. On the other hand, the normal component may be discontinuous according to ˆn·(ε2/vectorE2−ε1/vectorE1) =σ, whereεjis the dielectric constant of the medium jandσ is the density of surface charges at the interface. By Ohm’s l aw there must be a longitudinal component of /vectorEinside a resistive wire, even at its surface. As this component is continuous at an interface, it must also exist i n vacuum or in the air outside the conductor, not only close by, but also at meas urable distances from the wire. Many textbooks only consider an electric ﬁeld outside the cu rrent-carrying wire when discussing these boundary conditions. The ﬂux of e nergy in the electromagnetic ﬁeld is represented in classical electrom agnetism by Poynting’s vector/vectorS=/vectorE×/vectorB/µ0, where/vectorBis the magnetic ﬁeld and µ0= 4π×10−7 H/m is the magnetic permeability of the vacuum. The authors w ho deal with the electric ﬁeld outside wires by considering the boundary conditions normally present Poynting’s vector pointing radially inwards towar d the wire [168, pp. 180-181] [33, p. 27-8]. This goes back to Poynting himself in 1885 [169, 170]. Here is what Poynting wrote in this paper: “In the particular case of a steady current in a wire where the electrical level surfaces cut the wire perpendicularly to the axis, it appears that the energy dissipated in the wire as heat comes in from the surrounding medium, entering perpendicularly to t he surface.” (...) “In the neighbourhood of a wire containing a current, the ele ctric tubes may in general be taken as parallel to the wire while the magnetic tu bes encircle it.” In the ﬁrst paragraph of the section A straight wire carrying a steady current he wrote: “Let AB represent a wire in which is a steady current from A to B. The direction of the electric induction in the surrounding ﬁ eld near the wire, if the ﬁeld be homogeneous, is parallel to AB.” A typical representation found in the textbooks of the ﬁelds /vectorE,/vectorBand/vectorSin the vicinity of a current carrying cylindrical wire is that o f Figure 2.4. There are two points to make here. In the ﬁrst place, these dra wings and statements suggest that this electric ﬁeld should exist onl y close to the wire, while as a matter of fact it exists at all points in space. In th e second place, 26 Figure 2.4: Typical representation of the Poynting vector /vectorS=/vectorE×/vectorB/µ0outside a resistive wire carrying a steady current. they indicate that these authors are not concerned about the surface charges generating the ﬁeld. It is only at a few locations that /vectorSwill be orthogonal to the wire just outside it, namely, the locations where the sur face charge density goes to zero. These locations are an exception and not the rul e. In most other locations the density of surface charges will be either posi tive (closer to the positive terminal of the battery) or negative (closer to the negative terminal of the battery). The rule is that there will be a radial componen t which may be larger than the longitudinal one, pointing toward the wire o r away from it. One of the eﬀects of this radial component is that /vectorEand/vectorSwill usually be inclined just outside the wire and not orthogonal to it. These two misleading viewpoints are clearly represented by Feynman, Leighton and Sands’s statement in Section 27-5 (Examples of energy ﬂo w) of their book [33, p. 27-8], our emphasis in boldface: “As another example , we ask what hap- pens in a piece of resistance wire when it is carrying a curren t. Since the wire has resistance, there is an electric ﬁeld along it, driving t he current. Because there is a potential drop along the wire, there is also an electric ﬁeld just outside the wire, parallel to the surface (see Fig. 27-5 [our Figure 2.4]). There is, in addition, a magnetic ﬁeld which goes around the w ire because of the current. The EandBare at right angles; therefore there is a Poynting vector directed radially inward, as shown in the ﬁgure. There is a ﬂo w of energy into the wire all around. It is, of course, equal to the energy bein g lost in the wire in the form of heat. So our “crazy” theory says that the electr ons are getting their energy to generate heat because of the energy ﬂowing in to the wire from the ﬁeld outside. Intuition would seem to tell us that the ele ctrons get their energy from being pushed along the wire, so the energy should be ﬂowing down (or up) along the wire. But the theory says that the electrons are really being pushed by an electric ﬁeld, which has come from some charges v ery far away, and that the electrons get their energy for generating heat f rom these ﬁelds. The energy somehow ﬂows from the distant charges into a wide a rea of space and then inward to the wire.” As we have seen, the electric ﬁeld just outside the resistive wire is normally 27 not parallel to the wire. Moreover, the main contribution fo r the local electric ﬁeld at a speciﬁc point inside a wire is due to the charges alon g the surface of the wire around this point, contrary to their statement (who believed that it has come “from some charges very far away”). Probably they were t hinking here of the charges inside the battery. 28 Chapter 3 Experiments In this Chapter we present experiments that prove the existe nce of the electric ﬁeld outside resistive wires carrying steady currents. Man y experiments along these lines were probably performed in the second half of the XIXth century and in the early part of the XXth century, but they have been fo rgotten and are not quoted nowadays. Here we present only those which hav e come to our attention. We separate these experiments into three classes. The ﬁrst o ne is related to the zeroth order electric ﬁeld (due to electrostatic induct ion), which exists even when there is no current along the wire. The second class is re lated directly to the battery and to the current along the wire, being proporti onal to the emf of the battery or to the potential diﬀerence acting along the co nductor. The third class is related to the second order electric ﬁeld, proporti onal tov2 d/c2, where vdis the drifting velocity of the conduction electrons and cis light velocity in vacuum. 3.1 Zeroth Order Electric Field We are not aware of any speciﬁc experiments designed to measu re the force between a point charge and a nearby conductor. We are here con sidering a conductor which is initially neutral and has no currents ﬂow ing through it, until we bring a charge close to it and let both of them at rest relati ve to one another. After the electrostatic equilibrium is reached, the electr ical polarization of the conductor will cause a net force between the conductor and th e external charge. This zeroth order force will depend upon the shape of the cond uctor, upon its distance to the external charge, and upon the value of this ch arge. We can also express this by saying that a zeroth order electric ﬁeld will be created depending upon the external test charge, upon its distance t o the conductor and upon the shape of the conductor. Many quantitative exper iments along these lines were probably performed in the XIXth century. As we are not aware of them, we will not quote any speciﬁc experiment here. But we believe these 29 experiments, which were probably made with conductors havi ng many diﬀerent shapes, would have agreed with the predictions based upon Co ulomb’s force and upon the properties of conductors, otherwise this would have come to the attention of most scientists long ago. The basic properties of conductors in electrostatic equilibrium which we utilize in this book are : no electric ﬁeld on the interior, no net density of charges on the interior, any n et charge resides on the surface, the potential is constant throughout the inter ior and the surface of a conductor, and the electric ﬁeld is perpendicular to the su rface immediately outside it. Therefore, we will presume the calculations on t his topic to have been conﬁrmed by past observations. In the next section we di scuss Sansbury’s experiment, which has some qualitative aspects that touch u pon this subject. 3.2 Electric Field Proportional to the Voltage of the Battery We consider here the force between an external test charge an d a resistive wire carrying a steady current. We will consider the component of this force which is proportional to the voltage or to the emf of the battery con nected to the wire. That is, the component of the electric ﬁeld proportion al to the potential diﬀerence acting along the wire. The majority of the experim ents deal with voltages of the order of 104V, when the macroscopic eﬀects are more easily seen [171] [166, p. 653]. We present several kinds of experiments. Some map the lines o f electric ﬁeld outside resistive wires carrying steady currents. Others m ap the equipotential lines outside these conductors. Other experiments directl y measure the force between a charge test body and a wire carrying a steady curren t, when there is no motion between the wire and the test body. Anther experime nt measures the charging of an electroscope connected to diﬀerent points of a circuit carrying a steady current. Yet another experiment describes how to obt ain a part of the surface charge in diﬀerent points of the circuit, showing al so how to verify if it is positive or negative and also its magnitude. Bergmann and Schaefer present some experiments in which the y mapped the electric ﬁeld lines [172, pp. 164-167] [173, pp. 197-199 ]. They comment that due to the great conductivity of metals it is diﬃcult to u tilize metals as conductors in these experiments. Metals cannot sustain a great potential diﬀerence between their extremities, so that they produce o nly a very small external electric ﬁeld. For this reason they utilize graphi te paper strips of high resistivity and apply 20 000 to 40 000 volts between their ext remities in order to produce a steady current along the strip. They ground the cen ter of the strip to put it at zero potential, so that the lines of the electric ﬁel d are symmetrically distributed around it. They then spread semolina in castor o il around the strip, and obtained the result shown in Figure 3.1. The central stra ight dark line is the paper strip carrying a steady current. The particles of s emolina polarize due to the external electric ﬁeld and align themselves with i t, analogous to iron 30 ﬁlings mapping a magnetic ﬁeld. Figure 3.1: Experimental mapping of the external electric ﬁ eld of a straight conductor carrying a steady current. It should be observed that along the external surface of the c onductor there is a longitudinal component of the electric ﬁeld. This aspec t diﬀerentiates it from the electric ﬁeld outside conductors held at a constant potential (in which case the external electric ﬁeld in steady state is normal to t he conductor at every point of its surface), as has been pointed out by Bergma nn and Schaefer. By bending the conductor in a U-form they were able to show the lines of electric ﬁeld outside a transmission line or twin-lead, Fig ure 3.2. On the left side the electric ﬁeld lines are built into the plane of the co nductors, while on the right hand side they are built into a plane orthogonal to t he conductors. Figure 3.2: Experimental mapping of the external electric ﬁ eld of a transmission line. Bergmann and Schaefer also discuss the redistribution of ch arges along the surface of an open circuit connected to a battery when the cir cuit is closed. Another clear discussion of this situation can be found in th e recent book by Chabay and Sherwood [165, Chapter 6]. Experiments similar to those of Bergmann and Schaefer have b een performed by Jeﬁmenko [174] [175, pp. 295-312 and 508-511] [176, plate s 6 to 9 and pp. 299-319 and 508-511]. He utilized a transparent conducting ink to make a two- 31 dimensional printed circuit on glass plates of 10 inches ×12 inches. In the Figures, 3.3 to 3.7 the gray sections represent the current- carrying conducting strips. The power supply was capable of producing about 104V. He utilized a Du Mont high-voltage power supply type 263-A but mentioned that a small van de Graaﬀ generator might also be employed. After the powe r supply was turned on, he spread some ﬁne grass seeds (Redtop type) over t he plate and conducting system. The seeds lined up in the direction of the electric ﬁeld over and outside the conductors. Figure 3.3 depicts Jeﬁmenko’s experiment for a straight cur rent-carrying conductor. Figure 3.3: Straight current-carrying conductor. Figure 3.4 depicts his experiment for square-shaped (left) and circular (right) conducting rings. Figure 3.4: Square-shaped (left) and circular (right) cond ucting rings. Figure 3.5 depicts his experiment for shorted symmetric (le ft) and asymmet- ric (right) transmission lines. Figure 3.6 depicts his experiment for current-carrying wed ges with the two halves connected in parallel (left) and in series (right). And Figure 3.7 depicts his experiments involving current-c arrying rings on the left with two-pole (top) and four-pole (bottom) connect ions. On the right we have current-carrying discs with two-pole (top) and four -pole (bottom) con- nections. 32 Figure 3.5: Shorted symmetric (left) and asymmetric (right ) transmission lines. Figure 3.6: Current-carrying wedges with the two halves con nected in parallel (left) and in series (right). Figure 3.7: On the left are current-carrying rings with two- pole (top) and four- pole (bottom) connections. On the right are current-carryi ng discs with two-pole (top) and four-pole (bottom) connections. 33 In a private communication to one of the authors (AKTA), Jeﬁm enko men- tioned that he never measured the current in the grass seed ex periments. How- ever, he believed that they were of the order of a few microamp eres. He men- tioned that the patterns of the current-carrying conductor s were about 16 or 20 centimeters long. The experiments of Bergmann, Schaefer and Jeﬁmenko complem ent one an- other. After obtaining theoretical formulas for the equipo tentials and for the electric ﬁeld lines, we will compare them with some of these e xperimental results. In another type of experiment, Jeﬁmenko, Barnett and Kelly o btained the equipotential lines directly inside and outside conductor s with steady currents utilizing an electronic electrometer [177] [176, p. 301]. A radioactive alpha- source was utilized to ionize the air, in order to make it a con ductor of electricity, at the point where the ﬁeld was to be measured. The alpha-sour ce acquired the same potential as the ﬁeld at that point. The potential wa s measured (in relation to a reference point chosen at zero potential) w ith an electronic electrometer connected to the alpha-source. They utilized a hollow rectangular chamber with electrodes for end walls and semi-conducting s ide walls carrying uniform current. Graphite paper strips were used for the sid e walls, as in the experiments by Bergmann and Schaefer, and aluminum foil ser ved as electrodes with 80 V applied. The equipotentials were mapped experimen tally. Figure 3.8 depicts the conﬁguration of the problem. Figure 3.8: Conﬁguration of the system. Figure 3.9 depicts the equipotential lines measured in one o f the experiments. Figure 3.9: Measured equipotentials. In another experiment, undeveloped photographic ﬁlm was us ed in place of 34 the graphite paper, experimentally yielding the equipoten tial lines inside and outside the conductor carrying steady currents represente d in Figures 3.10 and 3.11. Figure 3.10: Measured equipotentials. Figure 3.11: Measured equipotentials with 80 Volt disc outs ide chamber. In this last experiment, Figure 3.11, they showed that an ext ernal charged body has no eﬀect on the ﬁeld inside the current carrying cond uctor. The current in the graphite paper was measured to be 5 ×10−2A, while in the photographic ﬁlm the current was measured to be only 4 ×10−6A [177]. A variety of qualitative experiments demonstrating the exi stence of an ex- ternal electric ﬁeld have been performed by Parker [164]. He utilized 5 to 10 kV power supply connected to a high resistance, low-current circuit, which was drawn on a ground-glass or Mylar surface with an IBM scoring p encil. When there was a steady current in the circuit he detected a force o n a charged pith ball located nearby, which varied from one end of the circuit to the other. This is a very interesting result, as it shows directly the force b etween a resistive cur- rent carrying circuit and a stationary charge located nearb y, the main question we asked in the beginning of this book. Unfortunately Parker did not men- tion the values of the current, the charge in the pith ball, th e distance between the pith ball and circuit, nor the detected force. He also uti lized a gold-leaf electroscope with a wire lead to probe quantitatively regio ns around the current carrying circuit. He could also map the lines of electric ﬁel d by dusting the glass 35 with plastic or felt ﬁbers while current was ﬂowing. In parti cular he utilized plastic ﬁbers of approximately 1 mm in length. This mapping i s similar to what Bergmann, Schaefer and Jeﬁmenko had done. There is also an interesting experiment by Sansbury in which he detected a force between a charged metal foil and a current-carrying c onductor directly by means of a torsion balance [178]. He placed a neutral 2 cm ×2 cm silver foil which was at the extremity of a torsion balance close to a U-shaped neutral conductor (length 50 cm, separation between the wires 10 cm) without current, Fig. 3.12. When he charged the foil with a charge which he esti mated to be approximately 0 .5×10−9C (by connecting it to a 3 kV voltage supply), he observed an attraction between the vane and the wire (the cha rged metal foil moved from atobin Fig. 3.13). This was probably due to the zeroth order force of electrostatic induction F0discussed above, i.e., a force due to image charges induced in the wire by the charged foil nearby. He then passed a steady current of 900 A through the wire by connecting it to a ±1000 A, 8 V, adjustable, regulated dc current supply. In this case he observed an extr a attraction or repulsion between the charged foil and the wire, depending o n the sign of the charge in the foil, Fig. 3.13. This extra force was greater th an 1.7×10−7N, although he was not able to make precise measurements. This e xtra force was probably due to the external electric ﬁeld being discussed h ere,i.e., to the elec- tric ﬁeld proportional to the voltage or emf of the battery. L ater on we analyze this experiment in more detail in connection with theoretic al calculations. Figure 3.12: Conﬁguration of Sansbury’s experiment. The force between Sansbury’s charged metal foil and current -carrying wire seems to be similar to the force between Parker’s charged pit h ball and current carrying circuit. Bartlett and Maglic considered the force detected by Sansbury an “anomalous electromagnetic eﬀect,” as suggested by the t itle of their paper [179]. They conducted a similar experiment. See Figs. 3.14 a nd 3.15. They utilized a rectangular 16-turn coil (4 wide ×4 high), with 30 cm width and 60 cm length. Each turn was made of a 1/8 in. copper tu bing (outer diameter of 0.3175 cm). In a private communication wi th one of the 36 Figure 3.13: In the beginning there is no current in the condu ctor and the uncharged silver foil remains at a. When the silver foil is charged, it moves fromatob. Then a steady current Iis passed through the conductor. In this case there appears an extra force of attraction or repulsion between the charged foil and the U-shaped conductor. authors (JAH) Bartlett reported that the conductor was wate r-cooled, with water ﬂowing through the hole in the center of the tubing. App roximately 50% of the cross-sectional area of the tubing was copper, and 50% was water. A current source connected to the coil maintained a steady cur rent of 50 A in each turn. They detected a force upon the charged metal foil with a n area of 2.54 cm×2.54 cm placed at a distance of 3.5 cm from the coil carrying a s teady current. This was similar to the eﬀect detected by Sansbury. Figure 3.14: Experiment performed by Bartlett and Maglic. But when the upper half of the current carrying circuit and th e test charge were shielded with an aluminum can, the eﬀect disappeared. T heir conclusion was that they could not ﬁnd this “anomalous” interaction, im plying that it did 37 Figure 3.15: Top view of Bartlett and Maglic’s experiment. not exist. However, they seem to have been unaware of one important poin t in connec- tion with Faraday cages. They are usually utilized to shield the system under consideration from external inﬂuences. But they aﬀect the n et force on each in- ternal test charge. For instance, if we have two charges q1andq2separated by a distanced, the coulombian force between them has a magnitude of q1q2/4πε0d2 and is directed along the line joining them, Figure 3.16. Figure 3.16: Electrostatic force between two charges far fr om other charges and conductors. When we surround both of them with a metallic shell, they will induce a distribution of charges along the surface of the shell. As a r esult of these induced charges, the shell will exert forces on q1and onq2, resulting in general in a net force on each one of them diﬀerent from the previous value of q1q2/4πε0d2. With a spherical shell the force upon the internal test charg es exerted by the induced charges along the wall can be easily calculated by th e method of images. Each charge qjlocated at a distance ajfrom the center of the shell of radius r0> aj, withj= 1 or 2, will induce charges equivalent to an image charge qij=−qjr0/ajat a distance aij=r2 0/aj> r0from the center of the shell, located along the straight line connecting the center of the shell andqj. The net force on q1, for instance, will be given by /vectorF21+/vectorFi1,1+/vectorFi2,1instead of simply /vectorF21. Here/vectorFm,nis the force exerted by the (image) charge mon the charge n. If 38 the straight line connecting the two charges q1andq2does not pass through the center of the shell, the net force on each one of them will chan ge its magnitude and also its direction in comparison with the previous value without the shell, Figure 3.17. Figure 3.17: Electrostatic forces on q1due toq2, to the image charge qi1and to the image charge qi2. If there are Ninternal charges, the net force on q1will be given by the sum of the forces due to the other N−1 charges on q1, plus theNforces of the image charges on q1. If the Faraday cage is not spherical, it will be very diﬃcult to calculate the new force on the test charges. In Bar tlett and Maglic’s case the Faraday cage was cylindrical with metallic lateral sides and dielectric extremities. This makes it very diﬃcult to estimate the eﬀec t of the shield upon the internal charged foil. Moreover, we have not just th e two charges q1andq2as discussed before, but the charged foil and a quantity of ch arges distributed along the surface of the resistive current-car rying wire. As the wire is made of a conducting material, the induced charges upon the F araday cage will change the distribution of charges spread along the surface of the wire (compared with the distribution of surface charges without the Farada y cage), such that even this new distribution is not yet known until they can be c alculated with Laplace’s equation and the appropriate boundary condition s. This enormously complicates the theoretical analysis of the expected net fo rce (exerted upon the test charge when the current-carrying wire and the test char ge are shielded). For this reason it seems preferable to perform this kind of ex periment without the metallic shield. The result obtained by Bartlett and Maglic is described as fo llows [179]: “Averaging the results of the runs with and without the shiel d we ﬁnd a signal of 0.3±0.3 mrad. (...) We multiply our measured rotation of 0.3 ±0.3 mrad by the sensitivity of the ﬁber (9 .1×10−5N/m) to obtain a force of (0 .27±0.27)10−7 N.” The impression we get from reading the paper is that the fo rce measured without the shield was 0 .27×10−7N, one order of magnitude smaller than that observed by Sansbury. On the other hand, we infer that wi th the shield Bartlett and Maglic measured no force (hence the ±sign given in the previous quotation). Our opinion is that the shield changes the distr ibution of charges 39 on the metallic shell. This changes the net force on each of th e internal charges, as we showed earlier. Therefore, it is very diﬃcult to compar e the two cases (with and without shielding). The ideal situation would be t o perform this kind of experiment without any shield. Unfortunately, this was n ot the procedure adopted by Bartlett and Maglic. Further discussions of Sansbury’s experiment with diﬀeren t approaches can be found in works by several authors [1, 4, 5, 82, 83, 180, 181, 182, 183]. Another kind of experiment was performed by Moreau, Ryan, Be uzenberg and Syme [184]. (See Figure 3.18.) Figure 3.18: Electroscope touching diﬀerent points of a hig h voltage resistive circuit carrying a steady current. They connected a 0- to 5-kV current-limited dc power supply t o a series circuit consisting of two resistances of 75 MΩ. The conductors were bare alu- minum bars and the resistances were strips of foam plastic im pregnated with Aqua Dac. The conductors and resistances had rectangular cr oss-sections of 12 mm×9 mm. Each one of the resistances was 50 cm in length. The condu cting bar between them was 30 cm in length. The main goal of the exper iment was to demonstrate directly that when a steady current ﬂows alon g a circuit, there is a gradient of surface charges along the conductors and alo ng the resistances, in such a way that this gradient of surface charge density pro duces an electric ﬁeld along the direction of the current. To indicate the char ge density at sev- eral points of the circuit, they connected these points to a h igh voltage probe. This probe was connected to a gold-leaf electroscope. Somet hing similar to this was done by Parker [164]. The electroscope was placed inside a nonconducting polystyrene case. When the power supply was turned up to 2 kV a microamme- ter in the circuit measured a current of about 13 µA. One side of the circuit was grounded. At this point the electroscope showed no deﬂectio n, indicating zero charge density. As the probe moved away from this point along the resistance strip, the electroscope deﬂected gradually with distance t raveled along the re- sistance, reaching a deﬂection of about 55oat its end. The deﬂection remained 40 constant as the probe was moved along the middle conductor, i ndicating that the charge density was essentially constant (or that it vari ed very little) along it. The deﬂection increased again along the second resistan ce, reaching a ﬁnal deﬂection of about 70oat the extremity. This indicated that the charge den- sity was relatively high on the upstream side of the resistor s, decreasing along them in the direction of the current. This produced an electr ic ﬁeld forcing the conduction charges through the resistances. Another very didactic experience was performed by Uri Ganie l and collabo- rators of the Science Education Group at the Weizmann Instit ute in Israel. This experiment has been cited and reproduced by Chabay and Sherw ood [171, 185] [166, Section 18.10, pp x and 652-654]. (See Figure 3.19.) Figure 3.19: Four identical high resistance resistors are c onnected in series in a closed circuit carrying a steady current. This Figure show s the qualitative distribution of surface charges. The thin metallized mylar strip is attracted by the bare (uninsulated) wires, touches them and is then repel led by them. By testing the charge gained by the metallized strip it is possi ble to determine the sign of the surface charges at each point along the circuit. A closed circuit is formed with four identical high resistan ce resistors con- nected in series, each with 80 MΩ. They are connected to two hi gh voltage power supplies, of 5 kV each. There is a grounded conductor be tween these two power supplies. One of the power supplies yields + 5 kV at o ne extremity of the ﬁrst resistor, while the other yields - 5 kV at the oppos ite extremity of the fourth resistor (relative to the ground). In other words , these two power supplies are connected in series. There are bare (uninsulat ed) wires between the resistors in order to allow the collection of their surfa ce charges by an ap- propriate probe. This is performed utilizing a ﬂexible, thi n metallized mylar strip. When the strip is brought close to the bare wire at the l eft side of the circuit, near location A, it is observed to be attracted to th e wire. It touches the wire and is then repelled by it. It is initially attracted due to polarization of the aluminized plastic strip created by the surface charg e on the wire. When it touches the wire, it jumps away, since it is charged by cont act with the bare wire and repelled by the surface charge at that location. Whe n the strip is tested it is found to be negatively charged. It is then discha rged, and the same 41 procedure is repeated, although this time it is brought clos e to the bare wire between the ﬁrst and second resistors, at B. Once more it is fo und to become negatively charged, but now with a smaller magnitude to the p revious case. No eﬀect is observed at C when the strip is ﬁrst discharged and th en brought close to this point. When the procedure is repeated at D, it is found to be posi- tively charged. The same happens at E, but now with a larger ma gnitude than at D. This experiment shows directly that diﬀerent points of the wire become charged when there is a constant current ﬂowing through it. T his surface charge density changes along the circuit. The largest gradient (va riation of the mag- nitude per unit length) occurs along the resistors. Along th e conductors there is a very small variation of the magnitude of the charge densi ty. Chabay and Sherwood point out that only at very high voltages is there en ough charge to observe electrostatic repulsion in a mechanical system [16 6, p. 654]. With a low voltage circuit (like in a ﬂashlight connected to ordinary 1 .5 V batteries), any charged body brought near the current-carrying wire will be initially attracted to it, regardless of the sign of the charged body. This will ha ppen both close to the positive and close to the negative terminals of the bat tery. The reason for this fact is that the zeroth order force will be much great er than the force proportional to the voltage of the battery. In conclusion we can infer that the experiments of Bergmann, Schaefer, Jeﬁmenko, Barnett, Kelly, Sansbury, Parker, Moreau, Ryan, Beuzenberg, Syme, Ganiel, Chabay and Sherwood prove the existence of the exter nal electric ﬁeld due to resistive, stationary wires carrying steady current s. They have also shown the existence of surface charges along the conductors and re sistors. These are the charges which produce the internal and external electri c ﬁelds. In order to have a complete proof, it would be necessary to rep eat each of these experiments with diﬀerent electromotive forces of the batteries and verify if the external electric ﬁeld is proportional to this voltage. To the best of our knowledge none of these experiments varied the emf in o rder to show or demonstrate unambiguously linearity of the force intensit y (or of the electric ﬁeld intensity) with applied voltage, unfortunately. In the following Chapters we compare these experiments with analytical so- lutions for the external potential and electric ﬁeld due to r esistive wires carrying steady currents. This will give further support to the infer ence that there is a force proportional to the voltage of the battery between an e xternal stationary point charge and a resistive wire carrying a steady current. 3.3 Second Order Electric Field All the previous experiments involved the electromotive fo rce of the battery. A completely diﬀerent set of experiments investigates a seco nd order electric ﬁeld, i.e., an electric ﬁeld proportional to v2 d/c2. Checking whether or not the second order electric ﬁeld exist s is much more diﬃcult than studying the electric ﬁeld discussed previous ly. The reason for this diﬃculty is that the order of magnitude of the second ord er electric ﬁeld, 42 E2, is normally much smaller than the electric ﬁeld associated with the electro- motive force of the battery, E1, and also much smaller than the zeroth order electric ﬁeld, E0. However, if the wire is a superconductor and a steady curren t is ﬂowing through (with no battery connected to the wire), th e external electric ﬁeldE1should go to zero. If we take into account (or neglect) the for ce due to electrostatic induction (related with image charges) as well, there remains in this case only the second order electric ﬁeld. This was the ap proach utilized by Edwards, Kenyon and Lemon in their experiments [14, 15], whi ch are the best known to us to analyze this eﬀect. They utilized type II super conductor (48% niobium and 52% titanium) cores of 2.5 mil radius with curren ts of the order of 16 A. They found an electric ﬁeld proportional to I2, independent of the direc- tion of the current, pointing toward the wire and with an orde r of magnitude compatible with that predicted by Weber’s law. What they act ually measured utilizing an electrometer was a potential diﬀerence betwee n the circuit and an electrostatic shield around the circuit. They measured pot ential diﬀerences with an order of magnitude of 10 mV. Bartlett and Ward made a number of diﬀerent experiments to de tect this second order electric ﬁeld (which they interpreted as due to a possible variation of the electron’s charge with its velocity), but failed to ﬁn d it [186]. They utilized normal resistive conductors, but in their analysi s they did not mention the electric ﬁeld proportional to the voltage of the battery discussed above. In another experiment, Kenyon and Edwards placed a beam-pow er radio tube within a Faraday cage [187]. They tried to measure the po tential diﬀerence between the system and the Faraday cage. They could not ﬁnd an y eﬀect with the order of magnitude of the earlier experiment of Edwa rds, Kenyon and Lemon. In any event attention must be called here to the Faraday cage around the system in these experiments, and also in most of those quo ted by Bartlett and Ward. As we mentioned previously, the charges induced in the walls of a Faraday cage due to internal charges will exert a net force on any internal test charge. If there are two or more internal charges, the net for ce in each will be diﬀerent in two cases: (A) without the Faraday cage (force du e only to the other internal charges), and (B) with Faraday cage (force due to th e other internal charges and to all induced charges in the walls of the cage). T his enormously complicates the analysis of all these experiments, and it is diﬃcult to reach a simple result. Beyond the complication of the Faraday cage, there is also the electric ﬁeld proportional to the voltage of the battery whi ch must be taken into account before discussing the second order electric ﬁeld. A nd this was not done by any of these authors when dealing with resistive conducto rs. The second order electric ﬁeld is usually much smaller than the electri c ﬁeld proportional to the voltage of the battery, as we will show later on. For this r eason the electric ﬁeld proportional to the voltage of the battery must be inclu ded in the analysis because it can mask the eﬀect which is being sought. More research is necessary before a ﬁnal conclusion can be dr awn on this second order electric ﬁeld. A great number of experimental a nd theoretical works in connection with this subject have been published in the last 25 years 43 [188, 189, 190, 191, 82, 192, 181, 193, 83, 4, 5, 182, 194, 195, 196, 197, 23, 6, 7, 198]. 44 Chapter 4 Force Due to Electrostatic Induction 4.1 Introduction The main subject of this book is the force between a stationar y wire carrying a steady current and an external charge at rest relative to the wire. In particular, we are interested in the component of this force which is prop ortional to the voltage or emf of the battery, or to the potential diﬀerence a cting along the wire. Before analyzing these cases we consider the force between a point charge and a conductor which has no current ﬂowing through it. We sup pose air or vacuum outside the conductor (and also inside hollow ones). We also consider only the equilibrium situation when the point charge and the conductor are at rest relative to one another and also at rest relative to an inertial frame of reference. It is also assumed that there are no other charges or conductors in the vicinity of the system, beyond the ones being considered here. The main material of this Chapter was discussed in 2005 [199]. 4.1.1 Point Charge and Inﬁnite Plane The simplest conﬁguration is that of a point charge qat a distance dfrom an inﬁnite conducting plane with zero net charge. Let us supp ose that the conducting plane is along the plane z= 0, while the charge qis located at (x,y,z ) = (0,0,z). The method of images yields in this case an attractive forc e acting upon qgiven by /vectorF0=∓q2 16πε0ˆz z2. (4.1) Here the top sign is valid for z>0, while the bottom sign is valid for z<0. 45 Expressing this force as /vectorF0=q/vectorE0yields a zeroth order electric ﬁeld given by /vectorE0=∓q 16πε0ˆz z2. (4.2) Even adding a ﬁnite charge Quniformly spread along the inﬁnite plane does not change these two results. That is, /vectorF0and/vectorE0are independent of Q. This force is always attractive and diverges to inﬁnity when d→0. 4.1.2 Point Charge and Spherical Shell Another simple case to consider is that of a point charge and a conducting spherical shell at rest relative to one another. We consider a spherical shell of radiusRcentered upon the origin 0 of a coordinate system. There is a n et chargeQon the conducting spherical shell, insulated from the earth . The test chargeqis located at /vector r=rˆrrelative to 0. The solution of this problem can also be obtained by the method of images and is found in most te xtbooks on electromagnetism. When r>R the force upon qis given by /vectorF0=q 4πε0/bracketleftbigg Q−qR3(2r2−R2) r(r2−R2)2/bracketrightbigg/vector r r3. (4.3) Whenr<R the force is independent of Qand is given by /vectorF0=q2 4πε0R/vector r (R2−r2)2. (4.4) These forces diverge to inﬁnity when r→R. Whenqis inside the shell, it always suﬀers an electrostatic force toward the closest wal l. Whenqis outside the shell, the force will be attractive not only when qQ < 0, but also when qQ> 0, provided qis at a very close distance to the shell. A detailed discussio n of this fact can be found, for instance, in Maxwell’s work [20 0, Chapter VII: Theory of electrical images, pp. 80-88], in a paper by Melehy [201] and in Jackson’s book [11, Section 2.3]. The zeroth order electric ﬁeld in these cases is given by (wit h/vectorF0=q/vectorE0): /vectorE0=1 4πε0/bracketleftbigg Q−qR3(2r2−R2) r(r2−R2)2/bracketrightbigg/vector r r3,ifr>R . (4.5) /vectorE0=q 4πε0R/vector r (R2−r2)2,ifr<R . (4.6) 4.2 Point Charge and Cylindrical Shell After considering these two simple cases we analyze the main subject of this chapter. The goal is to calculate the electrostatic force be tween an inﬁnite conducting cylinder of radius aheld at zero potential and an external point 46 chargeq. To the best of our knowledge this has never been done before. To this end we consider the Green function method [13, Chaps. 1 t o 3]. We begin reviewing a known solution of the potential inside a grounde d, closed, hollow and ﬁnite cylindrical shell with an internal point charge [1 3, p. 143]. We analyze the limit of an inﬁnite cylinder and explore the force exerte d on the point charge. We then perform a similar analysis for the case of an external point charge. We consider in detail the particular situation of a thin wire, i.e., with the point charge many radii away from the axis of the cylinder. These ca lculations were published in 2005 [199]. 4.3 Finite Conducting Cylindrical Shell with In- ternal Point Charge: Solution of Poisson’s Equation Consider a ﬁnite conducting cylindrical shell of radius aand length ℓ≫a, withzbeing its axis of symmetry. (See Fig. 4.1.) With cylindrical coordinates (ρ,ϕ,z ) the center of the shell is supposed to be at ( ρ,z) = (0,ℓ/2). We also consider a point charge qlocated at/vector r′= (ρ′<a,ϕ′,z′) inside the shell. We wish to calculate the electric potential of the system, the elect ric ﬁeld, the surface charge distribution induced by qand the net force between the cylinder and q. Figure 4.1: Finite conducting cylinder of length ℓand radius acentered at (ρ,z) = (0,ℓ/2), withzbeing its axis of symmetry. There are conducting covers atz= 0 and at z=ℓ. There is a point charge qlocated at ( ρ,ϕ,z ) = (ρ′< a,ϕ′,0<z′<ℓ). The electrostatic potential φobeys Poisson’s equation: ∇2 rφ=−ρ ε0. (4.7) By the standard Green function method, the solution of Poiss on’s equation for this case with Dirichlet boundary conditions (potentia l speciﬁed on a closed surface) is given by: 47 φ(/vector r) =1 4πε0/integraldisplay /integraldisplay /integraldisplay Vρ(/vector r′′)G(/vector r,/vector r′′)dV′′−1 4π/ci∇cleco√y∇t/integraldisplay/integraldisplay Sφ(/vector r′′)∂G ∂n′′da′′,(4.8) whereVis the volume of the cylindrical shell, Sits closed surface and ∂/∂n′′ is the normal derivative at the surface Sof the shell directed outwards. Here G(/vector r,/vector r′′) is a Green function satisfying the equation: ∇2 r′′G(/vector r,/vector r′′) =−4πδ(/vector r−/vector r′′). (4.9) As the surface of the cylinder in electrostatic equilibrium is at a constant po- tentialφ0, we stipulate that G(/vector r,/vector r′′) = 0 at this surface. We can expand the Dirac delta function in cylindrical coordi nates as given by: δ(/vector r−/vector r′′) =δ(ρ−ρ′′)δ(ϕ−ϕ′′) ρδ(z−z′′). (4.10) The delta functions for ϕandzcan be written in terms of orthonormal functions: δ(z−z′′) =2 ℓ/bracketleftBigg∞/summationdisplay n=1sinnπz ℓsinnπz′′ ℓ/bracketrightBigg , (4.11) δ(ϕ−ϕ′′) =1 2π/bracketleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′′)/bracketrightBigg . (4.12) Notice our particular choice of expansion for z, Eq. (4.11). This choice satisﬁes the condition G(/vector r,/vector r′′) = 0 in the covers of the cylindrical shell located atz= 0 and atz=ℓ. The Green function can be expanded in a similar fashion: G(/vector r,/vector r′′) =1 πℓ/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′′)/bracketleftBigg∞/summationdisplay n=1sinnπz ℓsinnπz′′ ℓgm(k,ρ,ρ′′)/bracketrightBigg/bracerightBigg , (4.13) wherek=nπ/ℓandgm(k,ρ,ρ′′) is the radial Green function to be determined. Substituting this expression into Eq. (4.9) and using (4.10 ) to (4.12) we obtain: 1 ρd dρ/parenleftbigg ρdgm dρ/parenrightbigg −/parenleftbigg k2+m2 ρ2/parenrightbigg gm=−4π ρδ(ρ−ρ′′). (4.14) Forρ/negationslash=ρ′′the right hand side of Eq. (4.14) is equal to zero. This means t hat gmis a linear combination of modiﬁed Bessel functions, Im(kρ) andKm(kρ). Suppose that ψ1(kρ) satisﬁes the boundary conditions for ρ < ρ′′and that ψ2(kρ) satisﬁes the boundary conditions for ρ>ρ′′: 48 ψ1(kρ<) =AIm(kρ<) +BKm(kρ<), (4.15) ψ2(kρ>) =CIm(kρ>) +DKm(kρ>). (4.16) HereA,B,CandDare coeﬃcients to be determined. The symmetry of the Green function in ρandρ′′requires that: gm(k,ρ,ρ′′) =ψ1(kρ<)ψ2(kρ>), (4.17) whereρ>andρ<are, respectively, the larger and the smaller of ρandρ′′. The potential must not diverge for ρ→0, so we must have B= 0. The Green function must vanish at ρ=a,i.e.,ψ2(ka) = 0. This yields C= −DKm(ka)/Im(ka). The function gmcan then be written as: gm(k,ρ,ρ′′) =HIm(kρ<)/bracketleftbigg Km(kρ>)−Im(kρ>)Km(ka) Im(ka)/bracketrightbigg . (4.18) The normalization coeﬃcient H=ACis determined by the discontinuity implied by the delta function in Eq. (4.14): dgm dρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle +−dgm dρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle −=−4π ρ′′=kW[ψ1,ψ2], (4.19) where the ±signs means evaluation at ρ=ρ′′±ǫand taking the limit ǫ→0. In the last equality, W[ψ1,ψ2] is the Wronskian of ψ1andψ2. Substituting gminto Eq. (4.19), and using W[Im(kρ′′),Km(kρ′′)] =−1/(kρ′′), we ﬁndH= 4π. The Green function for the problem of a ﬁnite conducting cylinde r with an internal charge can be ﬁnally written as: G(/vector r,/vector r′′) =4 ℓ/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′′)/braceleftBigg∞/summationdisplay n=1sin(kz)sin(kz′′)Im(kρ<)× ×/bracketleftbigg Km(kρ>)−Im(kρ>)Km(ka) Im(ka)/bracketrightbigg/bracerightbigg/bracerightbigg . (4.20) 4.3.1 Cylindrical Shell Held at Zero Potential Consider the cylinder to be held at zero potential, namely, φ(/vector r′′) = 0: φ(a,ϕ,0≤z≤ℓ) =φ(ρ≤a,ϕ,ℓ) =φ(ρ≤a,ϕ,0) = 0. (4.21) Substituting Eqs. (4.20) and (4.21) into Eq. (4.8) yields th e potential inside the cylinder as (with ρ(/vector r′′) =qδ(/vector r′−/vector r′′)): φ(/vector r,/vector r′) =q πε0ℓ/braceleftBigg∞/summationdisplay m=−∞/braceleftBigg∞/summationdisplay n=1eim(ϕ−ϕ′)sin/parenleftBignπz ℓ/parenrightBig sin/parenleftbiggnπz′ ℓ/parenrightbigg Im/parenleftBignπρ< ℓ/parenrightBig × 49 ×/bracketleftBigg Km/parenleftBignπρ> ℓ/parenrightBig −Im/parenleftBignπρ> ℓ/parenrightBigKm/parenleftbignπa ℓ/parenrightbig Im/parenleftbignπa ℓ/parenrightbig/bracketrightBigg/bracerightBigg/bracerightBigg . (4.22) Hereρ>(ρ<) is the larger (smaller) of ρandρ′. 4.4 Inﬁnite Conducting Cylindrical Shell with Internal Point Charge We now expand Jackson’s solution for the case of a cylindrica l shell of inﬁnite length. The solution for an inﬁnite cylinder diﬀers from the solutio n of the ﬁnite cylinder in that it essentially changes the expansion of the delta function in Eq. (4.11). In the inﬁnite cylinder there is no restriction o n the choice of n(or k): δ(z−z′′) =1 2π/integraldisplay∞ −∞eik(z−z′′)dk=1 π/integraldisplay∞ 0cos[k(z−z′′)]dk . (4.23) The Green function can be written as: G(/vector r,/vector r′′) =2 π/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′′)/integraldisplay∞ 0cos[k(z−z′′)]Im(kρ<)× ×/bracketleftbigg Km(kρ>)−Im(kρ>)Km(ka) Im(ka)/bracketrightbigg dk/bracerightbigg . (4.24) Note that we can pass from Eq. (4.11) to Eq. (4.23) by transfor ming the Fourier series into the Fourier transform, that is, by letti ngℓ→ ∞, setting nπ/ℓ=k,dk=π/ℓ,z→z+ℓ/2,z′′→z′′+ℓ/2 and by replacing the inﬁnite sum by the integral over k. 4.4.1 Cylindrical Shell Held at Zero Potential Consider the cylinder to be held at zero potential, namely, φ(a,ϕ,z ) = 0. Substi- tuting Eq. (4.24) into Eq. (4.8), the potential inside the cy linder can be written as (withρ(/vector r′′) =qδ(/vector r′−/vector r′′)): φ(/vector r,/vector r′) =q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0cos[k(z−z′)]Im(kρ<)× ×/bracketleftbigg Km(kρ>)−Im(kρ>)Km(ka) Im(ka)/bracketrightbigg dk/bracerightbigg . (4.25) Once more ρ>(ρ<) is the larger (smaller) of ρandρ′. 50 The zeroth order electric ﬁeld is given by /vectorE0=−∇φ, with components: Eρ(ρ<ρ′) =−∂φ ∂ρ=−q 2π2εo/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0kcos[k(z−z′)]Im′(kρ)× ×/bracketleftbigg Km(kρ′)−Im(kρ′)Km(ka) Im(ka)/bracketrightbigg dk/bracerightbigg , (4.26) Eρ(ρ>ρ′) =−∂φ ∂ρ=−q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0kcos[k(z−z′)]Im(kρ′)× ×/bracketleftbigg Km′(kρ)−Im′(kρ)Km(ka) Im(ka)/bracketrightbigg dk/bracerightbigg , (4.27) Eϕ=−1 ρ∂φ ∂ϕ=q π2ε0ρ/braceleftBigg∞/summationdisplay m=1msin[m(ϕ−ϕ′)]/integraldisplay∞ 0cos[k(z−z′)]Im(kρ<)× ×/bracketleftbigg Km(kρ>)−Im(kρ>)Km(ka) Im(ka)/bracketrightbigg dk/bracerightbigg , (4.28) Ez=−∂φ ∂z=q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0ksin[k(z−z′)]Im(kρ<)× ×/bracketleftbigg Km(kρ>)−Im(kρ>)Km(ka) Im(ka)/bracketrightbigg dk/bracerightbigg . (4.29) The zeroth order force /vectorF0=q/vectorE0(/vector r′) acting upon the charge qis given by Eq. (4.26) at /vector r=/vector r′without the ﬁrst term between brackets (which is the ﬁeld generated by the charge qitself). There is only a radial component when the cylinder has an inﬁnite length: /vectorF0(/vector r′) =q2 2π2ε0/braceleftBigg∞/summationdisplay m=−∞/integraldisplay∞ 0kIm(kρ′)Im′(kρ′)Km(ka) Im(ka)dk/bracerightBigg ˆρ =−q2 4π2ε0ρ′2/braceleftBigg∞/summationdisplay m=−∞/integraldisplay∞ 0I2 m(x)d dx/bracketleftbigg xKm(xa/ρ′) Im(xa/ρ′)/bracketrightbigg dx/bracerightBigg ˆρ . (4.30) In the last equation we integrated by parts. We plotted in Fig . 4.2 the zeroth order force of Eq. (4.30), normalized by the constant Fq≡q2/4πε0a2, as a function of ρ′/a. This force goes to zero when ρ′/a= 0 and diverges when ρ′→a, as expected. The surface charges can be calculated using Gauss’s law, yie lding: σ(a,ϕ,z ) =ε0Eρ(a,ϕ,z ) = 51 Figure 4.2: Zeroth order force F0between an inﬁnite grounded conducting cylin- der of radius aand a point charge qat a distance ρ′< afrom thezaxis (which is also the axis of symmetry of the cylinder), normali zed by the constant Fq≡q2/4πε0a2. =−q 2π2a/bracketleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0cos[k(z−z′)]Im(kρ′) Im(ka)dk/bracketrightBigg . (4.31) The charge per unit length λ(z) is given by: λ(a,z) =/integraldisplay2π 0σ(a,ϕ,z )adϕ=−q π/integraldisplay∞ 0cos[k(z−z′)]I0(kρ′) I0(ka)dk . (4.32) The total charge induced in the cylinder, supposing z′= 0, can be obtained integrating Eq. (4.32) from z=−∞to∞. Utilizing: δ(k) =1 2π/integraldisplay∞ −∞cos(kz)dz=1 π/integraldisplay∞ 0cos(kz)dz , (4.33) this yields: Q=/integraldisplay∞ −∞λ(a,z)dz=−q . (4.34) 4.5 Inﬁnite Conducting Cylindrical Shell with External Point Charge We can now consider a new case, i.e., a conducting cylinder with an external point charge. Suppose that the point charge qis located at /vector r′= (ρ′,ϕ′,z′), withρ′>a. Green function can be written analogously in this case as: G(/vector r,/vector r′′) =1 2π2/bracketleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′′)/integraldisplay∞ 0cos[k(z−z′′)]gm(k,ρ,ρ′′)dk/bracketrightBigg ,(4.35) 52 wheregmcan be written as the product ψ′ 1(ρ<ρ′′)ψ′ 2(ρ>ρ′′). The functions ψ′ 1andψ′ 2satisfy the modiﬁed Bessel equation. They can be written as a linear combination of the possible solutions: ψ′ 1(kρ<) =A′Im(kρ<) +B′Km(kρ<), (4.36) ψ′ 2(kρ>) =C′Im(kρ>) +D′Km(kρ>). (4.37) Forρ→ ∞ Green function must remain ﬁnite. This means that C′= 0. Additionally, Green function must be zero at the boundary su rface. That is, G= 0 at the surface of the cylinder ρ=a. This yields: ψ′ 1(a) =A′Im(ka) +B′Km(ka) = 0 →B′=−A′Im(ka) Km(ka).(4.38) In order to obtain the function gmwe still have to ﬁnd the constant H′: gm(k,ρ,ρ′′) =H′/bracketleftbigg Im(kρ<)−Km(kρ<)Im(ka) Km(ka)/bracketrightbigg Km(kρ>), (4.39) whereρ>(ρ<) is the larger (smaller) of ρandρ′′. From Eq. (4.19) we have that H′= 4π: gm(k,ρ,ρ′′) = 4π/bracketleftbigg Im(kρ<)−Km(kρ<)Im(ka) Km(ka)/bracketrightbigg Km(kρ>). (4.40) The Green function is then given by: G(/vector r,/vector r′′) =2 π/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′′)/integraldisplay∞ 0cos[k(z−z′′)]× ×/bracketleftbigg Im(kρ<)−Km(kρ<)Im(ka) Km(ka)/bracketrightbigg Km(kρ>)dk/bracerightbigg . (4.41) 4.5.1 Cylindrical Shell Held at Zero Potential Suppose that the surface of the cylinder is held at zero poten tial, namely: φ(a,ϕ,z ) = 0. (4.42) Applying Eqs. (4.41) and (4.42) in Eq. (4.8) with ρ(/vector r′′) =qδ(/vector r′−/vector r′′) yields: φ(/vector r,/vector r′) =q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0cos[k(z−z′)]× ×/bracketleftbigg Im(kρ<)−Km(kρ<)Im(ka) Km(ka)/bracketrightbigg Km(kρ>)dk/bracerightbigg . (4.43) Hereρ>(ρ<) is the larger (smaller) of ρandρ′. 53 Far from the origin, ρis much larger than ρ′, hence we can express Eq. (4.43) in approximate form. The ﬁrst term that appears between brac kets is given by Im(kρ<)Km(kρ>), withρ<=ρ′andρ>=ρ. Note the presence of the term Km(kρ), withρ≫ρ′, which decays rapidly for increasing k. This implies that the main contribution of the integrand is in the region 0 < k < 1/ρ. Then we can approximate Im(kρ′) for small arguments, i.e., forkρ′≪1, yielding Im(kρ′)≈(kρ′/2)m/m!. From this we can see that the most relevant term is the ﬁrst one, m= 0. The integral of the ﬁrst term between brackets in Eq. (4.4 3) is then given by: φ1(ρ≫ρ′)≈q 2π2ε0/integraldisplay∞ 0cos[k(z−z′)]K0(kρ)dk=q 4πε0ρ. (4.44) To arrive at the last equality we have used the identity given by [202, Prob. 11.5.11]: 2 π/integraldisplay∞ 0cos(xt)K0(yt)dt=1/radicalbig x2+y2. (4.45) The second term that appears between brackets in Eq. (4.43) c an be treated in a similar way. The main contribution of the integrand is in the region 0<k< 1/ρ. Again, the most relevant term is the ﬁrst one. Accordingly, w e approximate the functionK0(kρ′) for small arguments: K0(kρ′)≈ −ln(kρ′). The integral of the second term between brackets of Eq. (4.43) is then given by: φ2(ρ≫ρ′)≈ −q 2π2ε0/integraldisplay∞ 0cos[k(z−z′)]ln(kρ′) ln(ka)K0(kρ)dk . (4.46) From Eq. (4.43) the electric ﬁeld is given by /vectorE0=−∇φ, with components: Eρ(ρ<ρ′) =−q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0kcos[k(z−z′)]× ×/bracketleftbigg Im′(kρ)−K′ m(kρ)Im(ka) Km(ka)/bracketrightbigg Km(kρ′)dk/bracerightbigg , (4.47) Eρ(ρ>ρ′) =−q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0kcos[k(z−z′)]× ×/bracketleftbigg Im(kρ′)−Km(kρ′)Im(ka) Km(ka)/bracketrightbigg Km′(kρ)dk/bracerightbigg , (4.48) Eϕ=q π2ε0ρ/braceleftBigg∞/summationdisplay m=1msin[m(ϕ−ϕ′)]/integraldisplay∞ 0cos[k(z−z′)]× ×/bracketleftbigg Im(kρ<)−Km(kρ<)Im(ka) Km(ka)/bracketrightbigg Km(kρ>)dk/bracerightbigg , (4.49) Ez=q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0ksin[k(z−z′)]× 54 ×/bracketleftbigg Im(kρ<)−Km(kρ<)Im(ka) Km(ka)/bracketrightbigg Km(kρ>)dk/bracerightbigg . (4.50) The zeroth order force /vectorF0=q/vectorE0(/vector r′) acting upon the charge qis given by Eq. (4.47) at /vector r=/vector r′without the ﬁrst term between brackets (which is the ﬁeld generated by the charge qitself). There is only a radial component: /vectorF0(/vector r′) =q2 2π2ε0/bracketleftBigg∞/summationdisplay m=−∞/integraldisplay∞ 0kKm(kρ′)Km′(kρ′)Im(ka) Km(ka)dk/bracketrightBigg ˆρ =−q2 4π2ε0ρ′2/braceleftBigg∞/summationdisplay m=−∞/integraldisplay∞ 0K2 m(x)d dx/bracketleftbigg xIm(ax/ρ′) Km(ax/ρ′)/bracketrightbigg dx/bracerightBigg ˆρ . (4.51) In the last equation we integrated by parts. We plot the zerot h order force of Eq. (4.51) in Fig. 4.3, normalized by the constant Fq≡q2/4πε0a2, as a function ofρ′/a. This force goes to zero when ρ′/a→ ∞ and diverges when ρ′→a, as expected. Figure 4.3: Zeroth order force F0between an inﬁnite grounded conducting cylinder of radius aand a point charge qat a distance ρ′from thezaxis (which is also the axis of symmetry of the cylinder), normali zed by the constant Fq≡q2/4πε0a2. Forρ′< athe force comes from Eq. (4.30), while for ρ′> a the force is given by Eq. (4.51). The surface charges can be calculated using Gauss’s law, yie lding: σ(a,ϕ,z ) =ε0Eρ(a,ϕ,z ) = =−q 2π2a/bracketleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0cos[k(z−z′)]Km(kρ′) Km(ka)dk/bracketrightBigg . (4.52) The charge per unit length λ(z) is given by: λ(a,z) =/integraldisplay2π 0σ(a,ϕ,z )adϕ=−q π/integraldisplay∞ 0cos[k(z−z′)]K0(kρ′) K0(ka)dk . (4.53) 55 It is interesting to obtain the behaviour of λfor a thin wire, far from z′ (|z−z′| ≫ρ′≫a). Utilizing Eq. (3.150) of Jackson’s book [13] we obtain: λ≈ −q 2 ln(|z|/a)1/radicalbig ρ′2+z2. (4.54) The total charge induced in the cylinder can be obtained inte grating Eq. (4.53) fromz=−∞to∞. Utilizing Eq. (4.33) this yields: Q=/integraldisplay∞ −∞λ(a,z)dz=−q . (4.55) A plot ofλ(a,z) as a function of z, withz′= 0 and normalized by q/ρ′, is given in Fig. 4.4. The maximum value of λ(a,z) is given at z=z′, as expected. In Fig. 4.5 we plot λmaxas a function of ρ′/a, normalized by q/ρ′. From this Figure we can see that λmax→0 whenρ′/a→ ∞,i.e., for a conducting cylinder of zero thickness, a simple conducting straight line. Figure 4.4: Induced linear charge density λon the conducting cylinder with an external point charge, Eq. (4.53), as a function of z/a. We utilized z′= 0, ρ′/a= 2 and normalized by q/ρ′. 4.5.2 Thin Cylindrical Shell Held at Zero Potential Consider that the grounded conducting cylinder is very thin ,i.e.,a≪ρ′. The modiﬁed Bessel functions can be approximated for small argu ment by [203, Sec. 8.44]: Im(y≪1)≈1 m!ym 2m, (4.56) Km(y≪1)≈(m−1)!2m−1 ym, m> 0, (4.57) K0(y≪1)≈ − lny 2−γ . (4.58) 56 Figure 4.5: Maximum induced linear charge density λmax(z=z′) on the con- ducting cylinder with an external point charge, Eq. (4.53), as a function of ρ′/a. We normalized the plot by q/ρ′. Hereγ= 0.577 is the Euler-Mascheroni constant. The term between brackets in Eq. (4.51) for m= 0 and for m >0 can be approximated by, respectively: d dx/bracketleftBigg x1 −lnax 2ρ′−γ/bracketrightBigg ≈ −1 ln(a/ρ′), (4.59) d dx/bracketleftbigg x1 m!(ax/ρ′)m 2m(ax/ρ′)m (m−1)!2m−1/bracketrightbigg ≈(2m+ 1)x2m(a/ρ′)2m m!(m−1)!22m−1.(4.60) The most relevant term for ρ′≫ais therefore m= 0. Utilizing the identity/integraltext∞ 0K2 0(x)dx=π2/4 we have the zeroth order force acting upon the charge qas given by: /vectorF0(ρ′≫a)≈ −q2 4π2ε0ρ′2/integraldisplay∞ 0K2 0(x)ln(2ρ′/xa)−γ+ 1 [γ−ln(2ρ′/xa)]2dxˆρ ≈ −q2 4π2ε0ρ′2ln(ρ′/a)/integraldisplay∞ 0K2 0(x)dxˆρ=−q2 16ε0ρ′2ln(ρ′/a)ˆρ . (4.61) Alternatively, another expression for the force can be foun d by integrating the force exerted by the linear charge density of a thin cylin der,λ(a,z) of Eq. (4.53), acting upon the point charge q. Utilizing that /integraldisplay∞ −∞ρ′cos[k(z−z′)] [ρ′2+ (z−z′)2]3/2dz= 2kK1(kρ′), (4.62) we obtain: /vectorF0(/vector r′) =−q 4πε0/integraldisplay∞ −∞ρ′ /radicalbig ρ′2+z2λ(a,z) ρ′2+z2dzˆρ 57 =−q2 2π2ε0ρ′2/integraldisplay∞ 0xK0(x)K1(x) K0(xa/ρ′)dxˆρ . (4.63) To compare Eqs. (4.61) and (4.63) we can expand the latter usi ng the ap- proximation ρ′≫a. Utilizing that K1(x) =−dK0/dx, integrating by parts, and usingK0(xa/ρ′)≈ −ln(xa/2ρ′)−γ≈ln(ρ′/a) we obtain: /vectorF0=q2 2π2ε0ρ′2/integraldisplay∞ 0xK0(x)(dK0/dx) K0(xa/ρ′)dxˆρ ≈ −q2 4π2ε0ρ′/integraldisplay∞ 0K2 0(x)ln(2ρ′/xa)−γ+ 1 [γ−ln(2ρ′/xa)]2dxˆρ ≈ −q2 16ε0ρ′2ln(ρ′/a)ˆρ , (4.64) which is exactly Eq. (4.61). 4.5.3 Inﬁnite Cylindrical Shell Held at Constant Potential Suppose that the conducting cylinder is held at a constant po tential,φ(a,ϕ,z ) = φ0. From Eq. (4.41) we obtain (with n′′=ρ<andρ>=ρ): ∂G ∂n′′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ρ′′=a=−2 π/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′′)/integraldisplay∞ 0kcos[k(z−z′′)]Km(kρ) Km(ka)× ×/bracketleftbig Im′(kρ′′)Km(ka)−Im(ka)Km′(kρ′′)/bracketrightbig dk/bracerightBigg ρ′′=a =2 πa/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′′)/integraldisplay∞ 0cos[k(z−z′′)]Km(kρ) Km(ka)dk/bracerightBigg . (4.65) In the last equality we utilized the Wronskian relation W[Im(kρ′′),Km(kρ′′)] = −1/(kρ′′). The second term given by Eq. (4.8) can be written as: φ+=−1 4π/ci∇cleco√y∇t/integraldisplay/integraldisplay Sφ(/vector r′′)∂G ∂n′′da′′=φ0 2π2a/integraldisplay∞ −∞adz′′/integraldisplay2π 0dϕ′′× ×/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0cos[k(z−z′)]Km(kρ) Km(ka)dk/bracerightBigg =φ0 π/integraldisplay∞ −∞dz′′/integraldisplay∞ 0cos[k(z−z′)]K0(kρ) K0(ka)dk =2φ0 π/integraldisplay∞ 0dz′′/integraldisplay∞ 0cos[k(z−z′)]K0(kρ) K0(ka)dk . (4.66) 58 In the last equality we changed the limits of the integral ove rz′′. In order to calculate the last integral, we utilize Eq. (4.23 ). Changing vari- ables, we have:/integraldisplay∞ 0cos[k(z−z′)]dz′=πδ(k). (4.67) The approximation for small argument of K0(y), namely,K0(y)≈ −lny, is in this case inappropriate, because the term lim k→0K0(kρ)/K0(ka)→1 for any ρ. This is true for an inﬁnite cylinder, but gives no physical i nsight into the behaviour of the potential as a function of ρ. We should use instead k≪1/ρ< 1/a, yielding: φ+≈φ0ln(kρ) ln(ka),fork≪1/ρ<1/a. (4.68) The potential outside an inﬁnite conducting cylinder with a n external charge, held at a constant potential φ0, is then given by the summation of Eqs. (4.43) and (4.68). We can ﬁnd the potential of a cylinder held at a constant poten tialφ0by a diﬀerent method. Suppose we have a long straight line of len gthℓalong the zaxis, uniformly charged with a linear charge density λ. The potential at a distanceρfrom thezaxis, forℓ≫ρ, is given by: φline≈λ 2πε0lnℓ ρ. (4.69) At a distance ρ=afrom thezaxis, we have a constant potential φ0= 2λln(ℓ/a), which is the same boundary condition as before. This impli es that the solution is the same. Substituting λ, we obtain the potential as given by: φline=φ0ln(ℓ/ρ) ln(ℓ/a). (4.70) Note that Eq. (4.68) with k≪1/ρ < 1/aand Eq. (4.70) with ℓ≫a > ρ are essentially the same. Henceforth, we utilize Eq. (4.70) as t he solution for a long cylinder held at a constant potential. The ﬁnal potential of the problem of a long conducting cylind er held at a constant potential φ0with an external charge qis given by: φ(/vector r,/vector r′) =q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0cos[k(z−z′)]Km(kρ>)× ×/bracketleftbigg Im(kρ<)−Im(ka) Km(ka)Km(kρ<)/bracketrightbigg dk/bracerightbigg +φ0ln(ℓ/ρ) ln(ℓ/a). (4.71) The components of the zeroth order electric ﬁeld /vectorE0, the zeroth order force /vectorF0 exerted on q, the surface charge density σand the linear charge density λare given by, respectively: Eρ(ρ<ρ′) =−q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0kcos[k(z−z′)]× 59 ×/bracketleftbigg Im′(kρ)−Im(ka) Km(ka)K′ m(kρ)/bracketrightbigg Km(kρ′)dk/bracerightbigg , (4.72) Eρ(ρ>ρ′) =−q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0kcos[k(z−z′)]× ×/bracketleftbigg Im(kρ′)−Im(ka) Km(ka)Km(kρ′)/bracketrightbigg Km′(kρ)dk/bracerightbigg +φ0 ρln(ℓ/a), (4.73) Eϕ=q π2ε0ρ/braceleftBigg∞/summationdisplay m=1msin[m(ϕ−ϕ′)]/integraldisplay∞ 0cos[k(z−z′)]× ×/bracketleftbigg Im(kρ<)−Im(ka) Km(ka)Km(kρ<)/bracketrightbigg Km(kρ>)dk/bracerightbigg , (4.74) Ez=q 2π2ε0/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0ksin[k(z−z′)]× ×/bracketleftbigg Im(kρ<)−Im(ka) Km(ka)Km(kρ<)/bracketrightbigg Km(kρ>)dk/bracerightbigg , (4.75) /vectorF0(/vector r′) =−q2 4π2ε0ρ′2/braceleftBigg∞/summationdisplay m=−∞/integraldisplay∞ 0K2 m(x)d dx/bracketleftbigg xIm(ax/ρ′) Km(ax/ρ′)/bracketrightbigg dx/bracerightBigg ˆρ +qφ0 ρ′ln(ℓ/a)ˆρ , (4.76) σ(a,ϕ,z ) =−q 2π2/braceleftBigg∞/summationdisplay m=−∞eim(ϕ−ϕ′)/integraldisplay∞ 0kcos[k(z−z′)]Km(kρ′) Km(ka)dk/bracerightBigg +ε0φ0 aln(ℓ/a), (4.77) λ(a,z) =−q π/integraldisplay∞ 0cos[k(z−z′)]K0(kρ′) K0(ka)dk+2πε0φ0 ln(ℓ/a). (4.78) From Eq. (4.78) we can calculate the total charge on the cylin der: Q=/integraldisplay∞ −∞λ(a,z)dz=−q π/integraldisplay∞ −∞dz/integraldisplay∞ 0cos[k(z−z′)]K0(kρ′) K0(ka)dk+2πℓε0φ0 ln(ℓ/a) =−qlim k→0ln(kρ′) ln(ka)+2πℓε0φ0 ln(ℓ/a)=−q+2πℓε0φ0 ln(ℓ/a). (4.79) For a neutral charged cylinder, i.e.,Q= 0, we can relate the constant potential φ0with the charge qby: φ0=qln(ℓ/a) 2πε0ℓ. (4.80) 60 4.6 Discussion We can express the zeroth order force exerted by the grounded conducting inﬁnite cylinder of radius aupon the external point charge qat a distance ρ′ from the axis of the cylinder as given by: /vectorF0=−αLq2 4πε0ρ′2ˆρ , (4.81) whereαLis a dimensionless parameter. In this work we have obtained t hree diﬀerent expressions for this force, namely, Eqs. (4.51), ( 4.61) and (4.63). The parameterαLfor these three cases is given by, respectively: αL=1 π/braceleftBigg∞/summationdisplay m=−∞/integraldisplay∞ 0K2 m(x)d dx/bracketleftbigg xIm(ax/ρ′) Km(ax/ρ′)/bracketrightbigg dx/bracerightBigg , (4.82) αL≈1 π/integraldisplay∞ 0K2 0(x)ln(2ρ′/xa)−γ+ 1 [γ−ln(2ρ′/xa)]2dx≈π 4 ln(ρ′/a),(4.83) αL≈2 π/integraldisplay∞ 0xK0(x)K1(x) K0(xa/ρ′)dx . (4.84) We plot these three values of αLas functions of a/ρ′in Figs. 4.6 to 4.8. Figure 4.6: Dimensionless parameter αLgiven by Eq. (4.81) as a function of a/ρ′. The continuous line represents the parameter from Eq. (4.8 2); the tight- dashed line that of Eq. (4.83); and the light-dashed line tha t of Eq. (4.84). We can see that these three values of αLconverge to one another as a/ρ′→0. This was expected because Eq. (4.51) is valid for a cylinder o f ﬁnite thickness with arbitrary value of a/ρ′, while Eqs. (4.61) and (4.63) are valid only for a thin cylinder, i.e., fora≪ρ′. In table (4.85) we present the values of the exact αLgiven by Eq. (4.82) as a function of ρ′/a. 61 Figure 4.7: Dimensionless parameter αLgiven by Eq. (4.81) as a function of a/ρ′, for the region a/ρ′≪1. The continuous line represents the parameter from Eq. (4.82); the tight-dashed line represents the parameter of Eq. (4.83); and the light-dashed line (which in this interval of a/ρ′is overlaid on the continuous line) represents the parameter of Eq. (4.84). Figure 4.8: Dimensionless parameter αLgiven by Eq. (4.81) as a function of log10(a/ρ′), for the region a/ρ′≪1. The continuous line represents the parame- ter from Eq. (4.82); the tight-dashed line represents the pa rameter of Eq. (4.83); and the light-dashed line (which in this interval of a/ρ′is overlaid on the con- tinuous line) represents the parameter of Eq. (4.84). From Eq. (4.83) we can see that when a/ρ′≪1, the parameter αLbehaves asπ/[4 ln(ρ′/a)]. That is, it goes to zero when a/ρ′→0. According to these calculations we conclude that there is no force between a poi nt charge and an idealized grounded conducting line (of zero thickness). One of the authors (AKTA) [1] had expected 0 <αL<1, not speciﬁcally for a grounded conducting line, but for a conducting line with zero total charge. In par ticular he expected that 0.1<αL<0.9, by guessing the result based on dimensional analysis and in analogy with the case of a point charge qat a distance ρ′from an inﬁnite 62 conducting plane. In this last case, the net force upon the te st charge is given by αPq2/4πε0ρ′2, withαP= 1/4 = 0.25. The results of the calculations presented here, on the other hand, indicate that αL= 0 whena/ρ′= 0 (in the case of a grounded inﬁnite line). This is an interesting result indic ating that the existence of a force upon the external test charge requires not only tha t it is at a ﬁnite distance to the cylinder, but also the existence of a surface area diﬀerent from zero in the conductor with which it is interacting.  ρ′/a αL 1.1 29.1 1.2 8.94 1.5 2.20 2.0 0.944 5 0.322 10 0.228 100 0.130 1030.0930 1040.0727 10100.0318 (4.85) Later on we compare this zeroth order force with the force pro portional to the voltage of the battery arising when a constant current ﬂows upon the cylindrical wire. 63 64 Chapter 5 Relevant Topics 5.1 Properties of the Electrostatic Field We present here some properties of the scalar electric poten tial,φ, and of the electric ﬁeld, /vectorE. These properties are proved in detail in most books dealing with electromagnetism, so that we present here only the main aspects. Suppose that we are in an inertial frame of reference Swith origin 0. There are N point charges qjat rest in this reference frame, with j= 1,...,N . The position vector describing the location of charge qjrelative to 0 is represented by /vector rj. The electric potential at the point /vector rodue to these Ncharges, according to the principle of superposition, is deﬁned by: φ(/vector ro)≡N/summationdisplay j=1qj 4πε01 roj, (5.1) whereroj≡ |/vector ro−/vector rj|is the distance between the tip of the vector /vector roand the chargeqj. The electric ﬁeld at the point /vector rois given by /vectorE(/vector ro) =−∇oφ . (5.2) Performing the line integral between points AandBof the potential diﬀer- ence,dφ=−/vectorE·d/vectorℓ, yields /integraldisplayB A/vectorE·d/vectorℓ=−/integraldisplayB A(∇φ)·d/vectorℓ=−/integraldisplayB Adφ=φ(/vector rA)−φ(/vector rB). (5.3) That is, this integral is independent of the path of integrat ion, being a function only of the initial and ﬁnal points. If it is performed an integration around a closed path of arbi trary form, this yields a null value: /contintegraldisplay /vectorE·d/vectorℓ= 0. (5.4) 65 5.2 The Electric Field in Diﬀerent Points of the Cross-section of the Wire Let us suppose a rectilinear, resistive and homogeneous wir e, of uniform cross- section, conducting a steady current. It seems that Davy was the ﬁrst to prove in 1821 that the current ﬂows over the whole cross-section an d not only along the surface of the wire [63, p. 90]: As we have already seen, Cavendish investigated very comple tely the power of metals to conduct electrostatic discharges; th eir power of conducting voltaic currents was now examined by Davy.1His method was to connect the terminals of a voltaic battery by a p ath containing water (which it decomposed), and also by an alter na- tive path consisting of the metallic wire under examination . When the length of the wire was less than a certain quantity, the wa ter ceased to be decomposed; Davy measured the lengths and weigh ts of wires of diﬀerent materials and cross-sections under the se limiting circumstances; and, by comparing them, showed that the cond uct- ing power of a wire formed of any one metal is inversely propor tional to its length and directly proportional to its sectional are a, but in- dependent of the shape of the cross-section.2The latter fact, as he remarked, showed that voltaic currents pass through the sub stance of the conductor and not along its surface. A theoretical proof that the current ﬁlls the cross-section of the wire can be found in the book of Chabay and Sherwood [166, Section 18.2 .4, p. 631]. Suppose there is a solid, homogeneous, rectilinear and unif ormly resistive wire, with a cross-section of arbitrary form, carrying a steady cu rrent. In steady- state the electric ﬁeld must be parallel to the wire (to avoid transverse currents and transverse electrostatic polarizations). Imagine now a rectangular path ABCDA within the wire, with ABandCDparallel to the wire, while BCand DAare perpendicular to the wire. When we perform the line integ ration of the electric ﬁeld, we get a null value, as was shown in Section 5.1 . In this proof we utilized charges at rest. In the case of this Section we are co nsidering steady currents, so that the surface charges are also moving with a d rifting velocity of valuevdrelative to the bulk of the wire. But these drifting velociti es are very small compared to light velocity c. This means that the corrections of second order, of the type v2 d/c2, will be negligible in comparison to Coulomb’s force. Therefore, they will not be considered here. This means that the electric ﬁeld in the section ABmust be parallel to the wire, with its intensity equal to the intensity of the electric ﬁeld in the section CD. By the microscopic form of Ohm’s law we ﬁnd that the same result must be true for the volum e current density,/vectorJ. 1Phil. Trans. cxi (1821), p. 425. His results were conﬁrmed by Becquerel, Annales de Chimie , xxxii (1825), p. 423. 2These results had been known to Cavendish. 66 Even with the component of the electric ﬁeld arising from the radial Hall eﬀect, to be discussed in Section 6.4, the same result will be maintained. The reason for this is that the component of the electric ﬁeld poi nting toward the axis of the wire will have its line integral cancelled betwee n the sections BCand DA. Utilizing the same reasoning in the case of a circuit having t he shape of a solid and homogeneous ring, conducting an azimuthal curren t, we ﬁnd that the azimuthal electric ﬁeld (neglecting the small radial Hall e ﬀect) must decrease as 1/ρ, whereρis the distance of the observation point to the ring axis of symmetry. We will see an example of this fact in Chapter 13. We do not know any experience which tried to verify if the elec tric ﬁeld and volume current density are really constants in all points of the cross-section of a metallic rectilinear wire carrying a steady current. The s ame can be said of the 1/ρdependence in the case of a ring. But the experiences of Bergm an, Schaefer, Jeﬁmenko and Parker (see Chapter 3) show qualitat ively that these suppositions are reasonable. 5.3 Electromotive Force Versus Potential Diﬀer- ence In this book we will see several examples showing that the ele ctric ﬁeld outside a wire carrying a steady current is proportional to the elect romotive force (emf) of the battery connected to the wire. In this Section we empha size that the emf is a concept diﬀerent from the potential diﬀerence due to cha rges at rest. This topic has been discussed by a number of authors [204] [16, Sec tion 7.1.2, pp. 277-278] [171] [166, pp. 642-644]. In order to separate positive and negative charges it is nece ssary the existence of “non-Coulomb” forces, /vectorFnC,i.e., forces which are not of electrostatic origin. This must happen in all cases in which we separate these charg es: in frictional electricity; when two diﬀerent metals touch one another; in a chemical battery; in thermoelectric eﬀect; in piezoelectric eﬀect; in a Van de Graaﬀ generator; in a photoelectric cell, etc.The reason for this is that due to Coulomb’s force, charges of opposite sign attract one another and tend to get t ogether. What separate these charges (or prevent them from getting togeth er once they were separated) can then only be of non-electrostatic origin. Th at is, this interaction must be independent from Coulomb’s electrostatic force. The origin of the expression “electromotive force” is due to Volta (1745-1827) [204]. As we have seen in Section 5.1, the diﬀerence of electrostati c potential be- tween two points AandBis given by φB−φA=−/integraldisplayB A/vectorEC·d/vectorℓ . (5.5) Here/vectorECis the electrostatic ﬁeld due to charges at rest. This potent ial diﬀerence 67 does not depend upon the path of integration, being a functio n only of the initial and ﬁnal points. On the other hand, the electromotive force between two point sAandB, emfBA, is given by emfBA=/integraldisplayB A/vectorEnC·d/vectorℓ . (5.6) Here/vectorEnC=/vectorFnC/qis the impressed force acting upon the test charge q, divided by the value of this charge. This impressed force has a non-el ectrostatic origin. This line integral depends upon the path of integration. In a battery, for instance, the Coulomb and non-Coulomb forc es balance one another in an open circuit. This means that there will be a pot ential diﬀerence across the battery. Moreover, this potential diﬀerence is n umerically equal to the battery’s emf. The emf of a battery is also called its volt age. Analogously, the emf of a closed circuit is given by: emf=/contintegraldisplay /vectorEnC·d/vectorℓ . (5.7) If in the closed circuit there is a chemical battery, or anoth er force of non- electrostatic origin, this line integral upon a closed circ uit may have a net value diﬀerent from zero. Despite the term “force” in the expression emf, the emf is not a force in the Newtonian sense. The emf of a pile or chemical battery is nume rically equal to the potential diﬀerence generated between the terminals of the battery. It has the same units as potential diﬀerence, namely, volt or newto n/coulomb. Despite this fact, the emf is not a potential diﬀerence, as we emphasi zed in this Section. Its origin is due to a non-electrostatic force. And it is not a lways associated with a potential diﬀerence. For instance, in the case of a ring app roaching or moving away from a permanent magnet (example discussed by Weber, as we discuss in Appendix A), it is generated a current along the resistive ri ng, although there is no potential diﬀerence between any two points of the ring [ 204] [184]. 5.4 Russell’s Theorem Russell proved in an important short paper a general theorem related with straight parallel conductors of arbitrary cross-sections carrying steady currents [9]. He concluded that the density of surface charges σon the conductors vary linearly with distance along the direction of their common a xisz. The same was found valid for the potential φinside and outside the conductors. He considered homogeneous isotropic materials surrounded by an insulating medium of constant permittivity ε. His theorem is valid at great distances from their termination (in order to neglect edge eﬀects) and also far from the sources of electromotive force, emf, maintaining the current. 68 The essence of his proof is to consider that inside the conduc tors carrying steady currents the electric ﬁeld /vectorEhas everywhere the same longitudinal compo- nent. By/vectorE=−∇φthis means that the potential inside them must be a linear function of z. Outside the conductors the potential φmust satisfy Laplace’s equation ∇2φ= 0. But the solutions of Laplace’s equation which satisfy al l the boundary conditions are unique. As the boundary conditions in all conductors are linear functions of z, the same must be true outside them. That is, φ(x,y,z ) =F(x,y)(A+Bz), (5.8) whereF(x,y) is a function of the transverse coordinates, while AandBare constants. Analogously, the surface charge densities σare obtained by Gauss’s law as directly proportional to the normal component of the elec tric ﬁeld at the conductor boundaries. As /vectorE=−∇φ, Eq. (5.8) yields: σ(x,y,z ) =G(x,y)(A+Bz), (5.9) whereG(x,y) is a function of the transverse coordinates. This means that the solution of electrostatic problems can b e directly ap- plied to the solution of steady currents, by including a line ar dependence in the longitudinal component. In the following Chapters we will s ee many examples illustrating this theorem. But it should be kept in mind that it is valid only far from the t erminations of the conductors and also far from the batteries. Moreover, it is not valid as well close to the junction of two materials of diﬀerent condu ctivities. 69 70 Part II Straight Conductors 71 In this work the frame of reference will always be the laborat ory. When we speak of conductors and wires in general, it should be unders tood that they are usually uniformly resistive, unless stated otherwise. The medium outside the conductors or between them will be usually air or vacuum. No t ime variation of currents or potentials will be considered here. It is assume d that there are no conductors nor other external charges close to the current- carrying wire, so that we will consider it isolated from external inﬂuences (excep t for the test charge already mentioned). In this ﬁrst part we will consider one or more parallel resist ive conductors carrying steady currents along the zaxis. 73 74 Chapter 6 A Long Straight Wire of Circular Cross-section To our knowledge the ﬁrst to perform theoretical calculatio ns related to the elec- tric ﬁeld inside a wire of circular cross section due to surfa ce charges increasing linearly with the longitudinal coordinate has been Wilhelm Weber in 1852 [32], as we discuss in the Appendix A. Here we follow the treatment p ublished in 1999 [1]. 6.1 Conﬁguration of the Problem The situation considered here is that of a cylindrical and ho mogeneous resistive wire of length ℓand radius a≪ℓ, Figure 6.1. Figure 6.1: Conﬁguration of the problem. The axis of the wire coincides with the zdirection, with z= 0 at the center of the wire. A battery maintains constant potentials φLandφRat the extremities z=−ℓ/2 andz= +ℓ/2 of the wire, respectively. The wire carries a constant currentI, has a ﬁnite conductivity gand is at rest relative to the laboratory. There is air or vacuum outside the wire. At a distance ρ=/radicalbig x2+y2from the axis of the wire there is a stationary point charge q. We want to know the force exerted by the wire upon the charge q. In particular we wish to calculate 75 the component of this force which is proportional to the volt age or emf of the battery, or to the potential diﬀerence acting along the wire . To this end we will suppose the following approximation: ℓ≫ρ≥0, ℓ≫a>0 andℓ≫ |z| ≥0. (6.1) Herezis the longitudinal component of the vector position of q. See Fig. 6.1. We utilize throughout this chapter cylindrical coordinates ( ρ,ϕ,z ) and unit vectors ˆρ, ˆϕand ˆz. This wire must be closed somewhere. The calculations presen ted here with this approximation should be valid for the circuit of Figure 6.2. This is a square circuit with four sides of length ℓ, composed of cylindrical wires of radius a≪ℓ. There is an external point charge close to the middle of one of its sides (like AB,BCorCD) and far from the battery. The case when the point charge is close to the middle of the side ADhas been published in 2004 [205]. Figure 6.2: A closed square circuit made of a resistive wire o f circular cross- section. There is a point charge close to the middle of one of i ts sides. With this approximation we can consider that the three other sides will not contribute signiﬁcantly to the potential and ﬁeld near the c enter of the fourth side. Alternatively, it should also give approximate resul ts for a circular loop of larger radius R0=ℓ/2πand smaller radius a≪R0(a ring), if the point charge is close to the ring but far from the battery maintaining the c urrent. It might even be utilized as a ﬁrst gross approximation for the force o n the point charge of Figure 1.1 considering a generic circuit of large length a nd small curvatures (that is, with radii of curvature much larger than the diamet er of the wire and also much larger than the distance of the point charge to the w ire). We consider separately three components of the force exerte d by the wire onq: (A) The component due to the electrostatic induction, base d upon the charges induced along the surface of the wire by q, which has been considered in Chapter 4; (B) the component of the force due to the surface ch arges which exist in resistive wires carrying steady currents (force proport ional to the voltage or emf of the battery connected to the circuit); and (C) the forc e proportional to the 76 square of the drifting velocity, vd, of the conduction electrons, i.e., proportional tov2 d/c2. 6.2 Force Proportional to the Potential Diﬀer- ence Acting upon the Wire When a constant current ﬂows in a resistive wire connected to a battery, the elec- tric ﬁeld driving the conduction electrons against the resi stive friction exerted upon them by the lattice is due to free charges distributed al ong the surface of the wire, as we have seen before. We represent this surface ch arge density by σ(a,ϕ,z ). For steady currents, σis constant in time but varies along the length of the wire (that is, it is a function of z). Here we follow the approach of Weber and Kirchhoﬀ discussed in Chapter 2. The battery, due to the c hemical forces which maintain its terminals at diﬀerent potentials, is res ponsible for maintain- ing this distribution of charges along the surface of the wir e. But the battery does not generate directly the electric ﬁeld in all points al ong the circuit. The surface charges, on the other hand, generate not only the ele ctric ﬁeld inside the wire but also an electric ﬁeld outside it. The approach of this chapter is the following: We consider th e cylindrical wire carrying the constant current Iand calculate the potential φ1and electric ﬁeld/vectorE1inside and outside the wire due to these surface charges in th e absence of the test charge q. When we put the test charge at a distance ρfrom the wire the force on it due to the surface charges will be then giv en by/vectorF1=q/vectorE1, supposing that it is small enough such that it does not distur b the current nor the wire (except from the induction charges already conside red in Chapter 4, which will exert the force /vectorF0=q/vectorE0). We begin calculating the potential due to the surface charges. As there is a constant current in the wire, the electric ﬁeld i nside it and driving the current must be constant over the cross-section of the wire [63, p. 90]. Here we are disregarding the small radial Hall eﬀect ins ide the wire due to the azimuthal magnetic ﬁeld generated by the current to be di scussed in Section 6.4. This means that the potential and surface charge distri bution must be a linear function of z, as we saw in Section 5.4. Due to the axial symmetry of the wire it cannot depend on the azimuthal angle either. This mea ns that σ(a,ϕ,z ) =σA+σBz ℓ, (6.2) whereσAandσBare constants. Before proceeding we wish to discuss this expression. We may wish to con- sider the wire as globally neutral, i.e., no net charge as a whole. When we integrate the free charge density σover the whole surface of the wire we need to obtain a zero net value in this case. This will happen with E q. (6.2) after integrating from z=−ℓ/2 toz=ℓ/2 only in the symmetrical case in which σA= 0. This might represent, for instance, the top side BCof Figure 6.2. On the other hand, we will perform the calculations with a gener ic value ofσAso 77 that the calculation might be applicable, for instance, to t he left half of the top sideBCof Figure 6.2. The integration of σover this left side (from z=−ℓ/2 to zero) will yield a positive value, as it is closer to the pos itive terminal. This positive charge will be balanced by the negative charge lyin g on the right half of the top side of Figure 6.2 ( zgoing from zero to + ℓ/2). With a generic σAwe might also consider, for instance, the left side ABof Figure 6.2 with a positive charge, which will be balanced by the negative charge in the r ight sideCDof Figure 6.2. It should be emphasized that the point where σ= 0 is speciﬁed by the battery. The battery itself also speciﬁes where σwill be positive (portions of the wire closer to its positive terminal) or negative (porti ons of the wire closer to its negative terminal). Due to the axial symmetry of σwe can calculate φat the speciﬁc angle ϕ= 0 rad and then generalize the solution to all ϕ. The potential inside or outside the wire is then given by: φ1(ρ,z) =1 4πε0/integraldisplay2π ϕ2=0/integraldisplayℓ/2 z2=−ℓ/2σadϕ 2dz2/radicalbig ρ2+a2−2ρacosϕ2+ (z2−z)2 =1 4πε0/integraldisplay2π ϕ2=0/integraldisplayℓ/2 z2=−ℓ/2(σA+σBz2/ℓ)dϕ2dz2/radicalbigg/parenleftBig 1−2ρ acosϕ2+ρ2 a2/parenrightBig +/parenleftbigz2−z a/parenrightbig2.(6.3) Deﬁning the dimensionless variables s2≡1−2(ρ/a)cosϕ2+(ρ2/a2) andu≡ (z2−z)/awe are then led to: φ1(ρ,z) = (a/4πε0)[(σBa/ℓ)I1+(σA+σBz/ℓ)I2], where I1≡/integraldisplay2π ϕ2=0/integraldisplayℓ/2a−z/a u=−(ℓ/2a+z/a)udϕ2du√ s2+u2, (6.4) and I2≡/integraldisplay2π ϕ2=0/integraldisplayℓ/2a−z/a u=−(ℓ/2a+z/a)dϕ2du√ s2+u2. (6.5) These integrals can be solved with the approximation (6.1). The ﬁnal ap- proximate result is given by (generalizing for all ϕ): φ1(ρ≤a,ϕ,z )≈a ε0/bracketleftbigg σAlnℓ a+σBz ℓlnℓ ea/bracketrightbigg , (6.6) φ1(ρ≥a,ϕ,z )≈a ε0/bracketleftbigg σAlnℓ ρ+σBz ℓlnℓ eρ/bracketrightbigg . (6.7) From Eq. (6.1) we can neglect ln e= 1 in comparison with ln( ℓ/a) and ln(ℓ/ρ). This yields: φ1(ρ,ϕ,z )≈aσ(z) ε0lnℓ a=a(σA+σBz/ℓ) ε0lnℓ a,ifρ≤a , (6.8) 78 φ1(ρ,ϕ,z )≈aσ(z) ε0lnℓ ρ=a(σA+σBz/ℓ) ε0lnℓ ρ,ifρ≥a . (6.9) By writing the linear density of charges along the wire as λ(z)≡2πaσ(z) these last expressions can be written as φ1(ρ≤a,ϕ,z )≈λ(z) 2πε0lnℓ a, (6.10) φ1(ρ≥a,ϕ,z )≈λ(z) 2πε0lnℓ ρ. (6.11) From Eqs. (6.6) and (6.7) we can obtain the electric ﬁeld /vectorE1=−∇φ1: /vectorE1(ρ<a,ϕ,z )≈ −aσB ε0ℓ/parenleftbigg lnℓ ea/parenrightbigg ˆz , (6.12) /vectorE1(ρ>a,ϕ,z )≈a ε0/parenleftBig σA+σBz ℓ/parenrightBigˆρ ρ−aσB ε0ℓ/parenleftbigg lnℓ eρ/parenrightbigg ˆz . (6.13) To our knowledge the ﬁrst to obtain Eq. (6.12) beginning with the integration of Eq. (6.2) was Wilhelm Weber in 1852, as we discuss in Append ix A. From Eq. (6.1) we can neglect 1 in comparison with ln( ℓ/a) and ln(ℓ/ρ). This yields the coulombian force on a test charge qlocated at ( ρ,ϕ,z ) as given by (with/vectorF1=−q∇φ1): /vectorF1=q/vectorE1≈ −qa ε0∂σ(z) ∂z/parenleftbigg lnℓ a/parenrightbigg ˆz=−qaσB ℓε0/parenleftbigg lnℓ a/parenrightbigg ˆzifρ<a , (6.14) /vectorF1=q/vectorE1≈qaσ(z) ε0ˆρ ρ−qa ε0∂σ(z) ∂z/parenleftbigg lnℓ ρ/parenrightbigg ˆz =qa(σA+σBz/ℓ) ε0ˆρ ρ−qaσB ℓε0/parenleftbigg lnℓ ρ/parenrightbigg ˆzifρ>a . (6.15) We can relate these expressions with the current Iﬂowing in the wire. From Figure 6.1 and the fact that φ1is a linear function of zwe obtain φ1(ρ≤a,z) =φR+φL 2+ (φR−φL)z ℓ. (6.16) Equating this with Eq. (6.8) and utilizing Ohm’s law φL−φR=RI, whereR=ℓ/gπa2is the resistance of the wire, with gbeing its conductiv- ity, yieldsσB=−Rε0I/aln(ℓ/a) andσA=ε0(φR+φL)/2aln(ℓ/a) =ε0(RI+ 2φR)/2aln(ℓ/a). The density of free charges along the surface of the wire ca n then be written as: 79 σ(a,ϕ,z ) =ε0(φR+φL) 2aln(ℓ/a)−Rε0I aln(ℓ/a)z ℓ. (6.17) This means that the potential and the force on the test charge qare given by: φ1=φR+φL 2−RIz ℓifρ≤a , (6.18) φ1=φR+φL 2ln(ℓ/ρ) ln(ℓ/a)−RIln(ℓ/ρ) ln(ℓ/a)z ℓifρ≥a , (6.19) /vectorF1=q/vectorE1=qRI ℓˆzifρ<a , (6.20) /vectorF1=q/vectorE1=q/bracketleftbigg1 ln(ℓ/a)/parenleftbiggRI+ 2φR 2−RIz ℓ/parenrightbiggˆρ ρ+RI ℓln(ℓ/ρ) ln(ℓ/a)ˆz/bracketrightbigg ifρ>a . (6.21) Now that we have obtained the potential outside the wire we mi ght also invert the argument. That is, we might solve Laplace’s equat ion∇2φ= 0 in cylindrical coordinates inside and outside the wire (for a≤ρ≤ℓ) by the method of separation of variables, supposing a solution of t he formφ(ρ,ϕ,z ) = R(ρ)Φ(ϕ)Z(z). The arbitrary constants obtained by this method are found imposing the following boundary conditions: ﬁnite φ(0,ϕ,z),φ(a,ϕ,z ) = (φR+ φL)/2 + (φR−φL)z/ℓandφ(ℓ,ϕ,z) = 0. This last condition is not a trivial one and was obtained only after we found the solution in the or der presented in this work. See Eq. (6.9). The usual boundary condition tha t the potential goes to zero at inﬁnity does not work in the case of a long cylin der carrying a steady current. But the potential going to zero at ρ=ℓis a reasonable result. After all, this means that we are considering φ= 0 at a great distance from the wire. By this reverse method we obtain the potential insi de and outside the wire, then the electric ﬁeld by /vectorE=−∇φand lastly the surface charge density byε0times the normal component of the electric ﬁeld outside the w ire in the limit in which ρ→a. In this way we checked the calculations. If we putφL=φR=φ0orI= 0 in Eqs. (6.18) to (6.21) we recover the electrostatic solution (long wire charged uniformly wi th a constant charge densityσA, with total charge QA= 2πaℓσA), namely: φ1(ρ≤a) =φ0=aσA ε0lnℓ a, (6.22) φ1(ρ≥a) =φ0ln(ℓ/ρ) ln(ℓ/a)=aσA ε0lnℓ ρ, (6.23) /vectorE1(ρ<a) =/vector0, (6.24) /vectorE1(ρ>a) =φ0 ln(ℓ/a)ˆρ ρ=aσA ε0ˆρ ρ. (6.25) 80 We can also obtain the capacitance per unit length of this lon g cylindrical wire asC/ℓ= [QA/φ(a)]/ℓ= 2πε0/ln(ℓ/a). It is interesting to analyze here the solutions for points ex tremely close to the wire,ρ=a+dwithd≪a. In this approximation Eq. (6.9) yields: φ1≈aσ(z) ε0/parenleftbigg lnℓ a−d a/parenrightbigg . (6.26) The analysis presented here reﬁnes the previous work of Coom bes and Laue, who discussed in 1981 the limiting case of an inﬁnitely long w ire [17]. They arrived at the same uniform electric ﬁeld both inside and out side the wire. This is correct for an inﬁnitely long wire. In the present cas e we arrived at a uniform electric ﬁeld inside the wire and at an electric ﬁeld outside the wire with longitudinal and radial components depending on ρ, as we were considering a large but ﬁnite length ℓ. Eqs. (6.16), (6.20) and (6.21) show that the electric ﬁeld bo th inside and outside the wire is proportional to the potential diﬀerence φL−φR=RIacting along the wire. The same can be said of the force exerted upon a stationary external test charge by the resistive wire carrying a steady current. If we change the diameter of the wire, or its resistivity, we can change th e resistance of the wire. But if it is connected to the same battery, in such a way t hat it is under the action of the same potential diﬀerence, the current ﬂowi ng along the wire will change accordingly. But the density of surface charges and the external electric ﬁeld will not change. This important aspect has bee n emphasized by Chabay and Sherwood [165, 166, 171]. Moreover, there will be not only a tangential component of th e electric ﬁeld outside the wire (as might be expected from the continuity of this component at an interface between two media), but also a radial compone nt. In the sym- metrical case in which φL=−φR=RI/2 the ratio of the radial component of /vectorF1to its tangential component is given by z/[ρln(ℓ/ρ)]. For a wire with 1 meter length with z=ρ= 10 cm this ratio is given by 0 .4. This means that these two components are of the same order of magnitude. The longitudinal component of the electric ﬁeld is continuo us at an interface separating two media. From Eqs. (6.20) and (6.21) we can see t hat at the surface of the wire, ρ=a, the longitudinal component of the electric ﬁeld is given by : /vectorE1= (RI/ℓ)ˆz. This electric ﬁeld will act upon the surface conduction ele ctrons belonging to the wire, so that they will move with a constant t angential velocity in steady state, with the electric force balanced by the Ohmi c resistance. In equilibrium there will be the same number of free electrons e ntering and leaving a circular strip of length 2 πaand widthdz, so that the distribution of surface charges will not change with time, although being a function ofz. In any event, the surface charges will not remain stationary when there is a steady current, but will move due to this tangential electric ﬁeld existing a t the surface of the conductor. The drifting velocity of the conduction electro ns will be diﬀerent from zero not only in the bulk of the metal, but also along its s urface. The distribution of surface charges is actually a surface curre nt. The same will 81 happen for the other resistive conductors carrying steady c urrents discussed in this work. Eqs. (6.19) and (6.21) show that the external potential, ele ctric ﬁeld and force go to zero when ℓ/a→ ∞. This means that resistive electric currents exert forces upon external static charges, except in the idealize d case of ﬁlamentary current (zero cross-section conductor). The crucial aspec t for the existence of a force is not only that the current-carrying wires are resist ive but that they have ﬁnite cross-sections. Below we consider a force due to the square of the current. 6.3 Force Proportional to the Square of the Cur- rent Up to now we have only considered two components of the force e xerted by the resistive wire upon the external test charge: (A) the com ponent arising from electrostatic induction (due to induction charges alo ng the surface of the wire generated by the presence of q); and (B) the component arising from the potential diﬀerence acting along the wire (due to the charge s along the surface of the wire induced by the presence of the battery, when there is a steady current ﬂowing along the wire). This surface charge density and the a ccompanying electric ﬁeld are proportional to the emf of the battery or to the potential diﬀerence acting along the wire. We have not yet taken into ac count the force of the stationary lattice and mobile conduction electrons o n the stationary test charge. We consider it here in this Section, analyzing two di ﬀerent theoretical models: Lorentz’s force and Weber’s force. We ﬁrst consider Lorentz’s force (or Li´ enard-Schwarzschi ld’s force). In this case there are also components of the force exerted by a charg eq2belonging to the current carrying circuit on the test charge qwhich depend on the square of the velocity of q2,v2 d, and on its acceleration. If we have a steady current, the acceleration of q2will be its centripetal acceleration due to any curvature in the wire, proportional to v2 d/rc, wherercis the radius of curvature of the wire at each point. This might lead to a force proportional to v2 dor toI2. However, it has been shown that if we have a closed circuit carrying a cons tant current, there is no net eﬀect of the sum of all these terms on a stationary cha rge outside the wire [11, page 697, exercise 14.13] [15] [23, Section 6.6]. T he same result is valid for Clausius’s force law. In conclusion we might say the foll owing: According to Lorentz’s force, the stationary lattice creates an elect ric ﬁeld which is just balanced by the force due to the free electrons inside the clo sed wire, even when there is a constant current along the resistive wire. This mi ght be interpreted as considering the wire to be electrically neutral in its int erior (the radial Hall eﬀect will be considered later on). We now consider Weber’s electrodynamics [23]. As already st ated, we are disregarding the small radial Hall eﬀect inside the wire due to the azimuthal magnetic ﬁeld generated by the current. This means that the i nterior of the wire 82 can be considered essentially neutral. Despite this fact We ber’s electrodynamics predicts a force exerted by this neutral wire in a stationary charge nearby, even for closed circuits carrying constant currents. The re ason for this eﬀect is that the force exerted by the mobile electrons on the station ary test charge is diﬀerent from the force exerted by the stationary positive i ons of the lattice on the test charge. One of us has already performed these calcul ations in related situations, so that we present here only the ﬁnal result. The calculations have been published in 1991 [22] [23, Section 6.6, pages 161-168] . When we ﬁrst performed these calculations we were not completely consci ous of the surface charges discussed in this book (proportional to the emf of th e battery, or to the potential diﬀerence acting along the wire). For this reason the calculations were performed supposing wires electrically neutral in all inte rnal points and also along their surfaces. Despite the limitations of this suppo sition, we reproduce the ﬁnal results here in order to show that they are diﬀerent f rom the ﬁnal results obtained with Lorentz’s force when we assume the sam e conditions of neutrality. Once more we assume (6.1). For the situation of Figure 6.1, wi th a uniform current density /vectorJ= (I/πa2)ˆz, the force on the test charge is given by: /vectorF2=−qIvd 4πε0c2ˆρ ρ=−µ0 4π2qI2 a2enˆρ ρifρ>a , (6.27) wherevdis the drifting velocity of the electrons. We also utilized c2= 1/ε0µ0 andvd=I/πa2en, wheree= 1.6×10−19C is the elementary charge and nis the number of free electrons per unit volume. This force is proportional to the square of the current. The e lectric ﬁeld /vectorE2=/vectorF2/qpoints toward the current, as if the wire had become negative ly charged. Sometimes this second order ﬁeld is called motiona l electric ﬁeld. Suppose that we now bend the wire carrying a constant current (by letting its shape in the form of a ring, for instance). In this case Web er’s electrody- namics predicts another component of the force exerted by th is current upon a stationary charge outside the wire. This new component dep ends upon the acceleration of the source charges (in this case conduction electrons). As we are supposing a steady current which does not change with time, t he relevant accel- eration here is the centripetal one proportional to v2 d/rc, wherercis the radius of curvature of the wire at that location. This means that als o this component of the force will be proportional to v2 dor toI2. The order of magnitude is the same as the previous example. In 1991 [22] and in 1994 [23, Sec tion 6.6, pp. 161-168] it was calculated the net second order force acting upon a stationary charge outside the wire due to a circular closed circuit carr ying a steady az- imuthal current in the shape of a ring, utilizing Weber’s for ce. We showed that its net value had the order of magnitude of Eq. (6.27). To this end we have taken into account not only the component of the force which d epends upon the square of the velocity of the source charges, v2 d, but also the component of the force due to the centripetal acceleration of the source e lectrons. This means that Weber’s second order force does not go to zero even for cl osed circuits. 83 In the case of Lorentz’s force, on the other hand, this net sec ond order force is always null in the case of closed currents. This is an impor tant theoretical diﬀerence between these two theories. 6.4 Radial Hall Eﬀect Another simple question which might be asked is the followin g: Is a stationary resistive wire carrying a constant current electrically ne utral in its interior? Many authors quoted in Section 1.2 answered positively to th is question as this was one of their reasons for believing that this wire wou ld not generate any electric ﬁeld outside itself. However, we already showe d that there will be a longitudinal distribution of surface charges which wil l give rise to the longitudinal electric ﬁeld inside the wire and also to an ele ctric ﬁeld outside it. Here we show that there will also be a radial electric ﬁeld ins ide the wire due to the fact that its interior is negatively charged. To our knowledge the ﬁrst to consider this eﬀect and to presen t quantitative calculations were Matzed, Russell and Rosser [206, 167]. Sm ythe also discussed this subject brieﬂy [207, Section 6.04, pp. 250-252]. The usual Hall eﬀect is discussed in most textbooks on classi cal electromag- netism, so that we will not enter into details here. Normally they consider the eﬀects upon a current carrying conductor when placed in an ex ternal magnetic ﬁeld. These eﬀects include the so-called “Hall voltage” and related topics. However, what we discuss here is a similar eﬀect but due to the internal magnetic ﬁeld generated by the current-carrying wire itsel f, without the presence of any external magnetic ﬁeld. To distinguish this eﬀect fro m the usual Hall eﬀect, we utilize the expression radial Hall eﬀect (related to the case of a current ﬂowing along a cylindrical conductor). We here consider the radial Hall eﬀect due to the azimuthal ma gnetic ﬁeld inside the wire generated by the longitudinal current ﬂowin g in this wire. As is usually considered [63, p. 90], we will suppose the consta nt total current I to ﬂow uniformly over the cross-section of the cylindrical w ire with a current densityJ=I/πa2. With the magnetic circuital law/contintegraltext C/vectorB·d/vectorℓ=µ0IC, where Cis the circuit of integration and ICis the current passing through the surface enclosed by C, we obtain that the magnetic ﬁeld inside and outside the wire is given by: /vectorB(ρ≤a) =µ0Iρ 2πa2ˆϕ , (6.28) /vectorB(ρ≥a) =µ0I 2πρˆϕ . (6.29) The magnetic force on a speciﬁc conduction electron of charg eq=−einside the wire (due to the magnetic ﬁeld generated by all other cond uction electrons), at a distance ρ<a from the axis and moving with drifting velocity /vector v=−|vd|ˆz is given by: 84 /vectorF=q/vector v×/vectorB=−|µ0evdIρ| 2πa2ˆρ . (6.30) This radial force pointing inwards will create a concentrat ion of negative charges in the body of the conductor. This is like a pinch eﬀec t. In equilibrium there will be a radial force generated by these charges which will balance the magnetic force: qE=qvB. That is, there will be inside the wire, beyond the longitudinal electric ﬁeld E1driving the current, a radial electric ﬁeld pointing inwards given by: /vectorEρ(ρ≤a) =−|µ0vdIρ| 2πa2ˆρ . (6.31) The longitudinal electric ﬁeld inside the wire driving the c urrent is given by E1=RI/ℓ. In order to compare it with the magnitude of the radial elect ric ﬁeldEρdue to the Hall eﬀect we consider the maximum value of this las t ﬁeld very close to the surface of the wire, at ρ→a:Eρ→ |µ0vdI|/2πa. This means that (withR=ℓ/gπa2): |Eρ| |E1|=|µ0vdga| 2. (6.32) For a typical copper wire ( vd≈4×10−3m/s andg= 5.7×107Ω−1m−1) with 1 mm diameter this yields: Eρ/E1≈7×10−5. This shows that the radial electric ﬁeld inside the wire is negligible compared to the longitudi nal one. By Gauss’s law ∇ ·/vectorE=ρc/ε0we obtain that inside the wire there will be a constant negative charge density ρc−given by: ρc−=−|Ivd|/πa2c2. The total charge inside the wire is compensated by a positive charge spread over the surface of the wire with a constant surface density σ+=|ρc−a/2|= |Ivd|/2πac2. That is, the negative charge inside the wire in a small segme nt of lengthdz,ρc−πa2dz, is equal and opposite to the positive charge along its surface,σ+2πadz. This means that the radial Hall eﬀect will not generate any electric ﬁeld outside the wire, only inside it. For this reas on it is not relevant to the experiments discussed before. In any event it is impor tant to clarify this eﬀect. Contrary to the surface density of free charges σ(a,z), this constant charge densityσ+does not depend on the longitudinal component z. In conclusion we may say that the total surface charge densit y along the wire, not taking into account the motional electric ﬁeld and the induction of charges in the conductor due to external charges, is given by the constant σ+ added to the σgiven by Eq. (6.17). In our analysis of the radial Hall eﬀect we are not considerin g the motional electric ﬁeld already discussed as it is not yet completely c lear if it exists or not. The results of this Section are completely theoretical. The y are based upon the equilibrium of a magnetic force (due to the poloidal magn etic ﬁeld) and an electric force (orthogonal to the axis of the wire) acting up on a drifting electron moving along the axis of the wire. We are not aware of any exper iments which 85 tried to measure the internal density of charges ρc−in current carrying metallic conductors. We now compare all three components of the electric ﬁeld outs ide the wire with one another. 6.5 Discussion The solutions presented here will remain valid in the case of a hollow cylindrical shell of internal radius aiand external radius a. The internal density of surface charge atρ=aiwill be zero taking into account the approximations conside red here, while the external density of surface charge will be th e same as obtained before. The main diﬀerence is that the electric ﬁeld in the re gionρ < aiwill not produce any current as there is no conductor in this regio n. Although many authors forget about the zeroth order force F0due to elec- trostatic induction when dealing with a current-carrying w ire interacting with an external charge, there is no doubt it exists. Comparing th e three components of the force already discussed, it is the only one which diver ges as we approach the wire. If we are far away from the wire (at a distance ρ≫afrom it) this ze- roth order force falls as 1 /ρ2ln(ρ/a) (as we saw in Eqs. (4.81) and (4.83)), while the radial component of the force proportional to the voltag e of the battery and of the second order force, F1andF2, fall as 1/ρ(as we saw in Eqs. (6.21) and (6.27)). We now compare the three components of this force given by Eqs . (4.81) and (4.82), (6.21) and (6.27). To this end we consider a parti cular example with orders of magnitudes similar to those employed in Sansb ury’s experiment [178]. He utilized a U-shaped copper current conductor (50 c m long legs spaced 10 cm apart, with 0.95 cm diameter). As we will utilize his dim ensions in a diﬀerent conﬁguration (straight wire instead of a U-shaped conductor), we will consider our straight wire having a total length of ℓ= 1.20 m and a radius a= 4.75×10−3m. The conductivity of copper is g= 5.7×107Ω−1m−1and it has a number of free electrons per unit volume given by n= 8.5×1028m−3. The resistance of the wire is then given by R=ℓ/gπa2= 3.0×10−4Ω. He passed a current of 900 A in his wire, which means a potential diﬀeren ce between the extremities of the wire as given by φL−φR= 0.27 V. The drifting velocity in this case amounts to vd=I/πa2en= 0.9×10−3m/s. We will suppose moreover the symmetrical case in which φR=−φL=−0.135 V. The test charge will be the one estimated by Sansbury, namely, q≈5×10−10C, at a distance of ρ= 3.5 cm = 3.5×10−2m from the wire. This yields a/ρ′= 0.121. Although his test charge was spread over a 2 cm ×2 cm silver foil, here we suppose the test charge concentrated in a point. These values in Eqs. (4.81) and (4.82) yield αL= 0.247,F0= 4.5×10−7 N andE0=F0/q= 9.0×102V/m. Although Sansbury observed a zeroth order force, he did not measure it. For comparison we present here the zeroth order force upon an electron ( q=−1.6×10−19C) and upon a typical charge generated by friction ( q≈10−6C) at the same distance from the same wire, 86 namely:F0= 4.6×10−26N andF0= 1.8 N, respectively. The huge diﬀerence between these forces arises from the fact that F0is proportional to the square ofq. The corresponding zeroth order electric ﬁelds due to the el ectron and to the charge generated by friction are given by E0= 2.9×10−7V/m and E0= 1.8×106V/m, respectively. We now consider the force F1and electric ﬁeld E1=F1/qproportional to the voltage of the battery. We consider only the radial compo nent along the ˆ ρ direction given by Eq. (6.21). This component depends upon t he values of the potentials at the extremities of the wire and also upon the va lue ofz. With the given symmetrical potentials, φL+φR= 0 V, then the radial components of F1and ofE1go to zero at z= 0. The maximal magnitudes of F1and ofE1 happen atz=±ℓ/2. At these locations, with q= 5×10−10C and with the given conditions we obtain: F1= 3.5×10−10N andE1= 0.69 V/m. As regards the second order eﬀect, we utilize Eq. (6.27). Wit hq= 5×10−10 C and the given conditions we obtain: F2= 1.2×10−15N. This yields E2= F2/q= 2.4×10−6V/m. Finally we can compare the three force components along the r adial direction (forF1we consider only the maximal value). Utilizing q= 5×10−10C we obtained:F0= 4.5×10−7N,F1= 3.5×10−10N andF2= 1.2×10−15N. The corresponding components of the electric ﬁeld were given by :E0= 9.0×102 V/m,E1= 6.9×10−1V/m andE2= 2.4×10−6V/m. This yields F0/F1= 1.3×103,F1/F2= 2.9×105,E0/E1= 1.3×103andE1/E2= 2.9×105. That is, in this case F0≫F1≫F2andE0≫E1≫E2. Similar order of magnitudes are obtained in the experiment o f Bartlett and Maglic [179]. To facilitate the detection of the force F1it would be better not to place any test charge close to the wire. Instead of this, it would be ideal to bring a small neutral conductor close to the wire. In principle it wo uld not act upon the wire. But when we pass a current upon the resistive wire, t his wire should become charged along its surface. Therefore, it should gene rate an electric ﬁeld E1outside it. This electric ﬁeld would then polarize the small conductor outside it. Consequently, there would arise an attraction between t he conductor and the current-carrying wire. We have already seen experiment s of this kind in Chapter 3. The second possibility in order to facilitate the detection of the force F1even in the presence of the force F0(in the case in which we approach a charged body to the current-carrying wire) would be to increase the volta ge of the battery connected to the wire. As F1is proportional to the emf of the battery, we can makeF1greater than F0working with high resistance wires connected to high voltages. We also saw experiments of this kind in Chapter 3. In many cases we will have F0≫F1≫F2. Despite this fact the force /vectorF1has already been observed in the laboratory, as we saw in Chap ter 3. We consider the current ﬂowing in the top part of a circuit like t hat of our Figure 6.1, with symmetrical potentials: φR=−φL. In order to compare these theoretical results with the experiments, we need to obtain the lines of e lectric ﬁeld. To obtain these lines we follow the approach presented in Somme rfeld’s book [208, 87 pp. 125-130] (German original from 1948 based on lectures de livered in 1933- 1934). We obtain this in the plane xz(y= 0). Any plane containing the zaxis will yield a similar solution. The lines of electric ﬁel d are orthogonal trajectories to the equipotential lines. As /vectorE=−∇φ, the electric ﬁeld points along the direction of the maximum space rate of change of φ. We are then looking for a function ξ(ρ,z) such that ∇ξ(ρ,z)· ∇φ(ρ,z) = 0. (6.33) Forρ < a we haveφas a linear function of z, such that ξcan be found proportional to ρ. We write it as ξ(ρ < a,z ) =−Aℓρ, withAas a constant. The equipotential lines, φ(ρ,z) = constant, can be written as z1(ρ) =K1, whereK1is a constant (for each constant we have a diﬀerent equipoten tial line). Analogously, the lines of electric ﬁeld will be given byz2(ρ) =K2, where K2is another constant (for each K2we have a diﬀerent line of electric ﬁeld). From Eq. (6.33) we get dz2/dρ=−1/(dz1/dρ) = (∂φ/∂z )/(∂φ/∂ρ ). Integrating this equation we can obtain ξ(ρ,z). With Eq. (6.9) this yields the solution for ρ>a. We are then led to: ξ(ρ,z) = (φR−φL)ρ ℓifρ<a , (6.34) ξ(ρ,z) = (φR+φL)z ℓ+ (φR−φL)/parenleftbiggρ2 2ℓ2+z2 ℓ2−ρ2 ℓ2lnρ ℓ/parenrightbigg ifρ>a . (6.35) From these equations we can easily verify Eq. (6.33). In order to compare these results with the experiments of Ber gmann, Schae- fer, Jeﬁmenko, Barnett and Kelly we need essentially the val ue ofℓ/a. From Figure 3.1 we get ℓ/a≈33, from Figure 3.3 we get ℓ/a≈13, while from Fig- ure 3.10 we get ℓ/a≈4. The plots of the equipotentials between z=−ℓ/2 andℓ/2 given by Eqs. (6.8) and (6.9) with these values of ℓ/aare given in Figures 6.3, 6.4 and 6.5 (with the experimental results of Be rgmann, Schaefer, Jeﬁmenko, Barnett and Kelly overlaid on them). Plots of the lines of electric ﬁeld given by Eqs. (6.34) and (6 .35) with these values ofℓ/aare given in Figures 6.6, 6.7 and 6.8 (with the experimental r esults of Bergmann, Schaefer, Jeﬁmenko, Barnett and Kelly overlai d on them). These theoretical Figures overlaid on the experimental one s indicate a very good agreement between theory and experiment. We now consider Sansbury’s experiment discussed in Chapter 3. The ob- served force was of the order of 10−7N, although he was not able to make precise measurements. He analyzed brieﬂy the possibility that this extra force might be the forceF1discussed here, but only considered the longitudinal elect ric ﬁeld outside the wire. He then concluded that this force would be t hree orders of magnitude smaller than the eﬀect he measured. However, he wa s not aware of the radial component of /vectorE1, which can be larger than the longitudinal compo- nent, as we showed here. Moreover, his U-shaped wire was bent close to the foil 88 Figure 6.3: Theoretical equipotential lines overlaid on th e experimental lines of electric ﬁeld obtained by Bergmann and Schaefer. Figure 6.4: Theoretical equipotential lines overlaid on th e experimental lines of electric ﬁeld obtained by Jeﬁmenko. and thus the approximation to a long straight wire may not be a pplicable. Close to a corner the electric ﬁeld outside the wire is even larger t han the longitudinal one inside it [167]. Possibly what Sansbury detected direct ly was the force F1 discussed here. It would be important to repeat his experime nt carefully taking this into account. In this Chapter we have seen a ﬁrst example in which the electr ic ﬁeld inside and outside a resistive wire carrying a steady current is due to charges spread along the surface of the conductor. The density of these surf ace charges is con- stant in time but varies along the length of the wire. It is pro portional to the voltage generated by the battery connected to the wire. As th e internal and ex- ternal electric ﬁeld is produced by charges at the surface of the wire, we can see a direct connection between electrostatics (represented b y Gauss’s law) and cir- cuit theory (represented by Ohm’s law). This allows a connec tion between these 89 Figure 6.5: Theoretical equipotential lines overlaid on th e experimental ones obtained by Jeﬁmenko, Barnett and Kelly. Figure 6.6: Theoretical lines of electric ﬁeld overlaid on t he experimental ones obtained by Bergmann and Schaefer. two topics which are usually considered separately in the te xtbooks. Despite this fact some authors have called attention to the strong co nnection between these two branches of electromagnetism, beginning with Web er and Kirchhoﬀ, as we see in the Appendices. Some modern scientists mention t he same aspect [209, 210, 211, 171] [166, Chapter 18: A Microscopic View of E lectric Circuits, pp. 623-666]. The example discussed here is important to show clearly the e xistence of an external electric ﬁeld proportional to the potential diﬀ erence acting upon the resistive wire, even in the case of a straight wire carryi ng a steady current. This electric ﬁeld does not depend upon a variable current (w ith a longitudinal acceleration of the electrons along the direction of the wir e), nor of a centripetal acceleration of the conduction electrons (due to any curvat ure in the wire). That is, this external electric ﬁeld will exist even when there is no acceleration of the 90 Figure 6.7: Theoretical lines of electric ﬁeld overlaid on t he experimental ones obtained by Jeﬁmenko. Figure 6.8: Theoretical lines of electric ﬁeld overlaid on t he experimental equipo- tential lines obtained by Jeﬁmenko, Barnett and Kelly. conduction electrons. 91 92 Chapter 7 Coaxial Cable 7.1 Introduction Many authors studied the distribution of surface charges in resistive coaxial cables carrying steady currents, as well as the potential an d electric ﬁeld inside and outside the conductors [212, pp. 175-184] [170] [208, pp . 125-130] [213] [214] [176, pp. 318 and 509-511] [12] [215] [16, pp. 336-337] [216] [217]. Here we present the main results in this conﬁguration consid ering the general case of a return conductor of ﬁnite area and ﬁnite conductivi ty. In this case there will be an electric ﬁeld outside the external return co nductor, although the magnetic ﬁeld goes to zero in this region. The conﬁguration of the problem is that of Figure 7.1. Figure 7.1: Conﬁguration of the problem. A constant current Iﬂows uniformly in the zdirection along the inner conductor (radius aand conductivity g1), returning uniformly along the outer conductor (internal and external radii bandc, respectively, and conductivity g3). The conductors have uniform circular cross-sections and a l engthℓ≫c>b>a centered on z= 0. The medium outside the conductors is considered to be air or vacuum. The potentials at the extremities located at z=ℓ/2 of the inner and outer conductors are maintained at the constant values φAandφB, respectively. The potentials at the extremities located at z=−ℓ/2 of the outer and inner 93 conductors are maintained at the constant values φCandφD, respectively. 7.2 Potentials and Fields We are interested in calculating the potentials and ﬁelds in a point/vector r= (ρ,ϕ,z ) such thatℓ≫ρandℓ≫ |z|, so that we can neglect edge eﬀects. All solutions presented here were obtained with this approximation. With this approximation and conﬁguration we then have the potential as a linear funct ion ofz. See Section 5.4. In order to have uniform currents ﬂowing in the zdirection along the inner and outer conductors, with a potential satisfying the given values at the extremities, we have: φ(ρ≤a,ϕ,z ) =φA+φD 2+ (φA−φD)z ℓ, (7.1) φ(b≤ρ≤c,ϕ,z) =φC+φB 2+ (φB−φC)z ℓ. (7.2) By Ohm’s law (with R1andR3being the resistances of the inner and outer conductors, respectively) we obtain: φD−φA=R1I=ℓI πg1a2, (7.3) φB−φC=R3I=ℓI πg3(c2−b2). (7.4) In the four regions ( ρ<a,a<ρ<b ,b<ρ<c andc<ρ) the potential φ satisﬁes Laplace’s equation ∇2φ= 0. By Eqs. (7.1) and (7.2) we have the value ofφin the ﬁrst and third regions, which also supply the boundary conditions atρ=aand atρ=bin order to ﬁnd φin the second region. To ﬁnd φin the fourth region we need another boundary condition, in additi on to the value of φatρ=c, which is given by Eq. (7.2). We then impose the following bou ndary condition: φ(ρ=ℓ,ϕ,z) = 0 V. (7.5) This is the main non-trivial boundary condition for this pro blem. The same reasoning was utilized in Section 6.2 after Eq. (6.21). This equation says that the potential goes to zero at a radial distance ρ=ℓ, so that the length ℓof the cable appears in the solution. The usual condition φ(ρ→ ∞,ϕ,z) = 0 V does not work in the situation considered here. We ﬁrst tried this las t condition but could not obtain a correct solution for the potential, and only dis covered Eq. (7.5) working backwards. That is, from the work of Russell we knew t hat in general the density of the surface charges on a system of long paralle l homogeneous conductors in steady-state (as is the case of the coaxial cab le being considered here) varies linearly with distance along the direction of t heir common axis [9]. That is, ifdrepresentsa,borc, the surface charge densities at these surfaces 94 must be given by σd(z) =Ad+Bdz, with the constants AdandBdcharacterizing each surface. We then obtained the potential at all points in space by φ(/vector r) =1 4πε03/summationdisplay j=1/integraldisplay /integraldisplay Sjσ(/vector rj)daj |/vector r−/vector rj|. (7.6) Here the sum goes over the three surfaces ρ=a,bandc, extending from z=−ℓ/2 toz=ℓ/2. After solving these integrals we discovered that φwent to zero not at inﬁnity, but at ρ=ℓ. Although this diﬀerence is important mathematically in order to arrive at a working solution, phy sically we can say that the potential going to zero at ρ=ℓis equivalent to it going to zero at inﬁnity. As we suppose ℓ≫c > b > a , we are essentially imposing that the potential goes to zero at a large distance from the cable, whi ch is reasonable. Here we reverse the argument, as this is more straightforwar d. That is, we begin with the boundary conditions for φ, obtaining the solutions of Laplace’s equation, the electric ﬁeld /vectorE=−∇φand thenσby Gauss’s law. The boundary conditions are then the values of φatρ=a,ρ=b,ρ=cand ρ=ℓ. They are given by Eqs. (7.1), (7.2) and (7.5). The solutions of Laplace’s equation ∇2φ= 0 fora≤ρ≤band forc≤ρin cylindrical coordinates satisfying these boundary conditions yield: φ(a≤ρ≤b,ϕ,z) =φB+φC 2+ (φB−φC)z ℓ +/bracketleftbiggφA+φD−φC−φB 2+ (φA−φD+φC−φB)z ℓ/bracketrightbiggln(b/ρ) ln(b/a), (7.7) φ(c≤ρ,ϕ,z ) =/bracketleftbiggφC+φB 2+ (φB−φC)z ℓ/bracketrightbiggln(ℓ/ρ) ln(ℓ/c). (7.8) The lines of electric ﬁeld are given by a function ξ(ρ, z) such that ∇ξ·∇φ= 0. By the procedure described in the previous Chapter we obtain ξ(ρ<a,ϕ,z ) =−(φA−φD)ρ ℓ, (7.9) ξ(a<ρ<b,ϕ,z ) =φA+φD−φC−φB 2z ℓ+φB−φC 2ρ2 ℓ2lnb a +φA−φD+φC−φB 2/parenleftbiggz2 ℓ2+ρ2 2ℓ2−ρ2 ℓ2lnρ b/parenrightbigg , (7.10) ξ(b<ρ<c,ϕ,z ) =−(φB−φC)ρ ℓ, (7.11) ξ(c<ρ,ϕ,z ) =φB+φC 2z ℓ+φB−φC 2/parenleftbiggz2 ℓ2+ρ2 2ℓ2−ρ2 ℓ2lnρ ℓ/parenrightbigg .(7.12) 95 The electric ﬁeld /vectorE=−∇φis given by /vectorE(ρ<a,ϕ,z ) =φD−φA ℓˆz , (7.13) /vectorE(a<ρ<b,ϕ,z ) =/bracketleftbiggφA+φD−φC−φB 2 + (φA−φD+φC−φB)z ℓ/bracketrightBig1 ln(b/a)ˆρ ρ +/bracketleftbiggφC−φB ℓ+φD−φA+φB−φC ℓln(b/ρ) ln(b/a)/bracketrightbigg ˆz , (7.14) /vectorE(b<ρ<c,ϕ,z ) =φC−φB ℓˆz , (7.15) /vectorE(c<ρ,ϕ,z ) =/bracketleftbiggφC+φB 2+ (φB−φC)z ℓ/bracketrightbigg1 ln(ℓ/c)ˆρ ρ +φC−φB ℓln(ℓ/ρ) ln(ℓ/c)ˆz . (7.16) The main points to be emphasized here are the solutions (7.8) and (7.16). They show the existence of an electric ﬁeld outside the resis tive cable even when it is carrying a constant current. Here we do not consider the motional electric ﬁeld proportio nal to second order invd/c. Its order of magnitude is much smaller than the one consider ed here (proportional to the potential diﬀerence along the cab le). For this reason we do not need to take it into account here. The surface charge densities σalong the inner conductor ( ρ=a,σa(z)) and along the inner and outer surfaces of the return conductor ( ρ=b,σb(z) and ρ=c,σc(z)) can be obtained easily utilizing Gauss’s law: /integraldisplay /ci∇cleco√y∇t/integraldisplay S/vectorE·d/vector a=Q ε0, (7.17) whered/vector ais the surface element pointing normally outwards the close d surface SandQis the net charge inside S. This yields σa(z) =ε0E2ρ(ρ→a,z), σb(z) =−ε0E2ρ(ρ→b,z) andσc(z) =ε0E4ρ(ρ→c,z), where the subscripts 2ρand 4ρmean the radial component of /vectorEin the second and fourth regions, a<ρ<b andc<ρ, respectively. This means that: σa(z) =ε0 a1 ln(b/a)/bracketleftbiggφA+φD−φC−φB 2+ (φA−φD+φC−φB)z ℓ/bracketrightbigg ,(7.18) 96 σb(z) =−a bσa(z), (7.19) σc(z) =ε0 c1 ln(ℓ/c)/bracketleftbiggφC+φB 2+ (φB−φC)z ℓ/bracketrightbigg . (7.20) An alternative way of obtaining φand/vectorEis to begin with the surface charges as given by Eqs. (7.18) to (7.20). We then calculate the elect ric potential φ(and /vectorE=−∇φ) through Eq. (7.6). We checked the calculations with this pr ocedure. 7.3 The Symmetrical Case In order to visualize the equipotentials and lines of electr ic ﬁeld we consider ℓ/c= 5,ℓ/b= 15/2 andℓ/a= 15. There are two main cases of interest, the symmetrical and asymmetrical cases. In the symmetrical cas e there are two equal batteries located at both extremities of the cable, Fi gure 7.2. Figure 7.2: The symmetrical case. They generate potentials φB=φD=−φA=−φC≡φ0/2. In this case the surface charge densities go to zero at the center of the cable (z= 0) in all three surfaces (ρ=a,bandc). The equipotentials and lines of electric ﬁeld for this situation are shown in Figures 7.3 and 7.4, respectively. Figure 7.3: Equipotential lines for the symmetrical case. 97 Figure 7.4: Lines of electric ﬁeld for the symmetrical case. In this case the potential is simply proportional to zwithout any additive constant. We can then write it in terms of the currents and con ductivities as given by: φ(ρ≤a) =−Iz πg1a2, (7.21) φ(a≤ρ≤b) =−I πz ln(b/a)/bracketleftbiggln(b/ρ) g1a2−ln(ρ/a) g3(c2−b2)/bracketrightbigg , (7.22) φ(b≤ρ≤c) =Iz πg3(c2−b2), (7.23) φ(c≤ρ) =I πln(ℓ/ρ) ln(ℓ/c)z g3(c2−b2). (7.24) Particular cases include an equipotential outer conductor (φC=φB= 0) with an inﬁnite area ( c→ ∞ ) or with an inﬁnite conductivity ( g3→ ∞ ). These solutions are recovered taking g3(c2−b2)→ ∞, such that σc(z)→0, /vectorE(ρ > b )→0 andφ(ρ≥b)→0 for anyz. The opposite solution when the current ﬂows in an inner conductor of inﬁnite conductivity, returning in an outer conductor of ﬁnite area and ﬁnite conductivity, is also easi ly obtained from the previous result, yielding /vectorE(ρ<a)→0 andφ(ρ≤a)→0 for anyz. 7.4 The Asymmetrical Case In the asymmetrical case there is a battery at the left extrem ity and a load resistanceRLat the right extremity, Figure 7.5. We can represent the potentials generated by the battery pro ducing a voltage φ0between its terminals as φD=−φC≡φ0/2. By Ohm’s law the total current 98 Figure 7.5: The asymmetrical case. Iis related to the total resistance Rt≡R1+RL+R2byI=φ0/Rt. Analogously: φD−φA=φ0(R1/Rt),φA−φB=φ0(RL/Rt) andφB−φC=φ0(R2/Rt). These results in Eqs. (7.1) to (7.12) yield: φ(ρ≤a,ϕ,z ) =φ0/parenleftbiggR2+RL 2Rt−R1 Rtz ℓ/parenrightbigg , (7.25) φ(a≤ρ≤b,ϕ,z) =−φ0/bracketleftbigg/parenleftbiggR1+RL 2Rt−R1+R2+ 2RL 2Rtln(b/ρ) ln(b/a)/parenrightbigg −/parenleftbiggR2 Rt−R1+R2 Rtln(b/ρ) ln(b/a)/parenrightbiggz ℓ/bracketrightbigg , (7.26) φ(b≤ρ≤c,ϕ,z) =−φ0/parenleftbiggR1+RL 2Rt−R2 Rtz ℓ/parenrightbigg , (7.27) φ(c≤ρ,ϕ,z ) =−φ0/parenleftbiggR1+RL 2Rt−R2 Rtz ℓ/parenrightbiggln(ℓ/ρ) ln(ℓ/c). (7.28) ξ(ρ<a,ϕ,z ) =φ0R1 Rtρ ℓ, (7.29) ξ(a<ρ<b,ϕ,z ) =φ0/bracketleftbiggR1+R2+ 2RL 2Rtz ℓ+R2 2Rtρ2 ℓ2lnb a −1 2/parenleftbiggz2 ℓ2+ρ2 2ℓ2−ρ2 ℓ2lnρ b/parenrightbigg/bracketrightbigg , (7.30) ξ(b<ρ<c,ϕ,z ) =−φ0R2 Rtρ ℓ, (7.31) ξ(c<ρ,ϕ,z ) =−φ0/bracketleftbiggR1+RL 2Rtz ℓ−R2 2Rt/parenleftbiggz2 ℓ2+ρ2 2ℓ2−ρ2 ℓ2lnρ ℓ/parenrightbigg/bracketrightbigg .(7.32) These Equations are plotted in Figures 7.6 and 7.7 when ℓ/c= 5,ℓ/b= 15/2, ℓ/a= 15 andR1=R2=RL. As we obtained algebraic solutions for the ﬁelds, potential s and surface charges, it is easy to apply them for commercial cables. In th is way we can know the orders of magnitude of these quantities for several standard cables. 99 Figure 7.6: Equipotential lines for the asymmetrical case. Figure 7.7: Lines of electric ﬁeld for the asymmetrical case . 7.5 Discussion The distribution of charges given by Eqs. (7.18) to (7.20) is equivalent to equal and opposite charges in the facing surfaces. That is, the cha rge at the position ρ=a,z, in a length dz,dqa(z) = 2πaσa(z)dz, is equal and opposite to the charge at the position ρ=b,z, in the same length dz:dqb(z) = 2πbσb(z)dz=−dqa(z). The electric ﬁeld outside the coaxial cable then depends onl y on the surface charges at the external wall of the return conductor, σc(z): φ(c≤ρ,ϕ,z ) =c ε0σc(z)lnℓ ρ=/bracketleftbiggφB+φC 2+ (φB−φC)z ℓ/bracketrightbiggln(ρ/ℓ) ln(c/ℓ).(7.33) The main nontrivial conclusions of this analysis are Eqs. (7 .16) and (7.33). They show that although there is no vector potential or magne tic ﬁeld outside 100 a coaxial cable, the electric ﬁeld will be diﬀerent from zero when there is a ﬁnite resistivity in the outer conductor. To our knowledge the ﬁrs t to mention this external electric ﬁeld outside a resistive coaxial cable wa s Russell in his impor- tant paper of 1983 [213]. The solution of this Chapter presen ts an analytical calculation of this ﬁeld. This external electric ﬁeld indicates that there is no shiel ding in a coaxial cable with a resistive outer conductor (sheath). It is impor tant to realize this specially when dealing with interferences in telecommunic ation systems. Even with a long cable there will be this external electric ﬁeld, a s can be seen from Eq. (7.33). For this reason this resistive cable will inﬂuen ce other electrical systems nearby. This ﬁeld will be present even for variable c urrent. This is a relevant aspect neglected by most authors. 101 102 Chapter 8 Transmission Line 8.1 Introduction One of the most important electrical systems is that of a two- wire transmission line, usually called twin-leads. We consider here homogene ous resistive wires ﬁxed in the laboratory and carrying steady currents. The goa l here is to calculate the electric ﬁeld outside the wires. The case of twin-leads was ﬁrst considered by Stratton [218, p. 262]. Al- though he called attention to the electric ﬁeld outside the t ransmission line, this has been forgotten by most authors, as we have seen. We treate d this case in more detail in 1999 [219] and here we follow this latter appro ach. These are the only theoretical works dealing with this conﬁguration know n to us. 8.2 Two-Wire Transmission Line The conﬁguration of the system is given in Figure 8.1. We have two equal straight wires of circular cross-sections of radiiaand lengthℓ, surrounded by air. Their axes are separated by a distance band are parallel to the zaxis, symmetrically located relative to the zandxaxes. That is, the centers of the wires are located at ( x, y, z ) = (−b/2,0,0) and (+b/2,0,0). The conductivity of the wires is gand their extremities are located at z=−ℓ/2 andz= +ℓ/2. Here we calculate the electric potential φand the electric ﬁeld /vectorEat a point ( x, y, z ) such that ℓ≫r=/radicalbig x2+y2+z2. Moreover, we also assume that ℓ≫b/2>a, so that we can neglect edge eﬀects. We want to ﬁnd the potential and electric ﬁeld when a current Iﬂows uni- formly through one of the wires along the direction +ˆ zand returns uniformly through the other wire along the direction −ˆz. The current densities in both wires are then given by /vectorJ= (I/πa2)ˆzand/vectorJ=−(I/πa2)ˆz, respectively. As we are considering homogeneous wires with a constant conducti vityg, Ohm’s law yields the internal electric ﬁeld in the wires as /vectorE=±(I/gπa2)ˆz. We do not need to consider in /vectorEthe inﬂuence of the time variation of the vector potential 103 Figure 8.1: Two homogeneously resistive parallel wires of r adiiaseparated by a distanceb. The ﬁrst wire carries a steady current Ialong the positive zdirection while the second wire carries the return current Ialong the negative zdirection. as we are dealing with a steady current in stationary wires, s o that∂/vectorA/∂t=/vector0 everywhere. We can then write /vectorE=−∇φ. As we have a constant electric ﬁeld in each wire, this implies that the potential is constant ove r each cross-section and a linear function of z. In this work we consider a symmetrical situation for the potentials so that in the ﬁrst wire the current ﬂows from t he potential φLat z=−ℓ/2 toφRatz=ℓ/2 and returns in the second wire from −φRatz=ℓ/2 to−φLatz=−ℓ/2, Figure 8.1. We can then write: φF(z) =φR+φL 2+ (φR−φL)z ℓ=φR+φL 2+I gπa2z , (8.1) φS(z) =−φF(z). (8.2) In these equations φF(z) andφS(z) are the potentials as a function of zover the cross-section of the ﬁrst and second conductors, respec tively. In this Chapter we neglect the small Hall eﬀect due to the azim uthal mag- netic ﬁeld generated by these currents. See Section 6.4. Thi s eﬀect creates a redistribution of the charge density within the wires, and m odiﬁes the surface charges also. As these are usually small eﬀects, they will no t be considered here. We now ﬁnd the potential in space supposing there is air outsi de the conduc- tors. As the conductors are straight and the boundary condit ions (the potentials over the surface of the conductors) are linear functions of z, the same must be valid everywhere, as we saw in Section 5.4. That is, φ= (A+Bz)f(x, y), where AandBare constants and f(x, y) is a function of xandy. This function can be found by the method of images, imposing a constant potenti alφ0over the ﬁrst wire and −φ0over the second one [13, Section 2.1]. The ﬁnal solution for φand/vectorEsatisfying the given boundary conditions, valid for the reg ion outside the wires, is given by: 104 φ(x, y, z ) =−/parenleftbiggφR+φL 2+ (φR−φL)z ℓ/parenrightbigg1 2 lnb−√ b2−4a2 2a ×ln(x−√ b2−4a2/2)2+y2 (x+√ b2−4a2/2)2+y2, (8.3) /vectorE=−/parenleftbiggφR+φL 2+ (φR−φL)z ℓ/parenrightbigg√ b2−4a2 lnb+√ b2−4a2 2a ×(x2−y2+a2−b2/4)ˆx+ 2xyˆy D4 1 +φR−φL ℓ1 2 lnb−√ b2−4a2 2a/bracketleftBigg ln(x−√ b2−4a2/2)2+y2 (x+√ b2−4a2/2)2+y2/bracketrightBigg ˆz , (8.4) where: D4 1≡x4+y4+b4/16 +a4+ 2x2y2−b2x2/2 + 2a2x2+b2y2/2−2a2y2−b2a2/2. (8.5) The equipotentials at z= 0 are plotted in Figure 8.2. Figure 8.2: Equipotentials in the plane z= 0. It is also relevant to express these results in cylindrical c oordinates ( ρ, ϕ, z ) centered on the ﬁrst and second wires. See Figure 8.3. 105 Figure 8.3: Cylindrical coordinates centered on the ﬁrst an d second wires. For the ﬁrst wire this can be accomplished replacing xbyρFcosϕF−b/2, ybyρFsinϕF, ˆxby ˆρFcosϕF−ˆϕFsinϕFand ˆyby ˆρFsinϕF+ ˆϕFcosϕF, yielding: φ(ρF, ϕF, z) =−/parenleftbiggφR+φL 2+ (φR−φL)z ℓ/parenrightbigg1 2 lnb−√ b2−4a2 2aln/radicalBigg D2 2 D2 3,(8.6) where: D2 2≡ρ2 F−ρF(cosϕF)/parenleftBig b+/radicalbig b2−4a2/parenrightBig +b2 2−a2+b√ b2−4a2 2,(8.7) and D2 3≡ρ2 F−ρF(cosϕF)/parenleftBig b−/radicalbig b2−4a2/parenrightBig +b2 2−a2−b√ b2−4a2 2.(8.8) The electric ﬁeld is then given by: /vectorE=−/parenleftbiggφR+φL 2+ (φR−φL)z ℓ/parenrightbigg√ b2−4a2 lnb+√ b2−4a2 2a ×(ρ2 FcosϕF−ρFb+a2cosϕF)ˆρF+ (sinϕF)(ρ2 F−a2)ˆϕF D4 4 +φR−φL ℓ1 2 lnb−√ b2−4a2 2a ×/bracketleftBigg lnρ2 F−ρF(cosϕF)(b+√ b2−4a2) +b2/2−a2+b√ b2−4a2/2 ρ2 F−ρF(cosϕF)(b−√ b2−4a2) +b2/2−a2−b√ b2−4a2/2/bracketrightBigg ˆz ,(8.9) where: 106 D4 4≡ρ4 F−2ρ3 FbcosϕF+ρ2 Fb2+a4 + 2ρ2 Fa2(cos2ϕF−sin2ϕF)−2ρFba2cosϕF. (8.10) The density of surface charges over the ﬁrst and second wires ,σFandσS, can then be found by ε0times the radial component of the electric ﬁeld over the surface of each cylinder, yielding: σF=/parenleftbiggφR+φL 2+ (φR−φL)z ℓ/parenrightbiggε0 2alnb+√ b2−4a2 2a√ b2−4a2 b/2−acosϕF,(8.11) σS=−/parenleftbiggφR+φL 2+ (φR−φL)z ℓ/parenrightbiggε0 2alnb+√ b2−4a2 2a√ b2−4a2 b/2 +acosϕS.(8.12) In order to check these results we calculated the potential φinside each wire and in space, beginning with these surface charge densities and utilizing: φ(x, y, z ) =1 4πε0/bracketleftBigg/integraldisplayℓ/2 z′=−ℓ/2/integraldisplay2π ϕ′ F=0σF(ϕ′ F)adϕ′ Fdz′ |/vector r−/vector r′| +/integraldisplayℓ/2 z′=−ℓ/2/integraldisplay2π ϕ′ S=0σS(ϕ′ S)adϕ′ Sdz′ |/vector r−/vector r′|/bracketrightBigg . (8.13) Here we integrate over the surfaces of the ﬁrst and second cyl inders,SLand SR, respectively. We can then check these results assuming the correctness of the method of images for the electrostatic problem and utilizin g the approximations ℓ≫ |/vector r|andℓ≫b/2>a. Withb≫aandb≫ρF, Eqs. (8.11) and (8.6) yield: σF≈ε0 aln(b/a)/parenleftbiggφR+φL 2+ (φR−φL)z ℓ/parenrightbigg , (8.14) and φ(a<ρF≪b, ϕF, z)≈aσF(z) ε0lnb ρF. (8.15) These results are analogous to Eqs. (6.2) and (6.9). 107 8.3 Discussion The ﬁrst aspect to be discussed here is the qualitative inter pretation of these results. In all this Section we will assume φR= 0 in order to simplify the analysis. The distribution of surface charges for a given zis similar to the distribution of charges in the electrostatic problem given the potentials φ0and −φ0at the ﬁrst and second wires, without current. That is, σF(ϕF)>0 for any ϕFand its maximum value is at ϕF= 0 rad. The density of surface charges at the second wire, σS, has the same behaviour of σFwith an overall change of sign, with its maximum magnitude occurring at ϕS=πrad. A qualitative plot of the surface charges at z= 0 is given in Figure 8.4. Figure 8.4: Qualitative distribution of surface charges fo r two parallel wires in the planez= 0. A quantitative plot of σFis given in Figure 8.5 supposing b/2a= 10/3 and normalizing the surface charge density by the value of σFatϕF=πrad. Figure 8.5: Quantitative normalized distribution of the de nsity of surface charges in the ﬁrst wire in the plane z= 0 as a function of the azimuthal angle. It should also be remarked that for a ﬁxed ϕFthe surface density decreases linearly from z=−ℓ/2 toz=ℓ/2, the opposite happening with σSfor a ﬁxed ϕS. We can integrate the surface charges over the circumference of each wire, obtaining the integrated charge per unit length λ(z) as: 108 λF(z) =/integraldisplay2π ϕF=0aσF(ϕF)dϕF =−2πε0 ln [(b−√ b2−4a2)/2a]/bracketleftbiggφR+φL 2+ (φR−φL)z ℓ/bracketrightbigg . (8.16) λS(z) =−/integraldisplay2π ϕS=0aσS(ϕS)dϕS=−λF(z). (8.17) One important aspect to discuss is the experimental relevan ce of these sur- face charges in terms of forces. That is, as the wires have a ne t charge in each section, there will be an electrostatic force acting on them . We can then com- pare this force with the magnetic force. The latter is given e ssentially by (force per unit length): dFM dz=µ0I2 2πb, (8.18) where we are supposing b/2≫a. We now calculate the electric force per unit length on the ﬁrs t wire, integrat- ing the force over its circumference. We consider a typical r egion in the middle of the wire, around z= 0, and once more suppose b/2≫a: d/vectorFE dz=/integraldisplay2π ϕF=0aσF(ϕF)/vectorE(ρF=a,ϕF,z= 0)dϕF≈πε0φ2 L ln2b/a/parenleftbiggˆx b+ˆz ℓ/parenrightbigg .(8.19) From Eqs. (8.18) and (8.19) the ratio of the magnetic to the ra dial elec- tric force is given by (with Ohm’s law φ2 L/I2=R2= (ℓ/gπa2)2,Rbeing the resistance of each wire): FM FE≈µ0/ε0 2R2ln2b a. (8.20) Asµ0/ε0= 1.4×105Ω2this ratio will be usually many orders of magnitude greater than 1. This would be of the order of 1 when R≈370 Ω (supposing ln(b/a)≈1). This is a very large resistance for homogeneous wires. In order to compare this force with the magnetic force we supp ose typical copper wires of conductivities g= 5.7×107m−1Ω−1, lengthsℓ= 1 m, separated by a distance b= 6 mm and diameters 2 a= 1 mm. This means that by Ohm’s lawφ2 L/I2=R2≈5×10−4Ω2. With these values the ratio of the longitudinal electric force to the magnetic force is of the order of 7 ×10−11, while the ratio of the radial electric force to the magnetic force is of the orde r of 1×10−8. That is, the electric force between the wires due to these surface charges is typically 10−8times smaller than the magnetic force. This shows that we can usually neglect these electric forces. 109 Despite this fact it should be remarked that while the magnet ic force is re- pulsive in this situation (parallel wires carrying current s in opposite directions), the radial electric force is attractive, as we can see from th e charges in Figure 8.4. The situation described in this Chapter is very similar to th e experiments performed by Bergmann, Schaefer and Jeﬁmenko, whose result s are presented in Figures 3.2 and 3.5. We can compare these experiments with the theoretical calculations by plotting the equipotentials obtained here . We need essentially the values of ℓ/b,b/2aandℓ/2a. From Fig. 3.2 we obtain ℓ/b≈2.8,b/2a≈7.4 andℓ/2a≈20.7. From Fig. 3.5 we have ℓ/b≈1.9,b/2a≈3.0 andℓ/2a≈5.7. These values together with φA= 0 V and φB= 1 V yielded the equipotentials given by Eq. (8.3) at y= 0, Figures 8.6 and 8.7. Figure 8.6: Theoretical equipotential lines overlaid on th e experimental lines of electric ﬁeld obtained by Bergmann and Schaefer. The lines of electric ﬁeld orthogonal to the equipotentials can be obtained by the procedure described in Sommerfeld’s book, discussed in Section 6.5. This yields the following solutions in the plane y= 0 outside the wires: ξout(x,0,z) =−(φR+φL)z ℓ+ (φR−φL)/bracketleftbiggx(x2−3x2 o) 6xoℓ2ln(x−xo)2 (x+xo)2 +x2 o 3ℓ2ln(x−xo)2(x+xo)2 x4o−x2 3ℓ2−z2 ℓ2/bracketrightbigg , (8.21) wherexo≡√ b2−4a2/2. The lines of electric ﬁeld inside the ﬁrst and second wires ca n be written as, respectively: ξF(x,0,z) =−(φR−φL)|x+b/2| ℓ, (8.22) 110 Figure 8.7: Theoretical equipotential lines overlaid on th e experimental lines of electric ﬁeld obtained by Jeﬁmenko. ξS(x,0,z) = (φR−φL)|x−b/2| ℓ. (8.23) With the previous values of ℓ/b,b/2aandℓ/2afor the two experiments already mentioned we obtain the lines of electric ﬁeld by the se equations as given in Figures 8.8 and 8.9 (with Figure 3.2 and the left side of Figure 3.5 overlaid on them). Figure 8.8: Theoretical lines of electric ﬁeld overlaid on t he experimental lines obtained by Bergmann and Schaefer. These numerical plots are very similar to the experiments, e specially in the region between the wires. Although this calculation is stri ctly valid only for r≪ℓ, the numerical plots go from z=−ℓ/2 toℓ/2. As the result is in rea- 111 Figure 8.9: Theoretical lines of electric ﬁeld overlaid on t he experimental lines obtained by Jeﬁmenko. sonable agreement with the experiments, we conclude that th e exact boundary conditions at z=±ℓ/2 are not very relevant in these particular conﬁgurations. We can also estimate the ratio of the radial component of the e lectric ﬁeld to the axial component just outside the wire. We consider the ﬁrst wire at three diﬀerent values of z:z=−ℓ/2,z= 0 andz=ℓ/2. The axial component Ezis constant over the cross-section and does not depend on z. On the other hand, the radial component Exis a linear function of zand also depends on ϕF. In this comparison we consider ϕF= 0. With these values and Jeﬁmenko’s data in Eq. (8.4) we obtain Ex/Ez≈12 atz=−ℓ/2, 6 atz= 0 and 0 at z=ℓ/2. That is, the radial component of the electric ﬁeld just outside th e wire is typically one order of magnitude larger than the axial electric ﬁeld re sponsible for the current. Jeﬁmenko’s experiment gives a clear conﬁrmation o f this fact. 112 Chapter 9 Resistive Plates 9.1 Introduction In this Chapter we consider one or more resistive plates carr ying steady currents. We consider an ideal case of an inﬁnite resistive bidimensio nal plate (like an inﬁnite plane). The current is supposed to ﬂow uniformly ove r the plate along a straight direction. When there is no current ﬂowing in the conducting plate and we approximate a test charge, waiting until electrostatic equilibrium is r eached, with the test charge at a distance zfrom the plate, there will be an attraction between the plate and the charge given by Eq. (4.1). What happens when we now pass a constant current through the s tationary resistive plate connected to a battery? The electric ﬁeld th at maintains the current against Ohmic resistance is generated by a surface c harge distribution on the plate. Our goal is to calculate the potential and elect ric ﬁeld over the plate and in the space surrounding it when the plate carries a steady current. The subject of this Chapter was ﬁrst discussed by Jeﬁmenko [1 76, pp. 303- 304], and later by other authors [220, 221]. 9.2 Single Plate We consider the case of conducting plates from the point of vi ew of surface charge distributions generating the electric ﬁelds. The conﬁguration we are considering is that of a rectangular plate of length ℓyin theydirection and ℓzin thezdirection. The plate is located in the x= 0 plane with its center at ( x, y, z ) = (0,0,0). We assume that the current Iﬂows uniformly from −ℓz/2 to +ℓz/2 with a surface current density /vectorK= (I/ℓy)ˆz, Figure 9.1. We also assume that the surface charge density is linear along z, as we saw in Section 5.4: σ(z) =σA+σBz ℓz. (9.1) 113 Figure 9.1: A resistive plate in the plane x= 0 with a steady and uniform surface current density /vectorKalong the positive zdirection. Note that the surface charge should in general be a function o f theyandz coordinates, σ=σ(y, z). We neglect the dependence on yas an approximation forℓy≫ |/vector r|, where |/vector r|=/radicalbig x2+y2+z2is the distance from the observation point to the center of the plate. Moreover, we consider that t he test charge is far from the battery. The case in which the test charge or the o bservation point is close to the battery, in analogy with the case of a test char ge close to the middle point of side ADof Figure 6.2, was considered in 2005 [222]. In order to generate such steady and uniform (that is, indepe ndent of the variabley) longitudinal current along an inﬁnite plate, the ideal bat tery driving this current can be thought as an inﬁnite straight line along the plate and orthogonal to the direction of the current. The electric potential is readily given from the surface cha rgeσ(z) by: φ(/vector r) =1 4πε0/integraldisplay /integraldisplayσ(z′)da′ |/vector r−/vectorr′|. (9.2) This integral should be evaluated over the whole charge dist ribution. We are interested in the potential at the symmetric plane y= 0: φ(x,0, z) =1 4πε0/integraldisplayℓy/2 −ℓy/2/integraldisplayℓz/2 −ℓz/2σA+σBz′/ℓz/radicalbig x2+y′2+ (z−z′)2dy′dz′. (9.3) We solve these integrals utilizing three diﬀerent approxim ations: (A)ℓy≫ℓz≫/radicalbig x2+z2, (9.4) (B)ℓ≡ℓy=ℓz≫/radicalbig x2+z2, (9.5) (C)ℓz≫ℓy≫/radicalbig x2+z2. (9.6) For each case the potential is given by, respectively: φ(ℓy≫ℓz)≈σ(z) 2ε0/parenleftbiggℓz π− |x|/parenrightbigg +σA 2ε0ℓz πln2ℓy ℓz, (9.7) 114 φ(ℓy=ℓz≡ℓ)≈σ(z) 2ε0/parenleftbigg2ℓ πln(√ 2 + 1)− |x|/parenrightbigg +σA 2ε0ℓ πln(√ 2 + 1),(9.8) φ(ℓz≫ℓy)≈σ(z) 2ε0/parenleftbiggℓy πln2ℓz ℓy− |x|/parenrightbigg +σA 2ε0ℓy π. (9.9) For each approximation we deﬁne the constants λ1andλ2by the expressions: (A)λ1≡ℓz 2π, λ2≡ℓz 2πln2ℓy ℓz≫λ1, (9.10) (B)λ1≡ℓ 2πln(√ 2 + 1), λ2≡ℓ 2πln(√ 2 + 1) =λ1, (9.11) (C)λ1≡ℓy 2πln2ℓz ℓy, λ2≡ℓy 2π≪λ1. (9.12) The constants λ1andλ2have dimensions of length, are typically of the order of magnitude of the width or length of the plates, and are much larger than the distance to the point of interest r=√ x2+z2. With these constants we can write the electric potential for this single plate in the three given cases (A), (B) and (C) as: φ(x,0, z) =1 ε0/bracketleftbigg/parenleftbigg σA+σBz ℓz/parenrightbigg/parenleftbigg λ1−|x| 2/parenrightbigg +σAλ2/bracketrightbigg . (9.13) The electric ﬁeld /vectorE=−∇φis given by: /vectorE(x,0, z) =±1 ε0/bracketleftbiggσA+σBz/ℓz 2ˆx∓σBλ1− |x|/2 ℓzˆz/bracketrightbigg , (9.14) where the top (bottom) sign is for x>0 (x<0). In order to test the coherence of this procedure we invert the argument. Applying Gauss’s law to a small cylinder centered on the plat e we obtain the usual boundary condition relating the normal component of t he electric ﬁeld, Ex, to the surface charge density, σ, namely:ε0Ex(limx→0+)−ε0Ex(limx→ 0−) =σ(z). And this yields exactly the same charge distribution on th e plate as that given by the starting point, Eq. (9.1). We checked the calculations by a similar procedure in the other cases of two and four plates. The equipotentials given by Eq. (9.13) are shown in Figure 9. 2 in approxi- mation (A) with ℓy/ℓz= 3,φ(0,0,−ℓz/2) =φ0/2 andφ(0,0,ℓz/2) =−φ0/2. In this caseσB=−2πε0φ0/ℓzandσA= 0. The lines of electric ﬁeld are given by a function ξ(x,0,z) such that ∇ξ·∇φ= 0. Following the procedure described in Section 6.5 we obtai n in this case: ξ(x,0, z) =2σAz+σB(4λ1x−x2+z2)/ℓz ε0,ifx>0, (9.15) 115 Figure 9.2: Equipotential lines in a plane orthogonal to the plate. The arrows indicate the direction of the current. ξ(x,0, z) =2σAz−σB(4λ1x+x2−z2)/ℓz ε0,ifx<0. (9.16) This function presents a family of two hyperbolas in the regi ons above and below the plate. An example of this function ξis presented in Figure 9.3 in approximation (A) with ℓy/ℓz= 3,φ(0,0,−ℓz/2) =φ0/2 andφ(0,0,ℓz/2) = −φ0/2. Figure 9.3: Lines of electric ﬁeld in a plane orthogonal to th e plate. The arrows indicate the direction of the current. 9.3 Two Parallel Plates We now consider the experiments of Bergmann, Schaefer and Je ﬁmenko utilizing a diﬀerent model. We ﬁrst consider a single straight conduct or, Figures 3.2 and 3.3. Here we model these cases as that of a constant current ﬂo wing uniformly along thezaxis of a conductor of conductivity gin the form of a parallelepiped of lengthsℓy, 2aandℓz. Accordingly there will be free charges only along its outer surfaces located at x=±a(considering the thick conductor centered at (x, y, z ) = (0,0,0)). At both sides the free charges will be given by Equa- tion (9.1). The superposition of the two charged planes situ ated inx=aand x=−a, utilizing Eq. (9.13) and replacing xbyx±aappropriately yields the potential in the plane y= 0 as given by: φ(x,0, z) =1 ε0/bracketleftbigg/parenleftbigg σA+σBz ℓz/parenrightbigg/parenleftbigg 2λ1−|x−a|+|x+a| 2/parenrightbigg + 2σAλ2/bracketrightbigg .(9.17) 116 This potential can be seen in Figure 9.4 in approximation (A) withℓy/ℓz= ℓz/2a= 6.5. With the boundary conditions φ(±a,0,−ℓz/2) =φ0/2 and φ(±a,0, ℓz/2) =−φ0/2 we haveσB=−φ0ε0/(4λ1−2a) andσA= 0. Figure 9.4: Equipotential lines in a plane orthogonal to two parallel plates carrying steady currents along the positive zdirection. The electric ﬁeld is readily given by /vectorE=−∇φ: /vectorE(x>a, 0, z) =1 ε0/bracketleftbigg/parenleftbigg σA+σBz ℓz/parenrightbigg ˆx−σB2λ1−x ℓzˆz/bracketrightbigg , (9.18) /vectorE(−a<x<a, 0, z) =−1 ε0σB2λ1−a ℓzˆz , (9.19) /vectorE(x<−a,0, z) =−1 ε0/bracketleftbigg/parenleftbigg σA+σBz ℓz/parenrightbigg ˆx−σB2λ1+x ℓzˆz/bracketrightbigg . (9.20) As expected, the electric ﬁeld is constant in the region betw een the two plates. This fact allows us to utilize the situation of two pl ates to model also the parallelepiped of sides ℓyand 2acarrying a steady current along the z direction. The two plates already mentioned would be equiva lent to the top and bottom plates of the parallelepiped located in the plane sx=±a. The lines of electric ﬁeld ξ(x,0, z) such that ∇ξ· ∇φ= 0 can be obtained by the method described before. They are given by the followi ng equation: ξ(x,0, z) =  (2σAz+σB(4λ1x−x2+z2)/ℓz)/ε0, x>a , −σBax/ℓzε0, −a<x<a , (2σAz−σB(4λ1x+x2−z2)/ℓz)/ε0, x< −a .(9.21) In Figure 9.5 we plot this function with the approximation ℓy/ℓz=ℓz/2a= 6.5, in order to have similar dimensions as in Jeﬁmenko’s exper iment. This theoretical Figure is similar to Jeﬁmenko’s experimental o ne, Figure 3.3. 9.4 Four Parallel Plates We now wish to obtain plots similar to Figures 3.5 and 3.2 util izing the paral- lelepiped model of this Chapter. We have essentially a trans mission line in which 117 Figure 9.5: Lines of electric ﬁeld in a plane orthogonal to tw o parallel plates carrying steady currents along the positive zdirection. the current ﬂows uniformly along the zaxis of a parallelepiped of conductivity g1and thickness 2 a, returning uniformly along another parallel parallelepip ed of the same thickness but conductivity g2. The centers of the two conductors are separated by a distance b. In this case there will be free charges in the four planes situated at y=b/2±aandy=−b/2±a, withb/2>a> 0. 9.4.1 Opposite Potentials In this case both conductors have the same ﬁnite conductivit yg1=g2=g. We assume that the potentials are exactly opposite in the two th ick plates, for any z. The densities of surface charges for the plates located at x=±(b/2 +a) and x=±(b/2−a) are given by: σ(x=±(b/2 +a), y, z ) =±/parenleftbigg σAext+σBextz ℓz/parenrightbigg , (9.22) σ(x=±(b/2−a), y, z ) =±/parenleftbigg σAint+σBintz ℓz/parenrightbigg . (9.23) We can obtain the potential utilizing Eq. (9.3). To simplify the results we deﬁne two dimensionless constants with appropriate values for each one of the approximations (Eq. (9.4) to (9.6)), namely: (A)κ1≡4b−8a πℓz−4b+ 8a, κ2≡2b−4a πℓz−2b+ 4a, (9.24) (B)κ2≡3√ 2(b−2a) πℓz−3√ 2(b−2a), κ2≡2√ 2(b−2a) πℓz−2√ 2(b−2a), (9.25) (C)κ1≡2b−a πℓz−2b+a, κ2≡2b−a π(πℓz−2b+a). (9.26) With the given approximations we have κ1≪1 andκ2≪1. 118 In order to model the given experiments, the potential shoul d not depend onxin the regions b/2−a<x<b/ 2 +aand−b/2−a<x< −b/2 +a(as the current ﬂows only along the zdirection in these regions). This yields σAint=σAext κ2≡σA, σBint=σBext κ1≡σB. (9.27) The potential is then given by (in the plane y= 0 and in the following regions, respectively: x > b/ 2 +a, b/2−a < x < b/ 2 +a,−b/2 +a < x < b/2−a,−b/2−a<x< −b/2 +a, x< −b/2−a): φ=  ((b−2a)(σA+σBz/ℓz) + (b+ 2a−y)(σAκ2+σBzκ1/ℓz))/2ε0, (b−2a)(σA+σBz/ℓz)/2ε0, y(σA+σBz/ℓz)/ε0, −(b−2a)(σA+σBz/ℓz)/2ε0, −((b−2a)(σA+σBz/ℓz) + (b+ 2a+y)(σAκ2+σBzκ1/ℓz))/2ε0. (9.28) This potential can be seen in Figure 9.6 in approximation (A) withℓy/ℓz= ℓz/2a= 6.8. Figure 9.6: Equipotential lines in a plane orthogonal to the four plates. There is a current along the positive (negative) zdirection in the two top (bottom) plates. The electric ﬁeld /vectorE(x,0, z) =−∇φis given in the ﬁve regions by, respec- tively: /vectorE=  ((σAκ2+σBzκ1/ℓz)ˆx−σB[b−2a+κ1(b+ 2a−2x)]ˆz/ℓz)/2ε0, −(b−2a)σBˆz/2ε0ℓz, −((σA+σBz/ℓz)ˆx+xσBˆz/ℓz)/ε0, (b−2a)σBˆz/2ε0ℓz, ((σAκ2+σBzκ1/ℓz)ˆx+σB[b−2a+κ1(b+ 2a+ 2x)]ˆz/ℓz)/2ε0. (9.29) 119 The lines of electric ﬁeld, ξ(x,0, z), are given for each region in Eq. (9.30): ξ=  (2σAzκ2/κ1+σB[[b+ 2a+ (b−2a)/κ1]x−x2+z2]/ℓz)/ε0, −σB(b−2a)x/2ℓzε0, (2σAz−σB(x2−z2)/ℓz)/ε0, σB(b−2a)x/2ℓzε0, (2σAzκ2/κ1−σB[[b+ 2a+ (b−2a)/κ1]x−x2+z2]/ℓz)/ε0.(9.30) In Figure 9.7 we plot this function in the approximation (A) w ithℓy/ℓz= ℓz/2a= 6.8. The upper plate has the potential at its boundaries given b y φ(b/2−a < x < b/ 2 +a,0,−ℓz/2) =φ0/2 andφ(b/2−a < x < b/ 2 + a,0, ℓz/2) = 0, while the lower plate has the potential at its boundari es given byφ(−b/2−a < x < −b/2 +a,0,−ℓz/2) =−φ0/2 andφ(−b/2−a < x < −b/2 +a,0, ℓz/2) = 0. The relation between φ0and the surface charges for this case is given by σA=ε0φ0/2(b−2a) andσB=−ε0φ0/(b−2a). Figure 9.7: Lines of electric ﬁeld in a plane orthogonal to th e four plates. There is a current along the positive (negative) zdirection in the two top (bottom) plates. 9.4.2 Perfect Conductor Plate Now, suppose that the two lower plates (or the lower parallel epiped) are a perfect conductor, with zero resistivity. That is, suppose they are subjected to the same constant potential φ(−b/2−a < x < −b/2 +a,0, z) = Φ in the whole extension along the zaxis, but still conducting a steady current. This experimental result is shown at the right side of Figure 3.5 w ithg1≪g2. To model this case we consider four plates located at x=b/2 +a,x=b/2−a, x=−b/2+aandx=−b/2−a. Their surface charges are given by, respectively, σ(x=b/2 +a, y, z ) =σAb+σBbz/ℓz,σ(x=b/2−a, y, z ) =σAa+σBaz/ℓz, 120 σ(x=−b/2 +a, y, z ) =σ−Aa+σ−Baz/ℓzandσ(x=−b/2−a, y, z ) = σ−Ab+σ−Bbz/ℓz. The potential must not depend on xin the region b/2−a < x < b/ 2 +a, and must be a constant in the region −b/2−a<x< −b/2 +a. From this we ﬁnd: σAa=σAb(4λ1+ 4λ2−b−2a)−2Φε0 b−2a, σBa=σBb4λ1−b−2a b−2a, σ−Aa=−σAa, σ −Ba=−σBa, σ−Ab=σAb, σ −Bb=σBb. (9.31) With Eq. (9.13) and the appropriate replacements of xbyx±(b/2±a) we get in the ﬁve regions, respectively: φ(x,0, z) =  [(σAb+σBbz/ℓz)(4λ1−b/2−a−y) + 4λ2σAb]/ε0−Φ, (2b−4a)(σAa+σBaz/ℓz)/ε0+ Φ, (σAa+σBaz/ℓz)(b/2−a+y)/ε0+ Φ, Φ, (σAb+σBbz/ℓz)(b/2 +a+y)/ε0+ Φ. (9.32) The equipotentials are shown in Figure 9.8 in approximation (A) withℓy/ℓz= ℓz/2a= 6.8. Figure 9.8: Equipotential lines in a plane orthogonal to the four plates. The two top plates are uniformly resistive and carry currents al ong the positive z direction. The two bottom plates have zero resistivity and c arry currents along the negative zdirection. The electric ﬁeld in these ﬁve regions is given by, respectiv ely: 121 /vectorE=  [(σAb+σBbz/ℓz)ˆx−σBb(4λ1−b/2−a−x)ˆz/ℓz]/ε0, −(b−2a)σBaˆz/ℓzε0, −[(σAa+σBaz/ℓz)ˆx+σBa(b/2−a+x)ˆz/ℓz]/ε0, /vector0, −[(σAb+σBbz/ℓz)ˆx+σBb(b/2 +a+x)ˆz/ℓz]/ε0.(9.33) The lines of electric ﬁeld are given by: ξ(x,0, z) =  (2σAbz+σBb((8λ1−b−2a)x−x2+z2)/ℓz)/ε0, −σBb(b−2a)x/2ℓzε0, (2σAaz−σBa((b−2a)x+x2−z2)/ℓz)/ε0, −σBa(b−2a)2/4ℓzε0, (2σAbz−σBb((b+ 2a)x+x2−z2)/ℓz)/ε0.(9.34) They are shown in Figure 9.9 with the given approximation and the same dimensions as in Figure 9.7. The constant potential in the lo wer plate is Φ =−φ0/2. Once more there is a reasonable match with Jeﬁmenko’s expe ri- mental result, the right side of Figure 3.5, especially in th e region between the parallelepipeds. Figure 9.9: Lines of electric ﬁeld in a plane orthogonal to th e four plates. The two top plates are uniformly resistive and carry currents al ong the positive z direction. The two bottom plates have zero resistivity and c arry currents along the negative zdirection. 122 Chapter 10 Resistive Strip 10.1 The Problem Here we consider a constant current ﬂowing uniformly throug h the surface of a stationary and resistive straight strip. Our goal is to cal culate the potential φand electric ﬁeld /vectorEeverywhere in space and the surface charge distribution σalong the strip that creates this electric ﬁeld. We follow es sentially the work published in 2003 [223]. We consider a strip in the x= 0 plane localized in the region −a<y<a and −ℓ/2<z <ℓ/ 2, such that ℓ≫a>0. The medium around the strip is taken to be air or vacuum. The constant current Iﬂows uniformly along the positive z direction with a surface current density given by /vectorK=Iˆz/2a(see Fig. 10.1). By Ohm’s law this uniform current distribution is related to a s patially constant electric ﬁeld along the surface of the strip. In the steady st ate this electric ﬁeld can be related to the potential by /vectorE=−∇φ. This relation means that along the strip the potential is a linear function of zand independent of y. The problem can then be solved by ﬁnding the solution of Laplace’s equati on∇2φ= 0 in empty space and applying the boundary conditions. Figure 10.1: A resistive strip of width 2 aand lengthℓwith a steady and uniform surface current density /vectorKalong the positive zdirection. 123 10.2 The Solution Due to the symmetry of the problem, it is convenient to utiliz e elliptic-cylindrical coordinates ( ζ, ϑ, z ) see Figure 10.2 [224]. These variables can take the fol- lowing values: 0 ≤ζ≤ ∞, 0≤ϑ≤2πrad, and −∞ ≤z≤ ∞. The relation between cartesian ( x, y, z ) and elliptic-cylindrical coordinates is given by: x=asinhζsinϑ , (10.1) y=acoshζcosϑ , (10.2) z=z , (10.3) where 2ais the constant thickness of the strip. The inverse relation s are given by: ζ= tanh−1/radicalBigg y2−x2−a2+ Ω 2y2, (10.4) ϑ= tan−1/radicalBigg a2+x2−y2+ Ω 2y2, (10.5) z=z , (10.6) where Ω ≡/radicalbig (x2+y2+a2)2−4a2y2. Figure 10.2: Elliptic-cylindrical coordinates ( ζ, ϑ, z ). 124 Laplace’s equation in this coordinate system is given by: ∇2φ=1 a2(cosh2ζ−cos2ϑ)/parenleftbigg∂2φ ∂ζ2+∂2φ ∂ϑ2/parenrightbigg +∂2φ ∂z2= 0. (10.7) A solution of Eq. (10.7) can be obtained by separation of vari ables in the form φ(ζ,ϑ,z ) =H(ζ)Φ(ϑ)Z(z): H′′−(α2+α3a2cosh2ζ)H= 0, (10.8) Φ′′+ (α2+α3a2cos2ϑ)Φ = 0, (10.9) Z′′+α3Z= 0, (10.10) whereα2andα3are constants. For the long strip being considered here, it is possible to ne glect boundary eﬀects near z=±ℓ/2. It has already been proved that in this case the potential must be a linear function of z, not only over the strip, but also over all space. See Section 5.4. This condition means that α3= 0. There are then two possible solutions for Φ( ϑ). Ifα2= 0, then Φ = C1+C2ϑ; ifα2/negationslash= 0, then Φ = C3sin(√α2ϑ) +C4cos(√α2ϑ), whereC1toC4are constants. Along the strip we havex= 0, andy2≤a2, which means that Ω = a2−y2,ζ= 0 and ϑ= tan−1/radicalbig (a2−y2)/y2. We are assuming that the potential does not depend onyalong the strip. This independence and the relation between yandϑmeans that the potential will not depend on ϑas well. Thus a non-trivial solution for Φ can only exist if α2= 0,C2= 0, and Φ = constant for all ϑ. The solution for Hwithα2=α3= 0 will be then a linear function of ζ. The general solution of the problem is then given by: φ= (A1ζ−A2)/parenleftBig φA+φBz ℓ/parenrightBig =/parenleftBigg A1tanh−1/radicalBigg y2−x2−a2+ Ω 2y2−A2/parenrightBigg/parenleftBig φA+φBz ℓ/parenrightBig . (10.11) The electric ﬁeld /vectorE=−∇φtakes the form: /vectorE=−A1/parenleftbigg|y|x√ 2 Ω/radicalbig y2−x2−a2+ Ωˆx +|y|/radicalbig y2−x2−a2+ Ω y√ 2Ωˆy/parenrightbigg/parenleftBig φA+φBz ℓ/parenrightBig −φB ℓ/parenleftbigg A1tanh−1/radicalBigg y2−x2−a2+ Ω 2y2−A2/parenrightbigg ˆz , (10.12) 125 To ﬁnd the surface charge density, we utilize the approximat ion close to the strip (|y|<aand|x| ≪a): /vectorE≈ −A1/bracketleftBigg x |x|/radicalbig a2−y2ˆx+y|x| (a2−y2)3/2ˆy/bracketrightBigg/parenleftBig φA+φBz ℓ/parenrightBig −φB ℓ/parenleftBigg A1tanh−1|x|/radicalbig a2−y2−A2/parenrightBigg ˆz . (10.13) The surface charge density σ(y, z) can be obtained by the standard procedure utilizing Gauss’s law/integraltext ◦/integraltext S/vectorE·d/vector a=Q/ε0. The surface charge density is then obtained by considering the limit in which |x| →0 in Eq. (10.13) and a small cylindrical volume with its length much smaller than its dia meter, yielding: σ=ε0[/vectorE(x>0)·ˆx−/vectorE(x<0)·(−ˆx)]. If we use Eq. (10.13), the surface charge density is found to be given by: σ(x, z) =−2ε0A1(φA+φBz/ℓ)/radicalbig a2−y2. (10.14) The linear charge density λ(z) can be obtained as λ(z) =/integraltexta −aσ(y,z)dy, yielding λ(z) =−2πε0A1/parenleftBig φA+φBz ℓ/parenrightBig . (10.15) 10.3 Discussion In the plane x= 0 the current in the strip creates a magnetic ﬁeld /vectorBthat points along the positive (negative) xdirection for y >0 (y <0). Consider a speciﬁc conduction electron moving with drifting velocity /vector vd. The magnetic ﬁeld due to all other mobile conduction electrons will act on this speci ﬁc conduction electron with a force given by q/vector vd×/vectorB(see Fig. 10.3). This force will cause a redistribution of charges along the ydirection, with negative charges concentrating along the center of the strip and positive charges at the extremities y=±a. In the steady-state this redistribution of charges will create an electric ﬁeld along the ydirection,Ey, that will balance the magnetic force, namely, |qEy|=|qvdB|. We have disregarded this Hall electric ﬁeld because it is usu ally much smaller than the electric ﬁeld giving rise to the current, as was show n in 6.4. We now analyze some particular cases. We ﬁrst consider two li mits by com- paringawith the distance of the observation point ρ=/radicalbig x2+y2. Ifa2≫ρ2, we have Ω ≈a2+x2−y2+ 2x2y2/a2andζ≈ |x|/a, such that: φ≈/parenleftbigg A1|x| a−A2/parenrightbigg/parenleftBig φA+φBz ℓ/parenrightBig . (10.16) Combining this result with Eq. (10.14) in the approximation a2≫ρ2yields: 126 Figure 10.3: Magnetic ﬁeld /vectorBalong the strip due to the current along the positivezdirection. There is a magnetic force pointing toward the axi s acting upon the conduction electrons. φ≈σ(z) 2ε0/parenleftbigg aA2 A1− |x|/parenrightbigg . (10.17) This result coincides with Eq. (9.9) considering aA2/A1= (ℓy/π)ln(2ℓz/ℓy) and σA= 0. This was expected because a strip with ℓ2≫a2≫ρ2is equivalent to a large plate. This Equation is also equivalent to Eq. (6.26) with A2/A1= ln(ℓ/a) except by an overall factor of 2. This was once more to be expected. Th e electrostatic potential at a distance dfrom a charged plate is given by φ=φ0−σd/2ε0, whereφ0is an arbitrary constant and σis the total density of surface charge, half of it in each side of the charged plane. On the other hand, the electrostatic potential just outside a closed charged conductor (at a dist ancedfrom it) is givenφ=φ1−σd/ε 0, whereφ1is an arbitrary constant and σhere is the surface charge density at the point in which the potential is being estimated, while the internal potential has the constant value φ1. That is, when we close an open charged conducting surface, the charges in the interna l side migrate to the external side. The magnitude of the electric ﬁeld just outsi de the closed surface is twice the electric ﬁeld close to a large charged plane, sup posing the same local charge density in both cases. As we have seen here, the same ha ppens with the surface charges when a current ﬂows along the resistive surf ace. On the other hand, if a2≪ρ2, we have Ω ≈ρ2+a2−2a2y2/ρ2and ζ≈ln(ρ/a). Utilizing these results in Eq. (10.11) combined with Eq. ( 10.15) yields: φ≈λ(z) 2πε0/parenleftbiggA2 A1−lnρ a/parenrightbigg . (10.18) This result coincides with Eq. (6.11) with A2/A1= ln(ℓ/a), whereℓis the typ- ical length of the wire or strip being considered, with ℓ≫a. This is reasonable because Eq. (6.11) corresponds to the potential outside a lo ng straight cylindri- cal wire carrying a constant current. At a point far from the a xis of the strip both results coincide as they must. 127 10.4 Comparison with the Experimental Results These results indicate that there is an electric ﬁeld not onl y along the resistive strip carrying a steady current, but also in the space around the strip. As we have seen, Jeﬁmenko performed experiments which demonstra te the existence of this external electric ﬁeld [174] [176, Plate 6]. The conﬁ guration of Jeﬁ- menko’s experiment, Figure 3.3, is equivalent to what has be en considered here: a two-dimensional conducting strip made on a glass plate usi ng a transparent conducting ink. To compare our calculations with his experi mental results, we need the values of A2/A1andφA/φB. We takeA2/A1= 3.6 andφA/φB= 0. The condition φA/φB= 0 corresponds to the symmetrical case considered by Jeﬁmenko in which the electric ﬁeld is parallel to the conduc tor just outside of it atz= 0 (zero density of surface charges at z= 0). We ﬁrst consider the plane orthogonal to the strip, y= 0. In this case the potential reduces to: φ(x,0,z) =/parenleftBigg A1tanh−1/radicalbigg x2 x2+a2−A2/parenrightBigg/parenleftBig φA+φBz ℓ/parenrightBig . (10.19) The lines of the electric ﬁeld orthogonal to the equipotenti als can be obtained by the procedure described in Section 6.5. These lines are re presented by a functionξsuch that ∇ξ· ∇φ= 0. This yields: ξ(x,0,z) =A1φBz2 ℓ2+ 2A1φAz ℓ+A1φB 2x2 ℓ2 −A1φB|x| ℓ√ x2+a2 ℓcosh−1/radicalbigg x2+a2 a2 −A1φB 2a2 ℓ2/parenleftBig cosh−1/radicalbigg x2+a2 a2/parenrightBig2 −A2φB 4/parenleftBig|x| ℓ√ x2+a2 ℓ+a2 ℓ2ln|x|+√ x2+a2 a/parenrightBig .(10.20) A plot of Eqs. (10.19) and (10.20) is given in Fig. 10.4. We now consider the plane of the strip, x= 0. The potential reduces to: φ(0,|y| ≤a,z) = −A2/parenleftBig φA+φBz ℓ/parenrightBig , (10.21) φ(0,|y| ≥a,z) =/parenleftBigg A1tanh−1/radicalBigg y2−a2 y2−A2/parenrightBigg/parenleftBig φA+φBz ℓ/parenrightBig =/parenleftbigg A1cosh−1|y| a−A2/parenrightbigg/parenleftBig φA+φBz ℓ/parenrightBig . (10.22) When there is no current in the strip, the potential along it i s a constant for allz. From Eq. (10.21) this means φB= 0. This value of φBin Eqs. (10.11), 128 Figure 10.4: Equipotentials (dashed lines) and lines of ele ctric ﬁeld (continuous lines) in a plane orthogonal to the plane of the strip. (10.12) and (10.14) reduces these equations to the known ele ctrostatic solution of a strip charged to a constant potential [225]. By a similar procedure, the lines of electric ﬁeld for the pla nex= 0 are given by: ξ(0,|y| ≤a,z) =A2φBay ℓ2, (10.23) ξ(0,|y| ≥a,z) =A1φBz2 ℓ2+ 2A1φAz ℓ+A1φB 2y2 ℓ2 −A1φB|y| ℓ/radicalbig y2−a2 ℓcosh−1|y| a +A1φB 2a2 ℓ2/parenleftbigg cosh−1|y| a/parenrightbigg −A2φB 4/parenleftBig|y| ℓ/radicalbig y2−a2 ℓ −a2 ℓ2ln|y|+/radicalbig y2−a2 a/parenrightBig . (10.24) A plot of Eqs. (10.21) to (10.24) is presented in Fig. 10.5. Figure 10.6 presents the theoretical electric ﬁeld lines ov erlaid on the exper- imental result of Jeﬁmenko, Figure 3.3. In Fig. 10.7 the experimental result of Jeﬁmenko, Barnett an d Kelly, Fig- ure 3.10, is overlaid on the equipotential lines calculated utilizing Eqs. (10.23) and (10.24) with A2/A1= 3.0 andφA/φB= 0. The agreement is not as good 129 Figure 10.5: Equipotentials (dashed lines) and lines of ele ctric ﬁeld (continuous lines) in the plane of the strip. Figure 10.6: Theoretical lines of electric ﬁeld overlaid on the experimental lines obtained by Jeﬁmenko. as in our previous ﬁgure for two reasons: One reason is that ou r calculations are for a two-dimensional conﬁguration, while the experime nt of Jeﬁmenko, Bar- nett and Kelly [177] was performed in a three-dimensional re ctangular chamber. The second reason is that in the grass seed experiment [174], the ratio of the length to the width of the conductor was 7, but in the second ex periment [177], this ratio was only 2, which means that boundary eﬀects near z=ℓ/2 and z=−ℓ/2 are more important. These boundary eﬀects were not conside red in our calculations. 130 Figure 10.7: Theoretical equipotential lines overlaid on t he experimental lines obtained by Jeﬁmenko, Barnett and Kelly. 131 132 Part III Curved Conductors 133 In this third Part of the book we consider resistive conducto rs carrying steady currents along curved paths. Russell’s theorem, dis cussed in Section 5.4, is no longer valid due to the curvature of the wire. Three case s in particular will be discussed here, the azimuthal current in an inﬁnite c ylindrical shell, the azimuthal current over the surface of a spherical shell, and the azimuthal current in a toroidal conductor. These cases can still be sol ved analytically, and their solutions clarify some important aspects of surface c harges in conductors carrying steady currents. 135 136 Chapter 11 Resistive Cylindrical Shell with Azimuthal Current 11.1 Conﬁguration of the Problem The subject of this Chapter has been discussed mainly by Jeﬁm enko [175, Prob- lem 9.33 and Figure 14.7] [176, p. 318], Heald [226] and Griﬃt hs [16, p. 279]. We follow these approaches here. An inﬁnite homogeneous resistive cylindrical shell of radi usahas its axis coinciding with the zdirection. We utilize cylindrical coordinates ( ρ, ϕ, z ) with origin at the center of the shell, with ρ=/radicalbig x2+y2being the distance to thezaxis. There is a narrow slot along its entire length located a t (ρ, ϕ, z ) = (a, π, z ). An idealized line battery in the slot maintains its termin als located atϕ=±πrad with the constant potentials φ=±φB/2, respectively. See Figure 11.1. Figure 11.1: Conﬁguration of the problem. In accordance with Ohm’s law the potential along the surface of the cylin- drical shell is then given by 137 φ(a,ϕ,z ) =φBϕ 2π. (11.1) 11.2 Potential and Electric Field The potential inside and outside the cylindrical shell sati sﬁes Laplace’s equation ∇2φ= 0: 1 ρ∂ ∂ρ/parenleftbigg ρ∂φ ∂ρ/parenrightbigg +1 ρ2∂2φ ∂ϕ2+∂2φ ∂z2= 0. (11.2) The solution should be independent of z. Trying a solution in terms of separation of variables φ(ρ,ϕ,z ) =R(ρ)Φ(ϕ) yields Φ′′+m2Φ = 0, (11.3) ρd dρ/parenleftbigg ρdR dρ/parenrightbigg −m2R= 0, (11.4) wheremis a constant. The solutions of these equations if m= 0 are Φ(ϕ) = A0+B0ϕandR(ρ) =C0lnρ+D0. Ifm/negationslash= 0 we have φ(ϕ) =Amcos(mϕ) + Bmsin(mϕ) andR(ρ) =Cmρ−m+Dmρm. The solutions must be periodic in ϕ,i.e., Φ(ϕ+ 2π) = Φ(ϕ). This means that B0= 0 andm= 1,2,3,... The internal and external solutions, with appropriate coeﬃ cients, are then given by φ(ρ≤a,ϕ,z ) =A0i(C0ilnρ+D0i) +∞/summationdisplay m=1[Amicos(mϕ) +Bmisin(mϕ)]/bracketleftbig Cmiρ−m+Dmiρm/bracketrightbig , (11.5) φ(ρ≥a,ϕ,z ) =A0e(C0elnρ+D0e) +∞/summationdisplay m=1[Amecos(mϕ) +Bmesin(mϕ)]/bracketleftbig Cmeρ−m+Dmeρm/bracketrightbig , (11.6) We specify ﬁnite solutions when ρ→0. This means that C0i=C1i=C2i= ...= 0. We also specify solutions in which the potential goes to z ero when ρ→ ∞. This means that C0e=D0e=D1e=D2e=...= 0. Our solutions in these two regions reduce to φ(ρ≤a,ϕ,z ) =A0iD0i+∞/summationdisplay m=1[AmiDmicos(mϕ) +BmiDmisin(mϕ)]ρm, (11.7) 138 φ(ρ≥a,ϕ,z ) =∞/summationdisplay m=1[AmeCmecos(mϕ) +BmeCmesin(mϕ)]1 ρm.(11.8) The potential must be continuous in ρ=a. This means that A0iD0i= 0, AmiDmiam=AmeCmea−mandBmiDmiam=BmeCmea−m. DeﬁningAmiDmi≡ GmandBmiDmi≡Hmyields φ(ρ≤a,ϕ,z ) =∞/summationdisplay m=1[Gmcos(mϕ) +Hmsin(mϕ)]ρm, (11.9) φ(ρ≥a,ϕ,z ) =∞/summationdisplay m=1[Gmcos(mϕ) +Hmsin(mϕ)]a2m ρm. (11.10) Now we need to apply the boundary condition at ρ=a, Eq. (11.1). To this end we employ the Fourier expansion of ϕ, which is valid for −πrad<ϕ<π rad: ϕ= 2/bracketleftBigg∞/summationdisplay m=1(−1)m−1sin(mϕ) m/bracketrightBigg . (11.11) Comparing Eqs. (11.9) and (11.10) at ρ=awith Eqs. (11.1) and (11.11) yieldsGm= 0 andHmam=φB(−1)m+1/πm: φ(ρ≤a,ϕ,z ) =−φB π/bracketleftBigg∞/summationdisplay m=1(−1)m/parenleftBigρ a/parenrightBigmsin(mϕ) m/bracketrightBigg , (11.12) φ(ρ≥a,ϕ,z ) =−φB π/bracketleftBigg∞/summationdisplay m=1(−1)m/parenleftbigga ρ/parenrightbiggmsin(mϕ) m/bracketrightBigg . (11.13) These two series can be put in closed form [226]: φ(ρ≤a,ϕ, z ) =φB πtan−1ρsinϕ a+ρcosϕ=φB πtan−1y a+x=φB πψ ,(11.14) φ(ρ≥a,ϕ, z ) =φB πtan−1asinϕ ρ+acosϕ=φB πtan−1ay x2+ax+y2,(11.15) whereψis the polar angle about the slot as the axis. The equipotentials given by these equations are represente d in Figure 11.2. The lines of electric ﬁeld given by the function ξ(x,y) such that ∇ξ·∇φ= 0 can be obtained by the method described in Section 6.5. For th e regionρ <a this function is given by ξ(x,y) =φB2ax+x2+y2 a2. (11.16) These are circular arcs centered on the battery given by the f ollowing equation (for a particular ξo): 139 Figure 11.2: Equipotential lines. The battery is represent ed by the black spot. (x+a)2+y2=2ξo+φB φBa2. (11.17) Combining this result with Eqs. (8) and (10) of Heald’s paper [226] we can also obtain the function ξfor the region outside the cylinder (this result can be checked by observing that it satisﬁes ∇ξ· ∇φ= 0): ξ(x,y) =φBa2+ 2ax x2+y2. (11.18) These are also circular arcs with centers along the xaxis, given by (for a par- ticularξo): /parenleftbigg x−φB ξoa/parenrightbigg2 +y2=φB ξoξo+φB ξoa2. (11.19) From Eqs. (11.14), (11.15), (11.16) and (11.18) we can verif y that∇ξ·∇φ= 0. The electric ﬁeld can be readily obtained by /vectorE=−∇φ. Inside the shell it is given by: /vectorE(ρ<a, ϕ, z ) =−φB πa(sinϕ)ˆρ+ (ρ+acosϕ)ˆϕ a2+ρ2+ 2aρcosϕ =−φB πˆψ ρ′, (11.20) whereρ′≡/radicalbig ρ2+a2+ 2aρcosϕis the polar radius about the slot as the axis. That is, the lines of electric ﬁeld are circular arcs centere d on the battery. Outside the shell the electric ﬁeld is given by: 140 /vectorE(ρ>a, ϕ, z ) =φB πa ρρ(sinϕ)ˆρ−(a+ρcosϕ)ˆϕ a2+ρ2+ 2aρcosϕ, (11.21) with magnitude |/vectorE|=φBa/πρρ′. At the surface of the shell, ρ=a, Eqs. (11.20) and (11.21) yield the same tangential component: Eϕ(a,ϕ,z ) =−φB 2πa. (11.22) This is the correct result arising from Eq. (11.1), namely, E= ∆φ/L, where ∆φ≡φBis the potential diﬀerence generated by the battery and L≡2πais the length described by the electrons around the circuit. The lines of electric ﬁeld are represented in Figure 11.3. Figure 11.3: Lines of electric ﬁeld. The battery is represen ted by the black spot. 11.3 Surface Charge Densities The surface charge densities inside and outside the hollow s hell (that is, along the internal and external surfaces), σiandσo, are obtained utilizing Gauss’s law. They have the same value and are given by σi=σo=ε0φB 2πatanϕ 2=ε0φB 2πatanψ . (11.23) A plot of this surface charge density as a function of ϕis shown in Figure 11.4. The total charge density σtis given by σt=σi+σo=ε0φB πatanϕ 2=ε0φB πatanψ . (11.24) 141 Figure 11.4: Surface charge densities σ≡σi=σoinside and outside the hollow shell as a function of the azimuthal angle ϕ, according to Eq. (11.23) [226]. Expanding Eq. (11.23) for ϕ≪1 rad and utilizing Eqs. (11.1) and (11.11) yields: σi(ϕ≪1 rad) =σo(ϕ≪1 rad) ≈ε0φBϕ 4πa=ε0φ 2a. (11.25) This result coincides with Eq. (9.7) when we are over the plat e (x= 0) and σA= 0, if we equate the circumference of the cylindrical shell h ere (2πa) with the longitudinal length ℓzof the plate of Section 9.2, as expected. The surface charge density which appears in Eq. (9.7) is the total charge density due to the charges in both sides of the plate, analogous to σi+σoof Eq. (11.25). On the other hand, for ϕ= (π±δ) rad, with 0 < δ≪1 (that is, close to ϕ=πrad) we have: σi=σo≈ ∓ε0φB πaδ=∓ε0φB πs. (11.26) Heres≡aδis the distance along the surface of the cylindrical shell to the line battery. This is an important result which shows that th e surface charge density diverges inversely proportional to the distance fr om the line battery in this idealized case. Eq. (11.23) indicates that in regions close to the battery th e surface charge density is no longer a linear function of the longitudinal co ordinate (in this case the arcaϕ) of the resistive conductor. It is linear only close to ϕ= 0 rad but increases nonlinearly (that is, it is not proportional to aϕ) toward the battery. See Figure 11.4. This nonlinearity should also occur in stra ight conductors when we are close to the battery. This has been conﬁrmed in 2004 and 2005 [205, 222]. Eqs. (11.14) to (11.26) indicate that several functions are proportional to the emf of the battery, namely: the internal and external pot ential and electric ﬁeld, as well as the surface charge densities in the internal and external walls. That is, they are proportional to the voltage φ(π)−φ(−π) =φBbetween the terminals of the battery. Suppose we have two cylindrical sh ells 1 and 2 of the same radius but with diﬀerent resistivities. If we conne ct only shell 1 with batteryBand later on if we connect only shell 2 with the same battery B (assuming the battery has not lost its power), diﬀerent stea dy currents will ﬂow 142 in each shell, as they have diﬀerent resistivities. But the i nternal and external potentials, electric ﬁelds and surface charge densities wi ll be the same in both cases. This again illustrates that the electric ﬁeld outsid e a resistive conductor carrying a steady current is proportional to the voltage to w hich it is subjected. The importance of the present case is that this has been shown in a situation in which we were able to ﬁnd an exact analytical solution of all m agnitudes. That is, this external electric ﬁeld does not depend directly upo n the current ﬂowing in the circuit. After all, the electric ﬁeld was found to be th e same even when diﬀerent currents ﬂow in two circuits connected by the same e mf. In order to observe the eﬀects of the external electric ﬁeld, it is most i mportant to work with circuits connected to high voltage sources, as this ﬁel d is proportional to the applied emf. 11.4 Representation in Fourier Series Our solution of the potential in terms of Fourier series was p resented in Eqs. (11.12) and (11.13). These series can be put in closed form. See Eqs. ( 11.14) and (11.15). If this were not possible, we could continue to util ize the Fourier series representation obtaining the electric ﬁeld /vectorE=−∇φin the form /vectorE(ρ<a,ϕ,z ) =φB πρ/braceleftBigg∞/summationdisplay m=1/parenleftbigg−ρ a/parenrightbiggm [sin(mϕ)ˆρ+ cos(mϕ)ˆϕ]/bracerightBigg ,(11.27) /vectorE(ρ>a,ϕ,z ) =−φB πρ/braceleftBigg∞/summationdisplay m=1/parenleftbigg−a ρ/parenrightbiggm [sin(mϕ)ˆρ−cos(mϕ)ˆϕ]/bracerightBigg .(11.28) From these two equations the tangential component of the ele ctric ﬁeld at ρ=ais given by: Eϕ(a,ϕ,z ) =φB πa/bracketleftBigg∞/summationdisplay m=1(−1)mcos(mϕ)/bracketrightBigg . (11.29) This is a divergent series. This happens with the diﬀerentia tion of some Fourier series [202, Section 14.4]. By diﬀerentiating both sides of Eq. (11.11) we obtain 1 = 2/bracketleftBigg∞/summationdisplay m=1(−1)m−1cos(nϕ)/bracketrightBigg . (11.30) While Eq. (11.11) is convergent, the latter series is diverg ent. But if we disre- gard this and apply Eq. (11.30) into Eq. (11.29) we obtain the same result as Eq. (11.22). And this is a reasonable result. Figure 11.5 is a plot of 143 f(ϕ)≡∞/summationdisplay m=1(−1)m−1cos(nϕ), (11.31) including 100 terms in the summation. The oscillations are d ue to the divergent character of this series representation, although we can se e that the curve oscil- lates around the constant value of f(ϕ) = 1/2, which was expected according to Eq. (11.30). Figure 11.5: Plot of Eq. (11.31) including 100 terms in the su mmation. The oscillations are due to the divergent character of this seri es. The bold line is a plot of the constant 1 /2 as expected by Eq. (11.30). In order to deal with a divergent Fourier series, we thought o f applying an average approach. In particular, at each angle ϕiwe consider the average value of a generic function g(ϕ),g(ϕi), namely: g(ϕi)≡1 ∆ϕ/integraldisplayϕi+∆ϕ/2 ϕi−∆ϕ/2g(ϕ)dϕ . (11.32) The value of ∆ ϕis typically taken as the whole interval in which we are plott ing g(ϕ) divided by the number of terms we are including in the summat ion. For instance, if we are plotting a function g(ϕ) in the interval ϕ=−πrad toϕ=π rad and we include 100 terms in the Fourier series expansion o fg(ϕ), then ∆ϕ= (2π/100) rad. Figure 11.6 is a plot of f(ϕ) obtained from this averaging approach utilizing Eq. (11.31). From this Figure we can see that f(ϕ) coincides with the expected value of 1/2, indicating the correctness of this averaging p rocedure. The surface charge densities inside and outside the shell ca n be obtained from Gauss’s law. From Eqs. (11.27) and (11.28) this yields: 144 Figure 11.6: Plot of f(ϕ) obtained from Eqs. (11.32) and (11.31), overlaid on the constant value 1 /2. Asf(ϕ) coincides with this constant value, this indicates the correctness of this averaging procedure. σi=−lim ρ→aε0/vectorE(ρ<a)·ˆρ=−ε0φB πa/bracketleftBigg∞/summationdisplay m=1(−1)msin(mϕ)/bracketrightBigg , (11.33) σo= lim ρ→aε0/vectorE(ρ>a)·ˆρ=σi. (11.34) The same expressions are obtained from Eq. (11.23) by expand ing tan(ϕ/2) in Fourier series. The total charge density expressed in Fourier series is give n by σt=σi+σo=−2ε0φB πa/bracketleftBigg∞/summationdisplay m=1(−1)msin(mϕ)/bracketrightBigg . (11.35) In Figure 11.7 we present a plot of Eq. (11.35) including 100 t erms in the summation. The oscillations in this Figure are probably due to convergence problems of the Fourier series already discussed. Increasi ng the number of terms does not improve signiﬁcantly the plot or decrease the ampli tude of oscillation around each value of ϕ. The bold line in this Figure is given by Eq. (11.24). Fig. 11.8 is a plot of Eq. (11.24) overlaid on a plot of σt(ϕi) obtained from Eqs. (11.35) and (11.32). In this case we have considered a wh ole oscillation ofσt(ϕi) around each angle ϕi. The two plots coincided with one another (the two curves are indistinguishable in Fig. 11.8), indicating the correctness of this averaging procedure. 145 Figure 11.7: Total surface charge density σt=σi+σoof an inﬁnite resistive cylindrical shell carrying a steady azimuthal current as a f unction of the angle ϕ. The bold line is a plot of the closed form solution of σt(ϕ), Eq. (11.24), while the oscillatory line is a plot of σt(ϕ) expressed in a Fourier series, Eq. (11.35). Figure 11.8: Total surface charge density σt. The summation that appears as an oscillation in Fig. 11.7, given by Eq. (11.35), is smoothed o ut by taking the mean value for each point of its surroundings (in this case, a whol e oscillation around each point), utilizing Eq. (11.32). The closed analytical f orm, Eq. (11.24), is overlaid on it. Both plots coincide with one another, indica ting the correctness of this averaging procedure. 11.5 Lumped Resistor Heald also considered a lumped resistor, i.e., a cylindrical shell of ﬁnite resistiv- ity for −α<ϕ<α and zero resistivity outside this region [226]. The potenti al at the shell was given by 146 φ(a, ϕ, z ) =∞/summationdisplay k=1Aksin(kϕ). (11.36) Here the coeﬃcients Akare given by: Ak=φB π/bracketleftbigg/integraldisplayα 0ϕ αsin(kϕ)dϕ+/integraldisplayπ αsin(kϕ)dϕ/bracketrightbigg =φB π/bracketleftbigg(−1)k−1 k+sin(kϕ) k2α/bracketrightbigg . (11.37) The potential inside and outside the shell is given by, respe ctively: φ(ρ≤a, ϕ, z ) =φB π/bracketleftbigg tan−1ρsinϕ a+ρcosϕ +∞/summationdisplay k=1ρ aksin(kα) k2αsin(kϕ)/bracketrightBigg , (11.38) φ(ρ≥a, ϕ, z ) =φB π/bracketleftbigg tan−1asinϕ ρ+acosϕ +∞/summationdisplay k=1a ρksin(kα) k2αsin(kϕ)/bracketrightBigg . (11.39) The equipotentials for this case of lumped resistor are repr esented in Fig- ure 11.9. Figure 11.9: Equipotential lines for the lumped resistor. 147 Figure 11.10: Lines of electric ﬁeld for the lumped resistor . The electric ﬁeld lines are represented in Figure 11.10. The internal and external surface charge densities are agai n equal. In this case they are given by: σi=σo=ε0φB 2πa/bracketleftBigg tanϕ 2+∞/summationdisplay k=12sin(kα) kαsin(kϕ)/bracketrightBigg =ε0φB 2πa/bracketleftbigg tanϕ 2+1 αln|sin[(ϕ+α)/2] sin[(ϕ−α)/2]|/bracketrightbigg . (11.40) In this case the surface charge densities diverge not only at the battery but also at the discontinuity in the resistivity of the shell, as in Figure 11.11. Figure 11.11: Densities of surface charge σ≡σi=σoalong the internal and external surfaces of the hollow lumped resistor (continuou s line) as a function of the azimuthal angle, as given by Eq. (11.40). The dashed li ne represents the previous case of a uniformly resistive conductor. 148 Another qualitative discussion of lumped resistors can be f ound in the book of Chabay and Sherwood [166, Section 18.6]. Their analysis i s extremely didactic and helpful. 149 150 Chapter 12 Resistive Spherical Shell with Azimuthal Current 12.1 Introduction Our goal in this chapter is to consider a steady azimuthal cur rent ﬂowing in a resistive spherical shell [227]. The mathematical diﬃcult y is intermediate be- tween the inﬁnite cylindrical shell which we considered in t he previous Chapter and the toroidal conductor which is the subject of the next Ch apter. The im- portance of the present case is that we can obtain exact analy tical solutions for the external and internal distribution of surface charges, potential and electric ﬁeld which are not as complex as in the toroidal conductor. De spite this fact they show clearly the existence of an electric ﬁeld outside a resistive conductor bounded in a ﬁnite volume of space. To the best of our knowledg e this case has never been treated before by other authors. 12.2 Description of the Problem Consider a resistive spherical shell of radius a, centered at the origin. We suppose an idealized linear battery located along a meridia n of the shell (like Greenwich Meridian) and maintaining a constant potential d iﬀerence between its left and right sides. See Figure 12.1. That is, the battery is a semi-circumference in the plane y= 0 with its ex- tremities at ( x,y,z ) = (0,0,±a) and central point along the semi-circumference at (x,y,z ) = (−a,0,0). Utilizing spherical coordinates ( r,θ,ϕ) the linear bat- tery is then located at ( a,θ,π ). We suppose that the potential diﬀerence gen- erated by the battery does not depend upon the polar angle θ. The battery generates a steady current ﬂowing along the shell in the azim uthal direction −ˆϕ. See Figs. 12.1 and 12.2. The medium inside and outside the sp herical shell is supposed to be air or vacuum. 151 Figure 12.1: A resistive spherical shell of radius a(m) is centered at the origin. An idealized linear battery located at ( r,θ,ϕ) = (a,θ,π ) generates a steady currentI(A) ﬂowing along the surface of the shell in the azimuthal dir ection−ˆϕ. The bold line represents the battery, which has the form of a s emi-circumference. Figure 12.2: Projection of the resistive spherical shell wi th radiusain the plane z= 0. Notice that the battery, represented by the bold line, is seen as a straight line for −a≤x≤0. In this plane the current ﬂows in the clockwise direction −ˆϕ. According to Ohm’s law, the potential φalong the surface is given by (in- cluding also a constant potential for the sake of generality , so that we can return 152 to the situation of a charged shell without current as a speci al case): φ(a,θ,ϕ ) =φA+φBϕ 2π. (12.1) The goal is to ﬁnd solutions of Laplace’s equation ∇2φ= 0 outside and inside the spherical shell utilizing Eq. (12.1) as a boundar y condition, together with ﬁnite values of the potential at the center of the shell a nd at inﬁnity. The electric ﬁeld is then found by /vectorE=−∇φ. Lastly the surface charge density σ is obtained by the standard procedure of taking the radial co mponents of the external and internal electric ﬁelds when r→a. 12.3 General Solution Laplace’s equation in spherical coordinates can be written as: ∇2φ=∂2φ ∂r2+2 r∂φ ∂r+1 r2∂2φ ∂θ2+cotθ r2∂φ ∂θ+1 r2sin2θ∂2φ ∂ϕ2= 0. (12.2) The electric potential φcan be solved utilizing the method of separation of variables,φ(r,θ,ϕ) =R(r)Θ(θ)Φ(ϕ). This yields the following equations for the functions R, Θ and Φ [224, pp. 24–27]: R′′+2 rR′−α2 r2R= 0, (12.3) Θ′′+ Θ′cotθ+/parenleftbigg α2−α1 sin2θ/parenrightbigg Θ = 0, (12.4) Φ′′+α1Φ = 0, (12.5) whereα1andα2are constants. The function Φ( ϕ) must be periodic in ϕ, i.e., Φ(0) = Φ(2 π). This implies α1=q2, whereq= 0,1,2,...The solutions of Eq. (12.5) are then Φ(1) q= sin(qϕ) and Φ(2) q= cos(qϕ). Eq. (12.4) is the associated Legendre equation [202, Sec. 12.5]. In order to h ave ﬁnite solutions atθ= 0 rad and at θ=πrad the constant α2must have the form α2= p(p+ 1), withp= 0,1,2,...The solutions of Eq. (12.4) are then the associated Legendre functions of ﬁrst and second kind, namely, Θ(1) pq=Pq p(cosθ) and Θ(2) pq= Qq p(cosθ). Whenq= 0 they reduce to Legendre polynomial, Pp(cosθ), and to Legendre function of the second kind, Qp(cosθ), respectively. The solutions of Eq. (12.3) with α2=p(p+ 1) are given by R(1) p=rpandR(2) p=r−p−1. The potential must remain ﬁnite at every point in space. The s olution R(1) p=rpdiverges when r→ ∞ andp≥1. For this reason we eliminate this solution outside the shell. By specifying that the potentia l goes to zero when r→ ∞ we can also eliminate the solution with p= 0. Analogously we eliminate the solution R(2) p=r−p−1inside the shell as it diverges when r→0. The functionPq p(cosθ) is ﬁnite for 0 rad ≤θ≤πrad. On the other hand, Qq p(cosθ) diverges at θ= 0 rad and at θ=πrad. We then eliminate this solution both 153 inside and outside the shell. The ﬁnite solutions for the pot ential outside and inside the shell are then given by the combination of all poss ible values of Rp(r), Θpq(θ) and Φq(ϕ), respectively: φo(r≥a,θ,ϕ ) =∞/summationdisplay p=0r−(p+1)/braceleftBigg ApPp(cosθ) +∞/summationdisplay q=1[Bpqsin(qϕ) +Cpqcos(qϕ)]Pq p(cosθ)/bracerightBigg , (12.6) φi(r≤a,θ,ϕ ) =∞/summationdisplay p=0rp/braceleftBigg DpPp(cosθ) +∞/summationdisplay q=1[Epqsin(qϕ) +Fpqcos(qϕ)]Pq p(cosθ)/bracerightBigg . (12.7) In order to obtain the coeﬃcients Ap,Bpq,Cpq,Dp,EpqandFpqwe must apply the boundary condition at the surface of the shell, r=a. Expanding Eq. (12.1) in Fourier series [226]: φ(a,θ,ϕ ) =φA+φBϕ 2π=φA+φB π/bracketleftBigg∞/summationdisplay q=1(−1)q−1 qsin(qϕ)/bracketrightBigg . (12.8) As there are no terms in cos( qϕ) in Eq. (12.8) we obtain immediately Cpq= Fpq= 0. First we ﬁnd the coeﬃcients ApandBpqfor the region outside the shell (r≥a). Eq. (12.6) calculated at r=acombined with Eq. (12.8) yields the following equations: φA=∞/summationdisplay p=0a−(p+1)ApPp(cosθ), (12.9) φB π(−1)q−1 q=∞/summationdisplay p=0a−(p+1)BpqPq p(cosθ). (12.10) To ﬁnd the coeﬃcients ApandBpqwe multiply both sides of Eq. (12.9) by Pℓ(cosθ)sinθdθ, both sides of Eq. (12.10) by Pq ℓ(cosθ)sinθdθ, and integrate from 0 rad to πrad. We then utilize the orthogonality relation of Legendre polynomials [202, Eq. (12.104)]: /integraldisplayπ 0Pq p(cosθ)Pq ℓ(cosθ)sinθdθ=2 2p+ 1(p+q)! (p−q)!δpℓ, (12.11) whereδpℓis Kronecker’s delta function, which is 1 for p=qand 0 forp/negationslash=q. This yields: Ap=aφAδp0, (12.12) 154 and Bpq=φB πap+1(−1)q−1 q2p+ 1 2(p−q)! (p+q)!Ipq, (12.13) where we deﬁned: Ipq≡/integraldisplayπ 0Pq p(cosθ)sinθdθ . (12.14) Notice that Ipq= 0 forp+qodd due to the parity property of the associated Legendre functions [202, p. 725]. We can change the upper limit of the summation over qin Eq. (12.6) from ∞top, becausePq p(ξ) = 0 forq>p. The ﬁnal solution for the potential outside a spherical shell conducting a steady azimuthal current is g iven by: φo(r≥a,θ,ϕ ) =φAa r+φB 2π/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1ap+1 rp+1(−1)q−1 q(2p+ 1)(p−q)! (p+q)!× ×IpqPq p(cosθ)sin(qϕ)/bracketrightBigg . (12.15) It is useful to keep in mind that the summation order can be inv erted, from/summationtext∞ p=1/summationtextp q=1to/summationtext∞ q=1/summationtext∞ p=q. For the region far from the origin, r≫a, the two most relevant terms of Eq. (12.15) are: φo(r≫a,θ,ϕ )≈φAa r+φB3a2 8r2sinθsinϕ . (12.16) This can be understood as the potential of a point charge qsphere = 4πε0φAa at the center of the shell plus the potential of an electric di pole of moment /vector psphere located at the origin with /vector psphere = (3πε0φBa2/2)ˆy, namely: φo(r≫a,θ,ϕ ) =qsphere 4πε0r+/vector psphere·/vector r 4πε0r3. (12.17) The solution for the potential inside the sphere ( r≤a),φi, can be found by changing (a/r)p+1→(r/a)p, as discussed by Jackson [13, p. 101]: φi(r≤a,θ,ϕ ) =φA+φB 2π/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1rp ap(−1)q−1 q(2p+ 1)(p−q)! (p+q)!× ×IpqPq p(cosθ)sin(qϕ)/bracketrightBigg , (12.18) whereIpqis given by Eq. (12.14). 155 Utilizing that (as can be seen multiplying both sides of Eq. ( 12.19) by Pq ℓ(cosθ)sinθdθ, integrating from θ= 0 rad to θ=πrad and ﬁnally apply- ing Eqs. (12.14), (12.13) and (12.8)): ∞/summationdisplay p=1p/summationdisplay q=1(−1)q−1 q(2p+ 1)(p−q)! (p+q)!IpqPq p(cosθ)sin(qϕ) =ϕ , (12.19) we obtain from Eqs. (12.15) and (12.18) in the limit r→athatφo(a,θ,ϕ ) = φi(a,θ,ϕ ) =φA+φBϕ/2π, as expected. 12.4 Electric Field and Surface Charges The electric ﬁeld in spherical coordinates is given by: /vectorE=−∇φ=−∂φ ∂rˆr−1 r∂φ ∂θˆθ−1 rsinθ∂φ ∂ϕˆϕ . (12.20) This yields the following components outside and inside the shell, respectively: Er,o=φAa r2+φB 2π/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1ap+1 rp+2(−1)q−1 q(p+ 1)(2p+ 1)(p−q)! (p+q)!× ×IpqPq p(cosθ)sin(qϕ)/bracketrightBigg , (12.21) Eθ,o=φB 2π/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1ap+1 rp+2(−1)q−1 q(2p+ 1)(p−q)! (p+q)!IpqPq p′(cosθ)sinθsin(qϕ)/bracketrightBigg , (12.22) Eϕ,o=−φB 2π/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1ap+1 rp+2(−1)q−1(2p+ 1)(p−q)! (p+q)!IpqPq p(cosθ) sinθcos(qϕ)/bracketrightBigg , (12.23) Er,i=−φB 2π/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1rp−1 ap(−1)q−1 qp(2p+ 1)(p−q)! (p+q)!IpqPq p(cosθ)sin(qϕ)/bracketrightBigg , (12.24) Eθ,i=φB 2π/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1rp−1 ap(−1)q−1 q(2p+ 1)(p−q)! (p+q)!IpqPq p′(cosθ)sinθsin(qϕ)/bracketrightBigg , (12.25) Eϕ,i=−φB 2π/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1rp−1 ap(−1)q−1(2p+ 1)(p−q)! (p+q)!IpqPq p(cosθ) sinθcos(qϕ)/bracketrightBigg . (12.26) In Eqs. (12.22) and (12.25) Pq p′(ξ) is the derivative of the associated Legendre functionPq p(ξ) relative to its argument ξ. 156 From Eqs. (12.22), (12.25) and (12.19) we obtain in the limit r→athat: Eθ,o(a,θ,ϕ ) =Eθ,i(a,θ,ϕ ) =φB 2πa/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1(−1)q−1 q(2p+ 1)(p−q)! (p+q)!IpqPq p′(cosθ)sinθsin(qϕ)/bracketrightBigg =φB 2πad dθ/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1(−1)q−1 q(2p+ 1)(p−q)! (p+q)!IpqPq p(cosθ)sin(qϕ)/bracketrightBigg =φB 2πad dθϕ= 0. (12.27) From Eqs. (12.23), (12.26) and (12.19) we obtain in the limit r=athat: Eϕ,o(a,θ,ϕ ) =Eϕ,i(a,θ,ϕ ) =−φB 2πasinθ/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1(−1)q−1(2p+ 1)(p−q)! (p+q)!IpqPq p(cosθ)cos(qϕ)/bracketrightBigg =−φB 2πasinθd dϕ/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1(−1)q−1 q(2p+ 1)(p−q)! (p+q)!IpqPq p(cosθ)sin(qϕ)/bracketrightBigg =−φB 2πasinθd dϕϕ=−φB 2πasinθ. (12.28) Eq. (12.27) indicates that the non-radial electric ﬁeld at t he surface of the shell is only in the azimuthal direction, as expected from Eq . (12.1). The length of an azimuthal circumference at the polar angle θalong the surface of the shell is given by 2 πasinθ. Eq. (12.28) indicates that Eϕ(a,θ,ϕ ) at each polar angle θis given by the total electromotive force, ∆ φ=φB, over the length of the corresponding circuit at the polar angle θ, as expected. By Ohm’s law the same inverse proportionality with sin θwill be valid for the surface current density. That is,/vectorKshould be proportional to φB/sinθ. According to this model the current density should diverge at the poles (in θ= 0 rad and in θ=πrad). This indicates a limitation for the theoretical model which we are utilizing. This divergence arises due to the fact that we are utilizing a cond ucting spherical shell with an idealized linear battery along a meridian. In a real experiment this divergence should not occur. This means that our analyt ical theoretical solution obtained in this Section should not be valid close t o these two poles when compared with a real experiment. The reason for utilizi ng our theoretical model is that it yields an analytical solution for the import ant problem of a closed current ﬂowing in a ﬁnite volume of space. The surface charge distributions outside and inside the she ll are related to the electric ﬁeld through Gauss’s law: σo(a,θ,ϕ ) = lim r→aε0/vectorEo(r,θ,ϕ)·ˆr=ε0/braceleftBigg φA a+φB 2πa∞/summationdisplay p=1/bracketleftBiggp/summationdisplay q=1(−1)q−1 q× 157 ×(p+ 1)(2p+ 1)(p−q)! (p+q)!IpqPq p(cosθ)sin(qϕ)/bracketrightbigg/bracerightbigg , (12.29) σi(a,θ,ϕ ) =−lim r→aε0/vectorEi(r,θ,ϕ)·ˆr =ε0φB 2πa/bracketleftBigg∞/summationdisplay p=1p/summationdisplay q=1(−1)q−1 qp(2p+ 1)(p−q)! (p+q)!IpqPq p(cosθ)sin(qϕ)/bracketrightBigg .(12.30) In this case of a spherical shell we have σo(a,θ,ϕ )/negationslash=σi(a,θ,ϕ ). In the cylindrical case, on the other hand, we obtained the same surface charge d ensities both inside and outside the shell. The total surface charge densi ty is the sum of these two expressions, namely, σt=σo+σi. In Fig. 12.3 we plot the equipotentials in the plane z= 0 of the spherical shell withφA= 0 (no net charge in the shell). The current is in the clockwis e direction, the bold circumference represents the shell. Th e electric ﬁeld lines which are perpendicular to these equipotentials are also co ntained in the plane z= 0. This can be seen noting that for θ=π/2 rad we have Pq p′(cosθ) = 0 for p+qeven (see page 733 of the book by Arfken and Weber [202] combin ed with the recurrence relation (12.87) of the same work). Using the property that Ipq is null forp+qodd, we have that Eθ= 0 for both r<a andr>a. Figure 12.3: Equipotentials in the plane z= 0. The resistive spherical shell carries a clockwise steady current. The bold circumference represents the shell. The projection of the battery is represented by the bold stra ight line going from x=−atox= 0. The electric ﬁeld has no zcomponent, so the electric ﬁeld lines are orthogonal to the equipotentials in this plane. In Fig. 12.4 we plot the equipotentials in the plane x= 0. The current enters the plane of the paper on the left side of the bold circumferen ce and leaves the 158 plane of the paper on the right side. We utilized φA= 0. In this case the electric ﬁeld lines are not contained in this plane ( EϕorExare not null in the entire plane). Figure 12.4: Equipotentials in the plane x= 0 of the spherical shell with φA= 0. The bold circumference represents the shell. The current en ters the plane of the paper on the left side of the circumference and leaves the pap er on the right side. In Fig. 12.5 we plot the total surface charge density σtin the equatorial plane as a function of the azimuthal angle ϕ, normalized by the value of σt atϕ=π/4 rad. The presence of the term ( −1)qsin(qϕ) in Eqs. (12.29) and (12.30) causes a rapid variation in the calculation of σt. This can be seen in the oscillation of Fig. 12.5. The oscillations on the plot of σt(ϕ) shown in Figs. 12.5 and 11.7 probably occur because σtis proportional to the radial component of the electric ﬁeld that comes from diﬀerentiating a Fourier series. And someti mes there are con- vergence problems with the diﬀerentiation of Fourier serie s, as we saw in Section 11.4. By raising the number of terms in the Fourier series of σtwe increase only the number of oscillations in the curves. We did not succeed in putting the series solutions given by Eq s. (12.29) and (12.30) in closed analytical form. But utilizing the averag ing procedure pre- sented in Section 11.4, we obtained Fig. 12.6. The wiggles ar oundϕ=±π/2 rad should be due to numerical approximations without physical signiﬁcance. The real curve should be smooth like Fig. 11.8. Fig. 12.6 indicat es thatσt(ϕ) is linear with ϕfar from the battery ( i.e., aroundϕ= 0 rad), diverging close to it (whenϕ→ ±πrad). This is the important physical result. 159 Figure 12.5: Total surface charge density σt(ϕ) as a function of the azimuthal angleϕin the equatorial plane z= 0 of a resistive spherical shell carrying a steady azimuthal current, normalized by its value at ϕ=π/4 rad. We have utilized Eqs. (12.29) and (12.30) with the summation in pgoing from p= 1 to p= 100. Figure 12.6: Smoothed out plot of Fig. 12.5. 12.5 Conclusion We have obtained the surface charge density, σ, potential, φ, and electric ﬁeld, /vectorE, outside and inside a resistive spherical shell carrying a s teady azimuthal current. We have plotted the total surface charge density σtas a function of the azimuthal angle ϕ. We have found that σtis linear with ϕfar from the battery, diverging to inﬁnity close to it. At great distances from the spherical shell the potential is that of a point charge plus that of an electric di pole, Eq. (12.17). The total charge qand dipole moment /vector pof this system are given by Eq. (12.17) and 160 in the paragraph before it. Alternatively, they can also be f ound byq=/integraltext/integraltext σda and/vector p=/integraltext/integraltext σ/vector rda, wheredais an area element and the integration is over the surface of the system. The two approaches agree with one anot her, as expected. 161 162 Chapter 13 Resistive Toroidal Conductor with Azimuthal Current 13.1 Introduction The calculations of this Chapter were presented in 2003 and 2 004 [228, 229]. The only other attempt known to us to calculate the electric ﬁeld inside a resistive ring carrying a steady current due to charges distributed al ong the surface of the ring is that due to Weber in 1852 [32]. See the Appendix. Our goal is to ﬁnd a solution for the potential due to a current distributed in a ﬁnite volume of space, which creates an electric ﬁeld out side the Ohmic conductor. The only author who has fully solved a problem wit h the current bounded in a ﬁnite volume (beyond the case presented in the pr evious Chapter) is Jackson [12], who considered a coaxial cable of ﬁnite leng th. But as he considered a return conductor of zero resistivity, he obtai ned an electric ﬁeld only inside the cable, with no electric ﬁeld outside it. 13.2 Description of the Problem Consider a stationary toroidal Ohmic conductor (greater ra diusR0and smaller radiusr0) with a steady current I, constant over the length 2 πR0of the conduc- tor. We assume that the conductor has uniform resistivity, a nd the current is in the azimuthal direction, ﬂowing along the circular loop. Th e toroid is centered on the plane z= 0,zbeing its axis of symmetry. There is a battery located atϕ=πrad maintaining constant potentials at its extremities. Se e Fig. 13.1. We initially idealize the battery as of negligible thicknes s. Later on we consider the battery occupying a ﬁnite volume. The medium outside the conductor is supposed to be air or vacuum. 163 Figure 13.1: A toroidal Ohmic conductor with axis of symmetr y along the z axis, smaller radius r0and greater radius R0. A thin battery is located at ϕ=πrad maintaining constant potentials (represented by the + a nd - signs) in its extremities. A steady current ﬂows azimuthally in thi s circuit loop in the clockwise direction, from ϕ=πrad toϕ=−πrad. The goal here is to ﬁnd the electic potential φeverywhere in space, using the potential at the surface of the conductor as a boundary condi tion. The problem treated here can be applied to two cases: (a) the toroid is a fu ll homogeneous solid and the battery is a disc. See Fig. 13.2a. And (b) the tor oid is hollow and the battery is a circumference. See Fig. 13.2b. The symme try of this problem suggests the approach of toroidal coordinates ( η,χ,ϕ ) see Figure 13.3 [224, p. 112]. These coordinates were introduced by C. Neuma nn [230], who studied the distribution of surface charges in a metallic ri ng kept at a constant potential [231, p. 516]. Figure 13.2: The two cases being considered here: (a) a full s olid resistive toroidal conductor, with an azimuthal volume current densi ty/vectorJthrough the cross-section; (b) a hollow resistive toroidal conductor, with an azimuthal sur- face current density /vectorKthrough the circumference 2 πr0of the hollow toroidal shell. 164 Figure 13.3: Toroidal coordinates ( η,χ,ϕ ). These coordinates are deﬁned by: x≡asinhηcosϕ coshη−cosχ, y ≡asinhηsinϕ coshη−cosχ, z ≡asinχ coshη−cosχ. (13.1) Hereais a constant that gives the radius of a circumference in the z= 0 plane described by η→ ∞ (that is, when η→ ∞ we havex=acosϕ,y=asinϕ andz= 0). The values assumed by the toroidal coordinates are: 0 ≤η <∞, −πrad≤χ≤πrad and −πrad≤ϕ≤πrad. The inverse transformations are given by: η= arctanh2a/radicalbig x2+y2 x2+y2+z2+a2, χ = arctan2za x2+y2+z2−a2, ϕ= arctany x. (13.2) It is also convenient to present here the expressions for sin hη, coshηand cosχ: sinhη=2a/radicalbig x2+y2 /radicalbig (x2+y2+z2−a2)2+ 4a2z2, (13.3) coshη=x2+y2+z2+a2 /radicalbig (x2+y2+z2−a2)2+ 4a2z2, (13.4) cosχ=x2+y2+z2−a2 /radicalbig (x2+y2+z2−a2)2+ 4a2z2. (13.5) 165 The surface of the toroid is described by a constant η0. The internal (exter- nal) region of the toroid is characterized by η>η 0(η<η 0). The greater radius R0and the smaller radius r0are related to η0and toabyR0=acoshη0/sinhη0 andr0=a/sinhη0. See Figs. 13.1 and 13.3. That is, R0/r0= coshη0and η0= cosh−1(R0/r0). Laplace’s equation for the electric potential φ,∇2φ= 0, has the following form in toroidal coordinates: ∇2φ=(coshη−cosχ)2 a2sinhη/bracketleftbigg∂ ∂η/parenleftbiggsinhη coshη−cosχ∂φ ∂η/parenrightbigg + sinhη∂ ∂χ/parenleftbigg1 coshη−cosχ∂φ ∂χ/parenrightbigg/bracketrightbigg +(coshη−cosχ)2 a2sinh2η∂2φ ∂ϕ2= 0.(13.6) It can be solved in toroidal coordinates with the method of se paration of variables (by a procedure known as R-separation), leading t o a solution of the form [224, p. 112]: φ(η,χ,ϕ ) =/radicalbig coshη−cosχH(η)X(χ)Φ(ϕ). (13.7) The functions H,X, and Φ which appear here satisfy, respectively, the ordinar y equations (with Υ ≡coshη, and where pandqare constants): (Υ2−1)H′′+ 2ΥH′−[(p2−1/4) +q2/(Υ2−1)]H= 0,(13.8) X′′+p2X= 0,(13.9) Φ′′+q2Φ = 0.(13.10) 13.3 General Solution The solutions of Eqs. (13.9) and (13.10) for p/negationslash= 0 andq/negationslash= 0 are linear combinations of the general forms Xp(χ) =Cpχcos(pχ) +Dpχsin(pχ) and Φq(ϕ) =Cqϕcos(qϕ) +Dqϕsin(qϕ), respectively, where Cpχ,Dpχ,Cqϕand Dqϕare constants. When p=q= 0 the solutions reduce to, respectively, X0(χ) =C0χ+D0χχand Φo(ϕ) =C0ϕ+D0ϕϕ. Eq. (13.8) is the associ- ated Legendre equation, whose solutions are the associated Legendre functions Pq p−1 2(coshη) andQq p−1 2(coshη), known as toroidal Legendre polynomials [232, p. 173]. The solution must be periodic in ϕ,i.e.,φ(η,χ,ϕ + 2π) =φ(η,χ,ϕ ), and inχ,i.e.,φ(η,χ+ 2π,ϕ) =φ(η,χ,ϕ ). This condition implies that D0ϕ= 0, D0χ= 0,q= 1,2,3,..., andp= 1,2,3,... The functions Qq p−1 2(coshη) are irregular in η= 0 (which corresponds to the zaxis, or to great distances from the toroid). For this reason we eliminate them as physical solutions for this problem in the region outside the toroid (that is, η < η 0). The general solution consists of linear combinations of a ll possible 166 regular solutions of Pq p−1 2(coshη),Xp(χ) and Φq(ϕ): φ(η≤η0,χ,ϕ) =/radicalbig coshη−cosχ/braceleftBigg∞/summationdisplay q=0[Cqϕcos(qϕ) +Dqϕsin(qϕ)] ×/bracketleftBigg∞/summationdisplay p=0[Cpχcos(pχ) +Dpχsin(pχ)]Pq p−1 2(coshη)/bracketrightBigg/bracerightBigg . (13.11) We utilized the fact that sin 0 = 0 and cos0 = 1 to sum up from p=q= 0 to ∞. HereP0 p−1 2(coshη)≡Pp−1 2(coshη) are the Legendre functions [202, p. 724]. 13.4 Particular Solution for a Steady Azimuthal Current The surface of the toroid is described by a constant η0. Here we study the case of a steady current ﬂowing in the azimuthal ϕdirection along the Ohmic toroid. For this reason we suppose that the potential along the surfa ce of the toroid is linear in ϕ,φ(η0,χ,ϕ) =φA+φBϕ/2π. This potential can be expanded in Fourier series in ϕ: φ(η0,χ,ϕ) =φA+φBϕ 2π=φA+φB π/bracketleftBigg∞/summationdisplay q=1(−1)q−1 qsin(qϕ)/bracketrightBigg .(13.12) Fig. 13.4 shows the Fourier expansion of the potential along the conductor surface as a function of ϕ. The oscillations close to ϕ=±πrad are due to a Fourier series with a ﬁnite number of terms. The overshootin g is known as the Gibbs phenomenon, a peculiarity of the Fourier series at a si mple discontinuity [202, p. 783–7]. Figure 13.4: Fourier expansion of the potential along the co nductor surface as a function of the azimuthal angle ϕ, Eq. (13.12) with φA= 0 andφB=φ0. 167 We assume that the potential inside the full solid toroidal O hmic conductor (that is, for η>η 0), Fig. 13.2a, is also given by Eq. (13.12), namely: φ(η>η 0,χ,ϕ) =φA+φBϕ 2π. (13.13) The electric ﬁeld inside the solid toroid can be expressed in cylindrical coordi- nates (ρ,ϕ,z ) simply as: /vectorE=−∇φ=−φB 2πρˆϕ . (13.14) This electric ﬁeld does not lead to any accumulation of charg es inside a full solid conductor because ∇ ·/vectorE= 0. These are reasonable results. The potential satisﬁes Lapla ce’s equation ∇2φ= 0, as expected. The electric ﬁeld is inversely proportiona l to the distance ρ=/radicalbig x2+y2from thezaxis. This was to be expected as we are assuming a conductor of uniform resistivity. The diﬀerence of potent ial ∆φcreated by the battery at ϕ=πrad can be related to the azimuthal electric ﬁeld by a line integral: ∆φ=−/integraldisplay−π ϕ=π/vectorE·d/vectorℓ=−Eϕ2πρ . (13.15) Hereρis the radius of a circular path centered on the zaxis and located inside or along the surface of the toroid. This shows that Eϕshould be inversely proportional to ρ, as found in Eq. (13.14). Comparing Eqs. (13.14) and (13.15) yields: ∆φ=φB. (13.16) By Ohm’s law /vectorJ=g/vectorE(wheregis the uniform conductivity of the wire) we can see that /vectorJis also inversely proportional to the distance ρfrom thezaxis inside a full solid homogeneous toroidal conductor. We now consider the solution outside the conductor, valid fo r the cases of a solid and a hollow toroid. We calculate Eq. (13.11) with η=η0and utilize Eq. (13.12) as a boundary condition of this problem. As we do not have terms with cos( qϕ) in Eq. (13.12), this means that Cqϕ= 0 forq= 1,2,3,...Comparing Eq. (13.11) at η=η0 with Eq. (13.12) yields two equations connecting φAandφBto theC’s andD’s, namely: φA=C0ϕ/radicalbig coshη0−cosχ/braceleftBigg∞/summationdisplay p=0[Cpχcos(pχ) +Dpχsin(pχ)]Pp−1 2(coshη0)/bracerightBig , (13.17) φB=πqDqϕ (−1)q−1/radicalbig coshη0−cosχ/braceleftBigg∞/summationdisplay p=0[Cpχcos(pχ) 168 +Dpχsin(pχ)]Pq p−1 2(coshη0)/bracerightBig . (13.18) We now isolate the term 1 /√coshη0−cosχin Eqs. (13.17) and (13.18), expanding it in Fourier series. That is: 1√coshη0−cosχ=1 2π/braceleftBigg∞/summationdisplay p=0(2−δ0p)/bracketleftbigg/integraldisplayπ −πcos(pχ′)dχ′ √coshη0−cosχ′/bracketrightbigg cos(pχ)/bracerightBigg =√ 2 π/bracketleftBigg∞/summationdisplay p=0(2−δ0p)Qp−1 2(coshη0)cos(pχ)/bracketrightBigg , (13.19) whereδwpis the Kronecker delta, which is zero for w/negationslash=pand one for w=p. In the last passage we utilized an integral representation o fQp−1 2(coshη) [232, p. 156, Eq. (10)]: Qp−1 2(coshη0) =1 2√ 2/integraldisplayπ −πcos(pχ′)dχ′ √coshη0−cosχ′. (13.20) As in Eq. (13.19) we do not have terms of sin( pχ), this means that Dpχ= 0 in Eqs. (13.17) and (13.18). Using Eq. (13.19) with Eq. (13.1 7) yields (for p= 0,1,2,...): Ap≡C0ϕCpχ=φA(2−δ0p) 2πPp−1 2(coshη0)/integraldisplayπ −πcos(pχ′)dχ′ √coshη0−cosχ′ =√ 2φA(2−δ0p) πQp−1 2(coshη0) Pp−1 2(coshη0). (13.21) Using Eq. (13.19) with Eq. (13.18) yields: Bpq≡DqϕCpχ=φB(−1)q−1(2−δ0p) 2qπ2Pq p−1 2(coshη0)/integraldisplayπ −πcos(pχ′)dχ′ √coshη0−cosχ′ =√ 2φB(−1)q−1(2−δ0p) qπ2Qp−1 2(coshη0) Pq p−1 2(coshη0). (13.22) The ﬁnal solution outside the toroid is given by: φ(η≤η0,χ,ϕ) =/radicalbig coshη−cosχ/braceleftBigg∞/summationdisplay p=0Apcos(pχ)Pp−1 2(coshη) +∞/summationdisplay q=1sin(qϕ)/bracketleftBigg∞/summationdisplay p=0Bpqcos(pχ)Pq p−1 2(coshη)/bracketrightBigg/bracerightBigg , (13.23) where the coeﬃcients ApandBpqare given by Eqs. (13.21) and (13.22), respec- tively. 169 For the region inside the hollow toroid (that is, η > η 0), Fig. 13.2b, we havePq p−1 2(coshη→ ∞)→ ∞, whileQq p−1 2(coshη→ ∞)→0. For this reason we eliminate Pq p−1 2(coshη) as physical solutions for the region inside the hollow toroid. The potential is then given by: φ(η>η 0,χ,ϕ) =φA+/radicalbig coshη−cosχ ×/braceleftBigg∞/summationdisplay q=1sin(qϕ)/bracketleftBigg∞/summationdisplay p=0B′ pqcos(pχ)Qq p−1 2(coshη)/bracketrightBigg/bracerightBigg , (13.24) where the coeﬃcients B′ pqare deﬁned by: B′ pq≡φB(−1)q−1(2−δ0p) 2qπ2Qq p−1 2(coshη0)/integraldisplayπ −πcos(pχ′)dχ′ √coshη0−cosχ′ =√ 2φB(−1)q−1(2−δ0p) qπ2Qp−1 2(coshη0) Qq p−1 2(coshη0). (13.25) Note that the potential inside the solid toroid, Eq. (13.13) , and the potential inside the hollow toroid, Eq. (13.24), are diﬀerent. This ha ppens because the discontinuous boundary condition, Eq. (13.12), applies fo r anyη > η 0inside the solid toroid, particularly for ϕ→πrad (φ→φA+φB/2) andϕ→ −πrad (φ→φA−φB/2), where the disc battery is located. See Fig. 13.2a. This do es not happen to the hollow toroid, where the battery is a circum ference, and the potential must be continuous inside the hollow toroid. See F ig. 13.2b. We plotted the equipotentials of a full solid toroid on the pl anez= 0 in Fig. 13.5 with φA= 0 andφB=φ0. We utilized a toroidal surface described by η0= 2.187. Figure 13.6 shows a plot of the equipotentials of the full sol id toroid in the planex= 0 (perpendicular to the current), also with φA= 0,φB=φ0,R0= 1 andη0= 2.187. 13.5 Potential in Particular Cases We now analyze the potential outside the toroid, Eq. (13.23) , in four regions: (A) far away from the toroid, (B) close to the origin, (C) alon g thezaxis, and (D) along the circumference described by x2+y2=a2in the plane z= 0. (A) For great distances from the toroid (that is, r=/radicalbig x2+y2+z2≫a), Eqs. (13.2) to (13.5) yield: η≈2a/radicalbig x2+y2 r2≪1, (13.26) coshη≈1 +2a2(x2+y2) r4→1, (13.27) 170 Figure 13.5: Equipotentials for a resistive full solid toro idal conductor in the planez= 0. The bold circumferences represent the borders of the tor oid. The current runs in the azimuthal direction, from ϕ=πrad toϕ=−πrad. The thin battery is on the left ( ϕ=πrad). We have used R0= 1 andη0= 2.187. Figure 13.6: Equipotentials in the plane x= 0 for a resistive full solid toroidal conductor carrying a steady azimuthal current, Eq. (13.23) withφA= 0 and φB=φ0. The bold circumferences represent the conductor surface. We have usedR0= 1 andη0= 2.187. cosχ≈1−2a2z2 r4→1, (13.28) χ≈2az r2≪1, (13.29) /radicalbig coshη−cosχ≈a√ 2 r≪1. (13.30) For coshη≈1 +ǫ, where 0<ǫ≪1, we have the following expansion [232, pp. 163 and 173]: Pq p−1 2(1 +ǫ)≈Γ/parenleftbig p+q+1 2/parenrightbig 2q/2q! Γ/parenleftbig p−q+1 2/parenrightbigǫq/2/braceleftbigg 1 +ǫ/bracketleftbiggp2−1 4 2(1 +q)−q 4/bracketrightbigg/bracerightbigg .(13.31) 171 That is, for q= 0 and for q= 1,2,3,...we have, respectively: Pp−1 2(1 +ǫ)≈1 +ǫ/parenleftbiggp2 2−1 8/parenrightbigg →1, (13.32) Pq p−1 2(1 +ǫ)≈2−q/2Γ(p+q+1 2) q!Γ(p−q+1 2)ǫq/2≪1. (13.33) This means that the terms which appear in the potential for η≪1, up to the orderǫ1/2, are those which have the polynomials with q= 0 and with q= 1. That is,Pp−1 2(coshη≈1 +ǫ)≈1 andP1 p−1 2(coshη≈1 +ǫ)≈(p2−1/4)/radicalbig ǫ/2. The potential φ, Eq. (13.23), at great distances from the origin, is given in spherical coordinates ( r,θ,ϕ) by (where ǫ= 2a2sin2θ/r2): φ(r≫a,θ,ϕ )≈a√ 2 r/braceleftBigg∞/summationdisplay p=0cos/parenleftbigg p2acosθ r/parenrightbigg ×/bracketleftbigg Ap+Bp1/parenleftbigg p2−1 4/parenrightbigga rsinϕsinθ/bracketrightbigg/bracerightbigg , (13.34) so thatφ(r→ ∞)→0, as expected. (B) The potential close to the origin (that is, r≪a) can be calculated in the same manner. In this approximation: η≈2/radicalbig x2+y2 a≪1, (13.35) coshη≈1 +2(x2+y2) a2→1, (13.36) cosχ≈ −1 +2z2 a2→ −1, (13.37) χ≈π−2z a→π , (13.38) /radicalbig coshη−cosχ≈√ 2 +x2+y2−z2 √ 2a2→√ 2. (13.39) The potential (13.23) can be expressed as (with ǫ= 2r2sin2θ/a2): φ(r≪a,θ,ϕ )≈√ 2/braceleftBigg∞/summationdisplay p=0(−1)pcos/parenleftbigg p2rcosθ a/parenrightbigg ×/bracketleftbigg Ap+Bp1/parenleftbigg p2−1 4/parenrightbiggr asinϕsinθ/bracketrightbigg/bracerightbigg . (13.40) (C) Along the zaxis we have/radicalbig x2+y2= 0. From Eqs. (13.2) to (13.5) we have: η= 0, (13.41) 172 coshη= 1, (13.42) cosχ=z2−a2 z2+a2, (13.43) /radicalbig coshη−cosχ=a/radicalbigg 2 z2+a2. (13.44) The potential (13.23) along the zaxis can be written as: φ/parenleftBig r=/radicalbig x2+y2+z2=|z|,θ,ϕ/parenrightBig =a/radicalbigg 2 z2+a2/bracketleftBigg∞/summationdisplay p=0Apcos/parenleftbigg parccosz2−a2 z2+a2/parenrightbigg/bracketrightBigg . (13.45) (D) In the circumference described by x2+y2=a2, along the plane z= 0, we have η→ ∞ . The associated Legengre functions Pq p−1 2(coshη) and Qq p−1 2(coshη), forη≫1 (and, therefore, for cosh η≫1), can be approximated utilizing [232, p. 164]: Qq p−1 2(coshη≫1)≈(−1)q√πΓ/parenleftbig p+q+1 2/parenrightbig 2p+1 2p! coshp+1 2η,for anyp , (13.46) Pq p−1 2(coshη≫1)≈2p−1 2(p−1)! coshp−1 2η√πΓ/parenleftbig p−q+1 2/parenrightbig,forp>0, (13.47) where Γ is the gamma function [202, p. 591]. The potential ins ide the hollow toroid, Eq. (13.24), assumes the following form along this c ircumference: φ(η→ ∞,χ,ϕ) =φA−φB π3/2Q−1 2(coshη0)/bracketleftBigg∞/summationdisplay q=1sin (qϕ)Γ/parenleftbig q+1 2/parenrightbig qQq −1 2(coshη0)/bracketrightBigg .(13.48) 13.6 Electric Field and Surface Charges In toroidal coordinates the gradient is written as: ∇φ=1 a(coshη−cosχ)/parenleftbigg ˆη∂φ ∂η+ ˆχ∂φ ∂χ+ˆϕ sinhη∂φ ∂ϕ/parenrightbigg . (13.49) The electric ﬁeld can then be calculated by /vectorE=−∇φ, whose components for the region outside the toroid ( η<η 0) are given by: Eη=−sinhη√coshη−cosχ a/braceleftBigg∞/summationdisplay p=0cos(pχ)/braceleftbigg Ap/bracketleftbigg1 2Pp−1 2(coshη) 173 + (coshη−cosχ)Pp−1 2′(coshη)/bracketrightBig +∞/summationdisplay q=1sin(qϕ)Bpq/bracketleftbigg1 2Pq p−1 2(coshη) + (coshη−cosχ)Pq p−1 2′(coshη)/bracketrightbigg/bracerightBigg/bracerightBigg , (13.50) Eχ=−√coshη−cosχ a/braceleftBigg∞/summationdisplay p=0/bracketleftbiggsinχcos(pχ) 2−p(coshη−cosχ)sin(pχ)/bracketrightbigg ×/bracketleftBigg ApPp−1 2(coshη) +∞/summationdisplay q=1sin(qϕ)BpqPq p−1 2(coshη)/bracketrightBigg/bracerightBigg , (13.51) Eϕ=−(coshη−cosχ)3/2 asinhη/braceleftBigg∞/summationdisplay q=1qcos(qϕ)/bracketleftBigg∞/summationdisplay p=0Bpqcos(pχ)Pq p−1 2(coshη)/bracketrightBigg/bracerightBigg , (13.52) wherePq p−1 2′(coshη) are the derivatives of the Pq p−1 2(coshη) relative to cosh η. The electric ﬁeld inside the full solid toroid ( η>η 0) is given simply by: Eη= 0, Eχ= 0, Eϕ=−coshη−cosχ asinhηφB 2π=−φB 2π/radicalbig x2+y2. (13.53) The total surface charge distribution σtthat creates the electric ﬁeld inside (and outside of) the conductor, keeping the current ﬂowing, can be obtained with Gauss’s law (by choosing a Gaussian surface involving a small portion of the conductor surface) for the full solid toroid, Fig. 13.2a : σt(η0,χ,ϕ) =ε0/bracketleftBig /vectorE(η<η 0)·(−ˆη) +/vectorE(η>η 0)·ˆη/bracketrightBig η0 =ε0sinhη0 a/braceleftbiggφA+φBϕ/2π 2+ (coshη0−cosχ)3/2 ×/braceleftBigg∞/summationdisplay p=0cos(pχ)/bracketleftBig ApPp−1 2′(coshη0) +∞/summationdisplay q=1sin(qϕ)BpqPq p−1 2′(coshη0)/bracketrightBig/bracerightBigg/bracerightBigg . (13.54) 13.7 Thin Toroid Approximation Suppose that the toroid is very thin, with its radii describe d by a greater radius R0=acoshη0/sinhη0≈aand smaller radius r0=a/sinhη0, such that r0≪ 174 R0. See Fig. 13.1. The surface of the toroid is described by η0≫1 and, consequently, cosh η0≫1. In this approximation, the potential inside the hollow toro id and inside the full solid toroid is given by the same expression, Eq. (13.13 ). The electric ﬁeld is given by Eqs. (13.14) and (13.53). This means that there is no distribution of surface charges in the internal surface of a hollow thin to roid. The Legendre functions of the second kind calculated at η=η0, given by Qp−1 2(coshη0), appear in the coeﬃcients ApandBpqof the potential outside the toroid, Eqs. (13.21) and (13.22), respectively. As Eq. ( 13.46), calculated in η=η0and forq= 0, has a factor of cosh−p−1/2η0≪1, we can neglect all terms in Eq. (13.23) having p>0 compared with the term having p= 0. The potential outside the thin toroid ( η0≫1) can then be written as: φ(η≤η0,χ,ϕ) =/radicalBigg coshη−cosχ coshη0/braceleftBigg φAP−1 2(coshη) P−1 2(coshη0) +φB π/bracketleftBigg∞/summationdisplay q=1(−1)q−1 qsin(qϕ)Pq −1 2(coshη) Pq −1 2(coshη0)/bracketrightBigg/bracerightBigg . (13.55) It is interesting to ﬁnd the expressions for the potential an d electric ﬁeld outside but in the vicinity of the conductor (that is, η0> η≫1). A series expansion of the functions Pq −1 2(Υ) andPq −1 2′(Υ) around Υ → ∞ gives as the most relevant terms [232, p. 173]: Pq −1 2(Υ)≈/radicalbig 2/π Γ(1/2−q)ln(2Υ) −ψ(1/2−q)−γ√ Υ, (13.56) Pq −1 2′(Υ)≈/radicalbig 2/π Γ(1/2−q)1 Υ3/2/bracketleftbigg 1−ln(2Υ) −ψ(1/2−q)−γ 2/bracketrightbigg , (13.57) whereψ(z) = Γ′(z)/Γ(z) is the digamma function, and γ≈0.577216 is the Euler gamma. The potential just outside the thin toroid, Eq. (13.55), can then be written in this approximation as (utilizing that ψ(1/2) +γ=−ln4): φ(η0≥η≫1,χ,ϕ) =φAln(8 coshη) ln(8 coshη0) +φB π/bracketleftBigg∞/summationdisplay q=1(−1)q−1 qsin(qϕ)ln(2 coshη)−ψ/parenleftbig1 2−q/parenrightbig −γ ln(2 coshη0)−ψ/parenleftbig1 2−q/parenrightbig −γ/bracketrightBigg . (13.58) This equation is valid for −πrad≤ϕ≤πrad, even close to the battery. 175 The electric ﬁeld close to the surface of the thin toroid, jus t outside it, obtained from Eq. (13.58), is given by: Eη=−sinhη a/braceleftbiggφA ln(8 coshη0) +φB π/bracketleftBigg∞/summationdisplay q=1(−1)q−1 qsin(qϕ) ln(2 coshη0)−ψ/parenleftbig1 2−q/parenrightbig −γ/bracketrightBigg/bracerightBigg , (13.59) Eχ= 0, (13.60) Eϕ=−φB πa/bracketleftBigg∞/summationdisplay q=1(−1)q−1cos(qϕ)ln(2 coshη)−ψ/parenleftbig1 2−q/parenrightbig −γ ln(2 coshη0)−ψ/parenleftbig1 2−q/parenrightbig −γ/bracketrightBigg .(13.61) Note that Eϕ(η0) =−φB πa/bracketleftBigg∞/summationdisplay q=1(−1)q−1cos(qϕ)/bracketrightBigg =−φB 2πa. (13.62) That is, it coincides exactly with the electric ﬁeld inside t he solid toroid, Eq. (13.14). This is a divergent series presented in Eq. (11.30) which ari ses from diﬀerenti- ation of a convergent Fourier series, as we discussed in Sect ion 11.4. It can be handled by the average procedure presented in Eq. (11.32). The surface charge distribution in this thin toroid approxi mation is given by, from Eq. (13.59): σ(η0≫1,χ,ϕ) =−ε0Eη(η0) =ε0sinhη0 a/bracketleftbiggφA ln(8 coshη0) +φB π/parenleftBigg∞/summationdisplay q=1(−1)q−1 qsin(qϕ) ln(2 coshη0)−ψ/parenleftbig1 2−q/parenrightbig −γ/parenrightBigg/bracketrightBigg , (13.63) which is also valid for −πrad≤ϕ≤πrad. As there is no surface charge distribution in the internal surface of a hollow thin toroid , this expression means the total surface charge distribution which exists only in t he external surface of the (hollow or solid) thin toroid. In Fig. 13.7 we plotted the density of surface charges σas a function of the azimuthal angle ϕobtained from Eq. (13.63). We can see that σis linear with ϕclose toϕ= 0 rad. Close to the battery σdiverges to inﬁnity (that is, σ→ ∞ whenϕ→ ±πrad). To our knowledge the ﬁrst to conclude correctly that th e surface charge density in a resistive ring carrying a steady current grows toward the battery as a function of the azimuthal angle ϕin a pace faster than linearly was Weber in 1852. See the Appendix A. From Figure 13.7 and Eq. (11.11) we can then write the summati on of Eq. (13.63) for a thin toroid and far from the battery (that is , forη0≫1 andϕ≪πrad) as 176 Figure 13.7: Density of surface charges as a function of the a zimuthal angle ϕ obtained from Eq. (13.63) with φA= 0 andη0= 10 (R0/r0= 1.1×104). It is linear with ϕwhenϕ≈0 rad but then diverges to inﬁnity close to the battery. ∞/summationdisplay q=1(−1)q−1 qsin(qϕ) ln(2 coshη0)−ψ/parenleftbig1 2−q/parenrightbig −γ≡g(η0)ϕ 2. (13.64) Hereg(η0) is a dimensionless function of η0deﬁned by this equation. With this deﬁnition Eq. (13.63) can be written as σ(η0≫1,χ,ϕ≪π)≈ε0sinhη0 aφA ln(8 coshη0)+ε0sinhη0 ag(η0)φBϕ 2π ≡σA+σBϕ 2π. (13.65) The constants σAandσBare deﬁned by this equation, namely σA≡ε0sinhη0 aφA ln(8 coshη0), (13.66) σB≡ε0sinhη0 ag(η0)φB. (13.67) Combining Eq. (13.67) with Eq. (13.62) we can write the tange ntial compo- nent of the electric ﬁeld Eϕat the surface of the thin toroid as Eϕ(η0≫1) =−φB 2πa=−σB 2πε0sinhη0g(η0). (13.68) As we will see in Appendix A, Weber was the ﬁrst to obtain an ana logous to this result. His approach of dealing with this problem lea ds to a tangential component of the electric ﬁeld for a very thin toroid as given by Eq. (A.20), namely: Eϕ(η0≫1)≈ −r0σB 2πε0R0/parenleftbigg ln8R0 r0−π 2/parenrightbigg . (13.69) By comparing Eqs. (13.68) and (13.69) for a very thin toroid ( η0≫1, a≈R0, sinhη0=a/r0≈R0/r0) we can then try to ﬁt g(η0) as 177 g(η0)≡1 ln(R0/r0) +K0. (13.70) The constant K0deﬁned by this equation should be a function of η0and, ac- cording to Eqs. (13.68) and (13.69), should tend to ln 8 −π/2 = 0.509 when η0→ ∞. In Eq. (13.71) we present a least-square ﬁtting of g(η0) given by Eq. (13.64) with 10000 terms in the summation, for ϕvarying from −π/100 rad to π/100 rad, with steps of π/10000 rad. At the last column we present for each value of g(η0) the corresponding value of K0as given by Eq. (13.70).  η0R0/r0g(η0)K0 12.206 1050.0830051 0 .534 23.719 10100.0424678 0 .521 35.232 10150.0285259 0 .517 46.745 10200.0214746 0 .515 69.771 10300.0143698 0 .513 92.797 10400.0107974 0 .511 115.822 10500.00864749 0 .511 230.952 101000.00433335 0 .510 461.21 102000.00216907 0 .510 (13.71) This equation indicates that K0→ln 8−π/2, as expected if we apply Weber’s approach in order to deal with this problem. See Appendix A. S upposing that this is the case, we can then write the surface charge density for a thin toroid and far from the battery approximately as σ(η0≫1,χ,ϕ≪π)≈σA+σBϕ 2π ≈ε0 r0φA ln(8R0/r0)+ε0 r0φB ln(R0/r0) + ln 8 −π/2ϕ 2π. (13.72) In this approximation of a thin toroid, the surface charge de nsity given by Eq. (13.63) does not depend upon the angle χ. This means that the linear charge density λ(ϕ) is given simply by 2 πr0σ, namely: λ(η0≫1,ϕ) =2πr0ε0sinhη0 a/bracketleftbiggφA ln(8 coshη0) +φB π/parenleftBigg∞/summationdisplay q=1(−1)q−1 qsin(qϕ) ln(2 coshη0)−ψ/parenleftbig1 2−q/parenrightbig −γ/parenrightBigg/bracketrightBigg . (13.73) Far from the battery this reduces to, from Eq. (13.64): λ(η0≫1,ϕ≪π)≈2πr0ε0sinhη0 aφA ln(8 coshη0)+2πr0ε0sinhη0 ag(η0)φBϕ 2π 178 ≡λA+λBϕ 2π. (13.74) The constants λAandλBwere deﬁned by this equation. We can calculate the total charge qAof the thin toroid as a function of the constant electric potential φA. For this end, we integrate the surface charge densityσinχandϕ(in the approximation cosh η0≫1): qA=/integraldisplayπ −πhχdχ/integraldisplayπ −πhϕdϕσ(χ,ϕ) =4π2ε0φAR0 ln(8 coshη0)≈4π2ε0φAR0 ln(8R0/r0),(13.75) wherehη=hχ=a/(coshη−cosχ) andhϕ=asinhη/(coshη−cosχ) are the scale factors in toroidal coordinates [233]. Notice that fr om Eq. (13.75) we can obtain the capacitance of the thin toroid [234, p. 127]: C=qA φA=4π2ε0R0 ln(8 coshη0)=4π2ε0R0 ln(8R0/r0). (13.76) The potential along the zaxis is given by, from Eq. (13.45) in the thin toroid approximation: φ/parenleftBig r=/radicalbig x2+y2+z2=|z|,θ,ϕ/parenrightBig =qA 4πε01√ z2+a2. (13.77) Eq. (13.77) coincides with the coulombian result of a charge d thin toroid of radiusain thez= 0 plane and total charge qA. As we have seen, in the case of a thin toroid the term in the pote ntial with p= 0 is much larger than the terms with p >0. This means that Eq. (13.34) reduces to φ(r≫a,θ,ϕ )≈a√ 2 r/bracketleftbigg A0−B01 4a rsinϕsinθ/bracketrightbigg . (13.78) With Eqs. (13.21) and (13.22) we obtain φ(r≫a,θ,ϕ )≈a√ 2 r/bracketleftBigg√ 2φA πQ−1 2(coshη0) P−1 2(coshη0) −√ 2φB π2Q−1 2(coshη0) P1 −1 2(coshη0)a 4rsinϕsinθ/bracketrightBigg . (13.79) We now simplify the last two equations utilizing Eqs. (13.46 ), (13.56) and the relations Γ(1/2) =√π ,Γ(−1/2) =−2√π , (13.80) ψ(1/2) +γ=−ln4, ψ(−1/2) +γ= 2−ln 4. (13.81) 179 This yields: φ(r≫a,θ,ϕ )≈aπ r/braceleftbiggφA ln(8 coshη0)+φB 2π[ln(8 coshη0)−2]a rsinϕsinθ/bracerightbigg . (13.82) Utilizing a similar procedure beginning with Eq. (13.40) yi elds: φ(r≪a,θ,ϕ )≈π/braceleftbiggφA ln(8 coshη0)+φB 2π[ln(8 coshη0)−2]r asinϕsinθ/bracerightbigg . (13.83) 13.8 Comparison of the Thin Toroid Carrying a Steady Current with the Case of a Straight Cylindrical Wire Carrying a Steady Cur- rent It is useful to deﬁne a new coordinate system: s′=aϕ , ρ′=/radicalbigg/parenleftBig/radicalbig x2+y2−a/parenrightBig2 +z2. (13.84) We can interpret s′as a distance along the toroid surface in the ϕdirection, andρ′as the shortest distance from the circumference x2+y2=a2located in the planez= 0. When η0> η≫1 (that is,r0< ρ′≪a), Eqs. (13.84) and (13.4) result in cosh η≈a/ρ′≫1 and coshη0≈a/r0≫1. Forη0≥η≫1 we can approximate the term inside square brackets of Eq. (13.5 8) by (taking into account Eq. (11.11)): ∞/summationdisplay q=1(−1)q−1 qsin(qϕ)ln(2 coshη)−ψ/parenleftbig1 2−q/parenrightbig −γ ln(2 coshη0)−ψ/parenleftbig1 2−q/parenrightbig −γ ≈/parenleftBigg∞/summationdisplay q=1(−1)q−1 qsin(qϕ)/parenrightBigg ln(coshη) ln(coshη0)=ϕ 2ln(coshη) ln(coshη0). (13.85) Utilizing Eqs. (13.85) and (13.84) into Eq. (13.58) yields: φ(η0≥η≫1,χ,ϕ) =φAln(8a/ρ′) ln(8a/r0)+φBs′ 2πaln(a/ρ′) ln(a/r0). (13.86) Eq. (13.86) can be written in a slightly diﬀerent form. Consi der a certain piece of the toroid between the angles ϕ0and−ϕ0, with potentials in these 180 extremities given by φR=φA+φBϕ0/2πandφL=φA−φBϕ0/2π, respectively. This piece has a length of ℓ= 2aϕ0. The potential can then be written as: φ=φAln(ℓ/ρ′)−ln(ℓ/8a) ln(ℓ/r0)−ln(ℓ/8a)+φBϕ0s′ πℓln(ℓ/ρ′)−ln(ℓ/a) ln(ℓ/r0)−ln(ℓ/a) ≈/bracketleftbiggφR+φL 2+ (φR−φL)s′ ℓ/bracketrightbiggln(ℓ/ρ′) ln(ℓ/r0). (13.87) In the last approximation we neglected the terms ln( ℓ/8a) and ln(ℓ/a) in com- parison with the terms ln( ℓ/ρ′) and ln(ℓ/r0) utilizing the approximation r0< ρ′≪a(so thatℓ/r0>ℓ/ρ′≫ℓ/a). The electric ﬁeld can be expressed in this approximation as: /vectorE=−/bracketleftbiggφR+φL 2+ (φR−φL)s′ ℓ/bracketrightbiggˆη ρ′ln(ℓ/r0)−φR−φL ℓln(ℓ/ρ′) ln(ℓ/r0)ˆϕ .(13.88) Eqs. (13.87) and (13.88) can be compared to Eqs. (6.17) and (6 .18), re- produced as Eqs. (13.89) and (13.90), respectively. These e quations refer to a long straight cylindrical conductor of radius r0carrying a constant current, in cylindrical coordinates ( ρ′,ϕ,z) (note that the conversions from toroidal to cylindrical coordinates in this approximation are ˆ η≈ −ˆρ′and ˆϕ≈ˆz). In this case, the cylinder has a length ℓand radius r0≪ℓ, with potentials φLandφR in the extremities of the conductor, and RI=φL−φR: φ(r≥a) =/bracketleftbiggφR+φL 2+ (φR−φL)z ℓ/bracketrightbiggln(ℓ/ρ′) ln(ℓ/r0), (13.89) /vectorE(ρ′≥a) =/bracketleftbiggφR+φL 2+ (φR−φL)z ℓ/bracketrightbiggˆρ′ ρ′ln(ℓ/r0)−φR−φL ℓln(ℓ/ρ′) ln(ℓ/r0)ˆz . (13.90) The potential in the region close to the thin toroid coincide s with the cylin- drical solution, as expected. 13.9 Charged Toroid without Current Consider a toroid described by η0, without current but charged to a constant potentialφA. UsingφB= 0 in Eqs. (13.23), (13.13) and (13.24) we have the potential inside and outside the toroid, respectively: φ(η≥η0,χ,ϕ) =φA, (13.91) φ(η≤η0,χ,ϕ) =/radicalbig coshη−cosχ/bracketleftBigg∞/summationdisplay p=0Apcos(pχ)Pp−1 2(coshη)/bracketrightBigg ,(13.92) 181 wherePp−1 2(coshη0) are the Legendre functions, and the coeﬃcients Apare given by Eq. (13.21). This solution is already known in the li terature [235, p. 239] [236, p. 1304]. It is also possible to obtain the capacitance of the toroid. T o this end we com- pare the electrostatic potential at a distance rfar from the origin, Eq. (13.34), with the potential given by a point charge q,φ(r≫a)≈q/4πε0r: φ(r≫a,θ,ϕ )≈a√ 2 r/bracketleftBigg∞/summationdisplay p=0√ 2φA(2−δ0p) πQp−1 2(coshη0) Pp−1 2(coshη0)/bracketrightBigg =q 4πε0r.(13.93) The capacitance of the toroid with its surface at a constant p otentialφAcan be written as C=q/φA. From Eq. (13.93) this yields [235, p. 239] [237, p. 5-13] [238, p. 9] [239, p. 375]: C= 8ε0a/bracketleftBigg∞/summationdisplay p=0(2−δ0p)Qp−1 2(coshη0) Pp−1 2(coshη0)/bracketrightBigg . (13.94) Utilizing the thin toroid approximation, η0≫1, one can obtain the capacitance of a circular ring, Eq. (13.76). Another case of interest is that of a charged circular wire al ready discussed, which is the particular case of a toroid with r0→0. In this case the charged toroid reduces to an uniformly charged circumference of rad iusR0=a. With η0≫1 and coshη0≫1 we haveR0≈a. Keeping only the term with p= 0 in Eqs. (13.21) and (13.92) yields (with Eq. (13.75)): φ(η≤η0,χ,ϕ) =φA/radicalBigg coshη−cosχ coshη0P−1 2(coshη) P−1 2(coshη0) =qA 4π√ 2ε0a/radicalbig coshη−cosχP−1 2(coshη). (13.95) Expressed in spherical coordinates ( r,θ,ϕ), the potential for the thin toroid becomes: φ(r,θ,ϕ) =qA 4πε01 [(r2−a2)2+ 4a2r2cos2θ]1/4 ×P−1 2/parenleftBigg r2+a2 /radicalbig (r2−a2)2+ 4a2r2cos2θ/parenrightBigg . (13.96) From Eqs. (13.91) and (13.75) we can see that the constant ele ctrostatic potential along the thin toroid expressed in terms of its tot al chargeqAis given by: φ(r0≪R0,θ,ϕ) =qA/2πa 2πε0ln8a r0. (13.97) Even when the linear charge density qA/2πaremains constant, we can see from this expression that the potential diverges logarithmical ly whena/r0→ ∞. 182 We can expand Eq. (13.96) in powers of r</r>, wherer<(r>) is the lesser (greater) of aandr=/radicalbig x2+y2+z2. We present the ﬁrst three terms: φ(r,θ,ϕ)≈qA 4πε0/braceleftbigg1 r>−1 + 3 cos(2θ) 8r2 < r3 > +3 512/bracketleftBig 9 + 20 cos(2 θ) + 35 cos(4 θ)/bracketrightBigr4 < r5>/bracerightbigg . (13.98) Eqs. (13.95) to (13.98) can be compared with the solution giv en by Jackson [13, p. 104]. Jackson gives the exact electrostatic solutio n of the problem of a charged circular wire (that is, a toroid with radius r0= 0), in spherical coordinates ( r,θ,ϕ): φ(r,θ,ϕ) =qA 4πε0/bracketleftBigg∞/summationdisplay n=0r2n < r2n+1 >(−1)n(2n−1)!! 2nn!P2n(cosθ)/bracketrightBigg , (13.99) whereqAis the total charge of the wire. Eq. (13.99) expanded to n= 2 yields exactly Eq. (13.98). We have checked that Eqs. (13.96) and (1 3.99) are the same for at least n= 30. We plotted both Eqs. (13.95) and (13.99), in Fig. 13.8. They y ield the same result, as expected. It is worthwhile to note that in spheric al coordinates we have an inﬁnite sum, Eq. (13.99), while in toroidal coordina tes the solution is given by a single term, Eq. (13.95). The agreement shows that Eqs. (13.95) and (13.99) are the same solution only expressed in diﬀerent for ms. Figure 13.8: Equipotential lines on the plane x= 0 (perpendicular to the toroid) for the charged thin wire without current. Both Eqs. (13.95) and (13.99) coincide with one another. We utilized η0= 38 (coshη0= 1.6×1016) anda= 1. Notice the diﬀerence between this Figure and Figure 13.6: the left a nd right sides of the conductor here possess the same charge signs, while in Fi gure 13.6 they have opposite signs. Fig. 13.9 shows the potential as function of ρ(in cylindrical coordinates) in the planez= 0. Eqs. (13.95) and (13.99) give the same result. 183 Figure 13.9: Normalized potential as a function of ρ(distance to the zaxis) on the plane z= 0. Eqs. (13.95) and (13.99) give the same result. We utilize d η0= 38 (coshη0= 1.6×1016) anda= 1. Along the zaxis,i.e., for/radicalbig x2+y2= 0, the potential represented by Eq. (13.96) is given by: φ(r,θ,ϕ) =qA 4πε01√ z2+a2. (13.100) This is the same result which arises from a direct integratio n of the electro- static potential. That is, a charge qAuniformly distributed along a ﬁliform ring of radius a, located at the plane z= 0 and centered along the zaxis [202, Example 12.3.3]. 13.10 Comparison with Experimental Results Figure 13.5 can be compared with the experimental result fou nd by Jeﬁmenko [174, Fig. 3], reproduced here in Fig. 13.10 with Fig. 13.5 ov erlaid on it. The equipotential lines obtained here are orthogonal to the ele ctric ﬁeld lines. There is a very reasonable agreement between the theoretical resu lt and the experi- ment. In order to have a better ﬁt to his data we should consider an ex tended battery. As we can see from his account of the experiment, Jeﬁ menko painted two sections of his strip with a conducting ink of much smalle r resistivity than the remainder of the strip. These sections located at −ϕj< ϕ < −ϕiand ϕi<ϕ<ϕjwere charged to opposite potentials. Considering these sec tions as of zero resistivity we can model analytically the potential inside and along the surface of the toroid as: φ(η≥η0,χ,ϕ) =  −φBϕi 2ππ+ϕ π−ϕj,−π<ϕ< −ϕj, −φBϕi/2π,−ϕj<ϕ< −ϕi, φBϕ/2π, −ϕi<ϕ<ϕi, φBϕi/2π, ϕ i<ϕ<ϕj, φBϕi 2ππ−ϕ π−ϕj, ϕj<ϕ<π .(13.101) 184 Figure 13.10: Theoretical equipotential lines of Figure 13 .5 overlaid on the experimental lines of electric ﬁeld obtained by Jeﬁmenko. T he equipotential lines are orthogonal to the electric ﬁeld lines. Notice that the potential described by Eq. (13.101) no longe r has a discontinuity atϕ=πrad. The potential is linear between ϕ=−ϕiandϕ=ϕi, constant for−ϕj< ϕ < −ϕiandϕi< ϕ < ϕj, and linear for −πrad< ϕ < −ϕjand forϕj<ϕ<π rad. The boundary condition Eq. (13.12) is now replaced by: φ(η0,χ,ϕ) =φB π/braceleftBigg∞/summationdisplay q=1sin(qϕ) q2/bracketleftbiggsin(qϕj) π−ϕj+sin(qϕi) ϕi/bracketrightbigg/bracerightBigg . (13.102) The potential from Eq. (13.102) is represented in Fig. 13.11 with the values ϕi= 9π/10 rad = 2.83 rad and ϕj= 17π/18 rad = 2.97 rad. The equipotentials in the planez= 0 are plotted in Fig. 13.12. Fig. 13.13 represents Jeﬁmenko ’s experiment with Fig. 13.12 overlaid on it. The agreement is n ow even better than in Fig. 13.10. Despite this agreement it should be mentioned that Jeﬁmenko ’s experiment has a conducting strip painted on a glass plate. On the other h and, the the- oretical results presented in Figs. 13.5 and 13.12 represen t an equatorial slice through a three dimensional toroid. As we saw in Chapter 3, Je ﬁmenko, Bar- nett and Kelly succeeded in directly measuring the equipote ntial lines inside and outside a hollow rectangular conductor carrying a stead y current. If one day a similar experiment is performed with a toroid, it will b e possible to obtain a better comparison with the theoretical results of this Cha pter. The solution inside and along the surface of the full solid to roid yields only an azimuthal electric ﬁeld, namely, |Eϕ|= ∆φ/2πρ. But even for a steady current we must have a component of /vectorEpointing away from the zaxis,Eρ, due to the curvature of the wire. Here we disregard this compo nent due to its extremely small order of magnitude compared with the azimut hal component Eϕ. To grasp this, consider a conducting electron of charge −eand massm moving azimuthally with drifting velocity vdin a circumference of radius ρ 185 Figure 13.11: Fourier expansion of the potential along the c onductor surface as a function of the azimuthal angle ϕ, Eq. (13.102), with φB=πφ0/ϕi. Comparing this Figure with Figure 13.4 we can observe that the oscillat ions, as well as the overshooting, do not appear anymore, as the potential is now continuous for 0 rad≤ϕ≤2πrad. We have used ϕi= 9π/10 rad = 2.83 rad and ϕj= 17π/18 rad = 2.97 rad. around the zaxis. In a steady state situation there will be a redistribut ion of charges along the cross-section of the toroid creating an electric ﬁeld Eρ which will exert a centripetal force on the conduction elect rons. By Newton’s second law of motion we can equate the force eEρwith the mass of the electron times its centripetal acceleration, in such a way that eEρ=mv2 d/ρ. Suppose we have a 14 gauge copper wire ( r0= 8.14×10−4m) of 1 m length bent in a circumference of radius R0=ρ= (1/2π) m = 1.59×10−1m carrying a current of 1 A. The drifting velocity is given by vd= 3.55×10−5m/s, the resistance of the wire is 8 .13×10−3Ω and the potential diﬀerence created by the battery is ∆ φ= 8.13×10−3V. This yields Eϕ= 8.13×10−3V/m and Eρ= 4.5×10−20V/m. That is Eρ≪Eϕ, which justiﬁes disregarding the Eρ component of the electric ﬁeld in comparison with the Eϕcomponent. As we saw in Section 6.4, a stationary conductor carrying a st eady current which is uniform over its cross-section generates a charge d istribution inside the conductor. This charge distribution creates a radial el ectric ﬁeld inside the conductor. In steady state there is then an electric forc e acting upon any speciﬁc conduction electron which is counteracted by the ra dial magnetic force that arises due to the movement of the other conduction elect rons, the radial Hall eﬀect. However, this electric ﬁeld is rather small, (10−5smaller than the electric ﬁeld that maintains the current ﬂowing, supposing a typical copper conductor with 1 mm diameter and 4 ×10−3m/s drifting velocity). For this reason this electric ﬁeld and the corresponding charge redi stribution have been neglected in these calculations. In this Chapter we presented a solution for the potential ins ide and outside a resistive toroidal conductor carrying a steady azimuthal current. The current ﬂows in a ﬁnite volume of space and the solution obtained here indicates the existence of the electric ﬁeld outside the conductor. The th eoretical calculations were compared to the experimental results, indicating a ver y good agreement. 186 Figure 13.12: Equipotentials in the plane z= 0 for a resistive toroidal conductor carrying a steady azimuthal current, using Eq. (13.102) as b oundary condition andφB=πφ0/ϕi. The bold circumferences represent the conductor surface and the bold straight lines represent the angles ϕ=±ϕi=±9π/10 rad = 2.83 rad andϕ=±ϕj=±17π/18 rad = 2.97 rad. We have used η0= 2.187. Figure 13.13: Jeﬁmenko’s experiment with Figure 13.12 over laid on it – the equipotential lines are orthogonal to the electric ﬁeld lin es. 187 We also obtained the distribution of charges along the surfa ce of the resistive ring carrying a steady current, a subject which was ﬁrst cons idered by Wilhelm Weber 150 years ago, as we see in the Appendix. Weber made the ﬁ rst pre- liminary quantitative calculations related to this proble m, and this speciﬁc case has essentially been forgotten these many years. This Chapt er can be seen as a fulﬁllment of one of Weber’s goals. That is, to derive the dis tribution of surface charges in a ring which, together with the battery, creates a constant tangential electric ﬁeld for all azimuthal angles inside the ring. We ha ve also succeeded in deriving the force exerted upon a stationary and external po int charge by this stationary ring carrying a steady current. 188 Part IV Open Questions 189 Chapter 14 Future Prospects In this book we presented the main simple cases which can be tr eated analyt- ically. The goal now might be to consider theoretically othe r situations which have already been analyzed experimentally. Examples inclu de the current ﬂow- ing in a disc, Figure 3.7; current-carrying wedges with the t wo halves connected in parallel and in series, Figure 3.6; etc.The latter situation is interesting in order to know quantitatively the correct distribution of ch arges allowing the current to bend around a corner. Studies along these lines in clude Rosser [167], Jeﬁmenko [240] [176, pp. 302-303] and the book by Chabay and S herwood [165, Chapter 6]. Other interesting aspects are connected with the distribut ion of surface charges close to the battery and inside it. In Chapters 11 to 1 3 we discussed this in the cases of a cylindrical shell, a spherical shell and a ri ng with azimuthal currents. Another important discussion for the case of a coa xial cable of ﬁ- nite size has been given by Jackson [12]. Saslow considered a spherical battery surrounded by a conducting medium and analyzed the distribu tion of charges upon the surface of the battery [241]. The distribution of su rface charges close to a battery in the case of straight conductors carrying stea dy currents has been treated for two diﬀerent conﬁgurations in 2004 [205] an d 2005 [222]. Other cases should also be studied quantitatively in diﬀerent con ﬁgurations. Although it might be diﬃcult to obtain detailed information analytic ally about the distri- bution of surface charges in a battery of ﬁnite size, this mig ht be accomplished with computer calculations and numerical plots. It would also be important to analyze cases in which the curre nt is not generated by a chemical battery, but by the relative motion b etween a closed conducting circuit and a magnet, as in the ﬁrst case consider ed qualitatively by Weber and described in the Appendix. Calculations have be en performed relative to the surface charges in the case of a square circui t in the presence of a variable magnetic ﬂux [242], and also the case of a ring rotat ing in the presence of a magnetic ﬁeld [243]. It would be important to extend the c alculations to other spatial conﬁgurations and analogous situations. Another situation which has received little attention up to now is the dis- 191 tribution of charges in resistive conductors carrying stea dy currents when these conductors are composed of two or more diﬀerent materials. T hat is, the charges that accumulate on the interface between a conductor and a re sistor, or on the interface of two conductors with diﬀerent resistivities. S ome authors who have considered this problem include Jeﬁmenko [240], Heald [226 ], H¨ artel [244, 245], Chabay and Sherwood [165, 166], and Jackson [12]. Jackson’s work has an in- teresting comparison of the distribution of surface charge s in a circuit with a large resistance and in an equivalent open circuit, i.e., with the resistor removed from the circuit. Another relevant topic is to consider in detail the behavior of surface charges and the corresponding external electric ﬁeld in the transit ion from steady- currents to low and high frequency circuits with alternatin g currents. Important discussions of this subject have been given by Jackson [12] a nd Preyer [246]. We- ber and Kirchhoﬀ’s works related with the telegraphy equati on discussed in the Appendices should also be reconsidered and extended to diﬀe rent cases and conﬁgurations [30, 31]. Beyond these future extensions, there are a number of topics which still need to be clariﬁed. Consider a stationary point charge clos e to a stationary permanent magnet. Is there a net force between them beyond th e force due to electrostatic origin? That is, is there a force depending up on the magnetization of the magnet, or depending upon the magnetic ﬁeld it produce s? As we have seen in this book, there is a force between a stationary point charge and a stationary resistive circuit carrying a steady current. Th is force is proportional to the electromotive force of the battery. Is there a similar force between a stationary magnet and a stationary external charge? In this question we are not including the force due to electrostatic induction whic h must exist between a conducting magnet and the external charge, which is of elec trostatic origin (due to image charges, etc.) The magnet we are considering here has permanent magnetization. Its magnetic ﬁeld is due to permanent micros copic or molecular currents in its interior. The magnet is not connected to a che mical battery and for this reason it should not have a distribution of surface c harges as in the case of a resistive wire carrying a steady current. In any event th is subject should be better analyzed and careful experiments should be perfor med to answer this question. Analogously, there should not exist an electric ﬁeld outsid e a wire made of a superconducting material if it carries a steady current wi thout any external source of electromotive force, i.e., if there is no battery connected to the wire. As this wire has no resistance and is not connected to any batt ery, there should be no electric ﬁeld outside the wire (except for the zeroth or der electric ﬁeld if we approach a test charge to the wire). But it should be emphas ized once more that only experiments can decide this question. A possible connection between the external electric ﬁeld ar ound a resistive cylindrical conductor carrying a steady current and the Aha ronov-Bohm eﬀect was discussed in 2001 [247]. Although this idealized inﬁnit e conductor will not produce any external magnetic ﬁeld, it will produce an ex ternal electric ﬁeld if the solenoid is connected to a chemical battery. This electric ﬁeld is 192 not considered by most authors, as they are unaware of its exi stence. For this reason none of them considered the inﬂuence of this steady el ectric ﬁeld in the Aharonov-Bohm eﬀect, taking into account only the magnetic vector potential. The goal of our paper was to call attention to this external el ectric ﬁeld for the analysis of the Aharonov-Bohm eﬀect. In this book we have shown that a force must exist between a poi nt charge and a resistive wire carrying a steady current when they are a t rest relative to one another. It has been shown theoretically that this force (or the electric ﬁeld outside the wire) is proportional to the emf of the battery. B ut it is still necessary to show experimentally the proportionality between this fo rce and the voltage of the battery. This proportionality should appear accordi ng to the calculations presented here. They have yielded qualitative agreement wi th the experiments of Bergmann, Schaefer, Jeﬁmenko, Barnett and Kelly relatin g to equipotentials and electric ﬁeld lines. But we are not not aware of any experi ment showing directly the proportionality between this force and the vol tage generated by the battery. Another crucial question which still needs to be settled emp irically is related to the second order electric ﬁeld (proportional to the squar e of the current, or to the square of the drifting velocity of the mobile electron s). Alternatively we might ask if there is a second order force between a stationar y charge and a sta- tionary wire carrying a steady current. This electric ﬁeld a nd the corresponding force produced by it upon stationary charges are usually muc h smaller than the electric ﬁeld and forces discussed in this book (proportion al to the voltage of the battery). For this reason it is diﬃcult to decide unambig uously whether this eﬀect exists. Experiments to decide this question shou ld be performed sep- arately, considering three cases: (1) resistive wires conn ected to batteries and carrying steady currents, (2) superconductors carrying st eady currents without any external source of electromotive force, and (3) permane nt magnets. It may happen that this second order electric ﬁeld exists (or does n ot exist) for all three cases. It may also be that it exists for one or more of these cas es, but not for the other case(s). These three cases must be considered inde pendently from one another. The theoretical analysis of the experiments must t ake into account the force due to electrostatic induction (zeroth order elec tric ﬁeld) and also the component of the electric ﬁeld discussed in this book propor tional to the emf of the battery (for the case of resistive conductors). This i s not a simple task in complicated conﬁgurations. It is essential to be extreme ly careful with all possible inﬂuences in order to avoid misleading conclusion s. Another topic which has not been treated in this book is the co nvenience and importance of the surface charges and of microscopic asp ects of current conduction for the understanding of the macroscopic phenom ena associated with circuits carrying steady currents. This subject has great c onceptual and didactic relevance. Several studies relating to the teaching of elec tromagnetism have been developed through an exploration of this topic, as applied t o high school and to university courses [164, 244, 245, 248, 211, 165, 171, 249, 2 50, 166]. This book has shown how a very simple question of basic electr omagnetism has been answered incorrectly by many important authors alo ng several decades. 193 This has had a negative inﬂuence on the development of the sub ject for more than a century, and created many prejudices which are very di ﬃcult to eliminate. We should try to avoid the same mistake in the future. This was one of our reasons for writing this book. Another goal was to obtain the densities of charges spread up on the surfaces of resistive conductors carrying steady currents in severa l conﬁgurations. For long, straight conductors it was shown that these surface de nsities are a linear function of the longitudinal coordinate. For curved conduc tors, on the other hand, they grow faster than linearly along the length of the c onductor, increasing their magnitude toward both extremities of the battery. All of this can indeed be understood in terms of electrodynamical principles. Fol lowing French in the last page of his didactic book Newtonian Mechanics [251, p. 700], the best way to close this work is with a simple and fair statement, namely : “But Weber got there ﬁrst!” 194 Appendix A Wilhelm Weber and Surface Charges Wilhelm Eduard Weber (1804-1891) was one of the ﬁrst to menti on and ana- lyze quantitatively the surface charges in resistive condu ctors carrying steady currents. Here we discuss some parts of his papers dealing wi th this topic. In Section 1.4 we presented some important aspects of his life a nd work, quoting the publication of his collected papers and all of his works w hich have been translated into English. Weber wrote eight major memoirs between 1846 and 1878 under t he gen- eral title Electrodynamic Measurements , orDetermination of Electrodynamic Measures (the eighth memoir was published only posthumously in his co llected papers). The work which we discuss here is the second memoir of this ser ies, published in 1852: Electrodynamic Measurements Relating Specially t o Resistance Mea- surements [32]. To the best of our knowledge this work has nev er been translated into English or any other language. What we quote here is our t ranslation. The paper is divided into six parts and has ﬁve extra appendices. What interests us here is the ﬁfth part, which extends from Section 28 to Sectio n 36 (pp. 368 to 405 of Vol. 3 of Weber’s Werke [38]): On the Connection of the Theory of the Galvanic Circuit with the Electrical Fundamental Laws . Between square brack- ets we oﬀer our interpretation of expressions or sentences f rom Weber. We have produced the ﬁgures presented in this Appendix in order to il lustrate Weber’s reasoning. The footnotes presented here are also ours. Section 28 begins with the statement that until then there wa s no devel- opment of the relation between the theory of the galvanic cur rent [Ohm’s law] and the electrical fundamental laws [Coulomb’s force], as t hese two subjects were treated independently from one another. He says that th e main reason for this separate treatment lies in the mathematical diﬃculty o f connecting the two subjects in a complete manner.1His goal is to discuss some aspects which can 1An example of this mathematical diﬃculty can be seen in Chapt er 13 of this book, a 195 lead to a connection between both subjects. He mentions Ohm’ s law, valid for steady currents, relating the current intensity, the resis tance and the electro- motive force (or electro-motor force) [due to a chemical bat tery, for instance].2 He remarks that Ohm tried to base his law on the variable volum e density of charges in the conductor, in analogy with Fourier’s treatme nt of the propaga- tion of heat based on the variable distribution of the temper ature inside a body. That is, in a region of the wire carrying a steady current wher e there is no elec- tromotive force (no point of contact between two diﬀerent me tals, for instance), the force moving the charges against resistance would be due to a gradient in the volume density of charges.3Weber states that Ohm found the key to explain the law of the galvanic circuit based upon the distribution o f electric charges in the conductor. On the other hand, he mentions that Ohm’s ap proach is in contradiction with the fundamental laws of electrostatics , according to which free electricity can exist only along the surface of a conductor. Weber mentions that the same must be true in the case of a galvanic circuit wit h steady current, even disregarding the relative motion between the interact ing charges. While the local temperature gradient is a necessary condition for the local propagation of heat, the same does not need to be true for charges, as they a ct at a distance. Weber then considers an interesting example of a stationary homogeneous closed copper ring with overall equal cross-section. He ima gines a magnet mov- ing along the axis of the ring, perpendicular to its plane. Se e Figure A.1. Weber had already considered brieﬂy this situation in his ﬁrst maj or Memoir of 1846, [137, p. 203 of the Werke ]. According to Weber the magnet will exert the same electromotive force in all elements of the ring. As all eleme nts have the same resistance, the electromotive force will produce the same c urrent in all of them. In this case there will not appear any accumulation of charge s in any place of the ring.4According to Weber, only when there is a diﬀerence of the acti on of the electromotive force in diﬀerent parts of the circuit t here will appear ac- cumulation of charges.5The eﬀect of the distribution of free electricity along the surface of the wire will be to equalize [in all parts of the wire] the action of the electromotive force [due to the contact of two diﬀeren t metals or due to situation which Weber also considered quantitatively in hi s memoir, as we will see. 2In some examples it is possible to understand Weber’s elektromotorische Kraft as poten- tial diﬀerence or as electromotive force (emf) around a comp lete circuit carrying a steady current. On the other hand, in other situations it seems that Weber refers to the longitudinal component of the electric ﬁeld driving the conduction charg es along a wire carrying a steady current. The electric force associated with this electric ﬁ eld acts in the opposite direction of the resistive frictional force exerted upon the conduction electrons by the crystaline lattice of the metal. 3This can be seen in pp. 402 and 418 of Ohm’s work [252]. 4For an experimental proof of this fact, see the interesting p aper of Moreau and collabora- tors [184]. The relative motion between the magnet and the ci rcuit drives the current around the ring due to a non Coulomb force. In this case there is no pot ential diﬀerence between any two points on the ring [204]. 5This will be the case, for instance, when there is steady curr ent in a resistive wire connected to a chemical battery. The electromotive force of the batter y acts mainly inside itself and in the region close to its surface, so that along the other parts of the wire there must be forces of another origin moving the mobile charges against resisti ve forces. 196 a chemical battery, this electromotive force having diﬀeri ng intensities in dif- ferent portions of the circuit]. Weber then states that two t hings remain to be shown: (1) how this distribution of free electricity is poss ible according to the fundamental electrical laws, and what its properties shoul d be,6and (2) how the surface charges arise and are maintained. Figure A.1: Magnet moving along the axis of a copper ring. Acc ording to Weber the magnet will exert the same electromotive force in all ele ments of the ring. Section 29 is entitled “Proof of the possibility of a distrib ution of the free electricity in a conductor, through which is balanced the di ﬀerence in the ac- tion of given electromotive forces in diﬀerent parts of the c ircuit according to the proportionality of their resistances.” He begins by con sidering particles of free electricity along the surface of a conductor exerting e lectromotive forces [in this case electrostatic forces due to Coulomb’s law] upon al l charged particles of the conductor. These forces due to surface charges will de crease or increase the electromotive forces of the circuit [due to a chemical ba ttery, for instance]. He then asks if a distribution of surface charges is possible such that the [net] electromotive forces [that is, the resultant electric ﬁeld due to the chemical bat- tery and to the surface charges] will be equilibrated in all p arts of the circuit in proportion with the resistance of these parts. Disregard ing the eﬀect of the relative motion between the charges [relative motion betwe en the conduction charges and the ions of the lattice], Weber mentions that thi s question must be answered based upon the fundamental law of electrostatics. He then mentions the theorem proved by Poisson that there is one and only one po ssible distri- bution of charges on the surface of a conductor which equilib rates the electric forces exerted by external charges. He then applies this theorem conceptually to a cylindrical c onductor acted upon by an external point charge along its axis, at a great dis tance from the cylinder. See Figure A.2. This external charge exerts essen tially the same axial electrostatic force on all charges of the cylinder. There wi ll be a redistribution 6That is, how is it possible to derive from Coulomb’s force the distribution of the surface charges which will equalize the electric ﬁeld in all points i nside the resistive wire. 197 of charges along the surface of the cylinder, creating an opp osite electric ﬁeld and canceling this external force at all internal points of t he cylinder. If we now consider the presence of this ﬁxed distribution of surface c harges, without the presence of the external point charge [that is, as if the surf ace charges had been glued upon the surface of the cylinder and later on the extern al point charge were removed], there will be an axial electromotive force ac ting on all points of the cylinder.7 Next he considers a curved cylinder [like a piece of a ring in t he form of an arc of a circle] and a point charge at a great distance from i t, along the tangent to one element of the arc. This external point charge exerts a uniform force along the tangent of the arc, which is equilibrated by t he force due to the distribution of surface charges in the curved cylinder. Whe n this distribution of surface charges is kept ﬁxed at their places [by the applicat ion of other external forces to them] and the external point charge is removed, onl y the longitudinal electromotive force [acting upon all charges of the cylinde r] due to the surface charges will remain. He then generalizes this to all element s of the curved cylinder such that the surface charges on any speciﬁc elemen t will be a function of the surface charges on all other elements of the arc. Figure A.2: Point charge qalong the axis of a ﬁnite cylindrical conductor, at a great distance from it. This external charge exerts esse ntially the same electrostatic axial force upon all charges of the cylinder. In equilibrium there will be an electrical polarization of the cylinder, with the charges along its surface canceling exactly, at all internal points of the cyl inder, the electric ﬁeld due to the external charge. He then imagines this curved cylinder making a circle, like a ring, with a small separation between the initial and ﬁnal cross-sectio ns of the ring. Weber shows that these two surfaces should not touch one another; o therwise there will be an inﬁnite amount of opposite charges on these surfac es (supposing a uniform tangential electromotive force acting on all point s of the ring). He calls δthe distance between the extremities of the open ring and ±ethe charges of two elements of these opposite faces. Utilizing Coulomb’s l aw he shows that the force on a test charge inside the ring due to the two opposite f aces is proportional toδe. As he wants this electromotive force [or electric ﬁeld, as w e would say today] to remain constant as δ→0, it is necessary that simultaneously e→ ∞, which was what he wanted to prove. The electromotive force al ong the ring is 7He has proved in this ﬁrst simple case that there is a distribu tion of surface charges which exerts an equal longitudinal force on all points of a cylinde r, although he did not explicitly attempt to calculate the distribution of surface charges in this speciﬁc example. 198 then also proportional to δe. In the open region between the two extremities the electromotive force due to the charges in the end surfaces po ints in a direction opposite to the direction of the electromotive force acting on a test charge inside the ring and close to the extremities. He concludes that if we want the same electromotive force [net electric ﬁeld] at all points of the closed ring, then in the region between the extremities it is necessary for an ele ctromotive force to act, independent of the distribution of surface charges [ that is, a force of non-electrostatic origin]. As an example of such a force he m entions the case of copper and zinc touching one another [we might also mentio n the case of a chemical battery]. He draws three conclusions from these considerations: 1. It is not possible to have [steady] current in a closed ring due only to a distribution of surface charges on the ring. It is necessary to have an electro- motive force of diﬀerent origin in at least one cross-sectio n of the ring (like the contact of copper and zinc).8 2. The current in a circuit is proportional to the density of s urface charges along the circuit.9The electromotive force is proportional to δeand to the current in the circuit. 3. When we double all dimensions of a circuit but keep the same electro- motive force, then the density of surface charges should rem ain constant, even though the surface area is four times the previous one.10At the same time it follows that when we double all dimensions of a circuit, the d istanceδshould also double, but when the charge eremains constant, the electromotive force proportional to δeshould also double. This double electromotive force requir es the same motion [velocity] of the charges in a circuit with do ubled dimensions, as [the velocity] in a circuit of simple length and cross-sec tion. But this same motion [velocity] generates four times the current in a circ uit with doubled di- mensions (and four times the [area of the] cross-section). T hat is, a doubled electromotive force generates, in a circuit of doubled leng th and four times the cross-section [in comparison with the simple original circ uit], a current four times larger, which is in agreement with the laws of the galva nic circuit. Section 30 is entitled “On the law of the distribution of the f ree electricity 8This is similar to the theorem that/contintegraltext/vectorE·d/vectorℓ= 0, where /vectorEis the electrostatic ﬁeld of Coulomb’s law, d/vectorℓis an element of length and the line integral is over a closed c ircuit of arbitrary form. That is, in order to have an electromotive fo rce driving a current around a closed resistive circuit it is necessary to have a source of n on-electrostatic origin. See Section 5.1. 9An example of this general conclusion can be seen in Eq. (6.17 ) for the case of a straight wire. Combining it with Eq. (6.2), σ(z) =σA+σBz/ℓ, yields:I=−(aσB/Rε0)ln(ℓ/a). That is,Iis directly proportional to σB, as Weber concluded. 10From Eqs. (6.2), (6.14) and with the electric ﬁeld (Weber’s e lectromotive force in this case) given by E1= ∆φ/ℓ=RI/ℓ we obtain: σB=ε0E1 ln(ℓ/a)ℓ a. (A.1) That is,σBis proportional to E1and to (ℓ/a)/ln(ℓ/a). If at the same time we double ℓ anda, keeping a constant E1, thenσBwill remain constant. This is an example of Weber’s conclusion. 199 over the surface of a conductor carrying a constant and unifo rm current.” For a linear conductor he says that we can consider the surface cha rges as distributed along its axis.11He shows this considering a cylindrical conductor of length 2λ with a circular cross-section of radius α≪λ. See Figure A.3. Figure A.3: Cylindrical conductor of length 2 λand radius α≪λ. In the case of steady currents, the surface charge density is linear wit h the longitudinal x component, i.e., proportional to a+bx. He considers initially that in the case of a steady current th e density of surface charges is linear with the longitudinal xcomponent, i.e., proportional toa+bx.12He integrates the longitudinal electromotive force [our el ectric ﬁeld along the direction of the axis] due to these surface charges acting on a point located at the origin (the center of the cylinder), obtainin g the result (supposing λ≫α): /integraldisplayλ x=−λ2πα(a+bx)xdx (α2+x2)3/2≈4παb/parenleftBig logλ−logeα 2/parenrightBig , (A.2) wheree= 2.7183 is the natural logarithm base.13 He then shows that the same result is obtained when we conside r all the surface charges distributed along the axis of the cylinder, integrating from x= −λtox=λ, with the exception of the region between x=−eα/2 andx=eα/2. See Figure A.4. That is, he was able to derive Eq. (A.2) by assuming all surfac e charges concentrated along the axis of the wire and calculating the l ongitudinal electric ﬁeld at the origin integrating from x=−λtox=−eα/2 and from x=eα/2 tox=λ.14 11That is, the actual force exerted by the free charges distrib uted along the surface of a cylindrical conductor carrying a steady current upon a test charge can be replaced by the force upon this test charge due to an appropriate distributi on of charges along the axis of the cylinder. 12Weber’s 2λandαare equivalent, respectively, to our ℓandaof Figure 6.1. Weber’s a+bx is equivalent to our σ(z) =σA+σBz/ℓ, Eq. (6.2). 13This result is equivalent to Eq. (6.12), namely, E1= (aσB/ℓε0)ln(ℓ/ea). All results obtained by Weber in this Section can be put in the internatio nal system of units by dividing them by 4πε0. Weber’s log has base e, which means that his log can be written as our ln. In Chapter 6 we ﬁrst calculated the potential by integration , and then the electric ﬁeld by /vectorE=−∇φ. Here Weber has integrated the electric ﬁeld directly. The ﬁ nal result was the same, as expected. 14In other words, he considers the line having a linear charge d ensity given by 2 πα(a+bx). He considered the test charge at the origin. This is a very int eresting technique which greatly simpliﬁes the integrations. We have checked his integratio n and it is correct. We now generalize his calculation to obtain the longitudina l electric ﬁeld at an arbitrary 200 Figure A.4: Weber considered now all surface charges distri buted along the axis of the cylinder of length 2 λ. According to Weber, the linear integration that yields the same electric ﬁeld at the origin as that given by Figure A.3 and Eq. (A.2), now runs from x=−λtox=λ, except in the region between x=−eα/2 andx=eα/2. pointx′,Ex′(x′): Ex′(x′) =/parenleftBigg/integraldisplayx′−eα/2 x=−λ+/integraldisplayλ x=x′+eα/2/parenrightBigg 2πα(a+bx)(x′−x)dx [(x′−x)2]3/2 =−2πα/bracketleftbigg bln4(λ2−x′2) e2α2−2(a+bx′)x′ λ2−x′2/bracketrightbigg . (A.3) Atx′= 0 this yields Weber’s result, namely Ex′(0) =−4παbln2λ eα. (A.4) We now present an alternative way of obtaining the electric ﬁ eld. This alternative procedure will be followed by Weber in the calculation of the ring, as we will see shortly. If he had wished to obtain the electric ﬁeld from the potentia l, he would have had to calculate the potential at a generic point x′(and not only at the origin x′= 0). The integrals would need to go from x=−λtox=x′−eα/2 and from x=x′+eα/2 tox=λ. Let us write as Φ( x′) the function which would represent the potential at x′calculated in this way (later on we show that it is diﬀerent from the real potential φ(x′)). With a charge element dq= 2πα(a+bx)dxwe would obtain: Φ(x′) =/parenleftBigg/integraldisplayx′−eα/2 x=−λ+/integraldisplayλ x=x′+eα/2/parenrightBigg 2πα(a+bx)dx/radicalbig (x′−x)2 = 2πα/bracketleftbigg (a+bx′)ln4(λ2−x′2) e2α2−2bx′/bracketrightbigg . (A.5) From this expression we obtain −∂Φ(x′) ∂x′=−2πα/bracketleftbigg bln4(λ2−x′2) e2α2−2(a+bx′)x′ λ2−x′2−2b/bracketrightbigg . (A.6) And this is diﬀerent from the electric ﬁeld given by Eq. (A.3) ! For instance, in the limit whenx′→0 Eq. (A.6) yields −4παbln(2λ/e2α). And there is a diﬀerence of 1 /einside the logarithm as compared with the previous result which Weber o btained by direct integration of the electric ﬁeld. That is, although in general /vectorE=−∇φ, in this particular case we did not obtain Ex′(x′) = −∂Φ(x′)/∂x′, as might be expected. The origin of this diﬀerence is not eas y to locate but we need to clarify it before proceeding. Everything is due to Weber’s peculiar approximation method when we calculate Ex′(x′) or Φ(x′). In this method the location of the point of observation, x′, appears not only in the integrand, but also in the limits of t he integrals. The problem arises from the following mathematical result, valid for arbitrary functions and variables [253, p. 44]: ∂ ∂α/integraldisplayx=g(α) x=f(α)F(α,x)dx=/integraldisplayx=g(α) x=f(α)∂F(α,x) ∂αdx 201 Next he goes to his main calculation. He replaces the straigh t cylindrical conductor with a toroidal one, like a ring conducting an azim uthal current. He calls the greater radius of the ring rand its smaller radius α, supposing α≪r. He considers the electrostatic potential null at the azimut h angleψ=πrad and discontinuous at ψ= 0 rad. See Figure A.5. Figure A.5: Conducting ring with greater radius rand smaller radius α, with α≪r. The potential is discontinuous at ψ= 0 rad and null at ψ=πrad. Weber wants to calculate the tangential electric ﬁeld, Eψ, at the angle ψ,Eψ(ψ). The angleϕis the variable angle of integration. That is, the potential F(ψ) [represented by Weber as Fψ] is such that F(0) = −F(2π).15It is then given by F(ψ) =c(ψ−π), where [F(2π)−F(0)]/2πr=c/r is the value of the tangential electromotive force [our elec tric ﬁeld] assumed constant along the ring. He utilizes his linear approach in o rder to calculate, in +/braceleftBig∂g(α) ∂αF[α,g(α)]−∂f(α) ∂αF[α,f(α)]/bracerightBig . (A.7) In order to arrive at Eq. (A.3) beginning with Eq. (A.5) we wou ld need to utilize the following expression (obtained from Eq. (A.7)): Ex′(x′) =−∂Φ(x′) ∂x′+∂(x′−eα/2) ∂x′2πα[a+b(x′−eα/2)]/radicalbig [x′−(x′−eα/2)]2 −∂(x′+eα/2) ∂x′2πα[a+b(x′+eα/2)]/radicalbig [x′−(x′+eα/2)]2 =−∂Φ(x′) ∂x′−4παb . (A.8) And this equation coincides with Eq. (A.3) if we utilize Eq. ( A.6). This is the correct approach if we wish to obtain the electric ﬁeld E(x′) utilizing Weber’s approximate method and beginning with an equivalent to a potential function. Th at is, we need to follow this approach if we begin with the function Φ( x′) and wish to obtain E(x′) by diﬀerentiation. We will return to this point when considering Weber’s next calc ulation. 15Atψ= 0 rad there should be the point of contact between copper and zinc, or a chemical battery, or another non-electrostatic source of electromo tive force. Weber’s α,randFare equivalent to our r0,R0andφ, respectively. See Fig. 13.1. 202 a general way, the electrostatic potential at the angle ψalong the ring due to all surface charges, integrating from from ϕ=ψ+eα/2rtoϕ= 2π+ψ−eα/2r.16 He callsfϕdϕ the amount of free electricity in the arc element rdϕ, where fϕis the angular density of free charge (it is not yet speciﬁed w hether this charge density is a linear function of the angle ϕ). That is, fϕis the angular density of free electricity along the ring as a function of th e azimuth angle ϕ. [From now on we will call it f(ϕ). That is,f(ϕ) has units of charge per angle, or Coulomb per radian in the SI.] He mentions that according to O hm’s hypothesis, the density of charges along a uniformly resistive conducto r should be a linear function of the length along the circuit.17As we have a ring this would imply, according to Weber, that the angular density of charges shou ld be given by f(ϕ) =a(ϕ−π).18Weber then decides to test if this linear hypothesis is valid for a ring. Instead of calculating the electric ﬁeld directly, as he had done in the case of a linear conductor, he decided to calculate the electrostat ic potential.19To this end he divides the circumference of radius rinto two parts, ABD andDCA. See Figure A.6. The points A,B,DandCare located at ϕ= 0 rad,ϕ=ψ(where he wants 16That is, instead of performing an integration over the surfa ce of the ring, he performs only a linear integration replacing the ring by a circumference w ith an appropriate linear charge density. He calculates the potential at the angle ψ, where the test charge will be located. His integration can be thought of as going from ϕ= 0 rad to ϕ=πrad, except for the region betweenψ−eα/2randψ+eα/2r. 17This can be seen in p. 456 of Ohm’s work [252]. 18It should be observed that the ahere has no relation with the aof the previous surface charge density of a cylinder given by a+bx. Weber’s approach is analogous to the one utilized in Chapter 6. That is, he supposes a distribution of source ch arges and from them calculate the potential and electric ﬁeld. The approach utilized in Ch apter 13 was the opposite. In Chapter 13 it was given the potential along the surface of the conductor, Laplace’s equation was solved, yielding the potential everywhere in space. The n the electric ﬁeld was obtained as minus the gradient of the potential. And ﬁnally the surface c harges were obtained by applying Gauss’s law at the interface between the conductor and the ex ternal medium. For the ring we obtained a density of surface charges given by Eq. (13.63) . Far from the battery this is reduced to Eq. (13.65), namely, σ(ϕ) =σA+σBϕ/2π. Far from the battery the linear charge density which we obtained was given by Eq. (13.74). Weber’s a ngular density of charges, f(ϕ), is given by R0times the linear charge density. That is (far from the batter y and utilizing η0≫1,a≈R0,a/sinhη0=r0and coshη0=R0/r0): f(ϕ) =R0/parenleftBig λA+λBϕ 2π/parenrightBig = 2πr0R0/parenleftBig σA+σBϕ 2π/parenrightBig = 2πR0ε0/bracketleftBigφA ln(8R0/r0)+g(η0)φBϕ 2π/bracketrightBig . (A.9) Comparing this expression with Weber’s expression, f(ϕ) =a(ϕ−π), we ﬁnd that Weber’s ais equivalent to our r0R0σB=R0ε0g(η0)φB. 19In principle he would need to integrate Φ(ψ)≡/bracketleftbigg/integraldisplayψ−eα/2r ϕ=0+/integraldisplay2π ϕ=ψ+eα/2r/bracketrightbigg a(ϕ−π)dϕ r√ 2/radicalbig 1−cos(ψ−ϕ). (A.10) However, if he tried to perform this direct integration he wo uld end up needing to evaluate /integraltext xdx/sinx. The solution of this indeﬁnite integral yields an inﬁnite s eries, namely [203, p. 233]: 203 Figure A.6: Weber’s conﬁguration to integrate the potentia l. to know the value of the potential), ϕ= 2ψandϕ=ψ+π, respectively. In order to calculate the potential at Bdue to the charges spread along the arc ABD, with the exception of the small arc eα/r aroundB, he considers two charge elements symmetrically located around B, at angles ±χfromB. The charge elements located at ϕ=ψ±χ, in elementary arcs of length rdχ, are given bya(ψ±χ−π)dχ. Each of these charge elements is at the same distance 2rsin(χ/2) fromB. By adding the contributions of these two charge elements he obtains the diﬀerential potential at Bas given by a(ψ−π)dχ/rsin(χ/2). This was a very good idea in order to avoid the integral of xdx/sinx. After integration he obtains the potential a(ψ−π) r/integraldisplayψ χ=eα 2rdχ sinχ 2=2a(ψ−π) r/parenleftbigg log tanψ 4−log taneα 8r/parenrightbigg .(A.12) To obtain the potential at Bdue to the charges located around the arc DCA he proceeds in a similar way. He considers two charge element s located symmetrically around C, at angles ±χfromC. Relative to Athese two charge elements are located at ϕ=ψ+π±χ. Both are at the same distance 2 rsin[(π− ψ)/2)] fromB. The potential due to the sum of these two charge elements calculated at Bis then given by aψdχ/r cos(χ/2). After integration he obtains aψ r/integraldisplayπ−ψ χ=0dχ cosχ 2=−2aψ rlog tanψ 4. (A.13) /integraldisplay xdx sinx=x+∞/summationdisplay k=1(−1)k+12(22k−1−1) (2k+ 1)!B2kx2k+1. (A.11) It is diﬃcult to put this inﬁnite series in closed form. Instead of solving this integral directly, Weber utilizes a n ingenious approach by taking advantage of the symmetrical distribution of charges along the circumference, as we will show below. In this way he avoids this integral. 204 By adding Eqs. (A.12) and (A.13) he obtains the total potenti al atϕ=ψ as given by20 −2aψ rlog taneα 8r−2aπ r/parenleftbigg log tanψ 4−log taneα 8r/parenrightbigg . (A.15) By making the derivative of this expression with respect to t he arcrψ, namely,d/rdψ , Weber obtains the following expression for the magnitude o f the tangential component of the electromotive force [the ab solute value of our electric ﬁeld] at the angle ψdue to all surface charges along the ring, except for the charges in the arc eα/raroundψ: −2a r2log taneα 8r−aπ r2sin(ψ/2). (A.16) That is, Weber initially calculated the potential at the ang leψas given by Eq. (A.14). He then obtained absolute value of the tangentia l component of the 20This ﬁnal value obtained by Weber can be written as Φ(ψ)≡/bracketleftbigg/integraldisplayψ−eα/2r ϕ=0+/integraldisplay2π ϕ=ψ+eα/2r/bracketrightbigg a(ϕ−π)dϕ r√ 2/radicalbig 1−cos(ψ−ϕ) =−2aψ rlogtaneα 8r−2aπ r/parenleftBig log tanψ 4−log taneα 8r/parenrightBig . (A.14) 205 electric ﬁeld at ψas given by Eψ(ψ) =dΨψ(ψ)/rdψ, obtaining Eq. (A.16).21 He then mentions that this value is approximately constant o nly forψ≈ πrad,i.e., far from the point of discontinuity in the potential [far fr om the battery]. When we are close to ψ= 0 rad or to ψ=πrad, the magnitude of this longitudinal electromotive force is smaller than it s magnitude at ψ=π rad. He concludes that Ohm’s hypothesis is only valid for the middle part of the circuit (that is, for ψ≈πrad). He mentions that it is then also necessary to consider the cha rges which are located in the cross-sections of the ring where there is a dis continuity in the potential, charges which had not been considered by Ohm. He c alls±εthe amount of these opposite surface charges (which he consider s in his simpliﬁed model as concentrated at points) and δthe small distance separating them. See Figure A.7. After calculating the absolute value of the longitudinal el ectromotive force [that is, the tangential electric ﬁeld along the ring] actin g at the angle ψdue to this dipole, Weber obtains the result22 21Weber could have obtained the tangential component of the el ectric ﬁeld by direct inte- gration. That is, Eψ(ψ) =/parenleftbigg/integraldisplayψ−eα/2r ϕ=0+/integraldisplay2π ϕ=ψ+eα/2r/parenrightbigg a(ϕ−π)sin(ψ−ϕ)dϕ√ 8r2[1−cos(ψ−ϕ)]3/2 =2a r2ln taneα 8r+aπ r2sin(ψ/2)−eαa 2r31 sineα 4r. (A.17) The last term on the right hand side does not appear in Weber’s expression, Eq. (A.16). This is due to the same problem discussed in footnote 14. In order to arrive at Eq. (A.17) beginning with Eq. (A.14) and taking into account Eq. (A.7), Weber should have utilized: Eψ(ψ) =−1 r∂Φ ∂ψ+∂(ψ−eα/2r) ∂ψa(ψ−eα/2r−π) r2√ 2/radicalbig 1−cos(ψ−(ψ−eα/2r)) −∂(ψ+eα/2r) ∂ψa(ψ+eα/2r−π) r2√ 2/radicalbig 1−cos(ψ−(ψ+eα/2r)) =−1 r∂Φ ∂ψ−eαa 2r3sineα 4r. (A.18) And this coincides with Eq. (A.17) based on Eq. (A.14). Atψ=πrad and with α≪rwe obtain from Eq. (A.17): Eψ(ψ=πrad)≈2a r2/parenleftBig lnα 8r+π 2/parenrightBig . (A.19) Utilizing Weber’s aas ourr0R0σB(as we saw in footnote 18) and also his αandras our r0andR0, respectively, the latter equation can be written as (divid ing the right hand side by 4πε0in order to obtain the electric ﬁeld in the international sys tem of units): Eψ(ψ=πrad)≈ −r0σB 2πε0R0/parenleftBig ln8R0 r0−π 2/parenrightBig . (A.20) 22We have checked this result and it is correct. There should be an overall minus sign in front of this expression if we wish to express the algebraic v alue of the tangential electric ﬁeld due to this dipole. 206 Figure A.7: Simpliﬁed model proposed by Weber to consider th e opposite sur- face charges ±εin the discontinuity of the potential, separated by a small d is- tanceδ. 1 + cos2(ψ/2) sin3(ψ/2)δε 8r3. (A.21) To ﬁndδεhe considers the value of the net electromotive force [due to the surface charges along the ring and due to the dipole at the discontinuity]. He speciﬁes that this net electromotive force should be almo st constant with ψ. That is, he chooses δεsuch that the second and third derivatives of the electromotive force with regard to the azimuthal angle ψgo to zero at ψ=π rad. With this condition he obtains23 δε=8aπr 5. (A.22) Combining Eqs. (A.21) and (A.22) with the previous result ar ising from Ohm’s linear hypothesis yields as the ﬁnal result:24 2a r2log coteα 8r+2aπ 5r2sin3(ψ/2)/parenleftbigg 3 cos2ψ 2−2/parenrightbigg . (A.24) In this case the absolute value of the electromotive force fo rψ/negationslash=πrad is greater than its absolute value at ψ=πrad, while with only Ohm’s linear hypothesis he had found that the electromotive force was sma ller forψ/negationslash=π 23This result is also correct. 24Combining Eq. (A.22) with the negative of Eq. (A.21), as we di scussed in footnote 22, together with Eq. (A.17), we obtain: Eψ(ψ) =2a r2lntaneα 8r+aπ r2sin(ψ/2)−eαa 2r31 sineα 4r−aπ 5r21 + cos2(ψ/2) sin3(ψ/2). (A.23) 207 rad than at ψ=πrad. He then concluded [32] [38, p. 382] (our words in square brackets): “The correct hypothesis about the distri bution of the free electricity, from which should result an equal electromoti ve force [tangential or longitudinal electric ﬁeld] in all parts [along the ring] is then contained between both hypotheses above, which means the same as: the [surface ] electric charge of the circuit increases from the neutral point [ ψ=πrad, opposite to the battery] to the contact point [ ψ= 0 rad, where there is contact between copper and zinc, or the chemical battery, or another non-electrostatic sour ce of electromotive force] not uniformly, but accelerates gradually.”25He goes on to write: “The everywhere equal electromotive force which follows from th is [analysis] will be situated presumably between the two limiting values given b y the hypotheses above, namely 2a r2/parenleftBig log coteα 8r−π 2/parenrightBig (A.25) and 2a r2/parenleftbigg log coteα 8r−2π 5/parenrightbigg . (A.26) The factor ais related to the slope of the [surface] electric charge in the middle of the circuit [ ψ=πrad], when slope is understood, according to Ohm, as the diﬀerential quotient of the charge fϕ[that is, charge per angle f(ϕ)] in relation to the arc ϕ[in other words, a=df/dϕ ].” In Section 31 Weber presents a mathematical method to estima te the dis- tribution of surface charges in a linear conductor (that is, a ﬁliform conductor which can be straight or curved) carrying a steady current, i n diﬀerent cases. His method can also yield an estimation of the magnitude of th e correspond- ing electric ﬁeld inside the conductor produced by this dist ribution of surface charges. Section 32 is called “Proof of how a necessary distribution o f free electricity on the surface of a closed conductor arises when it carries a s teady and uniform current.” He considers a closed circuit with only one point a cted upon by an electromotive force [like the contact of copper and zinc]. O nly the charges in this point will begin to move, but according to Weber, this will ca use a distribution of free charges along the whole conductor. And there will be a sp eciﬁc distribution of free charges which will create an electromotive force [el ectric ﬁeld] at all other points of the circuit, allowing it to carry a steady cur rent. He goes on to mention that this distribution of surface charges does not p roduce electrostatic 25That is, he concluded that the surface charge density along t he resistive ring carrying a steady current grows linearly with the azimuthal angle ψonly close to ψ=πrad,i.e., opposite to the battery. When we approach the battery the den sity of surface charges must grow faster than linearly with the azimuthal angle ψ, in order to produce a uniform tangential electric ﬁeld at all points along the ring. That is, the surfa ce charge density cannot increase as a function of ψsimply asσ=C1+C2ψ. If it did increase linearly with ψ, the magnitude of the tangential electric ﬁeld would not be constant at all p oints along the ring. This is a remarkable prediction conﬁrmed by our calculations in Chap ter 13. See specially Figure 13.7. 208 equilibrium, otherwise the net electric force at any point a long the surface of the conductor would be orthogonal to the conductor. He says that this distribution of surface charges will create both a normal component of the electric force at the surface [of the conductor], and a tangential component. This means, in Weber’s view, that the free charges along the surface of the c onductor carrying a steady current cannot be stationary, but must participate in the motion of the internal current. But he also shows that the motion does not i mply a temporal variation of this distribution of surface charges. That is, this distribution will not change with time for steady currents, as at any section al ong the surface there will be an equal amount of charges entering and leaving the section. [The density of surface charges will then be a function of the long itudinal coordinate only, and not a function of time.] In Section 33 he mentions that during the printing of his work , Kirchhoﬀ’s paper dealing with the same subject was published [24] (this paper has been translated into English [27]). We discuss this paper in the n ext Appendix. Weber quotes the ﬁnal section of Kirchhoﬀ’s paper. This Sect ion of Weber’s work is important to indicate that Weber and Kirchhoﬀ arrive d at essentially the same ideas independently of one another, both trying to i mprove upon Ohm’s work and hypotheses. But Weber was the only one who atte mpted to calculate explicitly the distribution of surface charges i n speciﬁc conﬁgurations. Section 34 is called “To determine, through a comparison of e lectromotive and galvanometric observations of a galvanic circuit, the r elative velocity be- tween two electrical masses in which no attraction nor repul sion arises.” Weber derives here a theoretical relation of the fundamental cons tant which appears in his law of force (1846), with the current, resistance and ele ctromotive force in a circuit carrying a steady current. According to Weber’s fo rce law, Eq. (1.1), when two charges approach or separate from from one another w ith a constant relative velocity ˙ r=√ 2c(with the modern nomenclature that c= 3×108m/s), they will not aﬀect one another, regardless of the signs of th e charges. That is, the Coulombian component of the force will be balanced by the velocity com- ponent, yielding zero net force between them. Only in 1855-5 6 did Weber and Kohlrasch succeed in obtaining experimentally the value of this fundamental constant. See Section 1.4 for references. Section 35 is called “On the ratio of the velocity of the ﬂow to the velocity of the propagation of the current.” Here Weber presents a ﬁrs t theoretical comparison of the drifting velocity of charges in a conducto r carrying a constant current, with the velocity for the propagation of a variable current along this conductor. The numerical values of these two velocities wer e not yet known at that time, as no experiments had given their orders of magnit ude. Section 36 is called “On the origin of the resistance of condu ctors.” He begins by mentioning that for a complete understanding of th e resistance it is not enough to deﬁne it by its eﬀect (as the ratio between elect romotive force and current given by Ohm’s law). That is, it is also necessary to d eﬁne resistance by its origin. In particular we need to know if it comes throug h the ponderable part of the current or from its electric ﬂuid. Weber asks: Wha t is the origin of the force that creates resistance to the motion of the char ges against the 209 electromotive force accelerating them? He wants to know if t his force is purely electric, or if it acts upon the ponderable particles of the c urrent (due to forces having another origin, like molecular forces). In his reaso ning, he considers initially Fechner’s hypothesis, i.e., he assumes a double current with an equal amount of positive and negative charges moving relative to t he wire with equal and opposite velocities. He analyzes whether the encounter of these opposite charges might give rise to the resistive force, due only to el ectromagnetic forces between these charges. To this end he considers a simpliﬁed m odel in which only the negative charges move relative to the wire, while the pos itive charges remain ﬁxed in the lattice. He is here departing from Fecher’s hypot hesis and coming close to the modern model of a current in metallic conductors in which only the electrons move relative to the lattice. But at that time n o one knew about the existence of electrons and they also did not know the orde r of magnitude of the drift velocity of the mobile charges. Weber here imagi nes a negative charge making a Keplerian elliptical orbit around a positiv e charge due to a central force which falls as 1 /r2, disregarding the components of his fundamental force law (1.1) which depend on the relative velocity and rel ative acceleration between the charges (by considering that these components h ave a small value in comparison to the greater value of the Coulombian component ). When there is an electromotive force [like an external electric ﬁeld] act ing along the wire, it will perturb this orbit into a spiral form. The loops of this spira l will increase until the negative charges come into the sphere of action of anothe r positive charge along the wire. It will orbit this second positive charge unt il it comes into the sphere of action of the third positive charge along the line c omposing the wire. This transference of the negative charge to the following po sitive charges will continue as long as the electromotive force acts upon the con ductor. In the event this electromotive force stops acting, the negative charge will no longer move forward, but will continue to circle the speciﬁc positive ch arge around which it was moving when the electromotive force was interrupted. He concludes the Section by mentioning that it would be important to calculat e the time interval needed by the negative charge to move in its spiral orbit from one positive charge to the next, but that this calculation should be diﬃcu lt, as is shown by the perturbation theory of astronomy.26. This ﬁfth part of Weber’s paper is extremely important. Here we can see that he is one of the pioneers who pointed out the surface charges i n resistive con- ductors carrying steady currents. The chemical battery or c ontact between two diﬀerent metals, like copper and zinc, creates a diﬀerence o f potential between two points. But what creates the uniform electric ﬁeld tange ntial to the circuit at every point inside a resistive wire is the distribution of free charges along 26Weber’s idea that the resistive force might be due to a newton ian central force falling as 1/r2does not seem feasible to us for two main reasons. (1) The newt onian forces are conservative and (2) do not depend on the velocities of the in teracting bodies. The resistive force responsible for Ohm’s law, on the other hand, is non-co nservative and proportional to the drifting velocities of the mobile charges, acting again st the motion of these charges. The origin of this force must be sought somewhere else. The origi n of these resistive forces is a very diﬃcult topic in physics, and even today there is no clea r answer to this question. 210 the surface of this wire. He correctly pointed out that these surface charges must be in motion together with the current, as the tangentia l electric ﬁeld will act not only inside it, but also along the surface of the condu ctor. Moreover, he was probably the ﬁrst to try to calculate this distributio n of surface charges explicitly in a speciﬁc example. In particular he considere d a ring of ﬁnite cross- section, much smaller than the length of the ring, with a smal l gap at one point where a non-electrostatic electromotive force acts. With a n ingenious calcula- tion he showed that the distribution of surface charges incr eases linearly with the azimuthal angle only in the region opposite to the batter y. He showed that as we approach the gap the surface charge density must increa se faster than linearly with the azimuthal angle, a remarkable result conﬁ rmed 150 years later when this problem was completely solved analytically, as de scribed in Chapter 13 of this book. Weber goes even further, trying to understan d the origin of the resistive force in terms of microscopic forces of electroma gnetic origin between the interacting charges composing the current. This is a rem arkable piece of work which deserves to be more widely known. Weber produced another very important study in 1864 which co ntinues the study of surface charges: “Electrodynamic measurements re lating specially to electric oscillations” [254]. This is the ﬁfth work in the se ries of “Electrodynamic measurements.” To the best of our knowledge, it has also neve r been translated into English. The main theoretical derivations of this pape r were obtained in 1857 or prior to that, but were not published at this time. A si milar treatment was ﬁrst published by Kirchhoﬀ in 1857. As we discuss Kirchho ﬀ’s papers in the next Appendix, we will not enter into details here of We ber’s similar ﬁndings which were delayed in publication. Kirchhoﬀ’s pape r was published in Poggendorﬀ’s Annalen , now known as Annalen der Physik . Poggendorﬀ wrote a note after Kirchhoﬀ’s paper relating that after seeing it he had occasion to meet Weber in Berlin. Weber showed him the paper he intended to pub lish, with essentially the same results as Kirchhoﬀ’s. But Weber had no t yet sent it to print, as he was waiting for results of experiments on this to pic to be performed together with R. Kohlrasch [255]. This paper by Weber was pub lished in 1864. It deals with the propagation of electromagnetic signals al ong wires, taking into consideration variable currents and the eﬀects of all surfa ce charges upon the current. As we will see, Weber and Kirchhoﬀ arrived at the tel egraphy equation. 211 212 Appendix B Gustav Kirchhoﬀ and Surface Charges Here we discuss three papers by Kirchhoﬀ, one from 1849 and tw o from 1857 [24, 25, 26]. All of these papers have been translated into En glish [27, 28, 29]. For this reason we present only brief summaries of them. In the ﬁrst paper he pointed out a mistake in Ohm’s hypothesis according to which a uniform volume density of electricity could remai n at rest inside a conductor. Ohm assumed also that the electroscopic or elect romotive forces act- ing along a resistive conductor carrying a steady current wo uld be proportional to the variation of this volume density of charges as regards the longitudinal coordinate (in the case of a linear conductor). According to Kirchhoﬀ, on the other hand, what is constant inside a conductor in electrost atic equilibrium is its electric potential, but not its volume charge density. T he electromotive force inside a resistive conductor carrying a steady current is pr oportional to the vari- ation of this potential with the longitudinal coordinate. A nd the potential itself originates from free charges spread along the surface of the conductor. Kirchhoﬀ shows that even in the case of steady currents the potential w ill satisfy Laplace’s equation inside the conductor. He does not try to calculate t he distribution of these surface charges in any speciﬁc example. At the end of th is ﬁrst paper he shows that Weber’s law of force between point charges is also compatible with Ohm’s law, and with his reasoning of free charges along the su rface of resistive conductors. In his ﬁrst paper of 1857 Kirchhoﬀ derives the telegraphy equ ation for a sig- nal propagating along a thin conducting wire. We present her e his main results in vectorial notation and in the International System of Uni ts SI, following a paper of 1999 [256]. Weber’s simultaneous and more thorough work was delayed in publication, and was published only in 1864. Both worked i ndependently of one another and predicted the existence of periodic modes of oscillation of the electric current propagating at light velocity in a conduct ing circuit of negligible resistance. 213 In his ﬁrst paper of 1857, Kirchhoﬀ considered a conducting c ircuit of circular cross-section which might be open or closed. Kirchhoﬀ’s wir e could be straight or curved, provided the following assumption was satisﬁed: “that the form of the central line of the wire is such, that the distance betwee n two of its points, between which a ﬁnite portion of the wire lies, is never inﬁni tely small. By this supposition the case is excluded, that induction spira ls are contained in the circuit.” He wrote Ohm’s law taking into account the free ele ctricity along the surface of the wire and the induction due to the alteration of the strength of the current in all parts of the wire: /vectorJ=−g/parenleftBigg ∇φ+∂/vectorA ∂t/parenrightBigg . (B.1) Here/vectorJis the current density, gis the conductivity of the wire, φis the electric potential and /vectorAis a function analogous to the modern magnetic vector poten- tial (which Kirchhoﬀ will calculate from Weber’s force). He calculatesφby integrating the eﬀect of all free surface charges: φ(x,y,z,t ) =1 4πε0/integraldisplay /integraldisplayσ(x′,y′,z′,t)da′ |/vector r−/vector r′|. (B.2) Here/vector r=xˆx+yˆy+zˆzis the point where the potential is being calculated, tis the time, and σis the free surface charge. Kirchhoﬀ then performed a remarkable calculation, integrating this equation over th e whole surface of the wire of length ℓand radius αwithout specifying the behaviour of σwith regard to the variables x′,y′,z′ort, but only the requirement that α≪ℓ. Moreover, he supposed that the current density was the same at all point s of the periphery of a cross-section in the wire (that is, he neglected the eﬀec ts of curvatures in the wire) and that it was never inﬁnitely large. With only the se assumptions he arrived ﬁnally at: φ(s,t) =ασ(s,t) ε0lnℓ α. (B.3) Heresis a variable distance along the wire from a ﬁxed origin. See F igure B.1. This is equivalent to our Eq. (6.8). While our equation was de rived for a straight wire carrying a steady current, Kirchhoﬀ obtained it for a wi re which might be straight or slightly curved. Moreover, in his calculation t he potential, current and surface charge density could also be a function of time. T his was a remark- able result, also obtained by Weber and published in 1864 [25 4]. He obtains the vector potential /vectorAfrom Weber’s force, Eq. (1.1). That is, the component of this force which depends upon the accelerat ion of the charges can be written as −q∂/vectorA/∂t, with a vector potential given by /vectorA(x,y,z,t ) =µ0 4π/integraldisplay /integraldisplay /integraldisplay/bracketleftBig /vectorJ(x′,y′,z′,t)·(/vector r−/vector r′)/bracketrightBig (/vector r−/vector r′)dx′dy′dz′ |/vector r−/vector r′|3.(B.4) Here the integration is through the volume of the wire. 214 Figure B.1: A long curved conductor of length ℓand radius α. The variable s represents a distance along the wire from a ﬁxed origin O. After integrating this expression he arrived at /vectorA(s,t) =µ0 2πI(s,t)/parenleftbigg lnℓ α/parenrightbigg ˆs , (B.5) whereI(s,t) is the variable current. Given that I=Jπα2and thatR=ℓ/(πgα2) is the resistance of the wire, the longitudinal component of Ohm’s law could then be writte n as ∂σ ∂s+1 2πα1 c2∂I ∂t=−ε0R αℓln(ℓ/α)I . (B.6) In order to relate the two unknowns, σandI, Kirchhoﬀ utilized the equation for the conservation of charges, which he wrote as ∂I ∂s=−2πα∂σ ∂t. (B.7) To the best of our knowledge this was the ﬁrst time that this fu ndamental equation for the conservation of charges was published in th e literature. When these two relations are equated, they yield the equatio n of telegraphy, namely: ∂2ξ ∂s2−1 c2∂2ξ ∂t2=2πε0R ℓln(ℓ/α)∂ξ ∂t, (B.8) whereξcan represent I,σ,φor the longitudinal component of /vectorA. If the resistance is negligible, this equation predicts the propagation of sig- nals along the wire with light velocity. As Kirchhoﬀ put it, t he velocity of propagation of an electric wave “is independent of the cross -section, of the con- ductivity of the wire, also, ﬁnally, of the density of the ele ctricity: its value is 41950 German miles in a second, hence very nearly equal to the velocity of light in vacuo .” Equations similar to Eq. (B.8) can be found in pages 123 and 12 5 of the second part of Volume 4 of Weber’s Collected Papers (origina l paper of 1864) [254, 39]. In his second paper of 1857 Kirchhoﬀ generalizes this ﬁrst wo rk in order to consider three-dimensional conductors of arbitrary sha pe. We discussed this 215 brieﬂy in 1994 [29]. The results are essentially the same as b efore, but now he shows that it is possible to have free electricity distrib uted throughout the substance of the conductor, in the case of a current varying i n time and in space. Recently we developed Kirchhoﬀ’s ideas in the internationa l system of units and applied them to the propagation of electromagnetic sign als in a coaxial cable, a situation which was not considered by Kirchhoﬀ [30, 31, 257, 258]. It should be stressed that the works of Kirchhoﬀ of 1857 were p ublished before Maxwell wrote down his equations in 1861-64, establi shing the electro- magnetic theory of light. When Maxwell introduced the displ acement current (1/c2)∂/vectorE/∂t he was utilizing Weber’s constant c. He was also aware of Weber and Kohlrasch’s measurement of 1854-56 that chad the same value as light ve- locity in vacuum. He also knew Kirchhoﬀ’s derivation of the t elegraphy equation yielding the propagation of electromagnetic signals at lig ht velocity. Kirchhoﬀ’s work of 1857 was quoted only once by Maxwell in the note to para graph 805, p. 450 of his Treatise [158]. It should be remarked that this speciﬁc citation does not appear in the Index at the end of Maxwell’s Treatise . For this reason it may not have been noted by some authors. In note 26 of Schaﬀer’s pa per we ﬁnd the following important remark regarding Maxwell’s knowledge of Kirchhoﬀ’s ﬁrst paper of 1857: “In the early 1870s Maxwell made detailed note s on Kirchhoﬀ’s paper on electricity in wires: see Cambridge University Lib rary MSS ADD 7655 Vn/1, p. 44 ﬀ” [259]. As mentioned above, Kirchhoﬀ’s paper of 1857 was published e arlier than Weber’s paper of 1864, although both of them arrived at essen tially the same results independently of one another and at the same time, as pointed out by Poggendorﬀ, the editor of the Annalen der Physik (at that time called Poggen- dorﬀ’s Annalen ) in 1857 [255]. 216 Bibliography [1] A. K. T. Assis, W. A. Rodrigues Jr., and A. J. Mania. The ele ctric ﬁeld outside a stationary resistive wire carrying a constant cur rent.Founda- tions of Physics , 29:729–753, 1999. [2] R. Clausius. Ableitung eines neuen elektrodynamischen Grundgesetzes. Journal f¨ ur reine und angewandte Mathematik , 82:85–130, 1877. [3] A. O’Rahilly. Electromagnetic Theory — A Critical Examination of Fun- damentals . Dover, New York, 1965. [4] U. Bartocci and M. M. Capria. Some remarks on classical el ectromag- netism and the principle of relativity. American Journal of Physics , 59:1030–1032, 1991. [5] U. Bartocci and M. M. Capria. Symmetries and asymmetries in classi- cal and relativistic electrodynamics. Foundations of Physics , 21:787–801, 1991. [6] F. Cardone and R. Mignani. Observation of a DC voltage ind uced by a steady magnetic ﬁeld: A possible electromagnetic breakdow n of Lorentz invariance? Physics Essays , 13:645–656, 2000. [7] U. Bartocci, F. Cardone, and R. Mignani. Looking for a pos sible break- down of local Lorentz invariance for electromagnetic pheno mena: theory and ﬁrst experimental results. Foundations of Physics Letters , 14:51–64, 2001. [8] J. R. Reitz, F. J. Milford, and R. W. Christy. Foundations of Electromag- netic Theory . Addison-Wesley, Reading, 4th edition, 1993. [9] B. R. Russell. Surface charges on conductors carrying st eady currents. American Journal of Physics , 36:527–529, 1968. [10] J. D. Jackson. Classical Electrodynamics . John Wiley & Sons, New York, 1962. [11] J. D. Jackson. Classical Electrodynamics . John Wiley & Sons, New York, second edition, 1975. 217 [12] J. D. Jackson. Surface charges on circuit wires and resi stors play three roles. American Journal of Physics , 64:855–870, 1996. [13] J. D. Jackson. Classical Electrodynamics . John Wiley & Sons, New York, third edition, 1999. [14] W. F. Edwards. Inertia and an alternative approach to th e theory of inter- actions. Proceedings of the Utah Academy of Science, Arts, and Letter s, 51, Part 2:1–7, 1974. [15] W. F. Edwards, C. S. Kenyon, and D. K. Lemon. Continuing i nvestiga- tion into possible electric ﬁelds arising from steady condu ction currents. Physical Review D , 14:922–938, 1976. [16] D. J. Griﬃths. Introduction to Electrodynamics . Prentice Hall, Englewood Cliﬀs, second edition, 1989. [17] C. A. Coombes and H. Laue. Electric ﬁelds and charge dist ributions associated with steady currents. American Journal of Physics , 49:450– 451, 1981. [18] P. Lorrain, D. R. Corson, and F. Lorrain. Electromagnetic Fields and Waves . W. H. Freeman and Co., New York, 3rd edition, 1988. [19] S. Seely. Introduction to Electromagnetic Fields . McGraw-Hill, New York, 1958. [20] B. D. Popovi´ c. Introductory Engineering Electromagnetics . Addison- Wesley, Reading, 1971. [21] A. K. T. Assis. Deriving Amp` ere’s law from Weber’s law. Hadronic Journal , 13:441–451, 1990. [22] A. K. T. Assis. Can a steady current generate an electric ﬁeld? Physics Essays , 4:109–114, 1991. [23] A. K. T. Assis. Weber’s Electrodynamics . Kluwer Academic Publishers, Dordrecht, 1994. ISBN: 0-7923-3137-0. [24] G. Kirchhoﬀ. Ueber eine Ableitung der Ohm’schen Gesetz e, welche sich an die Theorie der Elektrostatik anschliesst. Annalen der Physik , 78:506– 513, 1849. Reprinted in G. Kirchhoﬀ’s Gesammelte Abhandlungen (Barth, Leipzig, 1882), pp. 49-55. [25] G. Kirchhoﬀ. Ueber die Bewegung der Elektricit¨ at in Dr ¨ ahten. Annalen der Physik , 100:193–217, 1857. Reprinted in G. Kirchhoﬀ’s Gesammelte Abhandlungen (Barth, Leipzig, 1882), pp. 131-154. [26] G. Kirchhoﬀ. Ueber die Bewegung der Elektricit¨ at in Le itern. Annalen der Physik , 102:529–544, 1857. Reprinted in G. Kirchhoﬀ’s Gesammelte Abhandlungen (Barth, Leipzig, 1882), pp. 154-168. 218 [27] G. Kirchhoﬀ. On a deduction of Ohm’s law in connexion wit h the theory of electrostatics. Philosophical Magazine , 37:463–468, 1850. [28] G. Kirchhoﬀ. On the motion of electricity in wires. Philosophical Maga- zine, 13:393–412, 1857. [29] P. Graneau and A. K. T. Assis. Kirchhoﬀ on the motion of el ectricity in conductors. Apeiron , 19:19–25, 1994. [30] A. K. T. Assis. On the propagation of electromagnetic si gnals in wires and coaxial cables according to Weber’s electrodynamics. Foundations of Physics , 30:1107–1121, 2000. [31] A. K. T. Assis and J. A. Hernandes. Telegraphy equation f rom Weber’s electrodynamics. IEEE Transactions on Systems and Circuits II , 52:289– 292, 2005. [32] W. Weber. Elektrodynamische Maassbestimmungen insbe sondere Wider- standsmessungen. Abhandlungen der K¨ onigl. S¨ achs. Gesellschaft der Wis- senschaften, mathematisch-physische Klasse , 1:199–381, 1852. Reprinted in Wilhelm Weber’s Werke , Vol. 3, H. Weber (ed.), (Springer, Berlin, 1893), pp. 301-471. [33] R. P. Feynman, R. B. Leighton, and M. Sands. The Feynman Lectures on Physics . Addison-Wesley, Reading, 1964. Volume 2: Mainly Electrom ag- netism and Matter. [34] E. M. Purcell. Electricity and Magnetism , volume 2 of Berkeley Physics Course . McGraw-Hill, New York, 1965. [35] O. D. Jeﬁmenko. Is magnetic ﬁeld due to an electric curre nt a relativistic eﬀect? European Journal of Physics , 17:180–182, 1996. [36] W. Weber. Wilhelm Weber’s Werke, W. Voigt, (Ed.), volume 1, Akustik, Mechanik, Optik und W¨ armelehre . Springer, Berlin, 1892. [37] W. Weber. Wilhelm Weber’s Werke, E. Riecke (Ed.), volume 2, Mag- netismus . Springer, Berlin, 1892. [38] W. Weber. Wilhelm Weber’s Werke, H. Weber (Ed.) , volume 3, Galvanis- mus und Elektrodynamik , ﬁrst part. Springer, Berlin, 1893. [39] W. Weber. Wilhelm Weber’s Werke, H. Weber, (Ed.), volume 4, Gal- vanismus und Elektrodynamik , second part. Springer, Berlin, 1894. [40] W. Weber and E. H. Weber. Wilhelm Weber’s Werke, E. Riecke (Ed.), volume 5, Wellenlehre auf Experimente gegr¨ undet oder ¨ uber die Well en tropfbarer Fl¨ ussigkeiten mit Anwendung auf die Schall- un d Lichtwellen . Springer, Berlin, 1893. 219 [41] W. Weber. Wilhelm Weber’s Werke, F. Merkel and O. Fischer (Eds.), volume 6, Mechanik der Menschlichen Gehwerkzeuge . Springer, Berlin, 1894. [42] W. Weber. On the theory of diamagnetism - Letter from Pro fessor Weber to Prof. Tyndall. Philosophical Magazine , 10:407–410, 1855. [43] W. Weber. On the measurement of electric resistance acc ording to an absolute standard. Philosophical Magazine , 22:226–240 and 261–269, 1861. [44] W. Weber. Electrodynamic measurements — Sixth memoir, relating spe- cially to the principle of the conservation of energy. Philosophical Maga- zine, 43:1–20 and 119–149, 1872. [45] C. F. Gauss and W. Weber. Results of the observations mad e by the Magnetic Association in the year 1836. In R. Taylor, editor, Scientiﬁc Memoirs, Vol. 2, pages 20–97, New York, 1966. Johnson Reprint Corpo- ration. [46] W. Weber. On the measurement of electro-dynamic forces . In R. Tay- lor, editor, Scientiﬁc Memoirs, Vol. 5, pages 489–529, New York, 1966. Johnson Reprint Corporation. [47] W. Weber. On the excitation and action of diamagnetism a ccording to the laws of induced currents. In R. Taylor, editor, Scientiﬁc Memoirs, Vol. 5, pages 477–488, New York, 1966. Johnson Reprint Corpo ration. [48] W. Weber. On the connexion of diamagnetism with magneti sm and elec- tricity. In J. Tyndall and W. Francis, editors, Scientiﬁc Memoirs, Vol. 7, pages 163–199, New York, 1966. Johnson Reprint Corporation . [49] W. Weber. Observations on the arrangment and use of the b iﬁlar magne- tometer. In R. Taylor, editor, Scientiﬁc Memoirs, Vol. 2, pages 268–280, New York, 1966. Johnson Reprint Corporation. [50] W. Weber. On a transportable magnetometer. In R. Taylor , editor, Sci- entiﬁc Memoirs, Vol. 2, pages 565–586, New York, 1966. Johnson Reprint Corporation. [51] W. Weber. An extract from remarks on the term-observati ons for 1839, of the German Magnetic Association. In R. Taylor, editor, Scientiﬁc Mem- oirs,Vol. 2, pages 587–588, New York, 1966. Johnson Reprint Corpo ra- tion. [52] W. Weber. Determinations of electrodynamic measure: p articularly in respect to the connection of the fundamental laws of electri city with the law of gravitation. English translation by G. Gregory, unpu blished, 21st Century Science and Technology, 1996. 220 [53] C. F. Gauss and W. E. Weber. Text of the Gauss-Weber corre spondence. 21st Century , 9(3):41–43, 1996. English translation by Susan P. Johnson . [54] W. Weber. Aphorisms. 21st Century , 10(2):50–53, 1997. English transla- tion by J. Tennenbaum. [55] W. Weber and R. Kohlrausch. On the amount of electricity which ﬂows through the cross-section of the circuit in galvanic curren ts. In F. Bevilac- qua and E. A. Giannetto, editors, Volta and the History of Electricity , pages 287–297. Universit` a degli Studi di Pavia and Editore Ulrico Hoepli, Milano, 2003. [56] K. H. Wiederkehr. Wilhelm Webers Stellung in der Entwic klung der Elek- trizit¨ atslehre. Dissertation, Hamburg, 1960. [57] K. H. Wiederkehr. Wilhelm Eduard Weber — Erforscher der Wellenbewe- gung und der Elektrizit¨ at (1804-1891) , volume 32 of Grosse Naturforscher, H. Degen (ed.). Wissenschaftliche Verlagsgesellschaft, S tuttgart, 1967. [58] K. H. Wiederkehr. Carl Friedrich Gauss (1777-1855) und Wilhelm Weber (1804-1891). In K. v. Me¨ yenn, editor, Die Grossen Physiker , Vol. 1, pages 357–370 and 522–524. Verlag C. H. Beck, M¨ unchen, 1997. [59] F. Kirchner. Determination of the velocity of light fro m electromagne- tic measurements according to W. Weber and R. Kohlrausch. American Journal of Physics , 25:623–629, 1957. [60] L. Rosenfeld. The velocity of light and the evolution of electrodynamics. Il Nuovo Cimento , Supplement to vol. 4:1630–1669, 1957. [61] A. E. Woodruﬀ. The contributions of Hermann von Helmhol tz to electro- dynamics. Isis, 59:300–311, 1968. [62] W. Weber and R. Kohlrausch. ¨Uber die Einf¨ uhrung absoluter elektrischer Maße. In S. Balke, H. Gericke, W. Hartner, G. Kerstein, F. Kle mm, A. Portmann, H. Schimank, and K. Vogel, editors, Ostwalds Klassiker der exakten Wissenschaften, new series, Vol. 5 . Friedrich-Vieweg & Sohn, Braunschweig, 1968. Commented by F. Kohlrausch and K. H. Wie derkehr. [63] E. T. Whittaker. A History of the Theories of Aether and Electricity , volume 1: The Classical Theories . Humanities Press, New York, 1973. [64] L. Rosenfeld. Kirchhoﬀ, Gustav Robert. In C. C. Gillisp ie, editor, Dic- tionary of Scientiﬁc Biography, Vol. 7, pages 379–383, New York, 1981. Charles Scribner’s Sons. [65] A. E. Woodruﬀ. Weber, Wilhelm Eduard. In C. C. Gillispie , editor, Dictionary of Scientiﬁc Biography, Vol. 14, pages 203–209, New York, 1981. Charles Scribner’s Sons. 221 [66] K. Reich. Carl Friedrich Gauß — 1777/1977 . Heinz Moos Verlag, M¨ unchen, 1977. [67] M. N. Wise. German concepts of force, energy, and the ele ctromagnetic ether: 1845–1880. In G. N. Cantor and M. J. S. Hodge, editors, Con- ceptions of Ether — Studies in the History of Ether Theories 1 740–1900 , pages 269–307, Cambridge, 1981. Cambridge University Pres s. [68] P. M. Harman. Energy, Force, and Matter – The Conceptual Development of Nineteenth-Century Physics . Cambridge University Press, Cambridge, 1982. [69] C. Jungnickel and R. McCormmach. Intellectual Mastery of Nature — Theoretical Physics from Ohm to Einstein , volume 1-2. University of Chicago Press, Chicago, 1986. [70] W. A. Atherton. Gauss and Weber: An unlikely partnershi p.Electronics and Wireless World , 95:521–522, 1989. [71] T. Archibald. Energy and the mathematization of electr odynamics in Germany, 1845-1875. Archives Internationales d’Histoire des Sciences , 39:276–308, 1989. [72] K. H. Wiederkehr. Wilhelm Weber und die Entwicklung in d er Geomag- netik und Elektrodynamik. In G. Berg, W. Eisenberg, W. Herge rt, and H. Thieme, editors, I. Weber - Symposium anl¨ aßlich des 100. Todestages W. Webers in Halle und Wittenberg am 20. und 21. Juni 1991 , pages 1– 14. Fachbereich Physik der Martin-Luther-Universit¨ at Ha lle-Wittenberg und Wilhelm-Weber-Gesellschaft e. V., 1993. [73] K. H. Wiederkehr. W. Weber und die Entwicklung der Elekt rodynamik. In G. Berg, W. Eisenberg, W. Hergert, and W. Schreier, editors, II. Weber - Symposium Die Gebr¨ uder Weber - Wegbereiter interdiszipl in¨ arer For- shcung in Halle und Leipzig am 16. Oktober und 18. November 19 93, pages 39–54. Fachbereich Physik der Martin-Luther-Univer sit¨ at Halle- Wittenberg und Wilhelm-Weber-Gesellschaft e. V., 1993. [74] K. H. Wiederkehr. Das Experiment von Wilhelm Weber und R udolf Kohlrausch 1855 und Maxwells elektromagnetische Lichtthe orie. In W. Schr¨ oder, editor, The Earth and the Universe (A Festschrift in hon- our of Hans-J¨ urgen Treder) , pages 452–463. Interdivisional Commission on History of the International Association of Geomagnetis m and Aeron- omy, Bremen-R¨ onnebeck, 1993. [75] K. H. Wiederkehr. Wilhelm Weber und Maxwells elektroma gnetische Lichttheorie. Gesnerus, Part. 3/4 , 51:256–267, 1994. [76] L. Hecht. Wilhelm Weber: giant of 19th century physics. 21st Century , 10(2):53–54, 1997. 222 [77] O. Darrigol. Electrodynamics from Amp` ere to Einstein . Oxford University Press, Oxford, 2000. [78] J. P. Wesley. Weber electrodynamics, Part I. General th eory, steady cur- rent eﬀects. Foundations of Physics Letters , 3:443–469, 1990. [79] T. E. Phipps Jr. Toward modernization of Weber’s force l aw.Physics Essays , 3:414–420, 1990. [80] T. E. Phipps Jr. Weber-type laws of action-at-a-distan ce in modern physics. Apeiron , 8:8–14, 1990. [81] J. P. Wesley. Weber electrodynamics, Part II. Unipolar induction, Z- antenna. Foundations of Physics Letters , 3:471–490, 1990. [82] J. P. Wesley. Weber electrodynamics, Part III. Mechani cs, gravitation. Foundations of Physics Letters , 3:581–605, 1990. [83] J. P. Wesley. Selected Topics in Advanced Fundamental Physics . Benjamin Wesley Publisher, Blumberg, 1991. [84] R. A. Clemente and A. K. T. Assis. Two-body problem for We ber-like interactions. International Journal of Theoretical Physics , 30:537–545, 1991. [85] A. K. T. Assis. Curso de Eletrodinˆ amica de Weber . Setor de Publica¸ c˜ oes do Instituto de F´ ısica da Universidade Estadual de Campina s - Unicamp, Campinas, 1992. Notas de F´ ısica IFGW N´ umero 5. Available o nline in PDF format at: http://www.ifi.unicamp.br/˜assis/. [86] T. E. Phipps Jr. Derivation of a modernized Weber force l aw.Physics Essays , 5:425–428, 1992. [87] A. K. T. Assis and R. A. Clemente. The ultimate speed impl ied by theories of Weber’s type. International Journal of Theoretical Physics , 31:1063– 1073, 1992. [88] S. Ragusa. Gravitation with a modiﬁed Weber force. Foundations of Physics Letters , 5:585–589, 1992. [89] J. Guala-Valverde. Gravitational redshift revisited .Galilean Electrody- namics , 3:58, 1992. [90] A. K. T. Assis and R. A. Clemente. The inﬂuence of tempera ture on gravitation. Il Nuovo Cimento B , 108:713–716, 1993. [91] G. Galeczki. The ultimate speed and Weber’s potential. Physics Essays , 6:448–450, 1993. [92] A. K. T. Assis. Eletrodinˆ amica de Weber - Teoria, Aplica¸ c˜ oes e Exerc´ ıc ios. Editora da Universidade Estadual de Campinas - UNICAMP, Cam pinas, 1995. ISBN: 85-268-0358-1. 223 [93] G. Galeczki and P. Marquardt. ¨Uber die longitudinalen Amp` ereschen Kr¨ afte und Webers Elektrodynamik. Fusion , 16:15–17, 1995. [94] J. Tennenbaum. An introduction to ‘The signiﬁcance of t he 1845 Gauss- Weber correspondence’. 21st Century , 9(3):2–5, 1996. [95] L. Hecht. The signiﬁcance of the 1845 Gauss-Weber corre spondence. 21st Century , 9(3):22–34, 1996. [96] S. D’Agostino. Absolute systems of units and dimension s of physical quan- tities: a link between Weber’s electrodynamics and Maxwell ’s electromag- netic theory of light. Physis , 33:5–51, 1996. [97] L. Hecht. Experimental apparatus and instrumentation .21st Century , 9(3):35–40, 1996. [98] T. E. Phipps, Jr. Clock rates in a machian universe. Toth-Maatian Review , 13:5910–5917, 1996. [99] E. T. Kinzer and J. Fukai. Weber’s force and Maxwell’s eq uations. Foun- dations of Physics Letters , 9:457–461, 1996. [100] F. Bunchaft and S. Carneiro. Weber-like interactions and energy conser- vation. Foundations of Physics Letters , 10:393–401, 1997. [101] J. Fukai and E. T. Kinzer. Compatibility of Weber’s for ce with Maxwell’s equations. Galilean Electrodynamics , 8:53–55, 1997. [102] A. K. T. Assis. Mecˆ anica Relacional . Editora do Centro de L´ ogica, Episte- mologia e Hist´ oria da Ciˆ encia da UNICAMP/FAPESP, Campina s, 1998. ISBN: 85-86497-01-0. [103] Marcelo Bueno and A. K. T. Assis. C´ alculo de Indutˆ ancia e de For- ¸ ca em Circuitos El´ etricos . Editora da UFSC/Editora da UEM, Flo- rian´ opolis/Maring´ a, 1998. ISBN: 85-328-0119-6. [104] J. Guala-Valverde. Gravitational mutual energy in We ber-type mechanics. Physics Essays , 11:164–165, 1998. [105] A. K. T. Assis. Uma Nova F´ ısica . Editora Perspectiva, S˜ ao Paulo, 1999. ISBN: 85-273-0199-7. [106] V. F. Mikhailov. The action of an electrostatic potent ial on the electron mass. Annales de la Fondation Louis de Broglie , 24:161–169, 1999. [107] A. K. T. Assis. Relational Mechanics . Apeiron, Montreal, 1999. ISBN: 0-9683689-2-1. [108] A. G. Gluckman. On W. E. Weber’s “Grundprincip der Elek trodynamik” and related earlier and contemporary studies. Physics Essays , 12:682–698, 1999. 224 [109] J. Guala-Valverde. Inercia y Gravitacion . Fundacion Julio Palacios, Neuquen, Argentina, 1999. In collabora- tion with J. Tramaglia and R. Rapacioli. Available at (ac- cessed in 2007): http://www.educ.ar/educar/servlet/Dow nloads/- SBDLIBROSI/INERCIAYGRAVITACION.PDF. [110] A. K. T. Assis and H. Torres Silva. Comparison between W eber’s elec- trodynamics and classical electrodynamics. Pramana Journal of Physics , 55:393–404, 2000. [111] A. K. T. Assis and J. Guala-Valverde. Mass in relationa l mechanics. Apeiron , 7:131–132, 2000. [112] S. D’Agostino. A History of the Ideas of Theoretical Physics . Kluwer, Dordrecht, 2000. [113] J. Guala-Valverde. A new theorem in relational mechan ics.Apeiron , 8:132–138, 2001. [114] M. d. A. Bueno and A. K. T. Assis. Inductance and Force Calculations in Electrical Circuits . Nova Science Publishers, Huntington, New York, 2001. ISBN: 1-56072-917-1. [115] V. F. Mikhailov. Inﬂuence of an electrostatic potenti al on the inertial electron mass. Annales de la Fondation Louis de Broglie , 26:33–38, 2001. [116] J. Guala-Valverde and P. Mazzoni. The unipolar dynamo tor: a genuine relational engine. Apeiron , 8:41–52, 2001. [117] R. A. Rapacioli. More about the claimed identity betwe en inertial and gravitational mass. Apeiron , 8:1–9, 2001. [118] J. P. Wesley. Selected Topics in Scientiﬁc Physics . Benjamin Wesley Publisher, Blumberg, 2002. [119] A. K. T. Assis, K. Reich, and K. H. Wiederkehr. Gauss and Weber’s creation of the absolute system of units in physics. 21st Century , 15, No. 3:40–48, 2002. [120] A. K. T. Assis and J. Guala-Valverde. On absolute and re lative motions in physics. Journal of New Energy , 6:8–12, 2002. [121] J. Guala-Valverde, P. Mazzoni, and R. Achilles. The ho mopolar motor: A true relativistic engine. American Journal of Physics , 70:1052–1055, 2002. [122] A. K. T. Assis and J. Guala-Valverde. Frequency in rela tional mechanics. Annales de la Fondation Louis de Broglie , 28:83–97, 2003. 225 [123] V. F. Mikhailov. Inﬂuence of a ﬁeld-less electrostati c potential on the inertial electron mass. Annales de la Fondation Louis de Broglie , 28:231– 236, 2003. [124] A. K. T. Assis and K. H. Wiederkehr. Weber quoting Maxwe ll.Mitteilun- gen der Gauss-Gesellschaft , 40:53–74, 2003. [125] J. Fukai. A Promenade Along Electrodynamics . Vales Lake Publishing, Pueblo West, 2003. [126] J. E. Junginger and Z. D. Popovic. An experimental inve stigation of the inﬂuence of an electrostatic potential on electron mass as p redicted by Weber’s force law. Canadian Journal of Physics , 82:731–735, 2004. [127] K. H. Wiederkehr. Ein bisher unver¨ oﬀentlichter Brie f von Rudolf Kohlrausch an Andreas v. Ettingshausen von 1854, das Kohlra usch- Weber-Experiment von 1854/55 und die Lichtgeschwindigkei t in Wil- helm Webers Elektrodynamik. NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine , 12:129–145, 2004. [128] D. F. Bartlett. Analogies between electricity and gra vity. Metrologia , 41:S115–S124, 2004. [129] A. K. T. Assis, K. Reich, and K. H. Wiederkehr. On the ele ctromagnetic and electrostatic units of current and the meaning of the abs olute system of units - For the 200th anniversary of Wilhelm Weber’s birth .Sudhoﬀs Archiv , 88:10–31, 2004. [130] J. Guala-Valverde. Distant matter in physics. Annales de la Fondation Louis de Broglie , 29:635–640, 2004. [131] J. Guala-Valverde, R. Achilles, and R. Blas. Inertial mass: a changing entity? — Weber vs. Einstein, Weber plus Einstein or none? Apeiron , 12:351–352, 2005. [132] J. P. Wesley and P. Marquardt. Light a Photon Flux and Other Topics . Benjamin Wesley Publisher, Blumberg, 2006. [133] T. E. Phipps Jr. Old Physics for New - a worldview alternative to Ein- stein’s relativity theory . Apeiron, Montreal, 2006. [134] W. Weber. Vorwort bei der ¨Ubergabe der Abhandlung: Elektrodynamische Maassbestimmungen, insbesondere Zur¨ uckf¨ uhrung der Str omintensit¨ ats- Messungen auf mechanisches Maass .Berichte ¨ uber die Verhandlungen der K¨ onigl. S¨ achs. Gesellschaft der Wissenschaften zu Leipz ig, mathematisch- physische Klasse , 17:55–61, 1855. Reprinted in Wilhelm Weber’s Werke , Vol. 3, H. Weber (ed.), (Springer, Berlin, 1893), pp. 591-59 6. 226 [135] W. Weber and R. Kohlrausch. ¨Uber die Elektricit¨ atsmenge, welche bei galvanischen Str¨ omen durch den Querschnitt der Kette ﬂies st.Annalen der Physik und Chemie, J. C. Poggendoﬀ (ed.), 99:10–25, 1856. Reprinted in Wilhelm Weber’s Werke , Vol. 3, H. Weber (ed.), (Springer, Berlin, 1893), pp. 597-608. [136] R. Kohlrausch and W. Weber. Elektrodynamische Maassb estimmungen insbesondere Zur¨ uckf¨ uhrung der Stromintensit¨ ats-Mes sungen auf mecha- nisches Maass. Abhandlungen der K¨ onigl. S¨ achs. Gesellschaft der Wis- senschaften, mathematisch-physische Klasse , 3:221–290, 1857. Reprinted in Wilhelm Weber’s Werke , Vol. 3, H. Weber (ed.), (Springer, Berlin, 1893), pp. 609-676. [137] W. Weber. Elektrodynamische Maassbestimmungen - ¨ ub er ein allgemei- nes Grundgesetz der elektrischen Wirkung. Abhandlungen bei Begr¨ undung der K¨ onigl. S¨ achs. Gesellschaft der Wissenschaften am Ta ge der zweihun- dertj¨ ahrigen Geburtstagfeier Leibnizen’s herausgegebe n von der F¨ urstl. Ja- blonowskischen Gesellschaft (Leipzig) , pages 211–378, 1846. Reprinted in Wilhelm Weber’s Werke , Vol. 3, H. Weber (ed.), (Springer, Berlin, 1893) pp. 25-214. [138] G. T. Fechner. Ueber die verkn¨ upfung der Faraday’sch en inductions- erscheinungen mit den Amp` ereschen elektro-dynamischen e rscheinungen. Annalen der Physik , 64:337–345, 1845. [139] H. C. Oersted. Experiments on the eﬀect of a current of e lectricity on the magnetic needle. Annals of Philosophy , 16:273–277, 1820. [140] H. C. Oersted. Experiments on the eﬀect of a current of e lectricity on the magnetic needle. In R. A. R. Tricker , Early Electrodynamics — The First Law of Circulation , pages 113–117, New York, 1965. Pergamon. [141] A. M. Amp` ere. Sur les eﬀets des courans ´ electriques. Annales de Chimie et de Physique , 15:59–76, 1820. Available at: http://www.ampere.cnrs.f r/. [142] A. M. Amp` ere. The eﬀects of electric currents. In R. A. R. Tricker , Early Electrodynamics - The First Law of Circulation , pages 140–154, New York, 1965. Pergamon. Translated by O. M. Blunn. [143] C. Blondel. A.-M. Amp` ere el la Cr´ eation de l’ ´Electrodynamique (1820- 1827). Biblioth` eque Nationale, Paris, 1982. [144] L. P. Williams. What were Amp` ere’s earliest discover ies in electrodynam- ics?Isis, 74:492–508, 1983. [145] P. Graneau. Ampere-Neumann Electrodynamics of Metals . Hadronic Press, Nonantum, 1985. [146] J. R. Hofmann. Amp` ere, electrodynamics, and experim ental evidence. Osiris , 3 (2nd Series):45–76, 1987. 227 [147] J. R. Hofmann. Amp` ere’s invention of equilibrium app aratus: a response to experimental anomaly. The British Journal for the History of Science , 20:309–341, 1987. [148] J. R. Hofmann. Amp` ere’s electrodynamics and the acce ptability of guiding assumptions. In A. Donovan et al., editor, Scrutinizing Science , pages 201–217. Kluwer Academic Publishers, Dordrecht, 1988. [149] L. P. Williams. Andr´ e-Marie Amp` ere. Scientiﬁc American , January:72– 79, 1989. [150] C. Blondel. Vision physique “´ eth´ erienne”, math´ em atisation “laplacienne”: l’´ electrodynamique d’Amp` ere. Revue d’Histoire des Sciences , 42:123–137, 1989. [151] P. Graneau and N. Graneau. Newton Versus Einstein – How Matter In- teracts with Matter . Carlton Press, New York, 1993. [152] P. Graneau. Ampere-Neumann Electrodynamics of Metals . Hadronic Press, Palm Harbor, 2nd edition, 1994. [153] J. R. Hofmann. Andr´ e-Marie Amp` ere, Enlightenment and Electrodynam- ics. Cambridge University Press, Cambridge, 1996. [154] P. Graneau and N. Graneau. Newtonian Electrodynamics . World Scien- tiﬁc, Singapore, 1996. [155] T. E. Phipps, Jr. and J. Guala Valverde. Simulation of A mp` erian current elements by magnetic toroids. 21st Century , 11:55–65, 1998. [156] J. L¨ uhr. Die Entstehung eines Demonstrationsexperi ments: Zur Geschichte der Amp` ere’schen Stromwaage. In P. Heering, F. Rieß, and C. Sichau, editors, Im Labor der Physikgeschichte — Zur Untersuchung historischer Experimentalpraxis , pages 135–156. BIS der Carl von Ossiet- zky, Oldenburg, 2000. [157] N. Graneau, T. Phipps Jr., and D. Roscoe. An experiment al conﬁrmation of longitudinal electrodynamic forces. European Journal of Physics D , 15:87–97, 2001. [158] J. C. Maxwell. A Treatise on Electricity and Magnetism . Dover, New York, 1954. [159] R. Clausius. Ueber die Vergleichung der elektrodynam ische Grundgesetze mit der Erfahrung. Annalen der Physik , 10:608–618, 1880. [160] F. Rohrlich. Classical Charged Particles — Foundations of Their Theory . Addison-Wesley, Reading, 1965. [161] R. Skinner. Mechanics . Blaisdell, Waltham, 1969. 228 [162] J. M. Pearson and A. Kilambi. Velocity-dependent nucl ear forces and Weber’s electrodynamics. American Journal of Physics , 42:971–975, 1974. [163] V. Bush. The force between moving charges. Journal of Mathematical Physics , 5:129–157, 1926. [164] S. Parker. Electrostatics and current ﬂow. American Journal of Physics , 38:720–723, 1970. [165] R. W. Chabay and B. A. Sherwood. Electric and Magnetic Interactions . Wiley, New York, 1995. [166] R. W. Chabay and B. A. Sherwood. Matter & Interactions , volume 2: Electric and Magnetic Interactions. Wiley, New York, 2002. [167] W. G. V. Rosser. Magnitudes of surface charge distribu tions associated with electric current ﬂow. American Journal of Physics , 38:265–266, 1970. [168] W. K. H. Panofsky and M. Phillips. Classical Electricity and Magnetism . Addison-Wesley, Reading, 2nd edition, 1962. [169] J. H. Poynting. On the connexion between electric curr ent and the electric and magnetic inductions in the surrounding ﬁeld. Transactions of the Royal Society of London A , 176:277–306, 1885. [170] A. Marcus. The electric ﬁeld associated with a steady c urrent in long cylindrical conductor. American Journal of Physics , 9:225–226, 1941. [171] R. W. Chabay and B. A. Sherwood. A uniﬁed treatment of el ectrostatics and circuits. Unpublished manuscript. Available at (acces sed in 2007): http://www4.ncsu.edu/˜rwchabay/mi/circuit.pdf. [172] L. Bergmann and C. Schaefer. Lehrbuch der Experimentalphysik , volume 2: Elektrizit¨ atslehre. Walter de Gruyter, Berlin, 1950. [173] L. Bergmann and C. Schaefer. Lehrbuch der Experimentalphysik , volume 2: Elektrizit¨ at und Magnetismus. Walter de Gruyter, Berli n, seventh edition, 1987. [174] O. Jeﬁmenko. Demonstration of the electric ﬁelds of cu rrent-carrying conductors. American Journal of Physics , 30:19–21, 1962. [175] O. D. Jeﬁmenko. Electricity and Magnetism . Appleton-Century-Crofts, New York, 1966. [176] O. D. Jeﬁmenko. Electricity and Magnetism . Electret Scientiﬁc Company, Star City, 2nd edition, 1989. [177] O. Jeﬁmenko, T. L. Barnett, and W. H. Kelly. Conﬁnement and shaping of electric ﬁelds by current-carrying conductors. Proceedings of the West Virginia Academy of Science , 34:163–167, 1962. 229 [178] R. Sansbury. Detection of a force between a charged met al foil and a current-carrying conductor. Review of Scientiﬁc Instruments , 56:415–417, 1985. [179] D. F. Bartlett and S. Maglic. Test of an anomalous elect romagnetic eﬀect. Review of Scientiﬁc Instruments , 61:2637–2639, 1990. [180] C. K. Whitney. Current elements in relativistic ﬁeld t heory. Physics Letters A , 128:232–234, 1988. [181] H. Hayden. Possible explanation for the Edwards eﬀect .Galilean Electro- dynamics , 1:33–35, 1990. [182] T. Ivezi´ c. Electric ﬁelds from steady currents and un explained electro- magnetic experiments. Physical Review A , 44:2682–2685, 1991. [183] R. C. Ritter and G. T. Gillies. Torsion balances, torsi on pendulums, and related devices. Review of Scientiﬁc Instruments , 64:283–309, 1993. [184] W. R. Moreau et al. Charge density in circuits. American Journal of Physics , 53:552–553, 1985. [185] R. W. Chabay and B. A. Sherwood. Video demonstrating su rface charge on a high-voltage circuit. Available at (accessed in 2007): http://www4.ncsu.edu/˜rwchabay/mi/. [186] D. F. Bartlett and B. F. L. Ward. Is an electron’s charge independent of its velocity? Physical Review D , 16:3453–3458, 1977. [187] C. S. Kenyon and W. F. Edwards. Test of current-depende nt electric ﬁelds. Physics Letters A , 156:391–394, 1991. [188] G. Bonnet. Electric ﬁeld arising from a steady current passing through a superconductor. Physics Letters A , 82:465–467, 1981. [189] J. C. Cur´ e. A modiﬁed version of the Millikan oil drop e xperiment to test the probable existence of a new electrodynamic ﬁeld. Physics Letters B , 116:158–160, 1982. [190] J. P. Wesley. Weber electrodynamics extended to inclu de radiation. Spec- ulations in Science and Technology , 10:47–61, 1987. [191] R. I. Gray. Uniﬁed Physics . Dahlgren, Virginia, 1988. [192] D. F. Bartlett and W. F. Edwards. Invariance of charge t o Lorentz trans- formation. Physics Letters A , 151:259–262, 1990. [193] T. Ivesi´ c. The relativistic electric ﬁelds arising f rom steady conduction currents. Physics Letters A , 144:427–431, 1990. [194] N. Bili´ c. Comment on Lorentz invariance/noninvaria nce of the electric charge. Physics Letters A , 162:87–90, 1992. 230 [195] D. K. Lemon, W. F. Edwards, and C. S. Kenyon. Electric po tentials asso- ciated with steady conduction currents in superconducting coils.Physics Letters A , 162:105–114, 1992. [196] A. K. Singal. On the charge invariance and relativisti c electric ﬁelds from a steady conduction current. Physics Letters A , 162:91–95, 1992. [197] V. N. Strel’tsov. The electric ﬁeld of a current-carry ing conductor. Hadronic Journal , 15:457–461, 1992. [198] G. G. Shishkin et al. Investigation of possible electric potential arising from a constant current through a superconductor coil. Journal of Physics D, 35:497–502, 2002. [199] J. A. Hernandes and A. K. T. Assis. Electric potential d ue to an inﬁnite conducting cylinder with internal or external point charge .Journal of Electrostatics , 63:1115–1131, 2005. [200] J. C. Maxwell. An Elementary Treatise on Electricity . Clarendon Press, Oxford, 1881. [201] M. A. Melehy. Thermal momentum in thermodynamics, par t 2. interfacial electriﬁcation: a new consequence of the ﬁrst and second law s.Physics Essays , 11:430–433, 1998. [202] G. B. Arfken and H. J. Weber. Mathematical Methods for Physicists . Academic Press, San Diego, fourth edition, 1995. [203] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products . Academic Press, New York, 5th edition, 1994. [204] R. N. Varney and L. H. Fisher. Electromotive force: Vol ta’s forgotten concept. American Journal of Physics , 48:405–408, 1980. [205] J. A. Hernandes, E. Capelas de Oliveira, and A. K. T. Ass is. Potential, electric ﬁeld and surface charges close to the battery for a r esistive cylin- drical shell carrying a steady longitudinal current. Revista de la Facultad de Ingenier´ ıa (Chile) , 12:13–20, 2004. [206] M. A. Matzek and B. R. Russell. On the transverse electr ic ﬁeld within a conductor carrying a steady current. American Journal of Physics , 36:905–907, 1968. [207] W. R. Smythe. Static and Dynamic Electricity . McGraw-Hill, New York, 3rd edition, 1968. [208] A. Sommerfeld. Electrodynamics . Academic Press, New York, 1964. [209] W. G. V. Rosser. What makes an electric current “ﬂow”. American Journal of Physics , 31:884–885, 1963. 231 [210] W. R. Moreau. Charge distributions on DC circuits and K irchhoﬀ’s laws. European Journal of Physics , 10:286–290, 1989. [211] B.-S. Eylon and U. Ganiel. Macro-micro relationships : the missing link between electrostatics and electrodynamics in students’ r easoning. Inter- national Journal of Science Education , 12:79–94, 1990. [212] C. Schaefer. Einf¨ uhrung in die theoretische Physik , volume 3, part 1. Walter de Gruyter, Berlin, 1932. [213] B. R. Russell. Surface charges and leaky shields. American Journal of Physics , 51:269–270, 1983. [214] R. N. Varney and L. H. Fisher. Electric ﬁeld associated with stationary currents. American Journal of Physics , 52:1097–1099, 1984. [215] A. K. T. Assis and J. I. Cisneros. The problem of surface charges and ﬁelds in coaxial cables and its importance for relativistic physi cs. In F. Selleri, editor, Open Questions in Relativistic Physics , pages 177–185, Montreal, 1998. Apeiron. [216] A. K. T. Assis and J. I. Cisneros. Surface charges and ﬁe lds in a resistive coaxial cable carrying a constant current. IEEE Transactions on Circuits and Systems I , 47:63–66, 2000. [217] G. F. Leal Ferreira and Mariˆ angela T. de Figueiredo. A eletrost´ atica na corrente el´ etrica em ﬁos. Revista Brasileira de Ensino de F´ ısica , 25:374– 377, 2003. [218] J. A. Stratton. Electromagnetic Theory . McGraw-Hill, New York, 1941. [219] A. K. T. Assis and A. J. Mania. Surface charges and elect ric ﬁeld in a two- wire resistive transmission line. Revista Brasileira de Ensino de F´ ısica , 21:469–475, 1999. [220] A. K. T. Assis, J. A. Hernandes, and J. E. Lamesa. Surfac e charges in conductor plates carrying constant currents. Foundations of Physics , 31:1501–1511, 2001. [221] R. Welti. Las cargas superﬁciales y el ﬂujo de energ´ ıa en un circuito simple. Revista Brasileira de Ensino de F´ ısica , 27:577–582, 2005. [222] J. A. Hernandes, E. Capelas de Oliveira, and A. K. T. Ass is. Resistive plates carrying a steady current: electric potential and su rface charges close to the battery. Foundations of Physics Letters , 18:275–289, 2005. [223] J. A. Hernandes and A. K. T. Assis. The potential, elect ric ﬁeld and surface charges for a resistive long straight strip carryin g a steady current. American Journal of Physics , 71:938–942, 2003. 232 [224] P. Moon and D. E. Spencer. Field Theory Handbook . Springer, Berlin, 2nd edition, 1988. [225] D. J. Griﬃths and Y. Li. Charge density on a conducting n eedle. American Journal of Physics , 64:706–714, 1996. [226] M. A. Heald. Electric ﬁelds and charges in elementary c ircuits. American Journal of Physics , 52:522–526, 1984. [227] J. A. Hernandes and A. K. T. Assis. The electric ﬁeld out side and inside a resistive spherical shell carrying a steady azimuthal cur rent. Physica Scripta , 72:212–217, 2005. [228] J. A. Hernandes and A. K. T. Assis. Electric potential f or a resistive toroidal conductor carrying a steady azimuthal current. Ph ysical Review E, Vol. 68, 046611, 2003. [229] J. A. Hernandes and A. K. T. Assis. Surface charges and e xternal electric ﬁeld in a toroid carrying a steady current. Brazilian Journal of Physics , 34:1738–1744, 2004. [230] C. Neumann. Theorie der Elektricit¨ ats- und W¨ arme-Vertheilung in ein em Ringe. Verlag der Buchhandlung des Waisenhauses, Halle, 1864. [231] E. Weber. Electromagnetic Theory – Static Fields and Their Mapping . Dover, New York, 1965. Formerly titled: Electromagnetic Fi elds, Theory and Applications. [232] H. Bateman. Higher Transcendental Functions , volume 1. McGraw-Hill, New York, 1953. [233] E. Ley-Koo and A. G´ ongora T. Desarrollo arm´ onico del potencial de Coulomb en coordenadas toroidales. Revista Mexicana de F´ ısica , 40:805– 813, 1994. [234] E. Weber. Electromagnetic Fields: Theory and Applications , volume 1. Wiley, New York, 1950. [235] W. R. Smythe. Static and Dynamic Electricity . Hemisphere, New York, 3rd edition, 1989. [236] P. M. Morse and H. Feshbach. Methods of Theoretical Physics , volume 2. McGraw-Hill, New York, 1953. [237] D. E. Gray (editor). American Institute of Physics Handbook . McGraw- Hill, New York, third edition, 1972. [238] C. Snow. Formulas for computing capacitance and induc tance. National Bureau of Standards Circular , 544:1–69, 1954. 233 [239] P. Moon and D. E. Spencer. Field Theory for Engineers . Van Nostrand, Princeton, 1961. [240] O. Jeﬁmenko. Electric ﬁelds in conductors. Physics Teacher , 15:52–53, 1977. [241] W. M. Saslow. Consider a spherical batery... American Journal of Physics , 62:495–501, 1994. [242] J. M. Aguirregabiria, A. Hernandez, and M. Rivas. An ex ample of surface- charge distribution on conductors carrying steady current s.American Journal of Physics , 60:138–141, 1992. [243] J. M. Aguirregabiria, A. Hernandez, and M. Rivas. Surf ace charges and energy ﬂow in a ring rotating in a magnetic ﬁeld. American Journal of Physics , 64:892–895, 1996. [244] H. H¨ artel. The electric voltage: What do students und erstand? What can be done to help for a better understanding? In W. Jung R. Duit and C. v. Rh¨ oneck, editors, Aspects of Understanding Elec- tricity. Proceedings of an International Workshop , pages 353–362, Kiel, 1985. IPN-Arbeitsberichte 59. Available at (accesse d in 2007): http://www.astrophysik.uni-kiel.de/˜hhaertel/PUB/vo ltageIPN.pdf. [245] H. H¨ artel. A qualitative approach to electricity. In stitute for Research on Learning, Palo Alto, CA, Report No. IRL87-0001. A revised pa rt of the full paper is available at (accessed in 2007): http://www.a strophysik.uni- kiel.de/˜hhaertel/PUB/voltage IRL.pdf, 1987. [246] N. W. Preyer. Transient behavior of simple RC circuits .American Journal of Physics , 70:1187–1193, 2002. [247] H. Torres S. and A. K. T. Assis. The inﬂuence of the elect ric ﬁeld outside a resistive solenoid on the Aharonov-Bohm eﬀect. Revista de la Facultad de Ingenieria de la Universidad de Tarapaca (Chile) , 9:29–34, 2001. [248] D. Psillos, P. Koumaras, and A. Tiberghien. Voltage pr esented as a pri- mary concept in an introductory teaching sequence on DC circ uits.Inter- national Journal of Science Education , 10:29–43, 1988. [249] A. Benseghir and J.-L. Closset. The electrostatics-e lectrokinetics tran- sition: historical and educational diﬃculties. International Journal of Science Education , 18:179–191, 1996. [250] B. A. Thacker, U. Ganiel, and D. Boys. Macroscopic phen omena and microscopic processes: Students understanding of transie nts in direct cur- rent electric circuits. American Journal of Physics , 67, Suppl. 1:S25–S31, 1999. 234 [251] A. P. French. Newtonian Mechanics . W. W. Norton & Company, New York, 1971. [252] G. S. Ohm. The galvanic circuit investigated mathemat ically. In R. Tay- lor, editor, Scientiﬁc Memoirs, Vol. 2, pages 401–506, New York, 1966. Johnson Reprint Corporation. English translation by W. Fra ncis. [253] R. K. Pathria. Statistical Mechanics . Pergamon Press, Oxford, 1977. [254] W. Weber. Elektrodynamische Maassbestimmungen insb esondere ¨ uber elektrische Schwingungen. Abh. s¨ achs. Ges. Wiss. , 6:105–241, 1864. Reprinted in Wilhelm Weber’s Werke , Vol. 4, H. Weber (ed.), (Springer, Berlin, 1894), pp. 105-241. [255] J. C. Poggendorﬀ. Bermerkung zu dem Aufsatz des Herrn P rof. Kirchhoﬀ. Annalen der Physik , 100:351–352, 1857. Reprinted in W. Weber’s Werke , Vol. 4, p. 242, H. Weber (ed.), (Springer, Berlin, 1894). [256] A. K. T. Assis. Arguments in favour of action at a distan ce. In A. E. Chubykalo, V. Pope, and R. Smirnov-Rueda, editors, Instantaneous Ac- tion at a Distance in Modern Physics - “Pro” and “Contra” , pages 45–56, Commack, 1999. Nova Science Publishers. [257] J. A. Hernandes and A. K. T. Assis. Propaga¸ c˜ ao de sina is em condutores com a eletrodinˆ amica de Weber e compara¸ c˜ ao com o eletroma gnetismo cl´ assico. In SBF, editor, Anais do XXI Encontro Nacional de F´ ısica de Part´ ıculas e Campos, S˜ ao Louren¸ co, 23 a 27/10/00 , page 1, S˜ ao Paulo, 2000. Sociedade Brasileira de F´ ısica. Published at (acces sed in 2007): http://www.sbf1.if.usp.br/eventos/enfpc/xxi/procs/r es89/. [258] J. A. Hernandes and A. K. T. Assis. Propaga¸ c˜ ao de sina is em condutores segundo a eletrodinˆ amica de Weber. Ciˆ encia e Natura , 23:7–26, 2001. [259] S. Schaﬀer. Accurate measurement is an english scienc e. In M. Norton Wise, editor, The Values of Precision , chapter 6, pages 137–172. Princeton University Press, Princeton, 1995. 235 Index Aharonov-Bohm eﬀect, 192, 193 Amp` ere’s force between current elements, 15, 16 law, 16, 19 Amp` ere, A. M., 12, 16 Anomalous electromagnetic eﬀect, 36, 37 Arfken, G. B., 158 Assis, A. K. T., 8, 12, 34, 62 Barnett, T. L., 34, 42, 88, 90, 91, 129–131, 185, 193 Bartlett, D. F., 36–40, 43, 87 Becquerel, 66 Bergmann, L., 30, 31, 34, 36, 42, 88–90, 110, 111, 116, 193 Bessel equation, 53 function, 48, 56 Beuzenberg, 40, 42 Biot-Savart’s law, 19 Capacitance, 81, 179, 182 Cavendish, H., 66 Chabay, R. W., 25, 31, 41, 42, 66, 81, 149, 191, 192 Charge density angular, 203 linear, 56, 57, 59, 79, 126, 178, 182, 200, 203 surface, 8, 11, 21, 24, 26, 27, 40–42, 59, 68, 77, 79–82, 85, 86, 89, 94, 107, 108, 113, 115, 126–128, 141, 142, 145, 146, 153, 158–160, 176– 179, 199, 200, 203, 208, 209,211, 214, 215 volume, 8–13, 30, 85, 86, 104, 196, 203, 213 Christy, R. W., 9 Classical electrodynamics, 3, 20, 22, 26, 84 Clausius’s force, 22, 82 postulate, 9 Clausius, R., 9, 10, 17–19 Closed circuit, 4, 9, 10, 17–20, 31, 41, 68, 76, 82–84, 157, 191, 196, 199, 208, 214 Conservation of charge, 215 Coombes, C. A., 11, 81 Corson, D. R., 11 Coulomb’s force, 15, 30, 66–68, 195, 197, 209, 210 law, 14, 198, 199 Current element, 9, 12, 15, 16 Davy, H., 66 Dirac delta function, 48 Dirichlet boundary condition, 47 Displacement current, 216 Du Mont power supply, 32 Edwards, W. F., 10, 43 Electromagnetic signal, 211, 216 Electromotive force, 3, 10, 18, 23, 42, 43, 67, 68, 157, 192, 193, 196–200, 202, 205–211, 213 Elliptic-cylindrical coordinates, 124 Emf, 23, 29, 30, 36, 42, 45, 67, 68, 76, 82, 83, 87, 142, 143, 236 193, 196 English translation, 12, 14, 15, 17, 24, 195, 209, 211, 213 Equipotential lines, 30, 34, 35, 88– 91, 97, 100, 105, 110, 111, 115–117, 119, 121, 128–131, 139, 140, 147, 158, 159, 170, 171, 183–185, 187, 193 Euler gamma, 175 Euler-Mascheroni constant, 57 Faraday cage, 38, 39, 43 Faraday’s law of induction, 15 Fechner’s hypothesis, 15, 16, 210 Fechner, G. T., 15, 19 Feynman, R. P., 13, 14, 27 Field electric, 3–5, 8–14, 17, 18, 20– 32, 34–36, 40–43, 46, 47, 51, 54, 55, 59, 65–67, 69, 75–77, 79–91, 93–97, 99– 101, 103, 104, 106, 107, 110– 113, 115, 117–123, 125–130, 139–143, 148, 151, 153, 156– 160, 163, 168, 173–177, 181, 184–188, 192, 193, 196–203, 205, 206, 208, 210, 211 electromagnetic, 26 electrostatic, 21, 67, 199 Li´ enard-Wiechert, 10 magnetic, 4, 8–10, 12–14, 26, 27, 31, 77, 82, 84, 85, 93, 100, 104, 126, 191, 192 Flux of energy, 26 Force electromagnetic, 210, 211 Li´ enard-Schwarzschild, 82 magnetic, 12, 84, 85, 109, 110, 126, 127, 186 Fourier series, 50, 139, 143–146, 154, 159, 167, 169, 176, 186 transform, 50 Fourier, J. B. J., 196 French, A. P., 194Galvanic circuit, 195, 196, 199, 209 current, 17, 18, 195 Ganiel, U., 41, 42 Gauss’s force, 22 law, 15, 51, 55, 69, 85, 89, 95, 96, 115, 126, 141, 144, 157, 174, 203 Gibbs phenomenon, 167 Green function, 47–50, 52, 53 Greenwich Meridian, 151 Griﬃths, D. J., 11, 137 H¨ artel, H., vi, 192 Hall eﬀect, 16, 67, 77, 82, 84, 85, 104, 186 electric ﬁeld, 126 voltage, 84 Hamburg University, iii Heald, M. A., 137, 140, 146, 192 Hernandes, J. A., 37 Humboldt Foundation, iii Inertial frame of reference, 4, 20, 45, 65 International System of Units SI, 4, 203, 213 Jackson, J. D., 9, 10, 14, 46, 50, 56, 155, 163, 183, 191, 192 Jeﬁmenko, O., 13, 14, 31, 32, 34, 36, 42, 67, 88–91, 110–113, 116, 117, 122, 128–131, 137, 184, 185, 187, 191–193 Kelly, W. H., 34, 42, 88, 90, 91, 129– 131, 185, 193 Kenyon, C. S., 10, 43 Keplerian elliptical orbit, 210 Kilambi, A., 19 Kirchhoﬀ, G., 4, 12, 24, 77, 90, 192, 209, 211, 213–216 Kohlrausch, R., 15, 209, 211, 216 Kronecker’s delta funcion, 154, 169 237 Laplace’s equation, 39, 69, 80, 94, 95, 123, 125, 138, 153, 166, 168, 203, 213 Laue, H., 11, 81 Legendre equation, 153, 166 function, 153, 155, 156, 166, 167, 175, 182 polynomial, 153, 154, 166 Leighton, R. B., 14, 27 Lemon, D. K., 10, 43 Li´ enard-Schwarzschild’s force, 82 Li´ enard-Wiechert ﬁeld, 10 Light velocity, 10, 29, 66, 213, 215, 216 Lines of electric ﬁeld, 30, 31, 34, 35, 87–91, 95, 97, 110–112, 115, 117, 120, 122, 128–130, 139–141, 148, 158, 159, 184, 185, 187, 193 Lorentz’s force, 22, 82–84 transformation, 14 Lorrain, F., 11 Lorrain, P., 11 Maglic, S., 36–40, 87 Magnetic ﬂux, 191 vector potential, 193, 214 Magnetic circuital law, 84 Matzek, M. A., 84 Maxwell, J. C., 17–19, 46, 216 Melehy, M. A., 46 Method of images, 3, 36, 38, 39, 43, 45, 46, 104, 107, 192 Milford, F. J., 9 Moreau, W. R., 40, 42, 196 Motional electric ﬁeld, 22, 83, 85, 96 Neumann, C., 164 Newton’s second law of motion, 186 Newtonian force, 68, 210 Mechanics, 194Non-Coulomb force, 67, 68, 196 Ohm’s law, 8, 9, 24, 26, 66, 79, 89, 94, 98, 103, 109, 123, 137, 152, 157, 168, 195, 196, 209, 210, 213–215 Ohm, G. S., 196, 203, 206–209, 213 Ohmic conductor, 163, 164, 168 resistance, 81, 113 toroid, 167 Parker, S., 25, 35, 36, 40, 42, 67 Pearson, J. M., 19 Poggendorﬀ, J. C., 211, 216 Poisson’s equation, 15, 47 Poisson, S. D., 197 Popovic, B. D., 11, 12 Potential diﬀerence, 3, 9, 10, 23, 29, 30, 43, 45, 65, 67, 68, 76, 81–83, 86, 90, 96, 141, 151, 186, 196, 210 Poynting’s vector, 26, 27 Poynting, J. H., 26 Preyer, N. W., 192 Principle of superposition, 65 Purcell, E. M., 13, 14 Ratio of electromagnetic and elec- trostatic units of charge, 15 Reitz, J. R., 9 Relativistic eﬀect, 13, 14 Riecke, 18 Riemann, B., 10, 22 Ritz, W., 10, 16, 22 Rohrlich, F., 19 Rosser, W. G. V., 26, 84, 191 Russell’s theorem, 68, 135 Russell, B. R., 9, 68, 84, 94, 101 Ryan, 40, 42 Sands, M., 14, 27 Sansbury, R., 30, 36, 37, 39, 40, 42, 86, 88, 89 Saslow, W. M., 191 Schaefer, C., 30, 31, 34, 36, 42, 67, 88–90, 110, 111, 116, 193 238 Second order components of Weber’s force, 15 correction, 66 eﬀect, 10, 21, 87 electric ﬁeld, 10, 12, 20–22, 29, 42, 43, 83, 96, 193 force, 20–22, 83, 84, 86, 193 Seely, S., 11 Sherwood, B. A., 25, 31, 41, 42, 66, 81, 149, 191, 192 Skinner, R., 19 Smythe, W. R., 84 Sommerfeld, A., 87, 110 State University of Campinas - UNI- CAMP, iii Stratton, J. A., 103 Superconductor, 43, 193 Syme, 40, 42 Telegraphy equation, 4, 12, 192, 211, 213, 215, 216 Toroidal coordinates, 164–166, 173, 179, 183 Van de Graaﬀ generator, 32, 67 Volta, A. G. A. A., 67 Voltage, 3, 7, 8, 10, 12, 17, 18, 20– 23, 25, 30, 32, 36, 40–43, 45, 63, 68, 76, 86, 87, 89, 98, 142, 143, 193 Ward, B. F. L., 43 Weber’s constant, 216 electrodynamics, 10, 12, 14, 16– 20, 22, 82, 83 force, 14–17, 19, 82, 83, 209, 213, 214 law, iii, 14–17, 19, 20, 43 Weber, H. J., 158 Weber, W. E., i, 4, 13–17, 19, 24, 68, 75, 77, 79, 90, 163, 176– 178, 188, 191, 192, 194–211, 213–216 Weizmann Institute, 41 Whittaker, E. T., 18, 19Wiederkehr, K. H., 14 Wronskian, 49, 58 Zeroth order eﬀect, 21 electric ﬁeld, 20–22, 29, 43, 46, 51, 59, 87, 192, 193 force, 10, 21, 22, 29, 36, 42, 51, 52, 55, 57, 59, 61, 63, 86 239 A. K. T. Assis and J. A. Hernandes , The Electric Force of a Current – Weber and the Surface Charges of Resistive Conductors Carrying Steady Currents (Apeiron, Montreal, 2007), ISBN: 978- 0-9732911- 5-5 Errata P. 166, the first line of Eq. (13.6) should rea d: ( )       ∂∂ − ∂∂ −=∇ηφ χηη ηηχηφcos coshsinh sinhcos cosh 23 2 a Andre Koch Torres Assis and Julio Akashi Hernandes The Electric Force of a Current The Electric Force of a Current Assis/Hernandes Apeiron About the Authors Andre Koch Torres Assis was born in Br azil (1962) and educated at the State University of Campinas – UNICAMP, BS (1983), PhD (1987). He spent the academic year of 1988 in England with a post-docto ral position at the Culham Laboratory (United Kingdom Atomic Energy Authorit y). He spent one year in 1991-92 as a Visiting Scholar at the Center for Elec tromagnetics Research of Northeastern University (Boston, USA). From August 2001 to November 2002 he worked at the Institute for the History of Natural Sc iences, Hamburg University (Hamburg, Germany) with a research fellowship awarded by the Alexander von Humboldt Foundation of Germany. He is the author of Weber’s Electrodynamics (1994), Relational Mechanics (1999); and (with M. A. Bueno) Inductance and Force Calculations in Electrical Circuits (2001). He has been Professor of physics at UNICAMP since 1989, working on the foundatio ns of electromagnetism, gravitation, and cosmology. Julio Akashi Hernandes was born in Brazil (1977) and educated at the State University of Campinas – UNICAMP, BS (1998), MS (2001), PhD (2005). He has always been keenly interested in basic physics, especially electromagnetism. He has published many papers on the electric field outside resistive wires carrying steady currents in ma jor international journals of physics. He is Professor of physics at Universidade Bandeirante de São Paulo, Brazil. The Electric Force of a Current analyzes the elec tric force between a charge and a circuit carr ying a steady current when they are at rest relative to one another. It presen ts experiments and analytical calcu- lations showing the existence of this force, contrary to the statements of many scientists. The force is pr oportional to the voltage of the bat- tery connected to the resistive circuit. It also includes calculations of the potential and electric field inside and outside resistive conductors carrying steady currents, and the di stribution of charges along the sur- face of the conductors that generate this field. It contains two appen- dices that discuss the pioneering and revolutionary works of Wilhelm Weber and Gustav Kirchhoff, and a substantial bibliography of mod- ern literature on the topic. 0-9732911-5-X Weber and the surface charges of resistive conductors carrying steady currents ,!7IA9H3-cjbbff!