Oleg D. Jefimenko WEST VIRGINIA UNIVERSITY ELECTROMAGNETIC RETARDATION AND THEORY OF RELATIVITY NEW CHAPTERS IN THE CLASSICAL THEORY OF FIELDS SECOND EDITION Electret Scientific Company Star City Copyright© 1997, 2004 by Oleg D. Jefimenko All rights reserved. Reproduction or translation of this book, or any part of it, beyond that permitted by the United States Copyright Act without written permission of the copyright owner is unlawful. Requests for permission or for further information should be addressed to the publisher, Electret Scientific Company, P.O. Box 4132, Star City, West Virginia 26504, USA. Library of Congress Control Number: 2004092642 ISBN 0-917406-24-9 (paperback) ISBN 0-917406-25-7 (hardcover) Printed in the United States of America www.ElectretScientific.com Electret Scientific 931 Rolling Meadows Road Waynesburg, PA 15370 PREFACE This book is a sequel to my Electricity and Magnetism, 2nd ed., (Electret Scientific, Star City, 1989) and Causality, Electromagnetic Induction, and Gravitation, (Electret Scientific, Star City, 1992). It is a result of a further exploration of the classical theory of fields in search of heretofore overlooked relations between physical quantities and heretofore overlooked applications of the theory. The book is divided into two parts. The first part, Chapters 1 to 5, presents the fundamentals of the theory of electromagnetic retardation with emphasis on recently discovered relations and recently developed mathematical techniques. The second part, Chapters 6 to 11, presents the fundamentals of the theory of relativity based entirely on the theory of electromagnetic retardation developed in the first part. Electromagnetic· retardation is as yet a fairly obscure concept, and therefore an explanation of what it is and why a book needs to be written about it is in order. Electric and magnetic fields propagate with finite velocity. Therefore there always is a time delay before a change in electromagnetic conditions initiated at a point of space can produce an effect at any other point of space. This time delay is called electromagnetic retardation. Recent studies have shown that electromagnetic retardation is of overriding importance for the general electromagnetic theory and, by extension, for the entire vi PREFACE classical theory of fields. We now know that electromagnetic retardation manifests itself in many different ways including, but not limited to, electromagnetic cause-and-effect relations, electromagnetic waves generated by oscillating electric charges and currents, electromagnetic fields and potentials of timedependent charge and current distributions, electromagnetic fields of moving charge distributions, mechanical relations between time-dependent or moving charges and currents, dynamics of atomic systems, time relations in moving electromagnetic systems, and the visual appearance of moving bodies. Perhaps the most important recently discovered aspect of the now evolving theory of electromagnetic retardation is that this theory leads to, and duplicates, many electromagnetic relations that are customarily considered to constitute consequences of relativistic electrodynamics. In fact, it is now clear that there exists an intimate relation between the theory of electromagnetic retardation and the theory of relativity. Obviously then, the phenomenon of electromagnetic retardation and its theoretical representation must be thoroughly understood and investigated. In contrast with the theory of electromagnetic retardation, the theory of relativity is fairly familiar. However, as far as its scientific essence is concerned, the theory of relativity means different things to different people. It is important therefore to give a clear definition of the expression "theory of relativity" as it is used in this book. In this book, "theory of relativity" (or "relativity theory," or simply "relativity") is used as a collective term for the body of equations, methods, and techniques whereby physical quantities measured in one inertial frame of reference can be correlated with physical quantities measured in any other inertial frame of reference. As already mentioned, there exists an intimate relation between the theory of electromagnetic retardation and the theory of relativity. On the basis of this relation, all the fundamental equations of the theory of relativity, including equations of relativistic electrodynamics and relativistic mechanics, are derived PREFACE vii in Chapters 6 to 8 in a natural and direct way from equations of the theory of electromagnetic retardation without any postulates, conjectures, or hypotheses. As a result, Maxwellian electromagnetism, electromagnetic retardation, and the theory of relativity are united in this book into one simple, clear, and harmonious theory of electromagnetic phenomena and of mechanical interactions between moving bodies. An important consequence of the theory of relativity developed in the above manner is the revelation of certain basic errors in the interpretation and use of Einstein's special relativity theory. The nature of these errors and the ways to avoid them are explained in Chapter 9. One of the most controversial elements of Einstein's special relativity theory is his idea of universal kinematic time dilation, according to which the rate of all moving physical and biological 11clocks II is uniformly dilated in consequence of nothing other than the relative motion of the clocks. As is shown in Chapter 10, moving elementary electromagnetic clocks indeed run slower than the same stationary clocks, but their slower rate is a consequence of dynamic interactions and depends on both the velocity and the construction of the clocks. An extension of the theory of relativity, as it is developed in this book, leads to a covariant theory of gravitation analogous to relativistic electrodynamics. This extension is presented in Chapter 11, the concluding chapter of the book. Although the book presents the results of original research, it is written in the style of a textbook and contains numerous illustrative examples demonstrating various applications of the theory developed in the book. Therefore it can be used not only for independent reading, but also as a supplementary textbook in courses on electromagnetic theory and on the theory of relativity. I am pleased to acknowledge with gratitude a stimulating exchange of correspondence with P. Hillion, J. J. Smulsky, V. N. Strel'tsov, and W. E. V. Rosser on some aspects of the theory of relativity, and with M. A. Heald on the subject of electromagnetic retardation. viii PREFACE I am very grateful to S. W. Durland and D. K. Walker for carefully reading the manuscript and for their most useful suggestions and recommendations. Special thanks are due to Yu. G. Kosarev who believes that retardation is a universal phenomenon that should be properly treated in a new branch of physics which he proposes to call "retardics." His comments are highly appreciated. Finally, I am very grateful to my wife Valentina for proofreading the numerous versions of the manuscript and for otherwise helping me to make the book ready for publication. Oleg D. Jefimenko April 14, 1997 PREFACE TO THE SECOND EDITION The second edition of this book is intended to update the presentation of the subject matter and to correct the misprints and other errors that appeared in the first edition. Sections 8-2, 9-4, and 11-3 have been rewritten. Two new Appendixes have been added. Particularly important is Appendix 3, containing an analysis of the physical nature of electric and magnetic forces and presenting a novel interpretation of the "near-action" mechanism of electromagnetic interactions. I am pleased to express my gratitude to my wife Valentina for her assistance in the preparation of this edition of the book. Oleg D. Jefimenko March 31, 2004 CONTENTS PREFACE V Part I RETARDATION 1 RETARDED INTEGRALS AND OPERATIONS WITH RETARDED QUANTITIES 1-1 Vector Wave Fields and Retarded Integrals 3 1-2 Mathematical Operations with Retarded Quantities 6 References and Remarks for Chapter 1 14 2 RETARDED INTEGRALS FOR ELECTROMAGNETIC FIELDS AND POTENTIALS 2-1. Maxwell's Equations and the Wave Field Theorem 15 2-2 Solution of Maxwell's Equations in Terms of Retarded Integrals 18 2-3. Surface Integrals for Retarded Electric and Magnetic Fields 28 2-4. Retarded Potentials for Electric and Magnetic Fields 34 2-5. Electromagnetic Induction 38 References and Remarks for Chapter 2 45 IX X CONTENTS 3 RETARDED INTEGRALS FOR ELECTRIC AND MAGNETIC FIELDS AND POTENTIALS OF MOVING CHARGES 3-1. Using Retarded Integrals for Finding Electric and Magnetic Fields and Potentials of Moving Charge Distributions 46 3-2. Correlation between the Electric and the Magnetic Field of a Moving Charge Distribution 58 References and Remarks for Chapter 3 61 4 ELECTRIC AND MAGNETIC FIELDS AND POTENTIALS OF MOVING POINT AND LINE CHARGES 4-1. The Electric Field of a Uniformly Moving Point Charge 62 4-2. The Magnetic Field of a Uniformly Moving Point Charge 70 4-3. Electric and Magnetic Fields of a Line Charge Uniformly Moving Along its Length 73 4-4. The Electric Field of a Point Charge in Arbitrary Motion 79 4-5. The Magnetic Field of a Point Charge in Arbitrary Motion 92 4-6. Electric and Magnetic Potentials of a Moving Point Charge 95 4-7. How Accurate are the Equations for the Fields and Potentials Obtained in this Chapter? 97 References and Remarks for Chapter 4 100 CONTENTS xi 5 ELECTRIC AND MAGNETIC FIELDS AND POTENTIALS OF AN ARBITRARY CHARGE DISTRIBUTION MOVING WITH CONSTANT VELOCITY 5-1. Converting Retarded Field Integrals for Uniformly Moving Charge Distributions into Present-Time (Present-Position) Integrals 103 5-2. Converting Retarded Potential Integrals for Uniformly Moving Charge Distributions into Present-Time (Present-Position) Integrals 115 5-3. Some Peculiarities of the Expressions for the Fields and Potentials Derived in this Chapter 120 References and Remarks for Chapter 5 126 Part II RELATIVITY 6 FROM ELECTROMAGNETIC RETARDATION TO RELATIVITY 6-1. Relativistic Electromagnetism, Relativistic Terminology, the Principle of Relativity, and Theories of Relativity 129 6-2. Equations for Transforming Electric and Magnetic Fields of Uniformly Moving Charge Distributions into Electric and Magnetic Fields of the Same Stationary Charge Distribution 131 6-3. Inverse Transformations 138 6-4. Equations for Transforming Electric and Magnetic Potentials of Uniformly Moving Charge Distributions into Electric and Magnetic Potentials of the Same Stationary Charge Distributions and Vice Versa 141 References and Remarks for Chapter 6 145 xii CONTENTS 7 THE ESSENTIALS OF RELATMSTIC ELECTRODYNAMICS 7-1. Basic Relativistic Transformation Equations 148 7-2. Transformation Equations for Velocity and Acceleration 153 7-3. Transformation Equations for Partial Derivatives with Respect to Coordinates and Time 156 7-4. The Invariance of the Cartesian Components of Maxwell's Equations under Relativistic Transformations 158 7-5. Testing Relativistic Transformations 165 7-6. The Method of Corresponding States 170 References and Remarks for Chapter 7 178 8 FROM RELATIVISTIC ELECTROMAGNETISM TO RELATIVISTIC MECHANICS 8-1. Transformation of the Lorentz Force 181 8-2. Transformation of Electromagnetic Energy and Momentum of a Parallel-Plate Capacitor 186 8-3. Relativistic Expression for Mechanical Momentum 190 8-4. Relativistic Mass, Longitudinal Mass, and Transverse Mass 193 8-5. Transformation Equations for Mechanical Force, Energy, and Momentum 196 8-6. Transformation of Torque 199 8-7. Rest Energy, Kinetic Energy, and the Relation between Relativistic and Classical Mechanics 200 References and Remarks for Chapter 8 205 CONTENTS xiii 9 COMMON MISCONCEPTIONS ABOUT RELATIVITY THEORY 9-1. The Lorentz Length Contraction 207 9-2. The Electric Field of a Moving Parallel-Plate Capacitor 212 9-3. Using Lorentz Transformations for Finding Electric and Magnetic Fields of a Moving Parallel-Plate Capacitor 217 9-4. The Right-Angle Lever Paradox 218 9-5. Is the Magnetic Field due to an Electric Current a Relativistic Effect? 228 References and Remarks for Chapter 9 231 10 THE RATE OF MOVING CLOCKS 10-1. The Idea of Time Dilation 235 10-2. Clocks Running in Accordance with Einstein's Special Relativity Theory 237 10-3. Clocks that do not Run in Accordance with Einstein's Special Relativity Theory 251 10-4. Reconciling the Theory with Experimental Data 262 References and Remarks for Chapter 10 264 11 GRAVITATION AND COVARIANCE 11-1. Analogy of Electromagnetism with Gravitation 267 11-2. Relativistic Transformation Equations for Gravitational and Cogravitational Fields 275 11-3. Relativistic Gravitation and Relativistic Mechanics 281 xiv CONTENTS 11-4. Covariant Formulation of the Electromagnetic and of the Gravitational-Cogravitational Theory 284 References and Remarks for Chapter 11 292 APPENDIXES Appendix 1. Vector Identities 296 Appendix 2. Transformation Equations for Momentum and Energy 299 Appendix 3. The Physical Nature of Electric and Magnetic Forces 302 INDEX 331 I ...____. RETARDATION 1 RETARDED INTEGRALS AND OPERATIONS WITH RETARDED QUANTITIES The fundamental laws of electromagnetism are represented mathematically by Maxwell's electromagnetic equations. The general solution of these equations for electromagnetic fields in a vacuum is expressed in terms of "retarded" field integrals which constitute the basic mathematical element in the general theory of time-dependent electromagnetic phenomena. A thorough understanding of the properties and use of retarded integrals is therefore indispensable for formulation and application of the theory. In this chapter we shall acquaint ourselves with retarded integrals and with operations involving quantities and expressions appearing in these integrals. 1-1. Vector Wave Fields and Retarded Integrals1 The vector wave field is the field of a vector V which satisfies the inhomogeneous wave equation (also known as the general wave equation) V XV xv+ _1 {_ J2V = K(x,y,z,t), c2 iJt2 (1-1.1) where K is some vector function of space and time which, for 3 4 CHAPTER 1 RETARDED INTEGRALS simplicity, will be assumed here to be zero outside a finite region of space (this differential equation constitutes a mathematical expression for a wave-like disturbance that propagates in space with the speed c). An important property of a vector wave field is that this field can be represented by the retarded field integral and retarded potentials, as explained in the following theorem. The Wave Field Theorem. A vector field V satisfying Eq. (1- 1.1) and vanishing at infinity can be represented by the retarded integral J V = - _1 41r -[V-'(-V'-r•V-)-dK]V, , (1-1.2) where the brackets are the "retardation symbol," to be explained below, and r is the distance from the source point P'(x', y', z') where the volume element of integration, dV', is located to the field point P(x, y, z) where V is being determined; the primed operator V' operates on the source-point coordinates only. (Note: The integration in the above integral is over all space; except when noted otherwise, the integration in all integrals that follow is also over all space.) The derivation of Eq. (1-1.2) is mostly of historical interest and will not be presented here.2 In lieu of the derivation we shall show in Example 1-2.3 that Eq. (1-1.1) is satisfied by V given by Eq. (1-1.2). Corollary I. A vector field V satisfying Eq. (1-1.1), vanishing at infinity, and having zero divergence outside a finite region of space can be represented by the retarded scalar potential cp and the retarded vector potential A as V = - Vcp +VxA, (1-1.3) with cp and A given by SECTION 1-1 WAVE FIELD THEOREM 5 J cp -_ - 1 41r [V' •V+Kl] , ----dV r + cp0 (1-1.4) and (1-1.5) where K1 and K2 are the ordinary potentials of the function K of Eq. (1-1.1) (so that K = - VK1 + V x Ki), both vanishing at infinity, and cp0 and Ao are arbitrary constants. Corollary II. A vector field V satisfying Eq. (1-1.1), vanishing at infinity, and having zero divergence outside a finite region of space can be represented by the retarded scalar potential cp and the retarded vector W as V = - Vcp +W, (1-1.6) with J cp = _1 [V' • V] dV' + 'Po 41r r (1-1. 7) and (1-1.8) where cp0 and W0 are arbitrary constants. The proof of these corollaries is presented in Examples 1-2.1 and 1-2.2. The retardation symbol [ ] indicates a special space and time dependence of the quantities to which it is applied and is defined by the identity [fl = f(x' ,y' ,z' ,t-rlc), (1-1.9) where tis the time for which the retarded integrals are evaluated. Thus the value of a function placed between the retardation 6 CHAPTER 1 RETARDED INTEGRALS symbol [ ] is not that which the function has at the time t for which the integrals are evaluated, but that which it had at some earlier time t' = t - rlc, or, as one says, the function is retarded. The integrals of retarded quantities, or retarded integrals, are mathematical expressions reflecting the phenomenon of "final signal speed" - that is, the fact that a certain time rlc must elapse before the results of some event at the point x', y', z' can produce an effect at the point x, y, z separated from the point x', y', z' by a distance r. Retarded integrals are closely associated with the principle of causality. According to this principle, all present phenomena are exclusively determined by past events. Therefore equations depicting causal relations between physical phenomena must, in general, be equations where a present-time quantity (the effect) relates to one or more quantities (causes) that existed at some previous time. As we shall presently see, in electromagnetic theory retarded integrals are "causal equations" expressing electric and magnetic fields and potentials in terms of their causative sources: the electric charge density p and the electric current density J. 3 1-2. Mathematical Operations with Retarded Quantities Mathematical manipulations with retarded integrals frequently require applications of the operator V to retarded quantities. When applying V to such functions, one should take into account that they depend on space coordinates not only explicitly, but also implicitly through r = {(x-x')2 +(y-y')2 +(z-z')2}'12 (1-2.1) appearing in the retarded time t' - rlc. One also should take into account that V may operate with respect to x, y, z coordinates as well as with respect to x', y', z' coordinates. Finally, one should SECTION 1-2 MATHEMATICAL OPERATIONS 7 take into account that a V operation may be performed upon a retarded quantity taken at the instant t = constant as well as at the instant t' = t - rlc = constant (the latter operation is identical with the corresponding operation upon the same unretarded quantity, combined with the subsequent "retardation" of the resulting expression by replacing in this expression t by t - rlc). Let us designate an unspecified scalar or vector function fix', y', z', t), together with an appropriate multiplication sign, if needed, by X. To avoid ambiguities with V operations involving X, we shall employ special notations, as follows. If an operation is to be performed with respect to primed coordinates, we shall use the primed operator V' in writing this operation, and we shall use the ordinary operator V for designating operations with respect to unprimed coordinates. If an operation upon a retarded X is to be performed considering the retarded time t - rlc as constant, we shall denote the operation as [VX] or [V'X], placing both the operator and the function upon which it operates between the retardation brackets, and we shall use the ordinary notations V[X] or V'[X] for operations upon retarded functions when these operations are to be performed considering the present time t, rather than t - rlc, as constant. We shall frequently use expressions aIKl operations involving the radius vector connecting a volume element dV' of an electric charge or current (the source point x', y', z') with the point of observation (the field point x, y, z). If this radius vector is directed toward the field point, we shall designate it as r, if it is directed toward the source point, we shall designate it as r' . Likewise, we shall designate the correspoIKling unit vectors as ru and r 'u· Observe that since r = (x - x ')i + (y - y ')j + (z - z')k and r' = (x' - x)i + (y' - y)j + (z' - z)k, the vector r' = - r, so that the result of any operation upon r' or r' with V or V' is the negative of the result of the same operation upon r or r, and the result of any operation upon r, r', r or r' with V is the negative of the result of the same operation with V'. 8 CHAPTER 1 RETARDED INTEGRALS We shall now derive several useful operational equations for I retarded functions. Let us consider the operation o[X]lox' y', z', 1, where [X] is some retarded scalar or vector function. 4 Taking into account that retarded functions depend on x', y ', and z' not only directly, but also indirectly through r, we can write o[X] I = ox' y',z',1 o[Xl I + ox' y',z',1-rlc o[X] I . o(t-rlc) x',y',z' o(t - rlc) ox' • ( l -2 _2) We can simplify the last expression by noting that o[X] o(t-rlc) I _ x',y',z'- ["o""xa]t I x',y',z'' (1-2.3) and that, by Eq. (1-2.1), O(t-r/c) = X -x I = -co-sa, ox' er C (1-2.4) where cos a is the direction cosine of vector r with respect to the x axis (Fig. 1.1). We then obtain Tt o[X]I =o[X]I +cosa[oX]I (1_25) OX/ Y',z',I OX/ y',z',t-rlc -C- x',y',z' • • Analogous expressions can be obtained also for o[X]loy' I x', z', 1 and for o[X]/oz' I x',y', 1• If we now multiply these expressions by the unit vectors i, j, and k, respectively, and then add them together, we obtain the following operational equation where V'[X] = [V'X] + r"[oX], C Ot (1-2.6) = _r = i(x-x') + j(y-y') + k(z-z') ru r r = icosa + j cos /3 + kcos 'Y (1-2.7) is the unit vector directed along r toward the point x, y, z (cos /3 SECTION 1-2 MATHEMATICAL OPERATIONS 9 y x,y,z x',y~· X z Fig. 1.1 The direction cosine of r with respect to the x axis is cosa = (x - x')lr. and cos 'Y are the direction cosines of r with respect to the y and z axis, respectively). In a similar manner we can obtain the corresponding equation for the unprimed V (assuming that X does not explicitly depend on x, y, z) V[X] = _ r"[ax]. C at (1-2.8) Combining Eqs. (1-2.8) and (1-2.6), we obtain an equation correlating one unprimed V operation with two primed V operations [V'X] = V[X] + V'[X]. (1-2.9) Differentiating V{[X]/r} and using Eq. (1-2.9), we obtain the correlation V [X] = _ rJX] + V[X] = r~[X] + [V'X] _ V'[X], (l-2 _10) r r2 r r2 r r and, combining the first and the last term of the last part of Eq. (1-2.10), we obtain a useful equation [V'X] = V [X] V' [X] -r- _r + -r - (1-2.11) 10 CHAPTER 1 RETARDED INTEGRALS Another useful equation is obtained by eliminating V[X] from the middle part of Eq. (1-2.10) by means of Eq. (1-2.8): (1-2.12) Finally we note that, since o[Xl _ o[Xl o(t-rlc) - 7ft' we have, by Eqs. (1-2.3) and (1-2.13), (1-2.13) [aaxt ] = o[Xl. ar (1-2.14) ~ Example 1-2.1 Prove Corollary I to the wave field theorem, assuming that V • V, K1, and K2 are zero outside a finite region of space. Expressing in Eq. (1-1.2) K as K = - VK1 + V x K 2 and using Eq. (1-2.11), we have V __l_f [V'(V' • V)-K] dV' 411" r = - - 1-f [V'(V' • V) +V'Kl -V' x Kil dV' 411" r (1-2.15) = - _l_f V [V'. V +Kl] dV' __l_f V' [V'. V +Kl] dV' _f411" r 411" r + _ 1 v x [Kil dV' + _l Jv' x [Kzl dV'. 411" r 411" r The second and the fourth integrals of the last expression can be transformed into surface integrals by using vector identities (V20) and (V-21) (see Appendix for a list of vector identities). But since, by supposition V • V, K1, and K2 are zero outside a finite region of space, while the surface integrals are taken over all space, the integrals vanish. We thus have SECTION 1-2 MATHEMATICAL OPERATIONS 11 V=-_1 Jv[V'•V+K1]dV1 +_1 Jvx [KJdV 1 . (1-2.16) ~ r ~ r Factoring V out from under the integral signs (we can do so because the integration is with respect to primed coordinates, while V operates upon the unprimed coordinates) and designating the resulting integrals as

0. . Fig. 2.1 Calculation of the electric and ma,gnetic fields X on the axis of a charged ring rotating with angular acceleration a. The current density J created by the rotating ring is J = pv = pwaOu = pataOu, where p is the charge density in the ring, w is the angular velocity of the ring, and Ou is a unit vector in the circular direction (right-handed with respect to x). The time derivative of J is {JJ/.. = 21rclw, the antenna is called a "Hertzian dipole." In a Hertzian dipole the current is the same along the entire length of the antenna. Find the magnetic and electric fields produced by the Hertzian dipole shown in Fig. 2.2, at a large distance r ► l from the dipole. r >> I Fig. 2.2 Calculation of the electric and magnetic fields generated by an electric dipole antenna. (The unit vector u/·E,B q// r ;o~ ~ a, S' x r>>a Fig. 2.5 Calculation of the electric and magnetic fields far from the charged ring rotating with constant angular acceleration. (The unit vector >R r R{\f • I I . X Fig. 2.6 Calculation of the voltage induced in a conducting ring placed outside a solenoid carrying a variable current. Let the axis of the solenoid be the x axis of a rectangular system of coordinates, let the ring be in the yz plane, and let the ends of the solenoid be at x = - L/2 and x = L/2. To find the electrokinetic field induced by the solenoid in the ring, we shall consider a point of the ring located on they axis. We can represent this point by the vector Rj. Consider next a point on the surface of the solenoid at a distance x from the yz plane. Combining cylindrical and rectangular coordinates, we can represent that point by the vector b = xi + a cos0j + a sin0k. The distance between the two points is then r = Rj - b = - xi + (R - a cos0)j - a sin0k, so that for r in Eq. (2-5.3) we have, by adding the squares of the components of rand taking the square root of the sum, r = (x2 + R2 + a2 - 2Ra cos0) 1'2. The current density in the solenoid can be written as J = (nl/Lw)Ou = (nl/Lw)(- sin0j + cos0k), 44 CHAPTER 2 RETARDED FIELDS AND POTENTIALS where I is the current in the solenoid, w is the thickness of the current sheet, and Ou is a unit vector in the direction of the current. The volume element to be used in Eq. (2-5.3) can be written as dV' = wad8dx. By the symmetry of the system, the contribution of the y component of J to the electrokinetic field is zero. Equation (2-5.3) becomes therefore (we replace lleoi2 by µ0) a( Ek= - _ ~ J 2.-Ju2 nI cos Ok wad8dx), (2-5.4) at 411" 0 -L/2 Lw(R 2 +a 2 -2Racos8 + x 2) 112 (2-5.5) Integrating by parts over 8, we obtain a( Ek= - - k µOn/Ra2f2"fu2 sm• 20 d8dx). (2-5.6) at 47rL O -u2 (R2 +a 2 -2Racos8 +x 2)3'2 Integrating over x and taking into account that L ;!!> R, a, we obtain f !...(k Ek = - µon/Ra2 2ir sin20 do). (2-5.7) at 27rL o (R2 +a 2 -2Racos8) The integral in Eq. (2-5.7) is just14 7r/R2. The electrokinetic field generated at the point ~ of the ring by the current in the solenoid is therefore (replacing k by Ou) E = - !...(o µonla2) 8 at u 2RL , (2-5.8) and the voltage induced in the ring is ... (2-5.9) REFERENCES AND REMARKS FOR CHAPTER 2 45 References and Remarks for Chapter 2 1. See, for example, Oleg D. Jefimenko, Electricity and Magnetism, 2nd ed., (Electret Scientific, Star City, 1989) pp. 500-503. 2. These equations constitute time-dependent counterparts of the corresponding time-independent equations. See Ref. 1, pp. 101-103 and 350-352. 3. Although we have derived Eqs. (2-2.4) and (2-2.5) from equations for electromagnetic waves, they are fundamental electromagnetic equations of general validity. In fact, Maxwell's equations and, hence, the entire Maxwellian electromagnetic field theory can be derived from them (in conjunction with Eq. 2-2.6). See Oleg D. Jefimenko, "Presenting electromagnetic theory in accordance with the principle of causality," Eur. J. Phys. 25, 287-296 (2004). 4. See, for example, Ref. 1, p. 497. 5. A similar equation is derived in W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed., (AddisonWesley, Reading, 1962) p. 248. 6. See, for example, Ref. 1, pp. 99-100. 7. See, for example, Ref. 1, pp. 346-347. 8. For a more detailed solution see Ref. 1, pp. 559-562. 9. Equations (2-4.5) and (2-4.2) were first obtained in 1867 by L. Lorenz. See E. T. Whittaker, A History of the Theories of Aether and Electricity (Thomas Nelson, London, 1953) Vol. I, Chapt. 8 ("Maxwell") pp. 267-268. 10. See, for example, Ref. 1, p. 55. 11. See, for example, Ref. 1, pp. 380-382. 12. See Oleg D. Jefimenko, Causality, Electromagnetic Induction, and Gravitation, 2nd Ed. (Electret Scientific, Star City, 2000) pp. 3-18. 13. The term "electrokinetic" is also used in reference to phenomena associated with the movement of charged particles through a continuous medium or with the movement of a continuous medium over a charged surface. These phenomena have no connection with the electrokinetic field discussed in this book. 14. It is best evaluated by using a computer program for symbolic integration. 3 RETARDED INTEGRALS FOR ELECTRIC AND MAGNETIC FIELDS AND POTENTIALS OF MOVING CHARGES In this chapter we shall learn how retarded integrals for electric and magnetic fields and potentials can be used for finding electric and magnetic fields and potentials of moving electric charge distributions. We shall also discover important relations between the electric and magnetic fields for two special cases of moving charge distributions: an arbitrary charge distribution moving with constant velocity and a point charge in arbitrary motion. 3-1. Using Retarded Integrals for Finding Electric and Magnetic Fields and Potentials of Moving Charge Distributions A time-variable electric charge distribution always involves a movement of electric charges. For example, if the density of a charge distribution changes with time, then some electric charges change their location within the charge distribution or move to or from the charge distribution. Conversely, a moving charge distribution is inevitably a time-variable charge distribution because it creates charge density in regions of space which it 46 SECTION 3-1 MOVING CHARGE DISTRIBUTIONS 47 enters and eliminates charge density from the regions of space which it leaves. Consequently, the electric and magnetic fields of a moving charge distribution can be determined from retarded field (or retarded potential) equations derived in Chapter 2 for the general case of time-dependent charge and current distributions. To use retarded field integrals for finding electric and magnetic fields of moving charge distributions, we need to express the time derivatives ap!iJt and aJ/at in terms of the velocity of the charge distribution under consideration. This can be done as follows. Consider a stationary charge distribution of density p as a function of x', y', z', p = p(x',y',z'). (3-1.1) If this charge distribution moves with velocity v without changing its density, the total time derivative of p is dp=ap+ ap dx' + ap dy' + ap dz.' =ap+v•V'p. dt at ax' dt ay' dt az' dt iJt (3-1.2) Since p remains the same as the charge moves, dp/dt = 0, so that ap = - v • V 1p. at (3-1.3) A moving charge distribution constitutes a current whose density is J = pv. Therefore v. aJ = a(p v) = -(v . V'p) v + p av = -(v • V'p) v + p (3-1.4) at at at Observe that in the retarded field integrals derived in Chapter 2, the denominator r representing the distance between the volume element dV' and the point of observation is not a function of time. Therefore it is not a function of time also in the case of moving charge distributions. A moving charge distribution must be considered as moving past different volume elements of space associated with different but fixed r's. The question arises, if dV' 48 CHAPTER 3 FIELDS OF MOVING CHARGES is a volume element of space, rather than a volume element of a moving charge distribution, how does one introduce the volume of the charge distribution into the field integrals? To answer this question, let us examine how the electric and magnetic fields of a moving charge distribution are created. The phenomenon of retardation indicates that time-dependent charge distributions send out electric (and magnetic) field "signals" that propagate in all directions with the velocity of light. The electric or magnetic field created by a time-variable charge distribution at the point of observation is the result of the signals sent out by all the individual charges within the distribution and simultaneously "received" at the point of observation at the instant t. But different charges within the distribution are at different distances from the point of observation, and the times needed for the signals originating from the different charges to arrive at the point of observation are different. Therefore the signals that are received at the point of observation simultaneously at the instant t are sent out from the different charges within the distribution at different retarded times t' = t - rlc. For a moving charge distribution these times are different not only because different charges within the distribution are located at different distances from the point of observation, but also because the location of these charges changes as the charge distribution moves. As a result, the region of space from which the field signals responsible for the field at the point of observation are sent is not equal to the region of space, or volume, occupied by the charge distribution when it is at rest. Consider a charge distribution of length l moving against the x axis with a constant velocity v. The electric field E of the charge is observed at the point O (Fig. 3.1). A field signal is sent from the trailing end of the distribution when this end is at the distance r1 from the point of observation. A field signal is sent from the leading end, when this end is at the distance r2 from the point of observation. Since the leading end is closer to the point SECTION 3-1 MOVING CHARGE DISTRIBUTIONS 49 of observation than the trailing end, the field signal from the leading end must be sent at a later time, if it is to arrive at the point of observation simultaneously with the signal sent from the trailing end. The difference in the times needed for the two signals to arrive at the point of observation is r1/c - rife. During this time the charge distribution moves a distance (r1/c - rzfc)v. Hence the distance l* between the two points from which the two signals are sent is (3-1.5) • -4--V I ~ r -V r. - 4 - - / - . . -" ' -,, I.. /* --- --~ "1 0. Fig. 3.1 For the two field signals to arrive simultaneously at 0, the field signal originating from the leading end of the moving charge must be sent later than the field signal originating from the trailing end of the charge. In this chapter we shall be mainly concerned with the special case of charge distributions for which ri,r2 ► l*. In this case (see Fig. 3.2), r1 - r2 = l* cos = l*(r • v)/rv, where r is the distance between the midpoint of l* and the point of observation, and is the angle between r and v. Substituting this expression for r1 - r2 in Eq. (3-1.5), we have l • = l * (r • v) /re+ l, (3-1.6) or l* = -e----,--1___,........... 1 -(r • v)/rc (3-1.7) Therefore, as already mentioned, the region of space from which 50 CHAPTER 3 FIELDS OF MOVING CHARGES Fig. 3.2 Geometrical relations between r, cp, and l* when r1, r2 ► l*. The significance of the vector l* will be explained later. ._v I* );',, r1-r2 r2 the moving charge sends out the field signals resulting in the electric and magnetic fields created at the point of observation is not equal to the region of space (volume) actually occupied by the charge. In the case of a charge distribution whose linear dimensions are small compared with the distance from the charge to the point of observation, this region of space, usually called the effective volume, or the retarded volume, AV',et is A V,eI, = AV' -,..--~-,..- ' 1 -(r • v)/rc (3-1.8) where AV' is the actual volume of the charge [this equation is obtained from Eq. (3-1. 7) by noting that the volume dimensions perpendicular to the direction of motion are not affected by retardation, and that the dimensions along the direction of motion change in accordance with Eq. (3-1.7)]. Although the distance l* given by Eq. (3-1.5) or Eq. (3-1.7) is a distance between two points in space rather than a length of an object, it is usually called the retarded length of the charge. In fact, it is actually the "visual" length of a rapidly moving body, as the length of the body would appear to a stationary observer. As follows from Eq. (3-1.7), the retarded length of a body moving toward the observer is longer, and the retarded length of a body moving away from the observer is shorter, than the actual length of the body. 1 It should be emphasized that Eqs. (3-1.6)-(31.8) hold only for charges or bodies observed from a distance SECTION 3-1 MOVING CHARGE DISTRIBUTIONS 51 much greater than the linear dimensions of the charge or body. For a general case, the retarded length or volume of a body cannot be expressed by a simple formula, but can be calculated in terms of the actual length of the body once the position of the body at the time of observation is given (Section 4-3). Another effect of retardation that needs to be taken into account when applying retarded field equations to moving charge distributions is an apparent distortion of the shape of a moving charge distribution. The distribution appears to change its shape because the retarded times for different points within the distribution are different. y • V 0 =-----------------tX► Fig. 3.3 Geometrical relations between the "present position vector" r0 and the "retarded position vector" r for a charge distribution moving with velocity v in the negative x direction. Consider a charge distribution moving against the x axis with a velocity v and observed from a point O (Fig. 3.3). The retarded volume element dV' of the charge distribution is at the point P and is represented by the vector r. The present position of the same volume element is at the point P0 and is represented by the vector r0. The distance .:ix' from P to P0 is the distance that the charge travels during the time that it takes the field signal to 52 CHAPTER 3 FIELDS OF MOVING CHARGES propagate from P to 0, that is, Lix' = v(r/c). We shall now show that, within the charge, any line parallel to they axis when the charge is at rest or at its present position appears to be slanted when the charge is moving and is at a retarded position. First, let us note that according to Fig. 3.3 the relation between the x component of the present position vector rO and the x and y components of the retarded position vector r is (as usual, we use primes to indicate source-point coordinates) x' = x~ + vrlc, (3-1.9) or (3-1.10) Differentiating Eq. (3-1.10) while keeping x0' constant, we have dx' = - -y'(-v/-c)- dy' r[I -(v/c)(x'Ir)] ' (3-1.11) which can be written as Y I.. ~-P,., V -+-,- r I -q- - , , _ - \0 '\ 0 rvlc .. I I - - -/d~ i -q-~ \rp p . Fig. 3.4 A charge at its retarded position appears to be elongated and its vertical lines appear to be slanted. SECTION 3-1 MOVING CHARGE DISTRIBUTIONS 53 y I* -acota X Fig. 3.5 Explanation of the vectors l* and a*. The vector l* represents the retarded length of the moving charge, the vector a* represents the "slanted" thickness of the charge. = y 1vlc -r[l--(-v/- c)c-os~ cp] =r[-l--(yr1v-•/cv-)/r-c] = (v/c)sincp 1-(r • v)/rc (3-1.12) Thus, according to Eq. (3-1.12), a vertical line (x0 ' = constant, dx0' l>a,b,l X Fig. 4.1 A charge of uniform density p is confined to a small rectangular prism. The total charge of the prism is q. The charge constitutes a point charge when viewed from a distance large compared to the linear dimensions of the prism. charge distribution constitutes a point charge. 3 Let the charge move with uniform velocity v = - vi. We want to find the electric and magnetic fields of this charge at the point of observation. To find the electric field produced by this charge, we shall use Eq. (2-2.1). First we eliminate from Eq. (2-2.1) the term with the current density J. We can do so with the help of Eq. (3-1.4). Since the velocity of our charge is v = v) = - vi, and since the charge moves without acceleration so that v = 0, Eq. (3-1.4) gives (4-1.1) Substituting Eq. (4-1.1) into Eq. (2-2.1), we then have for the electric field of the charge 64 CHAPTER 4 MOVING POINT AND LINE CHARGES - - - - I!---► Fig. 4.2 When the charge shown in Fig. 4.1 is moving and is at a retarded position, its apparent length, shape, and thickness of its front and back surface layers are no longer the same as for the stationary charge. (All r's meet at the origin). (4-1.2) Observe that in this equation V' and atax' operate on the unretarded p, so that in computing V' p and 8p/8x' we must use the ordinary, unretarded, shape and size of the prism. Since p is constant within the prism, V' p = 0 within it, and the only contribution to V'p comes from the surface layer of the prism, where p changes from p (inside the prism) to O(outside the prism). Let the thickness of the surface layer be w. Taking into account that V' p represents the rate of change of p in the direction of the greatest rate of change, we then have V'p = (plw)n;n, where n;n is a unit vector normal to the surface layer and pointing into the prism. Hence V' p for the right, left, top, bottom, front, and back surfaces of the charge (prism) are -(p/w)i, (p/w)i, -(p/w)j, (p/w)j, -(p/w)k, and (p/w)k, respectively. Likewise, 8p/8x' is zero in the interior of the charge and is different from zero only in the left and in the right surface layers of the charge, where SECTION 4-1 UNIFORMLY MOVING POINT CHARGE 65 V Fig. 4.3 The relations between r3, r4, and a* for the moving charge at a retarded position. (The two r's meet at the origin.) op!ox' = p!w in the left surface layer and op!ox' = - p!w in the right surface layer. The volume integral of Eq. (4-1.2) can be split therefore into six integrals, one over each of the six surface layers corresponding to the six surfaces of the charge (prism). However, since the center of the charge is in the xy plane (z' = 0), the integrals over the two surface layers parallel to the xy plane cancel each other, because V'p for one of the layers is opposite to that for the other layer, while r is the same for both layers. Thus only the four integrals over the layers parallel to the xz and yz planes remain. Let us designate the retarded distances from these layers to the point of observation as ri, ri, r3, and r4 (see Figs. 4.2 and 4.3). Since the linear dimensions of the charge are much smaller than r1, r2, r3, and r4, we can replace each integral over a surface layer by the product of the integrand and the volume of the corresponding layer. However, the integration in Eq. (4-1.2) is over the effective (retarded) volume of the charge, and therefore we must use not the true volume of the surface layers, but their effective volume. The effective volume of the surface layers is not the same as their actual volume, because, in accordance with Eq. (3-1.7), the length l of the two layers parallel to the xz plane must 66 CHAPTER 4 MOVING POINT AND LINE CHARGES be replaced by /*::: / 1-(r • v)/rc' (4-1.3) and because, also in accordance with Eq. (3-1.7), the thickness w of the two layers parallel to the yz plane must be replaced by w • ::: --:--,--w--,--,-- 1 -(r • v)/rc Equation (4-1.2) becomes therefore (4-1.4) E -- _-1- [-pl- wabW 1*(- I") +p_ /wabw2•·1 + p/_ wbl3• W (-"J) 41reo '1 '2 '3 (4-1.5) +p_ /wbl4• W• J(+v2- )(-plwabW•·1Ip+/w-abW2*(-")I)~ r4 c2 '1 '2 or, substituting I* and w* from Eqs. (4-1.3) and (4-1.4), E = - - -1 -[- -p-lw- , - abw( -I") + _ _ plw_.,...abW•I 41re0 r1 -r1 •v/c r2 -r2 •v/c + Plw blw( -j) + Plw blwj r3 -r3 • v /c r4 -r4 • v /c + (-v c 2 2 ) (- -pl- w - r1 -r1 • v/c ab W • Ir+ 2 --rp2/-w•-v,/-c abW ( - 1"))~ , (4-1.6) which simplifies to E = _ pb f1(1 _ v2)( 1 _ 1 )ai 41re0~ c 2 r2-r•2 v/c r1-r•1 v/c )t] + (r 1 -r • v /c - 1 r -r • v /c J • 4 4 3 3 (4-1.7) As can be seen from Figs. 4.2 and 4.3, the differences of the fractions in these equations are simply the increments of the function 1/(r - r • v/c) associated with the displacement of the tail SECTION 4-1 UNIFORMLY MOVING POINT CHARGE 67 of r over the distances represented by the vector I* [in the i component of Eq. (4-1.7)] and by the vector a* [in the j component of Eq. (4-1.7)]. Therefore we can write Eq. (4-1.7) as4 E = - t pb f1(1 41re0 - ev22)~l(v' 1 ) r-r•vle • I* tf i ~(v' + l r-r 1 •vl e ) • a ·]1j}. (4-1.8) Substituting the gradient from Eq. (3-1.17) (remembering that v = 0) and substituting I* and a* from Eqs. (3-1.14) and (3-1.15), we have E = pb f1(1 _ v2)( r - rvle • i) la i 411"t) e 2 r 3(l -r • v lre)2 1 -r • v Ire + (- r 3(-1r-- -r r•- vvlel-re• )2 · I ) - r( y'vle ----l l - r • v Ire) l1J• (4-1.9) + (- -r-- rvl-e - • J·) - -r--x'-vl-e l liJ·] . r 3(1 -r • v lre)2 r(l - r • v Ire) Simplifying and taking into account that r • i = - x', r • j = - y', v • i = - v, v • j = 0, and r • v = x'v, we obtain E = pabl r(1- vi)(-x'+rvle)i l 47rt:0r 3[1-r•vlre]3 e 2 + ( -x , +rv1e) -y '-viJe. + ( -y ') -r --x-'vJ le ·] r r = pabl ~(1- v2)(-x'i-rvle) +(1- v2)(-y)j], 47rt:0r 3[1 -r • vlre]3l e 2 e2 and finally, noting that r = - x'i - y'j, and that pabl = q, (4-1.11) 68 CHAPTER 4 MOVING POINT AND LINE CHARGES Equation (4-1.11) expresses E in terms of the retarded position of the charge specified by the retarded position vector r (see Fig. 3.4). Usually it is desirable to express E in terms of the present position of the charge specified by the present position vector r0 (see Fig. 3.4). We can convert Eq. (4-1.11) from r to rOby using Eqs. (3-1.19) and (3-1. 26). According to Eq. (3-1.19), r-rv/c=r0 , (4-1.12) so that the last factor in Eq. (4-1.11) is simply the present position vector r0. Substituting Eq. (4-1.12) and Eq. (3-1.26) into Eq. (41. 11), we obtain the desired equation for the electric field of a uniformly moving point charge expressed in terms of the present position of the charge E _- _ _ _q_(l -_v2_/ c 2_) _ _r0 . 41re0r5{1-(v 2/c 2)sin28}312 (4-1.13) This equation (in a different notation) was first derived by Oliver Heaviside in 1888 on the basis of Maxwell's equations by using the "operational calculus" that he invented.5 Fig. 4. 4 As was first noticed by Heaviside, the electric field of a . V moving point charge concentrates itself in the direction perpendicular to the direction of motion of the charge and decreases along the line of the motion. There are two interesting properties of Eq. (4-1.13). First, as was noted by Heaviside, with increasing velocity of the charge the electric field of the charge concentrates itself more and more SECTION 4-1 UNIFORMLY MOVING POINT CHARGE 69 about the equatorial plane, 8 = 1r/2, and decreases along the line of motion, 8 = 0. This effect is shown in Fig. 4.4. Second, the electric field appears to originate at the charge in its present position. This, of course, is merely an illusion, because by supposition the distance between the charge and the point of observation is much greater than the linear dimensions of the charge, so that neither Eq. (4-1.11) nor Eq. (4-1.13) gives us any information concerning the structure of the field close to the charge. Note also that because of the finite speed of the propagation of the field signals and light signals one can never observe the charge at its present position. In fact, the charge could have stopped after sending the field signal from its retarded position, and even then Eq. (4-1.13) would remain valid, although in this case Eq. (4-1.13) would apply to the "projected," or ."anticipated," present position of the charge. Example 4-1.1 Show that for a point charge moving without acceleration Eq. (3-2.13) reduces to (3-2.6). According to Eq. (4-1.12), the retarded position vector of the charge can be expressed in terms of the present position as r = r0 + rv/e. Substituting Eq. (4-1.14) into Eq. (3-2.13), we have (4-1.14) B = rxE = (r0 +rv/e)xE = r0 XE + (rv/e)xE. (4-1.lS) er er er er Since, by Eq. (4-1.13), Eis directed along r0, r0 x E = 0, and we are left with B = (v x E)/e2 , (4-1.16) which was to be proved. Example 4-1.2 Equation (4-1.13) represents a "snapshot" of the electric field of a moving point charge, since it does not express the 70 CHAPTER 4 MOVING POINT AND LINE CHARGES field as a function of time. Modify Eq. (4-1.13) so that it shows how the field changes as the charge moves. Let us assume that the "snapshot" is for t = 0. If the charge moves in the - x direction, the functional dependence of E on the x coordinate will be preserved for t ~ 0 if we express Eq. (4-1. 13) in terms of x0' and replace x0' by Xo' - vt. From Eqs. (3-1.26) and (3-1.25), we have , 0{1-(v2/c2)sin20}112 = 0c,~2+y'2-y,2v2/c2)112 = [x~2 +(1 -v2/c2)y'2]112. (4-1.17) Replacing in Eq. (4-1. 17) x0' by x0' - vt, we obtain where x0' is now the x coordinate of the point charge at t = 0. Expressing r0 in terms of its components and replacing x0' by x0' - vt, we similarly have r0 = -(?c,0' - vt)i - y'j. Therefore Eq. (4- 1.13) can be written as q(l-v 2/c 2){(x~ -vt)i +y'j} E = - ---,------- 4no{0c-o/ - vt)2 +(1-v2/c2)y' 2p12 ' (4-1.19) where the dependence of E on t is shown explicitly. This equation holds for the charge moving parallel to the x axis in the xy plane. If it moves parallel to the x axis anywhere in space, y'2 in this equation should be replaced by (y'2 + z'2). 4-2. The Magnetic Field of a Uniformly Moving Point Charge Although by using Eq. (2-2.2) or Eq. (2-2.5), we can find the magnetic field of a uniformly moving point charge in the same manner as we found the electric field in Section 4-1 (see Example 4-2.1), it is much easier to find it from the known electric field by using Eq. (3-2.5) or Eq. (3-2.6). SECTION 4-2 MAGNETIC FIELD OF POINT CHARGE 71 Applying Eq. (3-2.5) to Eq. (4-1.11), we obtain for the magnetic field in terms of the retarded position of the charge H = -...,q,....[1_-_vz_lc_z_J__,. [v x r] . 47rr3[1 -r • v /rc]3 (4-2.1) Applying Eq. (3-2.5) to Eq. (4-1.13), we obtain for the magnetic field in terms of the present position of the charge (4-2.2) ~ Example 4-2.1 Find the magnetic field of a uniformly moving point charge shown in Fig. 4.1 by using Eq. (2-2.2), (4-2.3) To use Eq. (4-2.3), we need to know V' x J associated with the charge under consideration. The moving charge constitutes a current density J = pv. Since vis not a function of x', y', z', we have V' x J = V'p x v. But pis constant within the charge, and therefore the only contribution to V' x J comes from the surface layer of the charge, where p changes from p (inside the charge) to 0 (outside the charge). Using the values for V' p obtained in Section 4-1, we then have for V' x J of the top, bottom, front, and back surface layers of the charge (prism) -pv/wk, pv/wk, pv/wj, and -p/wj, respectively; the left and right surface layers make no contribution to V' x J, because v and V'p are parallel (or antiparallel) there. Furthermore, since V' x J in the front surface layer is opposite to V' x J in the back surface layer, while both surface layers are at the same distance r from the point of observation, the contributions of these two layers to the integral in 72 CHAPTER 4 MOVING POINT AND LINE CHARGES Eq. (4-2.3) cancel each other, so that only the top and the bottom surface layers contribute to the magnetic field of the charge. Since the linear dimensions of the charge are much smaller than r3 and r4, (see Fig. 4.3), we can replace the integrals over the two surface layers by the product of the integrand and the volumes of the corresponding layers. Using Eq. (4-2.3) and taking into account the effective volume of the boundary layers (see Sections 3-1 and 4-1), we have, as in Eqs. (4-1.5)-(4-1.7), _l_[ H = - pv/w wblk + pv!w wblk] 41r r3 -r3 •v/e r4 -r4 •v/e = _ pvbl[ 1 _ 1 ]k 41r r3 -r3 •v/e r4 -r4 •vie • (4-2.4) The difference of the two fractions in the last expression is simply the increment of the function 1/(r - r • v/e) associated with the displacement of the tail of rover the distance represented by the vector a* (see Fig 4.3). Therefore, using Eqs. (3-1.17) and (31.15), we can write Eq. (4-2.4) as H = - pbvl[(- -r- -rv-/e- - • ·) y'vle I --,..,----,---,-ll 411" r 3(1 -r • v /re)2 r(l - r • v /re) +( r-rv/e ·j) r-x'vle a]k r 3(1 -r • v /re)2 r(l - r • v /re) • (4-2.5) Simplifying and taking into account that r • i = - x', r • j = y', v • i = - v, v • j = 0, and r • v = x'v, we obtain H = - qv [(-x' +rvfe)y'vlre +(-y')(l -x'v/re)]k 41rr 3[1 -r • v /re]3 = qv[l -v 2/e 2]y' k 41rr3[1 -r • v /re]3 ' (4-2.6) . which, noting that ry'k = v x r, is the same as Eq. (4-2.1). 6 SECTION 4-3 UNIFORMLY MOVING LINE CHARGE 73 4-3. The Electric and Magnetic Fields of a Line Charge Uniformly Moving Along its Length Consider a line charge of finite length L, cross-sectional area S, charge density p, and linear charge density A = pS moving with constant velocity v parallel to the x axis of a rectangular system of coordinates in the negative direction of the axis and at a distance R above the axis (Fig. 4.5). Let the point of observation O be at the origin. What is the electric field at O at the time t when the leading end of the charge is at a distance Li from the y axis? We can find the electric field of the moving charge by using Eq. (2-2.1) or Eq. (2-2.4) if we know the retarded position of the y -L ►I L ~ I .. 2 W/2"ff////////////A •:: --X2 ~-'I vr/c .. I v ·r x'1 ~" ~I R X a = - - - - - - - - - - - - - - - - - - - ' -... Fig. 4. 5 A line charge of linear density A is moving with constant velocity v. The retarded positions of the trailing and leading ends of the charge are x1' and xz', respectively. The present positions of the two ends are L1 and Li, respectively. The distance between the trajectory of the charge and the x axis is R. The point of observation O is at the origin. The "retarded, 11 or "effective, 11 length of the charge is longer than its true length. 74 CHAPTER 4 MOVING POINT AND LINE CHARGES charge corresponding to the time for which the field is computed. We can determine this position as follows. First, let us determine the retarded position xi' of the leading end of the charge corresponding to the time t, that is, the position from which the leading end sends out its field signal which arrives at O at the time t. If the retarded distance between O and the leading end is r2, then the time it takes for the signal to travel from the leading end to O is rife. During this time the charge travels a distance v(rife). Therefore at the moment when the leading end sends out its field signal, the position of the leading end is (4-3.1) Next, let us find the retarded position xi' of the trailing end of the charge corresponding to the time t. If the retarded distance between O and the trailing end is r1, then the time it takes for the signal to travel from the trailing end to O is rife. During this time the charge travels a distance v(rife). Hence, at the moment when the trailing end sends out its signal, the position of the trailing end is (4-3.2) The x component of the electric field. We are now ready to find the electric field of the charge by using Eq. (2-2.1) or Eq. (22.4). The easiest way to find the x component of the electric field of the charge under consideration is to use Eq. (2-2.1). According to this equation, the x component of the field is due to the x components of [V'p] and [aJtat] of the moving charge. For the line charge under consideration, these components exist only at the leading and trailing ends of the charge and are the same as for the moving charged prism discussed in the preceding sections of this chapter: [V'p]x = (p/w)i for the leading end, and [V'Plx = - (plw)i for the trailing end, [aJtat]x = - (v2plw)i for the leading end, and [aJ/at]x = (v2p/w)i for the trailing end, where w SECTION 4-3 UNIFORMLY MOVING LINE CHARGE 75 is the thickness of the surface layer of the charge (this is the actual thickness, not the retarded one). Since the surface layer of the charge may be assumed as thin as one wishes, the retarded volume integral in Eq. (2-2.1), as far as the x component of the field is concerned, reduces to the product of the integrand and the volume of the surface layers of the leading and trailing ends of the charge at their retarded positions. By Eq. (4-1.4), for the leading end, this volume is, using the asterisk to indicate values evaluated at retarded positions, (4-3.3) and for the trailing end it is wi' S = wS --,----,---.,--,--- 1 -(r1 • v)/r1c The x component of the electric field is therefore (4-3.4) or E A(l -v2/c2) ( 1 1 ) (4-3.6) x = - 41reo r2-xfv!c - r 1 -x(v!c • Equation (4-3.6) gives the electric field in terms of the retarded position of the charge. We shall now convert it to the present position of the charge (that is, the actual position of the charge at the time t). The calculations are similar to those used for deriving Eqs. (3-1.20)-(3-1.26). First, we note that, by Eq. (43.1), (4-3.7) Next, we write the denominator of the first fraction inside the parentheses of Eq. (4-3.6) as 76 CHAPTER 4 MOVING POINT AND LINE CHARGES Adding and subtracting x'2 and r/vlc- to the right side of Eq. (4- 3.8), we then have r2 -x{vlc (4-3.9) .x,; =(r{ -2r~{ vie +x{ 2v2/c 2+ 2-x{ 2 +r{v 2/ c 2-r{v2/c 2) 112 . Let us now collect the terms on the right of Eq. (4-3.9) into three groups: (4-3.10) (4-3.11) and (4-3.12) By Eq. (4-3.7), the first group represents L,/. The second group is simply R2(see Fig. 4.5). And the third group is - R2v2/c2. Similar relations hold for the denominator of the second fraction inside the parentheses of Eq. (4-3.6). Therefore Eq. (43. 6) transforms to E = A(l-v2/c 2)[ 1 _ 1 ] x 41recft (Lf IR2 + 1 -v21c2)112 (L;/R2 +1 -v2/c2)112 , (4-3.13) where only the present time quantities appear. The y component of the electric field. The easiest way to find the y component of the electric field of the charge under consideration is to use Eq. (2-2.4). Only the first integral of Eq. (2-2.4) makes a contribution to the y component of the field, because {JJ/>a,b,l X Fig. 4. 6 A charge of unifonn density p is confined to a small rectangular prism. The charge constitutes a point charge when viewed from a distance large compared to its linear dimensions. points of the prism are also different. If the prism is sufficiently far from the point of observation, which we assume to be the case, the difference between the retarded times corresponding to different points of the prism is very small, and therefore the retarded acceleration of the prism may be assumed to have the same value v for all points of the prism, even if in reality the acceleration is variable. Therefore the velocities of the different points of the prism can be calculated from velocity formulas for motion with constant acceleration. As we shall presently see, in addition to the velocity of the center of the prism, we only need the velocities of the right, left, top, and bottom surfaces of the prism. Let the distances of these surfaces from the point of observation be r1, r2, r3, and r4, as shown in Fig. 4.7. The time interval between the retarded time for the center of the prism and for its left or right surface is then approximately (r1-ri)/2c (see Section 3.1), and the time interval between the retarded time for the center of the prism and for its top or bottom surface is approximately (r3 -r4)/2c. Therefore the SECTION 4-4 ACCELERATING POINT CHARGE 81 Fig. 4. 7 When the charge shown in Fig. 4.6 is in a state of accelerated motion and is at a retarded position, its apparent length, shape, and thickness of its surface layers are no longer the same as for the stationary charge. The distances from the center of the charge and from the four surface layers to the point of observation are represented by the vectors r, r 1, r2, r3, and r4. All five r's meet at the point ofobservation (origin of coordinates). The acceleration vector is in the xy plane. (approximate) retarded velocities of the right, left, top, and bottom surfaces of the prism are, respectively, v1 = v - v(r1 r2)/2c, v2 = v + v(r1 - r;J/2c, v3 = v - v(r3 - r4)/2c, and v4 = v + v(r3 - r4)/2c. As was explained in Section 3-1, the apparent size and shape of the prism in its retarded position is not the same as that of the prism when it is at rest. In particular, if the prism moves in the - x direction, the prism appears to be longer, it appears to be slanted, and the effective volume of the prism and of its surface layers changes (Fig. 4.7). As a result, the following geometrical relations hold for the moving prism at its retarded position: The apparent length of the prism is, by Eq. (3-1. 7), l* l = -1--r •-v/- rc The apparent volume of the prism is, by Eq. (3-1.8), (4-4.1) 82 CHAPTER 4 MOVING POINT AND LINE CHARGES (abl) * = --,--a-b-l-,--1-r • v/rc (4-4.2) By the same equations, the apparent volume of the right surface layer (distance r1 from the origin) is (abw); = abw 1 -r1 • v1/r1c (4-4.3) the apparent volume of the left surface layer (distance r2 from the origin) is (abw)i• = - -a-bw- 1 -r2 • v/r2c (4-4.4) the apparent volume of the top surface layer (distance r3 from the origin) is (lbw)3• = lbw 1 -r3 • v/r3c (4-4.5) and the apparent volume of the bottom surface layer (distance r4 from the origin) is (lbw)4* = lbw 1 -r4 • vir4c (4-4.6) We shall find the electric field of our accelerating point charge by using Eq. (2-2.1) E = - _l_f [V'p + _!_ aJ] c2 at dV'. 41rt:0 r (2-2.1) Consider first the contribution of the gradient of the charge density, V 'p, to the field. Since p is constant within the charge, V'p = 0 within it, so that the only contribution to V'p comes from the surface layer of the charge, where p changes from 0 (outside the charge) to p (inside the charge). Let the actual thickness of the surface layer of the charge be w. Taking into account that V 'p represents the rate of change of p in the direction SECTION 4-4 ACCELERATING POINT CHARGE 83 of the greatest rate of change, we then have V'p = (p/w)nin• where nin is a unit vector normal to the surface layer and pointing into the charge.7 Since the center of the charge is in the xy plane (z' = 0), the integrals over the two surface layers parallel to the xy plane cancel each other, because V'p for one of the layers is opposite to that for the other layer, while r is the same for both layers. Thus, as far as V' p is concerned, only the four integrals over the layers parallel to the xz and yz planes remain. Referring to Figs. 4.6 and 4.7, they are the right, left, top, and bottom surface layers, and V' p associated with these surface layers is, respectively -(p/w)i, (p/w)i, -(p/w)j, and (p/w)j (these are the same relations that we used for finding the electric field of a uniformly moving point charge in Section 4.1). Assuming that r 1, r2, r3, and r4 are much larger than l*, we can replace the integrals over the four layers by the products of the integrands and the retarded volumes of the layers, which gives Let us designate the part of Eq. (4-4.7) which explicitly depends on p as EP. Using Eqs. (4-4.3)-(4-4.6) and cancelling w, we can write then The differences of the fractions in this equation are simply the increments of the function 1/(r - r • v/c) associated with the displacement of the tail of r over a small distance represented by 84 CHAPTER 4 MOVING POINT AND LINE CHARGES the vector I* [in the i component of Eq. (4-4.8)] and by the vector a* [in the j component of Eq. (4-4.8)]. Therefore, just as we did in the case of Eq. (4-1.7), we can write Eq. (4-4.8) as +[(v' E = P - J!.!!.....{r(v' 411'Bo [ l r-r•v/c ) ·I* lf_i l ) ·a* ]tj}. (4-4. 9) r-r•v/c Using Eqs. (3-1.16), (3-1.14), and (3-1.15), we now have E = pb [(r - rv/c + (r • v)r/c 2 • i) la i P 411'e0 r 3(1-r•v/rc)2 1-r•v/rc +(-r ~- r-v/-c -+-(r-•,v,)r-/c-2 • I · ) - - - - y 1vlc ,-,------, - - -,- alJ' r 3(1-r•v/rc)2 r(l-r•v/rc) (4-4.10) +(r-rv/c+(r•v)r/c 2 .j) r-x'vlc atj]. r 3(1-r • v/rc)2 r(l - r • v/rc) Simplifying and taking into account that r • i = - x', r • j = - y', v • i = - v, v • j = 0, and r • v = x'v, we obtain E = pabl { r - x ' + r v / c - ( r • v ) x ' / c 2] i P 411'e0r 3[l - r • v Ire]3 + [-x' + rvlc -(r • v)x' /c 2] y'vlc j r ,I } +[-y' -(r•v)y'/c2]r-: vcj (4-4.11) = pabl (-x'i-rv/c-(r•v)x'/c2i 4'll'e0r 3[1-r • v/rc]3 +(v2y, /cz)j -y, j -(r • v)y' /cZj]. Since we are not interested in the acceleration-independent field Ev (this field was found in Section 4-1), we shall drop in Eq. (4-4.11) the terms that do not contain the acceleration v, and shall designate the rest of the equations as E,4p, with the subscript "A" standing for "acceleration." Noting that r = - x'i - y'j, and that pabl = q, we then obtain SECTION 4-4 ACCELERATING POINT CHARGE 85 (4-4.12) Consider now the contribution of aJ1at to the field. By Eq. (31.4), we have -a=J -a-(p=v)-(v•V,p)v+pav_=-(v•V,p)v+pv· . (4-4.13) at at at However, because the retarded velocity is different in different regions (points) of the charge, we must evaluate Eq. (4-4.13) separately for each region under consideration. There are five such regions: the interior of the charge, the right surface, the left surface, the top surface, and the bottom surface. In the interior of the charge, V'p = 0. Therefore for the interior we have aJ at = pv. (4-4.14) At the right surface, V'p=(ap/ax')i= -(p/w)i, and the velocity is v 1• By Eq. (4-4.13), for the right surface we therefore have or TaJ1t = (plw)(v1xv1 + wv1), (4-4.16) and since we can make was small as we please, aJJ - at = (plw)v1 v1. X (4-4.17) At the left surface, V'p = ap1ax'i = p/wi, and the velocity is v2.Therefore, by the same reasoning as in the case of Eq. (44.16), (4-4.18) 86 CHAPTER 4 MOVING POINT AND LINE CHARGES At the top surface, V'p = op/oy'j = - plwj, and the velocity is v3. Therefore, (4-4.19) At the bottom surface, V'p = op/oy'j=plwj, and the velocity is v4. Therefore (4-4.20) Let us now designate the integral in Eq. (4-4.7) as EJ. Since, by supposition, all r's for the charge (prism) are much larger than the linear dimensions of the charge, we can replace the integration by the product of the respective integrands and the volumes of the five regions that contribute to oJ/ot. Using Eqs. (4-4.14), (44.17)-(4-4.20) and (4-4.2)-(4-4.6), we then have 41rec2E =P"( abl ) 0 J r 1-r • v/rc (4-4.21) or (4-4.22) Since the linear dimensions of the charge are very small compared to the r's, the difference of the fractions in the last two terms of Eq. (4-4.22) can be regarded as the total differential (increment) df = (oflox')dx' + (of/oy')dy' of the functions SECTION 4-4 ACCELERATING POINT CHARGE 87 vxv r-r•v/c (4-4.23) and r-r • vie (4-4.24) corresponding to the displacements of the tail of r by I* and by a*, respectively (see Fig. 4.7). Using Eq. (3-1.16), noting that r = - x'i - y'j, noting that vY = 0 (because vis parallel to the x axis), and remembering that v and v are functions of the retarded time t' = t - rlc, so that ov/ox' = (ov/ot')ot'/ox' = (ov/ot')x'lrc = vx'lrc with similar expressions for ov/oy', ovlox', and ov/oy', we have for the needed partial derivatives of the two functions a( vv ) -x'-rv/c-(r•v)x'/c2 ox' r[l -(/. v)/rc] = vxv r 3[1 x-(r • v)/rc]2 (4-4.25) (1\v + vxv)x' r 2c[l -(r • v)/rc]' and In evaluating Eq. (4-4.22) with the help of Eqs. (4-4.25)-(4-4.27), we shall omit from Eq. (4-4.25) the terms not containing v, since they only contribute to the acceleration-independent field Ev, which we already found in Section 4-1. Combining Eqs. (4-4.22), (4-4.25)-(4-4.27), (3-1.14), and (3-1.15), we then have, denoting the acceleration-dependent field as E1A , 88 CHAPTER 4 MOVING POINT AND LINE CHARGES 41re0e2EJA = --=---q-,v-....,.. r(l -r • v/re) +pab[ -vxv(r•v)x' _ (vxv+vx v)x'] ·----l-,-r 3e 2(1 -r • v /re)2 r 2e(l -r • v /re) (1 -r • v /re) -pbl [ v vxI Y ay 'vie • r 2e(l -r • v /re) r(l -r • v /re) (4-4.28) vyvy' a(r-r • v/e)] + r 2e(l -r • v/re) • r(l -r • v/re) ' or 41re0e2EJA qv = ----,-,----,-..,... r(l -r • v/re) + _ _ _q_ _ _ [ -vv(r·v)x' x -v. n , r 2e(l-r•v/re)2 re(l-r•v/re) x -vv. x'- v v Y x'y'v -v. vy 1 + v vy'(r Y • v)] . x re Y re (4-4.29) Since r • v = x'v = - x'vx and since - vx x' - vY y' = v • r (see Figs. 4.6 and 4.7), Eq. (4-4.29) reduces to 41re e 2E = qv 0 JA r(l -r · v/re) (4-4 •30) +--=----q-----=- [-v,(-r,-•-v-)-(-r-•-v,)- +(r • v)v +(r • v)v]' r 2e(l -r • v /re)2 re(l -r • v /re) which after elementary simplifications becomes E = qv + q~·~v JA 4,re0e 2r(l -r · v/re)2 41re0e 3r 2 (1 -r • v/re)3 (4-4.31) Finally, in accordance with Eq. (4-4.7), subtracting Eq. (4-4.31) from Eq. (4-4.12), we obtain for EA