Electromagnetic Field Theory BO THIDÉ Υ UPSILON BOOKS ELECTROMAGNETIC FIELD THEORY Electromagnetic Field Theory BO THIDÉ Swedish Institute of Space Physics and Department of Astronomy and Space Physics Uppsala University, Sweden and School of Mathematics and Systems Engineering Växjö University, Sweden Υ UPSILON BOOKS COMMUNA AB UPPSALA SWEDEN · · · Also available ELECTROMAGNETIC FIELD THEORY EXERCISES by Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik Freely downloadable from www.plasma.uu.se/CED This book was typeset in LATEX 2" (based on TEX 3.14159 and Web2C 7.4.2) on an HP Visualize 9000⁄360 workstation running HP-UX 11.11. Copyright c 1997, 1998, 1999, 2000, 2001, 2002, 2003 and 2004 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN X-XXX-XXXXX-X Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. Preface The current book is an outgrowth of the lecture notes that I prepared for the four-credit course Electrodynamics that was introduced in the Uppsala University curriculum in 1992, to become the five-credit course Classical Electrodynamics in 1997. To some extent, parts of these notes were based on lecture notes prepared, in Swedish, by BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty. Intended primarily as a textbook for physics students at the advanced undergradu- ate or beginning graduate level, it is hoped that the present book may be useful for research workers too. It provides a thorough treatment of the theory of electrodynam- ics, mainly from a classical field theoretical point of view, and includes such things as formal electrostatics and magnetostatics and their unification into electrodynam- ics, the electromagnetic potentials, gauge transformations, covariant formulation of classical electrodynamics, force, momentum and energy of the electromagnetic field, radiation and scattering phenomena, electromagnetic waves and their propagation in vacuum and in media, and covariant Lagrangian/Hamiltonian field theoretical meth- ods for electromagnetic fields, particles and interactions. The aim has been to write a book that can serve both as an advanced text in Classical Electrodynamics and as a preparation for studies in Quantum Electrodynamics and related subjects. In an attempt to encourage participation by other scientists and students in the authoring of this book, and to ensure its quality and scope to make it useful in higher university education anywhere in the world, it was produced within a World-Wide Web (WWW) project. This turned out to be a rather successful move. By making an electronic version of the book freely down-loadable on the net, comments have been only received from fellow Internet physicists around the world and from WWW ‘hit’ statistics it seems that the book serves as a frequently used Internet resource. This way it is hoped that it will be particularly useful for students and researchers working under financial or other circumstances that make it difficult to procure a printed copy of the book. Thanks are due not only to Bengt Lundborg for providing the inspiration to write this book, but also to professor CHRISTER WAHLBERG and professor GÖRAN FÄLDT, Uppsala University, and professor YAKOV ISTOMIN, Lebedev Institute, Moscow, for interesting discussions on electrodynamics and relativity in general and on this book in particular. Comments from former graduate students MATTIAS WALDENVIK, TOBIA CAROZZI and ROGER KARLSSON as well as ANDERS ERIKSSON, all at the Swedish Institute of Space Physics in Uppsala and who all have participated in the teaching, vii PREFACE on the material covered in the course and in this book are gratefully acknowledged. Thanks are also due to my long-term space physics colleague HELMUT KOPKA of the Max-Planck-Institut für Aeronomie, Lindau, Germany, who not only taught me about the practical aspects of the of high-power radio wave transmitters and trans- mission lines, but also about the more delicate aspects of typesetting a book in TEX and LATEX. I am particularly indebted to Academician professor VITALIY LAZAREV- ICH GINZBURG, 2003 Nobel Laureate in Physics, for his many fascinating and very elucidating lectures, comments and historical footnotes on electromagnetic radiation while cruising on the Volga river at our joint Russian-Swedish summer schools during the 1990s and for numerous private discussions. Finally, I would like to thank all students and Internet users who have downloaded and commented on the book during its life on the World-Wide Web. Uppsala, Sweden BO THIDÉ January, 2004 viii Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. Contents Preface vii Contents ix List of Figures xiii 1 Classical Electrodynamics 1 1.1 Electrostatics 2 1.1.1 Coulomb’s law 2 1.1.2 The electrostatic field 3 1.2 Magnetostatics 6 1.2.1 Ampère’s law 6 1.2.2 The magnetostatic field 7 1.3 Electrodynamics 9 1.3.1 Equation of continuity for electric charge 9 1.3.2 Maxwell’s displacement current 10 1.3.3 Electromotive force 10 1.3.4 Faraday’s law of induction 11 1.3.5 Maxwell’s microscopic equations 14 1.3.6 Maxwell’s macroscopic equations 14 1.4 Electromagnetic duality 15 1.5 Bibliography 22 2 Electromagnetic Waves 25 2.1 The wave equations 26 2.1.1 The wave equation for E 26 2.1.2 The wave equation for B 26 2.1.3 The time-independent wave equation for E 27 2.2 Plane waves 30 2.2.1 Telegrapher’s equation 31 2.2.2 Waves in conductive media 32 2.3 Observables and averages 33 ix CONTENTS 2.4 Bibliography 34 3 Electromagnetic Potentials 35 3.1 The electrostatic scalar potential 35 3.2 The magnetostatic vector potential 36 3.3 The electrodynamic potentials 36 3.3.1 Lorenz-Lorentz gauge 38 3.3.2 Coulomb gauge 42 3.3.3 Gauge transformations 42 3.4 Bibliography 45 4 Relativistic Electrodynamics 47 4.1 The special theory of relativity 47 4.1.1 The Lorentz transformation 48 4.1.2 Lorentz space 49 4.1.3 Minkowski space 54 4.2 Covariant classical mechanics 57 4.3 Covariant classical electrodynamics 58 4.3.1 The four-potential 58 4.3.2 The Liénard-Wiechert potentials 59 4.3.3 The electromagnetic field tensor 61 4.4 Bibliography 64 5 Electromagnetic Fields and Particles 67 5.1 Charged particles in an electromagnetic field 67 5.1.1 Covariant equations of motion 67 5.2 Covariant field theory 73 5.2.1 Lagrange-Hamilton formalism for fields and interactions 73 5.3 Bibliography 81 6 Electromagnetic Fields and Matter 83 6.1 Electric polarisation and displacement 83 6.1.1 Electric multipole moments 83 6.2 Magnetisation and the magnetising field 86 6.3 Energy and momentum 88 6.3.1 The energy theorem in Maxwell’s theory 88 6.3.2 The momentum theorem in Maxwell’s theory 89 6.4 Bibliography 91 7 Electromagnetic Fields from Arbitrary Source Distributions 93 7.1 The magnetic field 95 7.2 The electric field 96 7.3 The radiation fields 99 x Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book 7.4 Radiated energy 101 7.4.1 Monochromatic signals 101 7.4.2 Finite bandwidth signals 102 7.5 Bibliography 103 8 Electromagnetic Radiation and Radiating Systems 105 8.1 Radiation from extended sources 105 8.1.1 Radiation from a one-dimensional current distribution 106 8.1.2 Radiation from a two-dimensional current distribution 108 8.2 Multipole radiation 112 8.2.1 The Hertz potential 112 8.2.2 Electric dipole radiation 115 8.2.3 Magnetic dipole radiation 117 8.2.4 Electric quadrupole radiation 118 8.3 Radiation from a localised charge in arbitrary motion 119 8.3.1 The Liénard-Wiechert potentials 120 8.3.2 Radiation from an accelerated point charge 122 8.3.3 Bremsstrahlung 133 8.3.4 Cyclotron and synchrotron radiation 138 8.3.5 Radiation from charges moving in matter 145 8.4 Bibliography 152 F Formulae 155 F.1 The electromagnetic field 155 F.1.1 Maxwell’s equations 155 F.1.2 Fields and potentials 155 F.1.3 Force and energy 156 F.2 Electromagnetic radiation 156 F.2.1 Relationship between the field vectors in a plane wave 156 F.2.2 The far fields from an extended source distribution 156 F.2.3 The far fields from an electric dipole 156 F.2.4 The far fields from a magnetic dipole 157 F.2.5 The far fields from an electric quadrupole 157 F.2.6 The fields from a point charge in arbitrary motion 157 F.3 Special relativity 157 F.3.1 Metric tensor 157 F.3.2 Covariant and contravariant four-vectors 157 F.3.3 Lorentz transformation of a four-vector 158 F.3.4 Invariant line element 158 F.3.5 Four-velocity 158 F.3.6 Four-momentum 158 F.3.7 Four-current density 158 Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. xi CONTENTS F.3.8 Four-potential 158 F.3.9 Field tensor 158 F.4 Vector relations 159 F.4.1 Spherical polar coordinates 159 F.4.2 Vector formulae 160 F.5 Bibliography 161 M Mathematical Methods 163 M.1 Scalars, vectors and tensors 163 M.1.1 Vectors 163 M.1.2 Fields 165 M.1.3 Vector algebra 171 M.1.4 Vector analysis 174 M.2 Analytical mechanics 180 M.2.1 Lagrange’s equations 180 M.2.2 Hamilton’s equations 180 M.3 Bibliography 181 Index 183 xii Version released 19th June 2004 at 21:47.