Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 1. of 262 B¼ T; ELECTROMAGnETIC FIELD THEORY DRAFTSÁo¼ E;[;¼ Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 2. of 262 DRAFT Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 3. of 262 ELECTROMAGnETICFIELD THEORY SÁo¼ E;[;¼ DRAFT Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 4. of 262 DRAFT Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 5. of 262 ELECTROMAGnETIC FIELD THEORY SÁo¼ E;[;¼ Bo Thidé Swedish Institute of Space Physics Uppsala, Sweden and Department of Physics and Astronomy Uppsala University, Sweden DRAFT Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 6. of 262 Also available ELECTROMAGnETICFIELD 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 3.1415926 and Web2C 7.5.6 Copyright ©1997–2009 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN 978-0-486-4773-2 The cover graphicsDRAFT illustrates the linear momentum radiation pattern of a radio beam endowed with orbital angular momentum, generated by an array of tri-axial antennas. This graphics illustration was prepared by JOHAN SJÖHOLM and KRISTOFFER PALMER as part of their undergraduate Diploma Thesis work in En- gineering Physics at Uppsala University 2006–2007. Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 7. of 262 To the memory of professor LEV MIKHAILOVICH ERUKHIMOV (1936–1997) dear friend, great physicist, poet and a truly remarkable man. DRAFT Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 8. of 262 DRAFT Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 9. of 262 COnTEnTS Contents ix List of Figures xv Preface to the second edition xvii Preface to the first edition xix 1 Foundations of 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 The indestructibility of electric charge . 10 1.3.2 Maxwell’s displacement current . 10 1.3.3 Electromotive force . 11 1.3.4 Faraday’s law of induction . 12 1.3.5 The microscopic Maxwell equations . 15 1.3.6 Dirac’s symmetrised Maxwell equations . 15 1.4 Bibliography . 18 2 Electromagnetic Fields and Waves 19 2.1 Axiomatic classical electrodynamics . 19 2.2 Complex notation and physical observables . 20 2.2.1 Physical observables and averages . 21 2.2.2 DRAFTMaxwell equations in Majorana representation . 22 2.3 The wave equations for E and B .......................... 24 2.3.1 The time-independent wave equations for E and B . 27 2.4 Bibliography . 30 ix Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 10. of 262 x CONTENTS 3 Electromagnetic Potentials and Gauges 31 3.1 The electrostatic scalar potential . 31 3.2 The magnetostatic vector potential . 32 3.3 The electrodynamic potentials . 35 3.4 Gauge conditions . 36 3.4.1 Lorenz-Lorentz gauge . 37 3.4.2 Coulomb gauge . 40 3.4.3 Velocity gauge . 42 3.4.4 Other gauges . 42 3.5 Gauge transformations . 42 3.6 Bibliography . 47 4 Fundamental Properties of the Electromagnetic Field 49 4.1 Charge, space, and time inversion symmetries . 49 4.2 Electromagnetic duality . 51 4.3 Conservation laws . 54 4.3.1 Conservation of electric charge . 55 4.3.2 Conservation of total electric current . 55 4.3.3 Conservation of energy . 56 4.3.4 Conservation of linear momentum . 59 4.3.5 Conservation of angular momentum . 62 4.3.6 Conservation of centre of energy . 68 4.3.7 Electromagnetic virial theorem . 69 4.4 Bibliography . 69 5 Fields from Arbitrary Charge and Current Distributions 71 5.1 The magnetic field . 73 5.2 The electric field . 75 5.3 The radiation fields . 77 6 Radiation and Radiating Systems 81 6.1 Radiation of linear momentum and energy . 82 6.1.1 Monochromatic signals . 83 6.1.2 Finite bandwidth signals . 84 6.2 Radiation of angular momentum . 85 6.3 Radiation from a localised source volume at rest . 86 6.3.1 Electric multipole moments . 86 6.3.DRAFT2 The Hertz potential . 88 6.3.3 Electric dipole radiation . 91 6.3.4 Magnetic dipole radiation . 95 6.3.5 Electric quadrupole radiation . 97 Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 11. of 262 CONTENTS xi 6.4 Radiation from an extended source volume at rest . 97 6.4.1 Radiation from a one-dimensional current distribution . 98 6.5 Radiation from a localised charge in arbitrary motion . 105 6.5.1 The Liénard-Wiechert potentials . 105 6.5.2 Radiation from an accelerated point charge . 107 6.5.3 Bremsstrahlung . 118 6.5.4 Cyclotron and synchrotron radiation . 123 6.6 Bibliography . 131 7 Relativistic Electrodynamics 133 7.1 The special theory of relativity . 133 7.1.1 The Lorentz transformation . 134 7.1.2 Lorentz space . 135 7.1.3 Minkowski space . 140 7.2 Covariant classical mechanics . 143 7.3 Covariant classical electrodynamics . 144 7.3.1 The four-potential . 144 7.3.2 The Liénard-Wiechert potentials . 145 7.3.3 The electromagnetic field tensor . 148 7.4 Bibliography . 151 8 Electromagnetic Fields and Particles 153 8.1 Charged particles in an electromagnetic field . 153 8.1.1 Covariant equations of motion . 153 8.2 Covariant field theory . 160 8.2.1 Lagrange-Hamilton formalism for fields and interactions . 160 8.3 Bibliography . 168 9 Electromagnetic Fields and Matter 169 9.1 Maxwell’s macroscopic theory . 170 9.1.1 Polarisation and electric displacement . 170 9.1.2 Magnetisation and the magnetising field . 171 9.1.3 Macroscopic Maxwell equations . 173 9.2 Phase velocity, group velocity and dispersion . 174 9.3 Radiation from charges in a material medium . 175 9.3.1 Vavilov-Cerenkov radiation . 176 L 9.4 Electromagnetic waves in a medium . 180 9.4.1 Constitutive relations . 181 9.4.2 DRAFTElectromagnetic waves in a conducting medium . 183 9.5 Bibliography . 191 F Formulæ 193 Draft version released 13th September 2011 at 15:39 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 12. of 262 xii CONTENTS F.1 The electromagnetic field . 193 F.1.1 The microscopic Maxwell (Lorentz) equations . 193 F.1.2 Fields and potentials . 194 F.1.3 Force and energy . 194 F.2 Electromagnetic radiation . 194 F.2.1 Relationship between the field vectors in a plane wave . 194 F.2.2 The far fields from an extended source distribution . 195 F.2.3 The far fields from an electric dipole . 195 F.2.4 The far fields from a magnetic dipole . 195 F.2.5 The far fields from an electric quadrupole . 195 F.2.6 The fields from a point charge in arbitrary motion . 195 F.3 Special relativity . 196 F.3.1 Metric tensor for flat 4D space . 196 F.3.2 Covariant and contravariant four-vectors . 196 F.3.3 Lorentz transformation of a four-vector . ..