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Electromagnetic Field Theory Electromagnetic Field Theory BO THIDÉ ϒ UPSILON BOOKS ELECTROMAGNETIC FIELD THEORY Electromagnetic Field Theory BO THIDÉ Swedish Institute of Space Physics Uppsala, Sweden and Department of Astronomy and Space Physics Uppsala University, Sweden and LOIS Space Centre School of Mathematics and Systems Engineering Växjö University, Sweden ϒ UPSILON BOOKS 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 LATEX2ε (based on TEX 3.141592 and Web2C 7.4.4) on an HP Visu- alize 9000⁄3600 workstation running HP-UX 11.11. Copyright ©1997–2008 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN X-XXX-XXXXX-X To the memory of professor LEV MIKHAILOVICH ERUKHIMOV (1936–1997) dear friend, great physicist, poet and a truly remarkable man. Downloaded from http://www.plasma.uu.se/CED/Book Version released 1st July 2008 at 20:49. CONTENTS Contents ix List of Figures xiii Preface xv 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 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 Maxwell’s microscopic equations 15 1.3.6 Maxwell’s macroscopic equations 15 1.4 Electromagnetic duality 16 1.5 Bibliography 18 1.6 Examples 20 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 27 2.1.3 The time-independent wave equation for E 27 ix Contents 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 2.4 Bibliography 35 2.5 Example 36 3 Electromagnetic Potentials 39 3.1 The electrostatic scalar potential 39 3.2 The magnetostatic vector potential 40 3.3 The electrodynamic potentials 40 3.4 Gauge transformations 41 3.5 Gauge conditions 42 3.5.1 Lorenz-Lorentz gauge 43 3.5.2 Coulomb gauge 47 3.5.3 Velocity gauge 49 3.6 Bibliography 49 3.7 Examples 51 4 Electromagnetic Fields and Matter 53 4.1 Electric polarisation and displacement 53 4.1.1 Electric multipole moments 53 4.2 Magnetisation and the magnetising field 56 4.3 Energy and momentum 58 4.3.1 The energy theorem in Maxwell’s theory 58 4.3.2 The momentum theorem in Maxwell’s theory 59 4.4 Bibliography 62 4.5 Example 63 5 Electromagnetic Fields from Arbitrary Source Distributions 65 5.1 The magnetic field 67 5.2 The electric field 69 5.3 The radiation fields 71 5.4 Radiated energy 74 5.4.1 Monochromatic signals 74 5.4.2 Finite bandwidth signals 75 5.5 Bibliography 76 6 Electromagnetic Radiation and Radiating Systems 77 6.1 Radiation from an extended source volume at rest 77 6.1.1 Radiation from a one-dimensional current distribution 78 x Versionreleased1stJuly2008at20:49. Downloadedfromhttp://www.plasma.uu.se/CED/Book 6.1.2 Radiation from a two-dimensional current distribution 81 6.2 Radiation from a localised source volume at rest 85 6.2.1 The Hertz potential 85 6.2.2 Electric dipole radiation 90 6.2.3 Magnetic dipole radiation 91 6.2.4 Electric quadrupole radiation 93 6.3 Radiation from a localised charge in arbitrary motion 93 6.3.1 The Liénard-Wiechert potentials 94 6.3.2 Radiation from an accelerated point charge 97 6.3.3 Bremsstrahlung 105 6.3.4 Cyclotron and synchrotron radiation 108 6.3.5 Radiation from charges moving in matter 116 6.4 Bibliography 123 6.5 Examples 124 7 Relativistic Electrodynamics 133 7.1 The special theory of relativity 133 7.1.1 The Lorentz transformation 134 7.1.2 Lorentz space 136 7.1.3 Minkowski space 141 7.2 Covariant classical mechanics 143 7.3 Covariant classical electrodynamics 145 7.3.1 The four-potential 145 7.3.2 The Liénard-Wiechert potentials 146 7.3.3 The electromagnetic field tensor 148 7.4 Bibliography 152 8 Electromagnetic Fields and Particles 155 8.1 Charged particles in an electromagnetic field 155 8.1.1 Covariant equations of motion 155 8.2 Covariant field theory 161 8.2.1 Lagrange-Hamilton formalism for fields and interactions 162 8.3 Bibliography 169 8.4 Example 171 F Formulæ 173 F.1 The electromagnetic field 173 F.1.1 Maxwell’s equations 173 F.1.2 Fields and potentials 174 F.1.3 Force and energy 174 F.2 Electromagnetic radiation 174 Downloaded from http://www.plasma.uu.se/CED/Book Version released 1st July 2008 at 20:49. xi Contents F.2.1 Relationship between the field vectors in a plane wave 174 F.2.2 The far fields from an extended source distribution 174 F.2.3 The far fields from an electric dipole 175 F.2.4 The far fields from a magnetic dipole 175 F.2.5 The far fields from an electric quadrupole 175 F.2.6 The fields from a point charge in arbitrary motion 175 F.3 Special relativity 176 F.3.1 Metric tensor 176 F.3.2 Covariant and contravariant four-vectors 176 F.3.3 Lorentz transformation of a four-vector 176 F.3.4 Invariant line element 176 F.3.5 Four-velocity 176 F.3.6 Four-momentum 177 F.3.7 Four-current density 177 F.3.8 Four-potential 177 F.3.9 Field tensor 177 F.4 Vector relations 177 F.4.1 Spherical polar coordinates 178 F.4.2 Vector formulae 178 F.5 Bibliography 180 M Mathematical Methods 181 M.1 Scalars, vectors and tensors 181 M.1.1 Vectors 182 M.1.2 Fields 183 M.1.3 Vector algebra 187 M.1.4 Vector analysis 189 M.2 Analytical mechanics 191 M.2.1 Lagrange’s equations 191 M.2.2 Hamilton’s equations 192 M.3 Examples 194 M.4 Bibliography 202 Index 203 xii Versionreleased1stJuly2008at20:49. Downloadedfromhttp://www.plasma.uu.se/CED/Book Downloaded from http://www.plasma.uu.se/CED/Book Version released 1st July 2008 at 20:49. LIST OF FIGURES 1.1 Coulomb interaction between two electric charges 3 1.2 Coulomb interaction for a distribution of electric charges 5 1.3 Ampère interaction 7 1.4 Moving loop in a varying B field 13 5.1 Radiation in the far zone 73 6.1 Linear antenna 79 6.2 Electric dipole antenna geometry 80 6.3 Loop antenna 82 6.4 Multipole radiation geometry 87 6.5 Electric dipole geometry 89 6.6 Radiation from a moving charge in vacuum 94 6.7 An accelerated charge in vacuum 96 6.8 Angular distribution of radiation during bremsstrahlung 105 6.9 Location of radiation during bremsstrahlung 107 6.10 Radiation from a charge in circular motion 109 6.11 Synchrotron radiation lobe width 111 6.12 The perpendicular electric field of a moving charge 114 6.13 Electron-electron scattering 116 6.14 Vavilov-Cerenkovˇ cone 120 7.1 Relative motion of two inertial systems 135 7.2 Rotation in a 2D Euclidean space 141 7.3 Minkowski diagram 142 8.1 Linear one-dimensional mass chain 162 M.1 Tetrahedron-like volume element of matter 194 xiii Downloaded from http://www.plasma.uu.se/CED/Book Version released 1st July 2008 at 20:49. PREFACE This book is the result of a more than thirty-five year long love affair. In the autumn of 1972, I took my first advanced course in electrodynamics at the De- partment of Theoretical Physics, Uppsala University. A year later, I joined the research group there and took on the task of helping the late professor PER OLOF FRÖMAN, who one year later become my Ph.D. thesis advisor, with the prepa- ration of a new version of his lecture notes on the Theory of Electricity. These two things opened up my eyes for the beauty and intricacy of electrodynamics, already at the classical level, and I fell in love with it. Ever since that time, I have on and off had reason to return to electrodynamics, both in my studies, research and the teaching of a course in advanced electrodynamics at Uppsala University some twenty odd years after I experienced the first encounter with this subject. The current version of the book is an outgrowth of the lecture notes that I pre- pared for the four-credit course Electrodynamics that was introduced in the Up- psala University curriculum in 1992, to become the five-credit course Classical Electrodynamics in 1997. To some extent, parts of these notes were based on lec- ture notes prepared, in Swedish, by my friend and colleague 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 under- graduate 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 electrodynamics, mainly from a classical field theoretical point of view, and includes such things as formal electrostatics and magnetostatics and their uni- fication into electrodynamics, the electromagnetic potentials, gauge transforma- tions, covariant formulation of classical electrodynamics, force, momentum and energy of the electromagnetic field, radiation and scattering phenomena, electro- magnetic waves and their propagation in vacuum and in media, and covariant Lagrangian/Hamiltonian field theoretical methods for electromagnetic fields, par- ticles and interactions.
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