“main” 2000/11/13 page 1 ELECTRO MAGNETIC FIELD THEORY Υ Bo Thidé U P S I L O N M E D I A “main” 2000/11/13 page 2 “main” 2000/11/13 page 3 Bo Thidé ELECTROMAGNETIC FIELD THEORY “main” 2000/11/13 page 4 Also available ELECTROMAGNETIC FIELD THEORY EXERCISES by Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik “main” 2000/11/13 page 1 ELECTROMAGNETIC FIELD THEORY Bo Thidé Swedish Institute of Space Physics and Department of Astronomy and Space Physics Uppsala University, Sweden Υ U P S I L O N M E D I A U P P S A L A S W E D E N · · “main” 2000/11/13 page 2 This book was typeset in LATEX 2ε on an HP9000/700 series workstation and printed on an HP LaserJet 5000GN printer. Copyright ©1997, 1998, 1999 and 2000 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN X-XXX-XXXXX-X “main” 2000/11/13 page i Contents Preface xi 1 Classical Electrodynamics 1 1.1 Electrostatics . 1 1.1.1 Coulomb’s law . 1 1.1.2 The electrostatic field . 2 1.2 Magnetostatics . 4 1.2.1 Ampère’s law . 4 1.2.2 The magnetostatic field . 6 1.3 Electrodynamics . 8 1.3.1 Equation of continuity . 9 1.3.2 Maxwell’s displacement current . 9 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 Example 1.1 Duality of the electromagnetodynamic equations 16 Example 1.2 Maxwell from Dirac-Maxwell equations for a fixed mixing angle . 17 Example 1.3 The complex field six-vector . 18 Example 1.4 Duality expressed in the complex field six-vector 19 Bibliography . 20 2 Electromagnetic Waves 23 2.1 The wave equation . 24 2.1.1 The wave equation for E . 24 2.1.2 The wave equation for B . 24 2.1.3 The time-independent wave equation for E . 25 2.2 Plane waves . 26 2.2.1 Telegrapher’s equation . 27 i “main” 2000/11/13 page ii ii CONTENTS 2.2.2 Waves in conductive media . 29 2.3 Observables and averages . 30 Bibliography . 31 3 Electromagnetic Potentials 33 3.1 The electrostatic scalar potential . 33 3.2 The magnetostatic vector potential . 34 3.3 The electromagnetic scalar and vector potentials . 34 3.3.1 Electromagnetic gauges . 36 Lorentz equations for the electromagnetic potentials . 36 Gauge transformations . 36 3.3.2 Solution of the Lorentz equations for the electromag- netic potentials . 38 The retarded potentials . 41 Bibliography . 41 4 The Electromagnetic Fields 43 4.1 The magnetic field . 45 4.2 The electric field . 47 Bibliography . 49 5 Relativistic Electrodynamics 51 5.1 The special theory of relativity . 51 5.1.1 The Lorentz transformation . 52 5.1.2 Lorentz space . 53 Metric tensor . 54 Radius four-vector in contravariant and covariant form 54 Scalar product and norm . 55 Invariant line element and proper time . 56 Four-vector fields . 57 The Lorentz transformation matrix . 57 The Lorentz group . 58 5.1.3 Minkowski space . 58 5.2 Covariant classical mechanics . 61 5.3 Covariant classical electrodynamics . 62 5.3.1 The four-potential . 62 5.3.2 The Liénard-Wiechert potentials . 63 5.3.3 The electromagnetic field tensor . 65 Bibliography . 67 Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01. “main” 2000/11/13 page iii iii 6 Interactions of Fields and Particles 69 6.1 Charged Particles in an Electromagnetic Field . 69 6.1.1 Covariant equations of motion . 69 Lagrange formalism . 69 Hamiltonian formalism . 72 6.2 Covariant Field Theory . 76 6.2.1 Lagrange-Hamilton formalism for fields and interactions 77 The electromagnetic field . 80 Example 6.1 Field energy difference expressed in the field tensor . 81 Other fields . 84 Bibliography . 85 7 Interactions of Fields and Matter 87 7.1 Electric polarisation and the electric displacement vector . 87 7.1.1 Electric multipole moments . 87 7.2 Magnetisation and the magnetising field . 90 7.3 Energy and momentum . 91 7.3.1 The energy theorem in Maxwell’s theory . 92 7.3.2 The momentum theorem in Maxwell’s theory . 93 Bibliography . 95 8 Electromagnetic Radiation 97 8.1 The radiation fields . 97 8.2 Radiated energy . 99 8.2.1 Monochromatic signals . 100 8.2.2 Finite bandwidth signals . 100 8.3 Radiation from extended sources . 102 8.3.1 Linear antenna . 102 8.4 Multipole radiation . 104 8.4.1 The Hertz potential . 104 8.4.2 Electric dipole radiation . 108 8.4.3 Magnetic dipole radiation . 109 8.4.4 Electric quadrupole radiation . 110 8.5 Radiation from a localised charge in arbitrary motion . 111 8.5.1 The Liénard-Wiechert potentials . 112 8.5.2 Radiation from an accelerated point charge . 114 Example 8.1 The fields from a uniformly moving charge . 121 Example 8.2 The convection potential and the convection force . 123 Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book “main” 2000/11/13 page iv iv CONTENTS Radiation for small velocities . 125 8.5.3 Bremsstrahlung . 127 Example 8.3 Bremsstrahlung for low speeds and short ac- celeration times . 130 8.5.4 Cyclotron and synchrotron radiation . 132 Cyclotron radiation . 134 Synchrotron radiation . 134 Radiation in the general case . 137 Virtual photons . 137 8.5.5 Radiation from charges moving in matter . 139 Vavilov-Cerenkˇ ov radiation . 142 Bibliography . 147 F Formulae 149 F.1 The Electromagnetic Field . 149 F.1.1 Maxwell’s equations . 149 Constitutive relations . 149 F.1.2 Fields and potentials . 149 Vector and scalar potentials . 149 Lorentz’ gauge condition in vacuum . 150 F.1.3 Force and energy . 150 Poynting’s vector . 150 Maxwell’s stress tensor . 150 F.2 Electromagnetic Radiation . 150 F.2.1 Relationship between the field vectors in a plane wave 150 F.2.2 The far fields from an extended source distribution . 150 F.2.3 The far fields from an electric dipole . 150 F.2.4 The far fields from a magnetic dipole . 151 F.2.5 The far fields from an electric quadrupole . 151 F.2.6 The fields from a point charge in arbitrary motion . 151 F.2.7 The fields from a point charge in uniform motion . 151 F.3 Special Relativity . 152 F.3.1 Metric tensor . 152 F.3.2 Covariant and contravariant four-vectors . 152 F.3.3 Lorentz transformation of a four-vector . 152 F.3.4 Invariant line element . 152 F.3.5 Four-velocity . 152 F.3.6 Four-momentum . 153 F.3.7 Four-current density . 153 F.3.8 Four-potential . 153 Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01. “main” 2000/11/13 page v v F.3.9 Field tensor . 153 F.4 Vector Relations . 153 F.4.1 Spherical polar coordinates . 154 Base vectors . 154 Directed line element . 154 Solid angle element . 154 Directed area element . 154 Volume element . 154 F.4.2 Vector formulae . 154 General relations . 154 Special relations . 156 Integral relations . 157 Bibliography . 157 Appendices 148 M Mathematical Methods 159 M.1 Scalars, Vectors and Tensors . 159 M.1.1 Vectors . 159 Radius vector . 159 M.1.2 Fields . 161 Scalar fields . 161 Vector fields . 161 Tensor fields . 162 Example M.1 Tensors in 3D space . 164 M.1.3 Vector algebra . 167 Scalar product . 167 Example M.2 Inner products in complex vector space . 167 Example M.3 Scalar product, norm and metric in Lorentz space . 168 Example M.4 Metric in general.