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Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: of 252 B o Thidé R A FT ELECTROMAGNETIC FIELD THEORY D Second Edition D R A FT Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: of 252 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: of 252 ELECTROMAGNETIC FIELD THEORY D R A FT Second Edition D R A FT Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: of 252 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: of 252 ELECTROMAGNETIC FIELD THEORY Bo Thidé Swedish Institute of Space Physics Uppsala, Sweden and FT Second Edition D R A Department of Physics and Astronomy Uppsala University, Sweden Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: of 252 Also available ELECTROMAGNETIC FIELD THEORY EXERCISES by Freely downloadable from www.plasma.uu.se/CED FT Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik R A This book was typeset in LATEX 2" based on TEX 3.141592 and Web2C 7.5.6 Copyright ©1997–2009 by Bo Thidé Uppsala, Sweden All rights reserved D Electromagnetic Field Theory ISBN 978-0-486-4773-2 The cover graphics 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 J O H A N S J Ö H O L M and K R I S T O F F E R P A L M E R as part of their undergraduate Diploma Thesis work in Engineering Physics at Uppsala University 2006–2007 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: of 252 D R A FT To the memory of professor L E V M I K H A I L O V I C H E R U K H I M O V (1936–1997) dear friend, great physicist, poet and a truly remarkable man D R A FT Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: of 252 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: of 252 CONTENTS ix List of Figures xv Preface to the second edition Preface to the first edition FT Contents Foundations of Classical Electrodynamics D R A 1.1 Electrostatics 1.1.1 Coulomb’s law 1.1.2 The electrostatic field 1.2 Magnetostatics 1.2.1 Ampère’s law 1.2.2 The magnetostatic field 1.3 Electrodynamics 1.3.1 The indestructibility of electric charge 1.3.2 Maxwell’s displacement current 1.3.3 Electromotive force 1.3.4 Faraday’s law of induction 1.3.5 The microscopic Maxwell equations 1.3.6 Dirac’s symmetrised Maxwell equations 1.3.7 Maxwell-Chern-Simons equations 1.4 Bibliography ix xix 2 6 10 10 11 12 15 15 16 17 19 19 20 21 22 23 26 Electromagnetic Fields and Waves 2.1 Axiomatic classical electrodynamics 2.2 Complex notation and physical observables 2.2.1 Physical observables and averages 2.2.2 Maxwell equations in Majorana representation 2.3 The wave equations for E and B 2.3.1 The time-independent wave equations for E and B xvii Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 10 of 252 x CONTENTS 2.4 Bibliography Electromagnetic Potentials and Gauges 3.1 3.2 3.3 3.4 3.5 FT The electrostatic scalar potential The magnetostatic vector potential The electrodynamic potentials Gauge transformations Gauge conditions 3.5.1 Lorenz-Lorentz gauge 3.5.2 Coulomb gauge 3.5.3 Velocity gauge 3.6 Bibliography 29 29 30 33 34 35 36 40 41 44 Fields from Arbitrary Charge and Current Distributions 4.1 The retarded magnetic field 4.2 The retarded electric field 4.3 The fields at large distances from the sources 4.3.1 The far fields 4.4 Bibliography 28 45 47 49 53 56 57 Fundamental Properties of the Electromagnetic Field 59 59 61 63 63 63 65 68 71 71 76 6.1 Radiation of linear momentum and energy 6.1.1 Monochromatic signals 6.1.2 Finite bandwidth signals 6.2 Radiation of angular momentum 6.3 Radiation from a localised source volume at rest 6.3.1 Electric multipole moments 6.3.2 The Hertz potential 6.3.3 Electric dipole radiation 6.3.4 Magnetic dipole radiation 77 78 78 79 81 81 81 83 88 91 R A 5.1 Charge, space, and time inversion symmetries 5.2 Conservation laws 5.2.1 Conservation of charge 5.2.2 Conservation of current 5.2.3 Conservation of energy 5.2.4 Conservation of linear momentum 5.2.5 Conservation of angular momentum 5.2.6 Electromagnetic virial theorem 5.3 Electromagnetic duality 5.4 Bibliography D Radiation and Radiating Systems Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 238 of 252 216 j 13 MATHEMATICAL METHODS EXAMPLE M Divergence in 3D For an arbitrary R3 vector field a.x0 /, the following relation holds:  à  à a.x0 / r a.x0 / 0 r0 D C a.x / r jx x0 j jx x0 j jx x0 j (M.101) which demonstrates how the primed divergence, defined in terms of the primed del operator in equation (M.98) on the previous page, works End of example M.8 EXAMPLE M The Laplacian and the Dirac delta FT A very useful formula in 3D R3 is à  à  1 Dr D r r jx x0 j jx x0 j ı.x x0 / (M.102) where ı.x x0 / is the 3D Dirac delta ‘function’ This formula follows directly from the fact that  à Z  à I  à Z x x0 x x0 O d3x r r D d x r D d x n jx x0 j jx x0 j3 jx x0 j3 V V S (M.103) EXAMPLE M 10 A equals if the integration volume V S/, enclosed by the surface S.V /, includes x D x0 , and equals otherwise End of example M.9 The curl of a gradient R Using the definition of the R3 curl, equation (M.92) on page 214, and the gradient, equation (M.86) on page 213, we see that r Œr ˛.x/ D O i @j @k ˛.x/ ij k x (M.104) which, due to the assumed well-behavedness of ˛.x/, vanishes: D O i @j @k ˛.x/ ij k x D ij k @2 @x2 @x3 D @2 @x3 @x2 ! ˛.x/Ox1 @2 @x1 @x3 ! C @2 @x3 @x1 @2 @x2 @x1 ! C @2 @x1 @x2 Á0 We thus find that @ @ ˛.x/Oxi @xj @xk ˛.x/Ox2 ˛.x/Ox3 (M.105) Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 239 of 252 M r j 217 Scalars, vectors and tensors Œr ˛.x/ Á (M.106) for any arbitrary, well-behaved R3 scalar field ˛.x/ In 4D we note that for any well-behaved four-scalar field ˛.x Ä / @ @ @ @ /˛.x Ä / Á (M.107) so that the four-curl of a four-gradient vanishes just as does a curl of a gradient in R3 Hence, a gradient is always irrotational FT End of example M.10 The divergence of a curl EXAMPLE With the use of the definitions of the divergence (M.90) and the curl, equation (M.92) on page 214, we find that r Œr a.x/ D @i Œr a.x/i D ij k @i @j ak x/ (M.108) Using the definition for the Levi-Civita symbol, defined by equation (M.18) on page 202, we find that, due to the assumed well-behavedness of a.x/, ij k @j ak x/ D @ @xi ij k @ a @xj k @2 @x2 @x3 D @2 @x3 @x2 ! a1 x/ @2 @x3 @x1 @2 @x1 @x3 ! @2 @x1 @x2 @2 @x2 @x1 ! a2 x/ (M.109) R C A @i C a3 x/ Á0 D i.e., that r Œr a.x/ Á (M.110) for any arbitrary, well-behaved R3 vector field a.x/ In 4D, the four-divergence of the four-curl is not zero, for @ G D @ @ a x Ä / a x Ä / ¤ (M.111) End of example M.11 M 11 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 240 of 252 218 j EXAMPLE 13 MATHEMATICAL METHODS M 12 A non-trivial vector analytic triple product When differential operators appear inside multiple-product vector formulæ, one has to ensure that the range of the operator is taken into account in a proper and correct manner The following calculation of the triple product a r / b, where a and b are 3D vector fields, illustrates how this situation can be handled .a bD r/ D D D O i aj @k / ij k x ij k aj @k / bD O l aj @k bn ij k li n x ıj l ık n O i aj @i bj Dx xi ij k aj @k /.O O l ıi m bn / D lmn x D b/ O l aj @k bn ij k lmn ıi m x O l aj @k bn ij k i ln x O k aj @k bj ıj n ıkl /Oxl aj @k bn D x O j aj @j bk D r bi x (M.112) O j aj @j bk x a.r b/ FT This is formula (F.98) on page 194 End of example M.12 M Analytical mechanics M 2.1 Lagrange’s equations R A As is well known from elementary analytical mechanics, the Lagrange function or Lagrangian L is given by à  dqi ;t D T V (M.113) L.qi ; qP i ; t / D L qi ; dt where qi is the generalised coordinate, T the kinetic energy and V the potential energy of a mechanical system, Using the action Z t2 SD dt L.qi ; qP i ; t/ (M.114) t1 and the variational principle with fixed endpoints t1 and t2 , D ıS D one finds that the Lagrangian satisfies the Euler-Lagrange equations  à d @L @L D0 dt @qP i @qi (M.115) (M.116) To the generalised coordinate qi one defines a canonically conjugate momentum pi according to pi D @L @qP i (M.117) Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 241 of 252 M j 219 Analytical mechanics and note from equation (M.116) on the preceding page that @L D pPi @qi (M.118) If we introduce an arbitrary, differentiable function ˛ D ˛.qi ; t/ and a new Lagrangian L0 related to L in the following way L0 D L C @˛ d˛ @˛ C D L C qP i dt @qi @t (M.119) then @L0 @L @˛ D C @qP i @qP i @q @L d @˛ @L D C @qi @qi dt @q dt @qP i @qi D (M.120b) d @L Á dt @qP i @L @qi where and qi0 D M 2.2 @L @˛ @˛ @L0 D C D pi C @qP i @qP i @qi @qi (M.121) (M.122a) R A pi0 D FT Or, in other words, d @L0 Á @L0 (M.120a) @L0 @L D D qi @pPi @pP (M.122b) Hamilton’s equations From L, the Hamiltonian (Hamilton function) H can be defined via the Legendre transformation D H.pi ; qi ; t / D pi qP i L.qi ; qP i ; t / (M.123) After differentiating the left and right hand sides of this definition and setting them equal we obtain @H @H @H dpi C dqi C dt D qP i dpi C pi dqP i @pi @qi @t @L dqi @qi @L dqP i @qP i @L dt @t (M.124) According to the definition of pi , equation (M.117) on the facing page, the second and fourth terms on the right hand side cancel Furthermore, noting that according Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 242 of 252 220 j 13 MATHEMATICAL METHODS to equation (M.118) on the previous page the third term on the right hand side of equation (M.124) on the preceding page is equal to pPi dqi and identifying terms, we obtain the Hamilton equations: @H dqi D qP i D @pi dt @H D pPi D @qi dpi dt (M.125b) Bibliography FT M (M.125a) [72] M A BRAMOWITZ AND I A S TEGUN, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1972, Tenth Printing, with corrections [73] G B A RFKEN AND H J W EBER, Mathematical Methods for Physicists, fourth, international ed., Academic Press, Inc., San Diego, CA , 1995, ISBN 0-12-059816-7 [74] R A D EAN, Elements of Abstract Algebra, John Wiley & Sons, Inc., New York, NY , 1967, ISBN 0-471-20452-8 R A [75] A A E VETT, Permutation symbol approach to elementary vector analysis, American Journal of Physics, 34 (1965), pp 503–507 [76] A M ESSIAH, Quantum Mechanics, vol II, North-Holland Publishing Co., Amsterdam, 1970, Sixth printing [77] P M M ORSE AND H F ESHBACH, Methods of Theoretical Physics, Part I McGrawHill Book Company, Inc., New York, NY , 1953, ISBN 07-043316-8 [78] B S PAIN, Tensor Calculus, third ed., Oliver and Boyd, Ltd., Edinburgh and London, 1965, ISBN 05-001331-9 D [79] W E T HIRRING, Classical Mathematical Physics, Springer-Verlag, New York, Vienna, 1997, ISBN 0-387-94843-0 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 243 of 252 INDEX FT axiomatic foundation of classical electrodynamics, 19 axions, 16 Bengt Lundborg, xx Bessel functions, 99 Biot-Savart’s law, birefringent, 168 birefringent crystal, 165 braking radiation, 115 bremsstrahlung, 115, 123 R A Abdus Salam, acceleration field, 106 advanced time, 38 Albert Einstein, 1, Alfred North Whitehead, 130 Alfred-Marie Liénard, 102 Ampère’s law, Ampère-turn density, 166 Anders Eriksson, xx André-Marie Ampère, angular frequency, 26 angular momentum current density, 69 angular momentum flux tensor, 69 angular momentum theorem, 69 anisotropic, 168 anisotropic medium, 165 anomalous dispersion, 169 antecedent, 211 antenna, 93 antenna current, 93 antenna feed point, 94 antisymmetric tensor, 142 Arnold Johannes Wilhelm Sommerfeld, 172 arrow of time, 61 associated Legendre polynomial of the first kind, 85 associative, 134 axial gauge, 41 axial vector, 142, 210 D canonically conjugate four-momentum, 150 canonically conjugate momentum, 150, 218 canonically conjugate momentum density, 157 Carroll-Field-Jackiw electrodynamics, 16 Cecilia Jarlskog, xx Cesare Barbieri, xvii CGS units, 3, 15 characteristic impedance of vacuum, 79, 178 charge conjugation, 59 Charles-Augustin de Coulomb, Cherenkov radiation, 170 chiral media, 163 Christer Wahlberg, xx classical electrodynamics, 1, closed algebraic structure, 134 coherent radiation, 122 collision frequency, 175 collisional interaction, 182 221 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 244 of 252 222 INDEX curl, 214 cutoff, 117 cyclotron radiation, 119, 123 d’Alembert operator, 24, 36, 138, 215 David Bohm, 127 definiendum, 199 definiens, 199 del operator, 212 del squared, 215 demodulation, 165 dielectric permittivity, 164 differential distance, 132 differential vector operator, 212 diffusion coefficient, 184 dipole antennas, 93 Dirac delta, 216 Dirac’s symmetrised Maxwell equations, 16 direct product, 211 dispersive, 169 dispersive property, 165 displacement current, 11 divergence, 214 dot product, 208 dual electromagnetic tensor, 144 dual vector, 130 duality transformation, 71, 145 dummy index, 130 dyad, 211 dyadic product, 211 dyons, 72 D R A FT complete ˛-Lorenz gauge, 41 complex conjugate, 191 complex notation, 20, 199 complex vector, 208 complex-field six-vector, 22 component notation, 199 concentration, 215 conductivity, 174 consequent, 211 conservation law, 63 conservation law for angular momentum, 69 conservation law for linear momentum, 67 conservation law for the total current, 63 conservation laws, 61 conservative field, 12 conservative forces, 155 conserved quantities, 61 constants of motion, 61 constitutive relations, 15, 174 continuity equation, 62 contravariant component form, 130, 199 contravariant field tensor, 143 contravariant four-tensor field, 206 contravariant four-vector, 201 contravariant four-vector field, 133 contravariant vector, 130 control sphere, 61 convective derivative, 13 cosine integral, 96 Coulomb gauge, 40 Coulomb’s law, covariant, 128 covariant component form, 200 covariant field tensor, 143 covariant four-tensor field, 206 covariant four-vector, 202 covariant four-vector field, 133 covariant gauge, 139 covariant vector, 130 CPT theorem, 61 cross product, 210 E1 radiation, 89 E2 radiation, 93 Edwin Herbert Hall, 167 Einstein’s summation convention, 199 electric and magnetic field energy, 65 electric charge conservation law, 10 electric charge density, electric conductivity, 11 electric current density, Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 245 of 252 INDEX FT electrostatics, electroweak theory, EMF, 12 Emil Johann Wiechert, 102 energy theorem in Maxwell’s theory, 65 equation of continuity, 139 equation of continuity for electric charge, 10 equation of continuity for magnetic charge, 16 equations of classical electrostatics, equations of classical magnetostatics, Ettore Majorana, 22 Euclidean space, 135 Euclidean vector space, 131 Euler-Lagrange equation, 156 Euler-Lagrange equations, 157, 218 Euler-Mascheroni constant, 96 event, 134 Fabrizio Tamburini, xvii far field, 49, 52 far fields, 45, 56 far zone, 45, 53, 86 Faraday tensor, 143 Faraday’s law, 12 field, 200 field Lagrange density, 158 field momentum, 25 field point, field quantum, 117 fine structure constant, 117, 125 four-current, 138 four-del operator, 213 four-dimensional Hamilton equations, 150 four-dimensional vector space, 130 four-divergence, 214 four-gradient, 213 four-Hamiltonian, 150 four-Lagrangian, 148 four-momentum, 137 four-potential, 138 D R A electric dipole moment, 88 electric dipole moment vector, 82 electric dipole radiation, 89 electric displacement current, 75 electric displacement vector, 163, 164 electric field, 3, 164 electric field energy, 65 electric monopole moment, 82 electric permittivity, 182 electric polarisation, 82 electric quadrupole moment tensor, 82 electric quadrupole radiation, 93 electric quadrupole tensor, 92 electric susceptibility, 165 electricity, 1, electrodynamic potentials, 33 electromagnetic angular momentum density, 68, 90 electromagnetic energy current density, 64 electromagnetic energy flux, 64 electromagnetic field energy, 64 electromagnetic field energy density, 22, 64 electromagnetic field tensor, 143 electromagnetic linear momentum density, 66, 89 electromagnetic moment of momentum density, 68 electromagnetic orbital angular momentum, 70 electromagnetic scalar potential, 33 electromagnetic spin angular momentum, 70 electromagnetic vector potential, 33 electromagnetic virial density, 71 electromagnetic virial theorem, 71 electromagnetism, electromagnetodynamic equations, 16 electromagnetodynamics, 71 electromotive force, 12 electrostatic dipole moment vector, 31 electrostatic quadrupole moment tensor, 31 electrostatic scalar potential, 30 223 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 246 of 252 224 INDEX four-scalar, 200 four-tensor fields, 206 four-vector, 133, 201 four-velocity, 137 Fourier amplitude, 26 Fourier integral, 26 Fourier transform, 26, 36 free-free radiation, 115 frequency mixing, 11 functional derivative, 156 fundamental tensor, 130, 200, 206 FT Galileo Galilei, 198 Galileo’s law, 127 gauge fixing, 41 gauge function, 34 gauge invariant, 35 gauge theory, 35 gauge transformation, 35 Gauss’s law of electrostatics, general inhomogeneous wave equations, 35 general theory of relativity, 127 generalised coordinate, 150, 218 generalised four-coordinate, 150 Gerardus t’Hooft, xx Gibbs’ notation, 212 gradient, 213 Green function, 37, 85 group theory, 134 group velocity, 169 Göran Fäldt, xx Heaviside-Larmor-Rainich transformation, 71 Heaviside-Lorentz units, 15 Heinrich Rudolf Hertz, 77 Helmholtz equation, 94 Helmholtz’ theorem, 35 Helmholtz’s theorem, Helmut Kopka, xx help vector, 86 Hendrik Antoon Lorentz, 1, 36 Hermitian conjugate, 191 Hertz vector, 84 Hertz’s method, 81 heterodyning, 165 Hodge star operator, 71 homogeneous vector wave equations, 177 Hooke’s law, 154 Huygen’s principle, 37 D R A identity element, 134 Igor’ Evgen’evich Tamm, 172 Ilya Mikhailovich Frank, 172 in a medium, 171 incoherent radiation, 123 indefinite norm, 131 index contraction, 130 index lowering, 130 index of refraction, 164 inertial reference frame, 127 inertial system, 127 inhomogeneous Helmholtz equation, 37 inhomogeneous time-independent wave equation, 37 inhomogeneous wave equation, 36 inner product, 208 instantaneous, 112 interaction Lagrange density, 158 intermediate field, 52 intermediate zone, 45 invariant, 201 invariant line element, 132 inverse element, 134 Hall effect, 167 Hamilton density, 157 Hamilton density equations, 157 Hamilton equations, 150, 220 Hamilton function, 219 Hamilton gauge, 41 Hamilton operator, 43, 212 Hamiltonian, 43, 219 Hans Christian Ørsted, Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 247 of 252 INDEX Jacobi identity, 145 James Clerk Maxwell, xix, Jefimenko equations, 50 Johan Lindberg, xviii Johan Sjöholm, vi, xviii John Archibald Wheeler, 39 John Learned, xx Joule heat power, 65 Julian Seymour Schwinger, 1, 15 Kelvin function, 124 kinetic energy, 154, 218 kinetic momentum, 153 Kristoffer Palmer, vi, xviii Kronecker delta tensor, 202 Kronecker product, 211 M1 radiation, 92 Mach cone, 172 magnetic charge density, 15 magnetic current density, 15 magnetic dipole moment, 92, 166 magnetic dipole moment per unit volume, 166 magnetic dipole radiation, 92 magnetic displacement current, 75 magnetic field, 7, 164 magnetic field energy, 65 magnetic field intensity, 166 magnetic flux, 12 magnetic flux density, magnetic four-current, 145 magnetic induction, magnetic monopoles, 15 magnetic permeability, 182 magnetic susceptibility, 164, 167 magnetisation, 166 magnetisation currents, 166 magnetised plasma, 165 magnetising field, 163, 164, 166 magnetism, D R A Lagrange density, 155 Lagrange function, 154, 218 Lagrangian, 154, 218 Laplace operator, 214 Laplacian, 214 Larmor formula for radiated power, 112 law of inertia, 127 Legendre polynomial, 85 Legendre transformation, 219 Lev Mikahilovich Erukhimov, xx Levi-Civita tensor, 202 light cone, 133 light-like interval, 133 line broadening, 26 line element, 209 linear mass density, 155 linear momentum current density, 67 linear momentum density, 66 linear momentum flux tensor, 67 linear momentum operator, 25 linear momentum theorem in Maxwell’s theory, 67 Liénard-Wiechert potentials, 102, 141 longitudinal component, 179 loop antenna, 97 Lorentz boost parameter, 136 Lorentz force, 14, 67 Lorentz force density, 66 Lorentz power, 64 Lorentz power density, 64 Lorentz space, 131, 200 Lorentz transformation, 129 Lorenz-Lorentz gauge, 39 Lorenz-Lorentz gauge condition, 36, 139 lowering of index, 206 Ludvig Valentin Lorenz, 36 FT inverse Fourier transform, 27 ionosphere, 176 irrotational, 6, 214 225 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 248 of 252 226 INDEX multipole expansion, 81 Møller scattering, 125 natural units, 15 near zone, 45, 52 negative refractive index, 168 Newton’s first law, 127 Newton-Lorentz force equation, 150 Noether’s theorem, 61 non-Euclidean space, 131 non-linear effects, 11 norm, 130, 210 null vector, 133 FT magnetostatic vector potential, 30 magnetostatics, Majorana representation, 22 Marcus Eriksson, xviii Marie Sklodowska Curie, 172 mass density, 66 massive photons, 161 material derivative, 13 mathematical group, 134 matrix representation, 202 Mattias Waldenvik, xx Maxwell stress tensor, 67 Maxwell’s displacement current, 10 Maxwell’s macroscopic equations, 168 Maxwell’s microscopic equations, 15 Maxwell-Chern-Simons equations, 16 Maxwell-Lorentz equations, 15 Maxwell-Lorentz source equations, 20 mechanical angular momentum, 68 mechanical angular momentum density, 68 mechanical energy, 64 mechanical energy density, 64 mechanical Lagrange density, 158 mechanical linear momentum, 68 mechanical linear momentum density, 66 mechanical moment of momentum, 68 mechanical orbital angular momentum, 68 mechanical spin angular momentum, 68 mechanical torque, 68 metamaterials, 168 metric, 200, 209 metric tensor, 130, 200, 206 Michael Faraday, 197 Minkowski equation, 150 Minkowski space, 135 mixed four-tensor field, 206 mixing angle, 72 moment of velocity, 62 monad, 212 monochromatic, 46 monochromatic wave, 26 D R A observation point, Ohm’s law, 11, 174 Ohmic losses, 65 Oliver Heaviside, 109, 168, 172 one-dimensional wave equation, 179 orbital angular momentum (OAM), 70 orbital angular momentum operator, 70 outer product, 211 paraxial approximation, 55 parity transformation, 59, 210 Parseval’s identity, 80, 116, 124 Paul Adrien Maurice Dirac, 1, 39, 197 L Pavel Alekseevich Cerenkov, 172 Per Olof Fröman, xix phase velocity, 168 photon, 117, 184 photons, 163 physical observable, 20, 129 Pierre Duhem, Planck units, 15 plane wave, 180 plasma, 169 plasma frequency, 169, 176 plasma physics, 71 Poincaré gauge, 41 Poisson’s equation, 30 polar vector, 142, 210 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 249 of 252 INDEX FT reciprocal space, 28 refractive index, 164, 168, 176 relative dielectric permittivity, 165 relative permeability, 167, 182 relative permittivity, 182 Relativity principle, 128 relaxation time, 177 rest mass density, 158 retarded Coulomb field, 52 retarded induction field, 49 retarded potentials, 39 retarded relative distance, 102 retarded time, 38 Richard Phillips Feynman, 1, 39 Riemann-Silberstein vector, 22 Riemannian metric, 132 Riemannian space, 130, 200 Roger Karlsson, xx rotational degree of freedom, 81 Rudolf Kohlrausch, scalar, 198, 214 scalar field, 134, 200 scalar product, 208 Schrödinger equation, 43 self-force effects, 175 Sergey Ivanovich Vavilov, 172 Sheldon Glashow, shock front, 172 SI units, 3, 15 signature, 131, 207 simultaneous coordinate, 109 Sin-Itiro Tomonaga, Sir Isaac Newton, 197 Sir Rudolf Peierls, 197 skew-symmetric, 143 skin depth, 182 source equations, 167 source point, source terms, 20 space components, 131 R A polarisation charges, 164 polarisation currents, 166 polarisation potential, 84 polarisation vector, 84 positive definite, 135 positive definite norm, 131 postfactor, 211 postulates, 19 potential energy, 154, 218 potential theory, 85 Poynting vector, 64 Poynting’s theorem, 65 prefactor, 211 probability density, 43 Proca Lagrangian, 161 propagator, 37, 85 proper time, 133 pseudo-Riemannian space, 135 pseudoscalar, 198 pseudoscalars, 211 pseudotensor, 198 pseudotensors, 210 pseudovector, 60, 142, 198, 210 227 QCD, QED, 1, 163 quadratic differential form, 132, 209 quantum chromodynamics, quantum electrodynamics, 1, 40, 163 quantum mechanical non-linearity, D radial gauge, 41 radiated electromagnetic power, 65 radiation field, 106 radiation gauge, 40 radiation resistance, 96 radius four-vector, 129 radius vector, 199 raising of index, 206 rank, 202 rapidity, 136 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 250 of 252 228 INDEX time-independent diffusion equation, 178 time-independent telegrapher’s equation, 180 time-independent wave equation, 94, 178 time-like interval, 133 Tobia Carozzi, xx torque density, 68 total charge, 82 total electromagnetic angular momentum, 90 total electromagnetic linear momentum, 90 traceless, 32 translational degree of freedom, 78 transversality condition, 25 transverse components, 179 transverse gauge, 40 tryad, 212 FT space inversion, 59 space-like interval, 133 space-time, 131 spatial Fourier components, 27 spatial spectral components, 27 special theory of relativity, 127 spectral energy density, 80 spherical Bessel function of the first kind, 85 spherical Hankel function of the first kind, 85 spherical harmonic, 85 spherical waves, 57 spin angular momentum (SAM), 70 spin angular momentum operator, 70 standard configuration, 128 standing wave, 94 static net charge, 31 Steven Weinberg, super-potential, 84 superposition principle, 26 synchrotron radiation, 119, 123 synchrotron radiation lobe width, 120 R A uncertainty principle, 61 uncoupled inhomogeneous wave equations, 36 unit dyad, 212 unit tensor, 212 universal constant, 139 Unmagnetised plasma, 175 D telegrapher’s equation, 179, 182 temporal dispersive media, 12 temporal Fourier components, 26 temporal Fourier series, 26 temporal gauge, 41 temporal spectral components, 26 tensor, 198 tensor contraction, 206 tensor field, 202 tensor notation, 203 tensor product, 211 thermodynamic entropy, 61 three-dimensional functional derivative, 157 time component, 131 time reversal, 60 time-dependent diffusion equation, 184 time-dependent Poisson’s equation, 40 time-harmonic wave, 26 vacuum permeability, vacuum permittivity, vacuum polarisation effects, vacuum wave number, 178 variational principle, 218 L Vavilov-Cerenkov cone, 172 L Vavilov-Cerenkov radiation, 170, 172 vector, 198 vector product, 210 vector wave equations, 24 vector waves, 24 velocity field, 106 velocity gauge condition, 41 virtual simultaneous coordinate, 106 Vitaliy Lazarevich Ginzburg, xx wave equation, 63 wave equations, 23 Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 251 of 252 INDEX wave vector, 25, 28, 168, 180 Weber’s constant, Wilhelm Eduard Weber, William Rowen Hamilton, 212 world line, 134 D R A FT Yang-Mills theory, 35 Young’s modulus, 155 Yukawa meson field, 161 229 R A FT Draft version released 29th March 2010 at 16:35 CET—Downloaded from http://www.plasma.uu.se/CED/Book Sheet: 252 of 252 D E L E C T R O M AGNETIC FIELD THEORY ISBN 978-0-486-4773-2 [...]... 8.1 Charged particles in an electromagnetic field 8.1.1 Covariant equations of motion 8.2 Covariant field theory 8.2.1 Lagrange-Hamilton formalism for fields and interactions 8.3 Bibliography 9 Electromagnetic Fields and Matter D 9.1 Maxwell’s macroscopic theory 9.1.1 Polarisation... formulation of the theory; electromagnetic waves and their propagation; electromagnetic potentials and gauge transformations; analysis of symmetries and conservation laws describing the electromagnetic counterparts of the classical concepts of force, momentum and energy, plus other fundamental properties of the electromagnetic field; radiation phenomena; and covariant Lagrangian/Hamiltonian field- theoretical... strong, weak, and electromagnetic the latter has a special standing in the physical sciences Not only does it, together with gravitation, permanently make itself known to all of us in our everyday lives Electrodynamics is also by far the most accurate physical theory known, tested on scales running from sub-nuclear to galactic, and electromagnetic field theory is the prototype of all other field theories... FT 7.1 The special theory of relativity 7.1.1 The Lorentz transformation 7.1.2 Lorentz space 7.1.3 Minkowski space 7.2 Covariant classical mechanics 7.3 Covariant classical electrodynamics 7.3.1 The four-potential 7.3.2 The Liénard-Wiechert potentials 7.3.3 The electromagnetic field tensor 7.4 Bibliography 8 Electromagnetic Fields and Particles... equations 187 F 1.2 Fields and potentials 188 F 1.3 Force and energy 188 Electromagnetic radiation 188 F 2.1 Relationship between the field vectors in a plane wave 188 F 2.2 The far fields from an extended source distribution 189 F 2.3 The far fields from an electric dipole ... theories electricity and magnetism into a single super -theory, electromagnetism or classical electrodynamics (CED), and to realise that optics is a sub -field of this super -theory Early in the 20th century, H E N D R I K A N T O O N L O R E N T Z took the electrodynamics theory further to the microscopic scale and also laid the foundation for the special theory of relativity, formulated in its full extent... Lorentz—which we take as the axiomatic foundation for the theory of electromagnetic fields At the end of this chapter we present Dirac’s symmetrised form of the Maxwell equations by introducing (hypothetical) magnetic charges and magnetic currents into the theory While not identified unambiguously in experiments yet, magnetic charges and currents make the theory much more appealing, for instance by allowing... G L A S H O W , A B D U S S A L A M , and S T E V E N W E I N B E R G were able to unify electrodynamics with the weak interaction theory, creating yet another super -theory, electroweak theory, an achievement which rendered them the Nobel Prize in Physics 1979 The modern theory of strong interactions, quantum chromodynamics (QCD ), is heavily influenced by QED In this introductory chapter we start with... thorough treatment of the theory of electrodynamics, mainly from a classical field- theoretical point of view The first chapter is, by and large, a description of how Classical Electrodynamics was established by J A M E S C L E R K M A X W E L L as a fundamental theory of nature It does so by introducing electrostatics and magnetostatics and demonstrating how they can be unified into one theory, classical electrodynamics,... magnetostatic field In analogy with the electrostatic case, we may attribute the magnetostatic interaction to a static vectorial magnetic field Bstat The elemental Bstat from the elemental current element dI 0 D I 0 dl0 is defined as def dBstat x/ Á 0 4 dI 0 x jx x0 x0 j3 D 0I 4 0 dl0 x jx x0 x0 j3 (1.16) which expresses the small element dBstat x/ of the static magnetic field set up at the field point ... Liénard-Wiechert potentials 7.3.3 The electromagnetic field tensor 7.4 Bibliography Electromagnetic Fields and Particles 8.1 Charged particles in an electromagnetic field 8.1.1 Covariant... far the most accurate physical theory known, tested on scales running from sub-nuclear to galactic, and electromagnetic field theory is the prototype of all other field theories This book, E L... other fundamental properties of the electromagnetic field; radiation phenomena; and covariant Lagrangian/Hamiltonian field- theoretical methods for electromagnetic fields, particles and interactions