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ELECTROM AG NETIC FIELD THEORY

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ELECTROMAGNETICFIELD THEORY

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BoThid

Swedish InstituteofSpacePhysicsUppsala,Sweden

and

DepartmentofPhysicsand AstronomyUppsala University, Sweden

and

GalileanSchool ofHigherEducationUniversityofPadua

Padu

a,Italy


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Also available

ELECTROM AG NETIC FIELD THEORY

EXERCISES

by

Tobia Carozzi, Anders Eriksson, Bengt Lundborg,Bo Thid and Mattias Waldenvik

Freely downloadable from

www.plasma.uu.se/CEDThis book was typeset in LATEX 2"based on TEX 3.1415926 and Web2C 7.5.6

Copyright 19972011byBo ThidUppsala, SwedenAll rights reserved.

Electromagnetic Field TheoryISBN 978-0-486-4773-2

The cover graphics illustrates the linear momentum radiation pattern of a radio beam endowed with orbitalangular momentum, generated by an array of tri-axial antennas. This graphics illustration was prepared byJO H A N SJ H O LM and KR IS TO F F ER PA L M E Ras part of their undergraduate Diploma Thesis work in En-gineering Physics at Uppsala University 20062007.


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To the memory of professorLE V M I K H A I L O V I CH ERU K H I M O V(19361997)

dear friend, great physicist, poetand a truly remarkable man.


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CONTENTS

Contents ix

List of Figures xv

Preface to the second edition xvii

Preface to the first edition xix

1 Foundations of Classical Electrodynamics 11.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Coulombs law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 The electrostatic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Ampres law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 The magnetostatic field . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Electrodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.1 The indestructibility of electric charge . . . . . . . . . . . . . . . . . . . 101.3.2 Maxwells displacement current . . . . . . . . . . . . . . . . . . . . . . 101.3.3 Electromotive force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.4 Faradays law of induction . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.5 The microscopic Maxwell equations . . . . . . . . . . . . . . . . . . . . 15

1.3.6 Diracs symmetrised Maxwell equations . . . . . . . . . . . . . . . . . . 151.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Electromagnetic Fields and Waves 192.1 Axiomatic classical electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Complex notation and physical observables . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Physical observables and averages . . . . . . . . . . . . . . . . . . . . . 212.2.2 Maxwell equations in Majorana representation. . . . . . . . . . . . . . . 22

2.3 The wave equations forEandB . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 The time-independent wave equations forEandB . . . . . . . . . . . . 25

2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Electromagnetic Potentials and Gauges 333.1 The electrostatic scalar potential . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 The magnetostatic vector potential . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 The electrodynamic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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3.4 Gauge conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4.1 Lorenz-Lorentz gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.2 Coulomb gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.3 Velocity gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5.1 Other gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44


4 Fundamental Properties of the Electromagnetic Field 534.1 Discrete symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.1 Charge conjugation, spatial inversion, and time reversal . . . . . . . . . . 534.1.2 Csymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.3 Psymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.4 Tsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Continuous symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.1 General conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Conservation of electric charge . . . . . . . . . . . . . . . . . . . . . . . 584.2.3 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.4 Conservation of linear (translational) momentum . . . . . . . . . . . . . 61

4.2.4.1 Gauge-invariant operator formalism . . . . . . . . . . . . . . . 634.2.5 Conservation of angular (rotational) momentum . . . . . . . . . . . . . . 66

4.2.5.1 Gauge-invariant operator formalism . . . . . . . . . . . . . . . 68

4.2.6 Electromagnetic duality. . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2.7 Electromagnetic virial theorem . . . . . . . . . . . . . . . . . . . . . . . 714.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.4 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Fields from Arbitrary Charge and Current Distributions 855.1 Fourier component method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2 The retarded electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3 The retarded magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.4 The total electric and magnetic fields at large distances from the sources . . . . . 93

5.4.1 The far fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Radiation and Radiating Systems 1016.1 Radiation of linear momentum and energy . . . . . . . . . . . . . . . . . . . . . 102

6.1.1 Monochromatic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.1.2 Finite bandwidth signals . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Radiation of angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 105


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6.3 Radiation from a localised source volume at rest . . . . . . . . . . . . . . . . . . 1066.3.1 Electric multipole moments. . . . . . . . . . . . . . . . . . . . . . . . . 106

6.3.2 The Hertz potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.3.3 Electric dipole radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3.4 Magnetic dipole radiation. . . . . . . . . . . . . . . . . . . . . . . . . . 1136.3.5 Electric quadrupole radiation . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4 Radiation from an extended source volume at rest . . . . . . . . . . . . . . . . . 1156.4.1 Radiation from a one-dimensional current distribution. . . . . . . . . . . 115

6.5 Radiation from a localised charge in arbitrary motion . . . . . . . . . . . . . . . 1196.5.1 The Linard-Wiechert potentials . . . . . . . . . . . . . . . . . . . . . . 1216.5.2 Radiation from an accelerated point charge . . . . . . . . . . . . . . . . 123

6.5.2.1 The differential operator method . . . . . . . . . . . . . . . . . 1246.5.2.2 The direct method . . . . . . . . . . . . . . . . . . . . . . . . 1276.5.2.3 Small velocities. . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.5.3 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.5.4 Cyclotron and synchrotron radiation . . . . . . . . . . . . . . . . . . . . 133

6.5.4.1 Cyclotron radiation . . . . . . . . . . . . . . . . . . . . . . . . 1356.5.4.2 Synchrotron radiation . . . . . . . . . . . . . . . . . . . . . . 1366.5.4.3 Radiation in the general case . . . . . . . . . . . . . . . . . . . 1386.5.4.4 Virtual photons . . . . . . . . . . . . . . . . . . . . . . . . . . 139


7 Relativistic Electrodynamics 1537.1 The special theory of relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.1.1 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . 1547.1.2 Lorentz space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.1.2.1 Radius four-vector in contravariant and covariant form . . . . . 1567.1.2.2 Scalar product and norm . . . . . . . . . . . . . . . . . . . . . 1567.1.2.3 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.1.2.4 Invariant line element and proper time . . . . . . . . . . . . . . 1587.1.2.5 Four-vector fields. . . . . . . . . . . . . . . . . . . . . . . . . 1597.1.2.6 The Lorentz transformation matrix. . . . . . . . . . . . . . . . 1607.1.2.7 The Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . 160

7.1.3 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.2 Covariant classical mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.3 Covariant classical electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 164

7.3.1 The four-potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.3.2 The Linard-Wiechert potentials . . . . . . . . . . . . . . . . . . . . . . 1657.3.3 The electromagnetic field tensor . . . . . . . . . . . . . . . . . . . . . . 168

7.4 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171


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8 Electromagnetic Fields and Particles 1738.1 Charged particles in an electromagnetic field . . . . . . . . . . . . . . . . . . . . 173

8.1.1 Covariant equations of motion . . . . . . . . . . . . . . . . . . . . . . . 1738.1.1.1 Lagrangian formalism . . . . . . . . . . . . . . . . . . . . . . 1738.1.1.2 Hamiltonian formalism. . . . . . . . . . . . . . . . . . . . . . 176

8.2 Covariant field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1798.2.1 Lagrange-Hamilton formalism for fields and interactions . . . . . . . . . 180

8.2.1.1 The electromagnetic field . . . . . . . . . . . . . . . . . . . . 1838.2.1.2 Other fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.3 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

9 Electromagnetic Fields and Matter 1899.1 Maxwells macroscopic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

9.1.1 Polarisation and electric displacement . . . . . . . . . . . . . . . . . . . 1909.1.2 Magnetisation and the magnetising field . . . . . . . . . . . . . . . . . . 1919.1.3 Macroscopic Maxwell equations . . . . . . . . . . . . . . . . . . . . . . 193

9.2 Phase velocity, group velocity and dispersion . . . . . . . . . . . . . . . . . . . 1949.3 Radiation from charges in a material medium . . . . . . . . . . . . . . . . . . . 195

9.3.1 Vavilov- LCerenkov radiation. . . . . . . . . . . . . . . . . . . . . . . . . 1959.4 Electromagnetic waves in a medium . . . . . . . . . . . . . . . . . . . . . . . . 200

9.4.1 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2019.4.2 Electromagnetic waves in a conducting medium . . . . . . . . . . . . . . 203

9.4.2.1 The wave equations forEandB . . . . . . . . . . . . . . . . . 2039.4.2.2 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 2049.4.2.3 Telegraphers equation . . . . . . . . . . . . . . . . . . . . . . 205

9.5 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

F Formul 213F.1 Vector and tensor fields in 3D Euclidean space . . . . . . . . . . . . . . . . . . . 213

F.1.1 Cylindrical circular coordinates. . . . . . . . . . . . . . . . . . . . . . . 214F.1.1.1 Base vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 214F.1.1.2 Directed line element. . . . . . . . . . . . . . . . . . . . . . . 214F.1.1.3 Directed area element . . . . . . . . . . . . . . . . . . . . . . 214F.1.1.4 Volume element . . . . . . . . . . . . . . . . . . . . . . . . . 214

F.1.1.5 Spatial differential operators . . . . . . . . . . . . . . . . . . . 214F.1.2 Spherical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 215

F.1.2.1 Base vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 215F.1.2.2 Directed line element. . . . . . . . . . . . . . . . . . . . . . . 215F.1.2.3 Solid angle element . . . . . . . . . . . . . . . . . . . . . . . 215F.1.2.4 Directed area element . . . . . . . . . . . . . . . . . . . . . . 215F.1.2.5 Volume element . . . . . . . . . . . . . . . . . . . . . . . . . 216


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F.1.2.6 Spatial differential operators . . . . . . . . . . . . . . . . . . . 216F.1.3 Vector and tensor field formul. . . . . . . . . . . . . . . . . . . . . . . 216

F.1.3.1 The three-dimensional unit tensor of rank two. . . . . . . . . . 216F.1.3.2 The 3D Kronecker delta tensor. . . . . . . . . . . . . . . . . . 217F.1.3.3 The fully antisymmetric Levi-Civita tensor . . . . . . . . . . . 217F.1.3.4 Rotational matrices . . . . . . . . . . . . . . . . . . . . . . . . 217F.1.3.5 General vector and tensor algebra identities . . . . . . . . . . . 218F.1.3.6 Special vector and tensor algebra identities . . . . . . . . . . . 218F.1.3.7 General vector and tensor calculus identities . . . . . . . . . . 219F.1.3.8 Special vector and tensor calculus identities . . . . . . . . . . . 220F.1.3.9 Integral identities . . . . . . . . . . . . . . . . . . . . . . . . . 221

F.2 The electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223F.2.1 Microscopic Maxwell-Lorentz equations in Diracs symmetrised form . . 223

F.2.1.1 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 223F.2.2 Fields and potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

F.2.2.1 Vector and scalar potentials . . . . . . . . . . . . . . . . . . . 224F.2.2.2 The velocity gauge condition in free space . . . . . . . . . . . 224F.2.2.3 Gauge transformation . . . . . . . . . . . . . . . . . . . . . . 224

F.2.3 Energy and momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 224F.2.3.1 Electromagnetic field energy density in free space . . . . . . . 224F.2.3.2 Poynting vector in free space. . . . . . . . . . . . . . . . . . . 224F.2.3.3 Linear momentum density in free space . . . . . . . . . . . . . 224F.2.3.4 Linear momentum flux tensor in free space . . . . . . . . . . . 225F.2.3.5 Angular momentum density aroundx0in free space . . . . . . 225F.2.3.6 Angular momentum flux tensor aroundx0in free space . . . . 225

F.2.4 Electromagnetic radiation. . . . . . . . . . . . . . . . . . . . . . . . . . 225F.2.4.1 The far fields from an extended source distribution . . . . . . . 225F.2.4.2 The far fields from an electric dipole. . . . . . . . . . . . . . . 225F.2.4.3 The far fields from a magnetic dipole . . . . . . . . . . . . . . 226F.2.4.4 The far fields from an electric quadrupole . . . . . . . . . . . . 226F.2.4.5 Relationship between the field vectors in a plane wave . . . . . 226F.2.4.6 The fields from a point charge in arbitrary motion. . . . . . . . 226

F.3 Special relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227F.3.1 Metric tensor for flat 4D space . . . . . . . . . . . . . . . . . . . . . . . 227

F.3.2 Lorentz transformation of a four-vector . . . . . . . . . . . . . . . . . . 227F.3.3 Covariant and contravariant four-vectors . . . . . . . . . . . . . . . . . . 227

F.3.3.1 Position four-vector (radius four-vector) . . . . . . . . . . . . . 227F.3.3.2 Arbitrary four-vector field . . . . . . . . . . . . . . . . . . . . 227F.3.3.3 Four-del operator . . . . . . . . . . . . . . . . . . . . . . . . . 228F.3.3.4 Invariant line element . . . . . . . . . . . . . . . . . . . . . . 228F.3.3.5 Four-velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 228


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F.3.3.6 Four-momentum . . . . . . . . . . . . . . . . . . . . . . . . . 228F.3.3.7 Four-current density . . . . . . . . . . . . . . . . . . . . . . . 228

F.3.3.8 Four-potential . . . . . . . . . . . . . . . . . . . . . . . . . . 228F.3.4 Field tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

F.4 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

M Mathematical Methods 231M.1 Scalars, vectors and tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

M.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233M.1.1.1 Position vector . . . . . . . . . . . . . . . . . . . . . . . . . . 233

M.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234M.1.2.1 Scalar fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . 234M.1.2.2 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

M.1.2.3 Coordinate transformations . . . . . . . . . . . . . . . . . . . 235M.1.2.4 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

M.2 Vector algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239M.2.1 Scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239M.2.2 Vector product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239M.2.3 Dyadic product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

M.3 Vector calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242M.3.1 Thedeloperator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242M.3.2 The gradient of a scalar field . . . . . . . . . . . . . . . . . . . . . . . . 243M.3.3 The divergence of a vector field . . . . . . . . . . . . . . . . . . . . . . 243M.3.4 The curl of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . 243M.3.5 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244M.3.6 Vector and tensor integrals . . . . . . . . . . . . . . . . . . . . . . . . . 244

M.3.6.1 First order derivatives . . . . . . . . . . . . . . . . . . . . . . 244M.3.6.2 Second order derivatives . . . . . . . . . . . . . . . . . . . . . 246

M.3.7 Helmholtzs theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247M.4 Analytical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

M.4.1 Lagranges equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248M.4.2 Hamiltons equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

M.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250M.6 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

Index 267


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LIST OF FIGURES

1.1 Coulomb interaction between two electric charges . . . . . . . . . . . . . . . . . . . 31.2 Coulomb interaction for a distribution of electric charges . . . . . . . . . . . . . . . 51.3 Ampre interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Moving loop in a varyingBfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.1 Fields in the far zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.1 Multipole radiation geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Electric dipole geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.3 Linear antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.4 Electric dipole antenna geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.5 Radiation from a moving charge in vacuum . . . . . . . . . . . . . . . . . . . . . . 1206.6 An accelerated charge in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.7 Angular distribution of radiation during bremsstrahlung . . . . . . . . . . . . . . . . 1316.8 Location of radiation during bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . 1326.9 Radiation from a charge in circular motion . . . . . . . . . . . . . . . . . . . . . . . 1346.10 Synchrotron radiation lobe width . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.11 The perpendicular electric field of a moving charge . . . . . . . . . . . . . . . . . . 1396.12 Electron-electron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.13 Loop antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.1 Relative motion of two inertial systems. . . . . . . . . . . . . . . . . . . . . . . . . 154

7.2 Rotation in a 2D Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.3 Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.1 Linear one-dimensional mass chain. . . . . . . . . . . . . . . . . . . . . . . . . . . 180

9.1 Vavilov- LCerenkov cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

M.1 Tetrahedron-like volume element of matter. . . . . . . . . . . . . . . . . . . . . . . 251

xv


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PREFACE TO THE SECOND EDITION

This second edition of the book E L E CT RO M A G N E T I C F I E L D TH E O RY is a major revisionand expansion of the first edition that was published on the Internet (www.plasma.uu.se/CED/Book) in an organic growth process over the years 19972008. The main changes are an expan-sion of the material treated, an addition of a new chapter and several illustrative examples, and aslight reordering of the chapters.

The main reason for attempting to improve the presentation and to add more material is thatthis new edition is now being made available in printed form by Dover Publications and is used inan extended Classical Electrodynamics course at Uppsala University, at the last-year undergrad-uate, Master, and beginning post-graduate/doctoral level. It has also been used by the author in asimilar course at the Galilean School of Higher Education (Scuola Galileiana di Studi Superiori)at University of Padova. It is the authors hope that the second edition of his book will find a wid

use in Academia and elsewhere.The subject matter starts with a description of the properties of electromagnetism when the

charges and currents are located in otherwise free space, i.e., a space that is free of matter andexternal fields (e.g., gravitation). A rigorous analysis of the fundamental properties of the elec-tromagnetic fields and radiation phenomena follows. Only then the influence of matter on thefields and the pertinent interaction processes is taken into account. In the authors opinion, thisapproach is preferable since it avoids the formal logical inconsistency of introducing, very earlyin the derivations, the effect on the electric and mangetic fields when conductors and dielectricsare present (andvice versa) in anad hocmanner, before constitutive relations and physical mod-els for the electromagnetic properties of matter, including conductors and dielectrics, have beenderived from first principles. Curved-space effects on electromagnetism are not treated at all.

In addition to the Maxwell-Lorentz equations, which postulate the beaviour of electromag-netic fields due to electric charges and currents on a microscopic classical scale, chapter 1onpage 1also introduces Diracs symmetrised equations that incorporate the effects of magneticcharges and currents. In chapter 2on page 19,a stronger emphasis than before is put on theaxiomatic foundation of electrodynamics as provided by the Maxwell-Lorentz equations that aretaken as the postulates of the theory. Chapter3on page 33on potentials and gauges now providesa more comprehensive picture and discusses gauge invariance in a more satisfactory manner thanthe first edition did. Chapter 4 on page 53is new and deals with symmetries and conservedquantities in a more rigourous, profound and detailed way than in the first edition. For instance,the presentation of the theory of electromagnetic angular momentum and other observables (con-

stants of motion) has been substantially expanded and put on a firm basis. Chapter9on page 189is a complete rewrite and combines material that was scattered more or less all over the firstedition. It also contains new material on wave propagation in plasma and other media. When,in chapter 9 on page 189,the macroscopic Maxwell equations are introduced, the inherent ap-proximations in the derived field quantities are clearly pointed out. The collection of formul inappendixF on page 213has been augmented quite substantially. In appendixMon page 231,thetreatment of dyadic products and tensors has been expanded significantly and numerous examples

xvii
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xviii PREFACE TO THE SECOND EDITION

have been added throughout.I want to express my warm gratitude to professor C E S A R EBA R B I E R Iand his entire group,

particularly FA B R I Z I O TA M B U R I N I, at the Department of Astronomy, University of Padova,for stimulating discussions and the generous hospitality bestowed upon me during several shorterand longer visits in2008,2009, and2010that made it possible to prepare the current major revi-sion of the book. In this breathtakingly beautiful northern Italy where the cradle of renaissanceonce stood, intellectual titan G A L I L E O GA L I L E I worked for eighteen years and gave birth tomodern physics, astronomy and science as we know it today, by sweeping away Aristotelian dog-mas, misconceptions and mere superstition, thus most profoundly changing our conception of theworld and our place in it. In the process, Galileos new ideas transformed society and mankindirreversibly and changed our view of the Universe, including our own planet, forever. It is hopedthat this book may contribute in some small, humble way to further these, once upon a time,mind-bogglingand dangerousideas of intellectual freedom and enlightment.

Thanks are also due to JO H A N S J H O L M , KRI S T O F F E R PA L M E R, MA R C U S ER I K S-S O N, and J O H A N L I N D B E R Gwho during their work on their Diploma theses suggested im-provements and additions and to HO L G E RT H E Nand STAFFANY N G V Efor carefully checkingsome lengthy calculations and to the numerous undergraduate students, who have been exposedto various draft versions of this second edtion. In particular, I would like to mention B RU N OST R A N D B E R G.

This book is dedicated to my son MATTIAS, my daughter K A RO L I N A, my four grandsonsMA X , AL B I N , F I L I P and OS K A R, my high-school physics teacher, S TAFFAN R S B Y, andmy fellow members of the CA P E L L A PE D A G O G I CAUP S A L I E N S I S.

Padova, Italy BO TH I D

February,2011 www.physics.irfu.se/bt


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PREFACE TO THE FIRST EDITION

Of the four known fundamental interactions in naturegravitational, strong, weak, and electro-magneticthe latter has a special standing in the physical sciences. Not only does it, togetherwith gravitation, permanently make itself known to all of us in our everyday lives. Electrody-namics is also by far the most accurate physical theory known, tested on scales running fromsub-nuclear to galactic, and electromagnetic field theory is the prototype of all other field theo-ries.

This book, E L E CT RO M A G N E T I C F I E L D TH E O RY, which tries to give a modern view ofclassical electrodynamics, is the result of a more than thirty-five year long love affair. In theautumn of1972, I took my first advanced course in electrodynamics at the Department of Theo-retical Physics, Uppsala University. Soon I joined the research group there and took on the taskof helping the late professor P E R OL O F FR M A N, who was to become my Ph.D. thesis ad-

viser, with the preparation of a new version of his lecture notes on the Theory of Electricity. Thisopened my eyes to the beauty and intricacy of electrodynamics and I simply became intrigued byit. The teaching of a course in Classical Electrodynamics at Uppsala University, some twenty oddyears after I experienced the first encounter with the subject, provided the incentive and impetusto write this book.

Intended primarily as a textbook for physics and engineering students at the advanced under-graduate or beginning graduate level, it is hoped that the present book will be useful for researchworkers too. It aims at providing a thorough treatment of the theory of electrodynamics, mainlyfrom a classical field-theoretical point of view. The first chapter is, by and large, a descrip-tion of how Classical Electrodynamics was established by JA M E S C L E R K MA X W E L L as afundamental theory of nature. It does so by introducing electrostatics and magnetostatics anddemonstrating how they can be unified into one theory, classical electrodynamics, summarisedin Lorentzs microscopic formulation of the Maxwell equations. These equations are used asan axiomatic foundation for the treatment in the remainder of the book, which includes mod-ern formulation of the theory; electromagnetic waves and their propagation; electromagneticpotentials and gauge transformations; analysis of symmetries and conservation laws describingthe electromagnetic counterparts of the classical concepts of force, momentum and energy, plusother fundamental properties of the electromagnetic field; radiation phenomena; and covariantLagrangian/Hamiltonian field-theoretical methods for electromagnetic fields, particles and inter-actions. Emphasis has been put on modern electrodynamics concepts while the mathematicaltools used, some of them presented in an Appendix, are essentially the same kind of vector and

tensor analysis methods that are used in intermediate level textbooks on electromagnetics butperhaps a bit more advanced and far-reaching.

The aim has been to write a book that can serve both as an advanced text in Classical Elec-trodynamics and as a preparation for studies in Quantum Electrodynamics and Field Theory, aswell as more applied subjects such as Plasma Physics, Astrophysics, Condensed Matter Physics,Optics, Antenna Engineering, and Wireless Communications.

The current version of the book is a major revision of an earlier version, which in turn was an

xix


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xx PREFACE TO THE FIRST EDITION

outgrowth of the lecture notes that the author prepared for the four-credit course Electrodynam-ics 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 those notes were based onlecture notes prepared, in Swedish, by my friend and Theoretical Physics colleague B E N G TLU N D B O R G, who created, developed and taught an earlier, two-credit course called Electro-magnetic Radiation at our faculty. Thanks are due not only to Bengt Lundborg for providingthe inspiration to write this book, but also to professor C H R I S T E RWA H L B E R G, and professorG RA N F L D T, both at the Department of Physics and Astronomy, Uppsala University, forinsightful suggestions, to professor J O H N LE A RN E D , Department of Physics and Astronomy,University of Hawaii, for decisive encouragement at the early stage of this book project, to pro-fessor GE R A R D U S THO O F T, for recommending this book on his web page How to becomeagoodtheoretical physicist, and professor C E CI L I A J A RL S K O G, Lund Unversity, for pointingout a couple of errors and ambiguities.

I am particularly indebted to the late professor V ITALIY LA Z A R E V I C H G I N Z B U R G, forhis many fascinating and very elucidating lectures, comments and historical notes on plasmaphysics, electromagnetic radiation and cosmic electrodynamics while cruising up and down theVolga and Oka rivers in Russia at the ship-borne Russian-Swedish summer schools that wereorganised jointly by late professor L E V M I K A H I L O V I CH ERU K H I M O Vand the author duringthe1990s, and for numerous deep discussions over the years.

Helpful comments and suggestions for improvement from former PhD students TO B I AC A-RO Z Z I, RO G E R KA R L S S O N, and MATTIAS WA L D E N V I K , as well as AN D E R SERI K S S O Nat the Swedish Institute of Space Physics in Uppsala and who have all taught Uppsala studentson the material covered in this book, are gratefully acknowledged. Thanks are also due to the lateHE L M U TKO P K A, for more than twenty-five years a close friend and space physics colleagueworking at the Max-Planck-Institut fr Aeronomie, Lindau, Germany, who not only taught methe practical aspects of the use of high-power electromagnetic radiation for studying space, butalso some of the delicate aspects of typesetting in TEX and LATEX.

In an attempt to encourage the involvement of other scientists and students in the making ofthis book, thereby trying to ensure its quality and scope to make it useful in higher universityeducation anywhere in the world, it was produced as a World-Wide Web (WWW) project. Thisturned out to be a rather successful move. By making an electronic version of the book freelydownloadable on the Internet, comments have been received from fellow physicists around theworld. To judge from WWW hit statistics, it seems that the book serves as a frequently used In-ternet 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 ofthe book. I would like to thank all students and Internet users who have downloaded and com-mented on the book during its life on the World-Wide Web.

Uppsala, Sweden BO TH I D December,2008 www.physics.irfu.se/bt


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1

FOUNDATIONS OF CLASSICAL

ELECTRODYNAMICS

The classical theory of electromagnetism deals with electric and magnetic fieldsand their interaction with each other and with charges and currents. This theoryis classical in the sense that it assumes the validity of certain mathematical limit-ing processes in which it is considered possible for the charge and current distri-

butions to be localised in infinitesimally small volumes of space.1 Clearly, this 1 Accepting the mere existence ofan electrically charged particle re-quires some careful thinking. In hisexcellent bookClassical ChargedParticles, F R I T Z R O H R L I C Hwrites

To what extent does itmake sense to talk about anelectron, say, in classicalterms? These and similarquestions clearly indicatethat ignoring philosophyin physics means notunderstanding physics.For there is no theoretical

physics without somephilosophy; not admittingthis fact would be self-deception.

is in contradistinction to electromagnetism on an atomistic scale, where chargesand currents have to be described in a nonlocal quantum formalism. However,the limiting processes used in the classical domain, which, crudely speaking, as-sume that an elementary charge has a continuous distribution of charge density,will yield results that agree perfectly with experiments on non-atomistic scales,however small or large these scales may be.2

2 Electrodynamics has been testedexperimentally over a larger rangeof spatial scales than any otherexisting physical theory.

It took the genius of JA M E S CL E R K MA X W E L L to consistently unify, inthe mid-1800s, the theory ofElectricityand the then distinctively different the-oryMagnetisminto a single super-theory, ElectromagnetismorClassical Elec-trodynamics(CED), and also to realise that optics is a sub-field of this super-theory. Early in the 20th century, HE N D RI K AN T O O N LO R E N T Ztook theelectrodynamics theory further to the microscopic scale and also paved the wayfor the Special Theory of Relativity, formulated in its full extent by A L BE RTE I N S T E I N in 1905. In the 1930s PAU L AD R I E N MA U RI CE D I RA C ex-panded electrodynamics to a more symmetric form, including magnetic as wellas electric charges. With his relativistic quantum mechanics and field quantisa-tion concepts, Dirac had already in the 1920s laid the foundation for QuantumElectrodynamics (QED), the relativistic quantum theory for electromagneticfields and their interaction with matter for which R I C H A R DPH I L L I P S F E Y N-M A N, J U L I A NS E Y M O U RS CH W I N G E R , and S I N- IT I ROT O M O N A G Awere

awarded the Nobel Prize in Physics in1965. Around the same time, physicistssuch as SH E L D O N GL A S H O W, AB D U S SA L A M, and ST E V E NWE I N B E R Gwere able to unify electrodynamics with the weak interaction theory, thus cre-ating yet another successful super-theory,Electroweak Theory, an achievementwhich rendered them the Nobel Prize in Physics 1979. The modern theory ofstrong interactions, Quantum Chromodynamics (QCD), is heavily influencedby CED and QED.

1


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2 j 1. FOUNDATIONS OF CLASSICAL ELECTRODYNAMICS

In this introductory chapter we start with the force interactions in classicalelectrostatics and classical magnetostatics, introduce the corresponding static

electric and magnetic fields and postulate two uncoupled systems of differentialequations for them. We continue by showing that the conservation of electriccharge and its relation to electric current lead to a dynamic connection betweenelectricity and magnetism and how the two can be unified into Classical Elec-trodynamics. This theory is described by a system of coupled dynamic fieldequations the microscopic versions of Maxwells differential equations in-troduced by Lorentz which, in chapter chapter 2,we take as the axiomaticfoundation of the theory of electromagnetic fields and the basis for the treatmentin the rest of the book.

At the end of this chapterchapter1we present Diracs symmetrised form ofthe Maxwell-Lorentz equations that incorporate magnetic charges and magnetic

currents into the theory in a symmetric way. In practical work, such as in antennaengineering, magnetic currents have proved to be a very useful concept. We shallmake some use of this symmetrised theory of electricity and magnetism.

1.1 Electrostatics

The theory that describes physical phenomena related to the interaction betweenstationary electric charges or charge distributions in a finite space with station-ary boundaries is called electrostatics. For a long time, electrostatics, underthe nameelectricity, was considered an independent physical theory of its own,alongside other physical theories such as Magnetism, Mechanics, Optics, andThermodynamics.3

3 The physicist, mathematician andphilosopher P I E R R E M AURICE

MA R I E D U H E M(18611916)once wrote:

The whole theory ofelectrostatics constitutesa group of abstract ideasand general propositions,formulated in the clearand concise language ofgeometry and algebra,and connected with oneanother by the rules ofstrict logic. This wholefully satisfies the reason ofa French physicist and histaste for clarity, simplicityand order.. . .

1.1.1 Coulombs law

It has been found experimentally that the force interaction between stationary,electrically charged bodies can be described in terms of two-body mechani-

cal forces. Based on these experimental observations, Coulomb4

postulated,4 CH A R L E S-AU G U S T I N D ECO U L O M B(17361806) wasa French physicist who in 1775published three reports on theforces between electrically chargedbodies.

in 1775, that in the simple case depicted in figure 1.1on the facing page,themechanical force on a static electric charge q located at the field point (obser-vation point) x, due to the presence of another stationary electric charge q 0 atthesource pointx 0, is directed along the line connecting these two points, repul-sive for charges of equal signs and attractive for charges of opposite signs. Thispostulate is calledCoulombs law


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1.1. Electrostatics j 3

q0

q

O

x0

x x0

x

Figure 1.1: Coulombs law postu-lates that a static electric charge q ,located at a point xrelative to the

originO , will experience an elec-trostatic forceF es.x/from a staticelectric charge q0 located at x0.Note that this definition is indepen-dent of any particular choice of co-ordinate system since the mechani-cal force Fes is a true (polar) vector.

and can be formulated mathematically as

Fes.x/D qq0

4"0

x x0jx x0j3D

qq 0

4"0r

1jx x0j

D qq

0

4"0r

0 1

jx x0j

(1.1)

where, in the last step, formula (F.114) on page220was used. InSI units , whichwe shall use throughout, the electrostatic force5 Fes is measured in Newton (N), 5 Massive particles also interact

gravitationally but with a force thatis typically1036 times weaker.

the electric charges q and q0 in Coulomb (C), i.e.Ampere-seconds (As), andthe lengthjx x0j in metres (m). The constant "0D 107=.4c2/Farad permetre (Fm1) is the permittivity of free spaceandc ms1 is the speed of lightin vacuum.6 In CGS units, "0D 1=.4/and the force is measured in dyne, 6 The notationc for speed stems

from the Latin word celeritas

which means swiftness. Thisnotation seems to have beenintroduced by W I L H E L ME D-UARD W E B E R(18041891), andRU D O L FK O H L R A U S C H(18091858) andc is therefore sometimesreferred to asWebers constant.In all his works from 1907andonward, ALBERT E I N S T E I N(18791955) usedcto denote thespeed of light in free space.

electric charge in statcoulomb, and length in centimetres (cm).

1.1.2 The electrostatic field

Instead of describing the electrostatic interaction in terms of a force action at adistance, it turns out that for many purposes it is useful to introduce the conceptof a field. Thus we describe the electrostatic interaction in terms of a staticvectorial electric fieldEstat defined by the limiting process

Estat.x/def lim

q!0F es.x/

q (1.2)

where Fes

is the electrostatic force, as defined in equation (1.1)above, from a netelectric chargeq 0on the test particle with a small net electric chargeq. In the SIsystem of units, electric fields are therefore measured in NC1 or, equivalently,in Vm1. Since the purpose of the limiting process is to ascertain that the testcharge q does not distort the field set up by q0, the expression for Estat doesnot depend explicitly on q but only on the charge q0 and the relative positionvectorx x0. This means that we can say that any net electric charge produces


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an electric field in the space that surrounds it, regardless of the existence of asecond charge anywhere in this space.7 However, in order to experimentally7 In the preface to the first edition

of the first volume of his bookA Treatise on Electricity andMagnetism, first published in1873,Maxwell describes this in thefollowing almost poetic manner:

For instance, Faraday, inhis minds eye, saw lines offorce traversing all spacewhere the mathematicianssaw centres of forceattracting at a distance:Faraday saw a mediumwhere they saw nothingbut distance: Faradaysought the seat of thephenomena in real actions

going on in the medium,they were satisfied that theyhad found it in a powerof action at a distanceimpressed on the electricfluids.

detect a charge, a second (test) charge that senses the presence of the first one,must be introduced.

Using equations (1.1) and (1.2)on the previous page, and formula (F.114)on page220,we find that the electrostatic fieldEstat at the observation pointx(also known as thefield point), due to a field-producing electric chargeq 0at thesource pointx0, is given by

Estat.x/D q0

4"0

x x0jx x0j3D

q0

4"0r

1

jx x0j

D q0

4"0r

0

1

jx x0j

(1.3)

In the presence of several field producing discrete electric charges q0i , locatedat the points x0i , i D 1 ; 2 ; 3 ; : : : , respectively, in otherwise empty space, theassumption of linearity of vacuum8 allows us to superimpose their individual

8 In fact, a vacuum exhibits aquantum mechanical non-linearity

due tovacuum polarisationeffects, manifesting themselvesin the momentary creation andannihilation of electron-positron

pairs, but classically this non-linearity is negligible.

electrostatic fields into a total electrostatic field

Estat.x/D 14"0

Xi

q0ix x0ix x0i

3 (1.4)If the discrete electric charges are small and numerous enough, we can, in

a continuum limit, assume that the total charge q0 from an extended volumeto be built up by local infinitesimal elemental charges dq0, each producing anelemental electric field

dEstat.x/D 14"0

dq0 x x0jx x0j3 (1.5)

By introducing the electric charge density, measured in Cm3 in SI units, atthe point x 0 within the volume element d3x0D dx01dx02dx03 (measured in m3),the elemental charge can be expressed as dq0D d3x0 .x0/, and the elementalelectrostatic field as

dEstat.x/D 14"0

d3x0 .x0/ x x0jx x0j3 (1.6)

Integrating this over the entire source volumeV0, we obtain

Estat.x/D Z dEstat.x/D 14"0

ZV0

d3x0 .x0/ x x0jx x0j3

D 14"0

ZV0

d3x0 .x0/r

1

jx x0jD 1

4"0r

ZV0

d3x0 .x0/jx x0j (1.7)

where we used formula(F.114) on page220and the fact that .x0/does notdepend on the unprimed (field point) coordinates on which roperates.


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1.1. Electrostatics j 5

V0

q0i

q

O

x0i

x x0ix

Figure 1.2: Coulombs law for acontinuous charge density .x0/within a volume V 0 of limited ex-

tent. In particular, a charge den-sity .x0/D PNi q0i .x0 x0i /would represent a source distri-bution consisting of N discretecharges q0

i located at x0

i, where

iD 1 ; 2 ; 3 ; : : : ; N .

We emphasise that under the assumption of linear superposition, equation(1.7)on the facing page is valid for an arbitrary distribution of electric charges,including discrete charges, in which case is expressed in terms of Dirac deltadistributions:9 9 Since, by definition, the integralZ

V0d3x0 .x0 x0

i/

ZV0

d3x0 .x 0 x0i / .y0 y0

i/.z0 z0

i/ D 1

is dimensionless, andxhas the SIdimension m, the 3D Dirac deltadistribution .x0 x0

i/must have

the SI dimension m3

.

.x0/DX

i

q0i.x0 x0i / (1.8)

as illustrated in figure 1.2.Inserting this expression into expression (1.7)on thefacing page we recover expression (1.4)on the preceding page, as we should.

According toHelmholtzs theorem, discussed in subsectionM.3.7, any well-behaved vector field is completely known once we know its divergence and curlat all points xin 3D space.10 Taking the divergence of the general Estat ex-

10 HE R M A N NL U D W I GF E R-DINAND VONH ELMHOLTZ(18211894) was a physicist,physician and philosopher whocontributed to wide areas ofscience, ranging from electrody-namics to ophthalmology.

pression for an arbitrary electric charge distribution, equation (1.7)on the facingpage, and applying formula(F.126) on page222[see also equation (M.75) onpage246], we obtain

r Estat.x/D 14"0

r r

ZV0

d3x0 .x0/jx x0jD

.x/

"0(1.9a)

which is the differential form ofGausss law of electrostatics. Since, accordingto formula (F.100) on page220, r r.x/ 0for any R3 scalar field.x/,we immediately find that in electrostatics

rEstat.x/D 14"0

rr

ZV0d3x0

.x0/

jx

x0

j

D 0 (1.9b)

i.e.thatEstat is a purelyirrotationalfield.To summarise, electrostatics can be described in terms of two vector partial

differential equations

r Estat.x/D .x/"0

(1.10a)

r Estat.x/D0 (1.10b)


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representing four scalar partial differential equations.

1.2 Magnetostatics

Whereas electrostatics deals with static electric charges (electric charges that donot move), and the interaction between these charges,magnetostaticsdeals withstatic electric currents (electric charges moving with constant speeds), and theinteraction between these currents. Here we shall discuss the theory of magne-tostatics in some detail.

1.2.1 Ampres law

Experiments on the force interaction between two small loops that carry staticelectric currents I andI0(i.e.the currents I andI0 do not vary in time) haveshown that the loops interact via a mechanical force, much the same way thatstatic electric charges interact. Let Fms.x/denote the magnetostatic force ona loop C, with tangential line vector element dl, located at x and carrying acurrentIin the direction of dl, due to the presence of a loop C0, with tangentialline element dl0, located at x 0 and carrying a current I0 in the direction of dl0

in otherwise empty space. This spatial configuration is illustrated in graphical

form in figure1.3on the facing page.According toAmpres lawthe magnetostatic force in question is given bythe expression11

11 AN D R -MA R I E A M P R E(17751836) was a French mathe-matician and physicist who, only a

few days after he learned about thefindings by the Danish physicistand chemist HA N SC H R I S T I A NR S T E D(17771851) regardingthe magnetic effects of electriccurrents, presented a paper to theAcadmie des Sciences in Paris,postulating the force law that nowbears his name.

Fms.x/D 0II0

4

IC

dl I

C0dl0

x x0jx x0j3

D 0II0

4

IC

dl I

C0dl0 r

1

jx x0j (1.11)

In SI units,0D4 107 Henry per metre (Hm1) is thepermeability of freespace. From the definition of"0and0(in SI units) we observe that

"00D 107

4c2 (Fm1) 4 107 (Hm1)D 1

c2 (s2m2) (1.12)

which is a most useful relation.At first glance, equation (1.11) above may appear asymmetric in terms of the

loops and therefore be a force law that does not obey Newtons third law. How-ever, by applying the vector triple product bac-cab formula (F.53) on page218,


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1.2. Magnetostatics j 7

C0

C

I0dl0Idl

O

x0

xx0

x

Figure 1.3: Ampres law postu-lates how a small loop C, carry-ing a static electric current I di-

rected along the line element dlat x, experiences a magnetostaticforce Fms.x/ from a small loopC0, carrying a static electric currentI0 directed along the line elementdl0 located at x0.

we can rewrite(1.11) as

Fms.x/D 0II0

4

IC0

dl0I

C

dl r

1

jx x0j

0II0

4

IC

dl I

C0dl0 x x0jx x0j3

(1.13)

Since the integrand in the first integral over Cis an exact differential, this in-tegral vanishes and we can rewrite the force expression, formula (1.11)on thepreceding page, in the following symmetric way

Fms.x/D

0II0

4 IC dl IC0 dl0 x x0

jx x0j3 (1.14)

which clearly exhibits the expected interchange symmetry between loopsCandC0.

1.2.2 The magnetostatic field

In analogy with the electrostatic case, we may attribute the magnetostatic forceinteraction to a static vectorial magnetic fieldBstat. The small elemental staticmagnetic field dBstat.x/ at the field point x due to a line current element di0.x0/DI0dl0.x0/

Dd3x0j.x0/of static current I0with electric current densityj, mea-

sured in Am2 in SI units, directed along the local line element dl0of the loopatx0, is

dBstat.x/def lim

I!0dFms.x/

I D 0

4di0.x0/

x x0jx x0j3

D 04

d3x0j.x0/ x x0jx x0j3

(1.15)


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which is the differential form of theBiot-Savart law.The elemental field vector dBstat.x/at the field point x is perpendicular to

the plane spanned by the line current element vector di0.x0/at the source pointx0, and the relative position vector x x0. The corresponding local elementalforce dFms.x/is directed perpendicular to the local plane spanned by dBstat.x/and the line current element di.x/. The SI unit for the magnetic field, sometimescalled themagnetic flux densityor magnetic induction , is Tesla (T).

If we integrate expression (1.15) on the preceding page around the entireloop atx, we obtain

Bstat.x/DZ

dBstat.x/

D 04 ZV0d

3x0 j.x0/ x x0

jx x0j3

D 04

ZV0

d3x0 j.x0/ r 1

jx x0j

D 04

r

ZV0

d3x0 j.x0/jx x0j

(1.16)

where we used formula (F.114) on page220,formula (F.87) on page219, andthe fact that j.x0/does not depend on the unprimed coordinates on which roperates. Comparing equation (1.7)on page4with equation(1.16) above, wesee that there exists an analogy between the expressions for Estat andBstat butthat they differ in their vectorial characteristics. With this definition ofBstat,equation(1.11) on page6may be written

Fms.x/DII

C

dl Bstat.x/DI

C

di Bstat.x/ (1.17)

In order to assess the properties ofBstat, we determine its divergence andcurl. Taking the divergence of both sides of equation (1.16)above and utilisingformula(F.99) on page220,we obtain

r Bstat.x/D 04

r r

ZV0

d3x0 j.x0/jx x0jD 0 (1.18)

since, according to formula (F.99) on page220, r .r a/vanishes for anyvector fielda.x/.

With the use of formula (F.128) on page222, the curl of equation (1.16)above can be written

r Bstat.x/D 04

rr

ZV0

d3x0 j.x0/jx x0j

D0j.x/ 0

4

ZV0

d3x0 r0 j.x0/r0

1

jx x0j (1.19)


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1.3. Electrodynamics j 9

assuming that j.x0/ falls off sufficiently fast at large distances. For the stationarycurrents of magnetostatics, r jD0since there cannot be any charge accumu-lation in space. Hence, the last integral vanishes and we can conclude that

r Bstat.x/D0j.x/ (1.20)

We se that the static magnetic field Bstat is purelyrotational.

1.3 Electrodynamics

As we saw in the previous sections, the laws of electrostatics and magnetostaticscan be summarised in two pairs of time-independent, uncoupled partial differen-tial equations, namely theequations of classical electrostatics

r Estat.x/D .x/"0

(1.21a)

r Estat.x/D0 (1.21b)

and theequations of classical magnetostatics

r Bstat.x/D0 (1.21c)r Bstat.x/D0j.x/ (1.21d)

Since there is nothing a priorithat connects Estat directly with Bstat, we mustconsider classical electrostatics and classical magnetostatics as two separate andmutually independent physical theories.

However, when we include time-dependence, these theories are unified intoClassical Electrodynamics. This unification of the theories of electricity andmagnetism can be inferred from two empirically established facts:

1. Electric charge is a conserved quantity and electric current is a transport ofelectric charge. As we shall see, this fact manifests itself in the equation ofcontinuity and, as a consequence, inMaxwells displacement current.

2. A change in the magnetic flux through a loop will induce an electromotiveforce electric field in the loop. This is the celebrated Faradays law of induc-tion .


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1.3.1 The indestructibility of electric charge

Letj.t;x/denote the time-dependent electric current density. In the simplestcase it can be defined as jD vwherev is the velocity of the electric chargedensity.1212 A more accurate model is to

assume that the individual chargeand current elements obey somedistribution function that describestheir local variation of velocityin space and time. For instance,jcan be defined in statisticalmechanical terms as j.t; x/DPq

Rd3v vf .t; x; v/where

f .t; x; v/is the (normalised)distribution function for particlespeciescarrying an electricchargeq .

The electric charge conservation lawcan be formulated in the equation ofcontinuity for electric charge

@.t;x/

@t C r j.t;x/D0 (1.22)

[email protected];x/

@t D r j.t;x/ (1.23)

which states that the time rate of change of electric charge .t;x/is balanced

by a negative divergence in the electric current density j.t;x/, i.e.an influx ofcharge. Conservation laws will be studied in more detail in chapterchapter4.

1.3.2 Maxwells displacement current

We recall from the derivation of equation (1.20) on the previous page that therewe used the fact that in magnetostatics r j.x/ D 0. In the case of non-stationary sources and fields, we must, in accordance with the continuity equa-tion (1.22) above, set r j.t;x/D @.t;x/=@t . Doing so, and formally re-peating the steps in the derivation of equation (1.20) on the previous page, we

would obtain the result

r B.t;x/D0j.t; x/ C 0

4

ZV0

[email protected];x0/

@t r

0

1

jx x0j

(1.24)

If we assume that equation (1.7)on page4can be generalised to time-varyingfields, we can make the identification1313 Later, we will need to consider

this generalisation and formalidentification further. 1

4"0

@

@t

ZV0

d3x0 .t; x0/r0

1

jx x0j

D @@t

1

4"0

ZV0

d3x0 .t;x0/r

1

jx x0j

D @@t 14"0 r ZV0d3x0 .t;x0/jx x0jD @@t E.t; x/(1.25)

The result is Maxwells source equation for the Bfield

r B.t;x/D0j.t;x/ C @

@t"0E.t;x/

D0j.t; x/ C 0"0

@

@tE.t;x/

(1.26)


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where"[email protected];x/=@tis the famous displacement current. This, at the time, un-observed current was introduced by Maxwell, in a stroke of genius, in order to

make also the right-hand side of this equation divergence-free when j.t;x/isassumed to represent the density of the total electric current. This total currentcan be split up into ordinary conduction currents, polarisation currents andmagnetisation currents. This will be discussed in subsection9.1on page 190.

The displacement current behaves like a current density flowing in free space.As we shall see later, its existence has far-reaching physical consequences as itpredicts that such physical observables as electromagnetic energy, linear mo-mentum, and angular momentum can be transmitted over very long distances,even through empty space.

1.3.3 Electromotive forceIf an electric field E.t; x/is applied to a conducting medium, a current densityj.t;x/ will be set up in this medium. But also mechanical, hydrodynamicaland chemical processes can give rise to electric currents. Under certain physicalconditions, and for certain materials, one can assume that a linear relationshipexists between the electric current density jand E. This approximation is calledOhms law:14 14 In semiconductors this approx-

imation is in general applicableonly for a limited range ofE. Thisproperty is used in semiconductordiodes for rectifying alternatingcurrents.

j.t;x/DE.t;x/ (1.27)

whereis theelectric conductivitymeasured in Siemens per metre (Sm

1).We can view Ohms law equation (1.27) as the first term in a Taylor ex-

pansion of a general law jE.t; x/. This general law incorporatesnon-lineareffectssuch asfrequency mixingandfrequency conversion. Examples of mediathat are highly non-linear are semiconductors and plasma. We draw the atten-tion to the fact that even in cases when the linear relation between E and jisa good approximation, we still have to use Ohms law with care. The conduc-tivityis, in general, time-dependent (temporal dispersive media) but then it isoften the case that equation (1.27) above is valid for each individual temporalFourier (spectral) component of the field. In some media, such asmagnetised

plasmaand certain material, the conductivity is different in different directions.

For such electromagnetically anisotropic media (spatial dispersive media) thescalar electric conductivity in Ohms law equation (1.27) has to be replacedby a conductivity tensor. If the response of the medium is not only anisotropicbut also non-linear, higher-order tensorial terms have to be included.

If the current is caused by an applied electric field E.t;x/, this electric fieldwill exert work on the charges in the medium and, unless the medium is super-conducting, there will be some energy loss. The time rate at which this energy is


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expended isj Eper unit volume (Wm3). IfEis irrotational (conservative),jwill decay away with time. Stationary currents therefore require that an electric

field due to anelectromotive force(EMF) is present. In the presence of such afieldEemf, Ohms law, equation (1.27)on the previous page, takes the form

jD .Estat C Eemf/ (1.28)

The electromotive force is defined as

EDI

C

dl .Estat C Eemf/ (1.29)

where dlis a tangential line element of the closed loop C.1515 The term electromotive forceis something of a misnomer sinceErepresents a voltage,i.e.its SIdimension is V.

1.3.4 Faradays law of inductionIn subsection 1.1.2 we derived the differential equations for the electrostaticfield. Specifically, on page 5 we derived equation (1.9b) stating that rEstat D0and hence thatEstat is aconservative field(it can be expressed as a gradient of ascalar field). This implies that the closed line integral ofEstat in equation (1.29)above vanishes and that this equation becomes

EDI

C

dl Eemf (1.30)

It has been established experimentally that a non-conservative EMF field isproduced in a closed circuit Cat rest if the magnetic flux through this circuit

varies with time. This is formulated inFaradays lawwhich, in Maxwells gen-eralised form, reads

E.t /DI

C

dl E.t;x/D ddt

m.t /

D ddt

ZS

d2xOn B.t; x/D Z

S

d2xOn @@tB.t;x/

(1.31)

wheremis themagnetic fluxand Sis the surface encircled byC, interpretedas a generic stationary loop and not necessarily as a conducting circuit. Appli-cation of Stokes theorem on this integral equation, transforms it into the differ-ential equation

r E.t;x/D @@tB.t;x/ (1.32)

that is valid for arbitrary variations in the fields and constitutes the Maxwellequation which explicitly connects electricity with magnetism.

Any change of the magnetic flux mwill induce an EMF. Let us thereforeconsider the case, illustrated in figure1.4on the facing page, when the loop is


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d2xOn

B.x/ B.x/

v

dlC

Figure 1.4: A loop C that moveswith velocity v in a spatially vary-ing magnetic field B.x/will sense

a varying magnetic flux during themotion.

moved in such a way that it encircles a magnetic field which varies during themovement. The total time derivative is evaluated according to the well-knownoperator formula

d

dtD @

@tC dx

dt r (1.33)

This follows immediately from the multivariate chain rule for the differentiationof an arbitrary differentiable function f.t;x.t//. Here, dx=dtdescribes a chosenpath in space. We shall choose the flow path which means that dx=dtD v and

d

dtD @

@tC v r (1.34)where, in a continuum picture, v is the fluid velocity. For this particular choice,the convective derivativedx=dt is usually referred to as thematerial derivativeand is denoted Dx=Dt .

Applying the rule (1.34) to Faradays law, equation (1.31) on the precedingpage, we obtain

E.t /D ddt

ZS

d2xOn BD Z

S

d2xOn @B@t

Z

S

d2xOn .v r/B (1.35)

Furthermore, taking the divergence of equation (1.32)on the facing page, we see

thatr

@

@tB.t;x/D @

@tr B.t;x/D r rE.t;x/D0 (1.36)

where in the last step formula(F.99) on page220was used. Since this is true forall timest , we conclude that

r B.t;x/D0 (1.37)


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also for time-varying fields; this is in fact one of the Maxwell equations. Usingthis result and formula(F.89) on page219,we find that

r.B v/D.v r/B (1.38)

since, during spatial differentiation, vis to be considered as constant, This allowsus to rewrite equation (1.35) on the previous page in the following way:

E.t /DI

C

dl Eemf D ddt

ZS

d2xOn B

D Z

S

d2xOn @B@t

Z

S

d2xOn r.B v/(1.39)

With Stokes theorem applied to the last integral, we finally get

E.t /D IC

dl Eemf D ZS

d2xOn @B@t

IC

dl .B v/ (1.40)

or, rearranging the terms,IC

dl .Eemf v B/D Z

S

d2xOn @B@t

(1.41)

where Eemf is the field induced in the loop, i.e. in the moving system. Theapplication of Stokes theorem in reverse on equation (1.41) above yields

r.Eemf

v B/

D

@B

@t

(1.42)

An observer in a fixed frame of reference measures the electric field

EDEemf v B (1.43)

and an observer in the moving frame of reference measures the following Lorentzforceon a chargeq

FD qEemf DqEC q.v B/ (1.44)

corresponding to an effective electric field in the loop (moving observer)

Eemf DE C v B (1.45)

Hence, we conclude that for a stationary observer, the Maxwell equation

rED @B@t

(1.46)

is indeed valid even if the loop is moving.


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1.3.5 The microscopic Maxwell equations

We are now able to collect the results from the above considerations and formu-late the equations of classical electrodynamics, valid for arbitrary variations intime and space of the coupled electric and magnetic fields E.t; x/and B.t;x/.The equations are, in SI units,16 16 InCGS unitsthe Maxwell-

Lorentz equations are

r E D 4r B D 0

r EC 1c

@B

@tD 0

r B 1c

@E

@tD 4

c j

inHeaviside-Lorentz units (one of

several natural units )

r E D r B D 0

r EC 1c

@B

@tD 0

r B 1c

@E

@tD 1

cj

and inPlanck units(another set ofnatural units)

r E D 4r B D 0

r EC@B

@t D 0

r B @E@t

D 4j

r ED "0

(1.47a)

r BD0 (1.47b)

r EC @B@t

D0 (1.47c)

r B

1

c2

@E

@t D0j (1.47d)

In these equationsD.t;x/represents the total, possibly both time and spacedependent, electric charge density, with contributions from free as well as in-duced (polarisation) charges. Likewise, jDj.t;x/ represents the total, possiblyboth time and space dependent, electric current density, with contributions fromconduction currents (motion of free charges) as well as all atomistic (polarisationand magnetisation) currents. As they stand, the equations therefore incorporatethe classical interaction between all electric charges and currents, free or bound,in the system and are calledMaxwells microscopic equations. They were firstformulated by Lorentz and therefore another name frequently used for them is

the Maxwell-Lorentz equations and the name we shall use. Together with theappropriateconstitutive relations that relate and j to the fields, and the initialand boundary conditions pertinent to the physical situation at hand, they form asystem of well-posed partial differential equations that completely determineEandB.

1.3.6 Diracs symmetrised Maxwell equations

If we look more closely at the microscopic Maxwell equations (1.47), we see

that they exhibit a certain, albeit not complete, symmetry. Dirac therefore madethe ad hoc assumption that there exist magnetic monopoles represented by amagnetic charge density, which we denote by m D m.t;x/, and a magneticcurrent density, which we denote by jm Djm.t; x/.17

17 J U L I A NS E Y M O U RSC H W I N G E R(19181994)once put it:

...there are strong theo-retical reasons to believethat magnetic charge existsin nature, and may haveplayed an important rolein the development ofthe Universe. Searches

for magnetic charge con-tinue at the present time,emphasising that electro-magnetism is very far frombeing a closed object.

The magnetic monopole was firstpostulated by P I E R R EC U R I E(18591906) and inferred fromexperiments in2009.

With these new hypothetical physical entities included in the theory, and withthe electric charge density denoted e and the electric current density denotedje, the Maxwell-Lorentz equations will be symmetrised into the following two


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scalar and two coupled vectorial partial differential equations (SI units):

r ED e

"0 (1.48a)

r BD0m (1.48b)

r EC @B@t

D 0jm (1.48c)

r B 1c2

@E

@tD0je (1.48d)

We shall call these equations Diracs symmetrised Maxwell equations or theelectromagnetodynamic equations.

Taking the divergence of (1.48c), we find that

r .r E/D @@t

.r B/ 0r jm 0 (1.49)

where we used the fact that the divergence of a curl always vanishes. Using(1.48b) to rewrite this relation, we obtain theequation of continuity for magneticcharge

@m

@t C r jm D0 (1.50)

which has the same form as that for the electric charges (electric monopoles) andcurrents, equation(1.22) on page10.

1.4 Examples

Faradays law derived from the assumed conservation of magnetic chargeEX A M P LE 1.1

PO S TU LA TE1.1(I NDESTRUCTIBILITY OF MAGNETIC CHARGE) Magnetic charge ex-ists and is indestructible in the same way that electric charge exists and is indestructible.

In other words, wepostulate that there exists an equation of continuity for magnetic charges:

@m.t; x/

@t C r jm.t; x/D0 (1.51)

Use this postulate and Diracs symmetrised form of Maxwells equations to derive Faradayslaw.

The assumption of the existence of magnetic charges suggests a Coulomb-like law for mag-


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1.4. Examples j 17

netic fields:

B

stat

.x/D 0

4 ZV0d3x0 m.x0/ x x0

jx x0j3D 0

4 ZV0d3x0 m.x0/r 1jx x0jD 0

4r

ZV0

d3x0 m.x0/jx x0j

(1.52)

[cf. equation (1.7)on page4 for Estat] and, if magnetic currents exist, a Biot-Savart-likelaw for electric fields [cf.equation (1.16)on page8forBstat]:

Estat.x/D 04

ZV0

d3x0 jm.x0/ x x0jx x0j3

D 04

ZV0

d3x0 jm.x0/ r

1

jx x0j

D 04

r

ZV0

d3x0 jm.x0/jx x0j

(1.53)

Taking the curl of the latter and using formula (F.128)on page222

r Estat.x/D 0jm.x/ 0

4

ZV0

d3x0 r0 jm.x0/r0

1

jx x0j

(1.54)

assuming that jm falls off sufficiently fast at large distances. Stationarity means thatr jm D0so the last integral vanishes and we can conclude that

r Estat.x/D 0jm.x/ (1.55)

It is intriguing to note that if we assume that formula (1.53) above is valid also for time-varying magnetic currents, then, with the use of the representation of the Dirac delta func-

tion, equation(F.116)on page220,the equation of continuity for magnetic charge, equation(1.50) on the preceding page, and the assumption of the generalisation of equation (1.52)above to time-dependent magnetic charge distributions, we obtain, at least formally,

r E.t; x/D 0jm.t; x/ @

@tB.t; x/ (1.56)

[cf.equation (1.24) on page10] which we recognise as equation (1.48c)on the facing page.A transformation of this electromagnetodynamic result by rotating into the electric realmof charge space, thereby letting jm tend to zero, yields the electrodynamic equation (1.47c)on page15,i.e.the Faraday law in the ordinary Maxwell equations. This process would alsoprovide an alternative interpretation of the term @B=@tas a magnetic displacement current,dual to theelectric displacement current[cf.equation (1.26) on page10].

By postulating the indestructibility of a hypothetical magnetic charge, and assuming a di-rect extension of results from statics to dynamics, we have been able to replace Faradaysexperimental results on electromotive forces and induction in loops as a foundation for theMaxwell equations by a more fundamental one. At first sight, this result seems to be inconflict with the concept of retardation. Therefore a more detailed analysis of it is required.This analysis is left to the reader.

End of example1.1


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1.5 Bibliography

[1] T. W. BARRETT AND D . M . GRIMES, Advanced Electromagnetism. Founda-tions, Theory and Applications, World Scientific Publishing Co., Singapore, 1995,ISBN 981-02-2095-2.

[2] R. BECKER, Electromagnetic Fields and Interactions, Dover Publications, Inc.,New York, NY, 1982, ISBN 0-486-64290-9.

[3] W. GREINER, Classical Electrodynamics, Springer-Verlag, New York, Berlin, Hei-delberg, 1996, ISBN 0-387-94799-X.

[4] E. HALLN,Electromagnetic Theory, Chapman & Hall, Ltd., London, 1962.

[5] K. HUANG, Fundamental Forces of Nature. The Story of Gauge Fields , World Sci-entific Publishing Co. Pte. Ltd, New Jersey, London, Singapore, Beijing, Shanghai,

Hong Kong, Taipei, and Chennai, 2007, ISBN 13-978-981-250-654-4 (pbk).[6] J. D . JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc.,

New York, NY . . . , 1999, ISBN 0-471-30932-X.

[7] L . D . LANDAU AND E. M. LIFSHITZ, The Classical Theory of Fields, fourth re-vised English ed., vol. 2 ofCourse of Theoretical Physics, Pergamon Press, Ltd.,Oxford . . . , 1975, ISBN 0-08-025072-6.

[8] F. E . LOW, Classical Field Theory, John Wiley & Sons, Inc., New York, NY ...,1997, ISBN 0-471-59551-9.

[9] J . C . MAXWELL, A Treatise on Electricity and Magnetism, third ed., vol. 1, DoverPublications, Inc., New York, NY, 1954, ISBN 0-486-60636-8.

[10] J. C. MAXWELL, A Treatise on Electricity and Magnetism, third ed., vol. 2, DoverPublications, Inc., New York, NY, 1954, ISBN 0-486-60637-8.

[11] D. B. MELROSE ANDR. C. MCPHEDRAN,Electromagnetic Processes in DispersiveMedia, Cambridge University Press, Cambr

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