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E L EC T ROM AG N E T I C F I E L D T H E OR Y

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E L EC T ROM AG N E T I C

F I E L D T H E OR Y

S E

Bo ThidSwedish Institute of Space PhysicsUppsala, Sweden

and

Department of Physics and AstronomyUppsala University, Sweden


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Also available

E L E C T ROM AG N E T I C F I E L D T H E OR Y

E X E R C I S E S

byTobia 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 1997 2009 byBo 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 byJOHAN SJHOLMand KRISTOFFERPALMERas part of their undergraduate Diploma Thesis work inEngineering Physics at Uppsala University2006 2007 .


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To the memory of professorLEV MIKHAILOVICHERUKHIMOV(1936 1997 )

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


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C ON T E N TS

Contents ix

List of Figures xv

Preface to the second edition xviiPreface to the rst edition xix

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

1.1.1 Coulombs law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 The electrostatic eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Ampres law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 The magnetostatic eld. . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . 151.3.6 Diracs symmetrised Maxwell equations. . . . . . . . . . . . . . . . . . 151.3.7 Maxwell-Chern-Simons equations. . . . . . . . . . . . . . . . . . . . . 16

1.4 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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 forE andB . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 The time-independent wave equations forE andB . . . . . . . . . . . . 28

2.4 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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3 Electromagnetic Potentials and Gauges 333.1 The electrostatic scalar potential. . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 The magnetostatic vector potential. . . . . . . . . . . . . . . . . . . . . . . . . 343.3 The electrodynamic potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Gauge transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 Gauge conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5.1 Lorenz-Lorentz gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5.2 Coulomb gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.3 Velocity gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Fundamental Properties of the Electromagnetic Field 514.1 Charge, space, and time inversion symmetries. . . . . . . . . . . . . . . . . . . 514.2 Conservation laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 Conservation of charge. . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.2 Conservation of current. . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.3 Conservation of energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.4 Conservation of linear momentum. . . . . . . . . . . . . . . . . . . . . 584.2.5 Conservation of angular momentum. . . . . . . . . . . . . . . . . . . . 614.2.6 Electromagnetic virial theorem. . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Electromagnetic duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Fields from Arbitrary Charge and Current Distributions 715.1 The retarded magnetic eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 The retarded electric eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 The elds at large distances from the sources. . . . . . . . . . . . . . . . . . . . 79

5.3.1 The far elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6 Radiation and Radiating Systems 856.1 Radiation of linear momentum and energy. . . . . . . . . . . . . . . . . . . . . 86

6.1.1 Monochromatic signals. . . . . . . . . . . . . . . . . . . . . . . . . . . 876.1.2 Finite bandwidth signals. . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2 Radiation of angular momentum. . . . . . . . . . . . . . . . . . . . . . . . . . 896.3 Radiation from a localised source volume at rest. . . . . . . . . . . . . . . . . . 896.3.1 Electric multipole moments. . . . . . . . . . . . . . . . . . . . . . . . . 896.3.2 The Hertz potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.3.3 Electric dipole radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 966.3.4 Magnetic dipole radiation. . . . . . . . . . . . . . . . . . . . . . . . . . 996.3.5 Electric quadrupole radiation. . . . . . . . . . . . . . . . . . . . . . . . 100


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6.4 Radiation from an extended source volume at rest. . . . . . . . . . . . . . . . . 1016.4.1 Radiation from a one-dimensional current distribution. . . . . . . . . . . 101

6.5 Radiation from a localised charge in arbitrary motion. . . . . . . . . . . . . . . 1086.5.1 The Linard-Wiechert potentials. . . . . . . . . . . . . . . . . . . . . . 1096.5.2 Radiation from an accelerated point charge. . . . . . . . . . . . . . . . 1116.5.3 Bremsstrahlung. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.5.4 Cyclotron and synchrotron radiation. . . . . . . . . . . . . . . . . . . . 126

6.6 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7 Relativistic Electrodynamics 1357.1 The special theory of relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.1.1 The Lorentz transformation. . . . . . . . . . . . . . . . . . . . . . . . . 1367.1.2 Lorentz space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.1.3 Minkowski space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.2 Covariant classical mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.3 Covariant classical electrodynamics. . . . . . . . . . . . . . . . . . . . . . . . 146

7.3.1 The four-potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.3.2 The Linard-Wiechert potentials. . . . . . . . . . . . . . . . . . . . . . 1477.3.3 The electromagnetic eld tensor. . . . . . . . . . . . . . . . . . . . . . 150

7.4 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

8 Electromagnetic Fields and Particles 1558.1 Charged particles in an electromagnetic eld. . . . . . . . . . . . . . . . . . . . 155

8.1.1 Covariant equations of motion. . . . . . . . . . . . . . . . . . . . . . . 1558.2 Covariant eld theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.2.1 Lagrange-Hamilton formalism for elds and interactions. . . . . . . . . 1628.3 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

9 Electromagnetic Fields and Matter 1719.1 Maxwells macroscopic theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 172

9.1.1 Polarisation and electric displacement. . . . . . . . . . . . . . . . . . . 1729.1.2 Magnetisation and the magnetising eld. . . . . . . . . . . . . . . . . . 1739.1.3 Macroscopic Maxwell equations. . . . . . . . . . . . . . . . . . . . . . 175

9.2 Phase velocity, group velocity and dispersion. . . . . . . . . . . . . . . . . . . 1769.3 Radiation from charges in a material medium. . . . . . . . . . . . . . . . . . . 177

9.3.1 Vavilov-LCerenkov radiation. . . . . . . . . . . . . . . . . . . . . . . . . 1789.4 Electromagnetic waves in a medium. . . . . . . . . . . . . . . . . . . . . . . . 1829.4.1 Constitutive relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1839.4.2 Electromagnetic waves in a conducting medium. . . . . . . . . . . . . . 185

9.5 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

F Formul 195


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F.1 The electromagnetic eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195F.1.1 The microscopic Maxwell equations. . . . . . . . . . . . . . . . . . . . 195

F.1.2 Fields and potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . 196F.1.3 Force and energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

F.2 Electromagnetic radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196F.2.1 Relationship between the eld vectors in a plane wave. . . . . . . . . . 196F.2.2 The far elds from an extended source distribution. . . . . . . . . . . . 197F.2.3 The far elds from an electric dipole. . . . . . . . . . . . . . . . . . . . 197F.2.4 The far elds from a magnetic dipole. . . . . . . . . . . . . . . . . . . . 197F.2.5 The far elds from an electric quadrupole. . . . . . . . . . . . . . . . . 197F.2.6 The elds from a point charge in arbitrary motion. . . . . . . . . . . . . 197

F.3 Special relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198F.3.1 Metric tensor for at 4D space. . . . . . . . . . . . . . . . . . . . . . . 198F.3.2 Covariant and contravariant four-vectors. . . . . . . . . . . . . . . . . . 198F.3.3 Lorentz transformation of a four-vector. . . . . . . . . . . . . . . . . . 198F.3.4 Invariant line element. . . . . . . . . . . . . . . . . . . . . . . . . . . . 198F.3.5 Four-velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199F.3.6 Four-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199F.3.7 Four-current density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199F.3.8 Four-potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199F.3.9 Field tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

F.4 Vector relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199F.4.1 Spherical polar coordinates. . . . . . . . . . . . . . . . . . . . . . . . . 200F.4.2 Vector and tensor formul. . . . . . . . . . . . . . . . . . . . . . . . . 201

F.5 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

M Mathematical Methods 205M.1 Scalars, vectors and tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

M.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207M.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208M.1.3 Vector algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216M.1.4 Vector analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

M.2 Analytical mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226M.2.1 Lagranges equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 226M.2.2 Hamiltons equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

M.3 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228


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CONTENTS xiii

Index 229


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LI ST O F F I G U R E S

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 varyingB eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.1 Radiation in the far zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.1 Multipole radiation geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Electric dipole geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.3 Linear antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.4 Electric dipole antenna geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.5 Loop antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.6 Radiation from a moving charge in vacuum. . . . . . . . . . . . . . . . . . . . . . 1096.7 An accelerated charge in vacuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.8 Angular distribution of radiation during bremsstrahlung. . . . . . . . . . . . . . . . 121

6.9 Location of radiation during bremsstrahlung. . . . . . . . . . . . . . . . . . . . . . 1236.10 Radiation from a charge in circular motion. . . . . . . . . . . . . . . . . . . . . . . 1276.11 Synchrotron radiation lobe width. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.12 The perpendicular electric eld of a moving charge. . . . . . . . . . . . . . . . . . 1326.13 Electron-electron scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.1 Relative motion of two inertial systems. . . . . . . . . . . . . . . . . . . . . . . . . 1367.2 Rotation in a 2D Euclidean space. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.3 Minkowski diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.1 Linear one-dimensional mass chain. . . . . . . . . . . . . . . . . . . . . . . . . . . 162

9.1 Vavilov-LCerenkov cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179M.1 Tetrahedron-like volume element of matter. . . . . . . . . . . . . . . . . . . . . . . 212

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PR EFACE T O T H E S EC ON D ED I T ION

This second edition of the book ELECTROMAGNETICFIELD THEORY is a major revision of the rst edition that was published only on the Internet (www.plasma.uu.se/CED/Book ). Thereason for trying to improve the presentation and to add more material was mainly that this newedition is now being made available in printed form by Dover Publications and is to be used in anextended Classical Electrodynamics course at Uppsala University. Hopefully, this means that thebook will nd new uses in Academia and elsewhere.

The changes include a slight reordering of the chapters. First, the book describes the propertiesof electromagnetism when the charges and currents are located in otherwise free space. Only thenthe we go on to show how elds and charges interact with matter. In the authors opinion, thisapproach is preferable as it avoids the formal logical inconsistency of discussing, very early inthe book, such things as the effect of conductors and dielectrics on the elds and charges (andvice versa ), before constitutive relations and physical models for the electromagnetic properties of matter, including conductors and dielectrics, have been derived from rst principles.

In addition to the Maxwell-Lorentz equations and Diracs symmetrised version of these equa-tions (which assume the existence of magnetic monopoles), also the Maxwell-Chern-Simonsequations are introduced in Chapter1. In Chapter2, stronger emphasis is put on the axiomaticfoundation of electrodynamics as provided by the microscopic Maxwell-Lorentz equations. Chap-ter 4 is new and deals with symmetries and conserved quantities in a much more rigourous,profound and detailed way than in the rst edition. For instance, the presentation of the theoryof electromagnetic angular momentum and other observables (constants of motion) has beensubstantially expanded and put on a more rm physical basis. Chapter9 is a complete rewrite thatcombines material that was scattered more or less all over the rst edition. It also contains newmaterial on wave propagation in plasma and other media. When, in Chapter9, the macroscopicMaxwell equations are introduced, the inherent approximations in the derived eld quantities are

clearly pointed out. The collection of formul in AppendixF has been augmented. In AppendixM, the treatment of dyadic products and tensors has been expanded.I want to express my warm gratitude to professor CESAREBARBIERIand his entire group,

particularly FABRIZIOTAMBURINI, at the Department of Astronomy, University of Padova,for stimulating discussions and the generous hospitality bestowed upon me during several shorterand longer visits in2008 and2009 that made it possible to prepare the current major revision of the book. In this breathtakingly beautiful northern Italy, intellectual titan GALILEO GALILEI

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xviii PREFACE TO THE SECOND EDITION

worked for eighteen years and gave birth to modern physics, astronomy and science as we know ittoday, by sweeping away Aristotelian dogmas, misconceptions and mere superstition, thus most

profoundly changing our conception of the world and our place in it. In the process, Galileos newideas transformed society and mankind forever. It is hoped that this book may contribute in somesmall, humble way to further these, once upon a time, mind-bogglingand even dangerousideasof intellectual freedom and enlightment.

Thanks are also due to JOHANSJHOLM, KRISTOFFERPALMER, MARCUSERIKSSON,and JOHANLINDBERGwho during their work on their Diploma theses suggested improvementsand additions and to HOLGERTHEN for carefully checking some lengthy calculations.

This book is dedicated to my son MATTIAS, my daughter KAROLINA, my four grandsonsMAX, ALBIN, FILIPand OSKAR, my high-school physics teacher, STAFFANRSBY, and myfellow members of the CAPELLAPEDAGOGICAUPSALIENSIS.

Padova, Italy BO THIDOctober, 2009 www.physics.irfu.se/ bt


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PR EFACE T O T H E F IR ST ED I T ION

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, together withgravitation, permanently make itself known to all of us in our everyday lives. Electrodynamics isalso by far the most accurate physical theory known, tested on scales running from sub-nuclear togalactic, and electromagnetic eld theory is the prototype of all other eld theories.

This book, ELECTROMAGNETICFIELD THEORY, which tries to give a modern view of classical electrodynamics, is the result of a more than thirty-ve year long love affair. In the autumnof 1972 , I took my rst advanced course in electrodynamics at the Department of TheoreticalPhysics, Uppsala University. Soon I joined the research group there and took on the task of helpingthe late professor PER OLOF FRMAN, who was to become my Ph.D. thesis adviser, with thepreparation of a new version of his lecture notes on the Theory of Electricity. This opened my eyesto the beauty and intricacy of electrodynamics and I simply became intrigued by it. The teachingof a course in Classical Electrodynamics at Uppsala University, some twenty odd years after Iexperienced the rst encounter with the subject, provided the incentive and impetus to write thisbook.

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 eld-theoretical point of view. The rst chapter is, by and large, a description of how Classical Electrodynamics was established by JAMESCLERKMAXWELLas a fundamentaltheory of nature. It does so by introducing electrostatics and magnetostatics and demonstratinghow they can be unied into one theory, classical electrodynamics, summarised in Lorentzsmicroscopic formulation of the Maxwell equations. These equations are used as an axiomaticfoundation for the treatment in the remainder of the book, which includes modern formulation of

the theory; electromagnetic waves and their propagation; electromagnetic potentials and gaugetransformations; analysis of symmetries and conservation laws describing the electromagneticcounterparts of the classical concepts of force, momentum and energy, plus other fundamental prop-erties of the electromagnetic eld; radiation phenomena; and covariant Lagrangian/Hamiltonianeld-theoretical methods for electromagnetic elds, particles and interactions. Emphasis hasbeen put on modern electrodynamics concepts while the mathematical tools used, some of thempresented in an Appendix, are essentially the same kind of vector and tensor analysis methods that

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are used in intermediate level textbooks on electromagnetics but perhaps a bit more advanced andfar-reaching.

The aim has been to write a book that can serve both as an advanced text in ClassicalElectrodynamics and as a preparation for studies in Quantum Electrodynamics and Field Theory,as well 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 anoutgrowth of the lecture notes that the author prepared for the four-credit course Electrodynamicsthat was introduced in the Uppsala University curriculum in1992 , to become the ve-credit courseClassical Electrodynamics in1997 . To some extent, parts of those notes were based on lecturenotes prepared, in Swedish, by my friend and Theoretical Physics colleague BENGTLUNDBORG,who created, developed and taught an earlier, two-credit course called Electromagnetic Radiationat our faculty. Thanks are due not only to Bengt Lundborg for providing the inspiration to writethis book, but also to professor CHRISTERWAHLBERG, and professor GRAN FLDT, bothat the Department of Physics and Astronomy, Uppsala University, for insightful suggestions,to professor JOHN LEARNED, Department of Physics and Astronomy, University of Hawaii,for decisive encouragement at the early stage of this book project, to professor GERARDUSTHOOFT, for recommending this book on his web page How to become agood theoreticalphysicist, and professor CECILIA JARLSKOG, Lund Unversity for pointing out a couple of errors and ambiguities.

I am particularly indebted to the late professor VITALIYLAZAREVICHGINZBURG, for hismany fascinating and very elucidating lectures, comments and historical notes on plasma physics,electromagnetic radiation and cosmic electrodynamics while cruising up and down the Volgaand Oka rivers in Russia at the ship-borne Russian-Swedish summer schools that were organisedjointly by late professor LEV MIKAHILOVICHERUKHIMOVand the author during the1990 s,and for numerous deep discussions over the years.

Helpful comments and suggestions for improvement from former PhD students TOBIACAROZZI, ROGERKARLSSON, and MATTIASWALDENVIK, as well as ANDERSERIKSSONat the Swedish Institute of Space Physics in Uppsala and who have all taught Uppsala students onthe material covered in this book, are gratefully acknowledged. Thanks are also due to the lateHELMUTKOPKA, for more than twenty-ve years a close friend and space physics colleagueworking at the Max-Planck-Institut fr Aeronomie, Lindau, Germany, who not only taught me thepractical aspects of the use of high-power electromagnetic radiation for studying space, but alsosome 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 of thisbook, thereby trying to ensure its quality and scope to make it useful in higher university educationanywhere in the world, it was produced as a World-Wide Web (WWW) project. This turned out tobe a rather successful move. By making an electronic version of the book freely downloadableon the Internet, comments have been received from fellow physicists around the world. To judgefrom WWW hit statistics, it seems that the book serves as a frequently used Internet resource.This way it is hoped that it will be particularly useful for students and researchers working under


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PREFACE TO THE FIRST EDITION xxi

nancial or other circumstances that make it difcult to procure a printed copy of the book. Iwould like to thank all students and Internet users who have downloaded and commented on the

book during its life on the World-Wide Web.Uppsala, Sweden BO THIDDecember, 2008 www.physics.irfu.se/ bt


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E L EC T RO D Y N A MI C S

The classical theory of electromagnetism deals with electric and magnetic eldsand interactions caused by distributions of electric charges and currents. Thispresupposes that the concepts of localised electric charges and currents assumethe validity of certain mathematical limiting processes in which it is consideredpossible for the charge and current distributions to be localised in innitesimallysmall volumes of space. Clearly, this is in contradistinction to electromagnetismon an atomistic scale, where charges and currents have to be described in aquantum formalism. However, the limiting processes used in the classical domain,which, crudely speaking, assume that an elementary charge has a continuousdistribution of charge density, will yield results that agree with experiments onnon-atomistic scales, small or large.

It took the genius of JAMES CLERK MAXWELLto consistently unify thetwo distinct theorieselectricity andmagnetism into a single super-theory,elec-tromagnetism or classical electrodynamics (CED), and to realise that optics is asub-eld of this super-theory. Early in the20th century, HENDRIKANTOONLORENTZtook the electrodynamics theory further to the microscopic scale and

also laid the foundation for the special theory of relativity, formulated in its fullextent by ALBERT EINSTEINin 1905 . In the1930 s PAU L ADRIEN MAU-RICE DIRACexpanded electrodynamics to a more symmetric form, includingmagnetic as well as electric charges. With his relativistic quantum mechanicsand eld quantisation concepts, he also paved the way for the developmentof quantum electrodynamics (QED ) for which RICHARDPHILLIPSFEYN-MAN, JULIANSEYMOURSCHWINGER, and SIN-ITIRO TOMONAGAwere

1


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2 j 1. FOUNDATIONS OF CLAS SICAL ELEC TRODYNAMICS

awarded the Nobel Prize in Physics in1965 . Around the same time, physicistssuch as SHELDON GLASHOW, ABDU SALAM, and STEVE WEINBERG

were able to unify electrodynamics with the weak interaction theory, creatingyet another super-theory,electroweak theory , an achievement which renderedthem the Nobel Prize in Physics1979 . The modern theory of strong interactions,quantum chromodynamics (QCD ), is heavily inuenced by QED.

In this introductory chapter we start with the force interactions in classicalelectrostatics and classical magnetostatics and introduce the static electric andmagnetic elds to nd two uncoupled systems of equations for them. Then wesee how the conservation of electric charge and its relation to electric currentleads to the dynamic connection between electricity and magnetism and how thetwo can be unied into classical electrodynamics. This theory is described by asystem of coupled dynamic eld equationsthe microscopic Maxwell equationsintroduced by Lorentzwhich we take as the axiomatic foundation for the theoryof electromagnetic elds.

At the end of this chapter we present Diracs symmetrised formof the Maxwellequations by introducing (hypothetical) magnetic charges and magnetic currentsinto the theory. While not identied unambiguously in experiments yet, magneticcharges and currents make the theory much more appealing, for instance byallowing for duality transformations in a most natural way. Besides, in practicalwork, such as in antenna engineering, magnetic currents have proved to be a veryuseful concept. We shall make use of these symmetrised equations throughoutthe book. At the very end of this chapter, we present the Maxwell-Chern-Simonsequations that are of considerable interest to modern physics.

1.1 Electrostatics

The theory which describes physical phenomena related to the interaction betweenstationary electric charges or charge distributions in a nite space with stationaryboundaries is calledelectrostatics . For a long time, electrostatics, under thename electricity , was considered an independent physical theory of its own,alongside other physical theories such as Magnetism, Mechanics, Optics, andThermodynamics.1

1 The physicist and philosopherPIERR DUHEM(1861 1916 ) oncewrote:

The whole theory of electrostatics constitutesa group of abstract ideasand general propositions,formulated in the clearand concise language of geometry and algebra, andconnected with one anotherby the rules of strict logic.This whole fully satisesthe reason of a Frenchphysicist and his tastefor clarity, simplicity andorder....

1.1.1 Coulombs lawIt has been found experimentally that in classical electrostatics the interactionbetween stationary, electrically charged bodies can be described in terms of two-body mechanical forces. In the simple case depicted in Figure1.1 on the


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1.1. Electrostatics j3

q0

q

O

x 0

x x 0x

Figure 1.1: Coulombs law de-scribes how a static electric chargeq , located at a pointx relative tothe originO , experiences an elec-trostatic force from a static electricchargeq 0 located atx 0.

facing page, the forceF acting on the electrically charged particle with chargeq

located atx , due to the presence of the chargeq0located atx 0in an otherwiseempty space, is given byCoulombs law .2 This law postulates thatF is directed 2 CHARLES-AUGUSTIN DECOULOMB(1736 1806 ) wasa French physicist who in1775published three reports on theforces between electrically chargedbodies.

along the line connecting the two charges, repulsive for charges of equal signsand attractive for charges of opposite signs, and therefore can be formulatedmathematically as

F . x / Dqq 0

4 " 0

x x 0

jx x 0j3 D

qq 04 " 0

r 1jx x 0jD qq 04 " 0 r 01jx x0j(1.1)

where, in the last step, formula (F.117 ) on page203 was used. InSI units ,

which we shall use throughout, the forceF is measured in Newton (N), theelectric chargesq andq0in Coulomb (C) or Ampere-seconds (As), and the lengthjx x 0jin metres (m). The constant"0 D107 =.4 c 2 / 8:8542 10 12 Faradper metre (Fm1 ) is thevacuum permittivity andc 2:9979 108 m s 1 is thespeed of light in vacuum.3 In CGS units , "0 D1=.4 / and the force is measured3 The notationc for speed comesfrom the Latin word celeritas

which means swiftness. Thisnotation seems to have beenintroduced by WILHELMED-UARD WEBER (1804 1891 ), andRUDOLFKOHLRAUSCH(1809 1858 ) andc is therefore sometimesreferred to asWebers constant .In all his works from1907 andonward, ALBERT EINSTEIN(1879 1955 ) usedc to denote thespeed of light in vacuum.

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

1.1.2 The electrostatic eldInstead 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 eld. Therefore we describe the electrostatic interaction in terms of a staticvectorialelectric eld E stat dened by the limiting process

E statdef

limq! 0Fq

(1.2)

whereF is the electrostatic force, as dened in equation (1.1), from a net electricchargeq0on the test particle with a small electric net electric chargeq. Since


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the purpose of the limiting process is to assure that the test chargeq does notdistort the eld set up byq0, the expression forE stat does not depend explicitly

on q but only on the chargeq0and the relative radius vectorx x 0. This meansthat we can say that any net electric charge produces an electric eld in the spacethat surrounds it, regardless of the existence of a second charge anywhere inthis space.4 However, in order to experimentally detect a charge, a second (test)4 In the preface to the rst edition

of the rst volume of his bookA Treatise on Electricity and Magnetism , rst published in1873 ,James Clerk Maxwell describesthis in the following almost poeticmanner:

For instance, Faraday, inhis minds eye, saw lines of force traversing all spacewhere the mathematicians

saw centres of forceattracting at a distance:Faraday saw a mediumwhere they saw nothingbut distance: Faradaysought the seat of thephenomena in real actionsgoing on in the medium,they were satised that theyhad found it in a powerof action at a distanceimpressed on the electricuids.

charge that senses the presence of the rst one, must be introduced.Using (1.1) and equation (1.2) on the previous page, and formula (F.116 ) on

page203,we nd that the electrostatic eldE stat at theobservation point x (alsoknown as theeld point ), due to a eld-producing electric chargeq0at thesource point x 0, is given by

E stat. x / Dq0

4 " 0

x x 0

jx x0

j3 D

q04 " 0

r

1

jx x 0

jD

q04 " 0

r 0

1

jx x0

j(1.3)In the presence of several eld producing discrete electric chargesq0i , locatedat the pointsx 0i , i D1;2 ;3 ; : : : , respectively, in an otherwise empty space, theassumption of linearity of vacuum5 allows us to superimpose their individual

5 In fact, vacuum exhibits aquantum mechanical non-linearity due tovacuum polarisationeffects manifesting themselvesin the momentary creation andannihilation of electron-positronpairs, but classically this non-linearity is negligible.

electrostatic elds into a total electrostatic eld

E stat. x / D1

4 " 0 Xi q0i x x 0ix x 0i3(1.4)

If the discrete electric charges are small and numerous enough, we can, in acontinuum limit, assume that the total chargeq0from an extended volume to bebuilt up by local innitesimal elemental charges dq0, each producing an elementalelectric eld

dE stat. x / Ddq0

4 " 0

x x 0

jx x 0j3 (1.5)

By introducing theelectric charge density , measured in Cm3 in SI units, atthe pointx 0within the volume element d3x0Ddx01 dx02 dx03 (measured in m3 ),the elemental charge can be expressed as dq0Dd3x0 .x 0/ , and the elementalelectrostatic eld as

dE stat. x / Dd3x0 .x 0/

4 " 0

x x0

jx x0j3 (1.6)

Integrating this over the entire source volumeV 0, we obtain

E stat. x / D1

4 " 0 Z V 0d3x0 .x0/ x x 0jx x 0j3D

14 " 0 Z V 0d3x0 .x0/ r 1jx x 0j

D 1

4 " 0r Z V 0d3x0 .x 0/jx x 0j

(1.7)


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1.1. Electrostatics j5

V 0

q0i

q

O

x0i

x x 0i

x

Figure1.2: Coulombs law for a dis-tribution of individual chargesq 0i lo-calised within a volumeV 0 of lim-ited extent.

where we used formula (F.116 ) on page203 and the fact that .x0/ does notdepend on the unprimed (eld point) coordinates on whichr operates.

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 caseis expressed in terms of Dirac deltadistributions:6 6 Since, by denition, the integral

Z V 0d3x 0 . x 0 x 0i /Z V 0d3x 0 .x 0 x 0i /

.y 0 y 0i /.z0 z 0i / D1

is dimensionless, andx has thedimension m, the 3D Dirac deltadistribution. x 0 x 0i / must havethe dimension m3 .

.x 0/ DXi q0i . x0 x0i / (1.8)as illustrated in Figure1.2. Inserting this expression into expression(1.7) on the

facing page we recover expression (1.4) on the preceding page.According toHelmholtzs theorem , a sufciently well-behaved vector eld iscompletely determined by its divergence and curl.7 Taking the divergence of the

7 HERMANNLUDWIGFER-DINAND VONHELMHOLTZ(1821 1894 ) was a physicist,physician and philosopher whocontributed to wide areas of science, ranging from electrody-namics to ophtalmology.

generalE stat expression for an arbitrary electric charge distribution, equation (1.7)on the facing page, and using the representation of the Dirac delta distributiongiven by formula(F.119 ) on page203,one nds that

r E stat. x / Dr 1

4 " 0 Z V 0d3x0 .x0/ x x 0jx x 0j3D

14 " 0

Z V 0

d3x0 .x 0/ r r 1jx x 0

j

D 14 " 0 Z V 0d3x0 .x 0/ r 21jx x0jD

1"0 Z V 0d3x0 .x0/ . x x0/ D .x /"0

(1.9)

which is the differential form of Gausss law of electrostatics .Since, according to formula(F.105 ) on page203, r r . x / 0 for any


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R 3 scalar eld. x / , we immediately nd that in electrostatics

r E

stat

. x / D 1

4 " 0r

r

Z V 0d

3

x0.x 0/

jx x 0jD0 (1.10)i.e. , thatE stat is anirrotational eld.To summarise, electrostatics can be described in terms of two vector partial

differential equations

r E stat. x / D.x /"0

(1.11a)

r E stat. x / D0 (1.11b)representing four scalar partial differential equations.

1.2 MagnetostaticsWhile electrostatics deals with static electric charges,magnetostatics deals withstationary electric currents,i.e. , electric charges moving with constant speeds,and the interaction between these currents. Here we shall discuss this theory insome detail.

1.2.1 Ampres law

Experiments on the interaction between two small loops of electric current haveshown that they interact via a mechanical force, much the same way that electriccharges interact. In Figure1.3 on the next page, letF denote such a force actingon a small loopC , with tangential line element dl , located atx and carryinga currentI in the direction of dl , due to the presence of a small loopC 0, withtangential line element dl0, located atx 0and carrying a currentI 0in the directionof dl0in otherwise empty space. According toAmpres law this force is givenby the expression88 ANDR-MARIE AMPRE

(1775 1836 ) was a French mathe-matician and physicist who, only afew days after he learned about thendings by the Danish physicist

and chemist HANS CHRISTIANRSTED(1777 1851 ) regardingthe magnetic effects of electriccurrents, presented a paper to theAcadmie des Sciences in Paris,describing the law that now bearshis name.

F . x / D0 II 04 I C dl I C 0dl0 x x 0jx x 0j3

D 0 II 04 I C dl I C 0dl0 r 1jx x 0j(1.12)

In SI units, 0 D4 10 7 1:2566 10 6 H m 1 is thevacuum permeability .From the denition of "0 and 0 (in SI units) we observe that

"0 0 D107

4 c 2(F m 1 ) 4 10 7 (H m 1 ) D

1c2

(s2 m 2 ) (1.13)


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1.2. Magnetostatics j7

C 0

C

dj 0dj

O

x 0

x x0

x

Figure1.3: Ampres law describeshow a small loopC , carrying astatic electric line current density el-ement dj atx , experiences a magne-tostatic force from a small loopC 0,carrying a static electric current den-sity element dj 0 located atx 0. Theloops can have arbitrary shapes aslong as they are simple and closed.

which is a most useful relation.At rst glance, equation (1.12) on the facing page may appear unsymmetric in

terms of the loops and therefore be a force law that does not obey Newtons thirdlaw. However, by applying the vector triple product bac-cab formula(F.72) onpage201,we can rewrite(1.12) as

F . x / D 0 II 04 I C 0 dl0I C dl r 1jx x0j0 II 04 I C I C 0 x x 0jx x 0j3 dl dl0

(1.14)

Since the integrand in the rst integral is an exact differential, this integral

vanishes and we can rewrite the force expression, formula (1.12) on the facingpage, in the following symmetric way

F . x / D 0 II 04 I C I C 0 x x0jx x 0j3 dl dl0 (1.15)

which clearly exhibits the expected interchange symmetry between loopsC andC 0.

1.2.2 The magnetostatic eldIn analogy with the electrostatic case, we may attribute the magnetostatic in-teraction to a static vectorialmagnetic eld Bstat. The elementalBstat from theelemental current element dI 0DI 0dl0is dened as

dBstat. x / def 0

4dI 0 x x 0

jx x0j3 D

0 I 04

dl0 x x 0

jx x0j3 (1.16)

which expresses the small element dBstat. x / of the static magnetic eld set up atthe eld pointx by a small line current element dI 0DI 0dl0of stationary current


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I 0directed along the line element dl0at the source pointx0. The SI unit for themagnetic eld, sometimes called themagnetic ux density ormagnetic induction ,

is Tesla (T).If we generalise expression (1.16) on the previous page to an integrated steadystateelectric current density j . x / , measured in Am2 in SI units, we can writeI 0dl DdI 0Dd3x0j 0. x 0/ , and we obtainBiot-Savarts law

Bstat. x / D0

4 Z V 0d3x0j . x 0/ x x 0jx x0j3D

0

4 Z V 0d3x0j . x 0/ r 1jx x0jD

0

4r Z V 0d3x0 j . x0/jx x 0j

(1.17)

where we used formula (F.116 ) on page203,formula (F.95) on page202, andthe fact thatj . x 0/ does not depend on the unprimed coordinates on whichr operates. Comparing equation (1.7) on page4 with equation(1.17) above, wesee that there exists a close analogy between the expressions forE stat andBstatbut that they differ in their vectorial characteristics. With this denition of Bstat,equation(1.12) on page6 may we written

F . x / DI I C dl Bstat. x / (1.18)In order to assess the properties of Bstat, we determine its divergence and curl.

Taking the divergence of both sides of equation(1.17) above and utilising formula(F.104 ) on page203,we obtain

r Bstat. x / D0

4r r Z V 0d3x0 j . x0/jx x 0jD0 (1.19)

since, according to formula(F.104 ) on page203, r . r a / vanishes for anyvector elda . x / .

Applying the operator bac-cab rule, formula (F.101 ) on page203,the curl of equation(1.17) above can be written

r Bstat. x / D0

4r

r

Z V 0

d3x0 j . x 0/

jx x 0

jD 04 Z V 0d3x0j . x 0/ r 21jx x0jC

0

4 Z V 0d3x0j . x0/ r 0r 01jx x 0j(1.20)

In the rst of the two integrals on the right-hand side, we use the representationof the Dirac delta function given in formula (F.119 ) on page203,and integrate


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1.3. Electrodynamics j9

the second integral by parts, by utilising formula (F.92) on page202as follows:

Z V 0d

3

x0j . x0/ r 0r 01

jx x0jD Ox k Z V 0d3x0r 0 j . x 0/@@x0k 1jx x 0jZ V 0d3x0 r 0 j . x 0/ r 01jx x 0j

DOx k Z S 0d2x0On 0 j . x0/ @@x0k 1jx x0jZ V 0d3x0 r 0 j . x 0/ r 01jx x 0j

(1.21)

We note that the rst integral in the result, obtained by applying Gausss theorem,vanishes when integrated over a large sphere far away from the localised sourcej . x0/ , and that the second integral vanishes becauser j D0 for stationarycurrents (no charge accumulation in space). The net result is simply

r Bstat. x / D 0 Z V 0d3x0j . x0/. x x0/ D 0 j . x / (1.22)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 vector partialdifferential equations, namely theequations of classical electrostatics

r E stat. x / D.x /"0

(1.23a)

r E stat. x / D0 (1.23b)and theequations of classical magnetostatics

r Bstat. x / D0 (1.24a)r Bstat. x /

D0 j . x / (1.24b)

Since there is nothinga priori which connectsE stat directly withBstat, we mustconsider classical electrostatics and classical magnetostatics as two separate andmutually independent physical theories.

However, when we include time-dependence, these theories are unied into asingle super-theory,classical electrodynamics . This unication of the theories of electricity and magnetism can be inferred from two empirically established facts:


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10j 1. FOUNDATIONS OF CLAS SICAL ELEC TRODYNAMICS

1. Electric charge is a conserved quantity and electric current is a transport of electric charge. This fact manifests itself in the equation of continuity and, as

a consequence, ini, as we shall see,Maxwells displacement current .

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

1.3.1 The indestructibility of electric chargeLet j .t ; x / denote the time-dependent electric current density. In the simplestcase it can be dened asj

Dv wherev is the velocity of the electric charge

density .99 A more accurate model is toassume that the individual chargeelements obey some distributionfunction that describes their localvariation of velocity in space andtime.

The electric charge conservation law can be formulated in theequation of continuity for electric charge

@ .t;x /@t Cr j .t ; x / D0 (1.25)

which states that the time rate of change of electric charge.t; x / is balanced bya divergence in the electric current densityj .t ; x / .

1.3.2 Maxwells displacement currentWe recall from the derivation of equation(1.22) on the preceding page that therewe used the fact that in magnetostaticsr j . x / D0. In the case of non-stationarysources and elds, we must, in accordance with the continuity equation (1.25)above, setr j .t; x / D @ .t;x /=@t. Doing so, and formally repeating thesteps in the derivation of equation (1.22) on the preceding page, we would obtainthe formal result

r B .t ; x / D 0

Z V 0

d3x0j .t ; x 0/. x x 0/

C0

4@@tZ V 0d3x0 .t; x 0/ r 01jx x 0j

D 0 j .t ; x / C 0@@t

"0 E .t ; x /

(1.26)

where, in the last step, we have assumed that a generalisation of equation(1.7)on page4 to time-varying elds allows us to make the identication1010 Later, we will need to consider

this generalisation and formalidentication further.


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14 " 0

@@tZ V 0d3x0 .t; x0/ r 01jx x 0jD @@t14 " 0 Z V 0d

3x0 .t; x 0/ r 1jx x0jD @@t14 " 0 r Z V 0d3x0 .t; x0/jx x 0j D @@tE .t ; x /(1.27)

The result is Maxwells source equation for theB eld

r B .t ; x / D 0 j .t ; x / C @@t"0 E .t ; x /D 0 j .t ; x / C 0 "0 @@tE .t; x /(1.28)

where "0 @E .t ; x /=@t is the famousdisplacement current . This, at the time,unobserved current was introduced, in a stroke of genius, by Maxwell in orderto make also the right-hand side of this equation divergence free whenj .t ; x / isassumed to represent the density of the total electric current, which can be splitup in ordinary conduction currents, polarisation currents and magnetisationcurrents. This will be discussed in Subsection9.1 on page172.The displacementcurrent behaves like a current density owing in free space. As we shall seelater, its existence has far-reaching physical consequences as it predicts that suchphysical observables as electromagnetic energy, linear momentum, and angularmomentum can be transmitted over very long distances, even through emptyspace.

1.3.3 Electromotive forceIf an electric eldE .t; x / is applied to a conducting medium, a current densityj .t ; x / will be produced in this medium. But also mechanical, hydrodynami-cal and chemical processes can create electric currents. Under certain physicalconditions, and for certain materials, one can sometimes assume that a linear rela-tionship exists between the electric current densityj andE . This approximationis calledOhms law :11 11 In semiconductors this approxi-

mation is in general not applicableat all.j .t ; x / D E .t ; x / (1.29)

where is the electric conductivity (S m1

). In the case of an anisotropicconductor, is a tensor.We can view Ohms law, equation(1.29) above, as the rst term in a Taylor

expansion of the lawj E .t ; x / . This general law incorporatesnon-linear effectssuch asfrequency mixing . Examples of media which are highly non-linear aresemiconductors and plasma. We draw the attention to the fact that even in caseswhen the linear relation betweenE andj is a good approximation, we still have


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to use Ohms law with care. The conductivity is, in general, time-dependent(temporal dispersive media ) but then it is often the case that equation (1.29) on

the preceding page is valid for each individual Fourier (spectral) component of the eld.If the current is caused by an applied electric eldE .t; x / , this electric eld

will 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 isexpended isj E per unit volume (Wm3 ). If E is irrotational (conservative),jwill decay away with time. Stationary currents therefore require that an electriceld due to anelectromotive force (EMF ) is present. In the presence of such aeldE EMF, Ohms law, equation(1.29) on the previous page, takes the form

j D .E statCE EMF/ (1.30)The electromotive force is dened as

E DI C dl . E statCE EMF/ (1.31)where dl is a tangential line element of the closed loopC .1212 The term electromagnetic force

is something of a misnomer sinceE represents a voltage,i.e. , its SIdimension is V.

1.3.4 Faradays law of inductionIn Subsection1.1.2 we derived the differential equations for the electrostatic eld.Specically, on page6 we derived equation (1.10) stating thatr E stat

D0 and

thus thatE stat is aconservative eld (it can be expressed as a gradient of a scalareld). This implies that the closed line integral of E stat in equation (1.31) abovevanishes and that this equation becomes

E DI C dl E EMF (1.32)It has been established experimentally that a non-conservative EMF eld is

produced in a closed circuitC at rest if the magnetic ux through this circuitvaries with time. This is formulated inFaradays law which, in Maxwellsgeneralised form, reads

E .t / DI C dl E .t; x / D ddt m.t /D

ddtZ S d2x On B .t ; x / D Z S d2x On @@tB .t ; x / (1.33)

where m is themagnetic ux and S is the surface encircled byC which canbe interpreted as a generic stationary loop and not necessarily as a conducting


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d2x

On

B . x / B . x /

v

dlC

Figure1.4: A loopC which moveswith velocityv in a spatially vary-ing magnetic eldB . x / will sensea varying magnetic ux during themotion.

circuit. Application of Stokes theorem on this integral equation, transforms itinto the differential equation

r E .t ; x / D @@t

B .t ; x / (1.34)

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

Any change of the magnetic ux m will induce an EMF. Let us thereforeconsider the case, illustrated in Figure1.4, when the loop is moved in such away that it links a magnetic eld which varies during the movement. The totaltime derivative is evaluated according to the well-known operator formula

ddt D

@@tC

dxdt

r (1.35)

which follows immediately from the multivariate chain rule for the differentiationof an arbitrary differentiable functionf.t; x .t// . Here, dx =dt describes a chosenpath in space. We shall chose the ow path which means that dx =dt Dv and

ddt D

@@tCv r (1.36)

where, in a continuum picture,v is the uid velocity. For this particular choice,the convective derivative dx =dt is usually referred to as thematerial derivative and is denoted Dx =Dt .

Applying the rule(1.36) to Faradays law, equation(1.33) on the precedingpage, we obtain

E .t / D ddtZ S d2x On B D Z S d2x On @B@t Z S d2x On . v r /B (1.37)


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Furthermore, taking the divergence of equation(1.34) on the preceding page, wesee that

r @@t

B .t ; x / D @@tr B .t ; x / D r r E .t ; x / D0 (1.38)where in the last step formula (F.104 ) on page203was used. Since this is true

8t , we conclude thatr B .t ; x / D0 (1.39)

also for time-varying elds; this is in fact one of the Maxwell equations. Usingthis result and formula(F.96) on page202,we nd that

r . B v / D. v r /B (1.40)since, during spatial differentiation,v is to be considered as constant, This allowsus to rewrite equation (1.37) on the previous page in the following way:

E .t / DI C dl E EMFD ddtZ S d2x On BD Z S d2x On @B@t Z S d2x On r . B v / (1.41)

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

E .t / DI C dl E EMFD Z S d2x On @B@t I C dl . B v / (1.42)or, rearranging the terms,

I C

dl . E EMF v B / D

Z S

d2x On@B@t

(1.43)

whereE EMF is the eld which is induced in the loop,i.e. , in the moving system.The application of Stokes theorem in reverse on equation (1.43) yields

r . E EMF v B/ D @B@t

(1.44)

An observer in a xed frame of reference measures the electric eld

E DE EMF v B (1.45)and a observer in the moving frame of reference measures the followingLorentz force on a chargeq

F DqE EMFDqE Cq. v B / (1.46)corresponding to an effective electric eld in the loop (moving observer)

E EMFDE Cv B (1.47)Hence, we conclude that for a stationary observer, the Maxwell equation

r E D @B@t

(1.48)

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 formulatethe equations of classical electrodynamics valid for arbitrary variations in timeand space of the coupled electric and magnetic eldsE .t ; x / andB .t; x / . Theequations are, inSI units ,13 13 In CGS units the microscopic

Maxwell equations are

r E D4r E D

1c

@B@t

r B D0r B D

4c

j C1c

@E@t

in Heaviside-Lorentz units (one of severalnatural units )

r E Dr E D

1c

@B@t

r B D0r B D

1c

j C1c

@E@t

and inPlanck units (another set of natural units)

r E D4r E D

@B@t

r B

D0

r B D4 j C@E@t

r E D"0 (1.49a)r E D

@B@t

(1.49b)

r B D0 (1.49c)r B

D 0j

C"

0 0

@E

@t D 0j

C1

c2@E

@t(1.49d)

In these equationsD .t; x / represents the total, possibly both time and spacedependent, electric charge density, with contributions from free as well as induced(polarisation) charges. Likewise,j D j .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 wererst formulated by Lorentz and therefore another name often used for themis theMaxwell-Lorentz equations . Together with the appropriateconstitutive relations , which relate andj to theelds, and the initial andboundary conditionspertinent to the physical situation at hand, they form a system of well-posed partialdifferential equations which completely determineE andB.

1.3.6 Diracs symmetrised Maxwell equations

If we look more closely at the microscopic Maxwell equations (1.49), we see that

they exhibit a certain, albeit not complete, symmetry. Let us follow Dirac andmake thead hoc assumption that there existmagnetic monopoles represented bya magnetic charge density , which we denote bym D m.t; x / , and amagnetic current density , which we denote byj m Dj m.t ; x / .14 With these new hypotheti-

14 Nobel Physics Laureate JULIANSEYMOURSCHWINGER(1918 1994 ) once put it:

...there are strongtheoretical reasons tobelieve that magneticcharge exists in nature,and may have playedan important role inthe development of theUniverse. Searchesfor magnetic chargecontinue at the presenttime, emphasising thatelectromagnetism is veryfar from being a closedobject.

cal physical entities included in the theory, and with the electric charge densitydenoted e and the electric current density denotedj e, the Maxwell equationswill be symmetrised into the following two scalar and two vector, coupled, partial


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differential equations (SI units):

r E De

"0 (1.50a)r E D

@B@t

0 j m (1.50b)

r B D 0 m (1.50c)r B D 0 j e C"0 0

@E@t

(1.50d)

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

Taking the divergence of (1.50b), we nd that

r . r E / D @@t. r B / 0 r j m 0 (1.51)where we used the fact that, according to formula (F.104 ) on page203, thedivergence of a curl always vanishes. Using(1.50c) to rewrite this relation, weobtain theequation of continuity for magnetic charge

@ m

@t Cr jm D0 (1.52)

which has the same form as that for the electric charges (electric monopoles) andcurrents, equation(1.25) on page10.

1.3.7 Maxwell-Chern-Simons equationsDevelopments in condensed matter theory have motivated the study of an exten-sion of classical electrodynamics such that the microscopic Maxwell-Lorentzequations (1.49) on the previous page are replaced by theMaxwell-Chern-Simonsequations (MCS equations) which, in SI units, read

r E D.t; x /"0 Ca B (1.53a)

r E D @B@t

(1.53b)

r B D0 (1.53c)r B D 0 j .t ; x / C"0 0

@E@t

B a E (1.53d)

The MCS equations have been studied inCarroll-Field-Jackiw electrodynamicsand are used,e.g. , in the theory of axions .


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1.4. Bibliography j17

1.4 Bibliography

[1] T. W. BARRETT ANDD. M. GRIMES, Advanced Electromagnetism. Foundations,Theory and Applications , World Scientic 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, Heidel-berg, 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-entic 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 ANDE. M. LIFSHITZ, The Classical Theory of Fields , fourth revisedEnglish ed., vol. 2 of Course 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, Cambridge .. . , 1991, ISBN 0-521-41025-8.

[12] W. K. H. PANOFSKY ANDM. PHILLIPS, Classical Electricity and Magnetism ,second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962,ISBN 0-201-05702-6.

[13] F. ROHRLICH, Classical Charged Particles , third ed., World Scientic Publishing Co.Pte. Ltd., New Jersey, London, Singapore, . . . , 2007, ISBN 0-201-48300-9.

[14] J. A. STRATTON, Electromagnetic Theory , McGraw-Hill Book Company, Inc., New

York, NY and London, 1953, ISBN 07-062150-0.[15] J. VANDERLINDE, Classical Electromagnetic Theory , John Wiley & Sons, Inc., New

York, Chichester, Brisbane, Toronto, and Singapore, 1993, ISBN 0-471-57269-1.


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FTE L EC T ROM AG N E T I C F I E L D S A N D

W A V E S

As a rst step in the study of the dynamical properties of the classical electro-magnetic eld, we shall in this chapter, as alternatives to the rst-order Maxwell-Lorentz equations, derive a set of second-order differential equations for the eldsE andB. It turns out that these second-order equations are wave equations forEandB, indicating that electromagnetic wave modes are very natural and common

manifestations of classical electrodynamics.But before deriving these alternatives to the Maxwell-Lorentz equations, we

shall discuss the mathematical techniques of how to make use of complex vari-ables to represent real-valued physical observables in order to simplify the math-ematical work involved. We will also describe how to make use of the singlespectral component (Fourier component) technique, which simplies the algebra,at the same time as it claries the physical content.

2.1 Axiomatic classical electrodynamicsIn Chapter1 we described the historical route which led to the formulation of themicroscopic Maxwell equations. From now on we shall consider these equationsas postulates , i.e. , as theaxiomatic foundation of classical electrodynamics . Assuch, they describe, in scalar and vector-differential-equation form, the behaviourin timet 2R and in spacex 2R 3 of theelectric and magnetic eldsE .t ; x / 2R 3

19


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20j 2. ELE CTROMAGNETIC FIEL DS AND WAVES

andB .t ; x / 2R 3 , respectively [cf. equations (1.49) on page15]:r E

D"0(Gausss law) (2.1a)

r E D @B@t

(Faradays law) (2.1b)

r B D0 (No magnetic charges) (2.1c)r B D 0 j C"0 0

@E@t

(Maxwells law) (2.1d)

We reiterate that in these equations.t; x / and j .t ; x / are thetotal charge andcurrent densities, respectively. Hence, they are considered microscopic in thesense thatall charges and currents, including the intrinsic ones in matter, suchas bound charges in atoms and molecules, as well as magnetisation currents, areincluded (but macroscopic in the sense that quantisation effects are neglected).These charge and current densities may have prescribed arbitrary time and spacedependencies and be considered the sources of the elds, but they may also begenerated by the elds. Despite these dual rles, we shall refer to them as thesource terms of the microscopic Maxwell equations and the two equations wherethey appear as theMaxwell-Lorentz source equations .

2.2 Complex notation and physical observables

In order to simplify the mathematical treatment, we shall frequently allow themathematical variables representing the elds, the charge and current densities,and other physical quantities to be analytically continued into the complex do-main. However, when we use such acomplex notation we must be very carefulhow to interpret the results derived within this notation. This is because everyphysical observable is, by denition, real-valued.1 Consequently, the mathemati-

1 A physical observable is some-thing that can, at least in principle,be ultimately reduced to an inputto the human sensory system. Inother words, physical observablesquantify (our perception of) thephysical reality and as such theymust, of course, be described byreal-valued quantities.

cal expression for the observable under consideration must also be real-valued tobe physically meaningful.

If a physical scalar variable, or a component of a physical vector or tensor,is represented mathematically by the complex-valued number, i.e. , 2 C ,then in classical electrodynamics (in fact, in classical physics as a whole), one

makes the identicationobservable

DRemathematical

. Therefore, it is alwaysunderstood that one must take the real part of a complex mathematical variable inorder for it to represent a classical physical observable.22 This is at variance withquantum physics, where

observableDmathematical

.Letting denote complex conju-gation, the real part can be writtenRef g D12 . C / , i.e. , asthe arithmetic mean of and itscomplex conjugate . Similarly,the magnitude can be written

j j D. / 1=2 , i.e. , as thegeometric mean of and .

For mathematical convenience and ease of calculation, we shall in the follow-ing regularly useand tacitly assumecomplex notation, stating explicitly whenwe do not. One convenient property of the complex notation is that differentia-tions often become trivial to perform. However, care must always be exercised.


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2.2. Complex notation and physical observables j21

A typical situation being when products of two or more quantities are calculatedsince, for instance, for two complex-valued variables1 and 2 we know that

Ref 1 2g Ref 1gRef 2g. On the other hand,. 1 2 / D 1 2 .2.2.1 Physical observables and averagesAs just mentioned, it is important to be aware of the limitations of complexrepresentation of real observables when one has to calculate products of physicalquantities. Let us, for example, consider two physical vector eldsa .t ; x / andb .t; x / that are represented by their Fourier components,a 0 . x / exp. i!t / andb 0 . x / exp. i!t / , i.e. , by vectors in (a domain of) 3D complex spaceC 3 . Further-more, let be a binary operator for these vectors, representing either the scalarproduct operator (), the vector product operator (), the dyadic product operator(juxtaposition), or the outer tensor operator ( ). Then we make the interpretation

a .t ; x / b .t ; x / DRefag RefbgDRea 0 . x / e i!t Reb0 . x / e i!t (2.2)We can express the real part of the complex vectora as

Refag DRea 0 . x / e i!t D12 a 0 . x / e i!t Ca 0 . x / ei!t (2.3)and similarly forb . Hence, the physically acceptable interpretation of the scalarproduct of two complex vectors, representing classical physical observables, is

a .t ; x / b .t ; x / DRe

a 0 . x / e i!t

Re

b 0 . x / e i!t

D12 a 0 . x / e

i!t

Ca 0 . x / ei!t

12 b 0 . x / e

i!t

Cb 0 . x / ei!t

D14

a 0 b 0 Ca 0 b 0 Ca 0 b 0 e 2 i!t Ca 0 b 0 e2 i!t

D12

Refa 0 b 0 gC12

Rea 0 b0 e 2 i!t D

12

Rea 0 e i!t b 0 ei!t C12 Rea 0 e i!t b 0 e i!t D

12

Refa .t ; x / b .t ; x /gC12

Refa .t ; x / b .t ; x /g(2.4)

In physics, we are often forced to measure thetemporal average (cycle average )

of a physical observable. We use the notationh it , for such an average. If so,we see that the average of the product of the two physical quantities representedby a andb can be expressed as

ha bit hRefag Refbgit D12

Refa b gD12

Refa bgD

12

Refa 0 b 0 gD12

Refa 0 b0g(2.5)


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since the oscillating functionexp. 2i!t / in equation (2.4) on the previous pagevanishes when averaged in time over a complete period2 =! (or over innitely

many periods), and, therefore,ha .t ; x / b .t ; x /it gives no contribution.

2.2.2 Maxwell equations in Majorana representationIt is sometimes convenient to introduce thecomplex-eld six-vector , also knownas theRiemann-Silberstein vector

G .t ; x / DE .t ; x / CicB .t ; x / (2.6)whereG 2C 3 even if E ; B 2R 3 . This complex representation of the electro-magnetic elds is known as theMajorana representation .33 It so happens that ETTORE

MAJORANA(1906 -1938 ) used thedenitionG DE ic B , but thisis no essential difference from thedenition(2.6). One may say thatMajorana used the other branch of p 1 as the imaginary unit.

Inserting the Riemann-Silberstein vector into the Maxwell equations (2.1) onpage20, they transform into

r G .t ; x / D"0 (2.7a)r G .t; x / Dic 0 j C

1c2

@G@t (2.7b)

The length of the Riemann-Silberstein vector forE ; B 2R 3EXAMPLE2 .1One fundamental property of C 3 , to whichG belongs, is that inner (scalar) products inthis space are invariant under rotations just as they are inR 3 . However, as discussed inExampleM.3 on page216,the inner (scalar) product inC 3 can be dened in two differentways. Considering the special case of the scalar product of G with itself, assuming thatE 2R 3 andB 2R 3 , we have the following two possibilities of dening (the square of) thelength of G :1. The inner (scalar) product dened asG scalar multiplied with itself

G G D. E CicB/ . E CicB/ DE 2 c2 B 2 C2icE B (2.8)Since length is an scalar quantity which is invariant under rotations, we nd that

E 2 c2 B 2 DConst (2.9a)E B DConst (2.9b)

2. The inner (scalar) product dened asG scalar multiplied with the complex conjugate of itself

G G D. E CicB/ . E icB/ DE 2 Cc2 B 2 (2.10)which is also an invariant scalar quantity. As we shall see in Chapter4, this quantity isproportional to theelectromagnetic eld energy density .

3. As with any vector, the cross product of G with itself vanishes:


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2.3. The wave equations forE andB j23

G G D. E CicB/ . E CicB/DE E c2 B B Cic. E B/ Cic. B E /

D0 C0 Cic. E B/ ic. E B/ D0(2.11 )

4. The cross product of G with the complex conjugate of itself does, however, not vanish:

G G D. E CicB/ . E icB/DE E Cc2 B B ic. E B/ Cic. B E /D0 C0 ic. E B/ ic. E B/ D 2ic. E B/

(2.12)

is proportional to the electromagnetic power ux, to be introduced later.

End of example2.1

2.3 The wave equations forE andBThe Maxwell-Lorentz equations(2.1) on page20are four rst-order differentialequations that are coupled (bothE and B appear in the same equations). Twoof the equations are scalar [equation (2.1a) and equation (2.1c)], and two are in3D Euclidean vector form [equation (2.1b) and equation (2.1d)], representingthree scalar equations each. Hence, the Maxwell equations represent in fact eightscalar rst-order partial differential equations (1 C1 C3 C3). However, it is wellknown from the theory of differential equations that a set of rst-order, coupledpartial differential equations can be transformed into a smaller set of second order

partial differential equations, which sometimes become decoupled in the process.It turns out that in our case we will obtain one second-order differential equationfor E and one second-order differential equation forB. These second-orderpartial differential equations are, as we shall see,wave equations , and we shalldiscuss the implications of them. For certain media, theB wave eld can beeasily obtained from the solution of theE wave equation but in general this is notthe case.

To bring the rst-order differential equations(2.1) on page20into second orderone needs, of course, to operate on them with rst-order differential operators.If we apply the curl vector operator (r ) to both sides of the two Maxwellvector equations, equation (2.1b) and equation (2.1d), assuming that our physicalquantities vary in such a regular way that temporal and spatial differentiationcommute, we obtain the second order differential equations

r . r E / D @@t

. r B/ (2.13a)

r . r B / D 0 r j C"0 0@@t

. r E / (2.13b)


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As they stand, these second-order partial differential equations still appear to becoupled. However, by using the Maxwell equations once again we can formally

decouple them intor . r E / D 0

@j@t

"0 0@2 E@t2

(2.14a)

r . r B/ D 0 r j "0 0@2 B@t2

(2.14b)

Using to the operator triple product bac-cab rule equation(F.101 ) on page203,which gives

r . r E / Dr . r E / r 2 E (2.15)when applied toE and similarly toB, Gausss law equation (2.1a) on page20,and then rearranging the terms, recalling that"0 0

D1=c2 wherec is the speed

of light in free (empty) space, we obtain the two inhomogeneousvector wave equations

2 E D1c2

@2 E@t2 r

2 E D r

"00

@j@t

(2.16a)

2 B D1c2

@2 B@t2 r

2 B D 0 r j (2.16b)where 2 is thedAlembert operator , dened according to formula(M.97) onpage223.

These are the general wave equations for the electromagnetic elds, generatedin regions where there exist sources.t; x / and j .t ; x / of any kind. Simpleeveryday examples of such regions are electric conductors (e.g. , radio and TVantennas) or plasma (e.g. , the Sun and its surrounding corona). In principle, thesources and j can still cause the wave equations to be coupled, but in manyimportant situations this is not the case.44 Clearly, if the current density

in the RHS of equation(2.16b) isa function of E , as is the case if Ohms lawj D E is applicable,the coupling is not removed.

We notice that outside the source region,i.e. , in free space where D0and j D0, the inhomogeneous wave equations (2.16) above simplify to thewell-known uncoupled, homogeneous wave equations

1c2

@2 E@t2 r

2 E D0 (2.17a)1

c2

@2 B

@t2

r 2 B

D0 (2.17b)

These equations describe how the elds that were generated in the source region,e.g. , a transmitting antenna, propagate asvector waves through free space. Oncethese waves impinge upon another region which can sustain charges and/orcurrents,e.g. , a receiving antenna or other electromagnetic sensor, the eldsinteract with the charges and the currents in this second region in accordance withequations(2.16).


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2.3. The wave equations forE andB j25

Wave polarisation EXAMPLE2 .2Since electromagnetic waves are vector waves they may exhibitwave polarisation . Let us

consider a singleplane wave that propagates in free space.5 This means that the electric5 A single plane wave is a math-ematical idealisation. In reality,waves appear aswave packets , i.e. ,superpositions of a (possibly in-nite) number of plane waves.E.g. .,a radio beam from a transmittingantenna is a superposition (Fouriersum or integral) of many planewaves with slightly different anglesrelative to a xed, given axis or aplane.

and magnetic eld vectors are orthogonal to the propagation direction and are located in atwo-dimensional plane. Let us choose this to be thex1 x2 plane and thepropagation vector (wave vector) to be along thex3 axis. A generic temporal Fourier mode of the electric eldvectorE with (angular) frequency! is given by the real-valued expression

E .t; x / DE 1 cos.!t kx 3 C 1 / Ox 1 CE 2 cos.!t kx 3 C 2 / Ox 2 (2.18)In complex notation we can write

electromagnetic filed theory

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