electromagnetic field theory llys thuyết trường điện từ

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B o Thidé

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ELECTROMAGNETIC
FIELD THEORY

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Second Edition


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ELECTROMAGNETIC FIELD THEORY

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Second Edition


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ELECTROMAGNETIC
FIELD THEORY

Bo Thidé
Swedish Institute of Space Physics
Uppsala, Sweden
and

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Second Edition

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Department of Physics and Astronomy
Uppsala University, Sweden


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Also available

ELECTROMAGNETIC FIELD THEORY
EXERCISES
by

Freely downloadable from
www.plasma.uu.se/CED

FT

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

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This book was typeset in LATEX 2" based on TEX 3.141592 and Web2C 7.5.6

Copyright ©1997–2009 by
Bo Thidé
Uppsala, Sweden
All rights reserved.

D

Electromagnetic Field Theory
ISBN 978-0-486-4773-2

The cover graphics illustrates the linear momentum radiation pattern of a radio beam endowed with orbital
angular momentum, generated by an array of tri-axial antennas. This graphics illustration was prepared by
J O H A N S J Ö H O L M and K R I S T O F F E R P A L M E R as part of their undergraduate Diploma Thesis work in
Engineering Physics at Uppsala University 2006–2007.


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To the memory of professor

L E V M I K H A I L O V I C H E R U K H I M O V (1936–1997)
dear friend, great physicist, poet
and a truly remarkable man.


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CONTENTS

ix

List of Figures

xv

Preface to the second edition
Preface to the first edition

FT

Contents

1 Foundations of Classical Electrodynamics

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1.1 Electrostatics . . . . . . . . . . . . . . . . . .
1.1.1 Coulomb’s law . . . . . . . . . . . . .
1.1.2 The electrostatic field . . . . . . . . . .
1.2 Magnetostatics . . . . . . . . . . . . . . . . .
1.2.1 Ampère’s law . . . . . . . . . . . . . .

1.2.2 The magnetostatic field . . . . . . . . .
1.3 Electrodynamics . . . . . . . . . . . . . . . . .
1.3.1 The indestructibility of electric charge .
1.3.2 Maxwell’s displacement current . . . .
1.3.3 Electromotive force . . . . . . . . . . .
1.3.4 Faraday’s law of induction . . . . . . .
1.3.5 The microscopic Maxwell equations . .
1.3.6 Dirac’s symmetrised Maxwell equations
1.3.7 Maxwell-Chern-Simons equations . . .
1.4 Bibliography . . . . . . . . . . . . . . . . . . .

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2 Electromagnetic Fields and Waves
2.1 Axiomatic classical electrodynamics . . . . . . . . . . . .
2.2 Complex notation and physical observables . . . . . . . .
2.2.1 Physical observables and averages . . . . . . . . .
2.2.2 Maxwell equations in Majorana representation . . .
2.3 The wave equations for E and B . . . . . . . . . . . . . .
2.3.1 The time-independent wave equations for E and B

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CONTENTS

2.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Electromagnetic Potentials and Gauges
3.1
3.2
3.3
3.4

3.5

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The electrostatic scalar potential .
The magnetostatic vector potential
The electrodynamic potentials . .
Gauge transformations . . . . . .
Gauge conditions . . . . . . . . .
3.5.1 Lorenz-Lorentz gauge . . .

3.5.2 Coulomb gauge . . . . . .
3.5.3 Velocity gauge . . . . . .
3.6 Bibliography . . . . . . . . . . . .

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29
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4 Fields from Arbitrary Charge and Current Distributions
4.1 The retarded magnetic field . . . . . . . . . .
4.2 The retarded electric field . . . . . . . . . . .
4.3 The fields at large distances from the sources .
4.3.1 The far fields . . . . . . . . . . . . .
4.4 Bibliography . . . . . . . . . . . . . . . . . .

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5 Fundamental Properties of the Electromagnetic Field
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6.1 Radiation of linear momentum and energy . . .
6.1.1 Monochromatic signals . . . . . . . . .
6.1.2 Finite bandwidth signals . . . . . . . .

6.2 Radiation of angular momentum . . . . . . . .
6.3 Radiation from a localised source volume at rest
6.3.1 Electric multipole moments . . . . . . .
6.3.2 The Hertz potential . . . . . . . . . . .
6.3.3 Electric dipole radiation . . . . . . . . .
6.3.4 Magnetic dipole radiation . . . . . . . .

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5.1 Charge, space, and time inversion symmetries
5.2 Conservation laws . . . . . . . . . . . . . . .
5.2.1 Conservation of charge . . . . . . . .
5.2.2 Conservation of current . . . . . . . .
5.2.3 Conservation of energy . . . . . . . .
5.2.4 Conservation of linear momentum . .
5.2.5 Conservation of angular momentum .
5.2.6 Electromagnetic virial theorem . . . .
5.3 Electromagnetic duality . . . . . . . . . . . .

5.4 Bibliography . . . . . . . . . . . . . . . . . .

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6 Radiation and Radiating Systems


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6.3.5 Electric quadrupole radiation . . . . . . . . . . . . .
6.4 Radiation from an extended source volume at rest . . . . . .
6.4.1 Radiation from a one-dimensional current distribution
6.5 Radiation from a localised charge in arbitrary motion . . . .
6.5.1 The Liénard-Wiechert potentials . . . . . . . . . . .
6.5.2 Radiation from an accelerated point charge . . . . .
6.5.3 Bremsstrahlung . . . . . . . . . . . . . . . . . . . .
6.5.4 Cyclotron and synchrotron radiation . . . . . . . . .
6.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Relativistic Electrodynamics
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7.1 The special theory of relativity . . . . .
7.1.1 The Lorentz transformation . . .
7.1.2 Lorentz space . . . . . . . . . .
7.1.3 Minkowski space . . . . . . . .
7.2 Covariant classical mechanics . . . . . .
7.3 Covariant classical electrodynamics . .
7.3.1 The four-potential . . . . . . . .
7.3.2 The Liénard-Wiechert potentials
7.3.3 The electromagnetic field tensor

7.4 Bibliography . . . . . . . . . . . . . . .

8 Electromagnetic Fields and Particles

8.1 Charged particles in an electromagnetic field . . . . . . . . . . .
8.1.1 Covariant equations of motion . . . . . . . . . . . . . .
8.2 Covariant field theory . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Lagrange-Hamilton formalism for fields and interactions
8.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Electromagnetic Fields and Matter

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9.1 Maxwell’s macroscopic theory . . . . . . . . . . . . .
9.1.1 Polarisation and electric displacement . . . . .
9.1.2 Magnetisation and the magnetising field . . . .
9.1.3 Macroscopic Maxwell equations . . . . . . . .
9.2 Phase velocity, group velocity and dispersion . . . . .
9.3 Radiation from charges in a material medium . . . . .
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9.3.1 Vavilov-Cerenkov
radiation . . . . . . . . . . .
9.4 Electromagnetic waves in a medium . . . . . . . . . .
9.4.1 Constitutive relations . . . . . . . . . . . . . .
9.4.2 Electromagnetic waves in a conducting medium
9.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . .

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xii

Formulæ
F .1

F .2

F .4

F .5

F .1.1

The microscopic Maxwell equations . . . . . . . . . . . . . . . . . . . . 187

F .1.2

Fields and potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

F .1.3

Force and energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Electromagnetic radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
F .2.1

Relationship between the field vectors in a plane wave . . . . . . . . . . 188

F .2.2

The far fields from an extended source distribution . . . . . . . . . . . . 189

F .2.3

The far fields from an electric dipole . . . . . . . . . . . . . . . . . . . . 189

F .2.4

The far fields from a magnetic dipole . . . . . . . . . . . . . . . . . . . . 189

F .2.5

The far fields from an electric quadrupole . . . . . . . . . . . . . . . . . 189

F .2.6

The fields from a point charge in arbitrary motion . . . . . . . . . . . . . 189

Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
F .3.1

Metric tensor for flat 4D space . . . . . . . . . . . . . . . . . . . . . . . 190

F .3.2

Covariant and contravariant four-vectors . . . . . . . . . . . . . . . . . . 190

F .3.3

Lorentz transformation of a four-vector . . . . . . . . . . . . . . . . . . 190

F .3.4

Invariant line element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

F .3.5

Four-velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

F .3.6

Four-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

F .3.7

Four-current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

F .3.8

Four-potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

F .3.9

Field tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Vector relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
F .4.1

Spherical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 192

F .4.2

Vector and tensor formulæ . . . . . . . . . . . . . . . . . . . . . . . . . 193

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Mathematical Methods
M .1

197

Scalars, vectors and tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

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The electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

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F

CONTENTS

M .2

M .3

M .1.1

Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

M .1.2

Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

M .1.3

Vector algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

M .1.4

Vector analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Analytical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
M .2.1

Lagrange’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

M .2.2

Hamilton’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220


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CONTENTS

Index

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LIST OF FIGURES

1.1
1.2
1.3
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4.1 Radiation in the far zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

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Multipole radiation geometry . . . . . . . . . . . . . .
Electric dipole geometry . . . . . . . . . . . . . . . .
Linear antenna . . . . . . . . . . . . . . . . . . . . . .
Electric dipole antenna geometry . . . . . . . . . . . .
Loop antenna . . . . . . . . . . . . . . . . . . . . . .
Radiation from a moving charge in vacuum . . . . . .
An accelerated charge in vacuum . . . . . . . . . . . .
Angular distribution of radiation during bremsstrahlung
Location of radiation during bremsstrahlung . . . . . .
Radiation from a charge in circular motion . . . . . . .
Synchrotron radiation lobe width . . . . . . . . . . . .
The perpendicular electric field of a moving charge . .
Electron-electron scattering . . . . . . . . . . . . . . .

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6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11

6.12
6.13

Coulomb interaction between two electric charges . . . .
Coulomb interaction for a distribution of electric charges
Ampère interaction . . . . . . . . . . . . . . . . . . . .
Moving loop in a varying B field . . . . . . . . . . . . .

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7.1 Relative motion of two inertial systems . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2 Rotation in a 2D Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.3 Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.1 Linear one-dimensional mass chain . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

L
9.1 Vavilov-Cerenkov
cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
M .1

Tetrahedron-like volume element of matter . . . . . . . . . . . . . . . . . . . . . . . 204

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PREFACE TO THE SECOND EDITION

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This second edition of the book E L E C T R O M A G N E T I C F I E L D T H E O R Y is a major revision of
the first edition that was published only on the Internet (www.plasma.uu.se/CED/Book). The
reason for trying to improve the presentation and to add more material was mainly that this new
edition is now being made available in printed form by Dover Publications and is to be used in an
extended Classical Electrodynamics course at Uppsala University. Hopefully, this means that the
book will find new uses in Academia and elsewhere.
The changes include a slight reordering of the chapters. First, the book describes the properties
of electromagnetism when the charges and currents are located in otherwise free space. Only then
the we go on to show how fields and charges interact with matter. In the author’s opinion, this
approach is preferable as it avoids the formal logical inconsistency of discussing, very early in
the book, such things as the effect of conductors and dielectrics on the fields and charges (and
vice versa), before constitutive relations and physical models for the electromagnetic properties of
matter, including conductors and dielectrics, have been derived from first principles.
In addition to the Maxwell-Lorentz equations and Dirac’s symmetrised version of these equations (which assume the existence of magnetic monopoles), also the Maxwell-Chern-Simons
equations are introduced in Chapter 1. In Chapter 2, stronger emphasis is put on the axiomatic

foundation of electrodynamics as provided by the microscopic Maxwell-Lorentz equations. Chapter 5 is new and deals with symmetries and conserved quantities in a more rigourous and profound
way than in the first edition. For instance, the presentation of the theory of electromagnetic angular
momentum and other observables (constants of motion) has been substantially expanded and put
on a more firm physical basis. Chapter 9 is a complete rewrite that combines material that was
scattered more or less all over the first edition. It also contains new material on wave propagation in
plasma and other media. When, in Chapter 9, the macroscopic Maxwell equations are introduced,
the inherent approximations in the derived field quantities are clearly pointed out. The collection
of formulæ in Appendix F has been augmented. In Appendix M, the treatment of dyadic products
and tensors has been expanded.
I want to express my warm gratitude to professor C E S A R E B A R B I E R I and his entire group,
particularly F A B R I Z I O T A 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 shorter
and longer visits in 2008 and 2009 that made it possible to prepare the current major revision of
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October, 2009

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the book. In this breathtakingly beautiful northern Italy, intellectual titan G A L I L E O G A L I L E I
worked for eighteen years and gave birth to modern physics, astronomy and science as we know it
today, 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, Galileo’s new
ideas transformed society and mankind forever. It is hoped that this book may contribute in some
small, humble way to further these, once upon a time, mind-boggling—and even dangerous—ideas
of intellectual freedom and enlightment.
Thanks are also due to J O H A N S J Ö H O L M , K R I S T O F F E R P A L M E R , M A R C U S E R I K S S O N ,
and J O H A N L I N D B E R G who during their work on their Diploma theses suggested improvements
and additions.
This book is dedicated to my son M A T T I A S , my daughter K A R O L I N A , my four grandsons
M A X , A L B I N , F I L I P and O S K A R , my high-school physics teacher, S T A F F A N R Ö S B Y , and
my fellow members of the C A P E L L A P E D A G O G I C A U P S A L I E N S I S .
BO THIDÉ
www.physics.irfu.se/ bt


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Of the four known fundamental interactions in nature—gravitational, strong, weak, and electromagnetic—the latter has a special standing in the physical sciences. Not only does it, together with
gravitation, permanently make itself known to all of us in our everyday lives. Electrodynamics is
also by far the most accurate physical theory known, tested on scales running from sub-nuclear to
galactic, and electromagnetic field theory is the prototype of all other field theories.
This book, E L E C T R O M A G N E T I C F I E L D T H E O R Y , which tries to give a modern view of
classical electrodynamics, is the result of a more than thirty-five year long love affair. In the autumn
of 1972, I took my first advanced course in electrodynamics at the Department of Theoretical
Physics, Uppsala University. Soon I joined the research group there and took on the task of helping
the late professor P E R O L O F F R Ö M A N , who was to become my Ph.D. thesis adviser, with the
preparation of a new version of his lecture notes on the Theory of Electricity. This opened my eyes
to the beauty and intricacy of electrodynamics and I simply became intrigued by it. The teaching
of a course in Classical Electrodynamics at Uppsala University, some twenty odd years after I
experienced the first encounter with the subject, provided the incentive and impetus to write this
book.
Intended primarily as a textbook for physics and engineering students at the advanced undergraduate or beginning graduate level, it is hoped that the present book will be useful for research
workers too. It aims at providing a thorough treatment of the theory of electrodynamics, mainly
from a classical field-theoretical point of view. The first chapter is, by and large, a description of
how Classical Electrodynamics was established by J A M E S C L E R K M A X W E L L as a fundamental
theory of nature. It does so by introducing electrostatics and magnetostatics and demonstrating
how they can be unified into one theory, classical electrodynamics, summarised in Lorentz’s
microscopic formulation of the Maxwell equations. These equations are used as an axiomatic
foundation for the treatment in the remainder of the book, which includes modern formulation of
the theory; electromagnetic waves and their propagation; electromagnetic potentials and gauge
transformations; analysis of symmetries and conservation laws describing the electromagnetic
counterparts of the classical concepts of force, momentum and energy, plus other fundamental properties of the electromagnetic field; radiation phenomena; and covariant Lagrangian/Hamiltonian
field-theoretical methods for electromagnetic fields, particles and interactions. Emphasis has
been put on modern electrodynamics concepts while the mathematical tools used, some of them
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presented in an Appendix, are essentially the same kind of vector and tensor analysis methods that
are used in intermediate level textbooks on electromagnetics but perhaps 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
Electrodynamics 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 an
outgrowth of the lecture notes that the author prepared for the four-credit course Electrodynamics
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 on lecture
notes prepared, in Swedish, by my friend and Theoretical Physics colleague B E N G T L U N D B O R G ,
who created, developed and taught an earlier, two-credit course called Electromagnetic Radiation
at our faculty. Thanks are due not only to Bengt Lundborg for providing the inspiration to write
this book, but also to professor C H R I S T E R W A H L B E R G , and professor G Ö R A N F Ä L D T , both
at the Department of Physics and Astronomy, Uppsala University, for insightful suggestions,
to professor J O H N L E A R N E D , Department of Physics and Astronomy, University of Hawaii,

for decisive encouragement at the early stage of this book project, to professor G E R A R D U S
T ’ H O O F T , for recommending this book on his web page ‘How to become a good theoretical
physicist’, and professor C E C I L I A J A R L S K O G , Lund Unversity for pointing out a couple of
errors and ambiguities.
I am particularly indebted to the late professor V I T A L I Y L A Z A R E V I C H G I N Z B U R G , for his
many fascinating and very elucidating lectures, comments and historical notes on plasma physics,
electromagnetic radiation and cosmic electrodynamics while cruising up and down the Volga
and Oka rivers in Russia at the ship-borne Russian-Swedish summer schools that were organised
jointly by late professor L E V M I K A H I L O V I C H E R U K H I M O V and the author during the 1990’s,
and for numerous deep discussions over the years.
Helpful comments and suggestions for improvement from former PhD students T O B I A C A R O Z Z I , R O G E R K A R L S S O N , and M A T T I A S W A L D E N V I K , as well as A N D E R S E R I K S S O N
at the Swedish Institute of Space Physics in Uppsala and who have all taught Uppsala students on
the material covered in this book, are gratefully acknowledged. Thanks are also due to the late
H E L M U T K O P K A , for more than twenty-five years a close friend and space physics colleague
working at the Max-Planck-Institut für Aeronomie, Lindau, Germany, who not only taught me the
practical aspects of the use of high-power electromagnetic radiation for studying space, but also
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 of this
book, thereby trying to ensure its quality and scope to make it useful in higher university education
anywhere in the world, it was produced as a World-Wide Web (WWW) project. This turned out to
be a rather successful move. By making an electronic version of the book freely downloadable
on the Internet, comments have been received from fellow physicists around the world. To judge
from WWW ‘hit’ statistics, it seems that the book serves as a frequently used Internet resource.


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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 of the book. I
would like to thank all students and Internet users who have downloaded and commented on the
book during its life on the World-Wide Web.
BO THIDÉ
www.physics.irfu.se/ bt

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December, 2008


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FOUNDATIONS OF CLASSICAL
ELECTRODYNAMICS

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The classical theory of electromagnetism deals with electric and magnetic fields
and interactions caused by distributions of electric charges and currents. This
presupposes that the concepts of localised electric charges and currents assume
the validity of certain mathematical limiting processes in which it is considered
possible for the charge and current distributions to be localised in infinitesimally
small volumes of space. Clearly, this is in contradistinction to electromagnetism
on an atomistic scale, where charges and currents have to be described in a
quantum formalism. However, the limiting processes used in the classical domain,
which, crudely speaking, assume that an elementary charge has a continuous
distribution of charge density, will yield results that agree with experiments on
non-atomistic scales, small or large.
It took the genius of J A M E S C L E R K M A X W E L L to consistently unify the
two distinct theories electricity and magnetism into a single super-theory, electromagnetism or classical electrodynamics (CED), and to realise that optics is a
sub-field of this super-theory. Early in the 20th century, H E N D R I K A N T O O N
L O R E N T Z took the electrodynamics theory further to the microscopic scale and
also laid the foundation for the special theory of relativity, formulated in its full
extent by A L B E R T E I N S T E I N in 1905. In the 1930’s P A U L A D R I E N M A U R I C E D I R A C expanded electrodynamics to a more symmetric form, including

magnetic as well as electric charges. With his relativistic quantum mechanics
and field quantisation concepts, he also paved the way for the development of
quantum electrodynamics (QED ) for which R I C H A R D P H I L L I P S F E Y N M A N ,
J U L I A N S E Y M O U R S C H W I N G E R , and S I N - I T I R O T O M O N A G A in 1965
were awarded the Nobel Prize in Physics. Around the same time, physicists
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1 . F O U N D A TIONS OF CLASSICAL ELECTRODYNAMICS

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such as S H E L D O N G L A S H O W , A B D U S S A L A M , and S T E V E N W E I N B E R G
were able to unify electrodynamics with the weak interaction theory, creating
yet another super-theory, electroweak theory, an achievement which rendered
them the Nobel Prize in Physics 1979. The modern theory of strong interactions,
quantum chromodynamics (QCD ), is heavily influenced by QED.
In this introductory chapter we start with the force interactions in classical
electrostatics and classical magnetostatics and introduce the static electric and
magnetic fields to find two uncoupled systems of equations for them. Then we
see how the conservation of electric charge and its relation to electric current
leads to the dynamic connection between electricity and magnetism and how the
two can be unified into classical electrodynamics. This theory is described by a

system of coupled dynamic field equations—the microscopic Maxwell equations
introduced by Lorentz—which we take as the axiomatic foundation for the theory
of electromagnetic fields.
At the end of this chapter we present Dirac’s symmetrised form of the Maxwell
equations by introducing (hypothetical) magnetic charges and magnetic currents
into the theory. While not identified unambiguously in experiments yet, magnetic
charges and currents make the theory much more appealing, for instance by
allowing for duality transformations in a most natural way. Besides, in practical
work, such as in antenna engineering, magnetic currents have proved to be a very
useful concept. We shall make use of these symmetrised equations throughout
the book. At the very end of this chapter, we present the Maxwell-Chern-Simons
equations that are of considerable interest to modern physics.

1.1 Electrostatics

1

The physicist and philosopher
P I E R R E D U H E M (1861–1916)
once wrote:

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‘The whole theory of
electrostatics constitutes
a group of abstract ideas
and general propositions,
formulated in the clear
and concise language of
geometry and algebra, and

connected with one another
by the rules of strict logic.
This whole fully satisfies
the reason of a French
physicist and his taste
for clarity, simplicity and
order. . . .’

The theory which describes physical phenomena related to the interaction between
stationary electric charges or charge distributions in a finite space with stationary
boundaries is called electrostatics. For a long time, electrostatics, under the
name electricity, was considered an independent physical theory of its own,
alongside other physical theories such as Magnetism, Mechanics, Optics, and
Thermodynamics.1

1.1.1 Coulomb’s law
It has been found experimentally that in classical electrostatics the interaction
between stationary, electrically charged bodies can be described in terms of
two-body mechanical forces. In the simple case depicted in Figure 1.1 on the
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1.1. Electrostatics

q
x

j3

Figure 1.1: Coulomb’s law describes how a static electric charge
q, located at a point x relative to
the origin O, experiences an electrostatic force from a static electric
charge q 0 located at x0 .

x0

x
q0
x0

O

qq 0 x x0
D
4 "0 jx x0 j3

where, in the last step, formula (F.115) on page 195 was used. In SI units,
which we shall use throughout, the force F is measured in Newton (N), the
electric charges q and q 0 in Coulomb (C) or Ampere-seconds (A s), and the length
jx x0 j in metres (m). The constant "0 D 107 =.4 c 2 / 8:8542 10 12 Farad
per metre (F m 1 ) is the vacuum permittivity and c 2:9979 108 m s 1 is the
speed of light in vacuum.3 In CGS units, "0 D 1=.4 / and the force is measured
in dyne, electric charge in statcoulomb, and length in centimetres (cm).

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1.1.2 The electrostatic field

Instead of describing the electrostatic interaction in terms of a ‘force action at a
distance’, it turns out that for many purposes it is useful to introduce the concept
of a field. Therefore we describe the electrostatic interaction in terms of a static
vectorial electric field Estat defined by the limiting process
def

F
q!0 q

Estat Á lim

2

CHARLES-AUGUSTIN DE
C O U L O M B (1736–1806) was
a French physicist who in 1775
published three reports on the
forces between electrically charged
bodies.

Â
Â
Ã
Ã
qq 0
1
1
qq 0 0
r
r

D
4 "0
4 "0
jx x0 j
jx x0 j
(1.1)

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located at x, due to the presence of the charge q 0 located at x0 in an otherwise
empty space, is given by Coulomb’s law.2 This law postulates that F is directed
along the line connecting the two charges, repulsive for charges of equal signs
and attractive for charges of opposite signs, and therefore can be formulated
mathematically as

(1.2)

where F is the electrostatic force, as defined in equation (1.1), from a net electric
charge q 0 on the test particle with a small electric net electric charge q. Since

3

The notation c for speed comes
from the Latin word ‘celeritas’
which means ‘swiftness’. This notation seems to have been introduced

by W I L H E L M E D U A R D W E B E R (1804–1891), and R U D O L F
K O H L R A U S C H (1809–1858) and
c is therefore sometimes referred
to as Weber’s constant. In all his
works from 1907 and onward,
A L B E R T E I N S T E I N (1879–1955)
used c to denote the speed of light
in vacuum.