electromagnetic field theory llys thuyết trường điện từ

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 propa

DRAFT

E L E C T R O M AG N E T I C F I E L D T H E O RY

S e c o n d E d i t i o n

DRAFT

Department of Physics and Astronomy

Uppsala University, Sweden

Tobia Carozzi, Anders Eriksson, Bengt Lundborg,

Bo Thidé and Mattias Waldenvik

Freely downloadable from

All rights reserved.

Electromagnetic Field Theory

DRAFT

C O N T E N T S

1.1 Electrostatics 2

1.1.1 Coulomb’s law 2

1.1.2 The electrostatic field 3

1.2 Magnetostatics 6

1.2.1 Ampère’s law 6

1.2.2 The magnetostatic field 7

1.3 Electrodynamics 9

1.3.1 The indestructibility of electric charge 10

1.3.2 Maxwell’s displacement current 10

1.3.3 Electromotive force 11

1.3.4 Faraday’s law of induction 12

1.3.5 The microscopic Maxwell equations 15

1.3.6 Dirac’s symmetrised Maxwell equations 15

1.3.7 Maxwell-Chern-Simons equations 16

1.4 Bibliography 17

2 Electromagnetic Fields and Waves 19 2.1 Axiomatic classical electrodynamics 19

2.2 Complex notation and physical observables 20

2.2.1 Physical observables and averages 21

2.2.2 Maxwell equations in Majorana representation 22

2.3 The wave equations for E and B 23

2.3.1 The time-independent wave equations for E and B 26

ix

2.4 Bibliography 28

3 Electromagnetic Potentials and Gauges 29 3.1 The electrostatic scalar potential 29

3.2 The magnetostatic vector potential 30

3.3 The electrodynamic potentials 33

3.4 Gauge transformations 34

3.5 Gauge conditions 35

3.5.1 Lorenz-Lorentz gauge 36

3.5.2 Coulomb gauge 40

3.5.3 Velocity gauge 41

3.6 Bibliography 44

4 Fields from Arbitrary Charge and Current Distributions 45 4.1 The retarded magnetic field 47

4.2 The retarded electric field 49

4.3 The fields at large distances from the sources 53

4.3.1 The far fields 56

4.4 Bibliography 57

5 Fundamental Properties of the Electromagnetic Field 59 5.1 Charge, space, and time inversion symmetries 59

5.2 Conservation laws 61

5.2.1 Conservation of charge 63

5.2.2 Conservation of current 63

5.2.3 Conservation of energy 63

5.2.4 Conservation of linear momentum 65

5.2.5 Conservation of angular momentum 68

5.2.6 Electromagnetic virial theorem 71

5.3 Electromagnetic duality 71

5.4 Bibliography 76

6 Radiation and Radiating Systems 77 6.1 Radiation of linear momentum and energy 78

6.1.1 Monochromatic signals 78

6.1.2 Finite bandwidth signals 79

6.2 Radiation of angular momentum 81

6.3 Radiation from a localised source volume at rest 81

6.3.1 Electric multipole moments 81

6.3.2 The Hertz potential 83

6.3.3 Electric dipole radiation 88

6.3.4 Magnetic dipole radiation 91

6.3.5 Electric quadrupole radiation 92

6.4 Radiation from an extended source volume at rest 93

6.4.1 Radiation from a one-dimensional current distribution 93

6.5 Radiation from a localised charge in arbitrary motion 100

6.5.1 The Liénard-Wiechert potentials 100

6.5.2 Radiation from an accelerated point charge 102

6.5.3 Bremsstrahlung 112

6.5.4 Cyclotron and synchrotron radiation 118

6.6 Bibliography 125

7 Relativistic Electrodynamics 127 7.1 The special theory of relativity 127

7.1.1 The Lorentz transformation 128

7.1.2 Lorentz space 129

7.1.3 Minkowski space 134

7.2 Covariant classical mechanics 137

7.3 Covariant classical electrodynamics 138

7.3.1 The four-potential 138

7.3.2 The Liénard-Wiechert potentials 139

7.3.3 The electromagnetic field tensor 142

7.4 Bibliography 145

8 Electromagnetic Fields and Particles 147 8.1 Charged particles in an electromagnetic field 147

8.1.1 Covariant equations of motion 147

8.2 Covariant field theory 154

8.2.1 Lagrange-Hamilton formalism for fields and interactions 154

8.3 Bibliography 162

9 Electromagnetic Fields and Matter 163 9.1 Maxwell’s macroscopic theory 164

9.1.1 Polarisation and electric displacement 164

9.1.2 Magnetisation and the magnetising field 165

9.1.3 Macroscopic Maxwell equations 167

9.2 Phase velocity, group velocity and dispersion 168

9.3 Radiation from charges in a material medium 169

9.3.1 Vavilov- LCerenkov radiation 170

9.4 Electromagnetic waves in a medium 174

9.4.1 Constitutive relations 175

9.4.2 Electromagnetic waves in a conducting medium 177

9.5 Bibliography 185

F.1 The electromagnetic field 187

F.1.1 The microscopic Maxwell equations 187

F.1.2 Fields and potentials 188

F.1.3 Force and energy 188

F.2 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

F.3 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

F.4 Vector relations 191

F.4.1 Spherical polar coordinates 192

F.4.2 Vector and tensor formulæ 193

F.5 Bibliography 196

M Mathematical Methods 197 M.1 Scalars, vectors and tensors 198

M.1.1 Vectors 199

M.1.2 Fields 200

M.1.3 Vector algebra 208

M.1.4 Vector analysis 212

M.2 Analytical mechanics 218

M.2.1 Lagrange’s equations 218

M.2.2 Hamilton’s equations 219

M.3 Bibliography 220

DRAFT

DRAFT

L I S T O F F I G U R E S

1.1 Coulomb interaction between two electric charges 3

1.2 Coulomb interaction for a distribution of electric charges 5

1.3 Ampère interaction 7

1.4 Moving loop in a varying B field 13

4.1 Radiation in the far zone 55

6.1 Multipole radiation geometry 84

6.2 Electric dipole geometry 87

6.3 Linear antenna 94

6.4 Electric dipole antenna geometry 95

6.5 Loop antenna 97

6.6 Radiation from a moving charge in vacuum 101

6.7 An accelerated charge in vacuum 103

6.8 Angular distribution of radiation during bremsstrahlung 113

6.9 Location of radiation during bremsstrahlung 114

6.10 Radiation from a charge in circular motion 119

6.11 Synchrotron radiation lobe width 121

6.12 The perpendicular electric field of a moving charge 123

6.13 Electron-electron scattering 125

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

9.1 Vavilov- LCerenkov cone 171

M.1 Tetrahedron-like volume element of matter 204

xv

DRAFT

P R E FAC E T O T H E S E C O N D E D I T I O N

This second edition of the book EL E C T R O M A G N E T I C FI E L DTH E O R Yis 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

equa-tions (which assume the existence of magnetic monopoles), also the Maxwell-Chern-Simons

equations are introduced in Chapter1 In Chapter 2, stronger emphasis is put on the axiomatic

foundation of electrodynamics as provided by the microscopic Maxwell-Lorentz equations

Chap-ter5 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 Chapter9 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 Chapter9, the macroscopic Maxwell equations are introduced,

the inherent approximations in the derived field quantities are clearly pointed out The collection

of formulæ in AppendixFhas been augmented In AppendixM, the treatment of dyadic products

and tensors has been expanded

I want to express my warm gratitude to professor CE S A R EBA R B I E R Iand his entire group,

particularly FA B R I Z I O TA M B U R I N I, at the Department of Astronomy, University of Padova,

for stimulating discussions and the generous hospitality bestowed upon me during several shorter

and longer visits in2008 and 2009 that made it possible to prepare the current major revision of

xvii

xviii P R E F A C E T O T H E S E C O N D E D I T I O N

the book In this breathtakingly beautiful northern Italy, intellectual titan GA L I L E OGA L I L E I

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 mostprofoundly changing our conception of the world and our place in it In the process, Galileo’s 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-boggling—and even dangerous—ideas

of intellectual freedom and enlightment

Thanks are also due to JO H A N SJ Ö H O L M, KR I S T O F F E R PA L M E R, MA R C U S ER I K S S O N,and JO H A NLI N D B E R Gwho during their work on their Diploma theses suggested improvementsand additions

This book is dedicated to my son MA T T I A S, my daughter KA R O L I N A, my four grandsons

MA X, AL B I N, FI L I Pand OS K A R, my high-school physics teacher, ST A F F A N RÖ S B Y, and

my fellow members of the CA P E L L APE D A G O G I C AUP S A L I E N S I S

P R E FAC E T O T H E F I R S T E D I T I O N

Of the four known fundamental interactions in nature—gravitational, strong, weak, and

electro-magnetic—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, EL E C T R O M A G N E T I CFI E L D TH 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

of1972, 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 PE R OL O FFR 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

under-graduate or beginning under-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 JA M E SCL E R KMA X W E L Las 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

prop-erties 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

xix

xx P R E F A C E T O T H E F I R S T E D I T I O N

presented in an Appendix, are essentially the same kind of vector and tensor analysis methods thatare 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 five-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 BE N G T LU 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 writethis book, but also to professor CH R I S T E R WA H L B E R G, and professor GÖ R A N FÄ L D T, both

at the Department of Physics and Astronomy, Uppsala University, for insightful suggestions,

to professor JO H N LE 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 GE R A R D U S

physicist’, and professor CE C I L I A JA R L S K O G, Lund Unversity for pointing out a couple oferrors and ambiguities

I am particularly indebted to the late professor VI T A L I YLA Z A R E V I C H GI N Z B U R G, 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 LE VMI K A H I L O V I C H ER U K H I M O Vand the author during the1990’s,and for numerous deep discussions over the years

Helpful comments and suggestions for improvement from former PhD students TO B I ACA

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 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 judgefrom WWW ‘hit’ statistics, it seems that the book serves as a frequently used Internet resource

P R E F A C E T O T H E F I R S T E D I T I O N xxi

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

DRAFT

1

F O U N DAT I O N S O F C L A S S I C A L

E L E C T R O DY NA M I C S

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 JA M E S CL E R K MA X W E L Lto consistently unify the

two distinct theories electricity and magnetism into a single super-theory,

elec-tromagnetism or classical electrodynamics (CED), and to realise that optics is a

sub-field of this super-theory Early in the20th century, HE N D R I KAN T O O N

LO R E N T Ztook 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 AL B E R TEI N S T E I Nin1905 In the 1930’s PA U LAD R I E N MA U

-R I C E DI R A Cexpanded 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 RI C H A R D PH I L L I P SFE Y N M A N,

JU L I A N SE Y M O U R SC H W I N G E R, and SI N- IT I R O TO M O N A G A in 1965

were awarded the Nobel Prize in Physics Around the same time, physicists

1

In this introductory chapter we start with the force interactions in classicalelectrostatics and classical magnetostatics and introduce the static electric andmagnetic fields to find 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 unified into classical electrodynamics This theory is described by asystem of coupled dynamic field equations—the microscopic Maxwell equationsintroduced 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 Maxwellequations by introducing (hypothetical) magnetic charges and magnetic currentsinto the theory While not identified 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 ElectrostaticsThe theory which describes physical phenomena related to the interaction betweenstationary electric charges or charge distributions in a finite space with stationaryboundaries is called electrostatics 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 philosopher

PI E R R E DU H E M ( 1861–1916)

once wrote:

‘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 .’

1.1.1 Coulomb’s law

It has been found experimentally that in classical electrostatics the interactionbetween stationary, electrically charged bodies can be described in terms oftwo-body mechanical forces In the simple case depicted in Figure1.1 on thefacing page, the force F acting on the electrically charged particle with charge q

Figure 1.1: Coulomb’s law scribes how a static electric charge

de-q, located at a point x relative to the origin O, experiences an elec- trostatic force from a static electric charge q0located at x0.

located at x, due to the presence of the charge q0located at x0in an otherwise

empty space, is given by Coulomb’s law.2 This law postulates that F is directed 2 C H A R L E S- AU G U S T I N D E

CO U L O M B ( 1736–1806) was

a French physicist who in 1775 published three reports on the forces between electrically charged bodies.

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

1

jx x0j

(1.1)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 q0in Coulomb (C) or Ampere-seconds (A s), and the length

jx x0j in metres (m) The constant "0D 107=.4c2/ 8:8542  10 12Farad

per metre (F m 1) is the vacuum permittivity and c 2:9979  108m s 1is the

speed of light in vacuum.3In CGS units, "0D 1=.4/ and the force is measured 3 The notation c for speed comes

from the Latin word ‘celeritas’ which means ‘swiftness’ This nota- tion seems to have been introduced

by WI L H E L M E D U A R D

WE-B E R ( 1804–1891), and R U D O L F

KO 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,

AL B E R T EI N S T E I N ( 1879–1955) used c to denote the speed of light

in vacuum.

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

1.1.2 The electrostatic field

Instead of describing the electrostatic interaction in terms of a ‘force action at 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 Estatdefined by the limiting process

Estat def lim

q!0

F

where F is the electrostatic force, as defined in equation (1.1), from a net electric

charge q0on the test particle with a small electric net electric charge q Since

4 In the preface to the first edition

of the first volume of his book

A Treatise on Electricity and

Magnetism, first published in 1873,

James Clerk Maxwell describes

this in the following almost poetic

manner:

‘For instance, Faraday, in

his mind’s eye, saw lines of

force traversing all space

where the mathematicians

saw centres of force

attracting at a distance:

Faraday saw a medium

where they saw nothing but

distance: Faraday sought

the seat of the phenomena

in real actions going on

in the medium, they were

satisfied that they had found

it in a power of action at a

distance impressed on the

electric fluids.’

charge that senses the presence of the first one, must be introduced

Using (1.1) and equation (1.2) on the previous page, and formula (F.114) onpage 195, we find that the electrostatic field Estatat the observation point x (alsoknown as the field point ), due to a field-producing electric charge q0at the sourcepoint x0, is given by

1

jx x0j



(1.3)

In the presence of several field producing discrete electric charges qi0, located

at the points x0i, iD 1; 2; 3; : : : , respectively, in an otherwise empty space, theassumption of linearity of vacuum5allows us to superimpose their individual

5 In fact, vacuum exhibits a

quantum mechanical non-linearity

due to vacuum polarisation

effects manifesting themselves

in the momentary creation and

ˇ

If the discrete electric charges are small and numerous enough, we can, in acontinuum limit, assume that the total charge q0from an extended volume to bebuilt up by local infinitesimal elemental charges dq0, each producing an elementalelectric field

dEstat.x/D d

3x0.x0/4"0

V0

qi0q

O

x0i

x x0ix

Figure 1.2: Coulomb’s law for a tribution of individual charges qi0lo- calised within a volume V0of lim- ited extent.

dis-where we used formula (F.114) on page 195 and the fact that .x0/ does not

depend on the unprimed (field 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 case  is expressed in terms of Dirac delta

Z

V 0 d3x0ı.x0 x0i/

 Z

V 0 d3x0ı.x0 xi0/

 ı.y0 yi0/ı.z0 zi0/ D 1

is dimensionless, and x has the dimension m, the 3D Dirac delta distribution ı.x0 x0i/ must have the dimension m 3.

.x0/DX

i

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 to Helmholtz’s theorem, a sufficiently well-behaved vector field is

completely determined by its divergence and curl Taking the divergence of the

general Estatexpression for an arbitrary electric charge distribution, equation (1.7)

on the facing page, and using the representation of the Dirac delta distribution

given by formula (F.117) on page 195, one finds that

which is the differential form of Gauss’s law of electrostatics

Since, according to formula (F.103) on page 195, r  Œr ˛.x/  0 for any

i.e., that Estatis an irrotational field.

To summarise, electrostatics can be described in terms of two vector partialdifferential equations

1.2.1 Ampère’s 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, let F denote such a force acting

on a small loop C , with tangential line element dl, located at x and carrying

a current I in the direction of dl, due to the presence of a small loop C0, withtangential line element dl0, located at x0and carrying a current I0in the direction

of dl0in otherwise empty space According to Ampère’s law this force is given

by the expression7

7 AN D R É - MA R I E AM P È R E

( 1775–1836) was a French

mathe-matician and physicist who, only

a few days after he learned about

the findings by the Danish physicist

and chemist HA N S C H R I S T I A N

ØR S T E D ( 1777–1851) regarding

the magnetic effects of electric

currents, presented a paper to the

Académie des Sciences in Paris,

describing the law that now bears

x0

x x0x

Figure 1.3: Ampère’s law describes how a small loop C , carrying a static electric line current density el- ement dj at x, experiences a magne- tostatic force from a small loop C0, carrying a static electric current den- sity element dj0located at x0 The loops can have arbitrary shapes as long as they are simple and closed.

which is a most useful relation

At first 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 Newton’s third

law However, by applying the vector triple product ‘bac-cab’ formula (F.72) on

page 193, we can rewrite (1.12) as

C

dl
r

1

jx x0j



0II04

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

vanishes and we can rewrite the force expression, formula (1.12) on the facing

page, in the following symmetric way

1.2.2 The magnetostatic field

In analogy with the electrostatic case, we may attribute the magnetostatic

in-teraction to a static vectorial magnetic field Bstat The elemental Bstatfrom the

elemental current element dI0D I0dl0is defined as

0 x x

0

which expresses the small element dBstat.x/ of the static magnetic field set up at

the field point x by a small line current element dI0D I0dl0of stationary current

V 0d3x0j.x0/  r

1

jx x0j



D 04r Z

but that they differ in their vectorial characteristics With this definition of Bstat,equation (1.12) on page 6 may we written

r
Bstat.x/D 0

4r



r Z

Z

V 0d3x0j.x0/r2

1

jx x0j



C04

In the first of the two integrals on the right-hand side, we use the representation

of the Dirac delta function given in formula (F.117) on page 195, and integrate

@

@x0 k

1

1

jx x0j

Z

V 0d3x0r0
j.x0/ r0 1

jx x0j



(1.21)

We note that the first integral in the result, obtained by applying Gauss’s theorem,

vanishes when integrated over a large sphere far away from the localised source

j.x0/, and that the second integral vanishes because r
j D 0 for stationary

currents (no charge accumulation in space) The net result is simply

r  Bstat.x/D 0

Z

V 0d3x0j.x0/ı.x x0/D 0j.x/ (1.22)

1.3 Electrodynamics

As we saw in the previous sections, the laws of electrostatics and magnetostatics

can be summarised in two pairs of time-independent, uncoupled vector partial

differential equations, namely the equations of classical electrostatics

Since there is nothing a priori which connects Estatdirectly with Bstat, we must

consider classical electrostatics and classical magnetostatics as two independent

theories

However, when we include time-dependence, these theories are unified into a

single super-theory, classical electrodynamics This unification of the theories of

electricity and magnetism can be inferred from two empirically established facts:

10j 1 F O U N D A T I O N S O F C L A S S I C A L E L E C T R O D Y N A M I C S

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

a consequence, ini, as we shall see, Maxwell’s displacement current

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

1.3.1 The indestructibility of electric charge

Let j.t; x/ denote the time-dependent electric current density In the simplestcase it can be defined as j D v where v is the velocity of the electric chargedensity .8

8 A more accurate model is to

assume that the individual charge

elements obey some distribution

function that describes their local

variation of velocity in space and

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

a divergence in the electric current density j.t; x/

1.3.2 Maxwell’s displacement current

We recall from the derivation of equation (1.22) on the preceding page that there

we used the fact that in magnetostaticsr
j.x/ D 0 In the case of non-stationarysources and fields, 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 0d3x0j.t; x0/ı.x x0/

C 04

@

@tZ

where, in the last step, we have assumed that a generalisation of equation (1.7)

on page 4 to time-varying fields allows us to make the identification9

9 Later, we will need to consider

this generalisation and formal

identification further.

V 0

d3x0.t; x0/r

1

unobserved current was introduced, in a stroke of genius, by Maxwell in order

to make also the right-hand side of this equation divergence free when j.t; x/ is

assumed to represent the density of the total electric current, which can be split

up in ‘ordinary’ conduction currents, polarisation currents and magnetisation

currents This will be discussed in Subsection9.1 on page 164 The displacement

current behaves like a current density flowing in free space As we shall see

later, its existence has far-reaching physical consequences as it predicts that such

physical observables as electromagnetic energy, linear momentum, and angular

momentum can be transmitted over very long distances, even through empty

space

1.3.3 Electromotive force

If an electric field E.t; x/ is applied to a conducting medium, a current density

j.t; x/ will be produced in this medium But also mechanical,

hydrodynami-cal and chemihydrodynami-cal processes can create electric currents Under certain physihydrodynami-cal

conditions, and for certain materials, one can sometimes assume that a linear

rela-tionship exists between the electric current density j and E This approximation

approxi-mation is in general not applicable

at all.

where  is the electric conductivity (S m 1) In the case of an anisotropic

conductor,  is a tensor

We can view Ohm’s law, equation (1.29) above, as the first term in a Taylor

expansion of the law jŒE.t; x/ This general law incorporates non-linear effects

such as frequency mixing Examples of media which are highly non-linear are

semiconductors and plasma We draw the attention to the fact that even in cases

when the linear relation between E and j is a good approximation, we still have

12j 1 F O U N D A T I O N S O F C L A S S I C A L E L E C T R O D Y N A M I C S

to use Ohm’s law with care The conductivity  is, in general, time-dependent(temporal dispersive media) but then it is often the case that equation (1.29) onthe preceding page is valid for each individual Fourier (spectral) component ofthe field

If the current is caused by an applied electric field E.t; x/, this electric fieldwill exert work on the charges in the medium and, unless the medium is super-conducting, there will be some energy loss The time rate at which this energy isexpended is j
E per unit volume (W m 3) If E is irrotational (conservative), jwill decay away with time Stationary currents therefore require that an electricfield due to an electromotive force (EMF ) is present In the presence of such afield EEMF, Ohm’s law, equation (1.29) on the previous page, takes the form

The electromotive force is defined as

E DI

C

where dl is a tangential line element of the closed loop C 11

11 The term ‘electromagnetic force’

is something of a misnomer since

E represents a voltage, i.e., its SI

dimension is V.

1.3.4 Faraday’s law of induction

In Subsection1.1.2 we derived the differential equations for the electrostatic field.Specifically, on page 6 we derived equation (1.10) stating that r  EstatD 0 andthus that Estatis a conservative field (it can be expressed as a gradient of a scalarfield) This implies that the closed line integral of Estatin equation (1.31) abovevanishes and that this equation becomes

E DI

C

It has been established experimentally that a non-conservative EMF field isproduced in a closed circuit C at rest if the magnetic flux through this circuitvaries with time This is formulated in Faraday’s law which, in Maxwell’sgeneralised form, reads

E.t / DI

where ˆmis the magnetic flux and S is the surface encircled by C which can

be interpreted as a generic stationary ‘loop’ and not necessarily as a conducting

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

a varying magnetic flux during the motion.

circuit Application of Stokes’ theorem on this integral equation, transforms it

into the differential equation

r  E.t; x/ D @

which is valid for arbitrary variations in the fields and constitutes the Maxwell

equation which explicitly connects electricity with magnetism

Any change of the magnetic flux ˆm will induce an EMF Let us therefore

consider the case, illustrated in Figure1.4, when the ‘loop’ is moved in such a

way that it links a magnetic field which varies during the movement The total

time derivative is evaluated according to the well-known operator formula

d

dt D @

@t Cdx

which follows immediately from the multivariate chain rule for the differentiation

of an arbitrary differentiable function f t; x.t // Here, dx=dt describes a chosen

path in space We shall chose the flow path which means that dx=dt D v and

d

dt D @

where, in a continuum picture, v is the fluid velocity For this particular choice,

the convective derivative dx=dt is usually referred to as the material derivative

d2x On
v
r /B (1.37)

also for time-varying fields; this is in fact one of the Maxwell equations Usingthis result and formula (F.95) on page 194, we find that

since, during spatial differentiation, v is to be considered as constant, This allows

us to rewrite equation (1.37) on the previous page in the following way:

E.t / DI

C

dl
EEMFD d

dtZ

S

d2xOn
BD

Z

S

d2x On
@B

@tZ

1.3.5 The microscopic Maxwell equations

We are now able to collect the results from the above considerations and formulate

the equations of classical electrodynamics valid for arbitrary variations in time

and space of the coupled electric and magnetic fields E.t; x/ and B.t; x/ The

Maxwell equations are

In these equations D .t; x/ represents the total, possibly both time and space

dependent, electric charge density, with contributions from free as well as induced

(polarisation) charges Likewise, j D j.t; x/ represents the total, possibly

both time and space dependent, electric current density, with contributions from

conduction currents (motion of free charges) as well as all atomistic (polarisation

and magnetisation) currents As they stand, the equations therefore incorporate

the classical interaction between all electric charges and currents, free or bound,

in the system and are called Maxwell’s microscopic equations They were

first formulated by Lorentz and therefore another name often used for them

is the Maxwell-Lorentz equations Together with the appropriate constitutive

relations, which relate  and j to the fields, and the initial and boundary conditions

pertinent to the physical situation at hand, they form a system of well-posed partial

differential equations which completely determine E and B

1.3.6 Dirac’s 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 and

make the ad hoc assumption that there exist magnetic monopoles represented by

a magnetic charge density, which we denote by m D m.t; x/, and a magnetic

current density, which we denote by jmD jm.t; x/.13With these new

hypotheti-13 Nobel Physics Laureate JU L I A N

SE Y M O U R SC H W I N G E R ( 1918– 1994) once put it:

‘ there are strong theoretical reasons to believe that magnetic charge exists in nature, and may have played an important role in the development

of the Universe Searches for magnetic charge continue at the present time, emphasising that electromagnetism is very far from being a closed object’.

cal physical entities included in the theory, and with the electric charge density

denoted eand the electric current density denoted je, the Maxwell equations

will be symmetrised into the following two scalar and two vector, coupled, partial

1.4 Bibliography

[1] T W B ARRETT AND D M G RIMES , Advanced Electromagnetism Foundations,

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[2] R B ECKER , Electromagnetic Fields and Interactions, Dover Publications, Inc.,

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[3] W G REINER , Classical Electrodynamics, Springer-Verlag, New York, Berlin,

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1975, ISBN 0-08-025072-6.

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1997, ISBN 0-471-59551-9.

[9] J C M AXWELL , A Treatise on Electricity and Magnetism, third ed., vol 1, Dover

Publications, Inc., New York, NY, 1954, ISBN 0-486-60636-8.

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DRAFT