(Parte 1 de 3)

Υ Bo Thidé

Bo Thidé ELECTROMAGNETIC FIELD THEORY

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

Bo Thidé

Swedish Institute of Space Physics and

Department of Astronomy and Space Physics Uppsala University, Sweden

This book was typeset in LATEX2ε on an HP9000/700 series workstation and printed on an HP LaserJet 5000GN printer.

Copyright ©1997, 1998, 1999 and 2000 by Bo Thidé Uppsala, Sweden All rights reserved.

Electromagnetic Field Theory ISBN X-X-XXX-X

Contents

Preface xi

1.1 Electrostatics1
1.1.1 Coulomb’s law1
1.1.2 The electrostatic field2
1.2 Magnetostatics4
1.2.1 Ampère’s law4
1.2.2 The magnetostatic field6
1.3 Electrodynamics8
1.3.1 Equation of continuity9
1.3.2 Maxwell’s displacement current9
1.3.3 Electromotive force10
1.3.4 Faraday’s law of induction1
1.3.5 Maxwell’s microscopic equations14
1.3.6 Maxwell’s macroscopic equations14
1.4 Electromagnetic Duality15

1 Classical Electrodynamics 1

fixed mixing angle17
Example 1.3 The complex field six-vector18
Bibliography20

Example 1.1 Duality of the electromagnetodynamic equations 16 Example 1.2 Maxwell from Dirac-Maxwell equations for a Example 1.4 Duality expressed in the complex field six-vector 19

2.1 The wave equation24
2.1.1 The wave equation for E24
2.1.2 The wave equation for B24
2.1.3 The time-independent wave equation for E25
2.2 Plane waves26
2.2.1 Telegrapher’s equation27

2 Electromagnetic Waves 23 i

2.2.2 Waves in conductive media29
2.3 Observables and averages30
Bibliography31

i CONTENTS

3.1 The electrostatic scalar potential3
3.2 The magnetostatic vector potential34
3.3 The electromagnetic scalar and vector potentials34
3.3.1 Electromagnetic gauges36

3 Electromagnetic Potentials 3

Gauge transformations36

Lorentz equations for the electromagnetic potentials . 36

netic potentials38
The retarded potentials41
Bibliography41

3.3.2 Solution of the Lorentz equations for the electromag-

4.1 The magnetic field45
4.2 The electric field47
Bibliography49

4 The Electromagnetic Fields 43

5.1 The special theory of relativity51
5.1.1 The Lorentz transformation52
5.1.2 Lorentz space53
Metric tensor54
Scalar product and norm5
Invariant line element and proper time56
Four-vector fields57
The Lorentz transformation matrix57
The Lorentz group58
5.1.3 Minkowski space58
5.2 Covariant classical mechanics61
5.3 Covariant classical electrodynamics62
5.3.1 The four-potential62
5.3.2 The Liénard-Wiechert potentials63
5.3.3 The electromagnetic field tensor65
Bibliography67

5 Relativistic Electrodynamics 51 Radius four-vector in contravariant and covariant form 54 Downloaded fromhttp://w.plasma.u.se/CED/Book Draft version released 13th November 2000 at 2:01.

6.1 Charged Particles in an Electromagnetic Field69
6.1.1 Covariant equations of motion69
Lagrange formalism69
Hamiltonian formalism72
6.2 Covariant Field Theory76
The electromagnetic field80
tensor81
Other fields84
Bibliography85

6 Interactions of Fields and Particles 69 6.2.1 Lagrange-Hamilton formalism for fields and interactions 7 Example 6.1 Field energy difference expressed in the field

7.1 Electric polarisation and the electric displacement vector87
7.1.1 Electric multipole moments87
7.2 Magnetisation and the magnetising field90
7.3 Energy and momentum91
7.3.1 The energy theorem in Maxwell’s theory92
7.3.2 The momentum theorem in Maxwell’s theory93
Bibliography95

7 Interactions of Fields and Matter 87

8.1 The radiation fields97
8.2 Radiated energy9
8.2.1 Monochromatic signals100
8.2.2 Finite bandwidth signals100
8.3 Radiation from extended sources102
8.3.1 Linear antenna102
8.4 Multipole radiation104
8.4.1 The Hertz potential104
8.4.2 Electric dipole radiation108
8.4.3 Magnetic dipole radiation109
8.4.4 Electric quadrupole radiation110
8.5 Radiation from a localised charge in arbitrary motion1
8.5.1 The Liénard-Wiechert potentials112
8.5.2 Radiation from an accelerated point charge114

8 Electromagnetic Radiation 97

force123

Example 8.1 The fields from a uniformly moving charge . 121 Example 8.2 The convection potential and the convection Draft version released 13th November 2000 at 2:01. Downloaded fromhttp://w.plasma.u.se/CED/Book

Radiation for small velocities125
8.5.3 Bremsstrahlung127

iv CONTENTS

celeration times130
8.5.4 Cyclotron and synchrotron radiation132
Cyclotron radiation134
Synchrotron radiation134
Radiation in the general case137
Virtual photons137
8.5.5 Radiation from charges moving in matter139
Vavilov- Cerenkov radiation142
Bibliography147

Example 8.3 Bremsstrahlung for low speeds and short ac-

F.1 The Electromagnetic Field149
F.1.1 Maxwell’s equations149
Constitutive relations149
F.1.2 Fields and potentials149
Vector and scalar potentials149
Lorentz’ gauge condition in vacuum150
F.1.3 Force and energy150
Poynting’s vector150
Maxwell’s stress tensor150
F.2 Electromagnetic Radiation150

F Formulae 149

F.2.2 The far fields from an extended source distribution150
F.2.3 The far fields from an electric dipole150
F.2.4 The far fields from a magnetic dipole151
F.2.5 The far fields from an electric quadrupole151
F.2.6 The fields from a point charge in arbitrary motion151
F.2.7 The fields from a point charge in uniform motion151
F.3 Special Relativity152
F.3.1 Metric tensor152
F.3.2 Covariant and contravariant four-vectors152
F.3.3 Lorentz transformation of a four-vector152
F.3.4 Invariant line element152
F.3.5 Four-velocity152
F.3.6 Four-momentum153
F.3.7 Four-current density153
F.3.8 Four-potential153

F.2.1 Relationship between the field vectors in a plane wave 150 Downloaded fromhttp://w.plasma.u.se/CED/Book Draft version released 13th November 2000 at 2:01.

F.3.9 Field tensor153
F.4 Vector Relations153
F.4.1 Spherical polar coordinates154
Base vectors154
Directed line element154
Solid angle element154
Directed area element154
Volume element154
F.4.2 Vector formulae154
General relations154
Special relations156
Integral relations157
Bibliography157
M.1 Scalars, Vectors and Tensors159
M.1.1 Vectors159
Radius vector159
M.1.2 Fields161
Scalar fields161
Vector fields161
Tensor fields162
Example M.1 Tensors in 3D space164
M.1.3 Vector algebra167
Scalar product167
Example M.2 Inner products in complex vector space167
space168
Example M.4 Metric in general relativity168
Dyadic product169
Vector product170
M.1.4 Vector analysis170
The del operator170
Example M.5 The four-del operator in Lorentz space171
The gradient172
tances in 3D172
The divergence173

M Mathematical Methods 159 Example M.3 Scalar product, norm and metric in Lorentz Example M.6 Gradients of scalar functions of relative dis- Draft version released 13th November 2000 at 2:01. Downloaded fromhttp://w.plasma.u.se/CED/Book

Example M.7 Divergence in 3D173
The Laplacian173
Example M.8 The Laplacian and the Dirac delta173
The curl174
Example M.9 The curl of a gradient174
Example M.10 The divergence of a curl175
M.2 Analytical Mechanics176
M.2.1 Lagrange’s equations176
M.2.2 Hamilton’s equations176
Bibliography177

vi CONTENTS Downloaded fromhttp://w.plasma.u.se/CED/Book Draft version released 13th November 2000 at 2:01.

1.1 Coulomb interaction2
1.2 Ampère interaction5
1.3 Moving loop in a varying B field12
5.1 Relative motion of two inertial systems52
5.2 Rotation in a 2D Euclidean space59
5.3 Minkowski diagram59
6.1 Linear one-dimensional mass chain76
8.1 Radiation in the far zone98
8.2 Radiation from a moving charge in vacuum112
8.3 An accelerated charge in vacuum114
8.4 Angular distribution of radiation during bremsstrahlung128
8.5 Location of radiation during bremsstrahlung129
8.6 Radiation from a charge in circular motion133
8.7 Synchrotron radiation lobe width135
8.8 The perpendicular field of a moving charge138
8.9 Vavilov- Cerenkov cone144
M.1 Surface element of a material body164
M.2 Tetrahedron-like volume element of matter165

List of Figures vii

To the memory of LEV MIKHAILOVICH ERUKHIMOV dear friend, remarkable physicist and a truly great human.

Preface

This book is the result of a twenty-five year long love affair. In 1972, I took my first advanced course in electrodynamics at the Theoretical Physics department, Uppsala University. Shortly thereafter, I joined the research group there and took on the task of helping my supervisor, professor PER-OLOF FRÖMAN, with the preparation of a new version of his lecture notes on Electricity Theory. These two things opened up my eyes for the beauty and intricacy of electrodynamics, already at the classical level, and I fell in love with it.

Ever since that time, I have off and on had reason to return to electrodynamics, both in my studies, research and teaching, and the current book is the result of my own teaching of a course in advanced electrodynamics at Uppsala University some twenty odd years after I experienced the first encounter with this subject. The book is the outgrowth of the lecture notes that I 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 these notes were based on lecture notes prepared, in Swedish, by BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty.

Intended primarily as a textbook for physics students at the advanced undergraduate or beginning graduate level, I hope the book may be useful for research workers too. It provides a thorough treatment of the theory of electrodynamics, mainly from a classical field theoretical point of view, and includes such things as electrostatics and magnetostatics and their unification into electrodynamics, the electromagnetic potentials, gauge transformations, covariant formulation of classical electrodynamics, force, momentum and energy of the electromagnetic field, radiation and scattering phenomena, electromagnetic waves and their propagation in vacuum and in media, and covariant Lagrangian/Hamiltonian field theoretical methods for electromagnetic fields, particles and interactions. 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 related subjects. In an attempt to encourage participation by other scientists and students in xii PREFACE the authoring of this book, and to ensure its quality and scope to make it useful in higher university education anywhere in the world, it was produced within a World-Wide Web (W) project. This turned out to be a rather successful move. By making an electronic version of the book freely down-loadable on the net, I have not only received comments on it from fellow Internet physicists around the world, but know, from W ‘hit’ statistics that at the time of writing this, the book serves as a frequently used Internet resource. This way it is my hope 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 am grateful not only to Per-Olof Fröman and Bengt Lundborg for providing the inspirationfor my writing this book, but also to CHRISTER WAHLBERG at Uppsala University for interesting discussions on electrodynamics in general and on this book in particular, and to my former graduate students MATTIAS WALDENVIK and TOBIA CAROZZI as well as ANDERS ERIKSSON, all at the Swedish Institute of Space Physics, Uppsala Division, and who have participated in the teaching and commented on the material covered in the course and in this book. Thanks are also due to my long-term space physics colleague HELMUT KOPKA of the Max-Planck-Institut für Aeronomie, Lindau, Germany, who not only taught me about the practical aspects of the of highpower radio wave transmitters and transmission lines, but also about the more delicate aspects of typesetting a book in TEX and LATEX. I am particularly indebted to Academician professor VITALIY L. GINZBURG for his many fas- cinating and very elucidating lectures, comments and historical footnotes on electromagnetic radiation while cruising on the Volga river during our joint Russian-Swedish summer schools.

Finally, 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.

I dedicate this book to my son MATTIAS, my daughter KAROLINA, my high-school physics teacher, STAFFAN RÖSBY, and to my fellow members of the CAPELLA PEDAGOGICA UPSALIENSIS.

Uppsala, Sweden BO THIDÉ November, 2000

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Classical Electrodynamics

Classical electrodynamics deals with electric and magnetic fields and interactions caused by macroscopic distributions of electric charges and currents. This means that the concepts of localised 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 contradiction to electromagnetism on a truly microscopic scale, where charges and currents are known to be spatially extended objects. However, the limiting processes used will yield results which are correct on small as well as large macroscopic scales.

In this Chapter we start with the force interactions in classical electrostatics and classical magnetostatics and introduce the static electric and magnetic fields and 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 in one theory, classical electrodynamics, described by one system of coupled dynamic field equations.

1.1 Electrostatics

The theory that describes physical phenomena related to the interactionbetween stationary electric charges or charge distributions in space is called electrostatics.

1.1.1 Coulomb’s law

It has been found experimentally that in classical electrostatics the interaction between two stationary electrically charged bodies can be described in terms of a mechanical force. Let us consider the simple case described by Figure 1.1.1.

2 CLASSICAL ELECTRODYNAMICS

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 q0 located at x0 .

Let F denote the force acting on a charged particle with charge q located at x, due to the presence of a charge q′ located at x0. According to Coulomb’s law this force is, in vacuum, given by the expression

where we have used results from Example M.6 on page 172. In SI units, which we shall use throughout, the force F is measured in Newton (N), the charges q and q0 in Coulomb (C) [= Ampère-seconds (As)], and the length jx x0j in

in vacuum. In CGS units "0 = 1=(4 ) and the force is measured in dyne, the 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 it is often more convenient to introduce the concept of a field and to describe the electrostatic interaction in terms of a static vectorial electric field Estat defined by the limiting process

where F is the electrostatic force, as defined in Equation (1.1), from a net charge q0 on the test particle with a small electric net charge q. Since the

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1.1 ELECTROSTATICS 3 purpose of the limiting process is to assure that the test charge q does not influence the field, the expression for Estat does not depend explicitly on q but only on the charge q′ and the relative radius vector x−x0. This means that we can say that any net electric charge produces an electric field in the space that surrounds it, regardless of the existence of a second charge anywhere in this space.1

Using formulae (1.1) and (1.2), we find that the electrostatic field Estat at the field point x (also known as the observation point), due to a field-producing charge q0 at the source point x0, is given by

In the presence of several field producing discrete charges q0 respectively, the assumption of linearity of vacuum2 allows us to superimpose their individual E fields into a total E field

If the discrete charges are small and numerous enough, we introduce the charge density located at x0 and write the total field as

where, in the last step, we used formula Equation (M.68) on page 172. We emphasise that Equation (1.5) above is valid for an arbitrary distribution of charges, including discrete charges, in which case can be expressed in terms of one or more Dirac delta functions.

1In 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: [6]

“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.”

2In fact, vacuum exhibits a quantum mechanical nonlinearity due to vacuum polarisation effects manifesting themselves in the momentary creation and annihilation of electron-positron pairs, but classically this nonlinearity is negligible.

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4 CLASSICAL ELECTRODYNAMICS

Since, according to formula Equation (M.78) on page 175, ∇×[rα(x)] 0 for any 3D 3 scalar field (x), we immediately find that in electrostatics

I.e., Estat is an irrotational field.

Taking the divergence of the general Estat expression for an arbitrary charge distribution, Equation (1.5) on the preceding page, and using the representation of the Dirac delta function, Equation (M.73) on page 174, we find that

which is Gauss’s law in differential form.

1.2 Magnetostatics

While electrostatics deals with static charges, magnetostatics deals with stationary currents, i.e., charges moving with constant speeds, and the interaction between these currents.

1.2.1 Ampère’s law

Experiments on the interaction between two small current loops have shown that they interact via a mechanical force, much the same way that charges interact. Let F denote such a force acting on a small loop C carrying a current J located at x, due to the presence of a small loop C0 carrying a current J0

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1.2 MAGNETOSTATICS 5 dlC J x dl0

FIGURE 1.2: Ampère’s law describes how a small loop C, carrying a static electric current J through its tangential line element dl located at x, experiences a magnetostatic force from a small loop C0, carrying a static electric current J0 through the tangential line element dl0 located at x0. The loops can have arbitrary shapes as long as they are simple and closed.

located at x′. According to Ampère’s law this force is, in vacuum, given by the expression

Here dl and dl0 are tangential line elements of the loops C and C0, respectively,

which is a useful relation.

At first glance, Equation (1.8) above appears to be unsymmetric in terms of the loops and therefore to be a force law which is in contradiction with Newton’s third law. However, by applying the vector triple product “bac-cab”

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6 CLASSICAL ELECTRODYNAMICS formula (F.54) on page 155, we can rewrite (1.8) in the following way

Recognising the fact that the integrand in the first integral is an exact differential so that this integral vanishes, we can rewrite the force expression, Equation (1.8) on the previous page, in the following symmetric way

This clearly exhibits the expected symmetry in terms of loops C and C0 .

1.2.2 The magnetostatic field

In analogy with the electrostatic case, we may attribute the magnetostatic interaction to a vectorial magnetic field Bstat. I turns out that Bstat can be defined through

which expresses the small element dBstat(x) of the static magnetic field set up at the field point x by a small line element dl0 of stationary current J0 at the source point x0. The SI unit for the magnetic field, sometimes called the magnetic flux density or magnetic induction, is Tesla (T).

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