ElectroMagnetic Field Theory+Exercises

ϒ Bo Thidé

U













B









ϒ E LECTROMAGNETIC F IELD T HEORY Bo Thidé

Also available

E LECTROMAGNETIC F IELD T HEORY E XERCISES by Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik

Electromagnetic Field Theory Bo Thidé Swedish Institute of Space Physics and Department of Astronomy and Space Physics Uppsala University, Sweden

ϒ U

B

·

C

AB

·

U

·

S

This book was typeset in LATEX 2ε on an HP9000/700 series workstation and printed on an HP LaserJet 5000GN printer. Copyright ©1997, 1998, 1999, 2000 and 2001 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory ISBN X-XXX-XXXXX-X

Contents

Preface

xi

1 Classical Electrodynamics 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 Equation of continuity for 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 Maxwell’s microscopic equations . . . . 1.3.6 Maxwell’s macroscopic equations . . . . 1.4 Electromagnetic Duality . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

1 1 2 2 5 5 6 8 8 9 10 11 14 14 15

Example 1.1 Faraday’s law as a consequence of conservation of magnetic charge . . . . . . . . . . . . . 16 Example 1.2 Duality of the electromagnetodynamic equations 18 Example 1.3 Dirac’s symmetrised Maxwell equations for a fixed mixing angle . . . . . . . . . . . . . . . . 18 Example 1.4 The complex field six-vector . . . . . . . . . 20 Example 1.5 Duality expressed in the complex field six-vector 20

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Electromagnetic Waves 2.1 The Wave Equations . . . . . . . . . . . . . . . . 2.1.1 The wave equation for E . . . . . . . . . . 2.1.2 The wave equation for B . . . . . . . . . . 2.1.3 The time-independent wave equation for E

. . . .

. . . .

. . . .

. . . .

. . . .

Example 2.1 Wave equations in electromagnetodynamics

2.2

Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Telegrapher’s equation . . . . . . . . . . . . . . . . 2.2.2 Waves in conductive media . . . . . . . . . . . . . .

i

. . . . . . . .

. . . . . . . .

23 24 24 24 25 26 28 29 30

ii

C ONTENTS

2.3 Observables and Averages . . . . . . . . . . . . . . . . . . . . 32 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3

4

5

Electromagnetic Potentials 3.1 The Electrostatic Scalar Potential . . . . . . . . . . . . . . . . 3.2 The Magnetostatic Vector Potential . . . . . . . . . . . . . . . 3.3 The Electrodynamic Potentials . . . . . . . . . . . . . . . . . 3.3.1 Electrodynamic gauges . . . . . . . . . . . . . . . . . Lorentz equations for the electrodynamic potentials . . Gauge transformations . . . . . . . . . . . . . . . . . 3.3.2 Solution of the Lorentz equations for the electromagnetic potentials . . . . . . . . . . . . . . . . . . . . . The retarded potentials . . . . . . . . . . . . . . . . . Example 3.1 Electromagnetodynamic potentials . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

35 35 36 36 38 38 39

. . . .

40 44 44 45

Relativistic Electrodynamics 4.1 The Special Theory of Relativity . . . . . . . . . . . . . . . . 4.1.1 The Lorentz transformation . . . . . . . . . . . . . . 4.1.2 Lorentz space . . . . . . . . . . . . . . . . . . . . . . Radius four-vector in contravariant and covariant form Scalar product and norm . . . . . . . . . . . . . . . . Metric tensor . . . . . . . . . . . . . . . . . . . . . . Invariant line element and proper time . . . . . . . . . Four-vector fields . . . . . . . . . . . . . . . . . . . . The Lorentz transformation matrix . . . . . . . . . . . The Lorentz group . . . . . . . . . . . . . . . . . . . 4.1.3 Minkowski space . . . . . . . . . . . . . . . . . . . . 4.2 Covariant Classical Mechanics . . . . . . . . . . . . . . . . . 4.3 Covariant Classical Electrodynamics . . . . . . . . . . . . . . 4.3.1 The four-potential . . . . . . . . . . . . . . . . . . . 4.3.2 The Liénard-Wiechert potentials . . . . . . . . . . . . 4.3.3 The electromagnetic field tensor . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

47 47 48 49 50 50 51 52 54 54 54 55 57 59 59 60 62 65

Electromagnetic Fields and Particles 5.1 Charged Particles in an Electromagnetic Field 5.1.1 Covariant equations of motion . . . . Lagrange formalism . . . . . . . . . Hamiltonian formalism . . . . . . . .

. . . .

67 67 67 68 70

Downloaded from http://www.plasma.uu.se/CED/Book

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Draft version released 4th January 2002 at 18:55.

iii

5.2

Covariant Field Theory . . . . . . . . . . . . . . . . . . . . . . 74 5.2.1 Lagrange-Hamilton formalism for fields and interactions 74 The electromagnetic field . . . . . . . . . . . . . . . . . 78 Example 5.1 Field energy difference expressed in the field tensor . . . . . . . . . . . . . . . . . . . . . . 79

Other fields . . . . . . . . . . . . . . . . . . . . . . . . 82 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6 Electromagnetic Fields and Matter 6.1 Electric Polarisation and Displacement . . . . . . . . 6.1.1 Electric multipole moments . . . . . . . . . 6.2 Magnetisation and the Magnetising Field . . . . . . . 6.3 Energy and Momentum . . . . . . . . . . . . . . . . 6.3.1 The energy theorem in Maxwell’s theory . . 6.3.2 The momentum theorem in Maxwell’s theory Bibliography . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

85 85 85 88 90 90 91 93

7 Electromagnetic Fields from Arbitrary Source Distributions 7.1 The Magnetic Field . . . . . . . . . . . . . . . . . . . . . 7.2 The Electric Field . . . . . . . . . . . . . . . . . . . . . . 7.3 The Radiation Fields . . . . . . . . . . . . . . . . . . . . 7.4 Radiated Energy . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Monochromatic signals . . . . . . . . . . . . . . . 7.4.2 Finite bandwidth signals . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

95 97 99 101 104 104 104 106

8 Electromagnetic Radiation and Radiating Systems 8.1 Radiation from Extended Sources . . . . . . . . . . . . . . . 8.1.1 Radiation from a one-dimensional current distribution 8.1.2 Radiation from a two-dimensional current distribution 8.2 Multipole Radiation . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 The Hertz potential . . . . . . . . . . . . . . . . . . . 8.2.2 Electric dipole radiation . . . . . . . . . . . . . . . . 8.2.3 Magnetic dipole radiation . . . . . . . . . . . . . . . 8.2.4 Electric quadrupole radiation . . . . . . . . . . . . . . 8.3 Radiation from a Localised Charge in Arbitrary Motion . . . . 8.3.1 The Liénard-Wiechert potentials . . . . . . . . . . . . 8.3.2 Radiation from an accelerated point charge . . . . . . The differential operator method . . . . . . . . . . . . Example 8.1 The fields from a uniformly moving charge .

. . . . . . . . . . . . .

107 107 108 111 114 114 118 120 121 122 123 125 127 132


. . . . . . .

. . . . . . .


iv

C ONTENTS

8.3.3

Example 8.2 The convection potential and the convection force134 Radiation for small velocities . . . . . . . . . . . . . . 137 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . 138 Example 8.3 Bremsstrahlung for low speeds and short acceleration times . . . . . . . . . . . . . . . . . 141

8.3.4

Cyclotron and synchrotron radiation . . . Cyclotron radiation . . . . . . . . . . . . Synchrotron radiation . . . . . . . . . . . Radiation in the general case . . . . . . . Virtual photons . . . . . . . . . . . . . . 8.3.5 Radiation from charges moving in matter ˇ Vavilov-Cerenkov radiation . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

143 145 146 148 149 151 153 158

F Formulae F.1 The Electromagnetic Field . . . . . . . . . . . . . . . . . . . F.1.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . Constitutive relations . . . . . . . . . . . . . . . . . . F.1.2 Fields and potentials . . . . . . . . . . . . . . . . . . Vector and scalar potentials . . . . . . . . . . . . . . Lorentz’ gauge condition in vacuum . . . . . . . . . . F.1.3 Force and energy . . . . . . . . . . . . . . . . . . . . Poynting’s vector . . . . . . . . . . . . . . . . . . . . Maxwell’s stress tensor . . . . . . . . . . . . . . . . . F.2 Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . F.2.1 Relationship between the field vectors in a plane wave F.2.2 The far fields from an extended source distribution . . F.2.3 The far fields from an electric dipole . . . . . . . . . . F.2.4 The far fields from a magnetic dipole . . . . . . . . . F.2.5 The far fields from an electric quadrupole . . . . . . . F.2.6 The fields from a point charge in arbitrary motion . . . F.2.7 The fields from a point charge in uniform motion . . . F.3 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . F.3.1 Metric tensor . . . . . . . . . . . . . . . . . . . . . . F.3.2 Covariant and contravariant four-vectors . . . . . . . . F.3.3 Lorentz transformation of a four-vector . . . . . . . . F.3.4 Invariant line element . . . . . . . . . . . . . . . . . . F.3.5 Four-velocity . . . . . . . . . . . . . . . . . . . . . . F.3.6 Four-momentum . . . . . . . . . . . . . . . . . . . . F.3.7 Four-current density . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

159 159 159 159 159 159 160 160 160 160 160 160 160 160 161 161 161 161 162 162 162 162 162 162 162 163


. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .


v

F.3.8 Four-potential . . . . . . . . . . . F.3.9 Field tensor . . . . . . . . . . . . F.4 Vector Relations . . . . . . . . . . . . . . F.4.1 Spherical polar coordinates . . . . Base vectors . . . . . . . . . . . Directed line element . . . . . . . Solid angle element . . . . . . . . Directed area element . . . . . . Volume element . . . . . . . . . F.4.2 Vector formulae . . . . . . . . . . General vector algebraic identities General vector analytic identities . Special identities . . . . . . . . . Integral relations . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

Appendices

163 163 163 163 163 164 164 164 164 164 164 165 165 166 166 159

M Mathematical Methods M.1 Scalars, Vectors and Tensors M.1.1 Vectors . . . . . . . Radius vector . . . . M.1.2 Fields . . . . . . . . Scalar fields . . . . . Vector fields . . . . Tensor fields . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . . Example M.1 Tensors in 3D space .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

167 167 167 167 169 169 169 170 171

Example M.2 Contravariant and covariant vectors in flat Lorentz space . . . . . . . . . . . . . . . . . . 174

M.1.3 Vector algebra . . . . . . . . . . . . . . . . . . . . . . 176 Scalar product . . . . . . . . . . . . . . . . . . . . . . 176 Example M.3 Inner products in complex vector space . . . . 176 Example M.4 Scalar product, norm and metric in Lorentz space . . . . . . . . . . . . . . . . . . . . . . 177 Example M.5 Metric in general relativity . . . . . . . . . . 177

Dyadic product . Vector product . M.1.4 Vector analysis . The del operator

. . . . Example M.6 The four-del operator in Lorentz space .


. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . .

. . . . .

. . . . .

178 179 179 179 180


vi

C ONTENTS

The gradient . . . . . . . . . . . . . . . . . . . . . . . 181 Example M.7 Gradients of scalar functions of relative distances in 3D . . . . . . . . . . . . . . . . . . . 181

The divergence . . . . . . . . . . . . . . . . . . Example M.8 Divergence in 3D . . . . . . . . . . The Laplacian . . . . . . . . . . . . . . . . . . . Example M.9 The Laplacian and the Dirac delta . . The curl . . . . . . . . . . . . . . . . . . . . . . Example M.10 The curl of a gradient . . . . . . . . Example M.11 The divergence of a curl . . . . . . M.2 Analytical Mechanics . . . . . . . . . . . . . . . . . . . M.2.1 Lagrange’s equations . . . . . . . . . . . . . . . M.2.2 Hamilton’s equations . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .


. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

182 182 182 182 183 183 184 185 185 185 186


List of Figures

1.1 1.2 1.3

Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . 3 Ampère interaction . . . . . . . . . . . . . . . . . . . . . . . . 6 Moving loop in a varying B field . . . . . . . . . . . . . . . . . 12

4.1 4.2 4.3

Relative motion of two inertial systems . . . . . . . . . . . . . 48 Rotation in a 2D Euclidean space . . . . . . . . . . . . . . . . . 55 Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1

Linear one-dimensional mass chain . . . . . . . . . . . . . . . . 75

7.1

Radiation in the far zone . . . . . . . . . . . . . . . . . . . . . 103

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13

Linear antenna . . . . . . . . . . . . . . . . . . . . . Electric dipole geometry . . . . . . . . . . . . . . . . Loop antenna . . . . . . . . . . . . . . . . . . . . . . Multipole radiation geometry . . . . . . . . . . . . . . Electric dipole geometry . . . . . . . . . . . . . . . . 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 field of a moving charge . . . . . . ˇ Vavilov-Cerenkov cone . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

108 109 111 116 119 123 126 139 140 144 146 149 155

M.1 Tetrahedron-like volume element of matter . . . . . . . . . . . . 172

vii

To the memory of professor L EV M IKHAILOVICH E RUKHIMOV dear friend, great physicist and a truly remarkable human being.

If you understand, things are such as they are If you do not understand, things are such as they are G ENSHA

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 P ER -O LOF F RÖ MAN , 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 B ENGT L UNDBORG 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 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 (WWW) project. This turned out to be a rather successful

xi

xii

P REFACE

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 WWW ‘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 inspiration for my writing this book, but also to C HRISTER WAHLBERG and G ÖRAN FÄLDT, Uppsala University, and YAKOV I STOMIN, Lebedev Institute, Moscow, for interesting discussions on electrodynamics and relativity in general and on this book in particular. I also wish to thank my former graduate students M ATTIAS WALDENVIK and T OBIA C AROZZI as well as A NDERS E RIKSSON, all at the Swedish Institute of Space Physics, Uppsala Division, who all have participated in the teaching and commented on the material covered in the course and in this book. Thanks are also due to my longterm space physics colleague H ELMUT KOPKA of the Max-Planck-Institut für Aeronomie, Lindau, Germany, who not only taught me about the practical aspects of the of high-power 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 V ITALIY L. G INZBURG for his many fascinating 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 M ATTIAS, my daughter K AROLINA, my high-school physics teacher, S TAFFAN RÖSBY, and to my fellow members of the C APELLA P EDAGOGICA U PSALIENSIS. Uppsala, Sweden February, 2001


B O T HIDÉ


1 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 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 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—the Maxwell equations. At the end of the chapter we study Dirac’s symmetrised form of Maxwell’s 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.

1.1 Electrostatics The theory which describes physical phenomena related to the interaction between stationary electric charges or charge distributions in space is called electrostatics.1 For a long time electrostatics was considered an independent phys1 The

famous physicist and philosopher Pierre Duhem (1861–1916) once wrote:

1

2

C LASSICAL E LECTRODYNAMICS

ical theory of its own, alongside other physical theories such as mechanics and thermodynamics.

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 on the facing page. Let F denote the force acting on a electrically charged particle with charge q located at x, due to the presence of a charge q0 located at x0 . According to Coulomb’s law this force is, in vacuum, given by the expression F(x) =

1 1 qq0 qq0 0 qq0 x − x0 = − ∇ = ∇ 3 0 0 4πε0 |x − x | 4πε0 4πε0 |x − x | |x − x0 |

(1.1)

where in the last step Equation (M.80) on page 181 was used. In SI units, which we shall use throughout, the force F is measured in Newton (N), the electric charges q and q0 in Coulomb (C) [= Ampère-seconds (As)], and the length |x − x0 | in metres (m). The constant ε0 = 107 /(4πc2 ) ≈ 8.8542 × 10−12 Farad per metre (F/m) is the vacuum permittivity and c ≈ 2.9979 × 108 m/s is the speed of light in vacuum. In CGS units ε0 = 1/(4π) and the force is measured 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 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 def

Estat ≡ lim

q→0

F q

(1.2)

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

3

E LECTROSTATICS

q

x

x − x0 q0 x0

O F IGURE 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 .

Since the 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 q0 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 (1.1) and Equation (1.2) on the preceding page, and Formula (F.72) on page 165, we find that the electrostatic field Estat at the field point x (also known as the observation point), due to a field-producing electric charge q 0 at the source point x0 , is given by Estat (x) =

1 1 q0 0 q0 q0 x − x 0 ∇ = ∇ = − 4πε0 |x − x0 |3 4πε0 4πε0 |x − x0 | |x − x0 |

(1.3)

where in the last step Equation (M.80) on page 181 was used. In the presence of several field producing discrete electric charges q0i , located at the points x0i , i = 1, 2, 3, . . . , respectively, in an otherwise empty space, 1 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 [7]: “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.”



4


the assumption of linearity of vacuum1 allows us to superimpose their individual E fields into a total E field x − x0i 1 Estat (x) = q0i (1.4) ∑ x − x0 3 4πε0 i i

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

1 x − x0 3 0 1 1 d x =− ρ(x0 ) ρ(x0 )∇ d3x0 3 0 4πε0 V 0 4πε0 V 0 |x − x0 | |x − x | (1.5) ρ(x0 ) 3 0 1 =− d x ∇ 4πε0 V 0 |x − x0 | where we used Formula (F.72) on page 165 and the fact that ρ(x0 ) does not depend on the “unprimed” coordinates on which ∇ operates. We emphasise that Equation (1.5) above is valid for an arbitrary distribution of electric charges, including discrete charges, in which case ρ can be expressed in terms of one or more Dirac delta distributions. Since, according to formula Equation (M.90) on page 184, ∇ × [∇α(x)] ≡ 0 for any 3D 3 scalar field α(x), we immediately find that in electrostatics Estat (x) =

∇ × Estat (x) = −

1 ∇× ∇ 4πε0

ρ(x0 ) 3 0 d x =0 0 V 0 |x − x |

(1.6)

i.e., that Estat is an irrotational field. Taking the divergence of the general Estat expression for an arbitrary electric charge distribution, Equation (1.5), and using the representation of the Dirac delta distribution, Equation (M.85) on page 183, we find that 1 x − x0 3 0 ρ(x0 ) dx 4πε0 V 0 |x − x0 |3 1 1 ρ(x0 )∇ · ∇ d3x0 =− 0 4πε0 V |x − x0 | 1 1 =− ρ(x0 )∇2 d3x0 4πε0 V 0 |x − x0 | 1 = ρ(x0 )δ(x − x0 ) d3x0 ε0 V 0 ρ(x) = ε0 which is Gauss’s law in differential form. ∇ · Estat (x) = ∇ ·

(1.7)

1 In

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.



1.2

5

M AGNETOSTATICS

1.2 Magnetostatics While electrostatics deals with static electric charges, magnetostatics deals with stationary electric currents, i.e., electric charges moving with constant speeds, and the interaction between these currents. Let us discuss this theory in some detail.

1.2.1 Ampère’s law Experiments on the interaction between two small loops of electric current have shown that they interact via a mechanical force, much the same way that electric 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 C 0 carrying a current J 0 located at x0 . According to Ampère’s law this force is, in vacuum, given by the expression µ0 JJ 0 dl0 × (x − x0 ) dl × 4π C C 0 |x − x0 |3 0 µ0 JJ 1 =− dl × dl0 × ∇ 4π C C 0 |x − x0 |

F(x) =

(1.8)

Here dl and dl0 are tangential line elements of the loops C and C 0 , respectively, and, in SI units, µ0 = 4π × 10−7 ≈ 1.2566 × 10−6 H/m is the vacuum permeability. From the definition of ε0 and µ0 (in SI units) we observe that 107 1 (F/m) × 4π × 10−7 (H/m) = 2 (s2 /m2 ) (1.9) 2 4πc c which is a useful relation. At first glance, Equation (1.8) above may appear 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” Formula (F.54) on page 164, we can rewrite (1.8) as ε0 µ0 =

1 µ0 JJ 0 dl · ∇ dl0 4π C C 0 |x − x0 |

µ0 JJ 0 x − x0 − dl · dl0 4π C C 0 |x − x0 |3

F(x) = −

(1.10)

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) above, in the following symmetric way F(x) = −

µ0 JJ 0 4π

C C

0

x − x0 dl · dl0 3 0 |x − x |


(1.11)


6


J C

dl x − x0

dl0

x C0 J0

x0 O

F IGURE 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 C 0 , carrying a static electric current J 0 through the tangential line element dl0 located at x0 . The loops can have arbitrary shapes as long as they are simple and closed.

This clearly exhibits the expected symmetry in terms of loops C and C 0 .

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 def

dBstat (x) ≡

µ0 J 0 0 x − x 0 dl × 4π |x − x0 |3

(1.12)

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 J 0 at the source point x0 . The SI unit for the magnetic field, sometimes called the magnetic flux density or magnetic induction, is Tesla (T). If we generalise expression (1.12) to an integrated steady state electric current density j(x), we obtain Biot-Savart’s law: x − x0 3 0 µ0 µ0 j(x0 ) × d x =− 3 0 4π V 0 4π |x − x | j(x0 ) 3 0 µ0 = ∇× dx 4π V 0 x − x0

Bstat (x) =


V0

j(x0 ) × ∇

1 d3x0 |x − x0 |

(1.13)


1.2

7

M AGNETOSTATICS

where we used Formula (F.72) on page 165, Formula (F.60) on page 165, and the fact that j(x0 ) does not depend on the “unprimed” coordinates on which ∇ operates. Comparing Equation (1.5) on page 4 with Equation (1.13) on the facing page, we see that there exists a close analogy between the expressions for Estat and Bstat but that they differ in their vectorial characteristics. With this definition of Bstat , Equation (1.8) on page 5 may we written F(x) = J

C dl × B

stat

(x)

(1.14)

In order to assess the properties of Bstat , we determine its divergence and curl. Taking the divergence of both sides of Equation (1.13) on the facing page and utilising Formula (F.61) on page 165, we obtain ∇ · Bstat (x) =

µ0 ∇· ∇× 4π

j(x0 ) 3 0 d x =0 0 V0 x − x

(1.15)

since, according to Formula (F.66) on page 165, ∇ · (∇ × a) vanishes for any vector field a(x). Applying the operator “bac-cab” rule, Formula (F.67) on page 165, the curl of Equation (1.13) on the facing page can be written j(x0 ) 3 0 µ0 ∇× ∇× dx = 4π V 0 x − x0 µ0 1 µ0 =− j(x0 )∇2 d3x0 + [j(x0 ) · ∇0 ]∇0 0 4π V 0 4π V 0 |x − x |

∇ × Bstat (x) =

1 d3x0 |x − x0 | (1.16)

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.73) on page 165, and integrate the second one by parts, by utilising Formula (F.59) on page 165 as follows:

1 d3x0 |x − x0 | V0 ∂ 1 1 = xˆ k ∇0 · j(x0 ) 0 d3x0 − ∇0 · j(x0 ) ∇0 d3x0 0 | 0| 0 0 ∂x − x − x |x |x V V k ∂ 1 1 = xˆ k j(x0 ) 0 · dS − ∇0 · j(x0 ) ∇0 d3x0 0 |x − x0 | S ∂xk |x − x | V0 (1.17) [j(x0 ) · ∇0 ]∇0

Then 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 ∇ · j = 0 for



8


stationary currents (no charge accumulation in space). The net result is simply ∇ × Bstat (x) = µ0

V0

j(x0 )δ(x − x0 ) d3x0 = µ0 j(x)

(1.18)

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 differential equations, namely the equations of classical electrostatics ρ(x) ε0 stat ∇ × E (x) = 0 ∇ · Estat (x) =

(1.19a) (1.19b)

and the equations of classical magnetostatics ∇ · Bstat (x) = 0

∇×B

stat

(1.20a)

(x) = µ0 j(x)

(1.20b)

Since there is nothing a priori which connects Estat directly 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 one theory, classical electrodynamics. This unification of the theories of electricity and magnetism is motivated by two empirically established facts: 1. Electric charge is a conserved quantity and electric current is a transport of electric charge. This fact manifests itself in the equation of continuity and, as a consequence, in Maxwell’s displacement current. 2. A change in the magnetic flux through a loop will induce an EMF electric field in the loop. This is the celebrated Faraday’s law of induction.

1.3.1 Equation of continuity for electric charge Let j denote the electric current density (measured in A/m2 ). In the simplest case it can be defined as j = vρ where v is the velocity of the electric charge density ρ. In general, j has to be defined in statistical mechanical terms as



1.3

9

E LECTRODYNAMICS

j(t, x) = ∑α qeα v fα (t, x, v) d3v where fα (t, x, v) is the (normalised) distribution function for particle species α with electric charge qeα . The electric charge conservation law can be formulated in the equation of continuity ∂ρ(t, x) + ∇ · j(t, x) = 0 ∂t

(1.21)

which states that the time rate of change of electric charge ρ(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 (1.18) on the preceding page that there we used the fact that in magnetostatics ∇ · j(x) = 0. In the case of nonstationary sources and fields, we must, in accordance with the continuity Equation (1.21), set ∇ · j(t, x) = −∂ρ(t, x)/∂t. Doing so, and formally repeating the steps in the derivation of Equation (1.18) on the preceding page, we would obtain the formal result ∇ × B(t, x) = µ0

V0

j(t, x0 )δ(x − x0 ) d3x0 +

= µ0 j(t, x) + µ0

∂ ε0 E(t, x) ∂t

µ0 ∂ 4π ∂t

V0

ρ(t, x0 )∇0

1 d3x0 |x − x0 | (1.22)

where, in the last step, we have assumed that a generalisation of Equation (1.5) on page 4 to time-varying fields allows us to make the identification 1 ∂ 4πε0 ∂t

V0

ρ(t, x0 )∇0

1 ∂ 1 d3x0 = − 0 ∂t 4πε0 |x − x | ∂ = E(t, x) ∂t

V0

ρ(t, x0 )∇

1 d3x0 |x − x0 |

(1.23)

Later, we will need to consider this formal result further. The result is Maxwell’s source equation for the B field ∇ × B(t, x) = µ0 j(t, x) +

∂ ε0 E(t, x) ∂t

(1.24)

where the last term ∂ε0 E(t, x)/∂t is the famous displacement current. This term was introduced, in a stroke of genius, by Maxwell [6] in order to make



10


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. The displacement current is an extra term which behaves like a current density flowing in vacuum. As we shall see later, its existence has far-reaching physical consequences as it predicts the existence of electromagnetic radiation that can carry energy and momentum over very long distances, even in vacuum.

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. There exist also hydrodynamical and chemical processes which can create currents. Under certain physical conditions, and for certain materials, one can sometimes assume a linear relationship between the electric current density j and E, called Ohm’s law: j(t, x) = σE(t, x)

(1.25)

where σ is the electric conductivity (S/m). In the most general cases, for instance in an anisotropic conductor, σ is a tensor. We can view Ohm’s law, Equation (1.25) above, as the first term in a Taylor expansion of the law j[E(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 to use Ohm’s law with care. The conductivity σ is, in general, time-dependent (temporal dispersive media) but then it is often the case that Equation (1.25) is valid for each individual Fourier component of the field. If the current is caused by an applied electric field E(t, x), this electric field will exert work on the charges in the medium and, unless the medium is superconducting, there will be some energy loss. The rate at which this energy is expended is j · E per unit volume. If E is irrotational (conservative), j will decay away with time. Stationary currents therefore require that an electric field which corresponds to an electromotive force (EMF) is present. In the presence of such a field EEMF , Ohm’s law, Equation (1.25) above, takes the form j = σ(Estat + EEMF )

(1.26)

The electromotive force is defined as E=

C

(Estat + EEMF ) · dl


(1.27)


1.3

11

E LECTRODYNAMICS

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

1.3.4 Faraday’s law of induction In Subsection 1.1.2 we derived the differential equations for the electrostatic field. In particular, on page 4 we derived Equation (1.6) which states that ∇ × Estat (x) = 0 and thus that Estat is a conservative field (it can be expressed as a gradient of a scalar field). This implies that the closed line integral of Estat in Equation (1.27) on the preceding page vanishes and that this equation becomes E=

CE

EMF

· dl

(1.28)

It has been established experimentally that a nonconservative EMF field is produced in a closed circuit C if the magnetic flux through this circuit varies with time. This is formulated in Faraday’s law which, in Maxwell’s generalised form, reads E(t, x) =

C

=−

E(t, x) · dl = −

d dt

S

d Φm (t, x) dt

B(t, x) · dS = −

S

dS ·

∂ B(t, x) ∂t

(1.29)

where Φm is 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 circuit. Application of Stokes’ theorem on this integral equation, transforms it into the differential equation ∇ × E(t, x) = −

∂ B(t, x) ∂t

(1.30)

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 Φm will induce an EMF. Let us therefore consider the case, illustrated if Figure 1.3.4 on the following page, that the “loop” is moved in such a way that it links a magnetic field which varies during the movement. The convective derivative is evaluated according to the wellknown operator formula d ∂ = +v·∇ dt ∂t


(1.31)


12


dS

v

B(x)

v C dl

B(x)

F IGURE 1.3: A loop C which moves with velocity v in a spatially varying magnetic field B(x) will sense a varying magnetic flux during the motion.

which follows immediately from the rules of differentiation of an arbitrary differentiable function f (t, x(t)). Applying this rule to Faraday’s law, Equation (1.29) on the previous page, we obtain E(t, x) = −

d dt

S

B · dS = −

S

dS ·

∂B − (v · ∇)B · dS ∂t S

(1.32)

During spatial differentiation v is to be considered as constant, and Equation (1.15) on page 7 holds also for time-varying fields: ∇ · B(t, x) = 0

(1.33)

(it is one of Maxwell’s equations) so that, according to Formula (F.62) on page 165, ∇ × (B × v) = (v · ∇)B


(1.34)


1.3

13

E LECTRODYNAMICS

allowing us to rewrite Equation (1.32) on the facing page in the following way: E(t, x) =

C

=−

d dt

EEMF · dl = − S

∂B · dS − ∂t

S

S

B · dS

(1.35)

∇ × (B × v) · dS

With Stokes’ theorem applied to the last integral, we finally get E(t, x) =

C

EEMF · dl = −

S

∂B · dS − ∂t

S

∂B · dS ∂t

C

(B × v) · dl

(1.36)

or, rearranging the terms,

C

(EEMF − v × B) · dl = −

(1.37)

where EEMF is the field which is induced in the “loop,” i.e., in the moving system. The use of Stokes’ theorem “backwards” on Equation (1.37) above yields ∇ × (EEMF − v × B) = −

∂B ∂t

(1.38)

In the fixed system, an observer measures the electric field E = EEMF − v × B

(1.39)

Hence, a moving observer measures the following Lorentz force on a charge q qEEMF = qE + q(v × B)

(1.40)

corresponding to an “effective” electric field in the “loop” (moving observer) EEMF = E + (v × B)

(1.41)

Hence, we can conclude that for a stationary observer, the Maxwell equation ∇×E = −

∂B ∂t

(1.42)

is indeed valid even if the “loop” is moving.



14


1.3.5 Maxwell’s microscopic 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 equations are ρ(t, x) ε0 ∂B ∇×E = − ∂t ∇·B = 0 ∂E ∇ × B = ε 0 µ0 + µ0 j(t, x) ∂t ∇·E =

(1.43a) (1.43b) (1.43c) (1.43d)

In these equations ρ(t, x) represents the total, possibly both time and space dependent, electric charge, i.e., free as well as induced (polarisation) charges, and j(t, x) represents the total, possibly both time and space dependent, electric current, i.e., conduction currents (motion of free charges) as well as all atomistic (polarisation, magnetisation) currents. As they stand, the equations therefore incorporate the classical interaction between all electric charges and currents in the system and are called Maxwell’s microscopic equations. Another name often used for them is the Maxwell-Lorentz equations. Together with the appropriate constitutive relations, which relate ρ 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 Maxwell’s macroscopic equations The microscopic field equations (1.43) provide a correct classical picture for arbitrary field and source distributions, including both microscopic and macroscopic scales. However, for macroscopic substances it is sometimes convenient to introduce new derived fields which represent the electric and magnetic fields in which, in an average sense, the material properties of the substances are already included. These fields are the electric displacement D and the magnetising field H. In the most general case, these derived fields are complicated nonlocal, nonlinear functionals of the primary fields E and B: D = D[t, x; E, B]

(1.44a)

H = H[t, x; E, B]

(1.44b)



1.4

15

E LECTROMAGNETIC D UALITY

Under certain conditions, for instance for very low field strengths, we may assume that the response of a substance to the fields is linear so that D = εE

(1.45)

H = µ−1 B

(1.46)

i.e., that the derived fields are linearly proportional to the primary fields and that the electric displacement (magnetising field) is only dependent on the electric (magnetic) field. The field equations expressed in terms of the derived field quantities D and H are ∇ · D = ρ(t, x) ∂B ∇×E = − ∂t ∇·B = 0 ∂D ∇×H = + j(t, x) ∂t

(1.47a) (1.47b) (1.47c) (1.47d)

and are called Maxwell’s macroscopic equations. We will study them in more detail in Chapter 6.

1.4 Electromagnetic Duality If we look more closely at the microscopic Maxwell equations (1.48), we see that they exhibit a certain, albeit not a complete, symmetry. Let us further make the ad hoc assumption that there exist magnetic monopoles represented by a magnetic charge density, denoted ρm = ρm (t, x), and a magnetic current density, denoted jm = jm (t, x). With these new quantities included in the theory, and with the elecric charge density denoted ρe and the electric current density denoted je , the Maxwell equations can be written ρe ε0 ∂B ∇×E = − − µ0 jm ∂t 1 ρm ∇ · B = µ0 ρm = 2 c ε0 ∂E 1 ∂E ∇ × B = ε 0 µ0 + µ0 je = 2 + µ0 je ∂t c ∂t ∇·E =


(1.48a) (1.48b) (1.48c) (1.48d)


16


We shall call these equations Dirac’s symmetrised Maxwell equations or the electromagnetodynamic equations Taking the divergence of (1.48b), we find that ∇ · (∇ × E) = −

∂ (∇ · B) − µ0 ∇ · jm ≡ 0 ∂t

(1.49)

where we used the fact that, according to Formula (M.94) on page 184, the divergence of a curl always vanishes. Using (1.48c) to rewrite this relation, we obtain the equation of continuity for magnetic monopoles ∂ρm + ∇ · jm = 0 ∂t

(1.50)

which has the same form as that for the electric monopoles (electric charges) and currents, Equation (1.21) on page 9. We notice that the new Equations (1.48) on the preceding page exhibit the following symmetry (recall that ε0 µ0 = 1/c2 ): E → cB

(1.51a)

cB → −E e

cρ → ρ

(1.51b)

m

(1.51c)

m

ρ → −cρ e

cj → j

e

(1.51d)

m

m

j → −cj

(1.51e)

e

(1.51f)

which is a particular case (θ = π/2) of the general duality transformation (depicted by the Hodge star operator) ?

E = E cos θ + cB sin θ

?

(1.52a)

c B = −E sin θ + cB cos θ

(1.52b)

c ρ = cρ cos θ + ρ sin θ

(1.52c)

? m

(1.52d)

? e

e

m

e

m

ρ = −cρ sin θ + ρ cos θ

? e

e

m

c j = cj cos θ + j sin θ

(1.52e)

? m

(1.52f)

e

m

j = −cj sin θ + j cos θ

which leaves the symmetrised Maxwell equations, and hence the physics they describe (often referred to as electromagnetodynamics), invariant. Since E and je are (true or polar) vectors, B a pseudovector (axial vector), ρe a (true) scalar, then ρm and θ, which behaves as a mixing angle in a two-dimensional “charge space,” must be pseudoscalars and jm a pseudovector.



1.4

17


FARADAY ’ S LAW AS

E XAMPLE 1.1

A CONSEQUENCE OF CONSERVATION OF MAGNETIC CHARGE

Postulate 1.1 (Indestructibility of magnetic charge). Magnetic charge exists and is indestructible in the same way that electric charge exists and is indestructible. In other words we postulate that there exists an equation of continuity for magnetic charges. Use this postulate and Dirac’s symmetrised form of Maxwell’s equations to derive Faraday’s law. The assumption of existence of magnetic charges suggests that there exists a Coulomb law for magnetic fields: µ0 µ0 x − x0 3 0 d x =− ρm (x0 ) 3 0 4π V 0 4π − x | |x 1 µ0 − ∇ ρm (x0 ) d3x0 = 4π |x − x0 | V0

Bstat (x) =

[cf. Equation (1.5) on page 4 for law for electric fields: Estat (x) = −

µ0 4π

V

0

jm (x0 ) ×

Vρ

m

0

(x0 )∇

1 d3x0 |x − x0 |

(1.53)

Estat ]

and, if magnetic currents exist, a Biot-Savart

x − x 0 3 0 µ0 dx = 4π |x − x0 |3

Vj 0

m

(x0 ) × ∇

1 d3x0 |x − x0 | (1.54)

Taking the curl of the latter and using the operator “bac-cab” rule, Formula (F.62) on page 165, we find that ∇ × Estat (x) = µ0 =− 4π

V0

jm (x0 ) 3 0 dx = 0 V0 x − x 1 µ0 d3x0 + [jm (x0 ) · ∇0 ]∇0 0 4π V 0 |x − x |

µ0 ∇× ∇× 4π jm (x0 )∇2

1 d3x0 |x − x0 | (1.55)

Comparing with Equation (1.16) on page 7 for Estat and the evaluation of the integrals there we obtain ∇ × Estat (x) = µ0

Vj

m

0

(x0 )δ(x − x0 ) d3x0 = µ0 jm (x)

(1.56)

We assume that Formula (1.54) is valid also for time-varying magnetic currents. Then, with the use of the representation of the Dirac delta function, Equation (M.85) on page 183, the equation of continuity for magnetic charge, Equation (1.50) on the facing page, and the assumption of the generalisation of Equation (1.53) above to timedependent magnetic charge distributions, we obtain, formally, ∇ × E(t, x) = −µ0

Vj 0

m

(t, x0 )δ(x − x0 ) d3x0 −

= −µ0 jm (t, x) −

∂ B(t, x) ∂t

µ0 ∂ 4π ∂t

Vρ 0

m

(t, x0 )∇0

1 d3x0 |x − x0 |

(1.57)



18


[cf. Equation (1.22) on page 9] which we recognise as Equation (1.48b) on page 15. A transformation of this electromagnetodynamic result by rotating into the “electric realm” of charge space, thereby letting jm tend to zero, yields the electrodynamic Equation (1.48b) on page 15, i.e., the Faraday law in the ordinary Maxwell equations. By postulating the indestructibility of an hypothetical magnetic charge, we have thereby been able to replace Faraday’s experimental results on electromotive forces and induction in loops as a foundation for the Maxwell equations by a more appealing one. E ND

E XAMPLE 1.2

OF EXAMPLE

1.1C

D UALITY OF THE ELECTROMAGNETODYNAMIC EQUATIONS

Show that the symmetric, electromagnetodynamic form of Maxwell’s equations (Dirac’s symmetrised Maxwell equations), Equations (1.48) on page 15, are invariant under the duality transformation (1.52). Explicit application of the transformation yields ρe cos θ + cµ0 ρm sin θ ε0 ?ρe 1 1 = ρe cos θ + ρm sin θ = ε0 c ε0 ?B ∂ ∂ 1 = ∇ × (E cos θ + cB sin θ) + ∇ × ?E + − E sin θ + B cos θ ∂t ∂t c ∂B 1 ∂E = −µ0 jm cos θ − cos θ + cµ0 je sin θ + sin θ ∂t c ∂t 1 ∂E ∂B sin θ + cos θ = −µ0 jm cos θ + cµ0 je sin θ − c ∂t ∂t = −µ0 (−cje sin θ + jm cos θ) = −µ0 ?jm 1 ρe ∇ · ?B = ∇ · (− E sin θ + B cos θ) = − sin θ + µ0 ρm cos θ c cε0 = µ0 −cρe sin θ + ρm cos θ = µ0 ?ρm ∇ · ?E = ∇ · (E cos θ + cB sin θ) =

(1.58)

(1.59)

(1.60)

1 ∂?E 1 1 ∂ ∇ × ?B − 2 = ∇ × (− E sin θ + B cos θ) − 2 (E cos θ + cB sin θ) c ∂t c c ∂t 1 m 1 ∂B 1 ∂E = µ0 j sin θ + cos θ + µ0 je cos θ + 2 cos θ c c ∂t c ∂t (1.61) 1 ∂E 1 ∂B − 2 cos θ − sin θ c ∂t c ∂t 1 m = µ0 j sin θ + je cos θ = µ0 ?je c

QED E ND


OF EXAMPLE

1.2C


1.4

19


D IRAC ’ S SYMMETRISED M AXWELL EQUATIONS FOR A

E XAMPLE 1.3

FIXED MIXING ANGLE

Show that for a fixed mixing angle θ such that ρm = cρe tan θ jm = cje tan θ

(1.62a) (1.62b)

the symmetrised Maxwell equations reduce to the usual Maxwell equations. Explicit application of the fixed mixing angle conditions on the duality transformation (1.52) on page 16 yields 1 1 ρ = ρe cos θ + ρm sin θ = ρe cos θ + cρe tan θ sin θ c c 1 1 e e 2 e 2 (ρ cos θ + ρ sin θ) = ρ = cos θ cos θ ? m ρ = −cρe sin θ + cρe tan θ cos θ = −cρe sin θ + cρe sin θ = 0 1 e 1 e 2 ? e (j cos θ + je sin2 θ) = j j = je cos θ + je tan θ sin θ = cos θ cos θ ? m j = −cje sin θ + cje tan θ cos θ = −cje sin θ + cje sin θ = 0 ? e

(1.63a) (1.63b) (1.63c) (1.63d)

Hence, a fixed mixing angle, or, equivalently, a fixed ratio between the electric and magnetic charges/currents, “hides” the magnetic monopole influence (ρ m and jm ) on the dynamic equations. We notice that the inverse of the transformation given by Equation (1.52) on page 16 yields E = ?E cos θ − c?B sin θ

(1.64)

This means that ∇ · E = ∇ · ?E cos θ − c∇ · ?B sin θ

(1.65)

Furthermore, from the expressions for the transformed charges and currents above, we find that ?ρe 1 ρe ∇ · ?E = = (1.66) ε0 cos θ ε0 and ∇ · ?B = µ0 ?ρm = 0

(1.67)

so that ∇·E =

ρe 1 ρe cos θ − 0 = cos θ ε0 ε0

(1.68)

and so on for the other equations.

QED E ND OF


EXAMPLE

1.3C


20


The invariance of Dirac’s symmetrised Maxwell equations under the similarity transformation means that the amount of magnetic monopole density ρm is irrelevant for the physics as long as the ratio ρm /ρe = tan θ is kept constant. So whether we assume that the particles are only electrically charged or have also a magnetic charge with a given, fixed ratio between the two types of charges is a matter of convention, as long as we assume that this fraction is the same for all particles. Such particles are referred to as dyons. By varying the mixing angle θ we can change the fraction of magnetic monopoles at will without changing the laws of electrodynamics. For θ = 0 we recover the usual Maxwell electrodynamics as we know it. E XAMPLE 1.4

T HE

COMPLEX FIELD SIX - VECTOR

The complex field six-vector G(t, x) = E(t, x) + icB(t, x) where E, B ∈

3

and hence G ∈

(1.69) 3,

has a number of interesting properties:

1. The inner product of G with itself G · G = (E + icB) · (E + icB) = E 2 − c2 B2 + 2icE · B

(1.70)

is conserved. I.e., E 2 − c2 B2 = Const E · B = Const

(1.71a) (1.71b)

as we shall see later. 2. The inner product of G with the complex conjugate of itself G · G∗ = (E + icB) · (E − icB) = E 2 + c2 B2

(1.72)

is proportional to the electromagnetic field energy. 3. As with any vector, the cross product of G itself vanishes: G × G = (E + icB) × (E + icB)

= E × E − c2 B × B + ic(E × B) + ic(B × E) = 0 + 0 + ic(E × B) − ic(E × B) = 0

(1.73)

4. The cross product of G with the complex conjugate of itself G × G∗ = (E + icB) × (E − icB)

= E × E + c2 B × B − ic(E × B) + ic(B × E) = 0 + 0 − ic(E × B) − ic(E × B) = −2ic(E × B)

(1.74)

is proportional to the electromagnetic power flux.

E ND


OF EXAMPLE

1.4C


1.4

21

B IBLIOGRAPHY

D UALITY EXPRESSED IN

THE COMPLEX FIELD SIX - VECTOR

E XAMPLE 1.5

Expressed in the complex field vector, introduced in Example 1.4 on the facing page, the duality transformation Equations (1.52) on page 16 become ?

G = ?E + ic?B = E cos θ + cB sin θ − iE sin θ + icB cos θ

= E(cos θ − i sin θ) + icB(cos θ − i sin θ) = e−iθ (E + icB) = e−iθ G

(1.75)

from which it is easy to see that G · ?G∗ =

?

while

?F 2 = e−iθ G · eiθ G∗ = |F|2

(1.76)

?

G · ?G = e2iθ G · G

(1.77)

Furthermore, assuming that θ = θ(t, x), we see that the spatial and temporal differentiation of ?G leads to ∂?G ∂t ?G ≡ = −i(∂t θ)e−iθ G + e−iθ ∂t G (1.78a) ∂t ∂ · ?G ≡ ∇ · ?G = −ie−iθ ∇θ · G + e−iθ ∇ · G (1.78b) ∂ × ?G ≡ ∇ × ?G = −ie−iθ ∇θ × G + e−iθ ∇ × G

(1.78c)

which means that ∂t ?G transforms as ?G itself if θ is time-independent, and that ∇ · ?G and ∇ × ?G transform as ?G itself if θ is space-independent. E ND OF

EXAMPLE

1.5C

Bibliography [1] R. B ECKER, Electromagnetic Fields and Interactions, Dover Publications, Inc., New York, NY, 1982, ISBN 0-486-64290-9. [2] W. G REINER, Classical Electrodynamics, Springer-Verlag, New York, Berlin, Heidelberg, 1996, ISBN 0-387-94799-X. [3] E. H ALLÉN, Electromagnetic Theory, Chapman & Hall, Ltd., London, 1962. [4] J. D. JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, NY . . . , 1999, ISBN 0-471-30932-X. [5] L. D. L ANDAU AND E. M. L IFSHITZ, The Classical Theory of Fields, fourth revised English ed., vol. 2 of Course of Theoretical Physics, Pergamon Press, Ltd., Oxford . . . , 1975, ISBN 0-08-025072-6. [6] J. C. M AXWELL, A dynamical theory of the electromagnetic field, Royal Society Transactions, 155 (1864).



22


[7] 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. [8] J. C. M AXWELL, A Treatise on Electricity and Magnetism, third ed., vol. 2, Dover Publications, Inc., New York, NY, 1954, ISBN 0-486-60637-8. [9] D. B. M ELROSE AND R. C. M C P HEDRAN, Electromagnetic Processes in Dispersive Media, Cambridge University Press, Cambridge . . . , 1991, ISBN 0-52141025-8. [10] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6. [11] J. S CHWINGER, A magnetic model of matter, Science, 165 (1969), pp. 757–761. [12] J. S CHWINGER , L. L. D E R AAD , J R ., K. A. M ILTON , AND W. T SAI, Classical Electrodynamics, Perseus Books, Reading, MA, 1998, ISBN 0-7382-0056-5. [13] J. A. S TRATTON, Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, NY and London, 1953, ISBN 07-062150-0. [14] J. VANDERLINDE, Classical Electromagnetic Theory, John Wiley & Sons, Inc., New York, Chichester, Brisbane, Toronto, and Singapore, 1993, ISBN 0-47157269-1.



2 Electromagnetic Waves

In this chapter we investigate the dynamical properties of the electromagnetic field by deriving an set of equations which are alternatives to the Maxwell equations. It turns out that these alternative equations are wave equations, indicating that electromagnetic waves are natural and common manifestations of electrodynamics. Maxwell’s microscopic equations [cf. Equations (1.43) on page 14] are ρ(t, x) ε0 ∂B ∇×E = − ∂t ∇·B = 0 ∂E ∇ × B = ε 0 µ0 + µ0 j(t, x) ∂t ∇·E =

(2.1a) (2.1b) (2.1c) (2.1d)

and can be viewed as an axiomatic basis for classical electrodynamics. In particular, these equations are well suited for calculating the electric and magnetic fields E and B from given, prescribed charge distributions ρ(t, x) and current distributions j(t, x) of arbitrary time- and space-dependent form. However, as is well known from the theory of differential equations, these four first order, coupled partial differential vector equations can be rewritten as two un-coupled, second order partial equations, one for E and one for B. We shall derive these second order equations which, as we shall see are wave equations, and then discuss the implications of them. We shall also show how the B wave field can be easily calculated from the solution of the E wave equation.

23

24

E LECTROMAGNETIC WAVES

2.1 The Wave Equations We restrict ourselves to derive the wave equations for the electric field vector E and the magnetic field vector B in a volume with no net charge, ρ = 0, and no electromotive force EEMF = 0.

2.1.1 The wave equation for E In order to derive the wave equation for E we take the curl of (2.1b) and using (2.1d), to obtain ∇ × (∇ × E) = −

∂ ∂ ∂ (∇ × B) = −µ0 j + ε0 E ∂t ∂t ∂t

(2.2)

According to the operator triple product “bac-cab” rule Equation (F.67) on page 165 ∇ × (∇ × E) = ∇(∇ · E) − ∇2 E

(2.3)

Furthermore, since ρ = 0, Equation (2.1a) on the preceding page yields ∇·E = 0

(2.4)

and since EEMF = 0, Ohm’s law, Equation (1.26) on page 10, yields j = σE

(2.5)

we find that Equation (2.2) can be rewritten ∇2 E − µ 0

∂ σE + ε0 ∂t∂ E = 0 ∂t

(2.6)

or, also using Equation (1.9) on page 5 and rearranging, ∇2 E − µ 0 σ

∂E 1 ∂2 E − =0 ∂t c2 ∂t2

(2.7)

which is the homogeneous wave equation for E.

2.1.2 The wave equation for B The wave equation for B is derived in much the same way as the wave equation for E. Take the curl of (2.1d) and use Ohm’s law j = σE to obtain ∇ × (∇ × B) = µ0 ∇ × j + ε0 µ0


∂ ∂ (∇ × E) = µ0 σ∇ × E + ε0 µ0 (∇ × E) (2.8) ∂t ∂t


2.1

25

T HE WAVE E QUATIONS

which, with the use of Equation (F.67) on page 165 and Equation (2.1c) on page 23 can be rewritten ∇(∇ · B) − ∇2 B = −µ0 σ

∂2 ∂B − ε0 µ0 2 B ∂t ∂t

(2.9)

Using the fact that, according to (2.1c), ∇ · B = 0 for any medium and rearranging, we can rewrite this equation as ∂B 1 ∂2 B − =0 ∂t c2 ∂t2

∇2 B − µ 0 σ

(2.10)

This is the wave equation for the magnetic field. We notice that it is of exactly the same form as the wave equation for the electric field, Equation (2.7) on the facing page.

2.1.3 The time-independent wave equation for E We now look for a solution to any of the wave equations in the form of a timeharmonic wave. As is clear from the above, it suffices to consider only the E field, since the results for the B field follow trivially. We therefore make the following Fourier component Ansatz E = E0 (x)e−iωt

(2.11)

and insert this into Equation (2.7) on the preceding page. This yields 1 ∂2 ∂ E0 (x)e−iωt − 2 2 E0 (x)e−iωt ∂t c ∂t 1 = ∇2 E − µ0 σ(−iω)E0 (x)e−iωt − 2 (−iω)2 E0 (x)e−iωt c 1 2 2 = ∇ E − µ0 σ(−iω)E − 2 (−iω) E = c 2 σ ω E=0 = ∇2 E + 2 1 + i c ε0 ω

∇2 E − µ 0 σ

(2.12)

Introducing the relaxation time τ = ε0 /σ of the medium in question we can rewrite this equation as ∇2 E +

ω2 c2

1+

i E=0 τω

(2.13)

In the limit of long τ, Equation (2.13) tends to ∇2 E +

ω2 E=0 c2


(2.14)


26


which is a time-independent wave equation for E, representing weakly damped propagating waves. In the short τ limit we have instead ∇2 E + iωµ0 σE = 0

(2.15)

which is a time-independent diffusion equation for E. For most metals τ ∼ 10−14 s, which means that the diffusion picture is good for all frequencies lower than optical frequencies. Hence, in metallic conductors, the propagation term ∂2 E/c2 ∂t2 is negligible even for VHF, UHF, and SHF signals. Alternatively, we may say that the displacement current ε 0 ∂E/∂t is negligible relative to the conduction current j = σE. If we introduce the vacuum wave number ω (2.16) k= c √ we can write, using the fact that c = 1/ ε0 µ0 according to Equation (1.9) on page 5, 1 σ σ 1 σ = = = τω ε0 ω ε0 ck k

µ0 σ = R0 ε0 k

(2.17)

where in the last step we introduced the characteristic impedance for vacuum R0 =

E XAMPLE 2.1

µ0 ≈ 376.7 Ω ε0

WAVE EQUATIONS IN

(2.18)

ELECTROMAGNETODYNAMICS

Derive the wave equation for the E field described by the electromagnetodynamic equations (Dirac’s symmetrised Maxwell equations) [cf. Equations (1.48) on page 15] ρe (2.19a) ε0 ∂B ∇×E = − − µ 0 jm (2.19b) ∂t m ∇ · B = µ0 ρ (2.19c) ∂E + µ 0 je (2.19d) ∇ × B = ε 0 µ0 ∂t under the assumption of vanishing net electric and magnetic charge densities and in the absence of electromotive and magnetomotive forces. Interpret this equation physically. ∇·E =

Taking the curl of (2.19b) and using (2.19d), and assuming, for symmetry reasons, that there exists a linear relation between the magnetic current density jm and the magnetic



2.1

27

T HE WAVE E QUATIONS

field B (the analogue of Ohm’s law for electric currents, je = σe E) jm = σ m B

(2.20)

one finds, noting that ε0 µ0 =

1/c2 ,

∇ × (∇ × E) = −µ0 ∇ × jm −

that

∂ ∂ 1 ∂E (∇ × B) = −µ0 σm ∇ × B − µ0 j e + 2 ∂t ∂t c ∂t

= −µ0 σm µ0 σe E +

∂2 E

1 ∂E ∂E 1 − − µ0 σ e 2 c ∂t ∂t c2 ∂t2

(2.21) = ∇(∇ · E) − ∇2 E,

Using the vector operator identity ∇ × (∇ × E) and the fact that ∇ · E = 0 for a vanishing net electric charge, we can rewrite the wave equation as

∇2 E − µ 0 σe +

σm c2

∂E 1 ∂2 E − − µ20 σm σe E = 0 ∂t c2 ∂t2

(2.22)

This is the homogeneous electromagnetodynamic wave equation for E we were after. Compared to the ordinary electrodynamic wave equation for E, Equation (2.7) on page 24, we see that we pick up extra terms. In order to understand what these extra terms mean physically, we analyse the time-independent wave equation for a single Fourier component. Then our wave equation becomes

∇2 E + iωµ0 σe + ω2

= ∇2 E +

c2

σm c2

E+

ω2 E − µ20 σm σe E c2

1 µ0 m e σe + σm /c2 E=0 σ σ +i 2 ω ε0 ε0 ω

1−

(2.23)

Realising that, according to Formula (2.18) on the preceding page, µ 0 /ε0 is the square of the vacuum radiation resistance R0 , and rearranging a bit, we obtain the timeindependent wave equation in Dirac’s symmetrised electrodynamics ∇2 E +

ω2 c2

1−

R20 m e σ σ ω2

!"#

1+i

σe + σm /c2

$

R2

ε0 ω 1 − ω02 σm σe

* %'&)(

E=0

(2.24)

From this equation we conclude that the existence of magnetic charges (magnetic monopoles), and non-vanishing electric and magnetic conductivities would lead to a shift in the effective wave number of the wave. Furthermore, even if the electric conductivity vanishes, the imaginary term does not necessarily vanish and the wave might therefore experience damping (or growth) according as σ m is positive (or negative) in √ a perfect electric isolator. Finally, we note that in the particular case that ω = R0 σm σe , the wave equation becomes a (time-independent) diffusion equation

∇2 E + iωµ0 σe +

σm c2

E=0

(2.25)

and, hence, no waves exist at all! E ND OF


EXAMPLE

2.1C


28


2.2 Plane Waves Consider now the case where all fields depend only on the distance ζ to a given plane with unit normal n. ˆ Then the del operator becomes ∇ = nˆ

∂ ∂ζ

(2.26)

and Maxwell’s equations attain the form ∂E ∂ζ ∂E n× ˆ ∂ζ ∂B n· ˆ ∂ζ ∂B n× ˆ ∂ζ n· ˆ

=0 =−

(2.27a) ∂B ∂t

(2.27b)

=0

(2.27c)

= µ0 j(t, x) + ε0 µ0

∂E ∂E = µ0 σE + ε0 µ0 ∂t ∂t

(2.27d)

Scalar multiplying (2.27d) by n, ˆ we find that 0 = n· ˆ n× ˆ

∂B ∂ = n· ˆ µ0 σ + ε 0 µ0 E ∂ζ ∂t

(2.28)

which simplifies to the first-order ordinary differential equation for the normal component En of the electric field dEn σ + En = 0 dt ε0

(2.29)

with the solution En = En0 e−σt/ε0 = En0 e−t/τ

(2.30)

This, together with (2.27a), shows that the longitudinal component of E, i.e., the component which is perpendicular to the plane surface is independent of ζ and has a time dependence which exhibits an exponential decay, with a decrement given by the relaxation time τ in the medium. Scalar multiplying (2.27b) by n, ˆ we similarly find that 0 = n· ˆ n× ˆ

∂E ∂B = − n· ˆ ∂ζ ∂t

(2.31)

or n· ˆ

∂B =0 ∂t


(2.32)


2.2

29

P LANE WAVES

From this, and (2.27c), we conclude that the only longitudinal component of B must be constant in both time and space. In other words, the only non-static solution must consist of transverse components.

2.2.1 Telegrapher’s equation In analogy with Equation (2.7) on page 24, we can easily derive the equation ∂E 1 ∂2 E ∂2 E − µ σ − =0 0 ∂ζ 2 ∂t c2 ∂t2

(2.33)

This equation, which describes the propagation of plane waves in a conducting medium, is called the telegrapher’s equation. If the medium is an insulator so that σ = 0, then the equation takes the form of the one-dimensional wave equation ∂2 E 1 ∂2 E − =0 ∂ζ 2 c2 ∂t2

(2.34)

As is well known, each component of this equation has a solution which can be written Ei = f (ζ − ct) + g(ζ + ct),

i = 1, 2, 3

(2.35)

where f and g are arbitrary (non-pathological) functions of their respective arguments. This general solution represents perturbations which propagate along ζ, where the f perturbation propagates in the positive ζ direction and the g perturbation propagates in the negative ζ direction. If we assume that our electromagnetic fields E and B are time-harmonic, i.e., that they can each be represented by a Fourier component proportional to exp{−iωt}, the solution of Equation (2.34) becomes E = E0 e−i(ωt±kζ) = E0 ei(∓kζ−ωt)

(2.36)

By introducing the wave vector k = k nˆ =

ω ω nˆ = kˆ c c

(2.37)

this solution can be written as E = E0 ei(k·x−ωt)


(2.38)


30


Let us consider the lower sign in front of kζ in the exponent in (2.36). This corresponds to a wave which propagates in the direction of increasing ζ. Inserting this solution into Equation (2.27b) on page 28, gives n× ˆ

∂E = iωB = ik n× ˆ E ∂ζ

(2.39)

or, solving for B, B=

k 1 1 √ n× ˆ E = k × E = kˆ × E = ε0 µ0 n× ˆ E ω ω c

(2.40)

Hence, to each transverse component of E, there exists an associated magnetic field given by Equation (2.40). If E and/or B has a direction in space which is constant in time, we have a plane polarised wave (or linearly polarised wave).

2.2.2 Waves in conductive media Assuming that our medium has a finite conductivity σ, and making the timeharmonic wave Ansatz in Equation (2.33) on the previous page, we find that the time-independent telegrapher’s equation can be written ∂2 E ∂2 E 2 + ε µ ω E + iµ σωE = + K2E = 0 0 0 0 ∂ζ 2 ∂ζ 2

(2.41)

where K 2 = ε0 µ0 ω2 1 + i

σ ω2 = 2 ε0 ω c

1+i

σ σ = k2 1 + i ε0 ω ε0 ω

(2.42)

where, in the last step, Equation (2.16) on page 26 was used to introduce the wave number k. Taking the square root of this expression, we obtain K=k

1+i

σ = α + iβ ε0 ω

(2.43)

Squaring, one finds that k2 1 + i

σ = (α2 − β2 ) + 2iαβ ε0 ω

(2.44)

or β2 = α2 − k2

(2.45)

αβ =

(2.46)

k2 σ 2ε0 ω



2.2

31

P LANE WAVES

Squaring the latter and combining with the former, one obtains the second order algebraic equation (in α2 ) k 4 σ2 4ε20 ω2

α2 (α2 − k2 ) =

(2.47)

which can be easily solved and one finds that

+,, ,

α=k

- +,, ,

- β=k

.

/

2

/

2

σ ε0 ω

1+

+1 (2.48a)

2 1+

.

σ ε0 ω

−1

(2.48b)

2

As a consequence, the solution of the time-independent telegrapher’s equation, Equation (2.41) on the preceding page, can be written E = E0 e−βζ ei(αζ−ωt)

(2.49)

With the aid of Equation (2.40) on the facing page we can calculate the associated magnetic field, and find that it is given by B=

1 ˆ 1 1 K k × E = ( kˆ × E)(α + iβ) = ( kˆ × E) |A| eiγ ω ω ω

(2.50)

where we have, in the last step, rewritten α + iβ in the amplitude-phase form |A| exp{iγ}. From the above, we immediately see that E, and consequently also B, is damped, and that E and B in the wave are out of phase. In the case that ε0 ω σ, we can approximate K as follows: 1 2

σ K = k 1+i ε0 ω √ = ε0 µ0 ω(1 + i)

σ . 1 − i εσ0 ω / =k i ε0 ω σ = (1 + i) 2ε0 ω

1 2

≈ k(1 + i)

µ0 σω 2

σ 2ε0 ω

(2.51)

From this analysis we conclude that when the wave impinges perpendicularly upon the medium, the fields are given, inside this medium, by E0 = E0 exp − B0 = (1 + i)

µ0 σω ζ exp i 2

µ0 σ ( n× ˆ E0 ) 2ω


µ0 σω ζ − ωt 2 0

(2.52a) (2.52b)


32


Hence, both fields fall off by a factor 1/e at a distance δ=

2 µ0 σω

(2.53)

This distance δ is called the skin depth.

2.3 Observables and Averages In the above we have used complex notation quite extensively. This is for mathematical convenience only. For instance, in this notation differentiations are almost trivial to perform. However, every physical measurable quantity is always real valued. I.e., “Ephysical = Re {Emathematical }.” It is particularly important to remember this when one works with products of physical quantities. For instance, if we have two physical vectors F and G which both are time-harmonic, i.e., can be represented by Fourier components proportional to exp{−iωt}, then we must make the following interpretation F(t, x) · G(t, x) = Re {F} · Re {G} = Re 1 F0 (x) e−iωt 2 · Re 1 G0 (x) e−iωt 2 (2.54) Furthermore, letting ∗ denotes complex conjugate, we can express the real part of the complex vector F as 1 Re {F} = Re 1 F0 (x) e−iωt 2 = [F0 (x) e−iωt + F∗0 (x) eiωt ] 2

(2.55)

and similarly for G. Hence, the physically acceptable interpretation of the scalar product of two complex vectors, representing physical observables, is F(t, x) · G(t, x) = Re 1 F0 (x) e−iωt 2 · Re 1 G0 (x) e−iωt 2 1 1 = [F0 (x) e−iωt + F∗0 (x) eiωt ] · [G0 (x) e−iωt + G∗0 (x) eiωt ] 2 2 1 ∗ ∗ = F0 · G0 + F0 · G0 + F0 · G0 e−2iωt + F∗0 · G∗0 e2iωt 4 43 (2.56) 1 = Re 1 F0 · G∗0 + F0 · G0 e−2iωt 2 2 1 = Re 1 F0 e−iωt · G∗0 eiωt + F0 · G0 e−2iωt 2 2 1 = Re 1 F(t, x) · G∗ (t, x) + F0 · G0 e−2iωt 2 2



2.3

33

B IBLIOGRAPHY

Often in physics, we measure temporal averages (h i) of our physical observables. If so, we see that the average of the product of the two physical quantities represented by F and G can be expressed as 1 1 hF · Gi ≡ hRe {F} · Re {G}i = Re 1 F · G∗ 2 = Re 1 F∗ · G 2 2 2

(2.57)

since the temporal average of the oscillating function exp{−2iωt} vanishes.

Bibliography [1] J. D. JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, NY . . . , 1999, ISBN 0-471-30932-X. [2] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6.



34




3 Electromagnetic Potentials

Instead of expressing the laws of electrodynamics in terms of electric and magnetic fields, it turns out that it is often more convenient to express the theory in terms of potentials. In this chapter we will introduce and study the properties of such potentials.

3.1 The Electrostatic Scalar Potential As we saw in Equation (1.6) on page 4, the electrostatic field Estat (x) is irrotational. Hence, it may be expressed in terms of the gradient of a scalar field. If we denote this scalar field by −φstat (x), we get Estat (x) = −∇φstat (x)

(3.1)

Taking the divergence of this and using Equation (1.7) on page 4, we obtain Poisson’s equation ∇2 φstat (x) = −∇ · Estat (x) = −

ρ(x) ε0

(3.2)

A comparison with the definition of Estat , namely Equation (1.5) on page 4, shows that this equation has the solution φstat (x) =

1 4πε0

V

ρ(x0 ) 3 0 d x +α |x − x0 |

(3.3)

where the integration is taken over all source points x0 at which the charge density ρ(x0 ) is non-zero and α is an arbitrary quantity which has a vanishing gradient. An example of such a quantity is a scalar constant. The scalar function φstat (x) in Equation (3.3) is called the electrostatic scalar potential.

35

36

E LECTROMAGNETIC P OTENTIALS

3.2 The Magnetostatic Vector Potential Consider the equations of magnetostatics (1.20) on page 8. From Equation (M.94) on page 184 we know that any 3D vector a has the property that ∇ · (∇ × a) ≡ 0 and in the derivation of Equation (1.15) on page 7 in magnetostatics we found that ∇ · Bstat (x) = 0. We therefore realise that we can always write Bstat (x) = ∇ × Astat (x)

(3.4)

where Astat (x) is called the magnetostatic vector potential. We saw above that the electrostatic potential (as any scalar potential) is not unique: we may, without changing the physics, add to it a quantity whose spatial gradient vanishes. A similar arbitrariness is true also for the magnetostatic vector potential. In the magnetostatic case, we may start from Biot-Savart’s law as expressed by Equation (1.13) on page 6. Identifying this expression with Equation (3.4) allows us to define the static vector potential as Astat (x) =

µ0 4π

V

j(x0 ) 3 0 d x + a(x) |x − x0 |

(3.5)

where a(x) is an arbitrary vector field whose curl vanishes. From Equation (M.90) on page 184 we know that such a vector can always be written as the gradient of a scalar field.

3.3 The Electrodynamic Potentials Let us now generalise the static analysis above to the electrodynamic case, i.e., the case with temporal and spatial dependent sources ρ(t, x) and j(t, x), and corresponding fields E(t, x) and B(t, x), as described by Maxwell’s equations (1.43) on page 14. In other words, let us study the electrodynamic potentials φ(t, x) and A(t, x). From Equation (1.43c) on page 14 we note that also in electrodynamics the homogeneous equation ∇ · B(t, x) = 0 remains valid. Because of this divergence-free nature of the time- and space-dependent magnetic field, we can express it as the curl of an electromagnetic vector potential: B(t, x) = ∇ × A(t, x)


(3.6)


3.3

37

T HE E LECTRODYNAMIC P OTENTIALS

Inserting this expression into the other homogeneous Maxwell equation, Equation (1.30) on page 11, we obtain ∇ × E(t, x) = −

∂ ∂ [∇ × A(t, x)] = −∇ × A(t, x) ∂t ∂t

(3.7)

or, rearranging the terms, ∇ × E(t, x) +

∂ A(t, x) = 0 ∂t

(3.8)

As before we utilise the vanishing curl of a vector expression to write this vector expression as the gradient of a scalar function. If, in analogy with the electrostatic case, we introduce the electromagnetic scalar potential function −φ(t, x), Equation (3.8) becomes equivalent to E(t, x) +

∂ A(t, x) = −∇φ(t, x) ∂t

(3.9)

This means that in electrodynamics, E(t, x) can be calculated from the formula E(t, x) = −∇φ(t, x) −

∂ A(t, x) ∂t

(3.10)

and B(t, x) from Equation (3.6) on the preceding page. Hence, it is a matter of taste whether we want to express the laws of electrodynamics in terms of the potentials φ(t, x) and A(t, x), or in terms of the fields E(t, x) and B(t, x). However, there exists an important difference between the two approaches: in classical electrodynamics the only directly observable quantities are the fields themselves (and quantities derived from them) and not the potentials. On the other hand, the treatment becomes significantly simpler if we use the potentials in our calculations and then, at the final stage, use Equation (3.6) on the facing page and Equation (3.10) above to calculate the fields or physical quantities expressed in the fields. Inserting (3.10) and (3.6) on the facing page into Maxwell’s equations (1.43) on page 14 we obtain, after some simple algebra and the use of Equation (1.9) on page 5, the general inhomogeneous wave equations ρ(t, x) ∂ (∇ · A) = − ∂t ε0 2A 1 ∂ 1 ∂φ ∇2 A − 2 2 − ∇ ∇ · A + 2 = −µ0 j(t, x) c ∂t c ∂t ∇2 φ +


(3.11a) (3.11b)


38


which can be rewritten in the following, more symmetric, form 1 ∂2 φ ∂ 1 ∂φ ρ(t, x) − ∇2 φ − ∇ · A + 2 = 2 2 c ∂t ∂t c ∂t ε0 2 1∂ A 1 ∂φ = µ0 j(t, x) − ∇2 A + ∇ ∇ · A + 2 c2 ∂t2 c ∂t

(3.12a) (3.12b)

These two second order, coupled, partial differential equations, representing in all four scalar equations (one for φ and one each for the three components Ai , i = 1, 2, 3 of A) are completely equivalent to the formulation of electrodynamics in terms of Maxwell’s equations, which represent eight scalar firstorder, coupled, partial differential equations.

3.3.1 Electrodynamic gauges As they stand, Equations (3.11) on the preceding page and Equations (3.12) above look complicated and may seem to be of limited use. However, if we write Equation (3.6) on page 36 in the form ∇×A(t, x) = B(t, x) we can consider this as a specification of ∇ × A. But we know from Helmholtz’ theorem that in order to determine the (spatial) behaviour of A completely, we must also specify ∇ · A. Since this divergence does not enter the derivation above, we are free to choose ∇ · A in whatever way we like and still obtain the same physical results!

Lorentz equations for the electrodynamic potentials With a judicious choice of ∇ · A, the inhomogeneous wave equations can be simplified considerably. To this end, Lorentz introduced the so called Lorentz gauge condition1 ∇·A+

1 ∂ φ=0 c2 ∂t

(3.13)

because this condition simplifies the system of coupled equations Equation (3.12) above into the following set of uncoupled partial differential equa1 In

fact, the Dutch physicist Hendrik Antoon Lorentz, who in 1903 demonstrated the covariance of Maxwell’s equations, was not the original discoverer of this condition. It had been discovered by the Danish physicist Ludwig Lorenz already in 1867 [4].



3.3

39


tions which we call the Lorentz inhomogeneous wave equations:

5 5 5

2

φ ≡

def

2

def

A ≡

1 ∂2 φ ρ(t, x) 1 ∂2 − ∇2 φ = 2 2 − ∇2 φ = 2 2 c ∂t c ∂t ε0 2 2 1 ∂ 1∂ A − ∇2 A = 2 2 − ∇2 A = µ0 j(t, x) 2 2 c ∂t c ∂t

(3.14a) (3.14b)

where 2 is the d’Alembert operator discussed in Example M.6 on page 180. We shall call (3.14) the Lorentz potential equations for the electrodynamic potentials.

Gauge transformations We saw in Section 3.1 on page 35 and in Section 3.2 on page 36 that in electrostatics and magnetostatics we have a certain mathematical degree of freedom, up to terms of vanishing gradients and curls, to pick suitable forms for the potentials and still get the same physical result. In fact, the way the electromagnetic scalar potential φ(t, x) and the vector potential A(t, x) are related to the physically observables gives leeway for similar “manipulation” of them also in electrodynamics. If we transform φ(t, x) and A(t, x) simultaneously into new ones φ0 (t, x) and A0 (t, x) according to the mapping scheme ∂Γ(t, x) ∂t A(t, x) 7→ A0 (t, x) = A(t, x) − ∇Γ(t, x) φ(t, x) 7→ φ0 (t, x) = φ(t, x) +

(3.15a) (3.15b)

where Γ(t, x) is an arbitrary, differentiable scalar function called the gauge function, and insert the transformed potentials into Equation (3.10) on page 37 for the electric field and into Equation (3.6) on page 36 for the magnetic field, we obtain the transformed fields ∂A0 ∂(∇Γ) ∂A ∂(∇Γ) ∂A = −∇φ − − + = −∇φ − ∂t ∂t ∂t ∂t ∂t B0 = ∇ × A0 = ∇ × A − ∇ × (∇Γ) = ∇ × A E0 = −∇φ0 −

(3.16a) (3.16b)

where, once again Equation (M.90) on page 184 was used. Comparing these expressions with (3.10) and (3.6) we see that the fields are unaffected by the gauge transformation (3.15). A transformation of the potentials φ and A which leaves the fields, and hence Maxwell’s equations, invariant is called a gauge transformation. A physical law which does not change under a gauge transformation is said to be gauge invariant. By definition, the fields themselves are, of course, gauge invariant.



40


The potentials φ(t, x) and A(t, x) calculated from (3.11) on page 37, with an arbitrary choice of ∇ · A, can be further gauge transformed according to (3.15) on the previous page. If, in particular, we choose ∇·A according to the Lorentz condition, Equation (3.13) on page 38, and apply the gauge transformation (3.15) on the resulting Lorentz potential equations (3.14) on the previous page, these equations will be transformed into ∂ 1 ∂2 φ − ∇2 φ + 2 2 c ∂t ∂t 2 1∂ A − ∇2 A − ∇ c2 ∂t2

ρ(t, x) 1 ∂2 Γ − ∇2 Γ = 2 2 c ∂t ε0 2 1∂ Γ − ∇2 Γ = µ0 j(t, x) c2 ∂t2

(3.17a) (3.17b)

We notice that if we require that the gauge function Γ(t, x) itself be restricted to fulfil the wave equation 1 ∂2 Γ − ∇2 Γ = 0 c2 ∂t2

(3.18)

these transformed Lorentz equations will keep their original form. The set of potentials which have been gauge transformed according to Equation (3.15) on the preceding page with a gauge function Γ(t, x) which is restricted to fulfil Equation (3.18) above, i.e., those gauge transformed potentials for which the Lorentz equations (3.14) are invariant, comprises the Lorentz gauge. Other useful gauges are • The radiation gauge, also known as the transverse gauge, defined by ∇ · A = 0. • The Coulomb gauge, defined by φ = 0, ∇ · A = 0. • The temporal gauge, also known as the Hamilton gauge, defined by φ = 0. • The axial gauge, defined by A3 = 0. The process of choosing a particular gauge condition is referred to as gauge fixing.

3.3.2 Solution of the Lorentz equations for the electromagnetic potentials As we see, the Lorentz equations (3.14) on the preceding page for φ(t, x) and A(t, x) represent a set of uncoupled equations involving four scalar equations



3.3

41


(one equation for φ and one equation for each of the three components of A). Each of these four scalar equations is an inhomogeneous wave equation of the following generic form:

5

2

Ψ(t, x) = f (t, x)

(3.19)

where Ψ is a shorthand for either φ or one of the vector components of A, and f is the pertinent generic source component. We assume that our sources are well-behaved enough in time t so that the Fourier transform pair for the generic source function def

F −1 [ fω (x)] ≡ f (t, x) = def

F [ f (t, x)] ≡ fω (x) =

∞

fω (x) e−iωt dω

(3.20a)

f (t, x) eiωt dt

(3.20b)

−∞ ∞

1 2π

−∞

exists, and that the same is true for the generic potential component: Ψ(t, x) = Ψω (x) =

∞ −∞

1 2π

Ψω (x) e−iωt dω ∞ −∞

Ψ(t, x) eiωt dt

(3.21a) (3.21b)

Inserting the Fourier representations (3.20a) and (3.21a) into Equation (3.19) above, and using the vacuum dispersion relation for electromagnetic waves ω = ck

(3.22)

the generic 3D inhomogeneous wave equation, Equation (3.19), turns into ∇2 Ψω (x) + k2 Ψω (x) = − fω (x)

(3.23)

which is a 3D inhomogeneous time-independent wave equation, often called the 3D inhomogeneous Helmholtz equation. As postulated by Huygen’s principle, each point on a wave front acts as a point source for spherical waves which form a new wave from a superposition of the individual waves from each of the point sources on the old wave front. The solution of (3.23) can therefore be expressed as a superposition of solutions of an equation where the source term has been replaced by a point source: ∇2G(x, x0 ) + k2G(x, x0 ) = −δ(x − x0 )


(3.24)


42


and the solution of Equation (3.23) on the previous page which corresponds to the frequency ω is given by the superposition Ψω (x) =

fω (x0 )G(x, x0 ) d3x0

(3.25)

The function G(x, x0 ) is called the Green function or the propagator. In Equation (3.24) on the preceding page, the Dirac generalised function δ(x − x0 ), which represents the point source, depends only on x − x0 and there is no angular dependence in the equation. Hence, the solution can only be dependent on r = |x − x0 | and not on the direction of x − x0 . If we interpret r as the radial coordinate in a spherically polar coordinate system, and recall the expression for the Laplace operator in such a coordinate system, Equation (3.24) on the previous page becomes d2 (rG) + k2 (rG) = −rδ(r) dr2

(3.26)

Away from r = |x − x0 | = 0, i.e., away from the source point x0 , this equation takes the form d2 (rG) + k2 (rG) = 0 dr2

(3.27)

with the well-known general solution G = C+

0

0

−ik|x−x | eikr e−ikr eik|x−x | −e + C + C− ≡ C+ r r |x − x0 | |x − x0 |

(3.28)

where C ± are constants. In order to evaluate the constants C ± , we insert the general solution, Equation (3.28), into Equation (3.24) on the previous page and integrate over a small volume around r = |x − x0 | = 0. Since

G( x − x0 ) ∼ C +

1 1 + C− , 0 |x − x | |x − x0 |

x − x0 → 0

(3.29)

The volume integrated Equation (3.24) on the preceding page can under this assumption be approximated by

3

C+ + C− 4

∇2

=−


1 d3x0 + k2 C + + C − 4 3 |x − x0 |

0

3 0

1 d3x0 |x − x0 |

(3.30)

δ( x − x ) d x


3.3

43


In virtue of the fact that the volume element d3x0 in spherical polar coordinates is proportional to |x − x0 |2 , the second integral vanishes when |x − x0 | → 0. Furthermore, from Equation (M.85) on page 183, we find that the integrand in the first integral can be written as −4πδ(|x − x0 |) and, hence, that 1 4π

C+ + C− =

(3.31)

Insertion of the general solution Equation (3.28) on the preceding page into Equation (3.25) on the facing page gives Ψω (x) = C +

fω (x0 )

0

eik|x−x | 3 0 d x + C− |x − x0 |

fω (x0 )

0

e−ik|x−x | 3 0 dx |x − x0 |

(3.32)

The Fourier transform to ordinary t domain of this is obtained by inserting the above expression for Ψω (x) into Equation (3.21a) on page 41: | exp 7 −iω . t − k|x−x ω /98 0

Ψ(t, x) = C

+

0

fω (x )

6

+ C−

fω (x0 )

6

|x − x0 |

dω d3x0

| exp 7 −iω . t + k|x−x ω /98

(3.33)

0

|x − x0 |

dω d3x0

0 and the advanced time t 0 in the following If we introduce the retarded time tret adv way [using the fact that in vacuum k/ω = 1/c, according to Equation (3.22) on page 41]:

0 0 tret = tret (t, x − x0 ) = t −

k |x − x0 | |x − x0 | = t− ω c 0 k |x − x0 | |x − x0 | 0 0 = t+ tadv = tadv (t, x − x ) = t + ω c

(3.34a) (3.34b)

and use Equation (3.20a) on page 41, we obtain Ψ(t, x) = C +

0 , x0 ) f (tret d3x0 + C − |x − x0 |

0 , x0 ) f (tadv d3x0 |x − x0 |

(3.35)

This is a solution to the generic inhomogeneous wave equation for the potential components Equation (3.19) on page 41. We note that the solution at time t at the field point x is dependent on the behaviour at other times t 0 of the source at x0 and that both retarded and advanced t 0 are mathematically acceptable solutions. However, if we assume that causality requires that the potential at 0 , x0 ), we must in (t, x) is set up by the source at an earlier time, i.e., at (tret − Equation (3.35) above set C = 0 and therefore, according to Equation (3.31), C + = 1/(4π).



44


The retarded potentials From the above discussion on the solution of the inhomogeneous wave equation we conclude that, under the assumption of causality, the electrodynamic potentials in vacuum can be written 0 , x0 ) ρ(tret 1 d3x0 4πε0 |x − x0 | 0 , x0 ) µ0 j(tret A(t, x) = d3x0 4π |x − x0 |

φ(t, x) =

(3.36a) (3.36b)

Since these retarded potentials were obtained as solutions to the Lorentz equations (3.14) on page 39 they are valid in the Lorentz gauge but may be gauge transformed according to the scheme described in subsection 3.3.1 on page 39. As they stand, we shall use them frequently in the following. E XAMPLE 3.1

E LECTROMAGNETODYNAMIC POTENTIALS

In Dirac’s symmetrised form of electrodynamics (electromagnetodynamics), Maxwell’s equations are replaced by [see also Equations (1.48) on page 15]: ∇·E =

ρe ε0

(3.37a)

∇ × E = −µ0 jm − ∇ · B = µ0 ρ

∂B ∂t

m

∇ × B = µ 0 j e + ε 0 µ0

(3.37b) (3.37c)

∂E ∂t

(3.37d)

In this theory, one derives the inhomogeneous wave equations for the usual “electric” scalar and vector potentials (φe , Ae ) and their “magnetic” counterparts (φm , Am ) by assuming that the potentials are related to the fields in the following symmetrised form: ∂ e A (t, x) − ∇ × Am ∂t 1 1 ∂ B = − 2 ∇φm (t, x) − 2 Am (t, x) + ∇ × Ae c c ∂t

E = −∇φe (t, x) −

(3.38a) (3.38b)

In the absence of magnetic charges, or, equivalenty for φm ≡ 0 and Am ≡ 0, these formulae reduce to the usual Maxwell theory Formula (3.10) on page 37 and Formula (3.6) on page 36, respectively, as they should. Inserting the symmetrised expressions (3.38) into Equations (3.37) above, one obtains [cf., Equations (3.11) on page 37]



3.3

45

B IBLIOGRAPHY

∂ ρe (t, x) ∇ · Ae = − ∂t ε0 m (t, x) ρ ∂ ∇ · Am = − ∇ 2 φm + ∂t ε0 1 ∂ 2 Ae 1 ∂φe 2 e e − ∇ A + ∇ ∇ · A + c2 ∂t2 c2 ∂t

(3.39a)

(3.39b)

∇ 2 φe +

1 ∂ 2 Am 1 ∂φm − ∇2 Am + ∇ ∇ · A m + 2 2 2 c ∂t c ∂t

= µ0 je (t, x)

(3.39c)

= µ0 jm (t, x)

(3.39d)

By choosing the conditions on the vector potentials according to Lorentz’ prescripton [cf., Equation (3.13) on page 38] 1 ∂ e φ =0 c2 ∂t 1 ∂ ∇ · A m + 2 φm = 0 c ∂t these coupled wave equations simplify to ∇ · Ae +

(3.40) (3.41)

ρe (t, x) 1 ∂ 2 φe − ∇ 2 φe = (3.42a) 2 2 c ∂t ε0 1 ∂ 2 φm ρm (t, x) − ∇ 2 φm = (3.42b) 2 2 c ∂t ε0 1 ∂ 2 Ae − ∇2 Ae = µ0 je (t, x) (3.42c) c2 ∂t2 1 ∂ 2 Am − ∇2 Am = µ0 jm (t, x) (3.42d) c2 ∂t2 exhibiting once again, the striking properties of Dirac’s symmetrised Maxwell theory. E ND OF

EXAMPLE

3.1C

Bibliography [1] L. D. FADEEV AND A. A. S LAVNOV, Gauge Fields: Introduction to Quantum Theory, No. 50 in Frontiers in Physics: A Lecture Note and Reprint Series. Benjamin/Cummings Publishing Company, Inc., Reading, MA . . . , 1980, ISBN 08053-9016-2. [2] M. G UIDRY, Gauge Field Theories: An Introduction with Applications, John Wiley & Sons, Inc., New York, NY . . . , 1991, ISBN 0-471-63117-5. [3] J. D. JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, NY . . . , 1999, ISBN 0-471-30932-X. [4] L. L ORENZ, Philosophical Magazine (1867), pp. 287–301.



46


[5] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6.



4 Relativistic Electrodynamics

We saw in Chapter 3 how the derivation of the electrodynamic potentials led, in a most natural way, to the introduction of a characteristic, finite speed of √ propagation in vacuum that equals the speed of light c = 1/ ε0 µ0 and which can be considered as a constant of nature. To take this finite speed of propagation of information into account, and to ensure that our laws of physics be independent of any specific coordinate frame, requires a treatment of electrodynamics in a relativistically covariant (coordinate independent) form. This is the object of this chapter.

4.1 The Special Theory of Relativity An inertial system, or inertial reference frame, is a system of reference, or rigid coordinate system, in which the law of inertia (Galileo’s law, Newton’s first law) holds. In other words, an inertial system is a system in which free bodies move uniformly and do not experience any acceleration. The special theory of relativity1 describes how physical processes are interrelated when observed in different inertial systems in uniform, rectilinear motion relative to each other and is based on two postulates: 1 The Special Theory of Relativity, by the American physicist and philosopher David Bohm, opens with the following paragraph [4]:

“The theory of relativity is not merely a scientific development of great importance in its own right. It is even more significant as the first stage of a radical change in our basic concepts, which began in physics, and which is spreading into other fields of science, and indeed, even into a great deal of thinking outside of science. For as is well known, the modern trend is away from the notion of sure ‘absolute’ truth, (i.e., one which holds independently of all conditions, contexts, degrees, and types of approximation etc..) and toward the idea that a given concept has significance only in relation to suitable broader forms of reference, within which that concept can be given its full meaning.”

47

48

R ELATIVISTIC E LECTRODYNAMICS

vt y

Σ0

Σ

y0 v P(t, x, y, z) P(t0 , x0 , y0 , z0 )

O z

O0

x

x0

z0

F IGURE 4.1: Two inertial systems Σ and Σ0 in relative motion with velocity v along the x = x0 axis. At time t = t0 = 0 the origin O0 of Σ0 coincided with the origin O of Σ. At time t, the inertial system Σ0 has been translated a distance vt along the x axis in Σ. An event represented by P(t, x, y, z) in Σ is represented by P(t 0 , x0 , y0 , z0 ) in Σ0 .

Postulate 4.1 (Relativity principle; Poincaré, 1905). All laws of physics (except the laws of gravitation) are independent of the uniform translational motion of the system on which they operate. Postulate 4.2 (Einstein, 1905). The velocity of light in empty space is independent of the motion of the source that emits the light. A consequence of the first postulate is that all geometrical objects (vectors, tensors) in an equation describing a physical process must transform in a covariant manner, i.e., in the same way.

4.1.1 The Lorentz transformation Let us consider two three-dimensional inertial systems Σ and Σ0 in vacuum which are in rectilinear motion relative to each other in such a way that Σ 0 moves with constant velocity v along the x axis of the Σ system. The times and the spatial coordinates as measured in the two systems are t and (x, y, z), and t 0 and (x0 , y0 , z0 ), respectively. At time t = t0 = 0 the origins O and O0 and the x and x0 axes of the two inertial systems coincide and at a later time t they have the relative location as depicted in Figure 4.1. For convenience, let us introduce the two quantities v β= (4.1) c 1 γ= : (4.2) 1 − β2



4.1

T HE S PECIAL T HEORY

OF

49

R ELATIVITY

where v = |v|. In the following, we shall make frequent use of these shorthand notations. As shown by Einstein, the two postulates of special relativity require that the spatial coordinates and times as measured by an observer in Σ and Σ0 , respectively, are connected by the following transformation: ct0 = γ(ct − xβ)

(4.3a)

0

x = γ(x − vt)

(4.3b)

0

y =y

(4.3c)

z0 = z

(4.3d)

Taking the difference between the square of (4.3a) and the square of (4.3b) we find that c2 t02 − x02 = γ2 c2 t2 − 2xcβt + x2 β2 − x2 + 2xvt − v2 t2 4

3

=

1

c2 t 2

v2 c2 2 2 = c t − x2 1−

1−

v2 v2 2 − x 1 − c2 c2

(4.4)

From Equations (4.3) above we see that the y and z coordinates are unaffected by the translational motion of the inertial system Σ0 along the x axis of system Σ. Using this fact, we find that we can generalise the result in Equation (4.4) to c2 t2 − x2 − y2 − z2 = c2 t02 − x02 − y02 − z02

(4.5)

which means that if a light wave is transmitted from the coinciding origins O and O0 at time t = t0 = 0 it will arrive at an observer at (x, y, z) at time t in Σ and an observer at (x0 , y0 , z0 ) at time t0 in Σ0 in such a way that both observers conclude that the speed (spatial distance divided by time) of light in vacuum is c. Hence, the speed of light in Σ and Σ0 is the same. A linear coordinate transformation which has this property is called a (homogeneous) Lorentz transformation.

4.1.2 Lorentz space Let us introduce an ordered quadruple of real numbers, enumerated with the help of upper indices µ = 0, 1, 2, 3, where the zeroth component is ct (c is the



50


speed of light and t is time), and the remaining components are the components of the ordinary 3 radius vector x defined in Equation (M.1) on page 168: xµ = (x0 , x1 , x2 , x3 ) = (ct, x, y, z) ≡ (ct, x)

(4.6)

We want to interpret this quadruple xµ as (the component form of) a radius four-vector in a real, linear, four-dimensional vector space.1 We require that this four-dimensional space to be a Riemannian space, i.e., a space where a “distance” and a scalar product are defined. In this space we therefore define a metric tensor, also known as the fundamental tensor, which we denote by g µν .

Radius four-vector in contravariant and covariant form The radius four-vector xµ = (x0 , x1 , x2 , x3 ) = (ct, x), as defined in Equation (4.6), is, by definition, the prototype of a contravariant vector (or, more accurately, a vector in contravariant component form). To every such vector there exists a dual vector. The vector dual to xµ is the covariant vector xµ , obtained as (the upper index µ in xµ is summed over and is therefore a dummy index and may be replaced by another dummy index ν): xµ = gµν xν

(4.7)

This summation process is an example of index contraction and is often referred to as index lowering.

Scalar product and norm The scalar product of xµ with itself in a Riemann space is defined as gµν xν xµ = xµ xµ

(4.8)

This scalar product acts as an invariant “distance,” or norm, in this space. If we want the Lorentz transformation invariance, described by Equation (4.5) on the previous page, to be the manifestation of the conservation 1 The British mathematician and philosopher Alfred North Whitehead writes in his book The

Concept of Nature [13]: “I regret that it has been necessary for me in this lecture to administer a large dose of four-dimensional geometry. I do not apologise, because I am really not responsible for the fact that nature in its most fundamental aspect is fourdimensional. Things are what they are. . . ”



4.1


OF

51

R ELATIVITY

of the norm in a 4D Riemann space, then the explicit expression for the scalar product of xµ with itself in this space must be xµ x µ = c2 t 2 − x 2 − y2 − z2

(4.9)

We notice that our space will have an indefinite norm which means that we deal with a non-Euclidean space. We call the four-dimensional space (or spacetime) with this property Lorentz space and denote it ; 4 . A corresponding real, linear 4D space with a positive definite norm which is conserved during ordinary rotations is a Euclidean vector space. We denote such a space 4 .

Metric tensor By choosing the metric tensor in ; gµν =

> <= =?

as

1 if µ = ν = 0 −1 if µ = ν = i = j = 1, 2, 3 0 if µ = 6 ν

or, in matrix notation,

(gµν ) =

4

@AAB

(4.10)

CEDD

1 0 0 0 0 −1 0 0 0 0 −1 0 F 0 0 0 −1

(4.11)

i.e., a matrix with a main diagonal that has the sign sequence, or signature, {+, −, −, −}, the index lowering operation in our chosen flat 4D space becomes nearly trivial: xµ = gµν xν = (ct, −x)

(4.12)

Using matrix algebra, this can be written

CEDD

@AAB

x0 x1 x2 F x3

CEDD

=

@AAB

1 0 0 0 0 −1 0 0 0 0 −1 0 F 0 0 0 −1

@AAB

CEDD

x0 x1 x2 F x3

CEDD

=

@AAB

x0 −x1 −x2 F −x3

(4.13)

Hence, if the metric tensor is defined according to expression (4.10) the covariant radius four-vector xµ is obtained from the contravariant radius four-vector xµ simply by changing the sign of the last three components. These components are referred to as the space components; the zeroth component is referred to as the time component.



52


As we see, for this particular choice of metric, the scalar product of x µ with itself becomes xµ xµ = (ct, x) · (ct, −x) = c2 t2 − x2 − y2 − z2

(4.14)

which indeed is the desired Lorentz transformation invariance as required by Equation (4.9) on the previous page. Without changing the physics, one can alternatively choose a signature {−, +, +, +}. The latter has the advantage that the transition from 3D to 4D becomes smooth, while it will introduce some annoying minus signs in the theory. In current physics literature, the signature {+, −, −, −} seems to be the most commonly used one. The ; 4 metric tensor Equation (4.10) on the preceding page has a number of interesting properties: firstly, we see that this tensor has a trace Tr gµν 4 = −2 3 whereas in 4 , as in any vector space with definite norm, the trace equals the space dimensionality. Secondly, we find, after trivial algebra, that the following relations between the contravariant, covariant and mixed forms of the metric tensor hold: gµν = gνµ

(4.15a)

µν

g = gµν κµ

gνκ g = νκ

g gκµ =

gµν gνµ

= =

(4.15b) δµν δνµ

(4.15c) (4.15d) µ

Here we have introduced the 4D version of the Kronecker delta δν , a mixed four-tensor of rank 2 which fulfils δµν = δνµ =

G

1 0

if µ = ν if µ 6= ν

(4.16)

Invariant line element and proper time The differential distance ds between the two points xµ and xµ + dxµ in ; 4 can be calculated from the Riemannian metric, given by the quadratic differential form ds2 = gµν dxν dxµ = dxµ dxµ = (dx0 )2 − (dx1 )2 − (dx2 )2 − (dx3 )2

(4.17)

where the metric tensor is as in Equation (4.10) on the previous page. As we see, this form is indefinite as expected for a non-Euclidean space. The square



4.1


OF

53

R ELATIVITY

root of this expression is the invariant line element ds = c -

+,,

1 1− 2 c

H

dx 1 dt

2

dx 2 + dt

2

dx 3 + dt

2

I dt

1 v2 2 + (v )2 + (v )2 dt = c 1 − dt (v ) x y z c2 c2 : c = c 1 − β2 dt = dt = c dτ γ

= c 1−

(4.18)

where we introduced dτ = dt/γ

(4.19)

Since dτ measures the time when no spatial changes are present, it is called the proper time. Expressing Equation (4.5) on page 49 in terms of the differential interval ds and comparing with Equation (4.17) on the preceding page, we find that ds2 = c2 dt2 − dx2 − dy2 − dz2

(4.20)

is invariant during a Lorentz transformation. Conversely, we may say that every coordinate transformation which preserves this differential interval is a Lorentz transformation. If in some inertial system dx2 + dy2 + dz2 < c2 dt2

(4.21)

ds is a time-like interval, but if dx2 + dy2 + dz2 > c2 dt2

(4.22)

ds is a space-like interval, whereas dx2 + dy2 + dz2 = c2 dt2

(4.23)

is a light-like interval; we may also say that in this case we are on the light cone. A vector which has a light-like interval is called a null vector. The time-like, space-like or light-like aspects of an interval ds is invariant under a Lorentz transformation.



54


Four-vector fields Any quantity which relative to any coordinate system has a quadruple of real numbers and transforms in the same way as the radius four-vector x µ does, is called a four-vector. In analogy with the notation for the radius four-vector we introduce the notation aµ = (a0 , a) for a general contravariant four-vector field in ; 4 and find that the “lowering of index” rule, Equation (M.32) on page 174, for such an arbitrary four-vector yields the dual covariant four-vector field aµ (xκ ) = gµν aν (xκ ) = (a0 (xκ ), −a(xκ ))

(4.24)

The scalar product between this four-vector field and another one bµ (xκ ) is gµν aν (xκ )bµ (xκ ) = (a0 , −a) · (b0 , b) = a0 b0 − a · b

(4.25)

which is a scalar field, i.e., an invariant scalar quantity α(x κ ) which depends on time and space, as described by xκ = (ct, x, y, z).

The Lorentz transformation matrix Introducing the transformation matrix

µ

3Λν

4

CEDD

=

@AAB

γ −βγ 0 0 −βγ γ 0 0 0 0 1 0F 0 0 0 1

(4.26)

the linear Lorentz transformation (4.3) on page 49, i.e., the coordinate transformation xµ → x0µ = x0µ (x0 , x1 , x2 , x3 ), from one inertial system Σ to another inertial system Σ0 , can be written x0µ = Λµν xν

(4.27)

The inverse transform then takes the form xµ = (Λ−1 )µν x0ν

(4.28)

The Lorentz group It is easy to show, by means of direct algebra, that two successive Lorentz transformations of the type in Equation (4.28), and defined by the speed parameters β1 and β2 , respectively, correspond to a single transformation with speed parameter β=

β1 + β2 1 + β1 β2


(4.29)


4.1


OF

55

R ELATIVITY

X0

X 00

θ x01 θ x1 F IGURE 4.2: Minkowski space can be considered an ordinary Euclidean space where a Lorentz transformation from (x1 , X 0 = ict) to (x01 , X 00 = ict0 ) corresponds to an ordinary rotation through an angle θ. This rotation 2 2 leaves the Euclidean distance x1 + X 0 = x2 − c2 t2 invariant.

This means that the nonempty set of Lorentz transformations constitutes a closed algebraic structure with a binary operation which is associative. Furthermore, one can show that this set possesses at least one identity element and at least one inverse element. In other words, this set of Lorentz transformations constitutes a mathematical group. However tempting, we shall not make any further use of group theory.

4.1.3 Minkowski space Specifying a point xµ = (x0 , x1 , x2 , x3 ) in 4D space-time is a way of saying that “something takes place at a certain time t = x0 /c and at a certain place (x, y, z) = (x1 , x2 , x3 ).” Such a point is therefore called an event. The trajectory for an event as a function of time and space is called a world line. For instance, the world line for a light ray which propagates in vacuum is the trajectory x0 = x1 . Introducing X 0 = ix0 = ict

(4.30a)

X 1 = x1

(4.30b)

2

2

(4.30c)

3

3

(4.30d)

X =x X =x

dS = ids


(4.30e)


56


Σ x0

w

= ct

x00

x0 = x 1

ϕ P0 ct

ϕ O=

O0

x01

P

x1 = x

F IGURE 4.3: Minkowski diagram depicting geometrically the transformation (4.34) from the unprimed system to the primed system. Here w denotes the world line for an event and the line x0 = x1 ⇔ x = ct the world line for a light ray in vacuum. Note that the event P is simultaneous with all points on the x1 axis (t = 0), including the origin O while the event P0 , which is also simultaneous with all points on the x0 axis, including O0 = O, to an observer at rest in the primed system, is not simultaneous with O in the unprimed system but occurs there at time |P − P0 | /c.

√ where i = −1, we see that Equation (4.17) on page 52 transforms into dS 2 = (dX 0 )2 + (dX 1 )2 + (dX 2 )2 + (dX 3 )2

(4.31)

i.e., into a 4D differential form which is positive definite just as is ordinary 3D Euclidean space 3 . We shall call the 4D Euclidean space constructed in this way the Minkowski space J 4 .1 As before, it suffices to consider the simplified case where the relative motion between Σ and Σ0 is along the x axes. Then dS 2 = (dX 0 )2 + (dx1 )2

(4.32)

and we consider X 0 and x1 as orthogonal axes in an Euclidean space. As in all Euclidean spaces, every interval is invariant under a rotation of the X 0 x1 plane through an angle θ into X 00 x01 : X 00 = −x1 sin θ + X 0 cos θ x01 = x1 cos θ + X 0 sin θ

(4.33a) (4.33b)

1 The

fact that our Riemannian space can be transformed in this way into an Euclidean one means that it is, strictly speaking, a pseudo-Riemannian space.



4.2

57

C OVARIANT C LASSICAL M ECHANICS

See Figure 4.2 on page 55. If we introduce the angle ϕ = −iθ, often called the rapidity or the Lorentz boost parameter, and transform back to the original space and time variables by using Equation (4.30) on page 55 backwards, we obtain ct0 = −x sinhϕ + ct cosh ϕ

(4.34a)

0

x = x cosh ϕ − ct sinh ϕ

(4.34b)

which are identical to the transformation equations (4.3) on page 49 if we let sinh ϕ = γβ

(4.35a)

cosh ϕ = γ

(4.35b)

tanh ϕ = β

(4.35c)

It is therefore possible to envisage the Lorentz transformation as an “ordinary” rotation in the 4D Euclidean space J 4 This rotation i J 4 corresponds to a coordinate change in ; 4 as depicted in Figure 4.3 on the preceding page. Equation (4.29) on page 54 for successive Lorentz transformation then corresponds to the tanh addition formula tanh ϕ1 + tanh ϕ2 (4.36) tanh(ϕ1 + ϕ2 ) = 1 + tanh ϕ1 tanh ϕ2 The use of ict and J 4 , which leads to the interpretation of the Lorentz transformation as an “ordinary” rotation, may, at best, be illustrative, but is not very physical. Besides, if we leave the flat ; 4 space and enter the curved space of general relativity, the “ict” trick will turn out to be an impasse. Let us therefore immediately return to ; 4 where all components are real valued.

4.2 Covariant Classical Mechanics The invariance of the differential “distance” ds in ; 4 , and the associated differential proper time dτ [see Equation (4.18) on page 53] allows us to define the four-velocity uµ =

C

µ

dx = γ(c, v) = dτ

B@ K

c 1−

v2 c2

,

K

v 1−

v2 c2

F

= (u0 , u)

(4.37)

which, when multiplied with the scalar invariant m0 yields the four-momentum pµ = m0

µ

dx = m0 γ(c, v) = dτ


C

B@ K

m0 c 1−

v2 c2

,

K

m0 v 1−

v2 c2

F

= (p0 , p)

(4.38)


58


From this we see that we can write p = mv

(4.39)

where

K

m = γm0 =

m0

(4.40)

2

1 − vc2

We can interpret this such that the Lorentz covariance implies that the masslike term in the ordinary 3D linear momentum is not invariant. A better way to look at this is that p = mv = γm0 v is the covariantly correct expression for the kinetic three-momentum. Multiplying the zeroth (time) component of the four-momentum pµ with the scalar invariant c, we obtain cp0 = γm0 c2 =

m0 c2

K

1−

v2 c2

= mc2

(4.41)

Since this component has the dimension of energy and is the result of a covariant description of the motion of a particle with its kinetic momentum described by the spatial components of the four-momentum, Equation (4.38) on the preceding page, we interpret cp0 as the total energy E. Hence, cpµ = (cp0 , cp) = (E, cp)

(4.42)

Scalar multiplying this four-vector with itself, we obtain cpµ cpµ = c2 gµν pν pµ = c2 [(p0 )2 − (p1 )2 − (p2 )2 − (p3 )2 ] = (E, −cp) · (E, cp) = E 2 − c2 p2 =

(m0 c2 )2 2

1 − vc2

(4.43)

v2 1 − 2 = (m0 c2 )2 c

Since this is an invariant, this equation holds in any inertial frame, particularly in the frame where p = 0 and there we have E = m 0 c2

(4.44)

This is probably the most famous formula in physics history.



4.3

59

C OVARIANT C LASSICAL E LECTRODYNAMICS

4.3 Covariant Classical Electrodynamics In the rest inertial system the charge density is ρ0 . The four-vector (in contravariant component form) j µ = ρ0

dxµ = ρ0 uµ = ρ0 γ(c, v) = (ρc, ρv) dτ

(4.45)

where we introduced ρ = γρ0

(4.46)

is called the four-current. The contravariant form of the four-del operator ∂µ = ∂/∂xµ is defined in Equation (M.69) on page 180 and its covariant counterpart ∂µ = ∂/∂xµ in Equation (M.70) on page 180, respectively. As is shown in Example M.6 on page 180, the d’Alembert operator is the scalar product of the four-del with itself:

5

2

= ∂µ ∂µ = ∂µ ∂µ =

1 ∂2 − ∇2 c2 ∂t2

(4.47)

Since it has the characteristics of a four-scalar, the d’Alembert operator is invariant and, hence, the homogeneous wave equation is Lorentz covariant.

4.3.1 The four-potential If we introduce the four-potential Aµ =

φ ,A c

(4.48)

where φ is the scalar potential and A the vector potential, defined in Section 3.3 on page 36, we can write the inhomogeneous wave equations (Lorentz potential equations), Equations (3.14) on page 39, in the following compact (and covariant) way:

5

2 µ

A = µ0 jµ

(4.49)

With the help of the above, we can formulate our electrodynamic equations covariantly. For instance, the covariant form of the equation of continuity, Equation (1.21) on page 9 is ∂µ jµ = 0


(4.50)


60


and the Lorentz gauge condition, Equation (3.13) on page 38, can be written ∂µ Aµ = 0

(4.51)

The gauge transformations (3.15) on page 39 in covariant form are Aµ 7→ A0µ = Aµ + ∂µ Γ(xν )

(4.52)

If only one dimension Lorentz contracts (for instance, due to relative motion along the x direction), a 3D spatial volume transforms according to

: v2 1 dV = d3x = dV0 = dV0 1 − β2 = dV0 1 − 2 γ c

(4.53)

then from Equation (4.46) on the previous page we see that ρdV = ρ0 dV0

(4.54)

i.e., the charge in a given volume is conserved. We can therefore conclude that the elementary charge is a universal constant.

4.3.2 The Liénard-Wiechert potentials Let us now solve the Lorentz equation (the inhomogeneous wave equation) (3.14) on page 39 in vacuum for the case of a well-localised charge q0 at a source point defined by the radius four-vector x0µ = (x00 = ct0 , x01 , x02 , x03 ). The field point (observation point) is denoted by the radius four-vector xµ = (x0 = ct, x1 , x2 , x3 ). In the rest system we know that the solution is simply

3

Aµ 4

= 0

φ ,A c

= v=0

1 q0 ,0 4πε0 c |x − x0 |0

(4.55)

where |x − x0 |0 is the usual distance from the source point to the field point, evaluated in the rest system (signified by the index “0”). Let us introduce the relative radius four-vector between the source point and the field point: Rµ = xµ − x0µ = (c(t − t0 ), x − x0 )

(4.56)

Scalar multiplying this relative four-vector with itself, we obtain

2

Rµ Rµ = (c(t − t0), x − x0 ) · (c(t − t0), −(x − x0 )) = c2 (t − t0 )2 − x − x0


(4.57)


4.3

61


We know that in vacuum the signal (field) from the charge q0 at x0µ propagates to xµ with the speed of light c so that

x − x0 = c(t − t0 )

(4.58)

Inserting this into Equation (4.57) on the preceding page, we see that Rµ Rµ = 0

(4.59)

or that Equation (4.56) on the facing page can be written

Rµ = ( x − x 0 , x − x 0 )

(4.60)

Now we want to find the correspondence to the rest system solution, Equation (4.55) on the preceding page, in an arbitrary inertial system. We note from Equation (4.37) on page 57 that in the rest system

3u

µ

4

C

0

B@ K

=

and

c 1−

v2 c2

,

K

F

v 1−

v2 c2

= (c, 0)

(4.61)

v=0

(Rµ )0 = ( x − x0 , x − x0 )0 = ( x − x0 0 , (x − x0 )0 )

(4.62)

As all scalar products, uµ Rµ is invariant, which means that we can evaluate it in any inertial system and it will have the same value in all other inertial systems. If we evaluate it in the rest system the result is: uµ Rµ = uµ Rµ 4

3

= (uµ )0 (Rµ )0

0

= (c, 0) · ( x − x0 0 , −(x − x0 )0 ) = c x − x0 0

(4.63)

We therefore see that the expression Aµ =

q0 uµ 4πε0 cuν Rν

(4.64)

subject to the condition Rµ Rµ = 0 has the proper transformation properties (proper tensor form) and reduces, in the rest system, to the solution Equation (4.55) on the preceding page. It is therefore the correct solution, valid in any inertial system. According to Equation (4.37) on page 57 and Equation (4.60) above

uν Rν = γ(c, v) · x − x0 , −(x − x0 )4 = γ c x − x0 − v · (x − x0 )4

3


3

(4.65)


62


Generalising expression (4.1) on page 48 to vector form: def

β = β vˆ ≡

v c

(4.66)

and introducing

s ≡ x − x0 − def

v · (x − x0 ) ≡ x − x0 − β · (x − x0 ) c

(4.67)

we can write uν Rν = γcs

(4.68)

and uµ = cuν Rν

1 v , cs c2 s

(4.69)

from which we see that the solution (4.64) can be written Aµ (xκ ) =

q0 4πε0

1 v , = cs c2 s

φ ,A c

(4.70)

where in the last step the definition of the four-potential, Equation (4.48) on page 59, was used. Writing the solution in the ordinary 3D-way, we conclude that for a very localised charge volume, moving relative an observer with a velocity v, the scalar and vector potentials are given by the expressions 1 q0 1 q0 = 0 4πε0 s 4πε0 |x − x | − β · (x − x0 ) q0 v q0 v = A(t, x) = 0 2 2 4πε0 c s 4πε0 c |x − x | − β · (x − x0 ) φ(t, x) =

(4.71a) (4.71b)

These potentials are called the Liénard-Wiechert potentials.

4.3.3 The electromagnetic field tensor Consider a vectorial (cross) product c between two ordinary vectors a and b: c = a × b = i jk ai b j xˆ k = (a2 b3 − a3 b2 ) xˆ 1 + (a3 b1 − a1 b3 ) xˆ 2 + (a1 b2 − a2 b1 ) xˆ 3

(4.72)

We notice that the kth component of the vector c can be represented as ck = ai b j − a j bi = ci j = −c ji ,


i, j 6= k

(4.73)


4.3

63


In other words, the pseudovector c = a × b can be considered as an antisymmetric tensor of rank two. The same is true for the curl operator ∇×. For instance, the Maxwell equation ∂B (4.74) ∇×E = − ∂t can in this tensor notation be written ∂E j ∂Ei ∂Bi j − =− (4.75) ∂xi ∂x j ∂t We know from Chapter 3 that the fields can be derived from the electromagnetic potentials in the following way: B = ∇×A

(4.76a)

E = −∇φ −

(4.76b)

∂A ∂t In component form, this can be written

∂A j ∂Ai − = ∂i A j − ∂ j Ai (4.77a) ∂xi ∂x j ∂φ ∂Ai Ei = − i − = −∂i φ − ∂t Ai (4.77b) ∂x ∂t From this, we notice the clear difference between the axial vector (pseudovector) B and the polar vector (“ordinary vector”) E. Our goal is to express the electric and magnetic fields in a tensor form where the components are functions of the covariant form of the four-potential, Equation (4.48) on page 59: Bi j =

Aµ =

φ ,A c

(4.78)

Inspection of (4.78) and Equation (4.77) above makes it natural to define the four-tensor ∂Aν ∂Aµ F µν = − = ∂µ Aν − ∂ν Aµ (4.79) ∂xµ ∂xν This anti-symmetric (skew-symmetric), four-tensor of rank 2 is called the electromagnetic field tensor. In matrix representation, the contravariant field tensor can be written

3F

µν

4

CEDD

=

@AAB

0 −E x /c −Ey /c −Ez /c E x /c 0 −Bz By Ey /c Bz 0 −B x F Ez /c −By Bx 0


(4.80)


64


The covariant field tensor is obtained from the contravariant field tensor in the usual manner by index contraction (index lowering): Fµν = gµκ gνλ F κλ = ∂µ Aν − ∂ν Aµ

(4.81)

It is perhaps interesting to note that the field tensor is a sort of four-dimensional curl of the four-potential vector Aµ . The matrix representation for the covariant field tensor is

3 Fµν

4

CEDD

=

@AAB

0 E x /c Ey /c Ez /c −E x /c 0 −Bz By −Ey /c Bz 0 −B x F −Ez /c −By Bx 0

(4.82)

That the two Maxwell source equations can be written ∂ν F νµ = µ0 jµ

(4.83)

is immediately observed by explicitly setting µ = 0 in this covariant equation and using the matrix representation Formula (4.80) on the preceding page for the covariant component form of the electromagnetic field tensor F µν , to obtain 1 ∂F 00 ∂F 10 ∂F 20 ∂F 30 + + + = 0+ 0 1 2 3 ∂x ∂x ∂x ∂x c 1 = ∇ · E = µ0 j0 = µ0 cρ c

∂E x ∂Ey ∂Ez + + ∂x ∂y ∂z

(4.84)

or, equivalently, ∇ · E = µ 0 c2 ρ =

ρ ε0

(4.85)

which is the Maxwell source equation for the electric field, Equation (1.43a) on page 14. For µ = 1, Equation (4.84) yields ∂F 01 ∂F 11 ∂F 21 ∂F 31 ∂Bz ∂By 1 ∂E x +0− + = µ0 j1 = µ0 ρv x (4.86) + 1 + 2 + 3 =− 2 0 ∂x ∂x ∂x ∂x c ∂t ∂y ∂z or, using ε0 µ0 = 1/c2 , ∂By ∂Bz ∂E x − − ε0 µ0 = µ0 j x ∂z ∂y ∂t


(4.87)


4.3

65

B IBLIOGRAPHY

and similarly for µ = 2, 3. In summary, in three-vector form, we can write the result as ∇ × B − ε 0 µ0

∂E = µ0 j(t, x) ∂t

(4.88)

which is the Maxwell source equation for the magnetic field, Equation (1.43d) on page 14. The two Maxwell field equations ∇×E = −

∂B ∂t

∇·B = 0

(4.89) (4.90)

correspond to (no summation!) ∂κ Fµν + ∂µ Fνκ + ∂ν Fκµ = 0

(4.91)

Hence, Equation (4.83) on the facing page and Equation (4.91) above constitute Maxwell’s equations in four-dimensional formalism.

Bibliography [1] J. A HARONI, The Special Theory of Relativity, second, revised ed., Dover Publications, Inc., New York, 1985, ISBN 0-486-64870-2. [2] A. O. BARUT, Electrodynamics and Classical Theory of Fields and Particles, Dover Publications, Inc., New York, NY, 1980, ISBN 0-486-64038-8. [3] R. B ECKER, Electromagnetic Fields and Interactions, Dover Publications, Inc., New York, NY, 1982, ISBN 0-486-64290-9. [4] D. B OHM, The Special Theory of Relativity, Routledge, New York, NY, 1996, ISBN 0-415-14809-X. [5] W. T. G RANDY, Introduction to Electrodynamics and Radiation, Academic Press, New York and London, 1970, ISBN 0-12-295250-2. [6] L. D. L ANDAU AND E. M. L IFSHITZ, The Classical Theory of Fields, fourth revised English ed., vol. 2 of Course of Theoretical Physics, Pergamon Press, Ltd., Oxford . . . , 1975, ISBN 0-08-025072-6. [7] F. E. L OW, Classical Field Theory, John Wiley & Sons, Inc., New York, NY . . . , 1997, ISBN 0-471-59551-9.



66


[8] C. M ØLLER, The Theory of Relativity, second ed., Oxford University Press, Glasgow . . . , 1972. [9] H. M UIRHEAD, The Special Theory of Relativity, The Macmillan Press Ltd., London, Beccles and Colchester, 1973, ISBN 333-12845-1. [10] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6. [11] J. J. S AKURAI, Advanced Quantum Mechanics, Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1967, ISBN 0-201-06710-2. [12] B. S PAIN, Tensor Calculus, third ed., Oliver and Boyd, Ltd., Edinburgh and London, 1965, ISBN 05-001331-9. [13] A. N. W HITEHEAD, Concept of Nature, Cambridge University Press, Cambridge . . . , 1920, ISBN 0-521-09245-0.



5 Electromagnetic Fields and Particles

In previous chapters, we calculated the electromagnetic fields and potentials from arbitrary, but prescribed distributions of charges and currents. In this chapter we study the general problem of interaction between electric and magnetic fields and electrically charged particles. The analysis is based on Lagrangian and Hamiltonian methods, is fully covariant, and yields results which are relativistically correct.

5.1 Charged Particles in an Electromagnetic Field We first establish a relativistically correct theory describing the motion of charged particles in prescribed electric and magnetic fields. From these equations we may then calculate the charged particle dynamics in the most general case.

5.1.1 Covariant equations of motion We will show that for our problem we can derive the correct equations of motion by using in 4D ; 4 a function with similiar properties as a Lagrange function in 3D and then apply a variational principle. We will also show that we can find find a Hamiltonian-type function in 4D and solve the corresponding Hamilton-type equations to obtain the correct covariant formulation of classical electrodynamics.

67

68

E LECTROMAGNETIC F IELDS

AND

PARTICLES

Lagrange formalism Let us no introduce a function L(4) which fulfils the variational principle δ

τ1

L(4) (xµ , uµ ) dτ = 0

τ0

(5.1)

where dτ is the proper time defined via Equation (4.18) on page 53, and the endpoints are fixed. We shall show that L(4) acts as a kind of generalisation to the common 3D Lagrangian. We require that L(4) fulfils the following conditions: 1. The Lagrange function must be invariant. This implies that L(4) must be a scalar. 2. The Lagrange function must yield linear equations of motion. This implies that L(4) must not contain higher than the second power of the fourvelocity uµ . According to Formula (M.96) on page 185 the ordinary 3D Lagrangian is the difference between the kinetic and potential energies. A free particle has only kinetic energy. If the particle mass is m0 then in 3D the kinetic energy is m0 v2 /2. This suggests that in 4D the Lagrangian for a free particle should be 1 free L(4) = m0 uµ uµ 2

(5.2)

For an interaction with the electromagnetic field we can introduce the interaction with the help of the four-potential given by Equation (4.78) on page 63 in the following way 1 L(4) = m0 uµ uµ + quµ Aµ (xν ) 2

(5.3)

We call this the four-Lagrangian and shall now show how this function, together with the variation principle, Formula (5.1) above, yields covariant results which are physically correct. The variation principle (5.1) with the 4D Lagrangian (5.3) inserted, leads to δ

τ1 τ0

τ1

= =

τ1

τ0 τ1 τ0

.

m0 µ u u + quµ Aµ / dτ τ0 2 µ ∂Aµ m0 ∂(uµ uµ ) µ δu + q Aµ δuµ + uµ ν δxν dτ 2 ∂uµ ∂x

L(4) (xµ , uµ ) dτ = δ

m0 uµ δu


µ

(5.4)

+ q Aµ δuµ + uµ ∂ν Aµ δxν 4 dτ = 0

3


5.1

C HARGED PARTICLES

IN AN

69

E LECTROMAGNETIC F IELD

According to Equation (4.37) on page 57, the four-velocity is uµ =

dx µ dτ

(5.5)

which means that we can write the variation of uµ as a total derivative with respect to τ : δuµ = δ

dx µ d = δxµ 4 dτ dτ 3

(5.6)

Inserting this into the first two terms in the last integral in Equation (5.4) on the preceding page, we obtain τ1

δ

L(4) (xµ , uµ ) dτ

τ0

τ1

=

τ0

m0 uµ

d d δxµ 4 + qAµ δxµ 4 + quµ ∂ν Aµ δxν dτ dτ 3 dτ 3

(5.7)

Partial integration in the two first terms in the right hand member of (5.7) gives τ1

δ

L(4) (xµ , uµ ) dτ

τ0

τ1

=

τ0

−m0

duµ µ dAµ µ δx − q δx + quµ ∂ν Aµ δxν dτ dτ dτ

(5.8)

where the integrated parts do not contribute since the variations at the endpoints vanish. A change of irrelevant summation index from µ to ν in the first two terms of the right hand member of (5.8) yields, after moving the ensuing common factor δxν outside the partenthesis, the following expression: δ

τ1

L(4) (xµ , uµ ) dτ

τ0

τ1

=

τ0

duν dAν −m0 −q + quµ ∂ν Aµ δxν dτ dτ dτ

(5.9)

Applying well-known rules of differentiation and the expression (4.37) for the four-velocity, we can express dAν /dτ as follows: dAν ∂Aν dxµ = µ = ∂µ Aν uµ dτ ∂x dτ

(5.10)

By inserting this expression (5.10) into the second term in right-hand member of Equation (5.9) on the preceding page, and noting the common factor qu µ of



70


AND

PARTICLES

the resulting term and the last term, we obtain the final variational principle expression δ

τ1

L(4) (xµ , uµ ) dτ

τ0

τ1

=

τ0

ν −m0 du + quµ ∂ν Aµ − ∂µ Aν 4 3 dτ

(5.11) ν

δx dτ

Since, according to the variational principle, this expression shall vanish and δxν is arbitrary between the fixed end points τ0 and τ1 , the expression inside in the integrand in the right hand member of Equation (5.11) above must vanish. In other words, we have found an equation of motion for a charged particle in a prescribed electromagnetic field: duν = quµ ∂ν Aµ − ∂µ Aν 4 (5.12) dτ 3 With the help of Equation (4.79) on page 63 we can express this equation in terms of the electromagnetic field tensor in the following way: m0

duν = quµ Fνµ (5.13) dτ This is the sought-for covariant equation of motion for a particle in an electromagnetic field. It is often referred to as the Minkowski equation. As the reader can easily verify, the spatial part of this 4-vector equation is the covariant (relativistically correct) expression for the Newton-Lorentz force equation. m0

Hamiltonian formalism The usual Hamilton equations for a 3D space are given by Equation (M.101) on page 185 in Appendix M. These six first-order partial differential equations are ∂H dqi = (5.14a) ∂pi dt dpi ∂H =− (5.14b) ∂qi dt where H(pi , qi , t) = pi q˙ i − L(qi , q˙ i , t) is the ordinary 3D Hamiltonian, qi is a generalised coordinate and pi is its canonically conjugate momentum. We seek a similar set of equations in 4D space. To this end we introduce a canonically conjugate four-momentum pµ in an analogous way as the ordinary 3D conjugate momentum: pµ =

∂L(4) ∂uµ


(5.15)


5.1

C HARGED PARTICLES

IN AN

71


and utilise the four-velocity uµ , as given by Equation (4.37) on page 57, to define the four-Hamiltonian H(4) = pµ uµ − L(4)

(5.16)

With the help of these, the radius four-vector xµ , considered as the generalised four-coordinate, and the invariant line element ds, defined in Equation (4.18) on page 53, we introduce the following eight partial differential equations: ∂H(4) dxµ = (5.17a) ∂pµ dτ dpµ ∂H(4) =− (5.17b) µ ∂x dτ which form the four-dimensional Hamilton equations. Our strategy now is to use Equation (5.15) and Equations (5.17) above to derive an explicit algebraic expression for the canonically conjugate momentum four-vector. According to Equation (4.42) on page 58, c times a fourmomentum has a zeroth (time) component which we can identify with the total energy. Hence we require that the component p0 of the conjugate fourmomentum vector defined according to Equation (5.15) above be identical to the ordinary 3D Hamiltonian H divided by c and hence that this cp0 solves the Hamilton equations, Equations (5.14) on the facing page. This later consistency check is left as an exercise to the reader. Using the definition of H(4) , Equation (5.16) above, and the expression for L(4) , Equation (5.3) on page 68, we obtain 1 (5.18) H(4) = pµ uµ − L(4) = pµ uµ − m0 uµ uµ − quµ Aµ (xν ) 2 Furthermore, from the definition (5.15) of the canonically conjugate fourmomentum pµ , we see that pµ =

∂L(4) ∂ = ∂uµ ∂uµ

1 m0 uµ uµ + quµ Aµ (xν ) = m0 uµ + qAµ 2

(5.19)

Inserting this into (5.18), we obtain 1 1 H(4) = m0 uµ uµ + qAµ uµ − m0 uµ uµ − quµ Aµ (xν ) = m0 uµ uµ 2 2

(5.20)

Since the four-velocity scalar-multiplied by itself is uµ uµ = c2 , we clearly see from Equation (5.20) above that H(4) is indeed a scalar invariant, whose value is simply H(4) =

m0 c2 2


(5.21)


72


AND

PARTICLES

However, at the same time (5.19) provides the algebraic relationship uµ =

1 pµ − qAµ 4 m0 3

(5.22)

and if this is used in (5.20) to eliminate uµ , one gets m0 2 1 = 2m0 1 = 2m0

H(4) =

1 1 pµ − qAµ 4 pµ − qAµ 4 m0 3 m0 3

3p

µ

3p

µ

− qAµ 4

3 pµ − qAµ

4

(5.23)

pµ − 2qAµ pµ + q2 Aµ Aµ 4

That this four-Hamiltonian yields the correct covariant equation of motion can be seen by inserting it into the four-dimensional Hamilton’s equations (5.17) and using the relation (5.22): q ∂Aν ∂H(4) = − (pν − qAν ) µ µ ∂x m0 ∂x q ∂A ν = − m0 uν µ m0 ∂x (5.24) ν ∂Aν = −qu ∂xµ duµ ∂Aµ dpµ = −m0 − q ν uν =− dτ dτ ∂x where in the last step Equation (5.19) on the preceding page was used. Rearranging terms, and using Equation (4.80) on page 63, we obtain duµ (5.25) = quν ∂µ Aν − ∂ν Aµ 4 = quν Fµν dτ 3 which is identical to the covariant equation of motion Equation (5.13) on page 70. We can then safely conclude that the Hamiltonian in question is correct. Recalling expression (4.48) on page 59 and representing the canonically conjugate four-momentum as pµ = (p0 , p), we obtain the following scalar products: m0

pµ pµ = (p0 )2 − (p)2 1 Aµ pµ = φp0 − (p · A) c 1 Aµ Aµ = 2 φ2 − (A)2 c


(5.26a) (5.26b) (5.26c)


5.1

C HARGED PARTICLES

IN AN

73


Inserting these explicit expressions into Equation (5.23) on the facing page, and using the fact that for H(4) is equal to the scalar value m0 c2 /2, as derived in Equation (5.21) on the preceding page, we obtain the equation m0 c2 1 2 q2 = (p0 )2 − (p)2 − qφp0 + 2q(p · A) + 2 φ2 − q2 (A)2 2 2m0 c c

(5.27)

which is the second order algebraic equation in p0 : (p0 )2 −

2q 0 q2 φp − L (p)2 − 2qpME·N A + q2 (A)2O + 2 φ2 − m20 c2 = 0 c c

(5.28)

(p−qA)2

with two possible solutions q p = φ± c 0

K

(p − qA)2 + m20 c2

(5.29)

Since the fourth component (time component) p0 of a four-momentum vector pµ multiplied by c represents the energy [cf. Equation (4.42) on page 58], the positive solution in Equation (5.29) above must be identified with the ordinary Hamilton function H divided by c. Consequently, H ≡ cp0 = qφ + c

K

(p − qA)2 + m20 c2

(5.30)

is the ordinary 3D Hamilton function for a charged particle moving in scalar and vector potentials associated with prescribed electric and magnetic fields. The ordinary Lagrange and Hamilton functions L and H are related to each other by the 3D transformation [cf. the 4D transformation (5.16) between L (4) and H(4) ] L = p·v−H

(5.31)

Using the explicit expressions (Equation (5.30) on the preceding page) and (Equation (5.31) on the previous page), we obtain the explicit expression for the ordinary 3D Lagrange function L = p · v − qφ − c

K

(p − qA)2 + m20 c2

(5.32)

and if we make the identification p − qA =

K

m0 v 1−

v2 c2

= mv


(5.33)


74


AND

PARTICLES

where the quantity mv is the usual kinetic momentum, we can rewrite this expression for the ordinary Lagrangian as follows: 2

K

L = qA · v + mv − qφ − c m2 v2 + m20 c2 2

2

= mv − q(φ − A · v) − mc = −qφ + qA · v − m0 c

2

v2 1− 2 c

(5.34)

What we have obtained is the relativstically correct (covariant) expression for the Lagrangian describing the motion of a charged particle in scalar and vector potentials associated with prescribed electric and magnetic fields.

5.2 Covariant Field Theory So far, we have considered two classes of problems. Either we have calculated the fields from given, prescribed distributions of charges and currents, or we have derived the equations of motion for charged particles in given, prescribed fields. Let us now put the fields and the particles on an equal footing and present a theoretical description which treats the fields, the particles, and their interactions in a unified way. This involves transition to a field picture with an infinite number of degrees of freedom. We shall first consider a simple mechanical problem whose solution is well known. Then, drawing inferences from this model problem, we apply a similar view on the electromagnetic problem.

5.2.1 Lagrange-Hamilton formalism for fields and interactions Consider N identical mass points, each with mass m and connected to its neighbour along a one-dimensional straight line, which we choose to be the x axis, by identical ideal springs with spring constants k. At equilibrium the mass points are at rest, distributed evenly with a distance a to their two nearest neighbours. After perturbation, the motion of mass point i will be a one-dimensional oscillatory motion along xˆ . Let us denote the deviation for mass point i from its equilibrium position by ηi (t) xˆ . The solution to this mechanical problem can be obtained if we can find a Lagrangian (Lagrange function) L which satisfies the variational equation δ L(ηi , η˙ i , t) dt = 0


(5.35)


5.2

75

C OVARIANT F IELD T HEORY

ηi−1 m

m

k a

ηi m

k a

ηi+1

k a

m

m

k a

x

F IGURE 5.1: A one-dimensional chain consisting of N discrete, identical mass points m, connected to their neighbours with identical, ideal springs with spring constants k. The equilibrium distance between the neighbouring mass points is a and ηi−1 (t), ηi (t), ηi+1 (t) are the instantaneous deviations, along the x axis, of positions of the (i − 1)th, ith, and (i + 1)th mass point, respectively.

According to Equation (M.96) on page 185, the Lagrangian is L = T − V where T denotes the kinetic energy and V the potential energy of a classical mechanical system with conservative forces. In our case the Lagrangian is L=

1 N 2 2 m˙ηi − k(ηi+1 − ηi ) 2∑ i=1

(5.36)

Let us write the Lagrangian, as given by Equation (5.36), in the following way: N

L = ∑ aP

(5.37)

i

i=1

Here,

P

i

=

1 m 2 η˙ − ka . ηi+1a− ηi / 2 a i

2

(5.38)

is the so called linear Lagrange density. If we now let N → ∞ and, at the same time, let the springs become infinitesimally short according to the following scheme: a → dx m dm → =µ a dx ka → Y ηi+1 − ηi ∂η → a ∂x


(5.39a) linear mass density

(5.39b)

Young’s modulus

(5.39c) (5.39d)


76


AND

PARTICLES

we obtain L=

P

dx

(5.40)

where

PQ

∂η ∂η 1 ∂η η, , , t = H µ ∂t ∂x 2 ∂t

2

−Y

2

∂η ∂x

I (5.41)

Notice how we made a transition from a discrete description, in which the mass points were identified by a discrete integer variable i = 1, 2, . . . , N, to a continuous description, where the infinitesimal mass points were instead identified by a continuous real parameter x, namely their position along xˆ . A consequence of this transition is that the number of degrees of freedom for the system went from the finite number N to infinity! Another consequence is that P has now become dependent also on the partial derivative with respect to x of the “field coordinate” η. But, as we shall see, the transition is well worth the price because it allows us to treat all fields, be it classical scalar or vectorial fields, or wave functions, spinors and other fields that appear in quantum physics, on an equal footing. Under the assumption of time independence and fixed endpoints, the variation principle (5.35) on the previous page yields: δ L dt

=δ =

R

PS

RUTV

η,

∂η ∂η , dx dt ∂t ∂x

∂P ∂P ∂η ∂P ∂η δη + δ + δ dx dt ∂η ∂η ∂η ∂t ∂x WX ∂ . ∂t / ∂ . ∂x /

(5.42)

=0 The last integral can be integrated by parts. This results in the expression

RUTV

∂P ∂ − ∂η ∂t

C

∂P

B@

∂.

∂η ∂t

/

F

−

∂ ∂x

B@

C

∂P

∂.

∂η ∂x

/

F WX

δη dx dt = 0

(5.43)

where the variation is arbitrary (and the endpoints fixed). This means that the integrand itself must vanish. If we introduce the functional derivative δP ∂P ∂ = − δη ∂η ∂x


B@

C

∂P

∂.

∂η ∂x

/

F

(5.44)


5.2

77


we can express this as ∂ δP − δη ∂t

C

∂P

B@

∂.

∂η ∂t

F /

=0

(5.45)

which is the one-dimensional Euler-Lagrange equation. Inserting the linear mass point chain Lagrangian density, Equation (5.41) on the facing page, into Equation (5.45) above, we obtain the equation of motion for our one-dimensional linear mechanical structure. It is: ∂2 ∂2 η − Y η= ∂t2 ∂x2

µ

∂2 µ ∂2 − η=0 Y ∂t2 ∂x2

(5.46)

i.e., the one-dimensional wave equation for compression waves which propa√ gate with phase speed vφ = Y/µ along the linear structure. A generalisation of the above 1D results to a three-dimensional continuum is straightforward. For this 3D case we get the variational principle δ L dt = δ

=δ =

Y

P

d3x dt

PS

η,

∂η d4x ∂xµ

∂P ∂ − ∂η ∂xµ

R TV

C

B@

∂P

∂.

∂η ∂xµ

/

F WX

(5.47) 4

δη d x

=0 where the variation δη is arbitrary and the endpoints are fixed. This means that the integrand itself must vanish: ∂P ∂ − µ ∂η ∂x

B@

C

∂P

∂.

∂η ∂xµ

/

F

=0

(5.48)

This constitutes the four-dimensional Euler-Lagrange equations. Introducing the three-dimensional functional derivative δP ∂P ∂ = − δη ∂η ∂xi

B@

C

∂P

∂.

∂η ∂xi


/

F

(5.49)


78


we can express this as δP ∂ − δη ∂t

B@

PARTICLES

C

∂P

∂.

AND

∂η ∂t

F /

=0

(5.50)

In analogy with particle mechanics (finite number of degrees of freedom), we may introduce the canonically conjugate momentum density π(xµ ) = π(t, x) =

∂P

∂.

∂η ∂t

(5.51)

/

and define the Hamilton density

Z

π, η,

∂η ∂η ∂η ∂η ; t = π − PQ η, , i i ∂x ∂t ∂t ∂x

(5.52)

If, as usual, we differentiate this expression and identify terms, we obtain the following Hamilton density equations

Z

∂ ∂η = ∂π ∂t Z δ ∂π =− δη ∂t

(5.53a) (5.53b)

The Hamilton density functions are in many ways similar to the ordinary Hamilton functions and lead to similar results.

The electromagnetic field Above, when we described the mechanical field, we used a scalar field η(t, x). If we want to describe the electromagnetic field in terms of a Lagrange density P and Euler-Lagrange equations, it comes natural to express P in terms of the four-potential Aµ (xκ ). The entire system of particles and fields consists of a mechanical part, a field part and an interaction part. We therefore assume that the total Lagrange density P tot for this system can be expressed as

P

tot

=P

mech

+P

inter

+P

field

(5.54)

where the mechanical part has to do with the particle motion (kinetic energy). It is given by L(4) /V where L(4) is given by Equation (5.2) on page 68 and V is



5.2

79


the volume. Expressed in the rest mass density %0 , the mechanical Lagrange density can be written

P

mech

1 = %0 uµ uµ 2

(5.55)

The P inter part which describes the interaction between the charged particles and the external electromagnetic field. A convenient expression for this interaction Lagrange density is

P

inter

= jµ Aµ

(5.56)

For the field part P field we choose the difference between magnetic and electric energy density (in analogy with the difference between kinetic and potential energy in a mechanical field). Using the field tensor, we express this field Lagrange density as

P

field

=

1 µν F Fµν 4µ0

(5.57)

so that the total Lagrangian density can be written

P

tot

1 1 µν = %0 uµ uµ + jµ Aµ + F Fµν 2 4µ0

(5.58)

From this we can calculate all physical quantities.

F IELD ENERGY DIFFERENCE EXPRESSED IN THE FIELD TENSOR

E XAMPLE 5.1

Show, by explicit calculation, that 1 1 µν F F µν = 4µ0 2

B2 − ε0 E 2 µ0

(5.59)

i.e., the difference between the magnetic and electric field energy densities. From Formula (4.80) on page 63 we recall that

F µν

""#

=

0 E x /c E y /c E z /c

−E x /c −E y /c −E z /c 0 −Bz By Bz 0 −B x −By Bx 0

*& ( (

(5.60)

and from Formula (4.82) on page 64 that

Fµν

=

""#

0 −E x /c −E y /c −E z /c

E x /c 0 Bz −By

E y /c −Bz 0 Bx

E z /c By −B x 0

*& ( (

(5.61)

where µ denotes the row number and ν the column number. Then, Einstein summation



80


AND

PARTICLES

and direct substitution yields F µν F µν = F 00 F 00 + F 01 F 01 + F 02 F 02 + F 03 F 03 + F 10 F 10 + F 11 F 11 + F 12 F 12 + F 13 F 13 + F 20 F 20 + F 21 F 21 + F 22 F 22 + F 23 F 23 + F 30 F 30 + F 31 F 31 + F 32 F 32 + F 33 F 33 = 0 − E 2x /c2 − E y2 /c2 − E z2 /c2

(5.62)

− E 2x /c2 + 0 + B2z + B2y

− E y2 /c2 + B2z + 0 + B2x

− E z2 /c2 + B2y + B2x + 0

= −2E 2x /c2 − 2E y2 /c2 − 2E z2 /c2 + 2B2x + 2B2y + 2B2z

= −2E 2 /c2 + 2B2 = 2(B2 − E 2 /c2 ) or 1 1 µν F F µν = 4µ0 2

1 2 B2 − E µ0 c 2 µ0

=

1 2

B2 − ε0 E 2 µ0

(5.63)

where, in the last step, the identity ε0 µ0 = 1/c2 was used.

QED E ND

OF EXAMPLE

5.1C

Using P tot in the 3D Euler-Lagrange equations, Equation (5.48) on page 77 (with η replaced by Aν ), we can derive the dynamics for the whole system. For instance, the electromagnetic part of the Lagrangian density

P

EM

=P

inter

+P

field

= jν Aν +

1 µν F Fµν 4µ0

(5.64)

inserted into the Euler-Lagrange equations, expression (5.48) on page 77, yields two of Maxwell’s equations. To see this, we note from Equation (5.64) above and the results in Example 5.1 that ∂P

EM

= jν

(5.65)



∂Aν

5.2

81


Furthermore, ∂P EM ∂(∂ A )

∂µ

µ ν

∂ 1 ∂µ F κλ Fκλ 4 4µ0 ∂(∂µ Aν ) 3

∂ 1 = ∂µ (∂κ Aλ − ∂λ Aκ )(∂κ Aλ − ∂λ Aκ ) 4µ0 ∂(∂µ Aν ) =

=

1 ∂ ∂µ ∂κ Aλ ∂κ Aλ − ∂κ Aλ ∂λ Aκ 4µ0 ∂(∂µ Aν )

(5.66)

− ∂λ Aκ ∂κ Aλ + ∂λ Aκ ∂λ Aκ =

∂ 1 ∂µ ∂κ Aλ ∂κ Aλ − ∂κ Aλ ∂λ Aκ 4 2µ0 ∂(∂µ Aν ) 3

'

But ∂ ∂ ∂ ∂κ Aλ ∂κ Aλ 4 = ∂κ Aλ ∂κ Aλ + ∂κ Aλ ∂κ Aλ ∂(∂µ Aν ) 3 ∂(∂µ Aν ) ∂(∂µ Aν ) ∂ ∂ ∂κ Aλ + ∂κ Aλ gκα ∂α gλβ Aβ = ∂κ Aλ ∂(∂µ Aν ) ∂(∂µ Aν ) ∂ ∂ = ∂κ Aλ ∂κ Aλ + gκα gλβ ∂κ Aλ ∂α Aβ (5.67) ∂(∂µ Aν ) ∂(∂µ Aν ) ∂ ∂ = ∂κ Aλ ∂κ Aλ + ∂α Aβ ∂α Aβ ∂(∂µ Aν ) ∂(∂µ Aν ) = 2∂µ Aν Similarly, ∂ ∂κ Aλ ∂λ Aκ 4 = 2∂ν Aµ ∂(∂µ Aν ) 3

(5.68)

so that ∂µ

∂P EM ∂(∂ A )

µ ν

=

1 1 ∂µ ∂µ Aν − ∂ν Aµ 4 = ∂µ F µν µ0 3 µ0

(5.69)

This means that the Euler-Lagrange equations, expression (5.48) on page 77, for the Lagrangian density P EM and with Aν as the field quantity become ∂P

EM

∂Aν

− ∂µ

∂P EM ∂(∂ A )

µ ν


= jν −

1 ∂µ F µν = 0 µ0

(5.70)


82


AND

PARTICLES

or ∂µ F µν = µ0 jν

(5.71)

which, according to Equation (4.83) on page 64, is the covariant version of Maxwell’s source equations.

Other fields In general, the dynamic equations for most any fields, and not only electromagnetic ones, can be derived from a Lagrangian density together with a variational principle (the Euler-Lagrange equations). Both linear and non-linear fields are studied with this technique. As a simple example, consider a real, scalar field η which has the following Lagrange density:

P

=

1 ∂µ η∂µ η − m2 η2 4 23

(5.72)

Insertion into the 1D Euler-Lagrange equation, Equation (5.45) on page 77, yields the dynamic equation (

5

2

− m2 )η = 0

(5.73)

with the solution η = ei(k·x−ωt)

e−m|x| |x|

(5.74)

which describes the Yukawa meson field for a scalar meson with mass m. With π=

1 ∂η c2 ∂t

(5.75)

we obtain the Hamilton density

Z

=

1 2 2 c π + (∇η)2 + m2 η2 2

(5.76)

which is positive definite. Another Lagrangian density which has attracted quite some interest is the Proca Lagrangian

P

EM

=P

inter


+P

field

= jν Aν +

1 µν F Fµν + m2 Aµ Aµ 4µ0

(5.77)


5.2

83

B IBLIOGRAPHY

which leads to the dynamic equation ∂µ F µν + m2 Aν = µ0 jν

(5.78)

This equation describes an electromagnetic field with a mass, or, in other words, massive photons. If massive photons would exist, large-scale magnetic fields, including those of the earth and galactic spiral arms, would be significantly modified to yield measurable discrepances from their usual form. Space experiments of this kind onboard satellites have led to stringent upper bounds on the photon mass. If the photon really has a mass, it will have an impact on electrodynamics as well as on cosmology and astrophysics.

Bibliography [1] A. O. BARUT, Electrodynamics and Classical Theory of Fields and Particles, Dover Publications, Inc., New York, NY, 1980, ISBN 0-486-64038-8. [2] V. L. G INZBURG, Applications of Electrodynamics in Theoretical Physics and Astrophysics, Revised third ed., Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo and Melbourne, 1989, ISBN 288124-719-9. [3] H. G OLDSTEIN, Classical Mechanics, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1981, ISBN 0-201-02918-9. [4] W. T. G RANDY, Introduction to Electrodynamics and Radiation, Academic Press, New York and London, 1970, ISBN 0-12-295250-2. [5] L. D. L ANDAU AND E. M. L IFSHITZ, The Classical Theory of Fields, fourth revised English ed., vol. 2 of Course of Theoretical Physics, Pergamon Press, Ltd., Oxford . . . , 1975, ISBN 0-08-025072-6. [6] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6. [7] J. J. S AKURAI, Advanced Quantum Mechanics, Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1967, ISBN 0-201-06710-2. [8] D. E. S OPER, Classical Field Theory, John Wiley & Sons, Inc., New York, London, Sydney and Toronto, 1976, ISBN 0-471-81368-0.



84



AND

PARTICLES


6 Electromagnetic Fields and Matter The microscopic Maxwell equations (1.43) derived in Chapter 1 are valid on all scales where a classical description is good. However, when macroscopic matter is present, it is sometimes convenient to use the corresponding macroscopic Maxwell equations (in a statistical sense) in which auxiliary, derived fields are introduced in order to incorporate effects of macroscopic matter when this is immersed fully or partially in an electromagnetic field.

6.1 Electric Polarisation and Displacement In certain cases, for instance in engneering applications, it may be convenient to separate the influence of an external electric field on free charges and on neutral matter in bulk. This view, which, as we shall see, has certain limitations, leads to the introduction of (di)electric polarisation and (di)electric displacement.

6.1.1 Electric multipole moments The electrostatic properties of a spatial volume containing electric charges and located near a point x0 can be characterized in terms of the total charge or electric monopole moment q=

ρ(x0 ) d3x0

(6.1)

V

where the ρ is the charge density introduced in Equation (1.7) on page 4, the electric dipole moment vector p(x0 ) =

V

(x0 − x0 )ρ(x0 ) d3x0

(6.2)

85

86


AND

M ATTER

with components pi , i = 1, 2, 3, the electric quadrupole moment tensor Q(x0 ) =

V

(x0 − x0 )(x0 − x0 )ρ(x0 ) d3x0

(6.3)

with components Qi j , i, j = 1, 2, 3, and higher order electric moments. In particular, the electrostatic potential Equation (3.3) on page 35 from a charge distribution located near x0 can be Taylor expanded in the following way: φstat (x) =

q 1 1 (x − x0 )i + pi 2 4πε0 |x − x0 | |x − x0 | |x − x0 |

1 Qi j + |x − x0 |3

3 (x − x0 )i (x − x0 ) j 1 − δi j + . . . 2 |x − x0 | |x − x0 | 2

(6.4)

where Einstein’s summation convention over i and j is implied. As can be seen from this expression, only the first few terms are important if the field point (observation point) is far away from x0 . For a normal medium, the major contributions to the electrostatic interactions come from the net charge and the lowest order electric multipole moments induced by the polarisation due to an applied electric field. Particularly important is the dipole moment. Let P denote the electric dipole moment density (electric dipole moment per unit volume; unit: C/m2 ), also known as the electric polarisation, in some medium. In analogy with the second term in the expansion Equation (6.4) on page 86, the electric potential from this volume distribution P(x0 ) of electric dipole moments p at the source point x0 can be written 1 4πε0 1 = 4πε0

x − x0 3 0 1 d x =− 0 3 |x − x | 4πε0 V 1 P(x0 ) · ∇0 d3x0 |x − x0 | V P(x0 ) ·

φp (x) =

V

P(x0 ) · ∇

1 d3x0 |x − x0 | (6.5)

Using the expression Equation (M.83) on page 182 and applying the divergence theorem, we can rewrite this expression for the potential as follows: 1 4πε0 1 = 4πε0

φp (x) =


P(x0 ) ∇0 · P(x0 ) 3 0 3 0 d x − dx |x − x0 |

V V |x − x0 | 0 0 0 P(x ) · nˆ 2 ∇ · P(x ) 3 0 d x− dx 0| |x − x

S V |x − x0 | ∇0 ·

(6.6)


6.1

E LECTRIC P OLARISATION

AND

87

D ISPLACEMENT

where the first term, which describes the effects of the induced, non-cancelling dipole moment on the surface of the volume, can be neglected, unless there is a discontinuity in P · nˆ at the surface. Doing so, we find that the contribution from the electric dipole moments to the potential is given by φp =

1 4πε0

V

−∇0 · P(x0 ) 3 0 dx |x − x0 |

(6.7)

Comparing this expression with expression Equation (3.3) on page 35 for the electrostatic potential from a static charge distribution ρ, we see that −∇ · P(x) has the characteristics of a charge density and that, to the lowest order, the effective charge density becomes ρ(x) − ∇ · P(x), in which the second term is a polarisation term. The version of Equation (1.7) on page 4 where “true” and polarisation charges are separated thus becomes ∇·E =

ρ(x) − ∇ · P(x) ε0

(6.8)

Rewriting this equation, and at the same time introducing the electric displacement vector (C/m2 ) D = ε0 E + P

(6.9)

we obtain ∇ · (ε0 E + P) = ∇ · D = ρtrue (x)

(6.10)

where ρtrue is the “true” charge density in the medium. This is one of Maxwell’s equations and is valid also for time varying fields. By introducing the notation ρpol = −∇ · P for the “polarised” charge density in the medium, and ρtotal = ρtrue + ρpol for the “total” charge density, we can write down the following alternative version of Maxwell’s equation (6.23a) on page 90 ∇·E =

ρtotal (x) ε0

(6.11)

Often, for low enough field strengths |E|, the linear and isotropic relationship between P and E P = ε0 χE

(6.12)

is a good approximation. The quantity χ is the electric susceptibility which is material dependent. For electromagnetically anisotropic media such as a



88


AND

M ATTER

magnetised plasma or a birefringent crystal, the susceptibility is a tensor. In general, the relationship is not of a simple linear form as in Equation (6.12) on the preceding page but non-linear terms are important. In such a situation the principle of superposition is no longer valid and non-linear effects such as frequency conversion and mixing can be expected. Inserting the approximation (6.12) into Equation (6.9) on the previous page, we can write the latter D = εE

(6.13)

where, approximately, ε = ε0 (1 + χ)

(6.14)

6.2 Magnetisation and the Magnetising Field An analysis of the properties of stationary magnetic media and the associated currents shows that three such types of currents exist: 1. In analogy with “true” charges for the electric case, we may have “true” currents jtrue , i.e., a physical transport of true charges. 2. In analogy with electric polarisation P there may be a form of charge transport associated with the changes of the polarisation with time. We call such currents induced by an external field polarisation currents. We identify them with ∂P/∂t. 3. There may also be intrinsic currents of a microscopic, often atomic, nature that are inaccessible to direct observation, but which may produce net effects at discontinuities and boundaries. We shall call such currents magnetisation currents and denote them jM . No magnetic monopoles have been observed yet. So there is no correspondence in the magnetic case to the electric monopole moment (6.1). The lowest order magnetic moment, corresponding to the electric dipole moment (6.2), is the magnetic dipole moment m=

1 2

V

(x0 − x0 ) × j(x0 ) d3x0

(6.15)

For a distribution of magnetic dipole moments in a volume, we may describe this volume in terms of the magnetisation, or magnetic dipole moment per unit



6.2

M AGNETISATION

AND THE

89

M AGNETISING F IELD

volume, M. Via the definition of the vector potential one can show that the magnetisation current and the magnetisation is simply related: jM = ∇ × M

(6.16)

In a stationary medium we therefore have a total current which is (approximately) the sum of the three currents enumerated above: jtotal = jtrue +

∂P +∇×M ∂t

(6.17)

We might then, erroneously, be led to think that ∇ × B = µ0 jtrue +

∂P +∇×M ∂t

(6.18)

Moving the term ∇ × M to the left hand side and introducing the magnetising field (magnetic field intensity, Ampère-turn density) as H=

B −M µ0

(6.19)

and using the definition for D, Equation (6.9) on page 87, we can write this incorrect equation in the following form ∇ × H = jtrue +

∂E ∂P ∂D = jtrue + − ε0 ∂t ∂t ∂t

(6.20)

As we see, in this simplistic view, we would pick up a term which makes the equation inconsistent; the divergence of the left hand side vanishes while the divergence of the right hand side does not. Maxwell realised this and to overcome this inconsistency he was forced to add his famous displacement current term which precisely compensates for the last term in the right hand side. In Chapter 1, we discussed an alternative way, based on the postulate of conservation of electric charge, to introduce the displacement current. We may, in analogy with the electric case, introduce a magnetic susceptibility for the medium. Denoting it χm , we can write H=

B µ

(6.21)

where, approximately, µ = µ0 (1 + χm )


(6.22)


90


AND

M ATTER

Maxwell’s equations expressed in terms of the derived field quantities D and H are ∇ · D = ρ(t, x) ∇·B = 0

∇×E = −

(6.23a) (6.23b)

∂B ∂t

(6.23c)

∇ × H = j(t, x) +

∂ D ∂t

(6.23d)

and are called Maxwell’s macroscopic equations. These equations are convenient to use in certain simple cases. Together with the boundary conditions and the constitutive relations, they describe uniquely (but only approximately!) the properties of the electric and magnetic fields in matter.

6.3 Energy and Momentum We shall use Maxwell’s macroscopic equations in the following considerations on the energy and momentum of the electromagnetic field and its interaction with matter.

6.3.1 The energy theorem in Maxwell’s theory Scalar multiplying (6.23c) by H, (6.23d) by E and subtracting, we obtain H · (∇ × E) − E · (∇ × H) = ∇ · (E × H) ∂B ∂D 1∂ = −H · −E·j−E· =− (H · B + E · D) − j · E ∂t ∂t 2 ∂t

(6.24)

Integration over the entire volume V and using Gauss’s theorem (the divergence theorem), we obtain −

∂ ∂t

V

1 (H · B + E · D) d3x0 = 2

V

j · E d3x0 +

S

(E × H) · nd ˆ 2x0

(6.25)

But, according to Ohm’s law in the presence of an electromotive force field, Equation (1.26) on page 10: j = σ(E + EEMF )

(6.26)

which means that

V

j · E d3x0 =


V

j2 3 0 dx − σ

V

j · EEMF d3x0

(6.27)


6.3

E NERGY

AND

91

M OMENTUM

Inserting this into Equation (6.25) on the facing page

L

V

j · EEMF d3x0 =

MEN

O

Applied electric power

L

V

j2 3 0 ∂ dx + σ MEN O ∂t L

V

1 ˆ 2x0 (E · D + H · B) d3x0 + (E × H) · nd 2 MEN O L S M[N O

Joule heat

Field energy

Radiated power

(6.28) which is the energy theorem in Maxwell’s theory also known as Poynting’s theorem. It is convenient to introduce the following quantities: 1 E · D d3x0 2 V 1 H · B d3x0 Um = 2 V S = E×H Ue =

(6.29) (6.30) (6.31)

where U e is the electric field energy, U m is the magnetic field energy, both measured in J, and S is the Poynting vector (power flux), measured in W/m2 .

6.3.2 The momentum theorem in Maxwell’s theory Let us now investigate the momentum balance (force actions) in the case that a field interacts with matter in a non-relativistic way. For this purpose we consider the force density given by the Lorentz force per unit volume ρE+j×B. Using Maxwell’s equations (6.23) and symmetrising, we obtain ∂D ×B ∂t ∂D = E(∇ · D) + (∇ × H) × B − ×B ∂t = E(∇ · D) − B × (∇ × H) ∂B ∂ − (D × B) + D × ∂t ∂t = E(∇ · D) − B × (∇ × H) ∂ − (D × B) − D × (∇ × E) + H(∇ L ME·N BO ) ∂t

ρE + j × B = (∇ · D)E + ∇ × H −

(6.32)

=0

= [E(∇ · D) − D × (∇ × E)] + [H(∇ · B) − B × (∇ × H)] ∂ − (D × B) ∂t



92


AND

M ATTER

One verifies easily that the ith vector components of the two terms in square brackets in the right hand member of (6.32) can be expressed as [E(∇ · D) − D × (∇ × E)]i =

1 ∂D ∂E ∂ E· −D· + 2 ∂xi ∂xi ∂x j

1 E i D j − E · D δi j 2 (6.33)

and [H(∇ · B) − B × (∇× H)]i =

1 ∂B ∂H ∂ H· −B· + 2 ∂xi ∂xi ∂x j

1 Hi B j − B · H δ i j 2 (6.34)

respectively. Using these two expressions in the ith component of Equation (6.32) on the previous page and re-shuffling terms, we get ∂D ∂B ∂ ∂E ∂H + H· + (D × B)i −D· −B· ∂xi ∂xi ∂xi ∂xi \

∂t 1 1 E i D j − E · D δ i j + Hi B j − H · B δ i j 2 2 (6.35)

(ρE + j × B)i − =

∂ ∂x j

1 2

E·

Introducing the electric volume force Fev via its ith component (Fev )i = (ρE + j × B)i −

1 2

E·

∂D ∂E ∂B ∂H −D· + H· −B· ∂xi ∂xi ∂xi ∂xi 0

(6.36)

and the Maxwell stress tensor T with components 1 1 T i j = E i D j − E · D δ i j + Hi B j − H · B δ i j 2 2

(6.37)

we finally obtain the force equation

Fev + ∂t∂ (D × B)

= i

∂T i j = (∇ · T)i ∂x j

(6.38)

If we introduce the relative electric permittivity κ and the relative magnetic permeability κm as D = κε0 E = εE

(6.39)

B = κm µ0 H = µH

(6.40)



6.3

93

B IBLIOGRAPHY

we can rewrite (6.38) as ∂T i j κκm ∂S = Fev + 2 ∂x j c ∂t

(6.41) i

where S is the Poynting vector defined in Equation (6.29) on page 91. Integration over the entire volume V yields

L

V

Fev d3x0 +

MEN

O

Force on the matter

d dt L

V

κκm 3 0 Sd x = c2MEN O

Field momentum

L

S

T nˆ d2x0

MEN

O

(6.42)

Maxwell stress

which expresses the balance between the force on the matter, the rate of change of the electromagnetic field momentum and the Maxwell stress. This equation is called the momentum theorem in Maxwell’s theory. In vacuum (6.42) becomes

V

ρ(E + v × B) d3x0 +

1 d c2 dt

S d3x0 = V

S

T nˆ d2x0

(6.43)

or d mech d field p + p = dt dt

S

T nˆ d2x0

(6.44)

Bibliography [1] E. H ALLÉN, Electromagnetic Theory, Chapman & Hall, Ltd., London, 1962. [2] J. D. JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, NY . . . , 1999, ISBN 0-471-30932-X. [3] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6. [4] J. A. S TRATTON, Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, NY and London, 1953, ISBN 07-062150-0.



94



AND

M ATTER


7 Electromagnetic Fields from Arbitrary Source Distributions

While, in principle, the electric and magnetic fields can be calculated from the Maxwell equations in Chapter 1, or even from the wave equations in Chapter 2, it is often physically more lucid to calculate them from the electromagnetic potentials derived in Chapter 3. In this chapter we will derive the electric and magnetic fields from the potentials. We recall that in order to find the solution (3.35) for the generic inhomogeneous wave equation (3.19) on page 41 we presupposed the existence of a Fourier transform pair (3.20a) on page 41 for the generic source term Ψ(t, x) = Ψω (x) =

∞ −∞

1 2π

Ψω (x) e−iωt dω ∞ −∞

(7.1a)

Ψ(t, x) eiωt dt

(7.1b)

That such transform pairs exists is true for most physical variables which are neither strictly monotonically increasing nor strictly monotonically decreasing with time. For charge and current densities varying in time we can therefore, without loss of generality, work with individual Fourier components ρw (x) and jw (x), respectively. Strictly speaking, the existence of a single Fourier component assumes a monochromatic source (i.e., a source containing only one single frequency component), which in turn requires that the electric and magnetic fields exist for infinitely long times. However, by taking the proper limits, we may still use this approach even for sources and fields of finite duration. This is the method we shall utilise in this chapter in order to derive the electric and magnetic fields in vacuum from arbitrary given charge densities ρ(t, x) and current densities j(t, x), defined by the temporal Fourier transform

95

96


AND

M ATTER

pairs ρ(t, x) =

∞

ρω (x) e−iωt dω

−∞

∞

1 2π

ρω (x) =

(7.2a)

ρ(t, x) eiωt dt

−∞

(7.2b)

and j(t, x) = jω (x) =

∞ −∞

jω (x) e−iωt dω

1 2π

∞ −∞

(7.3a)

j(t, x) eiωt dt

(7.3b)

under the assumption that only retarded potentials produce physically acceptable solutions.1 The temporal Fourier transform pair for the retarded vector potential can then be written φ(t, x) =

∞ −∞

φω (x) e−iωt dω

1 φω (x) = 2π

∞

1 φ(t, x) e dt = 4πε −∞ 0 iωt

(7.4a) 0

eik|x−x | 3 0 ρω (x ) dx |x − x0 | 0

(7.4b)

where in the last step, we made use of the explicit expression for the temporal Fourier transform of the generic potential component Ψω (x), Equation (3.32) on page 43. Similarly, the following Fourier transform pair for the vector potential must exist: A(t, x) =

∞ −∞

1 Aω (x) = 2π

Aω (x) e−iωt dω

(7.5a) 0

eik|x−x | 3 0 µ0 jω (x0 ) dx A(t, x) e dt = 4π |x − x0 | −∞ ∞

iωt

(7.5b)

Clearly, we must require that Aω = A∗−ω ,

φω = φ∗−ω

(7.6)

in order that all physical quantities be real. Similar transform pairs and requirements of real-valuedness exist for the fields themselves. 1 In

fact, John A. Wheeler and Richard P. Feynman derived in 1945 a fully self-consistent electrodynamics using both the retarded and the advanced potentials [6]; See also [1].



7.1

97

T HE M AGNETIC F IELD

In the limit that the sources can be considered monochromatic containing only one single frequency ω0 , we have the much simpler expressions ρ(t, x) = ρ0 (x)e−iω0 t

(7.7a)

−iω0 t

(7.7b)

φ(t, x) = φ0 (x)e−iω0 t

(7.7c)

−iω0 t

(7.7d)

j(t, x) = j0 (x)e

A(t, x) = A0 (x)e

where again the real-valuedness of all these quantities is implied. As discussed above, we can safely assume that all formulae derived for a general temporal Fourier representation of the source (general distribution of frequencies in the source) are valid for these simple limiting cases. We note that in this context, we can make the formal identification ρω = ρ0 δ(ω − ω0 ), jω = j0 δ(ω − ω0 ) etc., and that we therefore, without any loss of stringence, let ρ0 mean the same as the Fourier amplitude ρω and so on.

7.1 The Magnetic Field Let us now compute the magnetic field from the vector potential, defined by Equation (7.5a) and Equation (7.5b) on the facing page, and Formula (3.6) on page 36: B(t, x) = ∇ × A(t, x)

(7.8)

The calculations are much simplified if we work in ω space and, at the final stage, Fourier transform back to ordinary t space. We are working in the Lorentz gauge and note that in ω space the Lorentz condition, Equation (3.13) on page 38, takes the form k ∇ · Aω − i φω = 0 c

(7.9)

which provides a relation between (the Fourier transforms of) the vector and scalar potentials. Using the Fourier transformed version of Equation (7.8) above and Equation (7.5b) on the facing page, we obtain µ0 Bω (x) = ∇ × Aω (x) = ∇ × 4π


0

eik|x−x | 3 0 jω (x ) dx |x − x0 | V 0

(7.10)


98


FROM

A RBITRARY S OURCE D ISTRIBUTIONS

Using Formula (F.60) on page 165, we can rewrite this as 0

eik|x−x | d3x0 |x − x0 |

V 0 x − x0 jω (x0 ) × − eik|x−x | d3x0 3 0 |x − x | V x − x0 ik|x−x0 | 1 3 0 + jω (x0 ) × ik e 0 |x − x | |x − x0 | d x

V

µ0 4π µ0 =− 4π

Bω (x) = −

jω (x0 ) × ∇

(7.11)

0

0 ik|x−x | × (x − x0 ) jω (x )e d3x0 |x − x0 |3 V 0 (−ik)jω(x0 )eik|x−x | × (x − x0 ) 3 0 + dx

|x − x0 |2 V

µ0 = 4π

From this expression for the magnetic field in the frequency (ω) domain, we obtain the total magnetic field in the temporal (t) domain by taking the inverse Fourier transform (using the identity −ik = −iω/c): B(t, x) = =

∞ −∞

µ0 4π

Bω (x) e−iωt dω

V

1 + c =

µ0 L 4π

V

jω (x0 )ei(k|x−x |−ωt) dω × (x − x0 )

V

0

|x − x0 |3

d3x0

(−iω)jω (x0 )ei(k|x−x |−ωt) dω × (x − x0 ) 0

|x − x0 |2

0 , x0 ) × (x − x0 ) j(tret µ0 d3x0 + 03 4πc

|xM[−N x |

Induction field

O L

V

d3x0

(7.12)

0 , x0 ) × (x − x0 ) ˙j(tret d3x0 2 0 |xM[− N x| O Radiation field

where def 0 ˙j(tret , x0 ) ≡

∂j ∂t

(7.13) 0 t=tret

The first term, the induction field, dominates near the current source but falls off rapidly with distance from it, is the electrodynamic version of the BiotSavart law in electrostatics, Formula (1.13) on page 6. The second term, the radiation field or the far field, dominates at large distances and represents energy that is transported out to infinity. Note how the spatial derivatives (∇) gave rise to a time derivative (˙)!



7.2

99

T HE E LECTRIC F IELD

7.2 The Electric Field In order to calculate the electric field, we use the temporally Fourier transformed version of Formula (3.10) on page 37, inserting Equations (7.4b) and (7.5b) as the explicit expressions for the Fourier transforms of φ and A: Eω (x) = −∇φω (x) + iωAω (x)

0

eik|x−x | 3 0 iµ0 ω 1 ∇ ρω (x0 ) dx + =− 4πε0 V 4π |x − x0 | 0 ρω (x0 )eik|x−x | (x − x0 ) 3 0 1 dx = 4πε0 V |x − x0 |3 − ik

V

0

eik|x−x | 3 0 jω (x ) dx |x − x0 | V 0

(7.14)

0

ρω (x0 )(x − x0 ) jω (x0 ) eik|x−x | 3 0 − dx c |x − x0 |

|x − x0 |

Using the Fourier transform of the continuity Equation (1.21) on page 9 ∇0 · jω (x0 ) − iωρω (x0 ) = 0

(7.15)

we see that we can express ρω in terms of jω as follows i ρω (x0 ) = − ∇0 · jω (x0 ) ω

(7.16)

Doing so in the last term of Equation (7.14) above, and also using the fact that k = ω/c, we can rewrite this Equation as 1 Eω (x) = 4πε0 −

1 c L

0

V

V

ρω (x0 )eik|x−x | (x − x0 ) 3 0 dx |x − x0 |3 0 [∇0 · jω (x0 )](x − x0 ) eik|x−x | 3 0 0 − ikjω (x ) |x − x0 | d xO

|x − x0 | MEN

(7.17)

Iω The last vector-valued integral can be further rewritten in the following way: Iω = =

V

V

0

[∇0 · jω (x0 )](x − x0 ) eik|x−x | 3 0 0 − ikj (x ) ω |x − x0 | |x − x0 | d x 0 eik|x−x | 3 0 ∂ jωm xl − xl0 0 − ik jωl (x ) xˆ l 0 |x − x0 | ∂xm |x − x0 | d x


(7.18)


100


FROM


But, since ∂ 0 ∂xm

xl − xl0 ik|x−x0 | e = |x − x0 |2

jωm

∂ jωm xl − xl0 ik|x−x0 | e 0 ∂xm |x − x0 |2 xl − xl0 ik|x−x0 | ∂ e + jωm 0 ∂xm |x − x0 |2

(7.19)

we can rewrite Iω as Iω = −

+

V

V

jωm

∂ 0 ∂xm

0

xl − xl0 eik|x−x | 3 0 ik|x−x0 | + ikj dx x ˆ e ω l 2 |x − x0 |

|x − x0 | xl − xl0 0 jωm xˆ eik|x−x | d3x0 2 l 0 |x − x |

∂ 0 ∂xm

(7.20)

where, according to Gauss’s theorem, the last term vanishes if jω is assumed to be limited and tends to zero at large distances. Further evaluation of the derivative in the first term makes it possible to write Iω = −

V

− ik

0

0 2 eik|x−x | + −jω j · (x − x0 ) (x − x0 )eik|x−x | d3x0 2 4 ω 0 0 |x − x | |x − x |

V]

−

jω · (x − x0 ) (x − x0 ) |x − x0 |3

e

ik|x−x0 |

0

eik|x−x | + jω d3x0 |x − x0 | ^

(7.21)

Using the triple product “bac-cab” Formula (F.54) on page 164 backwards, and inserting the resulting expression for Iω into Equation (7.17) on the previous page, we arrive at the following final expression for the Fourier transform of the total E field: 0

1 eik|x−x | 3 0 iµ0 ω Eω (x) = − ∇ ρω (x0 ) dx + 4πε0 V 4π |x − x0 | 0| 0 ik|x−x 0 1 ρω (x )e 0 3(x − x ) d3x0 = 4πε0 V |x − x | 0

[jω (x0 )eik|x−x | · (x − x0 )](x − x0 ) 3 0 dx V |x − x0 |4 0 1 [jω (x0 )eik|x−x | × (x − x0 )] × (x − x0 ) 3 0 + dx c V |x − x0 |4 0 [jω (x0 )eik|x−x | × (x − x0 )] × (x − x0 ) 3 0 ik dx − c V

|x − x0 |3 1 + c

0

eik|x−x | 3 0 jω (x ) dx |x − x0 | V 0

(7.22)

Taking the inverse Fourier transform of Equation (7.22), once again using the vacuum relation ω = kc, we find, at last, the expression in time domain for



7.3

101

T HE R ADIATION F IELDS

the total electric field: E(t, x) = =

∞ −∞

Eω (x) e−iωt dω

1 L 4πε0

V

0 , x0 )(x − x0 ) ρ(tret d3x0 |xMEN − x0 |3 O

Retarded Coulomb field

+

1 4πε L 0c

V

0 , x0 ) · (x − x0 )](x − x0 ) [j(tret d3x0 4 0 M[|xN − x | O

(7.23)

Intermediate field

+

1 L 4πε0 c

V

0 , x0 ) × (x − x0 )] × (x − x0 ) [j(tret d3x0 0 |4 − x |x MEN O

Intermediate field

1 + L 4πε0 c2

V

0 , x0 ) × (x − x0 )] × (x − x0 ) [˙j(tret d3x0 0 |3 − x |x M[N O

Radiation field

Here, the first term represents the retarded Coulomb field and the last term represents the radiation field which carries energy over very large distances. The other two terms represent an intermediate field which contributes only in the near zone and must be taken into account there. With this we have achieved our goal of finding closed-form analytic expressions for the electric and magnetic fields when the sources of the fields are completely arbitrary, prescribed distributions of charges and currents. The only assumption made is that the advanced potentials have been discarded; recall the discussion following Equation (3.35) on page 43 in Chapter 3.

7.3 The Radiation Fields In this section we study electromagnetic radiation, i.e., the part of the electric and magnetic fields fields, calculated above, which are capable of carrying energy and momentum over large distances. We shall therefore make the assumption that the observer is located in the far zone, i.e., very far away from the source region(s). The fields which are dominating in this zone are by definition the radiation fields. From Equation (7.12) on page 98 and Equation (7.23) above, which give



102


FROM


the total electric and magnetic fields, we obtain ∞

Brad (t, x) = Erad (t, x) =

=

−∞ ∞ −∞

−iωt Brad dω = ω (x) e −iωt Erad dω ω (x) e

1 4πε0 c2

V

where def 0 ˙j(tret , x0 ) ≡

∂j ∂t

µ0 4πc

V

0 , x0 ) × (x − x0 ) ˙j(tret d3x0 (7.24a) 2 0 |x − x |

0 , x0 ) × (x − x0 )] × (x − x0 ) [˙j(tret d3x0 |x − x0 |3

(7.24b)

(7.25) 0 t=tret

Instead of studying the fields in the time domain, we can often make a spectrum analysis into the frequency domain and study each Fourier component separately. A superposition of all these components and a transformation back to the time domain will then yield the complete solution. The Fourier representation of the radiation fields Equation (7.24a) and Equation (7.24b) above were included in Equation (7.11) on page 98 and Equation (7.22) on page 100, respectively and are explicitly given by 1 ∞ rad B (t, x) eiωt dt 2π −∞ jω (x0 ) × (x − x0 ) ik|x−x0 | 3 0 kµ0 dx = −i e 4π V |x − x0 |2 µ0 jω (x0 ) × k ik|x−x0 | 3 0 e = −i dx 4π V |x − x0 | 1 ∞ rad Erad E (t, x) eiωt dt ω (x) = 2π −∞ [jω (x0 ) × (x − x0 )] × (x − x0 ) ik|x−x0 | 3 0 k dx = −i e 4πε0 c V |x − x0 |3 1 [jω (x0 ) × k] × (x − x0 ) ik|x−x0 | 3 0 = −i dx e 4πε0 c V |x − x0 |2 Brad ω (x) =

(7.26a)

(7.26b)

where we used the fact that k = k kˆ = k(x − x0 )/ |x − x‘|. If the source is located inside a volume V near x0 and has such a limited spatial extent that max |x0 − x0 | |x − x0 |, and the integration surface S , centred on x0 , has a large enough radius |x − x0 | max |x0 − x0 |, we see from Figure 7.1 on the facing page that we can approximate

k x − x0 ≡ k · (x − x0 ) ≡ k · (x − x0 ) − k · (x0 − x0 ) ≈ k |x − x0 | − k · (x0 − x0 )


(7.27)


7.3

103

T HE R ADIATION F IELDS

S dS = nd ˆ 2x kˆ

x − x0

x − x0

x

x0

x0 − x 0 x0 V

O F IGURE 7.1: Relation between the surface normal and the k vector for radiation generated at source points x0 near the point x0 in the source volume V. At distances much larger than the extent of V, the unit vector n, ˆ normal to the surface S which has its centre at x0 , and the unit vector kˆ of the radiation k vector from x0 are nearly coincident.

Recalling from Formula (F.48) and Formula (F.49) on page 164 that dS = |x − x0 |2 dΩ = |x − x0 |2 sin θ dθ dϕ and noting from Figure 7.1 that kˆ and nˆ are nearly parallel, we see that we can approximate. kˆ · dS kˆ · nˆ = dS ≈ dΩ 2 |x − x0 | |x − x0 |2

(7.28)

Both these approximations will be used in the following. Within approximation (7.27) the expressions (7.26a) and (7.26b) for the radiation fields can be approximated as µ0 ik|x−x0 | jω (x0 ) × k −ik·(x0 −x0 ) 3 0 e e dx 4π V |x − x0 | (7.29a) µ0 eik|x−x0 | 0 −ik·(x0 −x0 ) 3 0 dx ≈ −i [jω (x ) × k] e 4π |x − x0 | V [jω (x0 ) × k] × (x − x0 ) −ik·(x0 −x0 ) 3 0 1 ik|x−x0 | Erad (x) ≈ −i e dx e ω 4πε0 c |x − x0 |2 V 0 1 eik|x−x0 | (x − x0 ) × [jω (x0 ) × k] e−ik·(x −x0 ) d3x0 ≈i 4πε0 c |x − x0 | |x − x0 | V (7.29b) Brad ω (x) ≈ −i



104


FROM


I.e., if max |x0 − x0 | |x − x0 |, then the fields can be approximated as spherical waves multiplied by dimensional and angular factors, with integrals over points in the source volume only.

7.4 Radiated Energy Let us consider the energy that is carried in the radiation fields Brad , Equation (7.26a), and Erad , Equation (7.26b) on page 102. We have to treat signals with limited lifetime and hence finite frequency bandwidth differently from monochromatic signals.

7.4.1 Monochromatic signals If the source is strictly monochromatic, we can obtain the temporal average of the radiated power P directly, simply by averaging over one period so that 1 1 Re 1 E × B∗ 2 = Re 1 Eω e−iωt × (Bω e−iωt )∗ 2 2µ0 2µ0 (7.30) 1 1 ∗ −iωt iωt 2 ∗ 2 = = Re 1 Eω × Bω e e Re 1 Eω × Bω 2µ0 2µ0

hSi = hE × Hi =

Using the far-field approximations (7.29a) and (7.29b) and the fact that 1/c = √ √ ε0 µ0 and R0 = µ0 /ε0 according to the definition (2.18) on page 26, we obtain 1 1 R0 hSi = 32π2 |x − x0 |2

V

(jω × k)e

−ik·(x0 −x0 ) 3 0

dx

2

x − x0 |x − x0 |

(7.31)

or, making use of (7.28) on the previous page,

dP 1 = R0 2 dΩ 32π

V

(jω × k)e

−ik·(x0 −x0 ) 3 0

dx

2

(7.32)

which is the radiated power per unit solid angle.

7.4.2 Finite bandwidth signals A signal with finite pulse width in time (t) domain has a certain spread in frequency (ω) domain. To calculate the total radiated energy we need to integrate



7.4

105

R ADIATED E NERGY

over the whole bandwidth. The total energy transmitted through a unit area is the time integral of the Poynting vector:

∞ −∞

∞

S(t) dt = =

−∞ ∞ −∞

(E × H) dt dω

∞

−∞

dω0

∞ −∞

(7.33)

0

(Eω × Hω0 )e−i(ω+ω )t dt

If we carry out the temporal integration first and use the fact that ∞

−∞

0

e−i(ω+ω )t dt = 2πδ(ω + ω0 )

(7.34)

Equation (7.33) can be written [cf. Parseval’s identity]

∞ −∞

S(t) dt = 2π

∞ −∞

= 2π

0

0

∞

2π µ0 2π = µ0

∞ ∞

= 2π = 2π

(Eω × H−ω ) dω

0 ∞

=

0 ∞ 0

(Eω × H−ω ) dω −

0

(Eω × H−ω ) dω +

(Eω × H−ω ) dω +


−∞ −∞

0 ∞


(E−ω × Hω ) dω

0

(7.35)

(Eω × B−ω + E−ω × Bω ) dω (Eω × B∗ω + E∗ω × Bω ) dω

where the last step follows from the real-valuedness of Eω and Bω . We insert the Fourier transforms of the field components which dominate at large distances, i.e., the radiation fields (7.26a) and (7.26b). The result, after integration over the area S of a large sphere which encloses the source, is 1 U= 4π

µ0 ε0

S

∞ 0

V

jω × k ik|x−x0 | 3 0 2 ˆ d x k · ndS ˆ dω e |x − x0 |

(7.36)

Inserting the approximations (7.27) and (7.28) into Equation (7.36) above and also introducing U=

∞ 0

Uω dω

(7.37)

and recalling the definition (2.18) on page 26 for the vacuum resistance R 0 we obtain

1 dUω dω ≈ R0 dΩ 4π

V

(jω × k)e


−ik·(x0 −x0 ) 3 0

dx

2

dω

(7.38)


106


FROM


which, at large distances, is a good approximation to the energy that is radiated per unit solid angle dΩ in a frequency band dω. It is important to notice that Formula (7.38) includes only source coordinates. This means that the amount of energy that is being radiated is independent on the distance to the source (as long as it is large).

Bibliography [1] F. H OYLE , S IR AND J. V. NARLIKAR, Lectures on Cosmology and Action at a Distance Electrodynamics, World Scientific Publishing Co. Pte. Ltd, Singapore, New Jersey, London and Hong Kong, 1996, ISBN 9810-02-2573-3(pbk). [2] J. D. JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, NY . . . , 1999, ISBN 0-471-30932-X. [3] L. D. L ANDAU AND E. M. L IFSHITZ, The Classical Theory of Fields, fourth revised English ed., vol. 2 of Course of Theoretical Physics, Pergamon Press, Ltd., Oxford . . . , 1975, ISBN 0-08-025072-6. [4] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6. [5] J. A. S TRATTON, Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, NY and London, 1953, ISBN 07-062150-0. [6] J. A. W HEELER AND R. P. F EYNMAN, Interaction with the absorber as a mechanism for radiation, Reviews of Modern Physics, 17 (1945), pp. 157–.



8 Electromagnetic Radiation and Radiating Systems

In Chapter 3 we were able to derive general expressions for the scalar and vector potentials from which we then, in Chapter 7, calculated the total electric and magnetic fields from arbitrary distributions of charge and current sources. The only limitation in the calculation of the fields was that the advanced potentials were discarded. Thus, one can, at least in principle, calculate the radiated fields, Poynting flux and energy for an arbitrary current density Fourier component and then add these Fourier components together to construct the complete electromagnetic field at any time at any point in space. However, in practice, it is often difficult to evaluate the source integrals unless the current has a simple distribution in space. In the general case, one has to resort to approximations. We shall consider both these situations.

8.1 Radiation from Extended Sources Certain radiation systems have a geometry which is one-dimensional, symmetric or in any other way simple enough that a direct calculation of the radiated fields and energy is possible. This is for instance the case when the current flows in one direction in space only and is limited in extent. An example of this is a linear antenna.

107

108

E LECTROMAGNETIC R ADIATION

AND

R ADIATING S YSTEMS

− L2

L 2

F IGURE 8.1: A linear antenna used for transmission. The current in the feeder and the antenna wire is set up by the EMF of the generator (the transmitter). At the ends of the wire, the current is reflected back with a 180◦ phase shift to produce a antenna current in the form of a standing wave.

8.1.1 Radiation from a one-dimensional current distribution Let us apply Equation (7.32) on page 104 to calculate the power from a linear, transmitting antenna, fed across a small gap at its centre with a monochromatic source. The antenna is a straight, thin conductor of length L which carries a one-dimensional time-varying current so that it produces electromagnetic radiation. We assume that the conductor resistance and the energy loss due to the electromagnetic radiation are negligible. The charges in this thin wire are set in motion due to the EMF of the generator (transmitter) to produce an antenna current which is the source of the EM radiation. Since we can assume that the antenna wire is infinitely thin, the current must vanish at the end points −L/2 and L/2 and. Furthermore, for a monochromatic signal, the current is sinusoidal and is reflected at the ends of the antenna wire and undergoes there a phase shift of π radians. The combined effect of this is that the antenna current forms a standing wave as indicated in Figure 8.1 For a Fourier component ω0 the standing wave current density can be written as j(t0 , x0 ) = j0 (x0 ) exp{−iω0 t0 } [cf. Equations (7.7) on page 97] where

sin[k(L/2 − x30 )] j0 (x0 ) = I0 δ(x10 )δ(x20 ) xˆ 3 sin(kL/2)


(8.1)


8.1

R ADIATION

FROM

109

E XTENDED S OURCES

rˆ ϕˆ

x3 = z x L 2

θˆ θ jω

kˆ

(x0 )

x2

ϕ x1 − L2 F IGURE 8.2: We choose a spherical polar coordinate system (r = |x| , θ, ϕ) and orient it so that the linear antenna axis (and thus the antenna current density jω ) is along the polar axis with the feed point at the origin.

where the current amplitude I0 is a constant (measured in A). In order to evaluate Formula (7.32) on page 104 with the explicit monochromatic current (8.1) inserted, we use a spherical polar coordinate system as in Figure 8.2 to evaluate the source integral

V0

j0 × k)e

=

=

dx

I02

2 2

sin[k(L/2 − x30 )] 0 k sin θe−ikx3 cos θ eikx0 cos θ dx30 I0 sin(kL/2) −L/2 L/2

=

−ik·(x0 −x0 ) 3 0

k2 sin2 θ ikx0 cos θ 2 e 2 sin2 (kL/2)

4I02

L/2 0

sin[k(L/2 − x30 )] cos(kx30 cos θ) dx30

cos[(kL/2) cosθ] − cos(kL/2) sin θ sin(kL/2)

2

2

(8.2) Inserting this expression and dΩ = 2π sin θ dθ into Formula (7.32) on page 104



110


AND

R ADIATING S YSTEMS

and integrating over θ, we find that the total radiated power from the antenna is P(L) = R0 I02

π

1 4π

cos[(kL/2) cosθ] − cos(kL/2) sin θ sin(kL/2)

0

2

sin θ dθ

(8.3)

One can show that lim P(L) =

kL→0

π 12

2

Lλ

R0 I02

(8.4)

where λ is the vacuum wavelength. The quantity Rrad (L) =

π P(L) P(L) = 1 2 = R0 2 6 Ieff 2 I0

Lλ

2

≈ 197

Lλ

2

Ω

(8.5)

is called the radiation resistance. For the technologically important case of a half-wave antenna, i.e., for L = λ/2 or kL = π, Formula (8.3) above reduces to P(λ/2) = R0 I02

1 4π

π 0

cos2

π

3 2 cos θ

sin θ

4

dθ

(8.6)

The integral in (8.6) can always be evaluated numerically. But, it can in fact also be evaluated analytically as follows: π

0

cos2

π

3 2 cos θ

sin θ

4

cos2 π2 u4 3 du = −1 1 − u2 cos2 . π2 u/ = 1 + cos(πu) 2

1 1 1 + cos(πu) = du 2 −1 (1 + u)(1 − u) (8.7) 1 1 1 + cos(πu) 1 1 1 + cos(πu) = du + du 4 −1 (1 + u) 4 −1 (1 − u) v 1 1 1 + cos(πu) du = 7 1 + u → 8 = 2 −1 (1 + u) π 1 2π 1 − cos v 1 = dv = [γ + ln 2π − Ci(2π)] 2 0 v 2 ≈ 1.22

dθ = [cos θ → u] =

1

where in the last step the Euler-Mascheroni constant γ = 0.5772 . . . and the cosine integral Ci(x) were introduced. Inserting this into the expression Equation (8.6) we obtain the value Rrad (λ/2) ≈ 73 Ω.



8.1

R ADIATION

111

E XTENDED S OURCES

FROM

rˆ x3 = z = z 0

ϕˆ x θˆ θ kˆ x2

zˆ 0 jω (x0 )

ϕ

x1

x0

ϕˆ 0

ϕ0 ρˆ 0

F IGURE 8.3: For the loop antenna the spherical coordinate system (r, θ, ϕ) describes the field point x (the radiation field) and the cylindrical coordinate system (ρ0 , ϕ0 , z0 ) describes the source point x0 (the antenna current).

8.1.2 Radiation from a two-dimensional current distribution As an example of a two-dimensional current distribution we consider a circular loop antenna and calculate the radiated fields from such an antenna. We choose the Cartesian coordinate system x1 x2 x3 with its origin at the centre of the loop as in Figure 8.3 According to Equation (7.29a) on page 103 in the formula collection the Fourier component of the radiation part of the magnetic field generated by an extended, monochromatic current source is Brad ω =

−iµ0 eik|x| 4π |x|

0

V0

e−ik·x jω × k d3x0

(8.8)

In our case the generator produces a single frequncy ω and we feed the antenna across a small gap where the loop crosses the positive x1 axis. The circumference of the loop is chosen to be exactly one wavelength λ = 2πc/ω. This means that the antenna current oscillates in the form of a sinusoidal standing current



112


AND

R ADIATING S YSTEMS

wave around the circular loop with a Fourier amplitude jω = I0 cos ϕ0 δ(ρ0 − a)δ(z0 )ϕˆ 0

(8.9)

For the spherical coordinate system of the field point, we recall from subsection F.4.1 on page 163 that the following relations between the base vectors hold: rˆ = sin θ cos ϕ xˆ 1 + sin θ sin ϕ xˆ 2 + cos θ xˆ 3 θˆ = cos θ cos ϕ xˆ 1 + cos θ sin ϕ xˆ 2 − sin θ xˆ 3

ϕˆ = − sin ϕ xˆ 1 + cos ϕ xˆ 2 and

xˆ 1 = sin θ cos ϕˆr + cos θ cos ϕθˆ − sin ϕϕˆ xˆ 2 = sin θ sin ϕˆr + cos θ sinϕθˆ + cos ϕϕˆ xˆ 3 = cos θˆr − sin θθˆ With the use of the above transformations and trigonometric identities, we obtain for the cylindrical coordinate system which describes the source: ρˆ 0 = cos ϕ0 xˆ 1 + sin ϕ0 xˆ 2 = sin θ cos(ϕ0 − ϕ)ˆr + cos θ cos(ϕ0 − ϕ)θˆ + sin(ϕ0 − ϕ)ϕˆ

(8.10)

= − sin θ sin(ϕ0 − ϕ)ˆr − cos θ sin(ϕ0 − ϕ)θˆ + cos(ϕ0 − ϕ)ϕˆ zˆ 0 = xˆ 3 = cos θˆr − sin θθˆ

(8.11)

ϕˆ 0 = − sin ϕ0 xˆ 1 + cos ϕ0 xˆ 2

(8.12)

This choice of coordinate systems means that k = kˆr and x0 = aρˆ 0 so that k · x0 = ka sin θ cos(ϕ0 − ϕ)

(8.13)

ϕˆ 0 × k = k[cos(ϕ0 − ϕ)θˆ + cos θ sin(ϕ0 − ϕ)ϕ] ˆ

(8.14)

and

With these expressions inserted and d3x0 = ρ0 dρ0 dϕ0 dz0 , the source integral becomes

0

V0

e−ik·x jω × k d3x0 = a 2π

= I0 ak

0

0

0

dϕ0 e−ika sin θ cos(ϕ −ϕ) I0 cos ϕ0 ϕˆ × k

0

e−ika sin θ cos(ϕ −ϕ) cos(ϕ0 − ϕ) cos ϕ0 dϕ0 θˆ

+ I0 ak cos θ


2π

2π 0

(8.15)

0

e−ika sin θ cos(ϕ −ϕ) sin(ϕ0 − ϕ) cos ϕ0 dϕ0 ϕˆ


8.1

R ADIATION

FROM

113

E XTENDED S OURCES

Utilising the periodicity of the integrands over the integration interval [0, 2π], introducing the auxiliary integration variable ϕ00 = ϕ0 −ϕ, and utilising standard trigonometric identities, the first integral in the RHS of (8.15) can be rewritten 2π

00

e−ika sin θ cos ϕ cos ϕ00 cos(ϕ00 + ϕ) dϕ00

0 2π

= cos ϕ = cos ϕ =

00

e−ika sin θ cos ϕ cos2 ϕ00 dϕ00 + a vanishing integral

0 2π

e−ika sin θ cos ϕ

0

00

1 1 + cos 2ϕ00 dϕ00 2 2

(8.16)

2π 1 00 e−ika sin θ cos ϕ dϕ00 cos ϕ 2 0 2π 00 1 e−ika sin θ cos ϕ cos 2ϕ00 dϕ00 + cos ϕ 2 0

Analogously, the second integral in the RHS of (8.15) can be rewritten 2π

00

e−ika sin θ cos ϕ sin ϕ00 cos(ϕ00 + ϕ) dϕ00

0

=

2π 00 1 e−ika sin θ cos ϕ dϕ00 sin ϕ 2 0 2π 00 1 − sin ϕ e−ika sin θ cos ϕ cos 2ϕ00 dϕ00 2 0

(8.17)

As is well-known from the theory of Bessel functions, Jn (−ξ) = (−1)n Jn (ξ) Jn (−ξ) =

i−n π

π

e−iξ cos ϕ cos nϕ dϕ =

0

i−n 2π

2π

e−iξ cos ϕ cos nϕ dϕ

(8.18)

0

which means that 2π

0

00

e−ika sin θ cos ϕ dϕ00 = 2πJ0 (ka sin θ) (8.19)

2π

e

−ika sin θ cos ϕ00

0

00

00

cos 2ϕ dϕ = −2πJ2 (ka sin θ)

Putting everything together, we find that

0

V0

e−ik·x jω × k d3x0 = Iθ θˆ + Iϕ ϕˆ

= I0 akπ cos ϕ [J0 (ka sin θ) − J2 (ka sin θ)] θˆ

(8.20)

+ I0 akπ cos θ sin ϕ [J0 (ka sin θ) + J2 (ka sin θ)] ϕˆ



114


AND

R ADIATING S YSTEMS

so that, in spherical coordinates where |x| = r,

−iµ0 eikr Iθ θˆ + Iϕ ϕˆ 4 (8.21) 4πr 3 To obtain the desired physical magnetic field in the radiation (far) zone we must Fourier transform back to t space and take the real part and evaluate it at the retarded time: Brad ω (x) =

Brad (t, x) = Re

−iµ0 e(ikr−ωt ) 4 ˆ 3 Iθ θ + Iϕ ϕˆ 4πr 0

µ0 sin(kr − ωt0 ) Iθ θˆ + Iϕ ϕˆ 4 = 4πr 3 I0 akµ0 = sin(kr − ωt0 ) cos ϕ [J0 (ka sin θ) − J2 (ka sinθ)] θˆ 4r + cos θ sin ϕ [J0 (ka sin θ) + J2 (ka sin θ)] ϕˆ

(8.22) From this expression for the radiated B field, we can obtain the radiated E field with the help of Maxwell’s equations.

8.2 Multipole Radiation In the general case, and when we are interested in evaluating the radiation far from the source volume, we can introduce an approximation which leads to a multipole expansion where individual terms can be evaluated analytically. We shall use Hertz’ method to obtain this expansion.

8.2.1 The Hertz potential Let us consider the equation of continuity, which, according to expression (1.21) on page 9, can be written ∂ρ(t, x) + ∇ · j(t, x) = 0 ∂t

(8.23)

In Section 6.1.1 we introduced the electric polarisation P such that ∇ · P = −ρ pol , the polarisation charge density. If we introduce a vector field π(t, x) such that ∇ · π = −ρtrue ∂π = jtrue ∂t


(8.24a) (8.24b)


8.2

115

M ULTIPOLE R ADIATION

and compare with Equation (8.23) on the preceding page, we see that π(t, x) satisfies this equation of continuity. Furthermore, if we compare with the electric polarisation [cf. Equation (6.9) on page 87], we see that the quantity π is related to the “true” charges in the same way as P is related to polarised charge. The quantity π is referred to as the polarisation vector since, formally, it treats also the “true” (free) charges as polarisation charges. We introduce a further potential Πe with the following property ∇ · Πe = −φ 1 ∂Πe =A c2 ∂t

(8.25a) (8.25b)

where φ and A are the electromagnetic scalar and vector potentials, respectively. As we see, Πe acts as a “super-potential” in the sense that it is a potential from which we can obtain other potentials. It is called the Hertz’ vector or polarisation potential and, as can be seen from (8.24) and (8.25), it satisfies the inhomogeneous wave equation

5

2

Πe =

1 ∂2 e π Π − ∇2 Πe = 2 2 c ∂t ε0

(8.26)

This equation is of the same type as Equation (3.19) on page 41, and has therefore the retarded solution Πe (t, x) =

1 4πε0

0 , x0 ) π(tret d3x0 0| 0 − x |x V

(8.27)

with Fourier components Πeω (x) =

1 4πε0

0

πω (x0 )eik|x−x | 3 0 dx |x − x0 | V0

(8.28)

If we introduce the help vector C such that C = ∇ × Πe

(8.29)

we see that we can calculate the magnetic and electric fields, respectively, as follows 1 ∂C c2 ∂t

(8.30a)

E = ∇×C

(8.30b)

B=



116


x − x0

AND

x0

x − x0 x

R ADIATING S YSTEMS

Θ x0 V0

O F IGURE 8.4: Geometry of a typical multipole radiation problem where the field point x is located some distance away from the finite source volume V 0 centered around x0 . If k |x0 − x0 | 1 k |x − x0 |, then the radiation at x is well approximated by a few terms in the multipole expansion.

Clearly, the last equation is valid only outside the source volume, where ∇ · E = 0. Since we are mainly interested in the fields in the far zone, a long distance from the source region, this is no essential limitation. Assume that the source region is a limited volume around some central point x0 far away from the field (observation) point x illustrated in Figure 8.4. Under these assumptions, we can expand the Hertz’ vector, expression (8.27) 0 , x0 ) in the on the previous page, due to the presence of non-vanishing π(tret vicinity of x0 , in a formal series. For this purpose we recall from potential theory that 0

0

eik|(x−x0 )−(x −x0 )| eik|x−x | ≡ |x − x0 | |(x − x0 ) − (x0 − x0 )| ∞

= ik ∑ (2n + 1)Pn (cos Θ) jn (k x n=0

0

(8.31)

− x0 )h(1) n (k |x − x0 |)

where 0

eik|x−x | is a Green function |x − x0 | Θ is the angle between x0 − x0 and x − x0 (see Figure 8.4 on page 116)

Pn (cos Θ) is the Legendre polynomial of order n

jn (k x0 − x0 ) is the spherical Bessel function of the first kind of order n

h(1) n (k |x − x0 |) is the spherical Hankel function of the first kind of order n



8.2

117


According to the addition theorem for Legendre polynomials, we can write n

0 im(ϕ−ϕ ) ∑ (−1)m Pmn (cos θ)P−m n (cos θ )e

Pn (cos Θ) =

0

(8.32)

m=−n

where Pm n is an associated Legendre polynomial and, in spherical polar coordinates,

x0 − x0 = ( x0 − x0 , θ 0 , φ0 )

(8.33a)

x − x0 = (|x − x0 | , θ, φ)

(8.33b)

Inserting Equation (8.31) on the facing page, together with Equation (8.32) above, into Equation (8.28) on page 115, we can in a formally exact way expand the Fourier component of the Hertz’ vector as ik ∞ n m imϕ ∑ ∑ (2n + 1)(−1)mh(1) n (k |x − x0 |) Pn (cos θ) e 4πε0 n=0 m=−n

Πeω = ×

V0

0

πω (x ) jn (k x

0

(8.34)

0 −imϕ0 3 0 − x0 ) P−m dx n (cos θ ) e

We notice that there is no dependence on x − x0 inside the integral; the integrand is only dependent on the relative source vector x0 − x0 . We are interested in the case where the field point is many wavelengths away from the well-localised sources, i.e., when the following inequalities

k x0 − x0 1 k |x − x0 |

(8.35)

hold. Then we may to a good approximation replace h(1) n with the first term in its asymptotic expansion: n+1 h(1) n (k |x − x0 |) ≈ (−i)

eik|x−x0 | k |x − x0 |

(8.36)

and replace jn with the first term in its power series expansion:

jn (k x0 − x0 ) ≈

2n n! k x0 − x0 4 (2n + 1)! 3

n

(8.37)

Inserting these expansions into Equation (8.34) above, we obtain the multipole expansion of the Fourier component of the Hertz’ vector Πeω ≈

∞

∑ Πeω(n)

(8.38a)

n=0



118


where Πeω(n) = (−i)n

1 eik|x−x0 | 2n n! 4πε0 |x − x0 | (2n)!

AND

R ADIATING S YSTEMS

V0

Vπω (x0 ) (k x0 − x0 )n Pn (cos Θ) d3x0 (8.38b)

This expression is approximately correct only if certain care is exercised; if many Πeω(n) terms are needed for an accurate result, the expansions of the spherical Hankel and Bessel functions used above may not be consistent and must be replaced by more accurate expressions. Taking the inverse Fourier transform of Πeω will yield the Hertz’ vector in time domain, which inserted into Equation (8.29) on page 115 will yield C. The resulting expression can then in turn be inserted into Equation (8.30) on page 115 in order to obtain the radiation fields. For a linear source distribution along the polar axis, Θ = θ in expression (8.38b), and Pn (cos θ) gives the angular distribution of the radiation. In the general case, however, the angular distribution must be computed with the help of Formula (8.32) on the previous page. Let us now study the lowest order contributions to the expansion of Hertz’ vector.

8.2.2 Electric dipole radiation Choosing n = 0 in expression (8.38b), we obtain Πeω(0) =

eik|x−x0 | 4πε0 |x − x0 |

V0

πω (x0 ) d3x0 =

1 eik|x−x0 | pω 4πε0 |x − x0 |

(8.39)

where pω = V 0 πω (x0 ) d3x0 is the Fourier component of the electric dipole moment p(t, x0 ) = V 0 (x0 − x0 )ρ(t, x0 ) d3x0 [cf. Equation (6.2) on page 85 which describes the static dipole moment]. If a spherical coordinate system is chosen with its polar axis along pω as in Figure 8.5 on the facing page, the components of Πeω(0) are 1 eik|x−x0 | pω cos θ 4πε0 |x − x0 | 1 eik|x−x0 | def Πeθ ≡ Πeω(0) · θˆ = − pω sin θ 4πε0 |x − x0 | def

Πer ≡ Πeω(0) · rˆ =

(8.40a) (8.40b)

def

Πeϕ ≡ Πeω(0) · ϕˆ = 0

(8.40c)

Evaluating Formula (8.29) on page 115 for the help vector C, with the spherically polar components (8.40) of Πeω(0) inserted, we obtain (0) Cω = Cω,ϕ ϕˆ =


1 4πε0

1 eik|x−x0 | − ik |x − x0 | pω sin θ ϕˆ |x − x0 |

(8.41)


8.2

119


kˆ x3

Brad

x

Erad

θ rˆ

p

x2

ϕ x1 F IGURE 8.5: If a spherical polar coordinate system (r, θ, ϕ) is chosesn such that the electric dipole moment p (and thus its Fourier transform pω ) is located at the origin and directed along the polar axis, the calculations are simplified.

Applying this to Equation (8.30) on page 115, we obtain directly the Fourier components of the fields ωµ0 1 eik|x−x0 | − ik 4π |x − x0 | |x − x0 | pω sin θ ϕˆ 1 ik x − x0 1 − 2 cos θ Eω = 2 4πε0 |x − x0 | |x − x0 | |x − x0 | ik|x−x0 | 1 ik 2 ˆ e − k sin θ θ + −

|x − x0 | pω |x − x0 |2 |x − x0 |

Bω = −i

(8.42a)

(8.42b)

Keeping only those parts of the fields which dominate at large distances (the radiation fields) and recalling that the wave vector k = k(x − x0 )/ |x − x0 | where k = ω/c, we can now write down the Fourier components of the radiation parts of the magnetic and electric fields from the dipole: ωµ0 eik|x−x0 | ωµ0 eik|x−x0 | pw k sin θ ϕˆ = − (pω × k) (8.43a) 4π |x − x0 | 4π |x − x0 | 1 eik|x−x0 | 1 eik|x−x0 | pω k2 sin θ θˆ = − [(pω × k) × k] Erad ω =− 4πε0 |x − x0 | 4πε0 |x − x0 | (8.43b) Brad ω =−



120


AND

R ADIATING S YSTEMS

These fields constitute the electric dipole radiation, also known as E1 radiation.

8.2.3 Magnetic dipole radiation The next term in the expression (8.38b) on page 118 for the expansion of the Fourier transform of the Hertz’ vector is for n = 1:

eik|x−x0 | k x0 − x0 πω (x0 ) cos Θ d3x0 4πε0 |x − x0 | V 0 1 eik|x−x0 | = −ik [(x − x0 ) · (x0 − x0 )] πω (x0 ) d3x0 4πε0 |x − x0 |2 V 0

Πeω(1) = −i

(8.44)

Here, the term [(x − x0 ) · (x0 − x0 )] πω (x0 ) can be rewritten

[(x − x0 ) · (x0 − x0 )] πω (x0 ) = (xi − x0,i )(xi0 − x0,i ) πω (x0 )

(8.45)

and introducing ηi = xi − x0,i

η0i

=

(8.46a)

xi0 − x0,i

(8.46b)

the jth component of the integrand in Πeω (1) can be broken up into 1 {[(x − x0 ) · (x0 − x0 )] πω (x0 )} j = ηi πω, j η0i + πω,i η0j 4 2 3 1 + ηi πω, j η0i − πω,i η0j 4 2 3

(8.47)

i.e., as the sum of two parts, the first being symmetric and the second antisymmetric in the indices i, j. We note that the antisymmetric part can be written as 1 1 ηi πω, j η0i − πω,i η0j 4 = [πω, j (ηi η0i ) − η0j (ηi πω,i )] 2 3 2 1 = [πω (η · η0 ) − η0 (η · πω )] j 2 1 = 1 (x − x0 ) × [πω × (x0 − x0 )] 2 2

(8.48) j

The utilisation of Equations (8.24) on page 114, and the fact that we are considering a single Fourier component, π(t, x) = πω e−iωt


(8.49)


8.2

121


allow us to express πω in jω as πω = i

jω ω

(8.50)

Hence, we can write the antisymmetric part of the integral in Formula (8.44) on the preceding page as 1 (x − x0 ) × πω (x0 ) × (x0 − x0 ) d3x0 2 V0 1 = i (x − x0 ) × jω (x0 ) × (x0 − x0 ) d3x0 2ω V0 1 = −i (x − x0 ) × mω ω

(8.51)

where we introduced the Fourier transform of the magnetic dipole moment mω =

1 2

V0

(x0 − x0 ) × jω (x0 ) d3x0

(8.52)

The final result is that the antisymmetric, magnetic dipole, part of Π eω(1) can be written Πe,antisym ω

(1)

=−

k eik|x−x0 | (x − x0 ) × mω 4πε0 ω |x − x0 |2

(8.53)

In analogy with the electric dipole case, we insert this expression into Equation (8.29) on page 115 to evaluate C, with which Equations (8.30) on page 115 then gives the B and E fields. Discarding, as before, all terms belonging to the near fields and transition fields and keeping only the terms that dominate at large distances, we obtain µ0 eik|x−x0 | (mω × k) × k 4π |x − x0 | k eik|x−x0 | Erad (x) = mω × k ω 4πε0 c |x − x0 |

Brad ω (x) = −

(8.54a) (8.54b)

which are the fields of the magnetic dipole radiation (M1 radiation).

8.2.4 Electric quadrupole radiation e,sym (1)

The symmetric part Πω of the n = 1 contribution in the Equation (8.38b) on page 118 for the expansion of the Hertz’ vector can be expressed in terms



122


AND

R ADIATING S YSTEMS

of the electric quadrupole tensor, which is defined in accordance with Equation (6.3) on page 86: Q(t, x0 ) =

V0

(x0 − x0 )(x0 − x0 )ρ(t, x0 ) d3x0

(8.55)

Again we use this expression in Equation (8.29) on page 115 to calculate the fields via Equations (8.30) on page 115. Tedious, but fairly straightforward algebra (which we will not present here), yields the resulting fields. The radiation components of the fields in the far field zone (wave zone) are given by iµ0 ω eik|x−x0 | k · Qω 4 × k 8π |x − x0 | 3 i eik|x−x0 | Erad k · Qω 4 × k × k ω (x) = 8πε0 |x − x0 | 3 Brad ω (x) =

(8.56a) (8.56b)

This type of radiation is called electric quadrupole radiation or E2 radiation.

8.3 Radiation from a Localised Charge in Arbitrary Motion The derivation of the radiation fields for the case of the source moving relative to the observer is considerably more complicated than the stationary cases studied above. In order to handle this non-stationary situation, we use the retarded potentials (3.36) on page 44 in Chapter 3 0 , x0 ) ρ(tret 1 d3x0 4πε0 V 0 |x − x0 | 0 , x0 ) j(tret µ0 d3x0 A(t, x) = 4π V 0 |x − x0 |

φ(t, x) =

(8.57a) (8.57b)

and consider a source region with such a limited spatial extent that the charges and currents are well localised. Specifically, we consider a charge q0 , for instance an electron, which, classically, can be thought of as a localised, unstructured and rigid “charge distribution” with a small, finite radius. The part of this “charge distribution” dq0 which we are considering is located in dV 0 = d3x0 in the sphere in Figure 8.6 on the next page. Since we assume that the electron (or any other other similar electric charge) is moving with a velocity v whose direction is arbitrary and whose magnitude can be almost comparable to the speed of light, we cannot say that the charge and current to be used in (8.57) is



8.3

R ADIATION

FROM A

L OCALISED C HARGE

x(t)

IN

123

A RBITRARY M OTION

c`b c`b c`b c`b c`b 0 c`b c`b c`b cb cb`xc`b −c`b x0c`b v(tc`b )c`b a`_ c`b a`_ c`b a`_dScb a_ 0 c`b c`b c`b c`b c`b c`b a`_ c`b a`_ c`b a`_ cb a_ x0 (t0 )

dr0

q0

dV 0

c

F IGURE 8.6: Signals which are observed at the field point x at time t were generated at source points x0 (t0 ) on a sphere, centred on x and expanding, as time increases, with the velocity c outward from the centre. The source charge element moves with an arbitrary velocity v and gives rise to a source “leakage” out of the source volume dV 0 = d3x0 .

0 , x0 ) d3x0 and 0 , x0 ) d3x0 , respectively, because in the finite time V ρ(tret V vρ(tret

interval during which the observed signal is generated, part of the charge distribution will “leak” out of the volume element d3x0 .

8.3.1 The Liénard-Wiechert potentials The charge distribution in Figure 8.6 on page 123 which contributes to the field at x(t) is located at x0 (t0 ) on a sphere with radius r = |x − x0 | = c(t − t0 ). The radius interval of this sphere from which radiation is received at the field point x during the time interval (t 0 , t0 + dt0 ) is (r0 , r0 + dr0 ) and the net amount of charge in this radial interval is 0 0 dq0 = ρ(tret , x0 ) dS 0 dr0 − ρ(tret , x0 )

(x − x0 ) · v 0 0 dS dt |x − x0 |

(8.58)

where the last term represents the amount of “source leakage” due to the fact that the charge distribution moves with velocity v(t 0 ). Since dt0 = dr0 /c and



124


AND

R ADIATING S YSTEMS

dS 0 dr0 = d3x0 we can rewrite this expression for the net charge as (x − x0 ) · v 3 0 dx c |x − x0 | (x − x0 ) · v 0 = ρ(tret , x0 ) 1 − d3x0 c |x − x0 |

0 0 dq0 = ρ(tret , x0 ) d3x0 − ρ(tret , x0 )

(8.59)

or 0 ρ(tret , x0 ) d3x0 =

dq0

(8.60)

0

)·v 1 − (x−x c|x−x0 |

which leads to the expression 0 , x0 ) dq0 ρ(tret 3 0 d x = 0 |x − x0 | |x − x0 | − (x−xc )·v

(8.61)

This is the expression to be used in the Formulae (8.57) on page 122 for the retarded potentials. The result is (recall that j = ρv) φ(t, x) =

1 4πε0

A(t, x) =

µ0 4π

dq0 |x − x0 | − (x−xc0 )·v v dq0 0 |x − x0 | − (x−xc )·v

(8.62a) (8.62b)

For a sufficiently small and well localised charge distribution we can, assuming that the integrands do not change sign in the integration volume, use the mean value theorem and the fact that V dq0 = q0 to evaluate these expressions to become 1 q0 1 q0 = 0 4πε0 |x − x0 | − (x−xc )·v 4πε0 s v q0 v v q0 A(t, x) = = = φ(t, x) 0 4πε0 c2 |x − x0 | − (x−xc )·v 4πε0 c2 s c2 φ(t, x) =

where

s = s(t0 , x) = x − x0 (t0 ) −

(x − x0 (t0 )) · v(t0 ) c 0 0 x − x0 (t0 ) v(t0 ) = x − x (t ) 1 − · |x − x0 (t0 )| c x − x0 (t0 ) v(t0 ) − = (x − x0 (t0 )) · c |x − x0 (t0 )|


(8.63a) (8.63b)

(8.64a) (8.64b) (8.64c)


8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

125

A RBITRARY M OTION

is the retarded relative distance. The potentials (8.63) are precisely the LiénardWiechert potentials which we derived in Section 4.3.2 on page 60 by using a covariant formalism. It is important to realise that in the complicated derivation presented here, the observer is in a coordinate system which has an “absolute” meaning and the velocity v is that of the particle, whereas in the covariant derivation two frames of equal standing were moving relative to each other with v. Expressed in the four-potential, Equation (4.48) on page 59, the Liénard-Wiechert potentials become Aµ (xκ ) =

q0 4πε0

1 v = (φ, cA) , s cs

(8.65)

The Liénard-Wiechert potentials are applicable to all problems where a spatially localised charge emits electromagnetic radiation, and we shall now study such emission problems. The electric and magnetic fields are calculated from the potentials in the usual way: B(t, x) = ∇ × A(t, x)

(8.66a)

∂A(t, x) E(t, x) = −∇φ(t, x) − ∂t

(8.66b)

8.3.2 Radiation from an accelerated point charge Consider a localised charge q0 and assume that its trajectory is known experimentally as a function of retarded time x0 = x0 (t0 )

(8.67)

(in the interest of simplifying our notation, we drop the subscript “ret” on t 0 from now on). This means that we know the trajectory of the charge q0 , i.e., x0 , for all times up to the time t0 at which a signal was emitted in order to precisely arrive at the field point x at time t. Because of the finite speed of propagation of the fields, the trajectory at times later than t 0 is not (yet) known. The retarded velocity and acceleration at time t 0 are given by v(t0 ) =

dx0 dt0

a(t0 ) = v˙ (t0 ) =

(8.68a) dv d2 x0 = dt0 dt0 2

(8.68b)

As for the charge coordinate x0 itself, we have in general no knowledge of the velocity and acceleration at times later than t 0 , in particular not at the time of



126


AND

R ADIATING S YSTEMS

? |x − x0 | v c

q0 x0 (t0 )

v(t0 )

x0 (t)

θ0

θ0

x − x0

x − x0 x(t)

F IGURE 8.7: Signals which are observed at the field point x at time t were generated at the source point x0 (t0 ). After time t0 the particle, which moves with nonuniform velocity, has followed a yet unknown trajectory. Extrapolating tangentially the trajectory from x0 (t0 ), based on the velocity v(t0 ), defines the virtual simultaneous coordinate x0 (t).

observation t. If we choose the field point x as fixed, application of (8.68) to the relative vector x − x0 yields d (x − x0 (t0 )) = −v(t0 ) dt0

(8.69a)

d2 (x − x0 (t0 )) = −˙v(t0 ) dt0 2

(8.69b)

The retarded time t0 can, at least in principle, be calculated from the implicit relation t0 = t0 (t, x) = t −

|x − x0 (t0 )| c

(8.70)

and we shall see later how this relation can be taken into account in the calculations. According to Formulae (8.66) on the previous page the electric and magnetic fields are determined via differentiation of the retarded potentials at the observation time t and at the observation point x. In these formulae the unprimed ∇, i.e., the spatial derivative differentiation operator ∇ = xˆ i ∂/∂xi means that we differentiate with respect to the coordinates x = (x1 , x2 , x3 ) while keeping t fixed, and the unprimed time derivative operator ∂/∂t means that we differentiate with respect to t while keeping x fixed. But the Liénard-Wiechert



8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

127

A RBITRARY M OTION

potentials φ and A, Equations (8.63) on page 124, are expressed in the charge velocity v(t0 ) given by Equation (8.68a) on page 125 and the retarded relative distance s(t0 , x) given by Equation (8.64) on page 124. This means that the expressions for the potentials φ and A contain terms which are expressed explicitly in t0 , which in turn is expressed implicitly in t via Equation (8.70) on the preceding page. Despite this complication it is possible, as we shall see below, to determine the electric and magnetic fields and associated quantities at the time of observation t. To this end, we need to investigate carefully the action of differentiation on the potentials.

The differential operator method We introduce the convention that a differential operator embraced by parentheses with an index x or t means that the operator in question is applied at constant x and t, respectively. With this convention, we find that

∂ ∂t0

x − x0 (t0 ) =

x

∂ x − x0 · |x − x0 | ∂t0

x

3

x − x0 (t0 )4 = −

(x − x0 ) · v(t0 ) (8.71) |x − x0 |

Furthermore, by applying the operator (∂/∂t)x to Equation (8.70) on the facing page we find that

∂t0 ∂t

∂ |x − x0 (t0 (t, x))| ∂t x c x ∂ ∂t0 |x − x0 | = 1− 0 ∂t x c

∂t 0 0 0 (x − x ) · v(t ) ∂t = 1+ ∂t c |x − x0 | x = 1−

(8.72) x

This is an algebraic equation in (∂t 0 /∂t)x which we can solve to obtain ∂t0 ∂t

|x − x0 | |x − x0 | = s |x − x0 | − (x − x0 ) · v(t0 )/c

= x

(8.73)

where s = s(t0 , x) is the retarded relative distance given by Equation (8.64) on page 124. Making use of Equation (8.73), we obtain the following useful operator identity

∂ ∂t

= x

∂t0 ∂t

x

∂ ∂t0


= x

|x − x0 | s

∂ ∂t0

(8.74) x


128


AND

R ADIATING S YSTEMS

Likewise, by applying (∇)t to Equation (8.70) on page 126 we obtain x − x0 |x − x0 (t0 (t, x))| =− · (∇)t (x − x0 ) c c |x − x0 | x − x0 (x − x0 ) · v(t0 ) =− + (∇)t t0 c |x − x0 | c |x − x0 |

(∇)t t0 = −(∇)t

(8.75)

This is an algebraic equation in (∇)t with the solution (∇)t t0 = −

x − x0 cs

(8.76)

which gives the following operator relation when (∇)t is acting on an arbitrary function of t0 and x: (∇)t = (∇)t t0

∂ ∂t0

x

+ (∇)t0 = −

x − x0 cs

∂ ∂t0

+ (∇)t0

(8.77)

x

With the help of the rules (8.77) and (8.74) we are now able to replace t by t 0 in the operations which we need to perform. We find, for instance, that 1 q0 4πε0 s 0 q x − x0 v(t0 ) x − x0 ∂s ∂t0

=− − − 4πε0 s2 |x − x0 | c cs x 0 0 ∂ µ0 q v(t ) ∂A ∂A ≡ = ∂t ∂t x ∂t 4π s 0 0 0 0 ∂s q0 = x − x s˙v(t ) − x − x v(t ) 0 2 3 4πε0 c s ∂t ∇φ ≡ (∇φ)t = ∇

(8.78a)

(8.78b) x

Utilising these relations in the calculation of the E field from the LiénardWiechert potentials, Equations (8.63) on page 124, we obtain ∂ A(t, x) ∂t q0 (x − x0 (t0 )) − |x − x0 (t0 )| v(t0 )/c = 4πε0 s2 (t0 , x) |x − x0 (t0 )|

E(t, x) = −∇φ(t, x) −

−


(x − x0 (t0 )) − |x − x0 | v(t0 )/c cs(t0 , x)

∂s(t0 , x) ∂t0

x

−

|x − x0 (t0 )| v˙ (t0 ) c2

(8.79)


8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

129

A RBITRARY M OTION

Starting from expression (8.64a) on page 124 for the retarded relative distance s(t0 , x), we see that we can evaluate (∂s/∂t 0 )x in the following way

∂s ∂t0

x − x0 − (x − x c) · v(t ) x x 0 1 ∂ x − x0 (t0 )4 0 ∂ 3 0 · v(t ) + (x − x0(t0 )) · ∂v(t0 ) = 0 x − x0 (t0 ) − ∂t c ] ∂t ∂t ^ =

=−

0

∂ ∂t0

0

(x − x0 ) · v(t0 ) v2 (t0 ) (x − x0 ) · v˙ (t0 ) + − c c |x − x0 |

(8.80)

where Equation (8.71) on page 127 and Equations (8.68) on page 125, respectively, were used. Hence, the electric field generated by an arbitrarily moving charged particle at x0 (t0 ) is given by the expression E(t, x) =

q0 |x − x0 (t0 )| v(t0 ) 0 0 (x − x (t )) − d 4πε0 s3 (t0 , x) L MEcN

1−

v2 (t0 ) c2 O

Coulomb field when v → 0

x − x0 (t0 ) + × L c2

(x − x0 (t0 )) −

M[N

|x − x0 (t0 )| v(t0 ) 0 c × v˙ (t ) O

Radiation field

(8.81) The first part of the field, the velocity field, tends to the ordinary Coulomb field when v → 0 and does not contribute to the radiation. The second part of the field, the acceleration field, is radiated into the far zone and is therefore also called the radiation field. From Figure 8.7 on page 126 we see that the position the charged particle would have had if at t0 all external forces would have been switched off so that the trajectory from then on would have been a straight line in the direction of the tangent at x0 (t0 ) is x0 (t), the virtual simultaneous coordinate. During the arbitrary motion, we interpret x − x0 as the coordinate of the field point x relative to the virtual simultaneous coordinate x0 (t). Since the time it takes from a signal to propagate (in the assumed vacuum) from x0 (t0 ) to x is |x − x0 | /c, this relative vector is given by x − x0 (t) = x − x0 (t0 ) −

|x − x0 (t0 )| v(t0 ) c


(8.82)


130


AND

R ADIATING S YSTEMS

This allows us to rewrite Equation (8.81) on the previous page in the following way E(t, x) =

v2 (x − x0 ) × v˙ q0 (x − x ) 1 − + (x − x0 ) × 0 3 2 4πε0 s c c2

(8.83)

In a similar manner we can compute the magnetic field: x − x0 ∂ × 0 cs ∂t 0 0 0 ∂A x−x q x−x ×v− =− × 0 2 2 0 4πε0 c s |x − x | c |x − x | ∂t x

B(t, x) = ∇ × A(t, x) ≡ (∇)t × A = (∇)t0 × A −

A x

(8.84)

where we made use of Equation (8.63) on page 124 and Formula (8.74) on page 127. But, according to (8.78a), q0 x − x0 x − x0 × (∇) φ = ×v t c |x − x0 | 4πε0 c2 s2 |x − x0 |

(8.85)

so that x − x0 ∂A × −(∇φ)t − c |x − x0 | ∂t 0 x−x = × E(t, x) c |x − x0 |

B(t, x) =

x

(8.86)

The radiation part of the electric field is obtained from the acceleration field in Formula (8.81) on the preceding page as Erad (t, x) = =

lim E(t, x)

|x−x0 |→∞ q0

(x − x0 ) × 3

(x − x0 ) −

4πε0 c2 s q0 = (x − x0 ) × [(x − x0 ) × v˙ ] 4πε0 c2 s3

|x − x0 | v c × v˙

(8.87)

where in the last step we again used Formula (8.82) on the previous page. Using this formula and Formula (8.86), the radiation part of the magnetic field can be written Brad (t, x) =


x − x0 × Erad (t, x) c |x − x0 |

(8.88)


8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

131

A RBITRARY M OTION

The direct method An alternative to the differential operator transformation technique just described is to try to express all quantities in the potentials directly in t and x. An example of such a quantity is the retarded relative distance s(t0 , x). According to Equation (8.64) on page 124, the square of this retarded relative distance can be written

2

s2 (t0 , x) = x − x0 − 2 x − x0

(x − x0 ) · v (x − x0 ) · v + c c

2

(8.89)

If we use the following handy identity

2

(x − x0 ) · v c

(x − x0 ) × v + c

2

02 2 |x − x0 |2 v2 2 0 |x − x | v cos θ + sin2 θ0 c2 c2 |x − x0 |2 v2 |x − x0 |2 v2 2 0 2 0 = (cos θ + sin θ ) = c2 c2

(8.90)

=

we find that (x − x0 ) · v c

2

=

(x − x0 ) × v |x − x0 |2 v2 − c2 c

2

(8.91)

Furthermore, from Equation (8.82) on page 129, we obtain the following identity: (x − x0 ) × v = (x − x0 ) × v

(8.92)

which, when inserted into Equation (8.91) above, yields the relation

2

(x − x0 ) · v c

=

(x − x0 ) × v |x − x0 |2 v2 − c2 c

2

(8.93)

Inserting the above into expression (8.89) for s2 , this expression becomes

2

s2 = x − x 0 − 2 x − x 0 = (x − x0 ) −

(x − x0 ) · v |x − x0 |2 v2 (x − x0 ) × v + − 2 c c c

|x − x0 | v c

2

−

(x − x0 ) × v c

(x − x0 (t)) × v = (x − x0 (t)) − c

2

2

2

(8.94)

2

(x − x0 ) × v ≡ |x − x0 | − c

2

2



132


AND

R ADIATING S YSTEMS

where in the penultimate step we used Equation (8.82) on page 129. What we have just demonstrated is that, in the case the particle velocity at time t can be calculated or projected, the retarded distance s in the LiénardWiechert potentials (8.63) can be expressed in terms of the virtual simultaneous coordinate x0 (t), viz., the point at which the particle will have arrived at time t, i.e., when we obtain the first knowledge of its existence at the source point x0 at the retarded time t0 , and in the field coordinate x(t), where we make our observations. We have, in other words, shown that all quantities in the definition of s, and hence s itself, can, when the motion of the charge is somehow known, be expressed in terms of the time t alone. I.e., in this special case we are able to express the retarded relative distance as s = s(t, x) and we do not have to involve the retarded time t 0 or transformed differential operators in our calculations. Taking the square root of both sides of Equation (8.94) on the preceding page, we obtain the following alternative final expressions for the retarded relative distance s in terms of the charge’s virtual simultaneous coordinate x 0 (t): s(t, x) =

(x − x0 ) × v |x − x0 | − c

= |x − x0 | 1 − =

2

2

|x − x0 |

2

(8.95a)

v2 2 sin θ0 c2

(8.95b)

v2 (x − x0 ) · v 1− 2 + c c

2

(8.95c)

Using Equation (8.95c) above and standard vector analytic formulae, we obtain ∇s2 = ∇ H |x − x0 |2 1 −

(x − x0 ) · v v2 − 2 c c

v2 vv + 2 · (x − x0 ) 2 c c

v v = 2 7 (x − x0 ) + × . × (x − x0 )/98 c c = 2 (x − x0 ) 1 −

2

I

(8.96)

which we shall use in the following example of a uniformly, unaccelerated motion of the charge. E XAMPLE 8.1

T HE

FIELDS FROM A UNIFORMLY MOVING CHARGE

In the special case of uniform motion, the localised charge moves in a field-free, iso-



8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

133

A RBITRARY M OTION

lated space and we know that it will not be affected by any external forces. It will therefore move uniformly in a straight line with the constant velocity v. This gives us the possibility to extrapolate its position at the observation time, x0 (t), from its position at the retarded time, x0 (t0 ). Since the particle is not accelerated, v˙ ≡ 0, the virtual simultaneous coordinate x0 will be identical to the actual simultaneous coordinate of the particle at time t, i.e., x0 (t) = x0 (t). As depicted in Figure 8.7 on page 126, the angle between x − x0 and v is θ0 while then angle between x − x0 and v is θ0 . We note that in the case of uniform velocity v, time and space derivatives are closely related in the following way when they operate on functions of x(t) : ∂ → −v · ∇ ∂t

(8.97)

Hence, the E and B fields can be obtained from Formulae (8.66) on page 125, with the potentials given by Equations (8.63) on page 124 as follows: ∂A 1 ∂vφ v ∂φ = −∇φ − 2 = −∇φ − 2 ∂t c ∂t c ∂t v v vv = −∇φ + · ∇φ = − 1 − 2 · ∇φ c c c vv = 2 − 1 · ∇φ c v v v B = ∇ × A = ∇ × 2 φ = ∇φ × 2 = − 2 × ∇φ c c c v v vv v v · ∇φ − ∇φ = 2 × 2 − 1 · ∇φ = 2× c c c c c v = 2 ×E c E = −∇φ −

%

$

%

$

$

ef$

%

$

(8.98a)

%

%

g

$

%

(8.98b)

Here 1 = xˆ i xˆ i is the unit dyad and we used the fact that v × v ≡ 0. What remains is just to express ∇φ in quantities evaluated at t and x. From Equation (8.63a) on page 124 and Equation (8.96) on the facing page we find that 1 q0 q0 =− ∇ ∇s2 4πε0 s 8πε0 s3 q0 v v =− (x − x0 ) + × × (x − x0 ) 3 4πε0 s c c

∇φ =

e

$

%g

(8.99)

When this expression for ∇φ is inserted into Equation (8.98a) above, the following result



134


%

AND

R ADIATING S YSTEMS

%

vv vv q0 − 1 − 1 · ∇s2 · ∇φ = − c2 8πε0 s3 c2 v v q0 × (x − x0 ) (x − x0 ) + × = 4πε0 s3 c c

E(t, x) =

$

$

− =

%

$

e

$

v v vv v v · (x − x0 ) − 2 · × × (x − x0 ) c c c c c

% gi

%

q0 v v v2 · (x − x0 ) − (x − x0 ) 2 (x − x0 ) + 3 4πε0 s c c c

$

− =

%

$

h

$

v v · (x − x0 ) c c

q0 4πε0 s

%

(x − x0 ) 1 − 3

(8.100)

v2 c2

follows. Of course, the same result also follows from Equation (8.83) on page 130 with v˙ ≡ 0 inserted. From Equation (8.100) above we conclude that E is directed along the vector from the simultaneous coordinate x0 (t) to the field (observation) coordinate x(t). In a similar way, the magnetic field can be calculated and one finds that B(t, x) =

µ0 q 0 4πs3

1−

v2 c2

v × (x − x0 ) =

1 v×E c2

(8.101)

From these explicit formulae for the E and B fields we can discern the following cases: 1. v → 0 ⇒ E goes over into the Coulomb field ECoulomb 2. v → 0 ⇒ B goes over into the Biot-Savart field 3. v → c ⇒ E becomes dependent on θ0

4. v → c, sin θ0 ≈ 0 ⇒ E → (1 − v2 /c2 )ECoulomb

5. v → c, sin θ0 ≈ 1 ⇒ E → (1 − v2 /c2 )−1/2 ECoulomb E ND

E XAMPLE 8.2

T HE

OF EXAMPLE

8.1C

CONVECTION POTENTIAL AND THE CONVECTION FORCE

Let us consider in more detail the treatment of the radiation from a uniformly moving rigid charge distribution. If we return to the original definition of the potentials and the inhomogeneous wave equation, Formula (3.19) on page 41, for a generic potential component Ψ(t, x) and a generic source component f (t, x),

j

2

Ψ(t, x) =


1 ∂2 − ∇2 Ψ(t, x) = f (t, x) c2 ∂t2

(8.102)


8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

135

A RBITRARY M OTION

we find that under the assumption that v = v xˆ 1 , this equation can be written

v2 c2

1−

∂2 Ψ ∂2 Ψ ∂2 Ψ + + = − f (x) ∂x21 ∂x22 ∂x23

(8.103)

i.e., in a time-independent form. Transforming x1 ξ1 = 1 − v2 /c2 ξ2 = x 2 ξ3 = x 3

k

(8.104a) (8.104b) (8.104c) def

and introducing the vectorial nabla operator in ξ space, ∇ξ ≡ (∂/∂ξ1 , ∂/∂ξ2 , ∂/∂ξ3 ), the time-independent equation (8.103) reduces to an ordinary Poisson equation ∇ξ2 Ψ(ξ) = − f (

k

1 − v2 /c2 ξ1 , ξ2 , ξ3 ) ≡ − f (ξ)

(8.105)

in this space. This equation has the well-known Coulomb potential solution 1 f (ξ 0 ) 3 0 dξ Ψ(ξ) = 4π V ξ − ξ 0

(8.106)

After inverse transformation back to the original coordinates, this becomes 1 f (x0 ) 3 0 dx Ψ(x) = 4π V s where, in the denominator,

(8.107)

1

2 v2 [(x2 − x02 )2 + (x3 − x03 )2 ] (8.108) 2 c Applying this to the explicit scalar and vector potential components, realising that for a rigid charge distribution ρ moving with velocity v the current is given by j = ρv, we obtain ρ(x0 ) 3 0 1 dx (8.109a) φ(t, x) = 4πε0 V s 1 vρ(x0 ) 3 0 v A(t, x) = d x = 2 φ(t, x) (8.109b) 2 4πε0 c V s c

s = (x1 − x01 )2 + 1 −

l

For a localised charge where ρ d3x0 = q0 , these expressions reduce to q0 (8.110a) φ(t, x) = 4πε0 s q0 v A(t, x) = (8.110b) 4πε0 c2 s which we recognise as the Liénard-Wiechert potentials; cf. Equations (8.63) on page 124. We notice, however, that the derivation here, based on a mathematical technique which in fact is a Lorentz transformation, is of more general validity than the one leading to Equations (8.63) on page 124. Let us now consider the action of the fields produced from a moving, rigid charge distribution represented by q0 moving with velocity v, on a charged particle q, also



8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

147

A RBITRARY M OTION

which vanishes for angles ϕ0 such that v c

cos ϕ0 = sin ϕ0 =

(8.164a) 1−

v2 c2

(8.164b)

Hence, the angle ϕ0 is a measure of the synchrotron radiation lobe width ∆θ; see Figure 8.11 on the preceding page. For ultra-relativistic particles, defined by γ=

K

1 2

1 − vc2

1,

1−

v2 1, c2

(8.165)

one can approximate ϕ0 ≈ sin ϕ0 =

1−

v2 1 = c2 γ

(8.166)

Hence, synchrotron radiation from ultra-relativistic charges is characterized by a radiation lobe width which is approximately ∆θ ≈

1 γ

(8.167)

This angular interval is swept by the charge during the time interval ∆t0 =

∆θ ω0

(8.168)

during which the particle moves a length interval ∆l = v∆t0 = v

∆θ ω0

(8.169)

in the direction toward the observer who therefore measures a pulse width of length ∆l v∆t0 v v ∆θ v 1 = ∆t0 − = . 1 − / ∆t0 = . 1 − / ≈ . 1− / c c c c ω0 c γω0 v4 v4 2 1− c 1+ c 1 v 1 1 1 3v = 3 ≈ 1− 2 = 3 γω0 L MEN c O 2γω0 2γ ω0 1+ L MEN cO 1/γ2 ≈2

∆t = ∆t0 −



148


AND

R ADIATING S YSTEMS

(8.170) As a general rule, the spectral width of a pulse of length ∆t is ∆ω o 1/∆t. In the ultra-relativistic synchrotron case one can therefore expect frequency components up to 1 = 2γ3 ω0 (8.171) ∆t A spectral analysis of the radiation pulse will exhibit Fourier components nω 0 from n = 1 up to n ≈ 2γ3 . When N electrons are contributing to the radiation, we can discern between three situations: ωmax ≈

1. All electrons are very close to each other so that the individual phase differences are negligible. The power will be multiplied by N 2 relative to a single electron and we talk about coherent radiation. 2. The electrons are perfectly evenly distributed in the orbit. This is the case, for instance, for electrons in a circular current in a conductor. In this case the radiation fields cancel completely and no far fields are generated. 3. The electrons are unevenly distributed in the orbit. This happens for an √ open ring current which is subject to fluctuations of order N as for all open systems. As a result we get incoherent radiation. Examples of this can be found both in earthly laboratories and under cosmic conditions.

Radiation in the general case We recall that the general expression for the radiation E field from a moving charge concentration is given by expression (8.87) on page 130. This expression in Equation (8.156) on page 144 yields the general formula dU rad (θ) µ0 q0 2 |x − x0 | = (x − x0 ) × dt0 16π2 cs5

(x − x0 ) −

|x − x0 | v c × v˙

2

(8.172)

Integration over the solid angle Ω gives the totally radiated power as 2

2 dU˜ rad µ0 q0 2 v˙ 2 1 − vc2 sin θ = 3 dt0 6πc . 1 − cv22 /

(8.173)

where θ is the angle between v and v˙ . In the limit v k v˙ , sin θ = 0, which corresponds to bremsstrahlung. For v ⊥ v˙ , sin θ = 1, which corresponds to cyclotron radiation or synchrotron radiation.



8.3

R ADIATION

L OCALISED C HARGE

FROM A

vt

IN

q0

149

A RBITRARY M OTION

v = v xˆ

θ0

b

|x − x0 | B E⊥ zˆ F IGURE 8.12: The perpendicular field of a charge q0 moving with velocity v = v xˆ is E ⊥ zˆ .

Virtual photons According to Formula (8.100) on page 134 and Figure 8.12, E⊥ = Ez =

q0 4πε0 s3

1−

v2 (x − x0 ) · xˆ 3 c2

(8.174)

Utilising expression (8.95a) on page 132 and simple geometrical relations, we can rewrite this as E⊥ =

b q0 4πε0 γ2 (vt)2 + b2 /γ2

(8.175)

3/2

This represents a contracted field, approaching the field of a plane wave. The passage of this field “pulse” corresponds to a frequency distribution of the field energy. Fourier transforming, we obtain Eω,⊥ =

1 2π

∞ −∞

E⊥ (t) eiωt dt =

q 4π2 ε

0 bv

bω K1 vγ

bω vγ

(8.176)

Here, K1 is the Kelvin function (Bessel function of the second kind with imaginary argument) which behaves in such a way for small and large arguments that Eω,⊥ ∼

q 4π2 ε

Eω,⊥ ∼ 0,

0 bv

,

bω vγ

bω vγ


(8.177a) (8.177b)


150


AND

R ADIATING S YSTEMS

showing that the “pulse” length is of the order b/(vγ). Due to the equipartition of the field energy into the electric and magnetic fields, the total field energy can be written U = ε0

V

2 3 E⊥ d x = ε0

∞

bmax

bmin

−∞

2 E⊥ vdt 2πb db

(8.178)

where the volume integration is over the plane perpendicular to v. With the use of Parseval’s identity for Fourier transforms, Formula (7.35) on page 105, we can rewrite this as U=

∞ 0

q2 ≈ 2 2π ε0 v

∞

0

vγ/ω

bmin

∞

bmax

Uω dω = 4πε0 v

0

2

Eω,⊥ dω 2πb db (8.179)

bmin

db dω b

from which we conclude that Uω ≈

q2 vγ ln 2 2π ε0 v bmin ω

(8.180)

where an explicit value of bmin can be calculated in quantum theory only. As in the case of bremsstrahlung, it is intriguing to quantise the energy into photons [cf. Equation (8.150) on page 143]. Then we find that Nω dω ≈

2α cγ dω ln π bmin ω ω

(8.181)

where α = e2 /(4πε0 h¯ c) ≈ 1/137 is the fine structure constant. Let us consider the interaction of two electrons, 1 and 2. The result of this interaction is that they change their linear momenta from p1 to p01 and p2 to 0 p2 , respectively. Heisenberg’s uncertainty principle gives bmin ∼ h¯ / p1 − p01 so that the number of photons exchanged in the process is of the order Nω dω ≈

dω cγ 2α ln . p1 − p01 / π h¯ ω ω

(8.182)

Since this change in momentum corresponds to a change in energy h¯ ω = E 1 − E10 and E1 = m0 γc2 , we see that 2α Nω dω ≈ ln π

]

E1 cp1 − cp01 m0 c2 E1 − E10

^

dω ω

(8.183)

a formula which gives a reasonable account of electron- and photon-induced processes.



8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

151

A RBITRARY M OTION

8.3.5 Radiation from charges moving in matter When electromagnetic radiation is propagating through matter, new phenomena may appear which are (at least classically) not present in vacuum. As mentioned earlier, one can under certain simplifying assumptions include, to some extent, the influence from matter on the electromagnetic fields by introducing new, derived field quantities D and H according to D = ε(t, x)E = κε0 E

(8.184)

B = µ(t, x)H = κm µ0 H

(8.185)

Expressed in terms of these derived field quantities, the Maxwell equations, often called macroscopic Maxwell equations, take the form ∇ · D = ρ(t, x) ∂ ∇×E+ B = 0 ∂t ∇·B = 0 ∂ ∇ × H − D = j(t, x) ∂t

(8.186a) (8.186b) (8.186c) (8.186d)

Assuming for simplicity that the electric permittivity ε and the magnetic permeability µ, and hence the relative permittivity κ and the relative permeability κm all have fixed values, independent on time and space, for each type of material we consider, we can derive the general telegrapher’s equation [cf. Equation (2.33) on page 29] ∂E ∂2 E ∂2 E − σµ − εµ =0 ∂ζ 2 ∂t ∂t2

(8.187)

describing (1D) wave propagation in a material medium. In Chapter 2 we concluded that the existence of a finite conductivity, manifesting itself in a collisional interaction between the charge carriers, causes the waves to decay exponentially with time and space. Let us therefore assume that in our medium σ = 0 so that the wave equation simplifies to ∂2 E ∂2 E − εµ =0 ∂ζ 2 ∂t2

(8.188)

If we introduce the phase velocity in the medium as 1 c 1 =√ vϕ = √ = √ εµ κε0 κm µ0 κκm


(8.189)


152


AND

R ADIATING S YSTEMS

√ where, according to Equation (1.9) on page 5, c = 1/ ε0 µ0 is the speed of light, i.e., the phase speed of electromagnetic waves in vacuum, then the general solution to each component of Equation (8.188) on the previous page Ei = f (ζ − vϕ t) + g(ζ + vϕ t),

i = 1, 2, 3

(8.190)

The ratio of the phase speed in vacuum and in the medium c √ √ def = κκm = c εµ ≡ n vϕ

(8.191)

is called the refractive index of the medium. In general n is a function of both time and space as are the quantities ε, µ, κ, and κm themselves. If, in addition, the medium is anisotropic or birefringent, all these quantities are rank-two tensor fields. Under our simplifying assumptions, in each medium we consider n = Const for each frequency component of the fields. Associated with the phase speed of a medium for a wave of a given frequency ω we have a wave vector, defined as def

k ≡ k kˆ = kˆvϕ =

ω vϕ vϕ vϕ

(8.192)

As in the vacuum case discussed in Chapter 2, assuming that E is time-harmonic, i.e., can be represented by a Fourier component proportional to exp{−iωt}, the solution of Equation (8.188) can be written E = E0 ei(k·x−ωt)

(8.193)

where now k is the wave vector in the medium given by Equation (8.192). With these definitions, the vacuum formula for the associated magnetic field, Equation (2.40) on page 30, 1 1 ˆ √ k×E = k×E B = εµ kˆ × E = vϕ ω

(8.194)

is valid also in a material medium (assuming, as mentioned, that n has a fixed constant scalar value). A consequence of a κ 6= 1 is that the electric field will, in general, have a longitudinal component. It is important to notice that depending on the electric and magnetic properties of a medium, and, hence, on the value of the refractive index n, the phase speed in the medium can be smaller or larger than the speed of light: c ω = n k

(8.195)



vϕ =

8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

153

A RBITRARY M OTION

where, in the last step, we used Equation (8.192) on the facing page. If the medium has a refractive index which, as is usually the case, dependent on frequency ω, we say that the medium is dispersive. Because in this case also k(ω) and ω(k), so that the group velocity vg =

∂ω ∂k

(8.196)

has a unique value for each frequency component, and is different from v ϕ . Except in regions of anomalous dispersion, vϕ is always smaller than c. In a gas of free charges, such as a plasma, the refractive index is given by the expression ω2p ω2

(8.197)

Nσ q2σ ε0 mσ

(8.198)

n2 (ω) = 1 − where ω2p = ∑ σ

is the plasma frequency. Here mσ and Nσ denote the mass and number density, respectively, of charged particle species σ. In an inhomogeneous plasma, N σ = Nσ (x) so that the refractive index and also the phase and group velocities are space dependent. As can be easily seen, for each given frequency, the phase and group velocities in a plasma are different from each other. If the frequency ω is such that it coincides with ωp at some point in the medium, then at that point vϕ → ∞ while vg → 0 and the wave Fourier component at ω is reflected there.

ˇ Vavilov-Cerenkov radiation As we saw in Subsection 8.1, a charge in uniform, rectilinear motion in vacuum does not give rise to any radiation; see in particular Equation (8.98a) on page 133. Let us now consider a charge in uniform, rectilinear motion in a medium with electric properties which are different from those of a (classical) vacuum. Specifically, consider a medium where ε = Const > ε0

(8.199a)

µ = µ0

(8.199b)

This implies that in this medium the phase speed is vϕ =

c 1 =√

(8.200)


154


AND

R ADIATING S YSTEMS

Hence, in this particular medium, the speed of propagation of (the phase planes of) electromagnetic waves is less than the speed of light in vacuum, which we know is an absolute limit for the motion of anything, including particles. A medium of this kind has the interesting property that particles, entering into the medium at high speeds |v|, which, of course, are below the phase speed in vacuum, can experience that the particle speeds are higher than the phase ˇ speed in the medium. This is the basis for the Vavilov-Cerenkov radiation that we shall now study. If we recall the general derivation, in the vacuum case, of the retarded (and advanced) potentials in Chapter 3 and the Liénard-Wiechert potentials, Equations (8.63) on page 124, we realise that we obtain the latter in the medium by a simple formal replacement c → c/n in the expression (8.64) on page 124 for s. Hence, the Liénard-Wiechert potentials in a medium characterized by a refractive index n, are φ(t, x) =

A(t, x) =

q0 1 q0 1 = 4πε0 |x − x0 | − n (x−x0 )·v 4πε0 s c

1 q0 v 1 q0 v = 4πε0 c2 |x − x0 | − n (x−x0 )·v 4πε0 c2 s c

(8.201a)

(8.201b)

where now

(x − x0 ) · v 0 s = x−x −n

(8.202)

c

The need for the absolute value of the expression for s is obvious in the case when v/c ≥ 1/n because then the second term can be larger than the first term; if v/c 1/n we recover the well-known vacuum case but with modified phase speed. We also note that the retarded and advanced times in the medium are [cf. Equation (3.34) on page 43]

0 0 tret = tret (t, x − x0 ) = t −

k |x − x0 | |x − x0 | n = t− ω c 0 k |x − x0 | |x − x0 | n 0 0 tadv = tadv (t, x − x ) = t + = t+ ω c

(8.203a) (8.203b)

so that the usual time interval t − t 0 between the time measured at the point of observation and the retarded time in a medium becomes t − t0 =

|x − x0 | n c


(8.204)


8.3

R ADIATION

FROM A

L OCALISED C HARGE

IN

A RBITRARY M OTION

αc

q0

155

x(t)

θc

v

x0 (t0 )

F IGURE 8.13: Instantaneous picture of the expanding field spheres from a point charge moving with constant speed v/c > 1/n in a medium where ˇ n > 1. This generates a Vavilov-Cerenkov shock wave in the form of a cone.

For v/c ≥ 1/n, the retarded distance s, and therefore the denominators in Equations (8.201) on the preceding page vanish when n(x − x0 ) ·

nv v = x − x0 cos θc = x − x0 c c

(8.205)

or, equivalently, when cos θc =

c nv

(8.206)

In the direction defined by this angle θc , the potentials become singular. During the time interval t − t0 given by expression (8.204) on the facing page, the field exists within a sphere of radius |x − x0 | around the particle while the particle moves a distance l = v(t − t0 )

(8.207)

along the direction of v. In the direction θc where the potentials are singular, all field spheres are tangent to a straight cone with its apex at the instantaneous position of the particle and with the apex half angle αc defined according to sin αc = cos θc =

c nv


(8.208)


156


AND

R ADIATING S YSTEMS

This cone of potential singularities and field sphere circumferences propagates ˇ with speed c/n in the form of a shock front, called Vavilov-Cerenkov radiation.1 ˇ The Vavilov-Cerenkov cone is similar in nature to the Mach cone in acoustics. In order to make some quantitative estimates of this radiation, we note that we can describe the motion of each charged particle q0 as a current density: j = q0 v δ(x0 − vt0 ) = q0 v δ(x0 − vt0 )δ(y0 )δ(z0 ) xˆ 1

(8.209)

which has the trivial Fourier transform q0 iωx0 /v 0 jω = e δ(y )δ(z0 ) xˆ 1 (8.210) 2π This Fourier component can be used in the formulae derived for a linear current in Subsection 8.1.1 if only we make the replacements ε0 → ε = n 2 ε0 (8.211a) nω k→ (8.211b) c In this manner, using jω from Equation (8.210) above, the resulting Fourier ˇ transforms of the Vavilov-Cerenkov magnetic and electric radiation fields can be calculated from the expressions (7.11) and (7.22) on page 100, respectively. The total energy content is then obtained from Equation (7.35) on page 105 (integrated over a closed sphere at large distances). For a Fourier component one obtains [cf. Equation (7.38) on page 105] Uωrad dΩ

1 ≈ 4πε0 nc

(j × k)e

−ik·x0 3 0

V ω q0 2 nω2 ∞ = exp i 16π3 ε0 c3 −∞

dx

2

dΩ

2

ωx0 − kx0 cos θ dx0 sin2 θ dΩ

v

(8.212)

1 The first systematic exploration of this radiation was made by P. A. Cerenkov ˇ in 1934, who was then a post-graduate student in S. I. Vavilov’s research group at the Lebedev Institute ˇ in Moscow. Vavilov wrote a manuscript with the experimental findings, put Cerenkov as the author, and submitted it to Nature. In the manuscript, Vavilov explained the results in terms of radioactive particles creating Compton electrons which gave rise to the radiation (which was the correct interpretation), but the paper was rejected. The paper was then sent to Physical Review and was, after some controversy with the American editors who claimed the results to be wrong, eventually published in 1937. In the same year, I. E. Tamm and I. M. Frank published the theory for the effect (“the singing electron”). In fact, predictions of a similar effect had been made as early as 1888 by Heaviside, and by Sommerfeld in his 1904 paper “Radiating body moving with velocity of light”. On May 8, 1937, Sommerfeld sent a letter to Tamm via Austria, saying that he was surprised that his old 1904 ideas were now becoming interesting. Tamm, ˇ Frank and Cerenkov received the Nobel Prize in 1958 “for the discovery and the interpretation ˇ of the Cerenkov effect” [V. L. Ginzburg, private communication]. The first observation of this type of radiation was reported by Marie Curie in 1910, but she never pursued the exploration of it [7].



8.3

R ADIATION

L OCALISED C HARGE

FROM A

IN

157

A RBITRARY M OTION

where θ is the angle between the direction of motion, xˆ 01 , and the direction to ˆ The integral in (8.212) is singular of a “Dirac delta type.” If the observer, k. we limit the spatial extent of the motion of the particle to the closed interval [−X, X] on the x0 axis we can evaluate the integral to obtain

4 Xω q0 2 nω2 sin2 θ sin2 1 − nv c cos θ v 3 dΩ 2 4π3 ε0 c3 1 − nvc cos θ 4 ωv

Uωrad dΩ =

(8.213)

3

which has a maximum in the direction θc as expected. The magnitude of this maximum grows and its width narrows as X → ∞. The integration of (8.213) over Ω therefore picks up the main contributions from θ ≈ θc . Consequently, we can set sin2 θ ≈ sin2 θc and the result of the integration is U˜ ωrad = 2π

π 0

Uωrad (θ) sinθ dθ = dcos θ = −ξc = 2π

q0 2 nω2 sin2 θc ≈ 2π2 ε0 c3

1

sin2

7p. 1 + nvξc /

7E. 1 + nvξc / ωv 8

−1

Xω v 2

8

1

−1

Uωrad (ξ) dξ (8.214)

dξ

The integrand in (8.214) is strongly peaked near ξ = −c/(nv), or, equivalently, near cos θc = c/(nv). This means that the integrand function is practically zero outside the integration interval ξ ∈ [−1, 1]. Hence, one can extend the ξ integration interval to (−∞, ∞) without introducing too much an error. Via yet another variable substitution we can therefore approximate 2

sin θc

1 −1

sin2

7E. 1 + nvξc /

7p. 1 + nvξc / ωv 8

Xω v 2

8

dξ ≈ 1 − =

c2 cX n2 v2 ωn

∞

sin2 x dx 2 −∞ x

(8.215)

c2

cXπ 1 − n2 v2 ωn

leading to the final approximate result for the total energy loss in the frequency interval (ω, ω + dω) q0 2 X U˜ ωrad dω = 2πε0 c2

1−

c2 ω dω n2 v2

(8.216)

As mentioned earlier, the refractive index is usually frequency dependent. Realising this, we find that the radiation energy per frequency unit and per unit length is U˜ ωrad dω q0 2 ω = 2X 4πε0 c2

1−


c2 dω n2 (ω)v2

(8.217)


158


AND

R ADIATING S YSTEMS

This result was derived under the assumption that v/c > 1/n(ω), i.e., under the condition that the expression inside the parentheses in the right hand side is positive. For all media it is true that n(ω) → 1 when ω → ∞, so there exist alˇ ways a highest frequency for which we can obtain Vavilov-Cerenkov radiation from a fast charge in a medium. Our derivation above for a fixed value of n is valid for each individual Fourier component.

Bibliography [1] R. B ECKER, Electromagnetic Fields and Interactions, Dover Publications, Inc., New York, NY, 1982, ISBN 0-486-64290-9. [2] M. B ORN AND E. W OLF, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light, sixth ed., Pergamon Press, Oxford,. . . , 1980, ISBN 0-08-026481-6. [3] V. L. G INZBURG, Applications of Electrodynamics in Theoretical Physics and Astrophysics, Revised third ed., Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo and Melbourne, 1989, ISBN 288124-719-9. [4] J. D. JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, NY . . . , 1999, ISBN 0-471-30932-X. [5] J. B. M ARION AND M. A. H EALD, Classical Electromagnetic Radiation, second ed., Academic Press, Inc. (London) Ltd., Orlando, . . . , 1980, ISBN 0-12472257-1. [6] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6. [7] J. S CHWINGER , L. L. D E R AAD , J R ., K. A. M ILTON , AND W. T SAI, Classical Electrodynamics, Perseus Books, Reading, MA, 1998, ISBN 0-7382-0056-5. [8] J. A. S TRATTON, Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, NY and London, 1953, ISBN 07-062150-0. [9] J. VANDERLINDE, Classical Electromagnetic Theory, John Wiley & Sons, Inc., New York, Chichester, Brisbane, Toronto, and Singapore, 1993, ISBN 0-47157269-1.



F Formulae

F.1 The Electromagnetic Field F.1.1 Maxwell’s equations ∇·D = ρ

(F.1)

∇·B = 0

(F.2)

∇×E = −

(F.3)

∂ B ∂t ∂ ∇×H = j+ D ∂t

(F.4)

Constitutive relations D = εE B H= µ

(F.5)

j = σE

(F.7)

P = ε0 χE

(F.8)

(F.6)

F.1.2 Fields and potentials Vector and scalar potentials B = ∇×A E = −∇φ −

(F.9) ∂ A ∂t

(F.10)

159

160

F ORMULAE

Lorentz’ gauge condition in vacuum ∇·A+

1 ∂ φ=0 c2 ∂t

(F.11)

F.1.3 Force and energy Poynting’s vector S = E×H

(F.12)

Maxwell’s stress tensor 1 T i j = Ei D j + Hi B j − δi j (Ek Dk + Hk Bk ) 2

(F.13)

F.2 Electromagnetic Radiation F.2.1 Relationship between the field vectors in a plane wave B=

kˆ × E c

(F.14)

F.2.2 The far fields from an extended source distribution 0 −iµ0 eik|x| d3 x0 e−ik·x jω × k 4π |x| V 0 0 i eik|x| Erad d3 x0 e−ik·x jω × k xˆ × ω (x) = 0 4πε0 c |x| V

Brad ω (x) =

(F.15) (F.16)

F.2.3 The far fields from an electric dipole ωµ0 eik|x| pω × k 4π |x| 1 eik|x| (pω × k) × k Erad ω (x) = − 4πε0 |x| Brad ω (x) = −


(F.17) (F.18)


F.3

161

S PECIAL R ELATIVITY

F.2.4 The far fields from a magnetic dipole µ0 eik|x| (mω × k) × k 4π |x| k eik|x| Erad mω × k ω (x) = 4πε0 c |x|

Brad ω (x) = −

(F.19) (F.20)

F.2.5 The far fields from an electric quadrupole iµ0 ω eik|x| k · Qω 4 × k 8π |x| 3 i eik|x| Erad k · Qω 4 × k × k ω (x) = 8πε0 |x| 3

Brad ω (x) =

(F.21) (F.22)

F.2.6 The fields from a point charge in arbitrary motion q v2 (x − x0 ) × v˙ (x − x ) 1 − + (x − x0 ) × 0 4πε0 s3 c2 c2

E(t, x) B(t, x) = (x − x0 ) × c|x − x0 |

E(t, x) =

s = x − x0 − (x − x0 ) ·

v c

x − x0 = (x − x0 ) − |x − x0 |

∂t0 ∂t

= x

|x − x0 | s

(F.23) (F.24)

(F.25) v c

(F.26) (F.27)

F.2.7 The fields from a point charge in uniform motion q v2 1 − (x − x ) 0 4πε0 s3 c2 v × E(t, x) B(t, x) = c2

E(t, x) =

s=

|x − x0 |2 −

(x − x0 ) × v c


(F.28) (F.29)

2

(F.30)


162

F ORMULAE

F.3 Special Relativity F.3.1 Metric tensor gµν =

@AAB

CEDD

1 0 0 0 0 −1 0 0 0 0 −1 0 F 0 0 0 −1

(F.31)

F.3.2 Covariant and contravariant four-vectors vµ = gµν vν

(F.32)

F.3.3 Lorentz transformation of a four-vector x0µ = Λµν xν

CEDD

γ −γβ 0 0 −γβ γ 0 0 Λµν = @A AB 0 0 1 0F 0 0 0 1 1 γ= : 1 − β2 v β= c

(F.33)

(F.34)

(F.35) (F.36)

F.3.4 Invariant line element ds = c

dt = c dτ γ

(F.37)

F.3.5 Four-velocity uµ =

dx µ = γ(c, v) dτ

(F.38)

F.3.6 Four-momentum pµ = m0 cuµ =


E ,p c

(F.39)


F.4

163

V ECTOR R ELATIONS

F.3.7 Four-current density j µ = ρ0 u µ

(F.40)

F.3.8 Four-potential Aµ =

φ ,A c

(F.41)

F.3.9 Field tensor F µν = ∂µ Aν − ∂ν Aµ =

CEDD

@AAB

0 −E x /c −Ey /c −Ez /c E x /c 0 −Bz By Ey /c Bz 0 −B x F Ez /c −By Bx 0

(F.42)

F.4 Vector Relations Let x be the radius vector (coordinate vector) from the origin to the point (x1 , x2 , x3 ) ≡ (x, y, z) and let |x| denote the magnitude (“length”) of x. Let further α(x), β(x), . . . be arbitrary scalar fields and a(x), b(x), c(x), d(x), . . . arbitrary vector fields. The differential vector operator ∇ is in Cartesian coordinates given by 3

∇ ≡ ∑ xˆ i i=1

∂ def ∂ def ≡ xˆ i ≡ ∂ ∂xi ∂xi

(F.43)

where xˆ i , i = 1, 2, 3 is the ith unit vector and xˆ 1 ≡ xˆ , xˆ 2 ≡ yˆ , and xˆ 3 ≡ zˆ . In component (tensor) notation ∇ can be written ∇i = ∂i =

∂ ∂ ∂ , , = ∂x1 ∂x2 ∂x3

∂ ∂ ∂ , , ∂x ∂y ∂z

(F.44)

F.4.1 Spherical polar coordinates Base vectors rˆ = sin θ cos ϕ xˆ 1 + sin θ sin ϕ xˆ 2 + cos θ xˆ 3 θˆ = cos θ cos ϕ xˆ 1 + cos θ sin ϕ xˆ 2 − sin θ xˆ 3

ϕˆ = − sin ϕ xˆ 1 + cos ϕ xˆ 2


(F.45a) (F.45b) (F.45c)


164

F ORMULAE

xˆ 1 = sin θ cos ϕˆr + cos θ cos ϕθˆ − sin ϕϕˆ xˆ 2 = sin θ sin ϕˆr + cos θ sin ϕθˆ + cos ϕϕˆ

(F.46a) (F.46b)

xˆ 3 = cos θˆr − sin θθˆ

(F.46c)

Directed line element dx xˆ = dl = dr rˆ + r dθ θˆ + r sin θ dϕ ϕˆ

(F.47)

Solid angle element dΩ = sin θ dθ dϕ

(F.48)

Directed area element d2x nˆ = dS = dS rˆ = r2 dΩ rˆ

(F.49)

Volume element d3x = dV = drdS = r2 dr dΩ

(F.50)

F.4.2 Vector formulae General vector algebraic identities a · b = b · a = δi j ai b j = ab cos θ

(F.51)

a · (b × c) = (a × b) · c

(F.53)

a × b = −b × a = i jk a j bk xˆ i

(F.52)

a × (b × c) = b(a · c) − c(a · b)

(F.54)

(a × b) · (c × d) = a · [b × (c × d)] = (a · c)(b · d) − (a · d)(b · c)

(F.56)

a × (b × c) + b × (c × a) + c × (a × b) = 0

(a × b) × (c × d) = (a × b · d)c − (a × b · c)d


(F.55) (F.57)


F.4

165

V ECTOR R ELATIONS

General vector analytic identities ∇(αβ) = α∇β + β∇α

(F.58)

∇ · (αa) = a · ∇α + α∇ · a

(F.59)

∇ · (a × b) = b · (∇ × a) − a · (∇ × b)

(F.61)

∇ × (αa) = α∇ × a − a × ∇α

(F.60)

∇ × (a × b) = a(∇ · b) − b(∇ · a) + (b · ∇)a − (a · ∇)b

(F.62)

∇ · ∇α = ∇2 α

(F.64)

∇(a · b) = a × (∇ × b) + b × (∇ × a) + (b · ∇)a + (a · ∇)b ∇ × ∇α = 0

∇ · (∇ × a) = 0

(F.63) (F.65) (F.66)

2

∇ × (∇ × a) = ∇(∇ · a) − ∇ a

(F.67)

Special identities In the following k is an arbitrary constant vector while x, as before, is the radius vector and a an arbitrary vector field. ∇·x = 3

(F.68)

∇(k · x) = k x ∇|x| = |x| 1 x ∇ =− 3 |x| |x| x 1 ∇· = −∇2 = 4πδ(x) 3 |x| |x| k 1 k·x ∇ = k· ∇ =− 3 |x| |x|

|x| x k·x = −∇ if |x| 6= 0 ∇× k× |x|3

|x|3 k 1 = k∇2 = −4πkδ(x) ∇2 |x| |x| ∇ × (k × a) = k(∇ · a) + k × (∇ × a) − ∇(k · a)

(F.70)

∇×x = 0


(F.69) (F.71) (F.72) (F.73) (F.74) (F.75) (F.76) (F.77)


166

F ORMULAE

Integral relations Let V(S ) be the volume bounded by the closed surface S (V). Denote the 3dimensional volume element by d3x(≡ dV) and the surface element, directed along the outward pointing surface normal unit vector n, ˆ by dS(≡ d2x n). ˆ Then

V

V

V

(∇ · a) d3x = (∇α) d3x =

S

dS · a

dS α

(F.78) (F.79)

S

(∇ × a) d3x =

S

dS × a

(F.80)

If S (C) is an open surface bounded by the contour C(S ), whose line element is dl, then

C α dl =

S

C a · dl =

dS × ∇α

S

dS · (∇ × a)

(F.81) (F.82)

Bibliography [1] G. B. A RFKEN AND H. J. W EBER, Mathematical Methods for Physicists, fourth, international ed., Academic Press, Inc., San Diego, CA . . . , 1995, ISBN 0-12059816-7. [2] P. M. M ORSE AND H. F ESHBACH, Methods of Theoretical Physics, Part I. McGraw-Hill Book Company, Inc., New York, NY . . . , 1953, ISBN 07-043316-8. [3] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6.



M Mathematical Methods M.1 Scalars, Vectors and Tensors Every physical observable can be described by a geometric object. We will describe the observables in classical electrodynamics mathematically in terms of scalars, pseudoscalars, vectors, pseudovectors, tensors or pseudotensors and will not exploit differential forms to any significant degree. A scalar describes a scalar quantity which may or may not be constant in time and/or space. A vector describes some kind of physical motion due to vection and a tensor describes the motion or deformation due to some form of tension. However, generalisations to more abstract notions of these quantities are commonplace. The difference between a scalar, vector and tensor and a pseudoscalar, pseudovector and a pseudotensor is that the latter behave differently under such coordinate transformations which cannot be reduced to pure rotations. Throughout we adopt the convention that Latin indices i, j, k, l, . . . run over the range 1, 2, 3 to denote vector or tensor components in the real Euclidean three-dimensional (3D) configuration space 3 , and Greek indices µ, ν, κ, λ, . . . , which are used in four-dimensional (4D) space, run over the range 0, 1, 2, 3.

M.1.1 Vectors Radius vector Any vector can be represented mathematically in several different ways. One suitable representation is in terms of an ordered N-tuple, or row vector, of the coordinates xN where N is the dimensionality of the space under consideration. The most basic vector is the radius vector which is the vector from the origin to the point of interest. Its N-tuple representation simply enumerates the coordinates which describe this point. In this sense, the radius vector from the origin to a point is synonymous with the coordinates of the point itself.

167

168

M ATHEMATICAL M ETHODS

In the 3D space 3 , we have N = 3 and the radius vector can be represented by the triplet (x1 , x2 , x3 ) of coordinates xi , i = 1, 2, 3. The coordinates xi are scalar quantities which describe the position along the unit base vectors xˆ i which span 3 . Therefore a representation of the radius vector in 3 is 3

x = ∑ xˆ i xi ≡ xˆ i xi def

(M.1)

i=1

where we have introduced Einstein’s summation convention (EΣ) which states that a repeated index in a term implies summation over the range of the index in question. Whenever possible and convenient we shall in the following always assume EΣ and suppress explicit summation in our formulae. Typographically, we represent a 3D vector by a boldface letter or symbol in a Roman font. Alternatively, we may describe the radius vector in component notation as follows: def

xi ≡ (x1 , x2 , x3 ) ≡ (x, y, z)

(M.2)

This component notation is particularly useful in 4D space where we can represent the radius vector either in its contravariant component form def

xµ ≡ (x0 , x1 , x2 , x3 )

(M.3)

or its covariant component form def

xµ ≡ (x0 , x1 , x2 , x3 )

(M.4)

The relation between the covariant and contravariant forms is determined by the metric tensor (also known as the fundamental tensor) whose actual form is dictated by the physics. The dual representation of vectors in contravariant and covariant forms is most convenient when we work in a non-Euclidean vector space with an indefinite metric. An example is Lorentz space ; 4 which is a 4D Riemannian space. ; 4 is often utilised to formulate the special theory of relativity. We note that for a change of coordinates xµ → x0µ = x0µ (x0 , x1 , x2 , x3 ), due to a transformation from a system Σ to another system Σ0 , the differential radius vector dxµ transforms as dx0µ =

∂x0µ ν dx ∂xν

(M.5)

which follows trivially from the rules of differentiation of x 0µ considered as functions of four variables xν .



M.1

S CALARS , V ECTORS

AND

169

T ENSORS

M.1.2 Fields A field is a physical entity which depends on one or more continuous parameters. Such a parameter can be viewed as a “continuous index” which enumerates the “coordinates” of the field. In particular, in a field which depends on the usual radius vector x of 3 , each point in this space can be considered as one degree of freedom so that a field is a representation of a physical entity which has an infinite number of degrees of freedom.

Scalar fields We denote an arbitrary scalar field in

3

by

def

α(x) = α(x1 , x2 , x3 ) ≡ α(xi )

(M.6)

This field describes how the scalar quantity α varies continuously in 3D space. In 4D, a four-scalar field is denoted def

α(x0 , x1 , x2 , x3 ) ≡ α(xµ )

3

(M.7)

which indicates that the four-scalar α depends on all four coordinates spanning this space. Since a four-scalar has the same value at a given point regardless of coordinate system, it is also called an invariant. Analogous to the transformation rule, Equation (M.5) on the preceding page, for the differential dxµ , the transformation rule for the differential operator ∂/∂xµ under a transformation xµ → x0µ becomes ∂ ∂xν ∂ = ∂x0µ ∂x0µ ∂xν

(M.8)

which, again, follows trivially from the rules of differentiation.

Vector fields We can represent an arbitrary vector field a(x) in

3

as follows:

a(x) = xˆ i ai (x)

(M.9)

In component notation this same vector can be represented as ai (x) = (a1 (x), a2 (x), a3 (x)) = ai (x j )


(M.10)


170


In 4D, an arbitrary four-vector field in contravariant component form can be represented as aµ (xν ) = (a0 (xν ), a1 (xν ), a2 (xν ), a3 (xν ))

(M.11)

or, in covariant component form, as aµ (xν ) = (a0 (xν ), a1 (xν ), a2 (xν ), a3 (xν ))

(M.12)

where xν is the radius four-vector. Again, the relation between aµ and aµ is determined by the metric of the physical 4D system under consideration. Whether an arbitrary N-tuple fulfils the requirement of being an (N-dimensional) contravariant vector or not, depends on its transformation properties during a change of coordinates. For instance, in 4D an assemblage yµ = (y0 , y1 , y2 , y3 ) constitutes a contravariant four-vector (or the contravariant components of a four-vector) if and only if, during a transformation from a system Σ with coordinates xµ to a system Σ0 with coordinates x0µ , it transforms to the new system according to the rule y0µ =

∂x0µ ν y ∂xν

(M.13)

i.e., in the same way as the differential coordinate element dx µ transforms according to Equation (M.5) on page 168. The analogous requirement for a covariant four-vector is that it transforms, during the change from Σ to Σ0 , according to the rule y0µ =

∂xν yν ∂x0µ

(M.14)

i.e., in the same way as the differential operator ∂/∂xµ transforms according to Equation (M.8) on the previous page.

Tensor fields We denote an arbitrary tensor field in 3 by A(x). This tensor field can be represented in a number of ways, for instance in the following matrix form:

C

A(x) =

B@

A11 (x) A12 (x) A13 (x) A21 (x) A22 (x) A23 (x)F A31 (x) A32 (x) A33 (x)

≡ Ai j (xk ) 4

def

3

(M.15)

where, in the last member, we have again used the more compact component notation. Strictly speaking, the tensor field described here is a tensor of rank two.



M.1

S CALARS , V ECTORS

AND

A particularly simple rank-two tensor in bol δi j , with the following properties: δi j =

171

T ENSORS

3

is the 3D Kronecker delta sym-

0 if i 6= j 1 if i = j

G

(M.16)

The 3D Kronecker delta has the following matrix representation

C

B@

(δi j ) =

1 0 0 0 1 0F 0 0 1

(M.17)

Another common and useful tensor is the fully antisymmetric tensor of rank 3, also known as the Levi-Civita tensor

i jk =

> =?

<=

1 0 −1

if i, j, k is an even permutation of 1,2,3 if at least two of i, j, k are equal if i, j, k is an odd permutation of 1,2,3

(M.18)

with the following further property i jk ilm = δ jl δkm − δ jm δkl

(M.19)

In fact, tensors may have any rank n. In this picture a scalar is considered to be a tensor of rank n = 0 and a vector a tensor of rank n = 1. Consequently, the notation where a vector (tensor) is represented in its component form is called the tensor notation. A tensor of rank n = 2 may be represented by a twodimensional array or matrix whereas higher rank tensors are best represented in their component forms (tensor notation).

T ENSORS IN 3D SPACE

E XAMPLE M.1

Consider a tetrahedron-like volume element V of a solid, fluid, or gaseous body, whose atomistic structure is irrelevant for the present analysis; figure M.1 on the following page indicates how this volume may look like. Let dS = d2x nˆ be the directed surface element of this volume element and let the vector T nˆ d2x be the force that matter, lying on the side of d2x toward which the unit normal vector nˆ points, acts on matter which lies on the opposite side of d2x. This force concept is meaningful only if the forces are short-range enough that they can be assumed to act only in the surface proper. According to Newton’s third law, this surface force fulfils T− nˆ = −T nˆ

(M.20)

Using (M.20) and Newton’s second law, we find that the matter of mass m, which at a



172


x3

nˆ

d2x x2 V

x1 F IGURE M.1: Terahedron-like volume element V containing matter. given instant is located in V obeys the equation of motion T nˆ d2x − cos θ1 T xˆ 1 d2x − cos θ2 T xˆ 2 d2x − cos θ3 T xˆ 3 d2x + Fext = ma

(M.21)

where Fext is the external force and a is the acceleration of the volume element. In other words m Fext T nˆ = n1 T xˆ 1 + n2 T xˆ 2 + n3 T xˆ 3 + 2 a − (M.22) dx m

Since both a and Fext /m remain finite whereas this limit

m/d2x

→ 0 as V → 0, one finds that in

3

T nˆ = ∑ ni T xˆ i ≡ ni T xˆ i

(M.23)

i=1

From the above derivation it is clear that Equation (M.23) above is valid not only in equilibrium but also when the matter in V is in motion. Introducing the notation

T i j = T xˆ i

j

(M.24)

for the jth component of the vector T xˆ i , we can write Equation (M.23) on the facing



M.1

S CALARS , V ECTORS

AND

173

T ENSORS

page in component form as follows 3

T nj ˆ = (T nˆ ) j = ∑ ni T i j ≡ ni T i j

(M.25)

i=1

Using Equation (M.25) above, we find that the component of the vector T nˆ in the direction of an arbitrary unit vector m ˆ is T nˆ mˆ = T nˆ · m ˆ 3

3

= ∑ T nj ˆ mj = ∑ j=1

j=1

m

3

∑ ni T i j n

(M.26)

m j ≡ ni T i j m j = n· ˆ T· m ˆ

i=1

Hence, the jth component of the vector T xˆ i , here denoted T i j , can be interpreted as the i jth component of a tensor T. Note that T nˆ mˆ is independent of the particular coordinate system used in the derivation. We shall now show how one can use the momentum law (force equation) to derive the equation of motion for an arbitrary element of mass in the body. To this end we consider a part V of the body. If the external force density (force per unit volume) is denoted by f and the velocity for a mass element dm is denoted by v, we obtain d dt

3

2

V v dm = V f d x + S Tnˆ d x

(M.27)

The jth component of this equation can be written d

3

2

3

2

V dt v j dm = V f j d x + S T njˆ d x = V f j d x + S ni Ti j d x

(M.28)

where, in the last step, Equation (M.25) was used. Setting dm = ρ d 3x and using the divergence theorem on the last term, we can rewrite the result as

V

ρ

d v j d3x = dt

V

f j d3x +

V

∂T i j 3 dx ∂xi

(M.29)

Since this formula is valid for any arbitrary volume, we must require that ρ

∂T i j d vj − fj − =0 dt ∂xi

(M.30)

or, equivalently ρ

∂T i j ∂v j + ρv · ∇v j − f j − =0 ∂t ∂xi

(M.31)

Note that ∂v j /∂t is the rate of change with time of the velocity component v j at a fixed point x = (x1 , x1 , x3 ). E ND OF


EXAMPLE

M.1C


174


In 4D, we have three forms of four-tensor fields of rank n. We speak of • a contravariant four-tensor field, denoted Aµ1 µ2 ...µn (xν ), • a covariant four-tensor field, denoted Aµ1 µ2 ...µn (xν ), µ µ ...µ

• a mixed four-tensor field, denoted Aµ1k+12...µnk (xν ). The 4D metric tensor (fundamental tensor) mentioned above is a particularly important four-tensor of rank 2. In covariant component form we shall denote it gµν . This metric tensor determines the relation between an arbitrary contravariant four-vector aµ and its covariant counterpart aµ according to the following rule: def

aµ (xκ ) ≡ gµν aν (xκ )

(M.32)

This rule is often called lowering of index. The raising of index analogue of the index lowering rule is: def

aµ (xκ ) ≡ gµν aν (xκ )

(M.33)

More generally, the following lowering and raising rules hold for arbitrary rank n mixed tensor fields: ν2 ...νk−1 νk κ ν1 ν2 ...νk−1 κ gµk νk Aνν1k+1 νk+2 ...νn (x ) = Aµk νk+1 ...νn (x )

(M.34)

κ ν1 ν2 ...νk−1 µk κ 2 ...νk−1 gµk νk Aνν1k ννk+1 ...νn (x ) = Aνk+1 νk+2 ...νn (x )

(M.35)

Successive lowering and raising of more than one index is achieved by a repeated application of this rule. For example, a dual application of the lowering operation on a rank 2 tensor in contravariant form yields Aµν = gµκ gλν Aκλ

(M.36)

i.e., the same rank 2 tensor in covariant form. This operation is also known as a tensor contraction. E XAMPLE M.2

C ONTRAVARIANT AND COVARIANT VECTORS IN

The 4D Lorentz space


q

4

FLAT

L ORENTZ SPACE

has a simple metric which can be described either by the


M.1

S CALARS , V ECTORS

AND

175

T ENSORS

metric tensor

t

1 if µ = ν = 0 −1 if µ = ν = i = j = 1, 2, 3 0 if µ 6= ν

rs

gµν =

su

(M.37)

or, in matrix notation,

""#

1 0 0 −1 0 0 0 0

(gµν ) =

0 0 0 0 −1 0 0 −1

*& (

(M.38)

(

i.e., a matrix with a main diagonal that has the sign sequence, or signature, {+, −, −, −} or gµν =

t rs su

−1 if µ = ν = 0 1 if µ = ν = i = j = 1, 2, 3 0 if µ = 6 ν

(M.39)

which, in matrix notation, can be represented as

(gµν ) =

""#

−1 0 0 1 0 0 0 0

0 0 1 0

0 0 0 1

*&)(

(M.40)

(

i.e., a matrix with signature {−, +, +, +}.

Consider an arbitrary contravariant four-vector aν in this space. In component form it can be written: def

aν ≡ (a0 , a1 , a2 , a3 ) = (a0 , a)

(M.41)

According to the index lowering rule, Equation (M.32) on the preceding page, we obtain the covariant version of this vector as def

aµ ≡ (a0 , a1 , a2 , a3 ) = gµν aν

(M.42)

In the {+, −, −, −} metric we obtain µ=0: µ=1: µ=2: µ=3:

a 0 = 1 · a 0 + 0 · a 1 + 0 · a 2 + 0 · a 3 = a0 0

1

2

3

(M.43) 1

(M.44)

a2 = 0 · a0 + 0 · a1 − 1 · a2 + 0 · a3 = −a2

(M.45)

a1 = 0 · a − 1 · a + 0 · a + 0 · a = −a 0

1

2

3

a3 = 0 · a + 0 · a + 0 · a + 1 · a = −a

3

(M.46)

or aµ = (a0 , a1 , a2 , a3 ) = (a0 , −a1 , −a2 , −a3 ) = (a0 , −a)

(M.47)

Analogously, using the {+, −, −, −} metric we obtain



176


aµ = (a0 , a1 , a2 , a3 ) = (−a0 , a1 , a2 , a3 ) = (−a0 , a)

(M.48) E ND OF

EXAMPLE

M.2C

M.1.3 Vector algebra Scalar product The scalar product (dot product, inner product) of two arbitrary 3D vectors a and b in ordinary 3 space is the scalar number a · b = xˆ i ai · xˆ j b j = xˆ i · xˆ j ai b j = δi j ai b j = ai bi

(M.49)

where we used the fact that the scalar product xˆ i · xˆ j is a representation of the Kronecker delta δi j defined in Equation (M.16) on page 171. In Russian literature, the scalar product is often denoted (ab). In 4D space we define the scalar product of two arbitrary four-vectors aµ and bµ in the following way aµ bµ = gνµ aν bµ = aν bν = gµν aµ bν

(M.50)

where we made use of the index lowering and raising rules (M.32) and (M.33). The result is a four-scalar, i.e., an invariant which is independent of in which inertial system it is measured. The quadratic differential form ds2 = gµν dxν dxµ = dxµ dxµ

(M.51)

i.e., the scalar product of the differential radius four-vector with itself, is an invariant called the metric. It is also the square of the line element ds which is the distance between neighbouring points with coordinates x µ and xµ + dxµ . E XAMPLE M.3

I NNER PRODUCTS IN COMPLEX VECTOR SPACE

A 3D complex vector A is a vector in 3 (or, if we like, in two real vectors aR and aI in 3 in the following way

def

def

A ≡ aR + iaI = aR aˆ R + iaI aˆ I ≡ A Aˆ ∈

3

6 ),

expressed in terms of (M.52)

The inner product of A with itself may be defined as def

def

A2 ≡ A · A = a2R − a2I + 2iaR · aI ≡ A2 ∈

(M.53)

from which we find that A=

v

a2R − a2I + 2iaR · aI ∈

(M.54)

Using this in Equation (M.52) above, we see that we can interpret this so that the



M.1

S CALARS , V ECTORS

AND

177

T ENSORS

complex unit vector is A Aˆ = = A

=

aR

v

a2R − a2I + 2iaR

aR

v

· aI

aˆ R + i

a2R − a2I − 2iaR · aI a2R + a2I

aI

v

a2R − a2I + 2iaR aI

aˆ R + i

v

· aI

aˆ I

a2R − a2I − 2iaR · aI a2R + a2I

aˆ I ∈

3

(M.55) On the other hand, the definition of the scalar product in terms of the inner product of complex vector with its own complex conjugate yields def

|A|2 ≡ A · A∗ = a2R + a2I = |A|2

(M.56)

with the help of which we can define the unit vector as A = Aˆ = |A| =

aR

v

a2R + a2I

aR

v

aˆ R + i

a2R + a2I

a2R + a2I

aI

v

a2R + a2I

aˆ R + i

aI

v

aˆ I (M.57)

a2R + a2I a2R + a2I

aˆ I ∈

3

E ND OF

S CALAR PRODUCT, NORM AND

METRIC IN

EXAMPLE

M.3C

L ORENTZ SPACE

E XAMPLE M.4

q

In 4 the metric tensor attains a simple form [see Equation (4.10) on page 51 for an example] and, hence, the scalar product in Equation (M.50) on the preceding page can be evaluated almost trivially and becomes aµ bµ = (a0 , −a) · (b0 , b) = a0 b0 − a · b The important scalar product of the

q

4

(M.58)

radius four-vector with itself becomes

xµ xµ = (x0 , −x) · (x0 , x) = (ct, −x) · (ct, x) = (ct)2 − (x1 )2 − (x2 )2 − (x3 )2 = s2

which is the indefinite, real norm of form

q

4.

The

q

(M.59) 4

metric is the quadratic differential

ds2 = dxµ dxµ = c2 (dt)2 − (dx1 )2 − (dx2 )2 − (dx3 )2

(M.60)

E ND OF


EXAMPLE

M.4C


178

E XAMPLE M.5


M ETRIC IN

GENERAL RELATIVITY

In the general theory of relativity, several important problems are treated in a 4D spherical polar coordinate system. Then the radius four-vector can be given as xµ = (ct, r, θ, φ) and the metric tensor is

""#

(gµν ) =

eκ 0 0 0

0 e−λ 0 0

0 0 −r2 0

0 0 0 −r2 sin2 θ

*& (

(M.61)

(

where κ = κ(ct, r, θ, φ) and λ = λ(ct, r, θ, φ). In such a space, the metric takes the form ds2 = c2 eκ (dt)2 − eλ (dr)2 − r2 (dθ)2 − r2 sin2 θ(dφ)2

(M.62)

In general relativity the metric tensor is not given a priori but is determined by the Einstein equations. E ND OF

EXAMPLE

M.5C

Dyadic product The dyadic product field A(x) ≡ a(x)b(x) with two juxtaposed vector fields a(x) and b(x) is the outer product of a and b. Operating on this dyad from the right and from the left with an inner product of an vector c one obtains def

def

def

def

A · c ≡ ab · c ≡ a(b · c)

(M.63a)

c · A ≡ c · ab ≡ (c · a)b

(M.63b)

i.e., new vectors, proportional to a and b, respectively. In mathematics, a dyadic product is often called tensor product and is frequently denoted a ⊗ b. In matrix notation the outer product of a and b is written

3

ab = xˆ 1

xˆ 2

xˆ 3 4

B@

C

a1 b1 a1 b2 a1 b3 a1 b2 a2 b2 a2 b3 F a1 b3 a3 b2 a3 b3

C

B@

xˆ 1 xˆ 2 F xˆ 3

(M.64)

which means that we can represent the tensor A(x) in matrix form as

3 Ai j (xk )

4

C

=

B@

a1 b1 a1 b2 a1 b3 a1 b2 a2 b2 a2 b3 F a1 b3 a3 b2 a3 b3

(M.65)

which we identify with expression (M.15) on page 170, viz. a tensor in matrix notation.



M.1

S CALARS , V ECTORS

AND

179

T ENSORS

Vector product The vector product or cross product of two arbitrary 3D vectors a and b in ordinary 3 space is the vector c = a × b = i jk a j bk xˆ i

(M.66)

Here i jk is the Levi-Civita tensor defined in Equation (M.18) on page 171. Sometimes the vector product of a and b is denoted a ∧ b or, particularly in the Russian literature, [ab]. A spatial reversal of the coordinate system (x10 , x20 , x30 ) = (−x1 , −x2 , −x3 ) changes sign of the components of the vectors a and b so that in the new coordinate system a0 = −a and b0 = −b, which is to say that the direction of an ordinary vector is not dependent on the choice of directions of the coordinate axes. On the other hand, as is seen from Equation (M.66), the cross product vector c does not change sign. Therefore a (or b) is an example of a “true” vector, or polar vector, whereas c is an example of an axial vector, or pseudovector. A prototype for a pseudovector is the angular momentum vector L = r × p and hence the attribute “axial.” Pseudovectors transform as ordinary vectors under translations and proper rotations, but reverse their sign relative to ordinary vectors for any coordinate change involving reflection. Tensors (of any rank) which transform analogously to pseudovectors are called pseudotensors. Scalars are tensors of rank zero, and zero-rank pseudotensors are therefore also called pseudoscalars, an example being the pseudoscalar xˆ i · ( xˆ j × xˆ k ). This triple product is a representation of the i jk component of the Levi-Civita tensor i jk which is a rank three pseudotensor.

M.1.4 Vector analysis The del operator In 3 the del operator is a differential vector operator, denoted in Gibbs’ notation by ∇ and defined as def

∇ ≡ xˆ i

∂ def ≡ ∂ ∂xi

(M.67)

where xˆ i is the ith unit vector in a Cartesian coordinate system. Since the operator in itself has vectorial properties, we denote it with a boldface nabla. In “component” notation we can write ∂i =

∂ ∂ ∂ , , ∂x1 ∂x2 ∂x3


(M.68)


180


In 4D, the contravariant component representation of the four-del operator is defined by ∂ ∂ ∂ ∂ , , , ∂x0 ∂x1 ∂x2 ∂x3

∂µ =

(M.69)

whereas the covariant four-del operator is ∂ ∂ ∂ ∂ , , , ∂x0 ∂x1 ∂x2 ∂x3

∂µ =

(M.70)

We can use this four-del operator to express the transformation properties (M.13) and (M.14) on page 170 as y0µ = ∂ν x0µ 4 yν

(M.71)

y0µ = ∂0µ xν 4 yν

(M.72)

3

and

3

respectively. E XAMPLE M.6

T HE

In

q

4

FOUR - DEL OPERATOR IN

L ORENTZ SPACE

the contravariant form of the four-del operator can be represented as

1∂ , −∂ = c ∂t

∂µ =

1∂ , −∇ c ∂t

(M.73)

and the covariant form as ∂µ =

1∂ ,∂ = c ∂t

1∂ ,∇ c ∂t

(M.74)

Taking the scalar product of these two, one obtains ∂µ ∂µ =

1 ∂2 − ∇2 = c2 ∂t2

j

2

which is the d’Alembert operator, sometimes denoted an opposite sign convention.

j

(M.75) , and sometimes defined with

E ND OF


EXAMPLE

M.6C


M.1

S CALARS , V ECTORS

AND

181

T ENSORS

With the help of the del operator we can define the gradient, divergence and curl of a tensor (in the generalised sense).

The gradient The gradient of an a(x):

3

scalar field α(x), denoted ∇α(x), is an

∇α(x) = ∂α(x) = xˆ i ∂i α(x) = a(x)

3

vector field

(M.76)

From this we see that the boldface notation for the nabla and del operators is very handy as it elucidates the 3D vectorial property of the gradient. In 4D, the four-gradient is a covariant vector, formed as a derivative of a four-scalar field α(xµ ), with the following component form: ∂µ α(xν ) =

∂α(xν ) ∂xµ

(M.77)

G RADIENTS OF SCALAR FUNCTIONS OF

RELATIVE DISTANCES IN

3D

E XAMPLE M.7

Very often electrodynamic quantities are dependent on the relative distance in 3 between two vectors x and x0 , i.e., on |x − x0 |. In analogy with Equation (M.67) on page 179, we can define the “primed” del operator in the following way: ∇0 = xˆ i

∂ = ∂0 ∂x0i

(M.78)

Using this, the “unprimed” version, Equation (M.67) on page 179, and elementary rules of differentiation, we obtain the following two very useful results:

∇ |x − x0 | = xˆ i

∂|x − x0 | x − x0 ∂|x − x0 | = = − x ˆ i ∂xi |x − x0 | ∂x0i

= −∇0 |x − x0 |

(M.79)

and ∇

1 |x − x0 |

=−

x − x0 = −∇0 |x − x0 |3

1 |x − x0 |

(M.80)

E ND OF


EXAMPLE

M.7C


182


The divergence We define the 3D divergence of a vector field in

3

∇ · a(x) = ∂ · xˆ j a j (x) = δi j ∂i a j (x) = ∂i ai (x) =

as

∂ai (x) = α(x) ∂xi

(M.81)

which, as indicated by the notation α(x), is a scalar field in 3 . We may think of the divergence as a scalar product between a vectorial operator and a vector. As is the case for any scalar product, the result of a divergence operation is a scalar. Again we see that the boldface notation for the 3D del operator is very convenient. The four-divergence of a four-vector aµ is the following four-scalar: ∂µ aµ (xν ) = ∂µ aµ (xν ) =

E XAMPLE M.8

∂aµ (xν ) ∂xµ

(M.82)

D IVERGENCE IN 3D

For an arbitrary ∇0 ·

3

vector field a(x0 ), the following relation holds:

a(x0 ) |x − x0 |

=

∇0 · a(x0 ) + a(x0 ) · ∇0 |x − x0 |

1 |x − x0 |

(M.83)

which demonstrates how the “primed” divergence, defined in terms of the “primed” del operator in Equation (M.78) on the preceding page, works. E ND OF

EXAMPLE

M.8C

The Laplacian The 3D Laplace operator or Laplacian can be described as the divergence of the gradient operator: ∇2 = ∆ = ∇ · ∇ =

3 ∂ ∂ ∂2 ∂2 xˆ i · xˆ j = δi j ∂i ∂ j = ∂2i = 2 ≡ ∑ 2 ∂xi ∂x j ∂xi i=1 ∂xi

(M.84)

The symbol ∇2 is sometimes read del squared. If, for a scalar field α(x), ∇2 α < 0 at some point in 3D space, it is a sign of concentration of α at that point.



M.1

S CALARS , V ECTORS

AND

183

T ENSORS

T HE L APLACIAN AND THE D IRAC DELTA

A very useful formula in 3D ∇·∇

1 − |x x0 |

= ∇2

3

E XAMPLE M.9

is

1 − |x x0 |

= −4πδ(x − x0 )

(M.85)

where δ(x − x0 ) is the 3D Dirac delta “function.” E ND OF

EXAMPLE

M.9C

The curl In 3 the curl of a vector field a(x), denoted ∇ × a(x), is another b(x) which can be defined in the following way: ∇ × a(x) = i jk xˆ i ∂ j ak (x) = i jk xˆ i

∂ak (x) = b(x) ∂x j

3

vector field

(M.86)

where use was made of the Levi-Civita tensor, introduced in Equation (M.18) on page 171. The covariant 4D generalisation of the curl of a four-vector field aµ (xν ) is the antisymmetric four-tensor field Gµν (xκ ) = ∂µ aν (xκ ) − ∂ν aµ (xκ ) = −Gνµ (xκ )

(M.87)

A vector with vanishing curl is said to be irrotational.

T HE CURL OF

E XAMPLE M.10

A GRADIENT

Using the definition of the 3 curl, Equation (M.86), and the gradient, Equation (M.76) on page 181, we see that ∇ × [∇α(x)] = i jk xˆ i ∂ j ∂k α(x)

(M.88)

which, due to the assumed well-behavedness of α(x), vanishes: i jk xˆ i ∂ j ∂k α(x) = i jk =

∂ ∂ α(x) xˆ i ∂x j ∂xk

∂2 ∂2 − ∂x2 ∂x3 ∂x3 ∂x2

+

∂2 ∂2 − ∂x3 ∂x1 ∂x1 ∂x3

α(x) xˆ 2

+

∂2 ∂2 − ∂x1 ∂x2 ∂x2 ∂x1

α(x) xˆ 3

α(x) xˆ 1 (M.89)

≡0



184


We thus find that ∇ × [∇α(x)] ≡ 0

(M.90)

for any arbitrary, well-behaved

3

scalar field α(x).

In 4D we note that for any well-behaved four-scalar field α(xκ ) (∂µ ∂ν − ∂ν ∂µ )α(xκ ) ≡ 0

(M.91)

so that the four-curl of a four-gradient vanishes just as does a curl of a gradient in

3.

Hence, a gradient is always irrotational. E ND OF

E XAMPLE M.11

T HE

EXAMPLE

M.10C

DIVERGENCE OF A CURL

With the use of the definitions of the divergence (M.81) and the curl, Equation (M.86) on the preceding page, we find that ∇ · [∇ × a(x)] = ∂i [∇ × a(x)]i = i jk ∂i ∂ j ak (x)

(M.92)

Using the definition for the Levi-Civita symbol, defined by Equation (M.18) on page 171, we find that, due to the assumed well-behavedness of a(x), ∂i i jk ∂ j ak (x) = =

∂ ∂ i jk ak ∂xi ∂x j

∂2 ∂2 − ∂x2 ∂x3 ∂x3 ∂x2

+

∂2 ∂2 − ∂x3 ∂x1 ∂x1 ∂x3

a2 (x)

+

∂2 ∂2 − ∂x1 ∂x2 ∂x2 ∂x1

a3 (x)

a1 (x) (M.93)

≡0 i.e., that ∇ · [∇ × a(x)] ≡ 0

(M.94)

for any arbitrary, well-behaved

3

vector field a(x).

In 4D, the four-divergence of the four-curl is not zero, for ∂ν Gµν = ∂µ ∂ν aν (xκ ) −

j

2 µ

a (xκ ) 6= 0

(M.95)

E ND OF


EXAMPLE

M.11C


M.2

185

A NALYTICAL M ECHANICS

Numerous vector algebra and vector analysis formulae are given in Chapter F. Those which are not found there can often be easily derived by using the component forms of the vectors and tensors, together with the Kronecker and Levi-Civita tensors and their generalisations to higher ranks. A short but very useful reference in this respect is the article by A. Evett [3].

M.2 Analytical Mechanics M.2.1 Lagrange’s equations As is well known from elementary analytical mechanics, the Lagrange function or Lagrangian L is given by L(qi , q˙ i , t) = L qi ,

dqi ,t = T −V dt

(M.96)

where qi is the generalised coordinate, T the kinetic energy and V the potential energy of a mechanical system, The Lagrangian satisfies the Lagrange equations ∂ ∂t

∂L ∂L − =0 ∂q˙ i ∂qi

(M.97)

To the generalised coordinate qi one defines a canonically conjugate momentum pi according to pi =

∂L ∂q˙ i

(M.98)

and note from Equation (M.97) that ∂L = p˙ i ∂qi

(M.99)

M.2.2 Hamilton’s equations From L, the Hamiltonian (Hamilton function) H can be defined via the Legendre transformation H(pi , qi , t) = pi q˙ i − L(qi , q˙ i , t)

(M.100)

After differentiating the left and right hand sides of this definition and setting them equal we obtain ∂H ∂H ∂H ∂L ∂L ∂L dpi + dqi + dt = q˙ i dpi + pi dq˙ i − dqi − dq˙ i − dt (M.101) ∂pi ∂qi ∂t ∂qi ∂q˙ i ∂t



186


According to the definition of pi , Equation (M.98) on the previous page, the second and fourth terms on the right hand side cancel. Furthermore, noting that according to Equation (M.99) on the preceding page the third term on the right hand side of Equation (M.101) on the previous page is equal to − p˙ i dqi and identifying terms, we obtain the Hamilton equations: ∂H dqi = q˙ i = ∂pi dt ∂H dpi = − p˙ i = − ∂qi dt

(M.102a) (M.102b)

Bibliography [1] G. B. A RFKEN AND H. J. W EBER, Mathematical Methods for Physicists, fourth, international ed., Academic Press, Inc., San Diego, CA . . . , 1995, ISBN 0-12059816-7. [2] R. A. D EAN, Elements of Abstract Algebra, John Wiley & Sons, Inc., New York, NY . . . , 1967, ISBN 0-471-20452-8. [3] A. A. E VETT, Permutation symbol approach to elementary vector analysis, American Journal of Physics, 34 (1965), pp. 503–507. [4] P. M. M ORSE AND H. F ESHBACH, Methods of Theoretical Physics, Part I. McGraw-Hill Book Company, Inc., New York, NY . . . , 1953, ISBN 07-043316-8. [5] B. S PAIN, Tensor Calculus, third ed., Oliver and Boyd, Ltd., Edinburgh and London, 1965, ISBN 05-001331-9. [6] W. E. T HIRRING, Classical Mathematical Physics, Springer-Verlag, New York, Vienna, 1997, ISBN 0-387-94843-0.



Index acceleration field, 129 advanced time, 43 Ampère’s law, 5 Ampère-turn density, 89 anisotropic, 152 anomalous dispersion, 153 antisymmetric tensor, 63 associated Legendre polynomial, 117 associative, 55 axial gauge, 40 axial vector, 63, 179

contravariant vector, 50 convection potential, 136 convective derivative, 11 cosine integral, 110 Coulomb gauge, 40 Coulomb’s law, 2 covariant, 48 covariant component form, 168 covariant field tensor, 64 covariant four-tensor field, 174 covariant four-vector, 170 covariant four-vector field, 54 covariant vector, 50 cross product, 179 curl, 183 cutoff, 142 cyclotron radiation, 145, 148

Bessel functions, 113 Biot-Savart’s law, 6 birefringent, 152 braking radiation, 141 bremsstrahlung, 141, 148 canonically conjugate four-momentum, 70 canonically conjugate momentum, 70, 185 canonically conjugate momentum density, 78 characteristic impedance, 26 classical electrodynamics, 8 closed algebraic structure, 55 coherent radiation, 148 collisional interaction, 151 complex field six-vector, 20 complex notation, 32 complex vector, 176 component notation, 168 concentration, 182 conservative field, 11 conservative forces, 75 constitutive relations, 14 contravariant component form, 50, 168 contravariant field tensor, 63 contravariant four-tensor field, 174 contravariant four-vector, 170 contravariant four-vector field, 54

d’Alembert operator, 39, 59, 180 del operator, 179 del squared, 182 differential distance, 52 differential vector operator, 179 Dirac delta, 183 Dirac’s symmetrised Maxwell equations, 16 dispersive, 153 displacement current, 9 divergence, 182 dot product, 176 dual vector, 50 duality transformation, 16 dummy index, 50 dyadic product, 178 dyons, 20 E1 radiation, 120 E2 radiation, 122 Einstein equations, 178 Einstein’s summation convention, 168 electric charge conservation law, 9 electric charge density, 4

187

188


electric conductivity, 10 field Lagrange density, 79 electric current density, 6 field point, 3 electric dipole moment, 118 field quantum, 143 electric dipole moment vector, 85 fine structure constant, 143, 150 electric dipole radiation, 120 four-current, 59 electric displacement, 14 four-del operator, 180 electric displacement vector, 87 four-dimensional Hamilton equations, electric field, 2 71 electric field energy, 91 four-dimensional vector space, 50 electric monopole moment, 85 four-divergence, 182 electric permittivity, 151 four-gradient, 181 electric polarisation, 86 four-Hamiltonian, 71 electric quadrupole moment tensor, 86 four-Lagrangian, 68 electric quadrupole radiation, 122 four-momentum, 57 electric quadrupole tensor, 122 four-potential, 59 electric susceptibility, 87 four-scalar, 169 electric volume force, 92 four-tensor fields, 174 electrodynamic potentials, 36 four-vector, 54, 170 electromagnetic field tensor, 63 four-velocity, 57 electromagnetic scalar potential, 37 Fourier component, 25 electromagnetic vector potential, 36 Fourier transform, 41 electromagnetodynamic equations, 16 functional derivative, 76 electromagnetodynamics, 16 fundamental tensor, 50, 168, 174 electromotive force (EMF), 10 Galileo’s law, 47 electrostatic scalar potential, 35 gauge fixing, 40 electrostatics, 1 gauge function, 39 energy theorem in Maxwell’s theory, gauge invariant, 39 91 gauge transformation, 39 equation of continuity, 9, 59 equation of continuity for magnetic monopoles, Gauss’s law, 4 general inhomogeneous wave equations, 16 37 equations of classical electrostatics, 8 generalised coordinate, 70, 185 equations of classical magnetostatics, generalised four-coordinate, 71 8 Gibbs’ notation, 179 Euclidean space, 56 gradient, 181 Euclidean vector space, 51 Green function, 42, 116 Euler-Lagrange equation, 77 group theory, 55 Euler-Lagrange equations, 77 group velocity, 153 Euler-Mascheroni constant, 110 event, 55 Hamilton density, 78 Hamilton density equations, 78 far field, 98 far zone, 101 Hamilton equations, 70, 186 Faraday’s law, 11 Hamilton function, 185 field, 169 Hamilton gauge, 40



189

Hamiltonian, 185 Heaviside potential, 136 Helmholtz’ theorem, 38 help vector, 115 Hertz’ method, 114 Hertz’ vector, 115 Hodge star operator, 16 homogeneous wave equation, 24 Huygen’s principle, 41 identity element, 55 in a medium, 154 incoherent radiation, 148 indefinite norm, 51 index contraction, 50 index lowering, 50 induction field, 98 inertial reference frame, 47 inertial system, 47 inhomogeneous Helmholtz equation, 41 inhomogeneous time-independent wave equation, 41 inhomogeneous wave equation, 41 inner product, 176 instantaneous, 138 interaction Lagrange density, 79 intermediate field, 101 invariant, 169 invariant line element, 53 inverse element, 55 irrotational, 4, 183 Kelvin function, 149 kinetic energy, 75, 185 kinetic momentum, 74 Kronecker delta, 171 Lagrange density, 75 Lagrange equations, 185 Lagrange function, 74, 185 Lagrangian, 74, 185 Laplace operator, 182 Laplacian, 182 Larmor formula for radiated power, 138 law of inertia, 47


Legendre polynomial, 116 Legendre transformation, 185 Levi-Civita tensor, 171 Liénard-Wiechert potentials, 62, 125, 135 light cone, 53 light-like interval, 53 line element, 176 linear mass density, 75 linearly polarised wave, 30 longitudinal component, 28 Lorentz boost parameter, 57 Lorentz force, 13, 91, 136 Lorentz gauge, 40 Lorentz gauge condition, 38, 60 Lorentz inhomogeneous wave equations, 39 Lorentz potential equations, 39 Lorentz space, 51, 168 Lorentz transformation, 49, 135 lowering of index, 174 M1 radiation, 121 Mach cone, 156 macroscopic Maxwell equations, 151 magnetic charge density, 15 magnetic current density, 15 magnetic dipole moment, 88, 121 magnetic dipole radiation, 121 magnetic field, 6 magnetic field energy, 91 magnetic field intensity, 89 magnetic flux, 11 magnetic flux density, 6 magnetic induction, 6 magnetic monopoles, 15 magnetic permeability, 151 magnetic susceptibility, 89 magnetisation, 88 magnetisation currents, 88 magnetising field, 14, 89 magnetostatic vector potential, 36 magnetostatics, 5 massive photons, 83 mathematical group, 55


190


matrix form, 170 Maxwell stress tensor, 92 Maxwell’s macroscopic equations, 15, 90 Maxwell’s microscopic equations, 14 Maxwell-Lorentz equations, 14 mechanical Lagrange density, 79 metric, 168, 176 metric tensor, 50, 168, 174 Minkowski equation, 70 Minkowski space, 56 mixed four-tensor field, 174 mixing angle, 16 momentum theorem in Maxwell’s theory, 93 monochromatic, 95 multipole expansion, 114, 117 near zone, 101 Newton’s first law, 47 Newton-Lorentz force equation, 70 non-Euclidean space, 51 non-linear effects, 10 norm, 50, 177 null vector, 53 observation point, 3 Ohm’s law, 10 one-dimensional wave equation, 29 outer product, 178 Parseval’s identity, 105, 142, 150 phase velocity, 151 photon, 143 physical measurable, 32 plane polarised wave, 30 plasma, 153 plasma frequency, 153 Poisson equation, 135 Poisson’s equation, 35 polar vector, 63, 179 polarisation charges, 87 polarisation currents, 88 polarisation potential, 115 polarisation vector, 115


positive definite, 56 positive definite norm, 51 potential energy, 75, 185 potential theory, 116 power flux, 91 Poynting vector, 91 Poynting’s theorem, 91 Proca Lagrangian, 82 propagator, 42 proper time, 53 pseudo-Riemannian space, 56 pseudoscalar, 167 pseudoscalars, 179 pseudotensor, 167 pseudotensors, 179 pseudovector, 63, 167, 179 quadratic differential form, 52, 176 quantum mechanical nonlinearity, 4 radiation field, 98, 101, 129 radiation fields, 101 radiation gauge, 40 radiation resistance, 110 radius four-vector, 50 radius vector, 167 raising of index, 174 rank, 170 rapidity, 57 refractive index, 152 relative electric permittivity, 92 relative magnetic permeability, 92 relative permeability, 151 relative permittivity, 151 Relativity principle, 48 relaxation time, 25 rest mass density, 79 retarded Coulomb field, 101 retarded potentials, 44 retarded relative distance, 125 retarded time, 43 Riemannian metric, 52 Riemannian space, 50, 168 row vector, 167 scalar, 167, 182


191

scalar field, 54, 169 scalar product, 176 shock front, 156 signature, 51, 175 simultaneous coordinate, 133 skew-symmetric, 63 skin depth, 32 source point, 3 space components, 51 space-like interval, 53 space-time, 51 special theory of relativity, 47 spherical Bessel function of the first kind, 116 spherical Hankel function of the first kind, 116 spherical waves, 104 standing wave, 108 super-potential, 115 synchrotron radiation, 145, 148 synchrotron radiation lobe width, 147

vacuum polarisation effects, 4 vacuum wave number, 26 ˇ Vavilov-Cerenkov radiation, 154, 156 vector, 167 vector product, 179 velocity field, 129 virtual simultaneous coordinate, 126, 129 wave equations, 23 wave vector, 29, 152 world line, 55 Young’s modulus, 75 Yukawa meson field, 82

telegrapher’s equation, 29, 151 temporal dispersive media, 10 temporal gauge, 40 tensor, 167 tensor contraction, 174 tensor field, 170 tensor notation, 171 tensor product, 178 three-dimensional functional derivative, 77 time component, 51 time-harmonic wave, 25 time-independent diffusion equation, 26 time-independent telegrapher’s equation, 30 time-independent wave equation, 26 time-like interval, 53 total charge, 85 transverse components, 29 transverse gauge, 40 vacuum permeability, 5 vacuum permittivity, 2



Companion volume to

E LECTROMAGNETIC F IELD T HEORY by Bo Thidé

E LECTROMAGNETIC F IELD T HEORY Exercises Please note that this is a VERY preliminary draft! Tobia Carozzi Anders Eriksson Bengt Lundborg Bo Thidé Mattias Waldenvik Department of Space and Plasma Physics Uppsala University and Swedish Institute of Space Physics Uppsala Division Sweden

Σ

Ipsum

This book was typeset in LATEX 2ε on an HP9000/700 series workstation and printed on an HP LaserJet 5000GN printer. Copyright c 1998 by Bo Thidé Uppsala, Sweden All rights reserved. Electromagnetic Field Theory Exercises ISBN X-XXX-XXXXX-X

C ONTENTS

Preface

vii

1 Maxwell’s Equations 1.1 1.2 1.3

Coverage . . . . . . . . . . . . . . . . . . . . . . Formulae used . . . . . . . . . . . . . . . . . . . . Solved examples . . . . . . . . . . . . . . . . . . Example 1.1 Macroscopic Maxwell equations . Solution . . . . . . . . . . . . . . . . . . .

1 . . . . .

. . . . .

. . . . .

Example 1.2 Maxwell’s equations in component form

Solution . . . . . . . . . . . . . . . . . . . . . . Example 1.3 The charge continuity equation . . . . Solution . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

2 Electromagnetic Potentials and Waves 2.1 2.2 2.3

Coverage . . . . . . . . . . . . . . . . . . . . . . Formulae used . . . . . . . . . . . . . . . . . . . . Solved examples . . . . . . . . . . . . . . . . . . Example 2.1 The Aharonov-Bohm effect . . . Solution . . . . . . . . . . . . . . . . . . . Example 2.2 Invent your own gauge . . . . . Solution . . . . . . . . . . . . . . . . . . .

9 . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

Example 2.3 Fourier transform of Maxwell’s equations

Solution . . . . . . . . . . . . . . . . . . . . . . . Example 2.4 Simple dispersion relation . . . . . . . .

Solution . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

3 Relativistic Electrodynamics 3.1

Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Draft version released 15th November 2000 at 20:39

1 1 1 1 2 4 4 5 5

9 9 9 9 10 11 11 13 13 15 15

17 17 i

ii

3.2 3.3

Formulae used . . . . . . . . . . . . . . . . . . . . . . Solved examples . . . . . . . . . . . . . . . . . . . . Example 3.1 Covariance of Maxwell’s equations . Solution . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

17 18 18 18 Example 3.2 Invariant quantities constructed from the field tensor 20 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Example 3.3 Covariant formulation of common electrodynamics formulas . . . . . . . . . . . . . . . . . . . .

Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3.4 Fields from uniformly moving charge via Lorentz transformation . . . . . . . . . . . . . . . . . . .

Solution . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Lagrangian and Hamiltonian Electrodynamics 4.1 4.2 4.3

Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4.1 Canonical quantities for a particle in an EM field . Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4.2 Gauge invariance of the Lagrangian density . . . Solution . . . . . . . . . . . . . . . . . . . . . . . . . . .

Coverage . . . . . . . . . . . . . . . . . . . Formulae used . . . . . . . . . . . . . . . . . Solved examples . . . . . . . . . . . . . . . Example 5.1 EM quantities potpourri . . Solution . . . . . . . . . . . . . . . . Example 5.2 Classical electron radius . Solution . . . . . . . . . . . . . . . . Example 5.3 Solar sailing . . . . . . . Solution . . . . . . . . . . . . . . . .

27 27 28 28 28 29 29

31 . . . . . . . . .

. . . . . . . . .

. . . . . . . . . Example 5.4 Magnetic pressure on the earth . Solution . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

6 Radiation from Extended Sources 6.1 6.2 6.3

23 23

27

5 Electromagnetic Energy, Momentum and Stress 5.1 5.2 5.3

21 21

Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 32 32 32 35 35 37 37 39 39

41

41 41 42 Example 6.1 Instantaneous current in an infinitely long conductor 42 Draft version released 15th November 2000 at 20:39

iii

Solution . . . . . . . . . . . . . . . . . Example 6.2 Multiple half-wave antenna . Solution . . . . . . . . . . . . . . . . . Example 6.3 Travelling wave antenna . . . Solution . . . . . . . . . . . . . . . . . Example 6.4 Microwave link design . . . Solution . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

7 Multipole Radiation 7.1 7.2 7.3

Coverage . . . . . . . . . . . . . . . . . . . Formulae used . . . . . . . . . . . . . . . . . Solved examples . . . . . . . . . . . . . . . Example 7.1 Rotating Electric Dipole . Solution . . . . . . . . . . . . . . . . Example 7.2 Rotating multipole . . . . Solution . . . . . . . . . . . . . . . . Example 7.3 Atomic radiation . . . . . Solution . . . . . . . . . . . . . . . . Example 7.4 Classical Positronium . . . Solution . . . . . . . . . . . . . . . .

53 . . . . . . . . . . .

8 Radiation from Moving Point Charges 8.1 8.2 8.3

Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8.2 Synchrotron radiation perpendicular to the acceleration . . . . . . . . . . . . . . . . . . . . . .

Solution . . . . . . . . . . . . . . . . . Example 8.3 The Larmor formula . . . . Solution . . . . . . . . . . . . . . . . . ˇ Example 8.4 Vavilov-Cerenkov emission . Solution . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

9 Radiation from Accelerated Particles Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulae used . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solved examples . . . . . . . . . . . . . . . . . . . . . . . . . . Draft version released 15th November 2000 at 20:39

53 53 54 54 54 56 56 58 58 59 59

63

Example 8.1 Poynting vector from a charge in uniform motion

9.1 9.2 9.3

42 47 47 50 50 51 51

63 63 64 64 64 66 66 67 67 69 69

71 71 71 72

Example 9.1 Motion of charged particles in homogeneous static EM fields . . . . . . . . . . . . . . . . . . . . .

Solution . . Example 9.2



Solution . .

72 . . . . . . . . . . . . . . . . . . . . . . . . . 72 Radiative reaction force from conservation of energy 74 . . . . . . . . . . . . . . . . . . . . . . . . . 74 Radiation and particle energy in a synchrotron . . 77 . . . . . . . . . . . . . . . . . . . . . . . . . 77 Radiation loss of an accelerated charged particle . 79 . . . . . . . . . . . . . . . . . . . . . . . . . 79


iv

L IST

6.1 6.2 6.3

The turn-on of a linear current at t 0 . . . . . . . . . . . . . . Snapshots of the field . . . . . . . . . . . . . . . . . . . . . . . . Multiple half-wave antenna standing current . . . . . . . . . . . .

43 44 47

9.1

Motion of a charge in an electric and a magnetic field . . . . . . .

74


v

OF

F IGURES


vi

P REFACE

This is a companion volume to the book Electromagnetic Field Theory by Bo Thidé. The problems and their solutions were created by the co-authors who all have taught this course or its predecessor. It should be noted that this is a preliminary draft version but it is being corrected and expanded with time. Uppsala, Sweden December, 1999


B. T.

vii

viii

P REFACE


L ESSON 1

Maxwell’s Equations 1.1

Coverage

In this lesson we examine Maxwell’s equations, the cornerstone of electrodynamics. We start by practising our math skill, refreshing our knowledge of vector analysis in vector form and in component form.

1.2

Formulae used E ρ ε0 B 0 ∂ E

B

1.3

∂t

B

µ0 j

(1.1a) (1.1b) (1.1c) 1 ∂ E c2 ∂ t

(1.1d)

Solved examples

M ACROSCOPIC M AXWELL EQUATIONS

E XAMPLE 1.1

The most fundamental form of Maxwell’s equations is

E ρ ε0 B 0 ∂ B E

∂t

B

µ0 j

(1.2a) (1.2b) (1.2c) 1 ∂ E c2 ∂ t


(1.2d)

1

2

L ESSON 1. M AXWELL’ S E QUATIONS

sometimes known as the microscopic Maxwell equations or the Maxwell-Lorentz equations. In the presence of a medium, these equations are still true, but it may sometimes be convenient to separate the sources of the fields (the charge and current densities) into an induced part, due to the response of the medium to the electromagnetic fields, and an extraneous, due to “free” charges and currents not caused by the material properties. One then writes j ρ

jind jext ρind ρext

(1.3) (1.4)

The electric and magnetic properties of the material are often described by the electric polarisation P (SI unit: C/m2 ) and the magnetisation M (SI unit: A/m). In terms of these, the induced sources are described by jind ρind

∂ P ∂ t P

M

(1.5) (1.6)

To fully describe a certain situation, one also needs constitutive relations telling how P and M depends on E and B. These are generally empirical relations, different for different media. Show that by introducing the fields

ε0 E P B µ0 M

D H

(1.7) (1.8)

the two Maxwell equations containing source terms (1.2a) and (??) reduce to

D

ρext

H

jext

(1.9)

∂ D ∂t

(1.10) (1.11)

known as the macroscopic Maxwell equations.

Solution If we insert j ρ

jind jext ρind ρext

(1.12) (1.13)

and


1.3. S OLVED

jind

ρind

3

EXAMPLES

∂ M P ∂ t P

(1.14) (1.15)

into

B

µ0 j

E

ρ ε0

1 ∂ E c2 ∂ t

(1.16) (1.17) (1.18)

we get

∂

µ0 jext P ∂t 1 E P ρext B

M

1 ∂ E c2 ∂ t

(1.19) (1.20)

ε0

which can be rewritten as

B

M!" µ0 ε0 E P #

jext

∂ P ε0 E ∂t

(1.21)

ρext

(1.22)

Now by introducing the D and the H fields such that D

ε0 E P

(1.23)

H

B M µ0

(1.24)

the Maxwell equations become

∂ H jext D ∂t D ρext

(1.25) (1.26) QED $

The reason these equations are known as “macroscopic” are that the material properties described by P and M generally are average quantities, not considering the atomic properties of matter. Thus E and D get the character of averages, not including details around single atoms etc. However, there is nothing in principle preventing us from using largescale averages of E and B, or even to use atomic-scale calculated D and H although this is a rather useless procedure, so the nomenclature “microscopic/macroscopic” is somewhat misleading. The inherent difference lies in how a material is treated, not in the spatial scales. E ND


OF EXAMPLE

1.1 %

4

E XAMPLE 1.2


M AXWELL’ S EQUATIONS IN COMPONENT FORM Express Maxwell’s equations in component form.

Solution Maxwell’s equations in vector form are written:

E ρ ε0 B 0 ∂ E B

∂t

µ0 j

B

(1.27) (1.28) (1.29) 1 ∂ E c2 ∂ t

(1.30)

In these equations, E, B, and j are vectors, while ρ is a scalar. Even though all the equations contain vectors, only the latter pair are true vector equations in the sense that the equations themselves have several components. When going to component notation, all scalar quantities are of course left as they are. Vector quantities, for example E, can always be expanded as E ∑3j & 1 E j xˆ j E j xˆ j , where the last step assumes Einstein’s summation convention: if an index appears twice in the same term, it is to be summed over. Such an index is called a summation index. Indices which only appear once are known as free indices, and are not to be summed over. What symbol is used for a summation index is immaterial: it is always true that a i bi ak bk , since both these expressions mean a1 b1 a2 b2 a3 b3 a b. On the other hand, the expression ai ak is in general not true or even meaningful, unless i k or if a is the null vector. The three E j are the components of the vector E in the coordinate system set by the three unit vectors xˆ j . The E j are real numbers, while the xˆ j are vectors, i.e. geometrical objects. Remember that though they are real numbers, the E j are not scalars. Vector equations are transformed into component form by forming the scalar product of both sides with the same unit vector. Let us go into ridiculous detail in a very simple case:

G

G xˆ k G j xˆ j xˆ k G j δ jk Gk

H

H xˆ k Hi xˆ i xˆ k Hi δik Hk

(1.31) (1.32) (1.33) (1.34) (1.35)

This is of course unnecessarily tedious algebra for an obvious result, but by using this careful procedure, we are certain to get the correct answer: the free index in the resulting equation necessarily comes out the same on both sides. Even if one does not follow this complicated way always, one should to some extent at least think in those terms. Nabla operations are translated into component form as follows:


1.3. S OLVED

5

EXAMPLES

∂ ' Φ( ∂ xi ∂ V ' V ( ∂ xi i ∂ ' V εi jk xˆ i V ( ∂xj k ∇φ

xˆ i

∂φ ∂ xi ∂ Vi ∂ xi ∂V εi jk k ∂xj

(1.36) (1.37) (1.38)

where V is a vector field and φ is a scalar field. Remember that in vector valued equations such as Ampère’s and Faraday’s laws, one must be careful to make sure that the free index on the left hand side of the equation is the same as the free index on the right hand side of the equation. As said above, an equation of the form Ai B j is almost invariably in error! With these things in mind we can now write Maxwell’s equations as

E

µ0 j

B

ρ )' ε0

B

0 )'

E

*

∂B )' ∂t

1 ∂E c2 ∂ t

)'

∂ Ei ρ ∂ xi ε0 ∂ Bi 0 ∂ xi ∂E ∂ εi jk k B ∂xj ∂t i εi jk

∂ Bk ∂xj

µ 0 ji

(1.39) (1.40) (1.41) 1 ∂ Ei c2 ∂ t

(1.42)

E ND

OF EXAMPLE

1.2 %

T HE CHARGE CONTINUITY EQUATION

E XAMPLE 1.3

Derive the continuity equation for charge density ρ from Maxwell’s equations using (a) vector notation and (b) component notation. Compare the usefulness of the two systems of notations. Also, discuss the physical meaning of the charge continuity equation.

Solution Vector notation In vector notation, a derivation of the continuity equation for charge looks like this: Compute

1. Apply

∂ ∂t E ∂ ∂t

in two ways:

to Gauss’s law:

∂ E + ∂t

1 ∂ ρ ε0 ∂ t Draft version released 15th November 2000 at 20:39

(1.43)

6


2. Take the divergence of the Ampère-Maxwell law:

1 ∂ B + µ0 j E c2 ∂t Use

(1.44)

, +- 0 and µ0 ε0 c2 1: .

∂

∂t

E

1 j ε0

(1.45)

Comparison shows that

∂ ρ j 0/ ∂t

(1.46)

Component notation In component notation, a derivation of the continuity equation for charge looks like this: Compute

∂ ∂ ∂ x i ∂ t Ei

1. Take

∂ ∂t

in two ways:

of Gauss’s law:

∂ ∂ Ei ∂ t ∂ xi

1 ∂ ρ ε0 ∂ t

(1.47)

2. Take the divergence of the Ampère-Maxwell law:

∂ Bk ∂ ε ∂ xi 0 i jk ∂ x j 1

µ0

Use that the relation εi jk Ai A j

('

∂ j ∂ xi i

1 ∂ ∂ Ei c2 ∂ x i ∂ t

- 0 is valid also if Ai

(1.48)

∂ ∂ xi ,

∂ ∂ Ei 1 ∂ j ∂ t ∂ xi ε0 ∂ x i i

and that µ0 ε0 c2

1: (1.49)

Comparison shows that

∂ ρ ∂t

∂ ji 0/ ∂ xi Draft version released 15th November 2000 at 20:39

(1.50)

1.3. S OLVED

7

EXAMPLES

Comparing the two notation systems We notice a few points in the derivations above:

2 It is sometimes difficult to see what one is calculating in the component system. The vector system with div, curl etc. may be closer to the physics, or at least to our picture of it.

2 In the vector notation system, we sometimes need to keep some vector formulas in memory or to consult a math handbook, while with the component system you need only the definitions of εi jk and δi j .

2 Although not seen here, the component system of notation is more explicit (read unambiguous) when dealing with tensors of higher rank, for which vector notation becomes cumbersome.

2 The vector notation system is independent of coordinate system, i.e., ∇φ is ∇φ in any coordinate system, while in the component notation, the components depend on the unit vectors chosen.

Interpreting the continuity equation The equation

∂ ρ j 0 (1.51) ∂t is known as a continuity equation. Why? Well, integrate the continuity equation over some volume V bounded by the surface S. By using Gauss’s theorem, we find that 3 dQ ∂ 43 ρ d3x 53 j d x 63 j dS (1.52) dt ∂ t V V S which says that the change in the total charge in the volume is due to the net inflow of electric current through the boundary surface S. Hence, the continuity equation is the field theory formulation of the physical law of charge conservation. E ND


OF EXAMPLE

1.3 %


8

L ESSON 2

Electromagnetic Potentials and Waves 2.1

Coverage

Here we study the vector and scalar potentials A and φ and the concept of gauge transformation. One of the most important physical manifestation of Maxwell’s equations is the EM wave. Seen as wave equations, the Maxwell equations can be reduced to algebraic equations via the Fourier transform and the physics is contained in so-called dispersion relations which set the kinematic restrictions on the fields.

2.2

Formulae used E ∇φ

7

B

2.3

∂A ∂t

A

Solved examples

T HE A HARONOV-B OHM EFFECT

E XAMPLE 2.1

Consider the magnetic field given in cylindrical coordinates, B r B r

8 r0 9 θ 9 z : Bˆz ; r0 9 θ 9 z : 0


(2.1) (2.2)

9

10

L ESSON 2. E LECTROMAGNETIC P OTENTIALS

AND

WAVES

Determine the vector potential A that “generated” this magnetic field.

Solution A interesting question in electrodynamics is whether the EM potentials φ and A are more than mathematical tools, and alternatives to the Maxwell equations, from which we can derive the EM fields. Could it be that the potentials and not Maxwell’s equations are more fundamental? Although the ultimate answer to these questions is somewhat metaphysical, it is exactly these questions that make the Aharonov-Bohm effect. Before we discuss this effect let us calculate the vector field from the given magnetic field. The equations connecting the potentials with the fields are

∂A φ ∂t

E B

(2.3)

A

(2.4)

In this problem we see that we have no boundary conditions for the potentials. Also, let us use the gauge φ 0. This problem naturally divides into two parts: the part within the magnetic field and the part outside the magnetic field. Let us start with the interior part:

1 r

1 ∂ Az r ∂θ ∂ Ar ∂z ∂ rAθ ∂r

∂A ∂t ∂ Aθ ∂z ∂ Az ∂r ∂ Ar ! ∂θ

0

(2.5a)

0

(2.5b)

0

(2.5c)

B

(2.5d)

The first equation tells us that A is time independent so A A r9 θ 9 z . Examining the other three we find that there is no dependence on θ or z either so A A r . All that remains is 1 ∂ rAθ B (2.6) r ∂r Integrating this equation we find that Aθ

Br 2

(2.7)

Moving to the outer problem, we see that the only difference compared with the inner problem is that B 0 so that we must consider 1 ∂ rAθ r ∂r

0

(2.8)


2.3. S OLVED

11

EXAMPLES

This time integration leads to C (2.9) r If we demand continuity for the function Aθ over all space we find by comparing with (2.7) the arbitrary constant C and can write in outer solution as Aθ

Aθ

Br02 2r

0! <

(2.10)

Now in electrodynamics (read: in this course) the only measurable quantities are the fields. So the situation above, where we have a region in which the magnetic field is zero but the potential is non-zero has no measurable consequence in classical electrodynamics. In quantum mechanics however, the Aharonov-Bohm effect shows that this situation does have a measurable consequence. Namely, when letting charged particles go around this magnetic field (the particles are do not enter the magnetic field region because of a impenetrable wall) the energy spectrum of the particles after passing the cylinder will have changed, even though there is no magnetic field along their path. The interpretation is that the potential is a more fundamental quantity than the field. E ND

OF EXAMPLE

2.1 %

I NVENT YOUR OWN GAUGE

E XAMPLE 2.2

Name some common gauge conditions and discuss the usefulness of each one. Then invent your own gauge and verify that it is indeed a gauge condition.

Solution Background The Maxwell equations that do not contain source terms can be “solved” by using the vector potential A and the scalar potential φ , defined through the relations B E

A

∇φ

(2.11)

∂ A ∂t

(2.12)

Assuming linear, isotropic and homogeneous media, we can use the constitutive relations D ε E H B µ , and j σ E j= (where j= is the free current density arising from other sources than conductivity) and definitions of the scalar and vector potentials in the remaining two Maxwell equations and find that ∇2 φ ∇2 A

∂ A ∂t

µσ

ρ ε

( ∂ ∂ 2A ∂ A µε φ µσ φ !> µ j= A µε 2 ∇ ∂t ∂t ∂t Draft version released 15th November 2000 at 20:39

(2.13) (2.14)

12


AND

WAVES

These equations are used to determine A and φ from the source terms. And once we have found A and φ it is straight forward to derive the E and B fields from (2.11) and (2.12). The definitions of the scalar and vector potentials are not enough to make A and φ unique, i.e. , if one is given A and φ then (2.11) and (2.12) determine B and E, but if one is given B and E there many ways of choosing A and φ . This can be seen through the fact that A and φ can be transformed according to A=

A ∇ψ

φ=

(2.15)

φ

∂ ψ ∂t

(2.16)

where ψ is an arbitrary scalar field, but the B and E fields do not change. This kind of transformation is called a gauge transformation and the fact that gauge transformations do not affect the physically observable fields is known as gauge invariance.

Gauge conditions The ambiguity in the definitions of A and φ can be used to introduce a gauge condition. In other words, since the definitions (2.11) and (2.12) do not completely define A and φ we are free to add certain conditions. Some common gauge conditions are Coulomb gauge Lorentz gauge Temporal gauge

A µε∂ φ ∂ t

A 0 µσ φ 0 φ 0

The Coulomb gauge is most useful when dealing with static fields. Using (2.13) and (2.14, for static fields, reduces to

ρ ε ∇2 A µj ∇2 φ

A

0 then (2.17) (2.18)

The Lorentz gauge is the most commonly used gauge for time-varying fields. In this case (2.13) and (2.14) reduce to

ρ ∂ ∂2 µε 2 ! φ ∂t ∂t ε 2 ∂ ∂2 µε 2 ! A µj ∇ µσ ∂t ∂t ∇2

µσ

(2.19) (2.20)

So the Lorentz transform decouples (2.13) and (2.14) and puts φ and A on equal footing. Furthermore, the resulting equations are manifestly covariant. In the temporal gauge one “discards” the scalar potential by setting φ (2.13) and (2.14) reduce to 1 ∂ 2A c2 ∂ t 2

6

A

µj


0. In this gauge (2.21)

2.3. S OLVED

13

EXAMPLES

Thus the single vector A describes both E and B in the temporal gauge.

How to invent your own gauge Gauges other than Coulomb, Lorentz and the temporal mentioned above are rarely used in introductory literature in Electrodynamics, but it is instructive to consider what constitutes a gauge condition by constructing ones own gauge. Of course, a gauge condition is at least a scalar equation containing at least one of the components of A or φ . Once you have an equation that you think might be a gauge, it must be verified that it is a gauge. To verify that a condition is a gauge condition it is sufficient to show that any given set of A and φ can be made to satisfy your condition. This is done through gauge transformations. So given a A and a φ which satisfy the physical conditions through (2.13) and (2.14) we try to see if it is possible (at least in principle) to find a gauge transformation to some new potential A= and φ = , which satisfy your condition. E ND

OF EXAMPLE

2.2 %

F OURIER TRANSFORM OF M AXWELL’ S EQUATIONS

E XAMPLE 2.3

Fourier transform Maxwell’s Equation. Use the Fourier version of Maxwell’s equations to investigate the possibility of waves that do not propagate energy; such waves are called static waves.

Solution Maxwell’s equations contain only linear operators in time and space. This makes it easy to Fourier transform them. By transforming them we get simple algebraic equations instead of differential equations. Furthermore, the Fourier transformed Maxwell equations are useful when working with waves or time-varying fields, especially since the response function, i.e. the dielectric function, is in many case more fundamentally described as a function of angular frequency ω than length x. To perform this derivation we need formulas on how to translate the operators and ∂ ∂ t in Maxwell’s equations.

? " ,

The Fourier transform in time, is defined by f˜ ω @-

3 A

∞ ∞

dt eiω t f t

(2.22)

and the Fourier transform in space, is defined analogously A˜ k +-

3 A

∞ ∞

d3x e

A ik B x A x

(2.23)

and so a combined spatial and time Fourier transform becomes F˜ ω 9 k @

3 A

∞ ∞

dt d3x e

A iC kB xA

ωt

D F t x 9


(2.24)

14


If we apply the last formula on

E

3 A '

E3 A

∞ ∞ ∞ ∞

AND

WAVES

E we get

dt d3x e

A iC kB xA

ωt

D E

dt d3x e

A iC kB xA

ωt

D ∂ Ei t 9 x ∂ xi

A A 3 dt F ∑ Ei t 9 x e i C k B x ω t DHG 3 dt d3x iki Ei t 9 x O i xI ∞ J MK L N & 0 iki 3 dt d3x Ei t 9 x iki E˜i ω 9 k + ik E˜ ω 9 k

where we have used partial integration. For

E

'

(2.25)

E we get

D E t ω 9 ∂ E t x 3 dt d3x e A i C k B x A ω t D εi jk j 9 ei 3 dt d3x e A i C k B x A

ωt

A A 3 dt P Ei /Q/M/ e i C k B x J KML & 0 iεi jk k j E˜k ω 9 k ei

∂ xk

ωt

A A DSR 5 3 dt d3x TU iεi jk k j Ek t 9 x e i C k B x N

ωt

De

i

V (2.26)

where we have once again used partial integration. One may proceed analogously for ∂ ∂ tE t 9 x . Trivially, one gets similar equations for the transformation of the D, H and B fields. Thus we have found that

˜ V t x #' ik V ω k 9 ˜ 9 V t 9 x #' ik V ω 9 k ∂ V t 9 x ' iω V˜ ω 9 k ∂t

(2.27a) (2.27b) (2.27c)

where V t 9 x is an arbitrary field and ' denotes here “Fourier transform”. These transformation rules are easy to remember by saying that roughly the Fourier transform of ∇ is ik and the Fourier transform of ∂ ∂ t is iω . Now we can use (2.27a), (2.27b) and (2.27c) on Maxwell’s equations. We then get, after some simple trimming

˜ E ω 9 ˜ ω H 9 ˜ ω D 9 B˜ ω 9

k ik k k

k : ω B˜ ω 9 k ˜ ω k k : ˜j ω 9 k W iω D 9 k : iρ˜ ω 9 k k : 0 Draft version released 15th November 2000 at 20:39

(2.28a) (2.28b) (2.28c) (2.28d)

2.3. S OLVED

15

EXAMPLES

where we have dropped the ˜ notation. These are the Fourier versions of Maxwell’s equation. As an example of the use of the Fourier transformed Maxwell’s equations let us derive static waves. Static waves are one possible oscillation mode for the E and H fields. Let’s say that we have a mode α such that the E Eα field is oscillating at ω ωα 0 and < that it has a k kα 0 which is parallel to the electric field, so kα X Eα . From (2.28a) < this implies that

ωα Bα kα Bα

0

Eα

0

(2.29) (2.30)

So, we see that S E H 0 trivially. The lesson here is that you can have time-varying fields that do not transmit energy! These waves are also called longitudinal waves for obvious reasons. E ND

OF EXAMPLE

2.3 %

S IMPLE DISPERSION RELATION

E XAMPLE 2.4

If a progressive wave is travelling in a linear, isotropic, homogenous, nonconducting dielectric medium with dielectricity parameter ε and permeability µ , what is the dispersion relation? And what is the group velocity in this case? Also, what is the dispersion relation in a conducting medium?

Solution A dispersion relation is a relation between ω and k, usually something like D ω 9 k +

0

(2.31)

From this one can solve for ω which is then a function of k1 9 k2 9 k3 . For isotropic media then ω will be a function of Y k YU k only. A dispersion relation determines what modes (i.e. what combinations of k and ω ) are possible. The dispersion relation is derivable in principal once one has explicit knowledge of the dielectricity function (or response function) for the medium in question. The two vector equations in Maxwell’s equations are

E

(2.32)

H

(2.33)

∂B ∂t ∂D j ∂t

so for a progressive wave characterized by ω and k propagating in a linear, isotropic, homogeneous medium with σ 0 (so j σ E 0), then these equations give Draft version released 15th November 2000 at 20:39

16


k k

E

B µ

(2.34)

ωε E

(2.35)

.

and then using (2.35) we get a single vector equation:

k E + ω k B k k2 E ω 2 ε µ E J kKML EN

. &

WAVES

ωB

Operating on (2.34) with k k

AND

(2.36) (2.37)

0 Progressive wave! 2 2 k ω εµ E 0

(2.38)

Since E is not assumed to be zero then k2

ω 2ε µ 0

(2.39)

k2 εµ

.

ω2

.

ω "Z [

(2.40) k εµ

"Z ku

(2.41)

where we have identified the phase velocity u

1

[

εµ.

The group velocity in this case is ∂ω ∂ vg uk + u ∂k ∂k so in this simple case the group velocity is the same as the phase velocity.

(2.42)

For the case of a conducting medium, in which j σ E, the two vector equations applied on a wave which at first resembles the progressive wave we used above gives k k

E

B µ

ωB

(2.43)

iσ E ωε E

(2.44)

Combining these two equation as done previously, we get

k2 E iσ µω E ω 2 ε µ E σ . T ω 2 i ω u2 k 2 V E 0 \

(2.45) (2.46)

ε

So that iσ 1 Z 4u2 k2 2ε 2] and the group velocity is

ω

σ2 ε2

(2.47)

∂ω 2u2 k "Z ^ ∂k 4u2 k2 σ 2 ε 2 If σ 0 then we are back again to the previous problem as can be verified. E ND


(2.48)

OF EXAMPLE

2.4 %

L ESSON 3

Relativistic Electrodynamics 3.1

Coverage

We examine the covariant formulation of electrodynamics. We take up the concept of 4-tensors and give examples of these. Also, show how 4-tensors are manipulated. We discuss the group of Lorentz transformations in the context of electrodynamics.

3.2

Formulae used

A Lorentz boost in the 3-direction Lµµ _ a`b

γ 0 0

γβ

bc

0 1 0 0

dfe

0 γβ e 0 0 1 0 g 0 γ

(3.1)

The field tensor (components are 0, 1, 2, 3)

F µν

bc

`b

0 E1 E2 E3

E1 0 cB3

cB2

E3

0 cB1

cB2

cB1 g 0

dfe

E2

cB3


e (3.2)

17

18

L ESSON 3. R ELATIVISTIC E LECTRODYNAMICS

3.3 E XAMPLE 3.1

Solved examples

C OVARIANCE OF M AXWELL’ S EQUATIONS Discuss the covariance of Maxwell’s equations by showing that the wave equation for electromagnetic fields is invariant with respect to Lorentz transformations but not Galilean transformations.

Solution The d’Alembert operator

h

1 ∂2 ∇2 (3.3) c2 ∂ t 2 is a fundamental operator in electrodynamics. The dynamics of EM fields is completely described using the d’Alembertian. Galilean transformations, even though closest to our intuitive picture of the fabric of space-time, does not leave the d’Alembertian invariant. A Galilean transformation is simply 2

k jjl

ijj

-

x1= x1 x2= x2 x3= x3 t =m t

(3.4)

vt

where the origin of the primed system is moving relative the unprimed along the 3-direction with velocity v. Now we introduce this transformation by expanding each differential in the unprimed coordinate system in terms of the differential in the primed system by using the chain rule of derivation, i.e. we evaluate ∂ ∂ x µ ∂ xν= ∂ xµ ∂ ∂ xν= , so

∂ ∂ x1

∂ ∂ x2

∂ ∂ x3

∂t= ∂ ∂ x1 ∂ t = ∂ ∂ x1=

∂ x1= ∂ ∂ x1 ∂ x1=

∂t= ∂ ∂ x2 ∂ t = ∂ ∂ x2=

∂ x1= ∂ ∂ x1 ∂ x1=

∂t= ∂ ∂ x3 ∂ t = ∂ ∂ x3=

∂ x1= ∂ ∂ x3 ∂ x1=

∂ x2= ∂ ∂ x1 ∂ x2=

∂ x3= ∂ ∂ x1 ∂ x3=

(3.5) (3.6)

∂ x2= ∂ ∂ x2 ∂ x2=

∂ x3= ∂ ∂ x2 ∂ x3=

(3.7) (3.8)

∂ x2= ∂ ∂ x3 ∂ x2=

∂ x3= ∂ ∂ x3 ∂ x3=

(3.9) (3.10)


3.3. S OLVED

19

EXAMPLES

and so ∂2 ∂ x21

∂2 ∂ x12 ;

n

∂2 ∂ x22

∂2 ∂ x22 ;

∂2 ∂ x23

n

∂2 ∂ x32

n

(3.11)

but

∂ t= ∂ ∂ x1= ∂ ∂t ∂t= ∂ t ∂ x1= ∂ ∂ v ∂ t= ∂ x3=

∂ ∂t

∂ x2= ∂ ∂ t ∂ x2=

∂ x3= ∂ ∂ t ∂ x3=

(3.12) (3.13)

and so

∂2 ∂ t2

∂

∂

∂

∂

∂2

∂2

∂2

2 (3.14) ∂ t = v ∂ x= ∂ t = v ∂ x= + ∂ t = 2 v ∂ x= 2 2v ∂ t = ∂ x= 3 3 3 3 where we have used the fact that the operators ∂ ∂ t = and ∂ ∂ x3= commute. Thus we have

found that

h

2

'

h 2= ∇= 2 Ψ

1 ∂ 2Ψ c2 ∂ t = 2

v2 ∂ 2 Ψ c2 ∂ x3= 2

2v ∂ 2 Ψ c2 ∂ t = ∂ x3=

0

(3.15)

Which obviously does not have the same form as the d’Alembertian in the unprimed system! Let us do the same calculations for the case of a Lorentz transformation; more specifically we consider a boost along the 3 axis which is given by xµ n

Lµµ n xµ

(3.16)

where Lµµ n

poqqr

γ 0 0 γβ

0 1 0 0

0 0 1 0

sMt γβ t 0 0 γ

u

(3.17)

(remember that µ runs over 0, 1, 2, 3). Since γ and β depend on v The 4-gradient, ∂µ - ∂ ∂ xµ transforms as

∂µ

∂ xµ n ∂ ∂ xµ µ n

∂ T Lνµ n xν V ∂µ n ∂ xµ

0

xν

∂ Lνµ n Lνµ n ∂µ n Lµµ n ∂µ n ∂ xµ 1

(3.18)

so

h

2

∂µ ∂ µ gµν Lµµ n Lνν n ∂µ n ∂ν n γ 2 γ 2 β 2 ∂02 ∂12 ∂22 γ 2 γ 2 β 2 ∂32 gµν ∂µ n ∂ν n ∂µ n ∂ µ n

(3.19)

In other words we have found that the d’Alembertian is invariant under Lorentz boosts. E ND


OF EXAMPLE

3.1 %

20

E XAMPLE 3.2


I NVARIANT QUANTITIES CONSTRUCTED FROM THE FIELD TENSOR Construct the dual tensor v αβ 12 ε αβ γδ Fγδ of the field tensor Fµν . What quantities

constructed solely with the field tensor and it’s dual tensor, are invariant under Lorentz transformations? Having found these quantities you should be able to answer the questions:

2 can a purely electric field in one inertial system be seen as a purely magnetic field in another?

2 and, can a progressive wave be seen as a purely electric or a purely magnetic field in an inertial system?

Solution The dual tensor of Fαβ is given by

v

1 αβ γδ ε Fγδ 2

αβ

1 αβ γδ ε gγ µ gδ ν F µν 2

where

qp q o

F µν

r

E1

0 E1 E2 E3

E2

0 cB3 cB2

cB2

cB1 u

0 cB1

gµν

qp q o r

0 1 0 0

0 0 1 0

0 0 0 1

(3.21)

0

and 1 0 0 0

s tt

E3

cB3

(3.20)

s tt u

(3.22)

We determine first the field tensor with two covariant indices through the formula

Fµν

gµα gνβ F αβ poqqr

so

v

αβ

r

qp q o

0 cB1 cB2 cB3

0 E1 E2 E3

E1 0 cB3 cB2

E2 cB3 0 cB1

E3 cB2 cB1 0

sMtt u

(3.23)

st cB1 cB2 cB3 t E2 0 E1 0 E1 u E3 E1 0 E2

We see that the dual tensor can be obtained from F µν by putting E

'

(3.24)

cB and B

'w E c.

From the formula for the dual tensor v we see that it is a 4-tensor since F is a four tensor and ε is easily shown to be an invariant under orthogonal transforms for which the Lorentz transform is a special case. What can we create from v µν and F µν which is invariant under Lorentz transformations? We consider the obviously invariant quantities µν


µν

3.3. S OLVED

v

µν

21

EXAMPLES

Fµν and F µν Fµν .

v

µν

Fµν

4cE B

(3.25)

2 c 2 B2 E 2 This means that E B and E 2 c2 B2 are Lorentz invariant scalars. F µν Fµν

(3.26)

Relation of EM fields in different inertial systems Now that we know that E B and

E c B are Lorentz invariant scalars, let see what they say about EM fields in different inertial systems. Let us say that X - E B and Y - E 2 c2 B2 . All inertial systems must have the same value for X and Y . A purely electric field in one inertial system means that B 0, so X 0 and Y x 0. A purely magnetic field would mean that E =y 0, so X 0 but Y z 0. In other words it does not seem that a purely electric field can be a purely magnetic field in any inertial system. 2

2 2

For a progressive wave E { B so X 0 and in a purely electric or a purely magnetic field X 0 also, but for a progressive wave E cB so Y 0 and if the other system has E =m 0 or B=m 0 then Y 0 force both the fields to be zero. So this is not possible. E ND

OF EXAMPLE

3.2 %

C OVARIANT FORMULATION OF COMMON ELECTRODYNAMICS FORMULAS Put the following well know formulas into a manifestly covariant form

2 The continuity equation 2 Lorentz force

2 The inhomogeneous Maxwell equations 2 The homogenous Maxwell equations Solution The Methodology To construct manifestly covariant formulas we have at our disposal the following “building blocks”: an event 4-velocity 4-momentum wave 4-vector 4-current density 4-potential 4-force

xµ uµ pµ kµ Jµ Aµ Fµ

ct 9 x , γ c 9 γ v , E c 9 p , ω c 9 k , ρ c 9 j , φ c 9 A , γ v F c 9 γ F

Also we have the 4-gradient


E XAMPLE 3.3

22


4-gradient

∂µ - ∂ ∂ xµ

∂ ∂ ct 9 ∂ ∂ x

as our operator building block and also the second rank 4-tensor

field tensor

F µν

qp q o r

0 E1 E2 E3

E1 0 cB3 cB2

E2

cB3 0 cB1

s tt

E3 cB2

cB1 u 0

Observe that we use indices which run 0, 1, 2, 3 where the 0-component is time-like component, there is also the system where indices run 1, 2, 3, 4 and the 4-component is the time-like component. Beware! A sufficient condition to formulate covariant electrodynamic formulas is that we make our formulas by combine the above 4-vectors. To make sure we have a covariant form we take outer product (i.e. simply combine the tensors so that all the indices are free) and then perform zero or more contractions, i.e. equate two indices and sum over this index (notationally this means we create a repeated index). In the notation we use here contractions must be between a contravariant (upper) index and a covariant (lower) index. One can always raise or lower a index by including a metric tensor g αβ . On top of this sufficient condition, we will need to use our knowledge of the formulas we will try to make covariant, to accomplish our goal.

The continuity equation We know that the continuity equation is a differential equation which includes the charge density and the current density and that it is a scalar equation. This leads us to calculate the contraction of the outer product between the 4-gradient ∂µ and the 4-current J ν ∂µ J µ 0

(3.27)

This is covariant version of the continuity equation, thus in space-time the continuity equation is simply stated as the 4-current density is divergence-free!

Lorentz force We know that the left hand side of Lorentz force equation is a 3-force. Obviously we should use the covariant 4-vector force instead. And on the right hand side of th Lorentz equation is a 3-vector quantity involving charge density and current density and the E and B fields. The EM fields are of course contained in the field tensor F µν . To get a vector quantity from F µν and J µ we contract these so our guess is Fµ

F µν Jν

(3.28)

Inhomogeneous Maxwell equations The inhomogeneous Maxwell may be written as the 4-divergence of the field tensor

∂α F αβ J β

(3.29)


3.3. S OLVED

23

EXAMPLES

Homogeneous Maxwell equations The homogenous Maxwell equations are written most compactly using the dual tensor of the field tensor. Using the dual tensor we have ∂α v

0

αβ

(3.30) E ND

OF EXAMPLE

3.3 %

F IELDS FROM UNIFORMLY MOVING CHARGE VIA L ORENTZ TRANSFORMATION In the relativistic formulation of classical electrodynamics the E and B field vector form the antisymmetic electrodynamic field tensor

qq o ∂µ Aν ∂ν Aµ p r

Fµν

0 Ex Ey Ez

Ex 0 cBz cBy

Ey cBz 0 cBx

s tt

Ez cBy cBx u 0

(3.31)

Show that the fields from a charge q in uniform, rectilinear motion q 1 v 2 c2 R0 E 4πε0 s3 B

v

E c2

are obtained via a Lorentz transformation of the corresponding fields in the rest system of the charge.

Solution We wish to transform the EM fields. The EM fields in a covariant formulation of electrodynamics is given by the electromagnetic field tensor F µν

oqqr

cB3

0 cB3 cB2 E1

0 cB1 E2

cB2 cB1 0 E3

E1 E2 E3 0

s tt u

(3.32)

where we are using components running as 1, 2, 3, 4. To transform the EM fields is to transform the field tensor. A Lorentz transformation of the field tensor can be written Fλσ F

Lλµ Lσν F ν µ

(3.33)

~LF0 L

(3.34)

where Lνµ

r

qp q o

γ 0 0 βγ

0 1 0 0

0 0 1 0

st βγ t 0 0 γ

u


(3.35)

E XAMPLE 3.4

24


^

1

where γ

1

β 2 and β v c where v v1 x. ˆ The fields in the rest system S0 are

q x0 xˆ y0 yˆ z0 zˆ ^ 4πε0 x0 2 y0 2 z0 2 3

E0

B0

0

(3.36) (3.37)

so that F µν

poqqr

0 0 0 E10

0 0 0 E20

E10 E20 E30 0

0 0 0 E30

sMtt u

(3.38)

A little matrix algebra gives

~L qorq

~ LFL

0 0 0 E10

qw q o

γ 0 0 βγ

βγ

r

qw qr o

0 1 0 0

2

0 0 0 E20 0 0 1 0

βγ

E10

γβ E20

0 0 0 E30

st βγ t 0 0 γ 2

E10

γβ E30 2 γ 1 β 2 E10

u

E10 E20 E30 0

s tt

γ 0 0 βγ

qoq u r

β γ E10

β γE0 oqqr β γ E20 3 γ E10

γβ E20

γβ E30

0 0 γ E20

0 0 γ E30

0 1 0 0 0 0 0 E20

0 0 1 0

st βγ t 0 0 γ 0 0 0 E30

u st

γ E10 t γ E20 γ E30 u β γ E10

(3.40)

st

t γ 1 β 2 E10 0 γ E2 u γ E30 2 0 2 0 β γ E1 β γ E1 2

(3.39)

(3.41)

E1 E2 E3

E10

B1

0

(3.45)

B2

(3.46)

(3.42) (3.43) (3.44)

γ E20 γ E30

γβ 0 E c 3 γβ 0 B3 E c 2

oqqr where s

x0 y0 z0 ct 0

vt

sMtt

u L oqrq

(3.47) x y z ct

sMtt u poqqr

sMt

t γ x s y u z γ ct β x


(3.48)

3.3. S OLVED

q γ O x s xˆ y yˆ zˆz ^ 4πε0 γ 2 x s 2 y2 z2 3

E

(3.49)

s x 2 y 2 z2

R20 s

25

EXAMPLES

x R0 sin θ q 4πε0 γ 2

E

2 x s

|

y2

γ2

π + R0 cos θ 2 R0

x s

z2

(3.50)

2

}

y2 z2 γ2

(3.51) (3.52)

3

R20 y2 z2

y2

z2

(3.53)

R20 y2 z2 2 1 ~ γ R20 1 β 2 sin2 θ

(3.54)

γ2 1

E

(3.55)

R0 q 1 β2 ^ 3 4πε0 3 R0 1 β 2 sin2 θ

(3.56) E ND


OF EXAMPLE

3.4 %


26

L ESSON 4

Lagrangian and Hamiltonian Electrodynamics 4.1

Coverage

We briefly touch the Lagrangian formulation of electrodynamics. We look at both the point Lagrangian for charges in EM fields and the Lagrangian density of the EM fields.

4.2

Formulae used

The Lagrangian for a charged particle in EM fields is L

mc2

qφ qv A γ

(4.1)

A useful Lagrangian density for EM field and its interaction with charged particles is given by

1 ε0 c2 B2 E 2 A j ρφ 2


(4.2)

27

28

4.3

L ESSON 4. L AGRANGIAN

E XAMPLE 4.1

AND

H AMILTONIAN E LECTRODYNAMICS

Solved examples

C ANONICAL QUANTITIES FOR A PARTICLE IN AN EM FIELD Derive the canonical momentum and the generalised force for the case of a charged particle 2 in EM field given by φ and A. The Lagrangian is L mcγ qφ qv A.

Solution We know from analytical mechanics that the canonical momentum P is found through ∂L Pi (4.3) ∂ vi so with mc2 qφ qv A (4.4) L γ we find that

∂L ∂ vi ∂ vv mc2 1 i 2 i qφ qv j A j ! ∂ vi c ] v i qAi mc2 | v v c2 1 cj 2 j

Pi

mγ vi qAi P p qA .

(4.5) (4.6)

And on the other hand the generalised force is ∂U d ∂U Qi ∂ xi dt ∂ x˙i where U is the generalised velocity dependent potential qφ

qv A, which gives us,

∂ d ∂ qφ qv j A j qφ qv j A j ∂ xi dt ∂ vi ∂v A dA ∂φ q j j q i q ∂ xi ∂ xi dt dA Q q φ q v A q dt

Qi

.

(4.7)

f 6 f ∂A A q q φ q v A q v q v A ∂t 5 ∂A A q q φ qv ∂t qE qv B Draft version released 15th November 2000 at 20:39

(4.8)

(4.9)

4.3. S OLVED

29

EXAMPLES

What does the canonical momentum P p qA represent physically? Consider a charge moving in a static magnetic field. This charge will perform gyro-harmonic motion. Obviously the momentum is not conserved but on the other hand we did not expect it to be conserved since there is a force on the charge. However we now from analytical mechanics, that it is the conservation of canonical momentum that is more general. Conservation of canonical momentum is found when the problem is translational invariant, which is true in the case we have here since the potentials do not depend on spatial coordinates. So we expect P p qA to be a constant of the motion, but what is it? E ND

OF EXAMPLE

4.1 %

G AUGE INVARIANCE OF THE L AGRANGIAN DENSITY

E XAMPLE 4.2

Consider the Lagrangian density of the EM fields in the form

1 ε c 2 B2 2 0

*

E 2 W A j ρφ

(4.10)

We know that EM field are invariant under gauge transformations

A=

A

φ=

φ

ψ ∂ψ ∂t

(4.11) (4.12)

Determine the Lagrangian density under a gauge transformation. Is it invariant? If not, discuss the consequences this would have.

Solution Let us insert the gauge transformation relations into the Lagrangian density. Remembering that E and B are invariant under gauge transformations we find that

'

∂ψ E 2 A= ψ j ρ φ= (4.13) ∂t ∂ψ 1 ε c2 B2 E 2 A= j ρφ = ψ j ρ (4.14) 2 0 ∂t 1 ∂ ρψ ∂ρ ψ (4.15) ε c2 B2 E 2 A= j ρφ = j ψ j ψ 2 0 ∂t ∂t 1 ∂ ρ ∂ ρψ ε0 c2 B2 E 2 A= j ρφ = ψ j ψ j (4.16) 2 ∂t ∂t ∂ ρψ (4.17) 0 ψ j ∂ t =

1 ε c 2 B2 2 0


30

L ESSON 4. L AGRANGIAN

AND

H AMILTONIAN E LECTRODYNAMICS

where we have used the continuity equation. We see that as , but what about the other two terms?

obviously has the same form

0

Some thought reveals that the neccessary condition for a Lagrangian to be physically acceptable is not the Lagrangian itself is invariant but rather that the variation of the action integral S p d3x dt is invariant. So now we would like to check that the gauge transformations indeed do not affect any the variation of the action. Now it is possible to proceed in two different ways to do this: one is simply carry out the integration in the definition of the action integral and check that its variation is zero, or two, remembering that the variation of the action is equivalent to the Euler-Lagrange equations, one could plug in the Lagrangian density (4.17) into Euler-Lagrange equations to check the resulting equations differ from the Maxwell equations. Let us use the first alternative. Since the action is linear in Sext

33

(

ψ j

∂ ρψ ! ∂t

it is sufficient to examine

d3x dt

33 ψ j dS dt 3 ρψ d3x

33 ψ j dS dt 3 ρψ d3x

t1 t0 t1

33 ψ

∂ρ ∂t

j!

d3x dt (4.18)

t0

where we have used the continuity equation. Furthermore, if we assume no flux source/sink at infinity then we can write Sext

3 ρψ d3x

t1

(4.19) t0

Now when taking the variation, we realize that we must hold t0 and t1 the end point of the particle path fixed, and thus

δ Sext 0

(4.20)

As one would expect, gauge transformations of the potentials do not effect the physics of the problem! E ND


OF EXAMPLE

4.2 %

L ESSON 5

Electromagnetic Energy, Momentum and Stress 5.1

Coverage

Here we study the force, energy, momentum and stress in an electromagnetic field.

5.2

Formulae used

Poynting’s vector S

E

H

(5.1)

Maxwell’s stress tensor Ti j

Ei D j Hi B j

1 δ E D Hk Bk 2 ij k k


(5.2)

31

32

L ESSON 5. E LECTROMAGNETIC E NERGY, M OMENTUM

AND

S TRESS

Table 5.1. The following table gives the relevant quantities. The field vectors in this table are assumed to be real. If the given fields are complex, use the real part in the formulas. Name Energy density Intensity Momentum density Stress

5.3 E XAMPLE 5.1

Symbol Uv S PEM T

Formula D 12 H B E H εµE H Hi B j 12 δi j Ek Dk 1 2E

Ei D j

Hk Bk

SI unit J/m3 W/m2 kg/s m2 Pa

Solved examples

EM QUANTITIES POTPOURRI Determine the instantaneous values of the energy density, momentum density, intensity and stress associated with the electromagnetic fields for the following cases:

A E E e ei C k 1 x 1 H k0 kˆ2 E µω

a)

ωt

D

b) same as in a) but for k1 c)

Br Bθ

for k1

[

ε µω , k2 k3 0.

iα

µ0 m cos θ 2π r 3 µ0 m sin θ 4π r 3

Also, identify these cases. Assume that D

ε E and B µ H.

Solution Background (a) Progressive wave This case is an example of a progressive or propagating wave. Since E and H are complex we must first take their real parts: Re E

E0 cos k1 x1 ω t e2

Re H

k1 E cos k1 x1 ω t e3 µω 0

The energy density is


(5.3) (5.4)

5.3. S OLVED

ε 2 k12 E 2 cos2 k1 x1 ω t ~ E0 cos2 k1 x1 ω t 2 2µω 2 0 [ ε E02 cos2 ω ε µ x1 ω t

Uv

33

EXAMPLES

(5.5) (5.6)

The intensity or power density is S

k E0 cos k1 x1 ω t e2 1 E0 cos k1 x1 ω t e3 + µω [ ε 2 E0 cos2 ω ε µ x1 ω t e2 e3 ] µ [ ε 2 E0 cos2 ω ε µ x1 ω t e1 ] µ

(5.7) (5.8) (5.9)

The momentum is PEM

[ [ ε µ S ε ε µ E02 cos2 ω ε µ x1 ω t

(5.10)

The stress is T11

T21 T22

T32 T33

ε 2 µ 2 E H 2 2 2 3 ε k12 2 * E02 cos2 k1 x1 ω t E cos2 k1 x1 ω t 2 µω 2 0 * ε E02 cos2 k1 x1 ω t T31 T12 0 ε µ 2 E22 H 2 2 3 ε k12 2 E cos2 k1 x1 ω t E02 cos2 k1 x1 ω t 2 µω 2 0 T13 T23 0 µ 2 ε 2 H E T22 0 2 3 2 2

*

(5.11) (5.12)

(5.13) (5.14) (5.15)

(b) Evanescent wave This case is a example of an evanescent wave. We take real part of the fields keeping in mind the fact that k1

A Re E E0 e

α x1

iα is imaginary:

sin ω t e2

α α A e1 Im E + e1 E0 e µω µω α E0 A α x 1 e cos ω t e3 µω

Re H

(5.16) α x1

The energy density is


cos ω t e2 (5.17)

34


AND

ε 2 A 2α x 1 2 α 2 E02 A 2α x1 e sin ω t cos2 ω t E0 e 2 2µω 2 A E02 e 2α x1 α2 ε sin2 ω t cos2 ω t ! 2 µω 2

Uv

S TRESS

(5.18) (5.19)

The intensity is

S

α E02 A e µω

2α x 1

sin ω t cos ω te1

(5.20)

The momentum is PEM

ε µ S

α E02 ε A e ω

2α x 1

sin ω t cos ω te1

(5.21)

The stress is

ε 2 µ 2 E H 2 2 2 3 T31 T12 0

T11

T21

T32

ε 2 µ 2 E H 2 2 2 3 T13 T23 0

T22

T33

µ 2 H 2 3

E02 e

A

2α x 1

2

ε sin2 ω t

α2 cos2 ω t ! µω 2

(5.22) (5.23)

A

E02 e 2α x1

ε sin2 ω t

2

ε 2 E T22 2 2

α cos2 ω t ! µω 2 2

(5.24) (5.25)

A

E02 e 2α x1

2

ε sin2 ω t

α cos2 ω t ! µω 2 2

(5.26)

(c) Magnetic dipole This case is a magnetic dipole. The fields are real and in spherical coordinates. The energy density is 1 B2 2 µ0 r

Uv

Bθ2 +

1 2 µ0

µ02 m2 4 cos2 θ 16π 2

r6

µ02 m2 sin2 θ ! 16π 2 r6

µ0 m 1 4 cos2 θ sin2 θ 6 32π 2 r µ 0 m2 1 3 cos2 θ 32π 2 r6 2

The intensity is S components are

(5.27)

0 since E 0, and likewise for the momentum. The stress tensor


5.3. S OLVED

1 2 B µ0 r

Trr

35

EXAMPLES

Uv

µ0 m2 cos2 θ 4π 2 r6

µ 0 m2 1 3 cos2 θ 32π 2 r6

Tφ r Trθ

µ 0 m2 8 cos2 θ 1 3 cos2 θ 32π 2 r6 µ m2 0 2 6 1 5 cos2 θ 32π r µ0 m2 sin θ cos θ 1 Br Bθ µ0 8π r6 0 Tθ r

Tθ θ

Tθ r

1 2 B µ0 θ

Uv

µ0 m2 sin2 θ 16π 2 r6

(5.28) (5.29) (5.30) (5.31)

µ 0 m2 1 3 cos2 θ 32π 2 r6

µ 0 m2 2 sin2 θ 1 3 cos2 θ 32π 2 r6 µ m2 0 2 6 1 3 cos2 θ 2 sin2 θ 32π r 0 0 0 0

Tφ θ Trφ Tθ φ Tφ φ

(5.32) (5.33) (5.34) (5.35) (5.36) E ND

OF EXAMPLE

5.1 %

C LASSICAL ELECTRON RADIUS Calculate the classical radius of the electron by assuming that the mass of an electron is the mass of it’s electric field and that the electron is a homogenous spherical charge distribution of radius re and total charge Y q Y e. Mass and energy are related through the equation E mc2 .

Solution One the “failures” of Maxwell’s equations or classical electrodynamics is on question of mass of point particles. From relativity one can show that fundamental particles should be point-like. However, if one calculates the electromagnetic mass from Maxwell’s equation one gets an infinite result due to the singularity in Gauss law E ρ ε0 . This points to the fact that Maxwell’s equations has a minimum length scale validity where quantum mechanics takes over. We will calculate this problem as follows: determine the electric field in all of space and then integrate the formula for the energy density of the electric field. This integral will contain the radius of the electron since it partitions the integration. We then relate this field energy to the mass of the electron, which is a known quantity. We use this relation to solve


E XAMPLE 5.2

36


AND

S TRESS

for the “electron radius”. We start by determining the electric field. Consider a homogeneous spherical charge distribution in a vacuum. Introduce spherical coordinates. We divide the whole space into two parts: the region outside of the sphere, i.e. r ; re , is given by Coulomb’s law: e E r ; re + (5.37) 4πε0 r2 and the region inside the sphere is given by Gauss law

E r

8 re + ρ ε0

(5.38)

which in this case gives 1 ∂ 2 r Er r2 ∂ r

e 4 3π re3 ε0

(5.39)

If we integrate this equation, i.e. (5.39), we find that e r Er r 8 re + 4π re3 ε0

(5.40)

We can easily verify that (5.40) is indeed a solution to (5.39) and furthermore it is continuous with (5.37), thus making the solution unique. We may now determine the energy density of the electric field due to the electron. It is simply 1 ε E2 2 0 Uv r

Uv

(5.41)

; re + Uv r 8 re +

.

2

1 e 32π 2 ε0 r4 2 e r2 32π 2 ε0 re6

(5.42)

Now we integrate Uv over all space. That is

3

U

all space 4π

3

3

0

re 0

8 re Uv r ; re d3x

Uv r

4π

e2 r2 r2 dr dΩ 3 32π 2 ε0 re6

e2 r5 8πε0 re6 5

re

0

e2 8πε0

1 r

∞ re

0

3

∞ re

e2 1 2 r dr dΩ 32π 2 ε0 r4

3e2 1 20πε0 re

(5.43)

Finally, we relate the total electric field energy to the rest mass of the electron and solve for the electron radius,

m e c2

U

2

.

3e 1 20πε0 re

.

re

(5.44)

me c2

3 e2 5 4πε0 me c2 Draft version released 15th November 2000 at 20:39

(5.45)

5.3. S OLVED

37

EXAMPLES

This last result can be compared with the de facto classical electron radius which is defined as re

e2 4πε0 me c2

(5.46)

and is found by calculating the scattering cross section of the electron. E ND

OF EXAMPLE

5.2 %

S OLAR SAILING

E XAMPLE 5.3

Investigate the feasibility of sailing in our solar system using the solar wind. One technical proposal uses kapton, 2 mm in thickness, with a 0.1 mm thick aluminium coating as material for a sail. It would weigh 1 g/m2 . At 1 AU (= astronomical unit distance between sun and earth) the intensity of the electromagnetic radiation from the sun is of the order 1 / 4 103 W/m2 . If we assume the sail to be a perfect reflector, what would the acceleration be for different incidence angles of the sun’s EM radiation on the sail?

Solution In this problem we will use the principle of momentum conservation. If the continuity equation for electromagnetic momentum ∂ PEM ∂ t T ∂ Pmech ∂ t is integrated over all space the stress term disappears and what is left says that a change in electromagnetic momentum is balanced by mechanical force. Imagine a localised pulse of a progressive electromagnetic wave incident on a plane. That the wave is progressive means simply that it is “purely radiation” (more about progressive waves in the next lesson). Let us characterize this wave by a Poynting vector S of duration ∆t. It travels in space with velocity c, it “lights up” the area ∆A on the surface on the sail, and, furthermore, S makes an angle 180 θ with the normal of the sail surface. The momentum carried by such a wave is 2 Pbefore EM ∆V + S c ∆A c ∆t cos θ

pbefore EM

(5.47)

so the momentum along the direction of the surface normal nˆ is

pbefore nˆ EM

Y SY c

∆A ∆t cos2 θ

(5.48)

After hitting the sail, the pulse will be characterized with all the same quantities as before the impact, except that the component along the surface normal will be opposite in sign. This is because the sail is assumed to be perfectly reflecting. Thus,

pafter n EM ^

n pbefore EM ^

Y SY c

∆A ∆t cos2 θ

Now the continuity equation for EM momentum says that ∂ PEM ∂ t the component along the sail surface normal this implies that


(5.49)

∂ Pmech F so for

38


pafter EM

pbefore EM

2

Y SY

∆A cos2 θ ∆t c The other force components are zero by symmetry. F ^ n

So finally, the pressure

n F ^ ∆A

2

n ^

AND

S TRESS

(5.50)

exerted by the pulse on the sail is simply

Y SY c

cos2 θ

(5.51)

We now have the pressure exerted by a pulse charcterized by S incident on a surface with an angle θ . The solar wind can be seen as a multitude of such pulses radiating radially outwards from the sun. At 1 AU, the solar constant is 1.3 kW m2 . The solar constant is the intensity of EM radiation or in other words, the magnitude of the Poynting vector Y S Y . From equation (5.51), we find that

2

1 / 3 03 W/m2 cos2 θ 3 108 m/s

(5.52)

Newton’s second law gives the acceleration a of the sail (which, as we recall, weighs A 3 10 3 kg/m2 ) due to the solar wind can at most be a

A

0 / 9 10 5 N/m2 3 10 A 3 kg/m2

3 mm/s2

(5.53)

which is half of the acceleration due to the sun’s gravitational field. E ND


OF EXAMPLE

5.3 %

5.3. S OLVED

39

EXAMPLES

M AGNETIC PRESSURE ON THE EARTH

E XAMPLE 5.4

Determine the magnetic pressure due to the Earth’s magnetic field at the magnetic A poles (take B 6 10 5 T) and compare this with the Earth’s atmospheric pressure 5 (1 atm 1 / 01 10 Pa). Now assume that the magnetic dipole moment is proportional to the angular velocity of the earth, how much faster would the angular velocity need to be for the magnetic pressure to be comparable to the atmospheric pressure?

Solution In this exercise we see that EMF can exert pressure not only via radiation pressure, but also through static fields. The static magnetic pressure is quantified in the Maxwellian stress tensor, which for this case is T

B2r µ0 0 0

¢¡£

0 0 0 0 0 0

¤¥

Uv

0

0

0

¦¡£ 0 Uv

0 0 Uv

¤¥

/

(5.54)

So by taking the inner product of T and the unit vector pointing in the direction the north pole, namely § 1 0 0¨ , we find the pressure in the radial direction to be B2r 2 µ0

A 52 A 6 10 2 1 / 3 10 1 / 3 10 A 6

6

1 / 4 10 A 3 Pa

(5.55)

Now we assume, according to the hypothesis in problem, that 1

∝m∝B∝

Tperiod

[

(5.56)

©

so that

©

ª

1 0

P0 P1

(5.57)

where we denote the current values of the pressure and rotational period time, respectively and the hypothetical values 1 and T1 . Solving for T1 we arrive at

. So with

T1

T1

1

T0 ª

0

and T0

(5.58)

1

105 Pa, we find that 1 / 4 10 A

24 3600

0

]

105

3

10 s

(5.59) E ND


OF EXAMPLE

5.4 %


40

L ESSON 6

Radiation from Extended Sources 6.1

Coverage

We will study the important general problem of how to calculate the EM fields induced by spatially extended, time-varying sources. This problem is solved in different ways depending on the explicit form of the source distribution. For truly extended bodies with non-monochromatic time dependence, we use the general expressions for the retarded potentials. And for monochromatic, one-dimensional current distributions, which we will call antennas here, we use the formulas given below.

6.2

Formulae used

General expressions for retarded potentials A t « x

µ0 4π ¬

φ t « x

1 4πε0 ¬

j x« t ® d3x ® x x

ρ x « t ® d3x ® x x

For “antennas”, we use Bω

iµ0k sin θ eik ¯ x ° x_ ¯ ® ²® ± ϕˆ 4π x x

Eω

ik sin θ eik ¯ x ° x³_ ¯ ˆ ® ²® ± θ 4πε0 c x x


41

42

L ESSON 6. R ADIATION

´

S µ

k2 sin2 θ µ0 c ® ® ® ± 32π 2 x x 2

®2

FROM

E XTENDED S OURCES

rˆ

where

±

6.3

E XAMPLE 6.1

l

¬ °

l

jω z e °

ikz_ cos θ

dz

Solved examples

I NSTANTANEOUS CURRENT IN AN INFINITELY LONG CONDUCTOR Consider the following idealised situation with an infinitely long, thin, conducting wire along the z axis. For t 8 0, it is current free, but at time t 0 a constant current J is applied simultaneously over the entire length of the wire. Consequently, the wire carries the current 0 t 8 0 j z +¶ 9 t x 0 J9 It is assumed that the conductor can be kept uncharged, i.e., ρ in whole space.

0 . Determine B, E and S

Hint: Calculate first the vector potential A .

Solution This problem belongs to the most general category of problems of the kind where given a source distribution one wants to find the EM fields. This is because the source is not monochromatic, so it is not an antenna, and furthermore it is an extended distribution, so multipole expansion analysis is not possible. So we must use the most general formula for calculating fields induced by time-varying sources, which in the Lorentz gauge take the form j t = 9 x=H Y x x= Y

A t 9 x @

µ0 3 4π

d3x=

φ t 9 x +

1 4πε0

ρ t = x= 3 d3x= 9 Y x x= Y Draft version released 15th November 2000 at 20:39

(6.1)

(6.2)

6.3. S OLVED

43

EXAMPLES

where the source time t = is to be replaced by t =y t ,Y x x=·Y¸ c so t = is seen as a function of t, x and x= . The function t Y x x= Y c is known as the retarded time tret = , an expression which is meaningful only relative the field point t 9 x . The algorithm of solution now that we have decided to use (6.1) is to find the explicit form for j t = 9 x=H and ρ t = 9 x=H , perform the integrations to obtain the potentials A t 9 x and φ t 9 x and, finally, derive the E and B fields from the potentials in the normal fashion and the Poynting vector from the fields. Let us find the explicit expression for the current density j x = 9 t =H , which is illustrated in Fig. 6.1. In many problems, the expressions for the sources consist of a time-dependent part times a space-dependent part. This is one such case. The “switching on” at t =) 0 can be written as a step function θ t =¹ . And if we orient the wire along the z axis, the space dependent part can be written δ x=H δ y=H J zˆ . On the other hand, the charge distribution ρ t 9 x + 0 as given in the problem formulation. This can be seen as and charges flowing in opposite directions such as to keep the total charge density ρ t 9 x + 0 but this could still have a total current density j t 9 x 0.

<

So we have for the current that j t = 9 x= +

δ x= δ y= J θ t = zˆ

(6.3)

and for the charge density

ρ t = 9 x= + 0

(6.4)

We insert (6.3) into (6.1), remembering to replace the source time t = with t and perform the integration. The integrations over x = and y= are trivial: µJ ∞ θ t Y x z= zˆ Y c A t 9 x + zˆ 0 3 dz= Y x z= zˆ Y 4π A ∞

Y x x=ºY c, (6.5)

z

r

Figure 6.1. The current density distribution j is along the z axis and is turned on at t = 0. We use cylindrical coordinates. For the remaining z= integration we use cylindrical coordinates (see Figure BLP1.2cyl) so we can write


44


x r rˆ Y x x= Y

.

[

r2

FROM

E XTENDED S OURCES

(6.6) (6.7)

= /

z2

The step function in the integrand is zero when its argument is less than zero. This means that when integrating over z= , only those z= contribute which satisfy

>Y r rˆ z= zˆ Y» c ; 0 .

t

;

ct

[

r2

z= 2

(6.8)

which can be written, if we assume

; 09

t

(6.9)

as z= 2

.

r[ 2 8 c2t 2 z= 8Y c2t 2 r2 Y .

or

[

c2 t 2

(6.10)

[

(6.11)

r 2 8 z= 8

r2 - a

c2 t 2

(6.12)

where we have introduced a simply as a shorthand. These limits can be understood as follows (cf. Fig. 6.2): after the current is switched on, EM fields are sent from each point along the wire and travel at the speed of light. Since the information (i.e., the current turnon) carried by the fields travel in the “line of sight”, or in other words in a straight line, each field point only sees those parts of the current which are close enough. This illustrates the concept of retarded time, which is only meaningful relative the field point, and also the information gathering sphere.

2¿

¿

P

P

P r

t

¼ r½ c

t

¼"¾ ¿ 2 À r2 ½ c

t

¼Á Â 2¿ÄÃ 2 À r2 ½ 2

Figure 6.2. This series of snapshots shows what the part of the current is seen at the field point P at different field times t. So we have now that µ J a A Å zˆ 0 4π Æ Ç a

Å

È

1

É Ê r2 1 Ê"Í 1 Î z2

µ J zˆ 0 ln ÌË 4π 1Î

Í 1Î

dzÉ r2 c2 t 2 r2 c2 t 2

ÏÐ


(6.13)

6.3. S OLVED

45

EXAMPLES

On the other hand, the scalar potential we may set so φ t 9 x +

0, since ρ t = 9 x=H+

0.

Now that we have the vector potential we can derive the E and B fields. The B field is B

A

(6.14)

and since A only has its zˆ -component different from zero in the cylindrical system, we have ∂ Az (6.15) B ϕˆ ∂r ∂ Az ∂r .

Thus we only need

Introducing f

-

^

1 r2 c2t 2 , which satisfies

∂f r 1 2 2 ∂r ct f

(6.16)

we then can write

µ0 J ∂ 1 ln 4π ∂ r 1 µ0 J 2 4π r f

∂ Az ∂r

.

B

ϕˆ

2π r

|

f f

!Ñ! (6.17)

µ0 J 1

(6.18)

r2 c2 t 2

Observe that for large t we have the “static” case: B

ϕˆ µ0 J 2π r

The electric field is derived from E

∂A ∂t 1 µJ ∂ zˆ 0 ln 4π ∂ t 1

f !! f

(6.19)

But

∂f ∂t

1 2r2 2 f c2 t 3

1 r2 f c2 t 3

(6.20)

and so

1 ∂ ln ∂t 1

1 1

f !Ñ!4 f

∂f

f 2 ∂t f 1 f

2

2 ft

(6.21)

and E

µ0 J | 1 2π t 1

zˆ

Notice that as t

'

(6.22) r2 c2 t 2

∞ then E

'

0!

All we have left is to determine the Poynting vector S


46


1 E B µ0

µ0 J2 rˆ 2 2π 2 rt 1 cr2t 2

S

1 µ J | 0 µ0 2π r 1

r2 c2 t 2

2π t

|

µ0 J 1

r2 c2 t 2

FROM

E XTENDED S OURCES

zˆ ϕˆ

(6.23)

(6.24)

From this expression we can for example calculate the radiated power per unit length by integrating over a cylindrical surface C enclosing the wire:

Ò S dC

P

µ0 J2 2π t 1 cr22t 2

(6.25)

We see that an infinite power is transmitted starting at t 0 and r 0 which travels out to infinite r. So in practice it is impossible to produce this physical setup. This is due to the quick “turn on”. In the physical world only gradual turn ons are possible. This many be seen as a consequence of what is known as Gibb’s phenomenom. In the above we tacitly employed the retarded potential without discussing the possibility of using the advanced potential. Let us see what happens if we apply the advanced potential to this problem. The only thing that changes from the outset is that the source time t = is replaced by the advanced time ta - t Y x x=ºY c instead of the retarded time tr - t Y x x=·Y c. With ta as the argument to the step function, the contribution to the integral comes only from those z= which satisfy

"Y r rˆ z= zˆ Y c ; 0 .Ó t 8Y r rˆ z= zˆ Y c

t

(6.26)

If we assume that

8 09

t

(6.27)

then we may write this as

. .

z = 2 r 2 ; c2 t 2 [ z= ;>Z c2 t 2 r 2

or z=

8"

[

c2 t 2

r2 9

[

(6.28) (6.29)

c2 t 2

r 2 8 z=

(6.30)

One can proceed further and calculate the resulting integral. But what is interesting is that now we see that the relation (6.9) seems to say that we have no information about what happened before turn on, while the relation (6.27) says we have no information about what happened after turn on. Physics seems to be conspiring on us in such a way that we cannot compare the advanced and the retarded potential at the same time! E ND


OF EXAMPLE

6.1 %

6.3. S OLVED

47

EXAMPLES

M ULTIPLE HALF - WAVE ANTENNA A wire antenna with a length of m »ÔÖÕ half wavelengths is assumed to have a current dis-

tribution in the form of a standing wave with current nodes at the its endpoints. Determine the angular distribution of the radiated electromagnetic power from the antenna. Hint: It can me convenient to treat even m and odd m separately.

Solution One realises that the setup in this problem is an antenna, since we have a monochromatic source and the current is an extended one-dimensional distribution. Thus, we may use the “antenna formulae,” but since only the radiated effect is ask for all we need is

×

S Ø+

k2 sin2 θ µ0 c YÙ 32π 2 r2

Y 2 rˆ

(6.31)

where l

Ù43 A jω z= e l

A

n

ikz cos θ

dz=

(6.32)

so we seek a form for the current distribution jω z= . It simplifies matters if we consider the cases for m an even and m an odd multiple of half wavelengths separately. As shown in Figure 6.3, for the case of m even, we will use sin kz =¹ and for m odd we use cos kz=¹ .

l

l

a) even .

z=

sin kz=¹

l

b) odd

.

l

z=

cos kz=¹

Figure 6.3. Depending on whether the length of the wire is an even (as seen in a) or odd (as seen in b) multiple of half the wave length, the current distribution is sin kz=¹ or cos kz=¹ . We perform both integrations over z= from l to l which is the total length L, which is a multiple m of λ 2. These facts give us a relation between the integration limits and k, namely πm L 2l mλ 2 . l / (6.33) 2 k Draft version released 15th November 2000 at 20:39

E XAMPLE 6.2

48


FROM

E XTENDED S OURCES

Let us consider the odd case. The current distribution is written jω z= + J0 cos kz=

(6.34)

and so

l u kz= ÙÚ 3 A J0 cos kz= e A ikzn cos θ dz= dz=Û 1 k du l mπ Ü 2 J A A 03 A eiu e iu e iu cos θ du 2k mπ Ü 2 mπ Ü 2 A A A J eiu C 1 cos θ D e iu C 1 cos θ D 0 2k i 1 cos θ i 1 cos θ A mπ Ü 2

mπ mπ 1 cos θ sin 1 cos θ 2 2 π π cos θ T sin 1 cos θ sin 1 cos θ V:Þ 2 2 J0 2 sin mπ 2 cos mπ 2 cos θ 2 cos θ cos mπ 2 sin mπ 2 cos θ ß k sin2 θ 2J0 cos mπ 2 cos θ (6.35) k sin2 θ J0 k sin2 θ

Ý

sin

Inserting this into (6.32) gives us

×

SØ

J02 µ0 c cos2 mπ 2 cos θ rˆ 8π 2 r2 sin2 θ

(6.36)

Let us consider the case when m is even so the current distribution is written jω z= + J0 sin kz=

/

(6.37)

We remember the relation (6.33) which is still valid but now m is an even number. So we have that


6.3. S OLVED

mπ

2k A ÙÚ J0 3 à mπ e

n

ikz cos θ

2k

Ü

mπ 2

J0 2ik J0 2ik

J0 ei C 1 2ik

3 A 3 A

Ü

mπ 2

Ü

mπ 2

Ü

mπ 2

J0 ik sin2 θ J0 ik sin2 θ

49

EXAMPLES

A

e

A

iu cos θ

A e C

u kz= dz= 1 du u A eiu e iu du

sin kz= dz=

iu 1 cos θ

D e A iu C 1}

eA iC 1A i 1 cos θ

cos θ

D

mπ 2

cos θ

Ý

.

×

SØ

ei C 1

}

e A i C 1} i 1 cos θ

cos θ

mπ R cos θ 2

2

2J0 cos T ik sin2 θ Ý 2J0 sin T i k sin2 θ

.áY Ù Y 2

mπ 2

D du D

mπ 2

cos θ

D m2π

Ü

mπ 2

A

Ü

mπ 2

mπ R 1 cos θ sin P 1 1 cos θ sin P 1 cos θ 2 Þ Ý mπ R mπ R T sin P 1 cos θ sin P 1 cos θ V

cos θ T sin P 1

D

cos θ

mπ R cos θ 2

mπ mπ sin T cos θ V 2 V 2 mπ cos θ V 2

2

mπ R 2 V~Þ mπ mπ cos θ sin T cos T cos θ V+Þ 2 V 2

sin P 1 cos θ

(6.38)

4J02 mπ sin2 T cos θ V 4 2 2 k sin θ

(6.39)

J02 µ0 c sin2 m2π cos θ rˆ sin2 θ 8π 2 r2

(6.40)

E ND


OF EXAMPLE

6.2 %

50

E XAMPLE 6.3


FROM

E XTENDED S OURCES

T RAVELLING WAVE ANTENNA A wire antenna of length â is fed at one of its endpoints by a transmitter signal and is at its other end terminated with a resistance to ground. The termination is adjusted such that no current is reflected back on the wire. This means that the current distribution comprises travelling waves emanating from the feed point so one can assume that j z =H~ J exp ikz=H along the wire. Determine the angular distribution of the electromagnetic radiation from this antenna.

Solution We need the formula

× where

k2 sin2 θ 32π 2 r2

SØ

]

µ0 YÙ ε0

LÜ 2 Ù43 A dz= e A LÜ 2

ik cos θ z

Y 2 rˆ

(6.41)

n j z=

(6.42)

In this case the distribution is given: j z=¹~ we get

J0 eikzn . If we insert this distribution into (6.41),

LÜ 2 ÙÚ J0 3 A dz= eik C 1 A cos θ D zn LÜ 2 LÜ 2 A eik C 1 cos θ D zn 1 iα A e e J0 sin α ik 1 cos θ A L Ü 2 2i kL 2 sin 1 cos θ ¸! J0 k 1 cos θ 2

4J02 k2 1 cos θ

YÙ Y2

2

sin2

kL 2

iα

(6.43)

1 cos θ !

(6.44)

Finally we have

×

SØ

J02 sin2 θ 8π 2 r2

]

µ0 sin2 kL 2 1 cos θ rˆ ε0 1 cos θ 2

(6.45)

E ND


OF EXAMPLE

6.3 %

6.3. S OLVED

51

EXAMPLES

M ICROWAVE LINK DESIGN

E XAMPLE 6.4

Microwave links are based on direct waves, i.e., propagation along the line of sight between the transmitter and receiver antennas. Reflections from the ground or a water surface may in unfavourable cases distort the transmission. Study this phenomenon using the following simple model: The transmitter antenna T is a horizontal half-wave dipole placed a distance h 1 above the ground level. The receiving antenna R is in the main lobe from T, at a horizontal distance D from T, and at height h2 . The signal at R is considered to be composed of the direct wave and a wave reflected from the ground. The reflection is assumed to cause a phase shift π in the wave, but no loss of power. The ground is considered flat over the distance D, and h1 9 h2 ã D. 1. Calculate the electric field E (magnitude and direction) in at R if the transmitter antenna is fed with a current j. 2. Discuss the meaning of the result. 3. Simplify the result for the case h1 h2

λ D.

ã

äê ç R ä ê ä ê ä ê ä ä ä ä ä ä ê ê ê ä éz ä r ä ä ê ê ä ä 1ä ä ê ä ä ê ê h2 ä ä ä ä ê ä ê T çè ëä ë è ê ë ë r2 ê ê ê ë h1 ê ë ë ê çè h1 å

æ D

Solution The Fourier transform of the E field in the far zone (radiation field) from a half-wave dipole antenna, i.e., a linearly extended current distribution with length λ 2 in the z direction, Eω ì rad r +

i eikr rˆ 4πε0 c r

λÜ 4 dz= eik B xn jω k 3 A λÜ 4

Since k k rˆ and we study the radiation in the maximum direction, i.e., perpendicular to jω z=¹~ J cos kz=¹ zˆ so that k x=m 0, this expressions simplifies to Draft version released 15th November 2000 at 20:39

52


FROM

E XTENDED S OURCES

λÜ 4 3 A dz= J cos kz= M§ rˆ zˆ rˆ ·¨ λÜ 4 ikr ik e J iJ eikr zˆ §J sin kzKML = ·¨ λA Ü λ4Ü N 4 zˆ 4πε0 c r k 2πε0 c r

ik eikr 4πε0 c r

Eω ì rad r ~

2 A superposition of the direct and reflected contributions (with the distance from the transmitter T to the receiver R equal to r1 and r2 , respectively), with due regard to the phase shift π (corresponding to a change of sign in the current), gives the Fourier transform of the total E field at the far zone point R : Eωtotì rad r @ Since h1 9 h2

^

ã

refl Edir ω ì rad r Eω ì rad r +

iJ 2πε0 c

eikr1 r1

eikr2 ! zˆ r2

(6.46)

D, Pythagoras’ theorem gives

h h 2 h2 h2 h h ∆r h1 h2 2 í D 1 2 D 1 2 1 2 r 2D 2D D 2 2 2 2 ^ h h h h h h ∆r 2 D 1 2 1 2 r r2 D2 h1 h2 2 í D 1 2D 2D D 2 r1

D2

where r

D

h21

h22 í D

(6.47)

2D

and 2h1 h2 (6.48) D is the difference in path distance. Insertion of (6.47) and (6.48) into (6.46), with k ω c 2π λ , gives the Fourier transform of the field at R ∆r

Etot ω ì rad r +

iJ eikr A Te 2πε0 c D

Ü eik∆r Ü 2 zˆ V

ik∆r 2

2π h1 h2 J eikr sin zˆ πε0 c D Dλ

from which we obtain the physical E field Etot rad r 9 t +

Re î Eωtotì rad e

A

iω t

ï

J 2π h1 h2 2π sin cos πε0 cD Dλ λ

D

h21

h22 2D

!ð ω t zˆ

We see that the received signal at R will be extinct if 2π h1 h2 Dλ

nπ E ND


OF EXAMPLE

6.4 %

L ESSON 7

Multipole Radiation 7.1

Coverage

We look at electric dipole, magnetic dipole and electric quadrupole radiation. Multipole radiation analysis is important since it simplifies the calculation of radiation fields from time-varying field and since EM multipoles exist in many fields of physics such as astrophysics, plasma physics, atomic physics, and nuclear physics.

7.2

Formulae used

Fields in far regions from an electric dipole Bω ñ rad x

ω µ0 eik ¯ x ¯ ® ® pω ñ 1 k 4π x

Eω ñ rad x

1 eik ¯ x ¯ ® ®ò pω ñ 1 kó k 4πε0 x

Fields in the far zone from a magnetic dipole Bω ñ rad x

Eω ñ rad x

µ0 eik ¯ x ¯ ® ® mω k k 4π x

k eik ¯ x ¯ ® ® mω k 4πε0 c x

Field in the far zone from an electric quadrupole Draft version released 15th November 2000 at 20:39

53

54

L ESSON 7. M ULTIPOLE R ADIATION

Bω ñ rad

i µ0 ω eik ¯ x ¯ ® ® k Q k 8π x

Eω ñ rad

i eik ¯ x ¯ ® ®ô k Q kõ k 8πε0 x

7.3 E XAMPLE 7.1

Solved examples

ROTATING E LECTRIC D IPOLE An electric dipole with constant electric dipole moment magnitude is located at a point in the xy plane and rotates with constant angular frequency.

(a) Determine the time-dependent electromagnetic fields at large distances from the dipole.

(b) Determine the radiated average power angular distribution and the total radiated power.

Solution (a) We can write the time-varying dipole momentum relative the location of the dipole as p t +

p0 cos ω t xˆ

sin ω t yˆ

(7.1)

which represents a constant dipole moment p0 times unit vector rotating with angular frequency ω . This can also be rewritten in complex form

.

p t +

Re î p0 e

.

p t +

A p0 e

or

iω t

A

iω t

xˆ

xˆ

ip0 e A

ip0 e A

iω t

iω t

yˆ

yˆ ï

(7.2) c/ c/

(7.3)

where c.c., stands for the complex conjugate of the term opposite the sign. In what follows we use the convention that we write the dipole expression as a complex quantity but we drop the c.c. term, which is commonplace when discussing harmonic oscillation. It is easy to identify the Fourier component of the dipole moment in this case pω t +

p0 xˆ

ip0 yˆ /

(7.4)

We notice, in this complex space variable space, that the yˆ component has the phase factor i eiπ Ü 2 relative the xˆ component, which is due the circular rotation. We would like to express (7.4) in spherical components rather than Cartesian components since the expressions for dipole fields in spherical components are simpler, so we transform


7.3. S OLVED

55

EXAMPLES

the base vectors xˆ and yˆ in the conventional fashion and get pω t #

cos θ cos ϕθˆ sin ϕϕˆ i sin θ sin ϕ rˆ cos θ sin ϕθˆ cos ϕϕˆ O+ p0 rˆ sin θ cos ϕ i sin ϕ θˆ cos θ cos ϕ i sin ϕ ϕˆ sin θ i cos ϕ O+ö cos ϕ i sin ϕ eiϕ ÷ p0 eiϕ rˆ sin θ θˆ cos θ iϕˆ p0 O sin θ cos ϕ rˆ

(7.5) (7.6) (7.7) (7.8) (7.9) (7.10)

Now that we have the Fourier component of the dipole moment expressed in spherical components we insert this into the dipole radiation fomulae:

µ0 ω eikr pω k 4π r 1 eikr Eω pω k k 4πε0 r

Bω

(7.11) (7.12)

First we calculate p

θˆ

rˆ

ϕˆ

k p0 ke øø sin θ cos θ i øø øø 1 0 0 øø ø øø p0 keiϕ ø iθˆ cos θ ϕˆ iϕ

(7.13) (7.14)

from which we get

p

k

rˆ θˆ

ϕˆ

k p0 ke øø 0 i cos θ øø 1 0 0 ø 2 iϕø p0 k e cos θθˆ iϕˆ iϕ

øø

(7.15)

øø

(7.16)

øø

so finally we can write the field in space and time coordinates (remember: A B t 9 x + Re î Bω x e iω t ï ),

A

µ0 ω ei C kr ω t D Re p0 keiϕ iθˆ cos θ ϕ úùû 4π r µ ωp k } 0 0 Re ei C kr A ω t ϕ D eiϕ iθˆ cos θ ϕ Þ 4π r Ý

B t 9 x :

(7.17) (7.18)

So our final expression is B

E

µ 0 ω 2 p0 sin kr ω t ϕ θˆ cos θ cos kr ω t ϕ ϕˆ 4π cr ω 2 p0 cos θ cos kr ω t ϕ θˆ sin kr ω t ϕ ϕˆ 4πε0 c2 r Draft version released 15th November 2000 at 20:39

(7.19) (7.20)

56


×

(b) We use the formula S Ø monochromatic so

×

1 2 µ0 E

B ü where the E and B fields are complex and

1 1 µ0 ω 1 1 E Bü 2 µ0 2µ0 4π 4πε0 r2 ω k p k pü k 2 2 32π ε0 r ω Y p k Y 2 rˆ 32π 2 ε0 r2

SØ

k

O p p

k k

k

p

pü

k +

k O~

(7.21) (7.22) (7.23)

µ0 p20 ω 4 1 cos2 θ rˆ 32π 2 cr2

(7.24)

The total power is then

3

P

× Ω

Sr Ø r2 dΩ

(7.25)

π µ0 p20 ω 4 2π 3 dθ sin ω 1 cos2 θ + 2 32π c 0 1 µ0 p20 ω 4 2 2π 3 2 A 1 dx 1 x ~ 32π c µ0 p20 ω 4 6π c

(7.26) (7.27) (7.28) E ND

E XAMPLE 7.2

OF EXAMPLE

7.1 %

ROTATING MULTIPOLE Two point charges of equal charge q are located in the xy plane at either end of the diameter of a circle of radius a. The particles rotate with a constant angular speed ω in the plane of the circle. Determine

(a) The Fourier components of p1 , m, and Q and

(b) The radiation diagram when ω a ã

c

Solution


7.3. S OLVED

r1= r2= v1= v2= p

57

EXAMPLES

a cos ω0t xˆ sin ω0t yˆ * r1= ω0 a sin ω t xˆ cos ω0t yˆ * v1=

(7.29) (7.30) (7.31) (7.32)

∑ qn rn= 0

(7.33)

n

m

1 rn= 2∑ n

qvn= 1 2π

1 r= 2 1

3 A

∞

qv1=

r1=

q

v1= O+ qω0 a2 zˆ

mω

zˆ qω0 a2

Qi j

∑ qn xin= x= jn

(7.36)

Qxx Qyy Qxy Qiz

q x21 x22 + 2qa2 cos2 ω0t qa2 1 cos 2ω0t 2qa2 sin2 ω0t qa2 1 cos 2ω0t Qyx qa2 sin 2ω0t Qzi 0

(7.37) (7.38) (7.39) (7.40)

Qxx Qyy Qxy

qa2 1 e A i2ω0t A qa2 1 e i2ω0t Qyx qa2 ie A i2ω0t

(7.41) (7.42) (7.43)

∞

e iω t d ω

qω0 a2 zˆ δ ω

(7.34) (7.35)

n

Fourier transform

Qxx ì ω

∞ 1 3 A 1 1 2ei2ω0t 1 2e A 2π ∞ 1 qa2 δ ω W δ ω 2ω0 O 2π

qa2

Qyyì ω

Qxyì ω

Qω

∞ 1 3 A qa2 1 cos 2ω0t eiω t dt 2π ∞ qa2 δ ω 1 2δ ω 2ω0 1 2δ ω

qa2

1 2π

3 A

qa2 δ ω ro

∞ ∞

e

i2ω0 t

(7.44)

2ω0 O

(7.45)

eiω t dt

(7.46) (7.47)

A i C 2ω 0 A ω D t dt qa2 iδ ω 2ω0

1 0 0 0 1 0 0 0 0

s u qa2 δ ω 2ω0 r o

(7.48) 1 0 0


0 0 1 0 0 0

u

s (7.49)

58


B

E

×

iµ0 ω eikr k Q k 8π r i eikr k Q k 8πε0 r O 1 E 2 µ0

SØ

Bü

(7.50)

k

(7.51)

(7.52) E ND

E XAMPLE 7.3

OF EXAMPLE

7.2 %

ATOMIC RADIATION A transition in an atom is described as quantum matrix element of a radiation operator between the un-normalised eigenstates Ψf Ψe

re A r exp i Eý 1 t cos θ e A r exp i Eý 0 t

(7.53)

At a certain moment, the atom is therefore described by

c f Ψ f ce Ψe

Ψ

(7.54)

where cf and ce can be viewed as given constants, chosen such that Ψ becomes normalised. According to semiclassical theory, one can interpret the magnitude squared of the wave function as a particle density function. Determine, according to this semiclassical interpretation, the power emitted by the atom via the dipole radiation which appears due to the transition between the two states. The power from an electric dipole is given by P

µ0 ω 4 Y p Y 2 12π c

(7.55)

Solution The charge density is qΨΨ ü

ρ

q TþY c f Ψ f Y 2 c f ceü Ψeü Ψ f c ü f ce Ψe Ψ ü f "Y ce Ψe Y 2 V

q TþY c f Y 2 r2 exp 2r cos2 θ "Y ce Y 2 exp 2r O

(7.56) (7.57)

P c f ceü exp i E1 E0 t c ü f ce exp i E1 E0 t R r cos θ exp 2r (7.58) V

Only the two last terms contribute because they are non-static. Via inspection we find the Fourier components for ω E1 E0 OUÿ

ρω r9 θ + qc ü f ce r cos θ exp 2r


(7.59)

7.3. S OLVED

59

EXAMPLES

Consequently, the corresponding Fourier component of the dipole moment p is p

3

qc ü f ce r cos θ exp

2r r rˆ r2 dr dΩ

qc ü f ce 3 r4 exp 2r dr ! 3 1 2

qc ü f ce 4! 3

2π 0

12qc ü f ce 0 9 0 9 3

3

π 0 2π 0

12qc ü f ce 0 9 0 9 2π 3

¶ ρω r d3x

2π 0

3

π

(7.60)

cos θ sin θ rˆ dθ dφ

(7.61)

0

cos θ sin θ sin θ cos φ 9 sin θ sin φ 9 cos θ dθ dφ

3 0

π 0 π

cos2 θ sin θ dθ dφ +

1 2 sin θ

(7.62) (7.63)

sin θ cos 2θ dθ +

(7.64)

π

12π qc ü f ce 0 9 0 9 1 1 2 3 sin 3θ sin θ dθ @ 0 12π qc ü f ce 0 9 0 9 1 1 2 1 3 1 O + 8π qc ü f ce 0 9 0 9 1 8π qc ü f ce zˆ

(7.65) (7.66) (7.67) (7.68)

and a similar integral for the complex conjugate term. The power from the electric dipole radiation due to the transition between the two states is given by P

µ0 ω 4 Y p Y 2 12π c

16π q2 µ0 E1 3c ÿ 4

E0 4

2 c ü f ce /

(7.69) E ND

OF EXAMPLE

7.3 %

C LASSICAL P OSITRONIUM

E XAMPLE 7.4

Calculate the radiation from a positron-electron head-on collision and subsequent annihilation, classically by assuming that the particles travel at a constant velocity v 0 ã c up until the time the annihilate.

Solution Background Consider a system of localised charges in motion. If we assume that we are observing at a distance Y x Ym R much greater than extension Y x = Y of the charge system and further that v ã c, it can be argued that the source time t = is approximately t R c (where R Y x Y ) instead of t ,Y x x=ºY c. This is because, due to the first assumption x (7.70) Y x x= Y í R x= R Since the timescale for the system is of the order T Y x =ºY v and since v ã c we have that

Y x=·Y c

ã

T

(7.71)


60


and this allows us to write t=

í t R c

(7.72)

The vector potential is this case µ0 3 j t R c d3x A (7.73) 4π R since the denominator in the integrand is now not dependent on the source coordinates. Substituting j ρ v, we rewrite the vector potential as µ0 A Σqv (7.74) 4π R Observe that the summation can be written as d Σqx= p˙ (7.75) Σqv dt where p is the electric dipole. Thus, µ0 A p˙ (7.76) 4π R Deriving the EM field in the usual manner we get

R R

1 p¨ 4πε0 rc2 O

E

R

(7.77)

R

R µ0 p¨ 4π rc2 R It can be shown that the angular and spectral distribution of energy is B

d dω dΩ

O p¨ ω

R R 2π c3

(7.78)

2

(7.79)

The Calculation The dipole electric moment of the positronium is p

2qe v0t zˆ

t t

0

z 0 ; 0

We take the second time derivative of this ∂p 2qe v0 θ t zˆ ∂t

∂ 2p 2qe v0 δ t zˆ ∂ t2 For the spectral and angular distributions of the radiation we have

(7.80)

(7.81) (7.82)

d R R 2 p¨ ω (7.83) dω dΩ 2π c3 where subscript ω denotes the Fourier transform of the dipole. Now the Fourier transform of the dipole is simply p¨ ω 2qv0 zˆ 2π 1 Ü 2 , so that Draft version released 15th November 2000 at 20:39

7.3. S OLVED

61

EXAMPLES

d q2e v20 zˆ Rˆ 2 π 2 c3 (7.84) dω dΩ In the final result we notice that there is no dependence on ω so the spectral density is white noise. E ND


OF EXAMPLE

7.4 %


62

L ESSON 8

Radiation from Moving Point Charges 8.1

Coverage

In our series on deriving fields from given sources we have come to the most fundamental case: the moving point charge. The fields are derived from the LiénardWiechert potentials. In what follows we will assume that the motion x t is known “in advance”. We find that accelerating charges radiate. We also look other mechanisms for a point charge to radiate such as Cerenkov emission.

8.2

Formulae used

According to the Formulae (F.23–26), the fields from a charge in arbitrary motion are given by

E t « x

q Rv 1

4πε0 s3

B t « x

x x

s

®

®

x x

v2 c2

E t « x ® ® c x x

x x

v c


Rv x x

c2

v˙

(8.1)

(8.2)

(8.3)

63

64


FROM

M OVING P OINT C HARGES

x x vc

x x

Rv

®

∂t ∂t

x

x x s

(8.4)

® (8.5)

Alternative formulae x x

r

v β« c

Rv

x x0

∂∂tt

E XAMPLE 8.1

β

r β r

s

8.3

(8.6)

x

s r

® ®

β

r rβ

v c

(8.7)

rv

r β r cos θ

r0

(8.8)

r 1 β cos θ

(8.9)

1 β cos θ

(8.10)

Solved examples

P OYNTING VECTOR FROM A CHARGE IN UNIFORM MOTION Determine the Poynting vector for the fields from a charge q which moves with constant velocity v. Show that no power is emitted from the charge during the motion.

Solution In general the fields due to a single point charge may be written as E B

Ev Erad Bv Brad

(8.11) (8.12)


8.3. S OLVED

65

EXAMPLES

Ev and Bv are known as the velocity fields and Erad and Brad are known as acceleration fields. These can be derived from the Liénard-Wiechert potentials and result in q r rβ 1 β 2 · 4πε0 s3 q T r P r rβ β˙ R 3 4πε0 cs

Ev

Erad

(8.13)

V

(8.14)

and

Bv Brad

q β r 1 β 2 3 4πε0 cs q r Pr rˆ 2 3 4πε0 c s Ý

(8.15)

˙R β Þ

rβ

(8.16)

- v c and s r β r

(8.17)

Brad - 0 and that 1 B β E

(8.18)

where β

The big outer square brackets P R indicate that one should evaluate their content at the retarded time t = t Y x x= t =HQY c, where x= t =H is the given motion of the charged particle. It is this that makes the equations for the fields difficult to evaluate in general. It may not be difficult to get an expression for x = t =H , but then to solve the equation for the retarded time t = t Y x x= t = QY c for t = t = t 9 x . In this case it is not necessary to perform this transformation of variables since we are not interested in the time evolution, so we drop the brackets. It is easy to verify that for uniform motion of the charge q, or in other words β˙ - 0, that Erad

c so the Poynting vector in this case is S

1 E µ0 c

β

E +

]

ε0 E β E µ0

(8.19)

Furthermore, it can be shown that s

| r β r + r02 r0 β 2

(8.20)

where we have introduced the virtual position vector r0 we may write E

q 1 1 β2 3 4πε0 r0 1 β 2 sin2 θ

- r rβ . With these last relations

r 3 2 0

Ü

(8.21)

where θ is the angle between r0 and β . Inserting this relation for E into the relation for S we obtain


66


S So

β and

]

(8.22)

r0

β r0 sin θ ϕˆ

(8.23)

r0


ε0 q 2 1 1 β2 2 6 r0 β r0 2 2 2 2 3 µ0 16π ε0 r0 1 β sin θ

FROM

β

r0 ~ β r02 sin θθˆ

(8.24)

So finally S

q2 v 1 β 2 2 sin θ ˆ θ 2 16π ε0 r04 1 β 2 sin2 θ 3

(8.25)

And now when we integrate the Poynting vector over a spherical surface A with radius R which encloses the moving charge which results in P

Ò S dA Ò S θˆ rˆ dΩ 0

(8.26)

since the Poynting vector is not radial, so a charge in uniform motion in vacuo, does not radiate energy. E ND

E XAMPLE 8.2

OF EXAMPLE

8.1 %

S YNCHROTRON RADIATION PERPENDICULAR TO THE ACCELERATION Determine the angular distribution of synchrotron radiation in the plane perpendicular to the acceleration v˙ for a charged particle moving with velocity v.

Solution We consider only the formulas for the radiation fields, for which the denominator is the cube of the retarded relative distance

r r β r rβ cos θ r 1 β cos θ Now, we have that v˙ r 0, so r rv v˙ : rv J rKML v˙N v˙ r rv & 0 v˙ § r r rβ ·¨ v˙ r2 r2 β cos θ v˙ r2 1 β cos θ v˙ rs s

where θ is the angle between the velocity and r. So that Draft version released 15th November 2000 at 20:39

(8.27)

(8.28a) (8.28b) (8.28c) (8.28d)

8.3. S OLVED

67

EXAMPLES

µ0 q µ q v˙ r v˙ rs 0 2 4π s3 4π s The Poynting vector is given by Erad

S

1 Y E Y 2 rˆ µ0 c

v˙2 rˆ µ02 q2 2 2 16π µ0 c r 1 β cos θ

(8.29)

(8.30)

4

Remember that the Poynting vector represents energy flux per unit time at the field point at the time t. The charge’s energy loss must be related to the time t = , i.e., the time when the energy was emitted!

∂U Sr r 2 ∂t

(8.31)

but

∂U ∂ t=

µ q2 ∂ t ∂U v˙2 r2 1 β cos θ 0 2 2 ∂ t= ∂ t 16π c r 1 β cos θ 4 µ0 q2 v˙2 1 16π 2 c 1 β cos θ 3

(8.32)

E ND

OF EXAMPLE

8.2 %

T HE L ARMOR FORMULA

E XAMPLE 8.3

Derive the Larmor formula by calculating the radiated power of an accelerating point charge due to electric dipole emission. Apply the Larmor formula to linear harmonic motion and circular motion.

Solution The Larmor formula is a very useful equation for deriving the power of emission from non-relativistic accelerating charged particles. It can be derived from the radiation fields of a non-relativistic (v ã c) charged particle, but it can also be seen as an electric dipole relative a co-moving coordinate system. We shall investigate the latter. First derive the power emitted by an electric dipole. Using a time domain (non-Fourier) version of the dipole fields: 1 p¨ 4πε0 rc2

E

rˆ

rˆ

µ0 p¨ rˆ 4π rc we find the Poynting vector to be B


(8.33) (8.34)

68

L ESSON 8. R ADIATION 1 E B µ0 1 16π 2 ε0 r2 c3 1 2 16π ε0 r2 c3 1 2 16π ε0 r2 c3

S

§ p¨ T Y p¨

rˆ

rˆ ¨

rˆ Y 2 rˆ

Y p¨

p¨

FROM


rˆ +

rˆ p¨ rˆ p¨ rˆ V

rˆ Y 2 rˆ

(8.35)

So integrating for the power P t P t #

3

π 0

3

2π

S rˆ r2 sin θ dφ dθ

0

1 16π 2 ε0 c3 1 16π 2 ε0 c3

Y p¨ Y 2

8πε0

c3

π

3

0

π

3

0

π

3

2π

3 3

0 2π 0

Y p¨

rˆ Y 2 sin θ dφ dθ

Y p¨ Y 2 sin3 θ dφ dθ

sin3 θ dθ

0

Y p¨ Y 2 4

8πε0 c3 3

Y p¨ Y 2

6πε0 c3

(8.36)

p t +

qx t

(8.37)

P t +

q2 Y x¨ Y 2 6πε0 c3

q2 Y a Y 2 6πε0 c3

where we have identified the acceleration a t +-

(8.38) x¨ t .

Linear harmonic motion x t +

x0 cos ω0t

2 9 a t + x¨ t + ω0 x0 cos ω0t / q2 ω04 x20 cos2 ω0t P t + 6πε c3

(8.39) (8.40)

0

×

PØ

q2 ω04 x20 12πε0 c3

(8.41)

Circular motion x t +

R0 cos ω0t

9 y t + R0 sin ω0t 9 Draft version released 15th November 2000 at 20:39

(8.42)

8.3. S OLVED

69

EXAMPLES

q2 ω04 R20 6πε0 c3

P t ~

(8.43)

Validity of the Larmor formula The Larmor formula although not covariant in form can indeed be extended such as to be valid for all inertial frames. One point that should definitely be raised is power radiated for more than one accelerating charge. It is not so simple that one may assume that power is proportional to the number of sources N. What must be understood is whether or not the sources are radiating coherently or incoherently. For example, consider the above case of circular motion. If we had a large collection of particles such as the case with in a particle storage ring or in circular wire, the radiation is not automatically proportional to N. If they are bunched the power is proportional to N 2 , this is coherent radiation. If the charges are distributed homogeneously the radiated power is 0. And if the charges are distributed evenly but fluctuate thermally then power is proportional to N, this is incoherent radiation. E ND

OF EXAMPLE

8.3 %

VAVILOV- Cˇ ERENKOV EMISSION

E XAMPLE 8.4

ˇ Show that the potentials at time t at a point inside the Vavilov-Cerenkov cone receive contributions from exactly two positions of the charged particle.

Solution The motion of the charge is given by x = t =H+

x x= 2 y2 § x v t = t ·¨ 2 y2 x2 v2 t t = 2 2vx t t = y2

r2

but r

c n

r2

v t =þ t so that

(8.44)

t t = is the retarded distance so that c2 t n2

t= 2 0

(8.45)

or, from the expression for r 2 above: x2

.

v2 t t = 2 2vx t t = y2

v2

c2 ! n2

c2 t n2

t= 2 0

2 2 2 t t = 2vx t t = y x 0

(8.46) (8.47)

This is a quadratic equation in t t =³ . For a fixed t we would in other words have two values of t = . It remains to show that the result is physically reasonable. Draft version released 15th November 2000 at 20:39

70


^

2vx Z

t t = #

|

vx Z

t t= Ô

4v2 x2 4 x2 y2 v2 2 v 2 c2 n 2

FROM


c2 n 2

x2 y2 v2 y2 2 v2 nc2

c2 n2

(8.48)

c2 2 x y2 #; v2 y2 n2 c2 y2 ; n 2 v2 x2 y2 2 sin αc ; sin2 α α 8 αc

. . . .

(8.49) (8.50) (8.51) (8.52)

ˇ where αc is the critical angle of the Cerenkov radiation; it is half of the opening angle of the shock wave of the radiation. So that, in other words, x 9 y is inside the cone! It remains to be shown that x 8 0

t t=

v2

c2 ! n2

Z ]

c2 2 x n2

Z vx ]

y2 v2 y2 vx

c2 x2 y2 n 2 v2 x2

y2 x2

vx

1

vx Z sin2 αc 2 tan2 α 1 cos α 0 ] 1 1 cos2 αc 1 cos2 α 1 vx Z cos2 α 0 ] 1 cos2 α

c 1 vx Z 1 cos2 α 0 ] 1

t t= J

v2

c2

(8.53)

cos2 α

c 2 ! vx Z 1 1 2α n cos ] 0J 1N MK L N KML

0

(8.54)

0

E ND


OF EXAMPLE

8.4 %

L ESSON 9

Radiation from Accelerated Particles 9.1

Coverage

The generation of EM fields via Liénard-Wiechert potentials are considered simultaneously with the Lorentz force to give a self-consistent treatment of radiation problems. In the previous lesson we solve radiation problems from given expressions for the motion. Now we consider how charges actually move in the EM fields and thus present the Lorentz force. We also discuss the effect of radiation on the motion of the radiating body itself known as radiative reaction

9.2

Formulae used

The covariant Lorentz force d mγ v dt

q E v

B


(9.1)

71

72


9.3 E XAMPLE 9.1

FROM

ACCELERATED PARTICLES

Solved examples

M OTION OF CHARGED PARTICLES IN HOMOGENEOUS STATIC EM FIELDS Solve the equations of motion for a charged particle in a static homogenous electric field E and magnetic field B. Hint: Separate the motion v into motion parallel and perpendicular to the magnetic field respectively.

Solution Background on equations of motion for charged particles As we know, the motion of a charged particle in electric and magnetic fields is given by the Lorentz force F

q E v

B

(9.2)

This can be seen as a definition of the E and B fields and also the fundamental equation for measuring the fields. The E field gives the force parallel to direction of motion and B gives the force perpendicular to the direction of motion. In this certain sense the Lorentz force is trivial: it is simply a definition of the EM fields. The equation for the Lorentz force is relativistically correct as it stands, as long as one interpretes d mγ v (9.3) F dt ^ where m is the mass of the particle, γ - 1 1 v2 c2 , v is the three-velocity, and t is the time. The fact that the Lorentz force in this form d mγ v q E v B (9.4) dt is Lorentz invariant is not immediately clear but can easily be shown. On the other hand this equation is difficult to solve for v t because γ contains v. In many cases one has conditions which are non-relativistic and under such conditions it is possible to simplify (9.3) and thus also (9.4). One simply uses the fact that as v c ' 0 then γ ' 1. In this case F mdv dt, (i.e. the Newtonian force definition), so the Lorentz force becomes q dv E v B (9.5) dt m This equation is the equation of motion for non-relativistic charge particles in EM fields.

The motion As is well known, charged particles are uniformly accelerated in a static and homogenous electric field, and in a static and homogenous magnetic field the charged particles perform circular motion. So what happens in a combined electric and magnetic fields?

The equations of motion are given in (9.5) The first step in solving this equation is to separate the motion into motion parallel with the B field which we will denote with v and


9.3. S OLVED

motion perpendicular to B, which we will denote v ; so v dv

dv

73

EXAMPLES

v y v and thus

dv q q E v B E y dt dt m m and since the X and { are mutually orthogonal the equation separates into

q E dt m and for the perpendicular direction

dv dt

(9.6)

(9.7)

q E v B (9.8) m dv q q v E B (9.9) . dt m m This is a first order linear ordinary differential equation in v . It is also inhomogeneous which means that we have both a solution to the homogenous equation and solution to the inhomogeneous equation. Let us call the solution to the inhomogeneous equation ω D and the solution to the homogenous equation ωR . For the inhomogeneous solution, we notice that neither the inhomogeneity nor the coefficient of the zeroth order term depends on t so the solution ωD itself cannot be time dependent. With this assumption, and that is left is ωD B E . This is easily solved, by taking the cross product of this equation with B we find

ωD B B E B 2 J ωDKML BN B B ωD E B & 0 E B ωD 2

. .

(9.10) (9.11)

(9.12)

B

The solution to the homogenous equation d ωr dt

q ωr m

B

(9.13)

A

on the other hand, can be shown to have the form ωr c e iω t where c is a constant vector perpendicular to B and satisfies c ic4 0 (where the denotes scalar product defined as the inner product of the vectors), so for example c 1 9 i 9 0 if B is in the 3 direction.

Interpretation of the Motion Having derived the solutions for the motion of the charged particle we are now in a position to describe it in words. First of all we see that the motion consists of three separate parts. First we have the motion along the B field which simply is not effected by the B field, so that part of the E field which is along the B field accelerates the charge as if there were no B field. The motion perpendicular to the B field can further be separated into two parts. One part, which here we have denoted ω r , represDraft version released 15th November 2000 at 20:39

74


FROM


y x

z

Figure 9.1. Motion of a charge in an electric and a magnetic field. Here the electric field is along the y axis and the magnetic field is along the z axis. There is a background velocity in the z axis for reasons of clarity. ents so called gyro-harmonic rotation, i.e. the particle moves in a circular orbit, with its axis of rotation parallel with the B field with period of rotation 2π ωc where ωc qB m is known as the cyclotron frequency. This gyro-harmonic motion is the complete solution to our problem if the E field was not considered. The second part of the motion in the perpendicular plane however, which we here denoted ωD involves both the B field and the E field. It is known as the drift velocity since this is the velocity of the centre of the gyro-harmonic motion. The motion is illustrated in Figure 9.1. E ND

E XAMPLE 9.2

OF EXAMPLE

9.1 %

R ADIATIVE REACTION FORCE FROM CONSERVATION OF ENERGY Correct the equations of motion for a charged particle in EM fields to include the effects of the energy loss due to radiation emitted when the particle accelerates. Assume nonrelativistic conditions, so radiated energy is given by Larmor formula.

Solution Background Among other things, Maxwell’s introduction of the electric and magnetic fields was a mathematical technique to divide the work of solving the motion of charge particles in EM fields. Instead of action-at-a-distance, the fields naturally divide the probDraft version released 15th November 2000 at 20:39

9.3. S OLVED

75

EXAMPLES

lem into two parts: the generation of fields from moving charged particles and the motion of charged particles in EM fields. In a certain sense this can be seen as a division into, on the one side a cause and on the other side, an effect of EM interaction, but the formulas governing these processes are not very symmetric with respect to each other in classical electrodynamics: the generation of fields from moving point charges is determined via the Liénard-Wiechert potentials and the motion of point charged particles is determined by the Lorentz force. Despite the lack of symmetry, this division seems successful at describing EM interaction except for one thing: it fails to describe the self-interaction of charged point particles. We know that the motion of charged particles is governed by the Lorentz force, but at the same time we know that acceleration of charged particles causes energy emission. These two facts have been, until now, treated separately, but taken together we realize something is missing. This can be seen for example in the case of a single charge under the influence of a mechanical force in a region of space with no EM fields except for the field from the charge itself. From the Liénard-Wiechert potentials we know that the mechanical force will cause the charge to radiate and if energy is to be conserved the emission must take its energy from the kinetic energy. Since the Lorentz force is zero and there are no other electromagnetic interactions that we know of, as of yet, we have no way of accounting for this radiative “friction”. This, as of yet not mentioned force, is known as radiative reaction or the radiative damping force. One question that comes to mind, after the above discussion, is how can so many problems be described by classical electrodynamics without considering radiative reaction? Obviously, it should have a negligible effect in most cases but what are the limiting conditions? Certainly, these should be determined by considering the conditions under which the energy emitted is of the same order as the kinetic energy of the charge. If we consider non-relativistic motion, the energy emitted by a charge accelerating at the order of a, under a period of duration of order T , is given by the Larmor formula and is of the order of q2 a2 T 6πε0 c3

Erad

(9.14)

On the other hand, the acceleration bestows the charge with kinetic energy on the order of Ekin

m aT 2

2

(9.15)

ã

So if we demand that Erad is equivalent to q2 a2 T 6πε0 c3

ã

m aT 2

Ekin and we wish to neglect the radiation reaction, then this

2

(9.16)

or T

q2 3πε0 c3

(9.17)


76


FROM


and if we define the characteristic time as τ - q2 3πε0 c3 we can say that the effects of the τ. radiative reaction are negligible in measurements made over timescales on order of T

Accounting for radiative reaction Having demonstrated the need for a force which accounts for radiative effects of accelerating charges we set out to determine its form. From the conservation of energy it is clear that the force we are looking for, which we will denote Frad , must satisfy

3 Frad v dt 3

q2 v˙ v˙ dt 6πε0 c3

(9.18)

where have integrated the Larmor formula over time on the right hand side. Partial integration yields

3 Frad v dt 3

q2 v¨ v dt 6πε0 c3

q2 v˙ v 6πε0 c3

(9.19)

if assume periodic motion then we find that

3

Frad

q2 v¨ ! 6πε0 c3

v dt

0

(9.20)

and so Frad

q2 v¨ 6πε0 c3

mτ v¨

(9.21)

We now can correct the equation of motion to include the radiative reaction m v˙ τ v¨ ~

Fext

(9.22)

This equation is know as Abraham-Lorentz equation of motion. Unfortunately, the Abraham-Lorentz equation of motion is not without its own inherent problems. E ND


OF EXAMPLE

9.2 %

9.3. S OLVED

77

EXAMPLES

R ADIATION AND PARTICLE ENERGY IN A SYNCHROTRON

E XAMPLE 9.3

An electron moves in a circular orbit in a synchrotron under the action of the homogeneous magnetic field B0 .

(a) Calculate the energy which is lost in electromagnetic radiation per revolution if the speed v of the electron is assumed to be constant.

(b) At which particle energy does the radiated energy per revolution become equal to the particle’s total energy, if B0 Hint:

1 / 5 T?

d m0 γ v dt =

dU dt =

F9

µ0 q2 v˙ 2 γ 4 6π c

Solution (a) Since we wish to consider synchrotron motion we use the relativistically correct equation of motion

dp qv B dt = where the right hand side is the Lorentz force with E momentum 4-vector so dp dγ d m γ v + m0 γ v˙ m0 v dt = dt = 0 dt = where dγ dt =

d dt =

But v v˙

1

^

v 2 c2 1 1 3d γ3 γ v v + 2 v v˙ 2

2c

0

1

dt

1 3 γ 2

d v2 ! dt = c2

(9.23)

0, and p is the space part of the (9.24)

(9.25) (9.26)

c

0, since we know that the electron moves in a circular orbit, so

dγ dt

0 and thus

dp (9.27) m0 γ v˙ dt = This means that the power necessary to keep the particle in a circular orbit, disregarding radiation losses, is v

dp γ2 m0 γ v v˙ 2 v2 v˙ v »!> 0 dt = c

(9.28)

which comes as no surprise. However, the circular motion does emit EM radiation. To find the expression for the power loss due to radiation we need an expression for the accelera-


78


FROM


tion Y v˙ Y . To this end we use what is left of (9.23), that is m0 γ v˙

qv

B

(9.29)

The acceleration is found by taking the norm of this last equation and since v have is the scalar equation m0 γ Y v˙ YY q Y vB0

X B all we (9.30)

This equation is easily solved to give

.

Y q Y B0

v˙

m0 γ

v

(9.31)

The factor m γ0 - ωc is known as the synchrotron angular frequency and is the relativistic 0 value of the angular frequency for gyro-harmonic motion or cyclotron angular frequency. qB

Now we can insert this expression into the relativistic generalisation of the Larmor formula

µ q2 v˙2 γ 4 µ q2 ω 2 γ 4 dU 0 0 c v2 dt = 6π c 6π c To find the energy per revolution, we need the period of revolution which is T so dU 2π µ0 q2 ωc2 γ 4 2 v dt = ωc 6π c µ 0 Y q Y 3 γ 3 B0 v 2 3cm0

T

Urev

µ0 q2 ωc2 γ 4 2 v 6π c

(9.32)

2π ω c , (9.33) (9.34)

(b) The task here is to equate the total energy and the radiated energy and solve for velocity and then see what total energy that velocity is associated with. The total energy is E m0 γ c2 and the radiated energy is Urev which when equated gives c2 1 v2 γ 2

µ 0 q 3 B0 3cm20

.

c2 v2

1

.

v2 c2

(9.35)

µ 0 q 3 B0 3cm20 1

1

µ0 Y q Y 3 γ 3 v2 B0 3cm0 ,

(9.36) (9.37)

µ0 q3 B0 3cm20

So, after a little algebra,

γ

ª 1

2cm20 µ 0 Y q Y 3 B0

(9.38)

Thus the particle energy for which the radiated energy is equal to the total particle energy is


9.3. S OLVED

E

79

EXAMPLES

m 0 c2 ª 1

2cm20 µ 0 Y q Y 3 B0

0 / 16 TeV

(9.39)

This is obviously the upper limit for which radiation effect can be neglected in the treatment of synchrotron motion of the particle. E ND

OF EXAMPLE

9.3 %

R ADIATION LOSS OF AN ACCELERATED CHARGED PARTICLE

E XAMPLE 9.4

A charged particle, initially at rest, is accelerated by a homogeneous electric field E 0 . Show that the radiation loss is negligible, even at relativistic speeds, compared to the particle’s own energy gain. Assume that under all circumstances E0 8 108 V/m.

Solution The relativistic equation of motion is dp dt =

d m γ v + dt = 0

γ2 v v˙ v ¸!, qE0 2 c

m0 γ v˙

(9.40)

X v˙ X E0 then

Since v

m0 γ v˙ 1 β 2 γ 2 +

m0 γ v˙ .

m0 γ 3 v˙

.

v˙

β2 ! 1 β2

m0 γ v˙ 1

1 !, m0 γ v˙γ 2 1 β2

Y q Y E0

(9.41)

Y q Y E0

(9.42)

Y q Y E0 m0 γ 3

(9.43)

Having found v˙ we try to derive the radiation field generated by this motion; we have that

µ0 q r rv v˙ 4π s3 µ0 q 1 r 4π r3 1 β cos θ 3

Erad

v

PßT r r V c

µ0 q 1 r 3 3 4π r 1 β cos θ

ro r

µ0 q 4π r3 1 µ0 q 4π r3 1 µ0 qv˙ 4π r 1

1 β cos θ 1 β cos θ sin θ β cos θ

r

v˙

v˙ R

s

r u Jv KML vN ˙ c

&

0

v˙

3

r

3

r2 v˙ sin θθˆ

θˆ

3


(9.44)

80


FROM


where have used the particular geometry of the vectors involved. Then we may determine the Poynting vector, which is S

1 2 Erad rˆ µ0 c ø ø

ø

ø

Sr rˆ

(9.45)

The radiated energy per unit area per unit time corresponding to this is

øø E

∂U Sr r 2 ∂t

rad 2

ø r2 ø µ0 c øø

(9.46)

but the energy radiated per unit area per unit time at the source point is

∂U ∂ t=

∂U ∂ t ∂t ∂t=

∂U s ∂t r

rs 2 Erad µ0 c ø ø

β cos θ µ02 q2 v˙2

r2 1

µ0 q2 v˙2 sin2 θ 2 16π c 1 β cos θ

µ0 c

16π 2

ø

ø

sin2 θ r2 1 β cos θ

6

(9.47)

5

The total radiated energy per unit time is π µ0 q2 v˙2 sin3 θ 3 π 2 dθ 5 16π 2 c 0 1 β cos θ µ q2 v˙2 1 1 x2 x cos θ 2π 0 2 3 dx dx dθ sin θ 8π c A 1 1 β x 5 µ q2 v˙2 1 1 x2 2π 0 2 3 A ! dx 8π c 1 1 β x 5 1 β x 5 µ q2 v˙2 4 µ0 q2 v˙2 4 6 µ0 q4 E02 1 2π 0 2 2 π γ 8π c 3 1 β 2 3 8π 2 c 3 6π m20 c

∂ U˜ ∂ t=

3

∂U dΩ ∂ t=

We compare this expression for the radiated energy with the total energy E dE dt =

m0 c2

dγ dt =

(9.48)

m0 γ c2 , so (9.49)

From the equation of motion we find that d dγ m γ v ~ m0 γ v˙ m0 v dt = 0 dt = and, from the expression for v, ˙ m0 γ v˙

Y q Y E0

Y q Y E0

(9.50)

(9.51)

γ2

Hence


9.3. S OLVED

m0 v

81

EXAMPLES

1 Y q Y E0 1 2 ! J KML γ N

dγ dt =

(9.52)

v2 c2 dγ dt =

.

Y q Y E0 v

(9.53)

m0 c2

and dE dt =

m0 c2

Finally, for E0 dU˜ dt =

Y q Y E0 v m0 c2

Y q Y E0 v

(9.54)

108 V/m, and an electron for which Y q YU e, we find that µ0 q4 E02 1 6π cm20 Y q Y E0 v

dE dt =

A

µ 0 Y e Y 3 E0 6π cm20 v

A

E0 4π 10 7 1 / 6 10 19 3 v 6π 2 / 998 108 9 / 11 10 A 31

2

1/ 1

Which of course is a very small ratio for a relativistic electron (v even for E 108 V/m.

10 c

A

(9.55)

v

2 / 998

E ND


12 E0

10 8 m/s)

OF EXAMPLE

9.4 %

82


FROM



ElectroMagnetic Field Theory+Exercises

Recommend Documents