B.Thide - Electromagnetic Field Theory

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Example M.7 Gradients of scalar functions of relative dis-

tances in 3D . . . . . . . . . . . . . . . . . . . 183

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List of Figures

1.1Coulomb interaction between two electric charges . . . . . . . . 3

1.4Moving loop in a varying B field . . . . . . . . . . . . . . . . . 13

4.1 Relative motion of two inertial systems . . . . . . . . . . . . . 50

4.2Rotation in a 2D Euclidean space . . . . . . . . . . . . . . . . . 57

4.3Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1Linear one-dimensional mass chain . . . . . . . . . . . . . . . . 77

8.4Multipole radiation geometry . . . . . . . . . . . . . . . . . . . 118

8.7An accelerated charge in vacuum . . . . . . . . . . . . . . . . . 128

8.8Angular distribution of radiation during bremsstrahlung . . . . . 141

8.9Location of radiation during bremsstrahlung . . . . . . . . . . . 142

8.10Radiation from a charge in circular motion . . . . . . . . . . . . 146

8.11Synchrotron radiation lobe width . . . . . . . . . . . . . . . . . 148

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To the memory of professor

LEV MIKHAILOVICH ERUKHIMOV

dear friend, great physicist

and a truly remarkable human being.

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If you understand, things are such as they are

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

GENSHA

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Preface

This book is the result of a twenty-five year long love affair. In 1972, I took my first advanced course in electrodynamics at the Theoretical Physics department, Uppsala University. Shortly thereafter, I joined the research group there and took on the task of helping my supervisor, professor PER-OLOF FRÖ- 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 BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty.

Intended primarily as a textbook for physics students at the advanced undergraduate or beginning graduate level, I hope the book may be useful for research workers too. It provides a thorough treatment of the theory of electrodynamics, mainly from a classical field theoretical point of view, and includes such things as electrostatics and magnetostatics and their unification into electrodynamics, the electromagnetic potentials, gauge transformations, covariant formulation of classical electrodynamics, force, momentum and energy of the electromagnetic field, radiation and scattering phenomena, electromagnetic waves and their propagation in vacuum and in media, and covariant Lagrangian/Hamiltonian field theoretical methods for electromagnetic fields, particles and interactions. The aim has been to write a book that can serve both as an advanced text in Classical Electrodynamics and as a preparation for studies in Quantum Electrodynamics and related subjects.

In an attempt to encourage participation by other scientists and students in 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

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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 CHRISTER WAHLBERG and GÖRAN FÄLDT, Uppsala University, and YAKOV ISTOMIN, 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 MATTIAS WALDENVIK and TOBIA CAROZZI as well as ANDERS ERIKSSON, 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 HELMUT 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 VITALIY L. GINZBURG 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 MATTIAS, my daughter KAROLINA, my high-school physics teacher, STAFFAN RÖSBY, and to my fellow members of

the CAPELLA PEDAGOGICA UPSALIENSIS.

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Classical Electrodynamics

Classical electrodynamics deals with electric and magnetic fields and interactions caused by macroscopic distributions of electric charges and currents. This means that the concepts of localised 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 have to be treated as spatially extended objects and quantum corrections must be included. However, the limiting processes used will yield results which are correct on small as well as large macroscopic scales.

It took the genius of James Clerk Maxwell to unify electricity and magnetism into a super-theory, electromagnetism or classical electrodynamics (CED), and to realise that optics is a subfield of this new super-theory. Early in the 20th century, Nobel laureate Hendrik Antoon Lorentz took the electrodynamics theory further to the microscopic scale and also laid the foundation for the special theory of relativity, formulated by Albert Einstein in 1905. In the 1930s Paul A. M. Dirac expanded electrodynamics to a more symmetric form, including magnetic as well as electric charges and also laid the foundation for the development of quantum electrodynamics (QED) for which Sin-Itiro Tomonaga, Julian Schwinger, and Richard P. Feynman earned their Nobel prizes in 1965. Around the same time, physicists such as the Nobel laureates Sheldon Glashow, Abdus Salam, and Steven Weinberg managed to unify electrodynamics with the weak interaction theory to yet another super-theory, electroweak theory.

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

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can be unified into one “super-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 with stationary boundaries is called electrostatics. For a long time electrostatics, under the name electricity, was considered an independent physical theory of its own, alongside other physical theories such as magnetism, mechanics, optics and thermodynamics.1

1.1.1 Coulomb's law

It has been found experimentally that in classical electrostatics the interaction between stationary, electrically charged bodies can be described in terms of 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

where in the last step Equation (M.81) on page 184 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 jx x0j in metres (m). The constant "0 = 107=(4 c2) 8:8542 10 12

1The famous physicist and philosopher Pierre Duhem (1861–1916) once wrote:

“The whole theory of electrostatics constitutes a group of abstract ideas and general propositions, formulated in the clear and concise language of geometry and algebra, and connected with one another by the rules of strict logic. This whole fully satisfies the reason of a French physicist and his taste for clarity, simplicity and order. . . ”

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