Author: Masud Mansuripur

Field, Force, Energy and Momentum in Classical Electrodynamics

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Library License: US $356
ISBN: 978-1-60805-321-6 (Print)
ISBN: 978-1-60805-253-0 (Online)
Year of Publication: 2011
DOI: 10.2174/97816080525301110101

Introduction

The classical theory of electrodynamics is based on Maxwell's equations and the Lorentz law of force. This book begins with a detailed analysis of these equations, and proceeds to examine their far-reaching consequences. The traditional approach to electrodynamics treats the "microscopic" equations of Maxwell as fundamental, with electric charge and electric current as the sole sources of the electric and magnetic fields. Subsequently, polarization and magnetization are introduced into Maxwell's equations to account for the observed behavior of material media. The augmented equations, known as Maxwell's "macroscopic" equations, are considered useful for practical applications, but ultimately reducible to the more fundamental "microscopic" equations. In contrast, the present book takes Maxwell's "macroscopic" equations as the foundation of classical electrodynamics, and treats electrical charge, electrical current, polarization, and magnetization as the basic constituents of material media. These constituents not only produce the electromagnetic fields, but also interact with these fields and exchange energy and momentum with them. The laws that govern the distribution of energy and momentum in space-time are introduced and discussed in great detail. Interactions of electromagnetic fields with matter involve exchanges of energy with electrical currents, with polarization, and with magnetization, and also exchanges of linear and angular momenta via electromagnetic force and torque exerted on the aforementioned constituents of matter. Throughout the book, a large number of examples demonstrate the solution of Maxwell's equations in diverse situations, and examine the flow of energy and momentum as well as the distribution of force and torque throughout the matter-field systems under consideration.

Preface

This book grew out of a graduate-level course in electrodynamics that I have taught at the University of Arizona's College of Optical Sciences over the past six years. A typical student enrolled in the course is a first year graduate student in Optical Sciences, Electrical Engineering, or Physics, who has had some prior exposure to electromagnetic theory. The level of mathematics required for this subject is not particularly advanced; students are expected to be familiar with calculus, vector algebra, complex numbers, ordinary differential equations, and elementary aspects of the Fourier transform theory. Most of the mathematical tools and techniques needed for developing the theory of electrodynamics are in fact interwoven with the course material in the form of a section here, a chapter there, or a few problems at the end of each chapter. The student is thus motivated to learn the required mathematics in the relevant physical context whenever the need arises.

The approach of this book to classical electrodynamics is rather unconventional. It begins with a minimum set of postulates that are considered fundamental in the sense that they cannot be derived from each other or from other laws of classical physics. The set of postulates, of course, must be selfconsistent, as well as consistent with the conservation laws and with special relativity. These postulates are described in their most general form at the outset, with no apologies for their sudden appearance and no attempt to motivate them, say, by tracing the historical path that led to their discovery. The laws of nature are what they are; it may have taken man a long and tortuous path to their discovery, but once the laws are known, one should simply accept them and try to understand their consequences.

In this context, an analogy with a board game such as chess is constructive. Before one sets out to play the game, one must learn the configuration of the board, the identity of the pieces, and the governing set of rules in their detailed and complete form. For most practical purposes, it is irrelevant how the game has evolved over the years, how the rules may have changed, and whether or not there is any justification for the rules. The important thing is to learn the rules and play the game. In the case of physics, of course, the postulates are justified because their consequences agree with observations. This, however, is something that one will appreciate later, as one begins to understand the subject and learns how to deduce the logical consequences that flow from the basic principles. The task before the student, therefore, is to master the nomenclature and learn the basic rules of electrodynamics, then try to deduce their consequences.

In the presence of known sources of radiation (i.e., sources whose spatio-temporal distributions are given a priori) we will use the method of plane-wave decomposition and superposition to derive general expressions for electromagnetic fields and potentials. This will enable us to examine several idealized situations in Chapters 4 and 5, thereby gaining insight into the nature of electromagnetic fields and radiation. In the course of this analysis, I find it useful to move back and forth between the space-time domain, where the fields and their sources reside, and the Fourier domain, which is home to various plane-waves whose superposition reproduces the fields and the sources. The mathematical methods used in the two domains may differ, but the final results pertaining to physical observables of any given system are invariably the same.

Throughout the book I have striven to be brief yet precise. Whenever possible, I develop a general formalism to tackle a given class of problems, then specialize the solution to examine specific problems within that class. For example, in dealing with plane-wave propagation in isotropic, homogeneous, linear media (Chapter 7), Maxwell's equations are solved in a way that is applicable to transparent as well as absorptive media, encompassing both propagating and evanescent waves while accommodating arbitrary states of polarization (i.e., linear, circular, elliptical). Once the general solution is at hand, a few specific examples show its application to problems such as propagation in transparent or absorptive media, total internal reflection, incidence at Brewster's angle, etc. The student is thus equipped with the tools needed to tackle problems within a broad class, without having to learn each specific case as an isolated instance.

The theory of electrodynamics is too broad, and its applications too diverse, to allow coverage in a brief textbook such as this one. My goal, therefore, is not to be comprehensive, but rather to build a foundation upon which one could base future learning and further investigations. Throughout the book, fundamental notions are laid out in their most general form and described in sufficient detail to give the reader a firm grasp of their content. Examples are then used to bring out important logical consequences of these concepts and to showcase certain practical applications. I have opted for idealized examples with exact analytical solutions, as such solutions can be relied upon to provide valid answers to physical questions even when one or more parameters are pushed to extreme limits. At the same time, each idealized example corresponds to some physical setting in a recognizable limit (e.g., large diameter, vanishing thickness, point-particle, uniform charge distribution, etc.), so that, in principle at least, the predictions made in the context of an idealized situation could be subjected to experimental verification.

The exercise problems at the end of each chapter elaborate the concepts developed in the chapter, providing the student with the means to test his/her understanding of the subject as well as extending the methods and ideas in new directions. One gains broad insight into electrodynamics and appreciates the richness of its various applications by solving these problems and trying to understand the physical meaning behind each solution. In the case of problems marked with an asterisk, detailed solutions are provided at the end of the book.

In a one-semester course, I have been able to cover the first seven chapters, with selected sections from the remaining chapters (e.g., Chapters 8 and 10) assigned for self study. The more advanced topics such as solving Maxwell's equations in cylindrical coordinates (Chapter 9), plane-wave propagation in spatially-dispersive media (Chapter 11), and the reciprocity theorem (Chapter 12) are better left for a second semester. If the course is extended to a second semester (or third quarter), it would be desirable to supplement the present text with other standard topics such as diffraction, spatial and temporal coherence, wave propagation in dispersive media, reflection and refraction in the presence of birefringence and optical activity, time-reversal symmetry, the Ewald-Oseen extinction theorem, and the Lorentz transformation of electromagnetic fields and sources between inertial frames.

I thank many students who, in the course of the past few years, have asked penetrating questions, corrected my mistakes, and generally motivated me by their enthusiasm for the subject. I am grateful to a number of colleagues and associates who have guided me along the path of learning and provided answers to my numerous questions. Special thanks are thus due to Brian Anderson, Stephen Barnett, Jean-Pierre Delville, Poul Jessen, Miroslav Kolesik, Henri Lezec, Rodney Loudon, Jerome Moloney, Miles Padgett, Pavel Polynkin, Din Ping Tsai, John Weiner, Ewan Wright, and Armis Zakharian. I spent the Spring semester of 2010 as a visiting professor at the Physics Department of the National Taiwan University in Taipei, where I wrote several chapters of this book while teaching a group of bright and motivated students. I take this opportunity to thank Taiwan's National Science Council for supporting my sabbatical leave, and also express my gratitude to Professor Din Ping Tsai for being a warm and gracious host. Last but not least, I am grateful to my wife, Annegret, without her loving care and patient support this book would not have become a reality.

Tucson, July 2011
Masud Mansuripur

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