An Introduction to Classical Electromagnetic Radiation, Manuais, Projetos, Pesquisas de Física

Baixe An Introduction to Classical Electromagnetic Radiation e outras Manuais, Projetos, Pesquisas em PDF para Física, somente na Docsity! & im = [96] A E G E) Ee Ú  | i | This book provides a thorough description of classical electromagnetic radiation, starting from Maxwell's equations and moving on to show how fundairiental con- cepts are applied in a wide variety of examples from areas such as classical optios, , antenna analysis, electromagnetic scattering, and particle accelerators. 'Theoretical - and experimental results are interwoven throughout to help give insipht into the Physical and historical foundations of the subject. ro Following introductory chapters covering the basic theory of clássical electro- magnetism and the properties of plane waves, the concept of a plane-wave spectrum is developed and applied to the radiation from apertures. Radiation from a moving point charge is described in depth, às is that from a variety of thin-wire antennas. A key feature of the boók is that pulsed and time-harmonic fields are presented on an equal footing. Mathematical and physical explanations are enhanced by a wealth of illustrations (over 300), and the book includes more than 140 problems. Ifwill be of, greatinterest to advanced undergraduate and graduate students of electrical engineering and Physics, as well as to scientists and engineers working in applied electromagnetics. = " An introduction to classical electromagnetic . radiation GLENN S. SMITH Regehts' Prófessor of Electrical Engingering * Georgia Institute of Technology Atlanta, GA CAMBRIDGE UNIVERSITY PRESS PUBLISHED BY THB PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE “The Pit Building, Trumpington Street, Cambridge CB2 IRP, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, United Kingdom "40 West 20th Street, New York, NY 100114211, USA io Stamford Road, Oakdeigh, Melboume 3166, Australia O Glen S. Smith 1997 “This book is in copyright. Subject to statutory exception ani to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without : the written permission of Cambridge University Press. First published 1997 Printed in the United States of America Typeset in Times Roman Library of Congress Cataloging-in-Publication Data Smith, Glenn S. (Glenn Stanley), 1945- “An introduction to classical electromagnetic radiation / Glenn S. Smith. - pm Includes bibliographical references ISBN O 521 58093 5 hardback ISBN O 521 58698 4 paperback QCS6L.S6S 1997 96-25107 539.2 - do20 cr. iab A catalog recond for this book is available from the British Library ISBN 521 58093 5 hardback ISBN 0 521 58698 4 paperback To the memory of'my parents : Stanley and Florence, to my wife Linda, and to the future of our children Geoff and Elie Contents Preface 1 Basic theory of classical electromagnetism 1.1 Historical introduction: Maxwell's equations in integral form 1.2 Curl and divergence: Maxwell's equations in differential form 1,3 Surface densities, boundary conditions, and perfect conductors 13.1 Surface densities of charge and current 1.3.2 -Electromagnetic boundary conditions “1.3.3 The concept óf a perfect conductor - 1:4 Energy of the electromagnetic field — Poynting's theorem '1.5 Electromagnetic boundary value problem - uniqueness theorem 1.6 Numerical solution of Maxwell's equations using finite differences: An example a “1.7 Harmonictime dependence and the Fourier transform 17.1 Linear systems 1.7.2 Maxwell's equations 17.3 Poynting's theorem 1.7.4 Uniqueness theorem 1.7.5 Fourier transform References Problems 2 Electromagnetic plane waves in free space: Polarized waves 2.1 General time dependence 2.2 Harmonic time dependence: Monochromatic plane waves 2.3 The polarization ellipse in the coordinate system of the principal axes 2.4 The Poincaré sphere and the Stokes parameters 2.5 Optical elements for processing polarized light 2.6 Transmission and reception of polarized waves with antenas 128 134 140 142 153 vii xii Preface For the sake of persons of... different types, scientific truth should be presented in different forms, and should be regarded as equally scientific, whether it appears in robust form and: the vivid coloring of a physical illustration, or in the tenuity and paleness of a symbolic expression. A glance through the book shows that there are many more illustrations (over 300) than normally found in a book on this subject at this level. Experience has shown that students appreciate these pictures. A physical concept is often more easily grasped and remembered when presented with an illustration that supports the mathematics. Tables sumrnarizing the important equations are included throughout the book. Their púrpose is to keep. the student from having to search through the text to find a useful equation, eithér when solving a homework problem or when using the text as a reference in the future, Experimental results have been included in almost every chapter. “These ré: sults serve several functions. For example, in Chapter 1 the numerical solution of Maxwell's partial -differential equations using finite differences is described, and the theoretically determined radiation of a pulse by an antenna is presented. These * , theoretical results are compared with experimental measurements, and the excellent agreement shows the student the accuracy that can be expected from this beautiful theory. In Chapter 3 experimental results are used for a different purpose. The field in an electrically large aperture is assumed to be uniform when the aperture is il- luminated by a normally incident plane wave. The experimental results are used to show the approximate naturé of this assumption. And in Chapter 7, measurements for. the color and polarization of skylight are used to motivate an examination of molecular scattering in the atmosphere. Many texts at this level concentrate on electromagnetic fields that vary ipi ically in time (usually the Fourier transform of the field). In this book, however, solutions in the.time domain and the frequency domain are considered on an equal footing. The discussion of the former often precedes the discussion of the latter, because the radiation of time-varying signals (pulses) often is more easily under- stood from a physical point of view. An example is the radiation from thin-wire antennas presented in Chapter 8. A pulse of charge traveling along the antenna is considered first. Radiation ocçurs each time the pulse encounters a discontinuity ora bend, a situation analogous to a moving point charge undergoing acceleration. The infinite duration of time-harmonic excitation complicates this simple physical picture for the radiation. Historical information is included in most chapters, with the dates of birth and death given at the first mention of most prominent scientists. This provides the student with some appreciation of the long and rather interesting development of. the subject, as well as a rough idea of the period of time when various discoveries were made. When the students see the array of great scientists who developed this subject over a long period of time, they are often less exasperated by their struggle to become proficient in the subject during their first exposure. The starting point for Chapter 1 is Maxwell's equations in integral form, which are familiar to most students. These equations are used to analyze a few of the basic experiments performed by early investigators - Coulomb, Oersted, Ampêre, Preface y xiii and Faraday. In addition to providing the student with a review, this material adds some historical perspective. The definitions for the vector operations of curl and divergence are introduced, and they are used to 'convert Maxwell's equations in integral form to those in differential form. Only electromagrietic fields in free space or “simple materials” are considered, that is, materials whose constitutive parameters are scalar constants in the time domain (linear, homogeneous, isotropic, and nondispersive materials). Next, the mathematical concept of à surface density of charge or current is described, and the electromagnetic Pound conditions are obtained. The energy of the electromagnetic field is defined, and Poynting's theorem for the conservation of energy is obtained from Maxwell's equations. This theorem is applied to a problem studied by both Poynting and Heaviside, the transport of energy along a straight wire, and the results arerused to illustrate the physical interpretation for the Poynting vector. Next, the elements of an electromagnetié boundary value problem aré described, and a uniqueness theorem for the elettromagnetic field (general time dependence) is obtained by manipulation of Poynting's theorem. At this point, presentation of the basic equations of electromagnetism is com- plete, and the student is in a position to understand the formulation and solution ofa practical problem. What is lacking is a technique for obtaining a solution to the equa- tions. The technique presented is the finite-difference time-dómairi (FDTD) method, a conceptually simple, numerical method that involves the direct discretization'of Maxvwel!'s equations. After demonstrating the method for à one-dimensional prob- lem, we use it to analyze a cylindrical monopole antenná with pulsé excitation. The boundary value problem for the monopole is formulated based on the earlier discus- sion of the uniqueness theorem, the numerical solution for the electromagnetic field is obtained, and the theoretical results are compared with accurate measurements. Chapter 1 ends with a discussion of harmonic time dependence and the Fourier transform. The special properties of linear systems are briefly stated, and the equa- tions of electromagnetism are specialized to the case of harmonic time dependence. Itis my experience that, even though most students are familiar with harmonic time dependence (phasors) from courses on linear circuits, they find this review helpful. Chapter 2 is concerned with the propagation of homogeneous plane waves in free space. For many students this is partly a review. The Poincaré sphere and Stokes parameters are introduced to describe the state of polarization of a monochromatic wave. This material is illustrated with two practical examples: a discussion of optical elements for processing polárized light (the Jones calculus) and the reception of polarized waves by antennas. The chapter ends with a historical note describing Hertz's early experiments with electromagnetic waves, which confirmed Maxwell's equations. Chapter 3 begins with a discussion of inhomogeneous plané wavésin freé space. A physical argument is then used to show that the complete set of homogeneous and inhomogeneous plane waves, i.e., the plane-wave spectrum, can be used to con- struct a solution to Maxwell's.equations. Two-dimensional fields are treated first, with the problem of a uniformly illuminated slit as'an example. The asymptotic or radiated field is obtained by a physical argument based on the method of stationary xiv Prefacç phase. The procedure is extended to three-dimensional fields and illustrated with additional examples: the uniformly illuminated circular aperture, the circular aper- ture with tapered illumination (reflector antennas), and the Gaussian beam (paraxial approximation). Electromagnetic analogues of some of the famous principles from optics, such as Huygens principle, Fresnel zoónes, Babinet's principle, and the method of images, are the focis of Chapter 4. Huygens" principle is obtained by directly manipulating results obtained in Chapter 3 for the plane-wave spectrum, and it is shown to be an alternate representation for an electromagnetic field. The principles are illustrated with examples. These'examplés introduce additional concepts, e.g.. transmission coefficients and scattéring cross sections for planar apertures and obstacles, the forward scattering theorem, Kirchhoff's approximation for the field in an aperture and the physical optics approximation for the current on an obstacle, and comple- mentary antennas. The remaining chapters, 5-8, áre concerned with the radiation from distributions of current and charge. In Chapter 5; following discussions of the scalar electric and vector magnetic potentials and the Dirac delta function, the retarded'potentials for a general distribution of charge and current are derived. The presentation is conven- tional, with the electromagnetic field being determined from the retarded potentials. The general results from Chapter 5 are applied to radiation from a moving point charge in Chapter 6, radiation from infinitesimal dipoles (electric and magnetic) in Chapter 7, and radiation from simple, thin-wire antennas in Chapter 8: In Chapter 6, the electromagnetic field of a moving point charge is derived using the historically impórtant approach based on the Liénard-Wiechert potentials. These . results are then specialized for the cases of low velocity, general velocity with the, acceleration parallel to the velocity (bremsstrahlung), and general velocity with the acceleration normal to the:velocity. Graphics are used extensively to illustrate these cases. Sections of this chapter are devoted to synchrotron radiation (important for circular particle accelerators) and Cherenkov radiation (radiation from a particle whose velocity is greater than the speed of light in the surrounding dielectric). The chapter ends with a brief, qualitative discussion of the self force for a charged particle. Chapter 7 deals with dipole radiation. After a derivation of the electromagnetic field for the infinitesimal electric dipole or current element (general time depen- dence), duality is introduced and used to obtain the field of the infinitesimal magnetic dipole or current loóp. Brief discussions of the electrically short linear antenna qnd the electrically small loop. antenna illustrate the use of these results. Simple arrays of electrically short linear antennas (dipoles) are first dealt with in the time domain. Later, the case of harmonic time dependence is presented, and conventional topics associated with arrays are introduced: pattern muliplication, end-fire and broadside arrays, superdirectivity, etc. The scattering from electrically small objects (Rayleigh scattering) is described in terms of induced electric and magnetic dipole moments. Polarizabilities and scattering cross sections are given for the perfectly conducting sphere,: disc, and thin wire. Babinet's principle is used with the results for the disc to obtain the Preface kv transmission coefficient for an electrically small, circular apeiture. These results, together with those from Chapter 4, cover the two extremes for-electromagnetic scattering and aperture transmission, i.e., scattering (transmission) for electrically small and electrically large objects (apertures). Chapter 7 ends with a discussion of the color and polarization of skylight — the famous “blue sky problem.” First, experimental results for these phenomena are presented, and then a reasonable explanation is sought based on the scattering of sunlight in the atmosphere. Brief discussions of natural light (unpolarized light) and molecular scattering accompany the explanation. The last chapter, Chapter 8, is devoted to radiation from simple, thin-wire an- tennas. First there is a review of the numerical results from the accurate analysis of'the monopole antenna presented in Chapter 1. These results are then used to ob- tain an approximation, based on an “educated guess," for the pulse of chargê on à basic traveling-wave element. More complicated antennas (standing-wave dipole, traveling-wave loop, etc.) are viewed as a superposition of básic traveling-wave elements, and their radiated fields are easily obtained as the sum of the fields of the elements. Radiation is shown to occur when the pulse of Charge encounters the discontinuities and bends in the wire. The analogy to the radiation from-a mov- ing point charge undergoing acceleration is stressed throughout. The chapter ends with the specialization of results to the practically important case of harmonic Was dependence. There are two appendices. Appendix A contains a discussion Of units and dimen- sions, restricted to the International System (SD). It is mainly intênded to show that the mysterjous factors, such as 89, (Lo, and the 4 in Coulomb's law, have a rational explânation associated with measurement. Appendix B is a verybrief discussion of vector analysis that includes a set of problems. Its purpose, apart from providing a list of vector relations, is to provide students with a review of a'subject that is often “fuzzy” in their memories. The maturity of this subject necessarily means that the source of all basic infor- mation is the writings of others. The references at the end of each chapter are the sources I consulted in preparing the material for the chapter. In most cases, these include the works of the original investigators of a subject as well as a number of interpretations of later writers. Of course, any errors in the presentation are my own, and I ask the readers to please bring these to my attention. A list of supplementary references with brief annotation, included at the end of the book, is intended to give the reader guidance in selecting material for further study of the subject. 1 made no attempt to make this list comprehensive, so am sure that there are many more, excellent books with equivalent content that could have been included. There is a set of problems (over 140 in all) at the end of each chapter. A solutions manual is available (to instructors) ftom the publisher. I would like to thank the many people who have contributed to my understanding “of this subject. Almost thirty years ago, 1 was privileged, as a graduate student at Harvard University, to receive the instruction of Professors Ronold W. P. King and Tai Tsun Wu. I hope that Professor King's insistence on a balance of theory with experiment is evident in this book. xvi 2) Preface My colleaguês àt the Georgia Institute of Technology have been most gra- cious in providing comments on the manuscript. Their suggestions have greatly improved the clarity of the presentation. Professor Waymond R, Scott, Jr. and Dr. William B. McFarland read the entire first draft of the manuscript. In ad- dition, on a number of occasions, Professor Scott provided assistance with the computer-generated graphics. Others who have provided comments on selected chapters are: J. A. Buck, D. L. Brundrett, R. F. Fox, M. P. Kesler, and T. P. Montoya. Dr. James G. Maloney provided assistance with the material on the FDTD method and comments on the chapters associated with antenna analysis. Mrs. A, B, Powell expertly typed early versions of some chapters that originated as class notes. My former colleague Professor John D. Nordgárd (University of Colorado at Colorado Springs) read several chapters from the final draft of the manuscript, His comments were móst helpful. During the last eleven years, most of the time I spent writing this book was taken from the time 1 would normally spend with my family. I am very grateful to my wife Linda and children Geoff and Elie for their patience and understanding during this period. Writing this book and the associated teaching have been both enjoyable and rewarding experiences. In this regard, 1 can not do better than quote the physicist J. Robert Oppenheimer:? Whatever trouble life holds for you, that part of your lives which you spend finding out about things, things that you can tell others about, and that you can leam from them, that part will be essentially a gay, a sunny, a happy life... Glenn Stanley Smith - Atlanta, GA May 1996 2 From Uncommon Sense / J. Robert Oppenheimer, N. Metropolis, G.-C. Rota, and D. Sharp, Editors, p: 170, Birkhiuser Boston, Boston, MA, 1984. . et . 1 Basic theory of classical electromagnetism 11 Historical introduction: Maxwell's equations in integral form The development of electromagnetic theory has à long history, beginning perhaps with the ancients” experimentation with the elecítical properties of amber and the magnetic properties of lodestone. The modem elements of the theory, however, had their origin in the investigations of 17th-, 18th-, and 19th-century scientists, or natural philosophers as they were called. The names of soime of the more próminent of these scientists are listed in Table 1.1 in chronological order according to their dates of birth [1-3]. Priorto the early nineteenth century, the phenomena associated with electrostatics, magnetism, and optics were largely thought to be independent; hence, the nâmes in Table 1.1 are divided into these categories.! By the middle of the nineteenth century, Hans Christian Oersted (1777-1851) had discovered that an electric current produces a magnetic field, and the connection between steady currents and magnetism had been examined extensively by André Marie Ampêre (1775-1836) and others. In addition, the experiments of Michael Faraday (1791-1867) had shown that a time-varying magnetic field could produce an electric'current. In 1864, building on these earlier investigations, James Clerk Maxwell (1831-1879) presented à theory that provided a complete and unified explanation for electric, magnetic, and optical phenomena [4, 5]. A striking fea- ture of Maxwell's theory was its prediction of electromagnetic waves with all the characteristics theretofore associated with optical waves. The efforts during the remainder of the nineteenth ceritury were devoted to ob- taining additional experimental verification and refining Maxwell's theory. Heinrich Rudolf Hertz (1857-1894) demonstrated, unequivocally; the existence of electro- magnetic waves, and Hertz, Oliver Heaviside (1850-1925), and Hêndrik Antoon Lorentz (1853-1928), among others, put Maxwell's theory for classical electro magnetism in the form we use today. : In the twéntieth century, classical electromagnetism has served as the origin for other fields of study, such as relativity and atomic physics. Thé unification accomplished by Maxwell for electromagnetism now serves ás a model for the unification sought for all'areas of physics. : ! Many of the séientists listed in Table 1.1 made contributions to more than One arés, although they ate listed under a single area in the table. For example, Faraday experimented with electrostatics, magnetism, and opties. 6 Basic theory of classical electromagnetism Table 1.2. Electromagnetic quantities and their units SL unit In terms of Im terms of Quantity . Terminology other units base units E Electric field strength . vm. m.kg.s3.A- B Magnetic field strength T kg-s2.4-1 D Electric excitatión cjmê m2.s.A (electric displacement) A Magnetic excitation Ajm mA ã Volúme density of current Ajmê m2.A p Volume density of charge Cc/mê m.s.A & Surface density of current Alm mia pe Surface density of charge c/m? m2.s.A £o Permittivity of free'space F/m (8.8541... x 10-12) to Permeability of freé space H/m m.kg:s 2.42 (4mx 1057) 5 c Speed of light in free space - mes (2.9999... x 108) . 5 “Poyúting vector w/m kg:s3 space (1.6, 1.7): . DF, 9) =cÊG,0) = e eoÊG, 0), a.9) HG) = Izç, )= 1 Br. (1.10) . H Helo where 5, = €/e, is the rélative permittivity or dielectric constant and pi, = ft/Hto is the relative permeability. In a conductor, the volume density of current will also be related to the electric field: Je,0 = If.) =0ÊG.0, É (1) where o is the electrical. conductivity of the material. The subscript c indicates a conduction current; it will be omitted unless it is needed to resolve an ambiguity. Notice that e, 41, and o in these equations are independent of position (7) and time (t)andthate, >'1,m, >1,ando > 0. Materials obeying the.constitutive relations (1.9)1.11) will be called simple materials. We must emphasize that these. simple constitutive relations are accu- rate representations for the electromagnetic behavior of actual materials only in restricted cases. More complicated relations are often required ir practical applica- tions. It is easy to understand one of the limitations of these relations: They require 1,1 Historical introduction: Maxwell's equations in integral form | 7 . quantities like É and D to have the sâme temporal variation, Since the relationship between £ and D is the result of physical mechanisms within the material, such as the orienting of microscopic dipoles, the simple constitutive relation (1.9) implies thãt the mechanisms respond instantaneously to any temporal variation in thé field. Clearly, this iwill not be true for fields with extremely rapid terporal variations, In this book we will limit our discussion to electromagnetism in free space or-in simple materials. . The equations we have presented so far, (1.1)-(1.11), provide a self-contained description of electromagnetism. The connection between electromagnetism and mechanics is provided by an additional relation — the Lorentz force expression [7]. For a volume V of free space at rest containing charge and current described by the volume densities p and 7, the net mechanical force of electromagnetic origin acting on the charge and current is P= ff 08+3xByv. (1.12) y The integrand of (1.12), which is a force per unit volume, F=08+JxB, . (113) is often referred to as the Lorentz force density. If the volume V is a material body that carries the charge and the current, then the forces on the charge and the current are forces on the body, and (1.12) is the net mechanical forcé of electromagnetic origin acting on the body. We cam use Equation (1.12) to obtain an expression for the force on a charged particle moving in an applied electromagnetic field &,, B, in free space. The charged particle is a small body with charge density p and total charge e= [por The velocity of the particle is 7, making the current density within the particle J=põ The total field at a point within the particle is the applied field Ê,, B, plus the field due to the charge in the particle itself É, B,, which is called the self field: E-E+ê, B=B,+5. From (1.12) the total force on the particle is P=h+Ã, . (14) with the applied force : E f= E) (ps + oi x BodV, v and the self force A = Hi (Êê + pix Bdv. v 8 Basic theory of classical electromagnetism Now we will assume that the particle is so small that the applied field is essentially uniform over its extent. The force due to the applied field is then A=(E+ix £o fl pdv, ; =q(ê+ixB). “ (115) This is the familiar Lorentz force for a moving particle in an applied electromagnetic field. The total electromagnetic force on the particle (1,14) is the applied force F plus the self force 7, 2 In practical situations the self force is often negligible, and the total force on the particle is o. the applied force ; =q(E+UxB). (116) This equation can be considered an operational definition for the electric and mag- netic fields in free space. The applied electric field is defined by the force 7, expe- rienced by a stationary test charge q: E = Fela. “The additional force jy that results from uniform motion of the test charge is used to define the applied magnetic field. The component of the magnetic field normal to the velocity d of the charge is which can be written a -Px Fi B a. “grp The equations of electromagnetism, (1.1)-(1.12), are based on extensive.exper- imental observation, Historically, deducing these equations from experiment was a long and arduous task. It involved many of the famous names in physics, as a glance at Table 1.1 shows. For our purposes, it would be of little benefit to re- producé the arguments that led from experimental observation to the equations of electromagnetism. What we will do instead is describe a few of the more impor- tant early experiments and show that the results of.these experiments are predicted by the equátions of electromagnetism. We will be using the theory to analyze the experiments rather than using the experiments to synthesize the theory, since the Jatter is a much more difficult process. Our approach provides an additional benefit: We will get to review the application of the integral form of Maxwell's equations to determine the fields, forces, etc. for a few simple geometries. For the establishment of a physical law, which is expressed as a mathemati- cal relationship, experimental observations of a quantitative nature are generally 2 The self force first aroge in the development of early classical models for the electron, in which the electron was assumed to be a small charged sphere. H. A. Lorentz was an important contributor to these early models. A brief qualitative argument for the origin of the self force is given in Section 6.5. “The history of the classical models for the electron and detailed derivations of the self fores are presented in References [8-10]. - 1.1 Historical introduction: Maxwell's equations in integral form 9 required. Until the middle of the eighteenth century, experimental observations of electric and magnetic phenomena were at best semiquantitative. For example, observations suggested that the electrostatic force between two charged objects in- creased as they came closer together, but the mathematical dependence of the force on the separation was not established. : In the latter half of the eighteenth century, several iúvestigators, including Henry Cavendish (1731-1810) and Joseph Priestley (1733-1804), inferred from experi- ments that the electrostatic force between charged objects varies inversely as thê square of the distance separating the objects [1]. In the modem literature, however, Charles Augustin de Coulomb (1736-1806) is generally credited with establishing this result, and itis Coulomb's experiment that we will describe [11, 12]. Coulomb had previously developed a torsion balance for measuring small forces, and he made use of it in his electrical studies. A sketch of his apparatus is shown in Figure 1.2.) The torsion balance is formed from a thin, vertical, silver wire held taut by a weight attached to its lower end. When the wire is twisted it produces a force of torsion. Coulomb had shown carlier that this forcé was directly proportiorial to the angle of twist W. Two small, gilded, elder wood pith balls are located in a plane normal to this wire. One is held fixed; the other is mounted on, a horizontal arm attached to theswire and is free to moye as the, wire is twisted. The balance was initially set so that there was no twistin the wire when the two uncharged balls were in contact. The pointer fixed to the upper end of the wire was zeroed. Then a small charged pin was brought in contact with the balls to give them equal charge of the same sign. The movable ball was repelled through the angle Ys (a counterclockwise rotation in Figure 1.2). The distance between the centers of the balls r and the angle 1, could be reduced by rotating;the poiriter through the angle yp (a clockwise rotation in Figure 1.2), making the;total angle of twist for the wire = Vit Yo. Using two values of the angle of twist, y and >, Coulomb was able to show that the electrostatic force of repulsion F behaves as Ano (2): AR da An)" that is, the force is inversely proportional to the square of the distance separating the balis. Additional. experimentation led to the following result, known as Coulomb's law, ag Fa 3) where q and'q” are the charges on the two balls, * We should mention that Coulomb also vetified this relationship for attractive forces (ie., for balls with charges of opposite sign). Initially. he tried to use the apparatus in Figure 1.2 for his purpose but found that the mechanistm Was instable, 3 Allofthe divas showing the apparatus used in the early experiments are this áuthor's rendition, e.g, Figures 1.2, 1.4, 1.5a, 17, 1.9, and 1,12, In most cases details of the construction have been omitted to allow a clearer description of the mechanism. 10 Basic theory of classical electromagnetism ZEROOF TORSION Fig. 1.2. Coulomb's apparátus for electrical studies, The apparatus was enclosed in glass tubes, not shown, to prevent air currents from disturbing the measurements. with the pith balls often coming together. He then devised a new apparatus, which also used his torsion balance, to study the attractive forces [11, 12]. Coulomb's observations are predicted by the equations of electromagnetism, as we will show by applying these equations to the geometry in Figure 1.3. Here two spheres (1 and 2) in free space are separated by a distance r12 that is assumed to be much larger than the fádius of either sphere. As a result, the charge on sphere 1 is approximately uniformly distributed over its surface, which means that its electric field is in the radial direction, É, and is spherically symmetric. On applying Gauss” electric law (1.3) to a spherical surface of rádius r surróunding sphere 1, we find that ph Divas = ant fp as = estatome) = fJf mav an Ss s v 1.1 Historical introduction: Maxwell's equations in integral form n SPHERICAL - SURFACE S VOLUME V ELECTRIC FELD É SPHERE 1, 91 Fig. 1.3. Geometry for analyzing Coulomb's experiment. or E No ê= p. “E amor? Since the distance r12 between the spheres is much larger than the radius of sphere 2, the electric field of sphere 1 is approximately uniform' over the volume of sphere 2: . Bom fu. Z 4meord Now the force on sphere 2 due to the electric field of sphere 1 is obtained by applying “the Lorentz force expression (1.12) to the volume V containing Sphére 2: F= fl, mênav = Ea [ff mav = aêm. f= to, ' qum or which is a mathematical statement of Coulomb's observations. Coulomb also used his torsion balance to measure the force between magnets." By using a long, thin, magnetized needle, he was able to study the force near one” . pole of the magnet. From these measurements he concluded that the force between two magnetic poles of strength zm and 74, separated by the distance r was ml, : r2 Note the similarity of this expression to the one he obtained for the force between electric charges. When we write Coulomb's results in terms of magnetic charge Fa 16º Basic theory of classical eleciromagnetism Fig. 1.8. Geometry for analyzing Anis s experiment to observe the force on parallel, current-carrying wires. steel points fastened to its ends. In the arrangement shown, the series connection - causes the cnrrents in the two parallel sections of wire AB and DE to be in opposite directions. The connections could easily be rearranged to make the currents in the same direction. With this apparatus Ampêre observed that the parallel wires were attracted when the currents were in the same direction and répelled when the currents were in opposite directions. This effect is opposite to what is observed for electric charges and rhagnetic poles, end like elementa repel and unlike elements attract. Ampêre's observations are contained in the Loren force expression (1.12). Con- sider the two parallel, infinitely long, current-carrying wirgs shown in Figure 1.8. The wires are separated by the distance d. From (1.21) the magnetic field at the center of wire 2 due to the current in wire 1 is Holt 4 Bn= 2d > We will assume that the field is approximately uniform over the small cross section of wire 2; the force on a length £ of wire 2 due to the magnetic field of wire 1 is then 4 = ff fxBudv= [E (12) x Bda! = — —Hoh ht, = For equal currents in the same direction (1 = h), the wires are attracted; for (1.22) equal currents in opposite directions (11 = =), the wires are repelled, as Ampêre, observed, 1.1 Historical introduction: Maxwell's equations in integral form 17 In experiments like the one shown in Figure 1.7, the conductors essentially short circuit the battery. The battery then cannot maintain a constant voltage, so the current in the circuit varies with time, which makes quantitative méasurements difficult. Ampêre avoided this problem in his later experiments by basing them on the null principle. In these experiments, the same current passed through all conductors of the circuit. The physical arrangement of the conductors was chosen so that they pioduced-no force - a null — at the location of a detector. Temporal variations in the current did not affect the. null condition. By using different physical arrangements for the conductors, Ampêre was able to infer the mathematical behavior for the force - between current elements from these experiments, a feat described by Maxwell [5] as “one of the most brilliant achievements in science.” . We will look at one of the four experiments, based on the null principle, that Ampêre described in his famous memoir [17]. This particular experiment shows that the force of inagnetic origin on a small element of current is always normal to the direction of the; current. This result is obyiously contained in. thg Lorentz force expression (1.12), since the cross product 7 x B is always normal to 7: Ampêre's apparatus for this experiment is shown in Figure 1,9..A battery pro- duces a current 1 in the'conducting circuit AOBEB/OA'D, which rests on the top - ofatable. There are four radial conductors with common ends at 0: AO, BO, A'O, and B'Ô, The circular arc of wire CC" rests on the mercury-filled troughs at B and B'. A radial support arm EF is connected to this arc by a hinge at its midpoint E. The radial support arm is also attached to a vertical rod that ís tree to rotate about the axis OP. The current-carrying conductors on the table are viewed as ihê two separate closed circuits shown in Figure 1.10. Circuit 1 is the loop AOA'DA; circuit 2 is the loop BOB'B, Ciróuit 1 produces a magnetic field By at circuit 2, and it is the force that this field causes on the movable arc CC” that is observed in the experiment. When the arc CC” is positioned so that the circle of which it is a part has its center O coincident with O, as in Figure 1.1 la, there is no movement of the arc. This null result implies that the force “acting on the arc is normal to the arc and acts through O. The motion of the arc is thus prevented by-the radial support am EF. When the are CC' is positioned so that the circle of which it is a part has its center O' away from O, as in Figure 1.11b, the arc moves. In this case the force Fi is normal to the arc, but it does not act through O; thus, the radial support arm EF does not prevent it from moving. Ampêre obtained the same results for arcs of different length, and he concluded [18], “.. that the action of a closed circuit, or an assembly of closed circuits (a magnetic field) on an infinitesimal element of an electric current is perpendicular to this element” Note that the currents in the two segments BO and B/O of circuit 2 also produce a magnetic field at the arc CC”, For the case in Figure 1.1 1a, Ampêre correctly argued that, due to symmetry, the forces caused by these currents would not move the arc. For the case in Figure 1.11b, the forces caused by these currents do move the arc. The additional effect due to the magnetic field of circuit 1 is then observed by changing the shape of the wire ADA. Basic theory of classical electromagnetism 1.1 Historical introduction: Maxwell's equations in integral form 19 POINT FREE ROTATION er k ir TOP CIRCUIT 2 Fig. 1.10. The current-carrying conductors of Ampêre's experiment viewed as forming two closed circuits. After Oersted showéd that an electric current produced à magnetic field, many scientists attempted to demonstrate the reciprocal effect, that is, an electric current produced by magnetism. There were a few near successes, but it was Michael Faraday in 1831 who finally obtained conclusive experimental proof for this effect. Faraday, like Ampêre, based his theoretical reasoning on a series of experiments [19-21]. We will describe one of these experiments, the one with which Faraday first observed a magnetic field producing an electric current [22]. A drawing of Faraday's apparatus is shown in Figure 1.12. Two coils, A and B, are wound on an iron ring with a diameter of about 15 cm. The terminals of coil A are connected to a battery, while the terminais of coil B are connected to a loop of wire, a portion of which passes over à magnetic needle. Faraday observed that the magnetic needle moved momentarily whenever the battery was connected to or disconnected from the circuit. The needie-was deflected, oscillated, and eventually returned to its initial position. Faraday had discovered electromagnetic induction. When the battery was connected, it produced a rapidly increasing current Za in coil A and, consequently, a rapidly increasing magnetic field in the iron ring. This time-varying magnetic field induced a transient current Ts in coil B and the connected wire loop. The current in the wire loop, in tum, produced a transient magnetic field that deflected the magnetic needle. Once the current in coil A was Steady, there was no effect on'the magnetic needle. . Ampêre's apparatus for demonstrating that the force of magnetic origin on a small element of curreat is always normal to the direction 20 Basic theory of classical electromagnetism a) b) . Fig. 1.11, Two orientations for the circular are of wire CC. Itis clear why the investigators before Faraday failed to discover electromagnetic induction. They used apparatus not significantly different from Faraday's, but they restricted their observations to the time when the current was steady and missed the small momentary effects produced when the battery was connected or disconnected. The theoretical explanation for Faraday's observations rests on Equation (1.1), appropriately called Faraday's law, First, we will consider the cross section of the iron ring shown in Figure 1.13a. We will apply Equation (1.2) to the surface S4 on this cross section. The circular contour CA, which is on the axis of the ring, bounds Sa: After neglecting the second term on the right-hand side of this equation, we have Radio ff Jucdã. Ca Sa The magnetic excitation Fa is approximately uniform on the axis of the ring, 1.1 Historical introduction: Maxwell's equations in integral form 2 IRON RING 4 : MAGNETIC % FA E OSCILLATORY MOTION OF - NEEDLE “e NEEDLE WHEN BATTERY + CONNECTED OR : DISCONNECTED Fig. 1.12, The apparatus with which Faraday first observed electromagnetic.induction. Ha = Mao Ô, and there are Ny turns, each carsying current 4 in coil A; thus 2m i Hudi= | Haôbdo =2xbHyo JC 9=0 = ff ducaS= mam Sa or simply Na = Ta. Has = sopa The magnetic field Byp is related to H 49 by the permeability of the iron gu (1.10) uNa Bag = UMa = mb Now we will apply Faraday's law (1.1) to the surface Sp, whieh is bounded by the contour Ca formed by the coil B and the connected wire loop: TA: Es-dê=— ff 3Ba gp (1.23) ca so dt For the right-hand side of this equation, we note that the portion of Sp that couples the magnetic field B, is the Ny turns, cachof area Sy = 7a? (see the lower drawing 26 Basic theory of classical electromagnetism (OPEN SURFACE) S, a, (xD Gov) LETSDE | (7—49/2,2) ea ha RIGHT SIDE z Gy+ Ay/2,2)) > b) “x Fig. 1.15. Geometry ustd for defining the curl. a) Element of surface area AS bounded by curve C. b) Example for rectangular Cartesian coordinates. the vector field À around the curve C is definedto be” r =f À-dê, (1.28) where dÊ is the vector differential length along C (an infinitesimal vector locally tangentto C). The sense in which the curve C is traversed relative to À is determined by the right-hand rule, The component in the direction À of the curl of the vector field at P is the limiting value of the circulation of À divided by the ani area AS, as the area approaches zero about P: àrculÃA= jim 165) = dm, (=e£4 Ã-. ai). (1.29): This result is independent of the coordinate system. 1.2 Curl and divêrgence; Maxwell's equations in differential form 2 Now we will consider the curli in a particular coordinate System — the Tectangular Cartesian coordinate system (x, , 2). The component of curl À in the direction £ is to be determinéd at the point P(x, y, z). The element of surface area is a rectangle with sides Ay and Az and normal £ centered at P. The geometry is shown in Figure 1.15b. The line integral (1.29) now consists of four segments, one for each side of the rectangle. The integrals for the right and left sides, tam Ir= f Atx, y + Ay/2,2) -Bde! (1.302) r=a-nepa Ê and Ai = h= f Ã(x, y — Ay/2,2)-2dy, (1.30b) V=rtáopo combine to give. 2+Az/2 R+h= Í [ate + 49/20) — Adi, y = Ay/2 2)]de. =p, . (1.31) TE À, is continuous over the interval 2 — Az/2 < 2 <z + Az/2, the mean value * theorem for integrais can be used to write (1.31) )asé In tio = [A+ Ay/2,2) — Ad, Ay EJA, (139) where z — Az/2 <2* < 2 + Az/2. Now if A, is also continuous for the interval y-4y/2<y E y+4y/2andhas a first partial derivative 9.4,/9) for the interval y—Ay/2<Y <y+Ay/2, the mean value theorem can be used to write (1.32) as À A, s IrtlL= AyAz, (1.33) E where y — Ay/2'<y!-< y + Ay/2. ; The line integrals over the other pair of sides (top and bottom) can be handled in asimilar manner: (1.34) The mean value theorem for integras states that if (6) is continuous for a < $i< b, then LafQE = 1600), a<pab The mean value theotem states that if g($) is continuous for a < ÉS b and g'($) exists for ax Ea -< b, then eb)-gla)=g(ENb—a), a<s <b. “The relationship between these two theorem is seen om substituta /(8) = g'() [28]. 28 Basic theory of classical electromagnetism The component of the curl ih the direction % is then É 1 1 Mira ( R+IL+Ir+ 2) E -cudÃ(,y,2) Ay>0 AyAZz B:50 1º |[94 DA = ao VAÃZ [5 - Java ; a Aybz LO lago DZ ley ze (1.35) Assuming that the partial derivatives are continuous, in the mit Ay > 0, we have >)! > yete. and E : dA, DA -nÃgp)= E -D j . 2 -curl Á(x, 7,2) 3» % . (1.36) “This is the x component óf the vector curl À; the y and z components, determined in a similar manner, are : a dA, DA, : PreunlÃa,yy)=2- , 5 -culA(x,y,2) dx 3x (1.37) and A, DA lÃt,p)= 2a, 1.38 :2 cul A(x, y, 2) = Ta 3y (1.38) which makes DC o/DA, DA [BA DA) | (DA, DA culÃ(g,y )= 5 3» de )es( e dx +2 Em » - dà ' = ix (1.39) When the notation : Yxã= DO ixié (1.402) : isrpa i o is introduced, we have. : . cuil Atx, 20) = V x Ary, 0. : (1.40b) Divergence “The components of the vector field À are continuous in the neigh- borhood of the point P located by' the position vector 7 shown in Figure 1.16a. The element'of volume AV containing P has the piecewise smooth surface S with outward-pointing, unit normal vector à. The divergence of the vector field atP is the limiting value of the net outward flux of À through the surface S divided by the volume AV, as the volume approaches zero about P: div ÀG) = (lim av fes das). (1.41) This result is independent of the coordinate system. 1.2 Curl and divergence: Maxwell's equations in differential form 29 a) - Getz Ga) 1 q k Az RIGHTFACE. r (+ 8x/2yi2) A a 1 > dá ! e (6-4x/2,y,2) z Ax > Bo. x, Fig. L.16. Geomeiryused for defining the divergence and the gradient. a) Element of volume AV with surface 5, b) Example for rectangular Cartesian coordinates. Now we will consider the divergence in the rectangular Cartésian coordinate system (x, y, 2). The volume element A V is a rectangular box with sides Ax, Ay, and Az centered at P(x, y, 2) (Figure 1.16b). The surface integral (1.41) consists of six integrals, one over cach face of the box. For the right and left faces, the integrals are THA? pyrayia . E Ir = / / 2 A(x+Ax/2,y, 2Ndy dy =3- 2/2 dy'=y—by/2 30 Babic theory of classical electromagnetism and 2h pray Di no [born E Me Bafo y eae, t'=1-62/2 dy'=y—by/2 . which combine to give + +A2/2 V+rAy/2 , Ir + Ir =[ / [Asa + Ax/2,y,2)) = A2/2 2 —Adx— 6x2, y 2)]dy'de. 147" If Az is continuous for the intervals y — Ay/2 << y + Ay/2andz— Az/2 < 7 £2+ Az/2, the mean value theorem for integrals can be used to write (1.42) as Int = [AME+ Ax/2, 9,2) Ade — Ax/2, 7º e)JAyhe, (143) wherey— Ay/2<)*<y+Ay/2andz— Az/2<zt<z+ A/Z As is also continuous for the interval x — Ax/2 < x! < x + Ax/2 and has a first partial derivative 34;/8x for the interval x — Ax/2 < x! < x + Ax/2, the mean value theorem can be used to write (1.43) as . Ar AxAyAZ, (1.44) ôx ) h+h= [E wherex — Ax/2<x"<x+Ax/2. | . The surface integrals over the other pairs of faces (front and back: Ir, Ig; top. and bottom: Ir, Iy) can be handled in similar manner: AxAyAZz (1.45) AxhyAz. (1.46) The divergence is then E co flrth Alto tir+y, div Ãte,9,2,) = lim (frtitirta trt dx) A AxAyAz dead s7>0 % 1 IA Ex > amoo | AxAyãz || dx Brad AO By Joxavae)) DX asisgoio, po Assuming that the partial derivatives áre continuous, in the limit Ax — 0, we have 1.2 Curl and divergence: Maxwell's equations in differential form 31 xt > x,x** > xçete. and: Jd DA, , DA, GA) = + (1:47) or, when the notation . 1 Wo. dà : v.Ã= fd : d= St ; (1.484) ima is introduced, we have divã, 2) = Vo À(x, 9,2). 5 (1.48) Note that the integral definitions we started with for the: curl and divergence, (1.29) and (1.41); are more general than the differential forms that we subsequently obtained, (1.39) and (1.47). The use of the former Tequires-the-vector field to be continuous, whereas the use of the latter requires both the -vêctor field and its first partial derivatives to be continuous. We next apply the definitions of the cur] and the divergence to convert Maxwell's equations in'integral form to their differential form, Where required; quantities and their derivatives will be assumed continuous. First, we will consider Faraday's law (1.1) applied to thé surface ASS in Figure 1.15a: La 1 2 1 9B a asp ócio 75 hã dê * After taking'the-limit AS — O and introducing the definitión (1.29) for the curl, this equation becomes da a ” 1 aB oB im | dê di|=a. = lim |-— dE] = =p. SE delas fs a ] Rae dim, AS JJas dt q Poa or simply : a 2... 38 h-culê =. 3º ; (1.49) The orientation for the surface AS (or the unit vector à) is arbitrary in the definition for the curl; thus, (1.49) implies that É cdêç, o) = Sto D, or using (1.40b) we obtain vxêr0)= eo, (1.50) This is the differential form of Faraday's law; the differential form for the Ampêre- Maxwell law (1.2), obtained in a similar manner, is . E Ê cute) = 7,1) + EO, 36 re": Básic theory of classical electromagnetism FRÉE SPACE E=0 a) a 10 £ E E , > 1 s Ê os . , ã y a < Litissis 14 . -8:-6 -4 —2 4 b) zà Fig. 1.18. a) Neutral metal sheet with no applied field. b) Charge densities of and pj near right-hand surface of sheet. (Densities are based on quantum-mechanical calculations from Lang and Kohn (33].) hence, the net charge in the vicinity ofeach surface must be zero; for examiple, for the right-hand surface L L / paia = / [oito + ertoJdz = 0, e Das where L is lárge comipared to tie lattice spacing of the ions. The charge density 97 is seen to be nearly uniform in the metal away from the surface, to oscillaté as the surface is approached, then to rapidly decay to zero. When the metal sheet is pláced in a uniform electrostatic field normal to its surface, as in Figure 1.19a, the uniform positive charge density is. assumed to remain unchanged, p*(z) = p*(z), and the electrons are allowed to redistribute to produce the new negative charge density p-(z) [34]. There is now an excess of positive charge at the right-hand surface of the sheet and an excess of negative charge at the left-hand surface óf the sheet. The excess charge produces an electric field that cancels the applied field in the interior of the metal sheet. 1.3 Surface densities, boundary conditions, and perfect conductors y a) Ap, ARBITRARY UNITS b) Fig. 1.19, a) Metal sheet placed in a uniform electrostatic field, b) Excess charge density Ap near right-hand surface of sheet. (Based on quantum-mechanical calculations from Lang and Kohn [34].) With the excess charge density Ap(2z) defined to be the difference in the charge densities with and without the applied field, we have Solo) = 0(2) — pole) = 02) + pit) — [of (o) + oz (o)] = p-(2) — pita), and, for the right-hand surface, - L L . f plajdz = f Aple)dz = e,E., 1=-—L 1=-L where E, 2 is the electric field outside the sheet. Figure 1.19b shows the excess charge density Ap(z) for a sheet of metal like sodium placed in a weak external electric field [34]. The excess charge is seen to be located within a very small depth near the surface: —10 À sz g 10 À (1à = 10-10 m). . 38 Basic theory of classical electromagnetism These results show that there are situations where charge is confined within a layer of microscopic thickness at the surface of a body. For macroscopic electromagnetic calculations, the charge is then effectively on the surface of the body, and the volume density of charge p can be replaced by a surface density of charge, which we will call ps. The surface density of charge has the units of charge per unit area (C/m?). For our example, the plane metal sheet, the surface charge density on the right-hand surface is simply L L Ps =[ plaldz =[ Aplejdz. t=-L 1=—L Since a surface charge density arises in electrostatic problems, we might expect a surface current density to arise in magnetostatic problems (problems that involve only steady currents). This, however, is generally not the case. In Figure 1.20 we compare similar electrostatic and magnetostatic problems. A static line charge parallel to a metal half space (Figure 1.208) induces a charge at the surface of the metal. The combined electrostatic field of the line charge and surface charge is zero within the metal. A-steady line current over a metal half space (nonmagnetic, | = fo), as shown in Figure 1.20b, induces no current within the metal. In fact, the magnetic field produced by the steady line current is the same as when the metal half space is absent. The metal is a shield for the electrostatic field but not for the magnetostatic field. , We must consider time-varying electromagnetic fields to observe current con- centrated near the surface of a metal. If we let the line current of Figure 1.21a oscillate with a very high angular frequency w =.27f, so that T(t) = Iocoswt, the volume density of current in the metal can be shown to be approximately [35] FO t) = AQ" cos [wt — 2/8 — WO). The parameter à is called the skin depth; it is simply related to the electrical con- stitutive parameters of the metal (g, 1,0) by (1.55) Note that the current for this example is independent of the coordinate x. The terms A(y) and y(y) are the amplitude and phase of the current at the surface of the metal. The behavior of the current is made clear when it is evaluated at the time 1 = W(9)/w and normalized by dividing by the value at the surface (z = 0): = Brut= vo) DO T01=vJo) = ef cos(/8). ?.We use the term microscopic to refer to observations made on a scale comparable to atomic di- mensions, The contrasting term macroscopie is used to refer to observations made on a scale much larger than atomic dimeásions. For example, the probe used in a microscopic measurement is capa- be of resolving the field betiveen atoms, molecules, etc,, whereas, the probe used in a macroscopic measurement is enrmous compared to atomic dimensions and measures a spatial average of the microscopic field. E 1.3 Surfhce densities, boundary conditions, and perfect conductors 39 METAL LINE ê=0 CHARGE “RA à ——s 4 ( N Dm — nan e a, NON MAGNETIC METAL (u=po) “'B b) - Fig. 1.20. a) Electrostatic problem: line charge over à half space of metal. b) Magnetostatic problem: line current over a half space of nonmagnetic métal ( = 19). The current in the metal (Figure 1.21b) is seen to decay exponentially from its value at the surface. The characteristic length for the decay is the skin depth 8. A detailed analysis shows that the electromagnetic field also decays with the same character- istic length; thus, the electromagnetic field (É, B) is essentially zero in the metal at a distance greater thanafew skin depths from the surface. The metal is now a shield for the time-varying electromagnetic field; cômpare B in Figures 1.20b and 1.21a. The phenomenon we have described is known as the skin-effecr approximation. [35, 36]. It is an approximation that holds when o/we > 1 and the skin depth ” is small compáred to all physical dimensions of the problem.!º For our example !º For a metallic conductor of general shape, the skin depth must be small com; j e pared to the dimensions of the conductor and the radii of curvature of its surface. Also, the electromagnetio feld must vary negiigibly along she surface over distances comparable to a few skin depths [35].  “0 Basic theory of classical electromagnetism By METAL ê=0 OSCILLATING se LINE CURRENT zo — XL , a) 4 b) Fig. 1.21. a) Electrodynamic problem: rapidly oscillating line current over a half space of metal. b) Normalized volume density of current near surface of metal. Results are for 82 « d?, kod = wdfc & 1, where à is the skin depth. (Figure 1.21) the skin depth 8 must be small compared to the spacing d between the line current and the metal half.space (5? « d?). For good conductors like copper (E =, = Ho, 0 = 5:8 x 10º S/m), the approximation is very useful, because the skin depth can be Extremely small at practical frequencies; for example, at the radio frequency f = 100 kHz, o/we = 1.04 x 10" and 6 = 2,09 x 107! m, atthe microwave frequency f = 1 GHz, o/we = 1.04 x 10º and 5 = 2.09 x 10º m. Of course, for à material that is not as good a conductor as copper and for low frequencies the approximation may be of little value; for example, for a typical soil 1,3 Surface densities, boundary conditions, and perfect conductors 41 (6 = 20,1 = oO = 5 x 10-2 S/m) at the power frequency f = 60 Hz, o/we = 1.0 x 10º and é = 2.9 x 102m! Insituations where the skin effect approximation holds and the current is confined to a very thin layer at the surface of a body, the current is effectively on the surface of the body. For macroscopic electromagnetic calculations, the volume density of current j can then be replaced by a surface density of current, which we will call Js. The surface density of current has the units of current per unit length (A/m). For our example (Figure 1.21) the surface density of current on the metal is simply ta 108 Joo= [ , Tone nde n where the upper limit is chosen large enough that 7 = O at that point. It is important to remember that the surface densities of charge and current, ps and js, are models — idealizations, used to represent charge and current that are actually confined to a thin layer near a surface. Simple mathematical statements for * these models result when the surface is the plane z = 0, as in Figures 1,18-1.21. | Let the actual volume densities of charge and current be p(x, y, 2) and J(x,), 9. In the model these densities are replaced by the volume densities p'(x, y, 2) and Toyo GR : . Pe = pl, nto ' (1.56) Ty = Te yo, . (1.57) where ô(z) is the Dirac delta function (Paul Adrien Maurice Dirac, 1902-1984) or the impulse function with the following properties (see Section 5.2): D=0, 50, as . Í as TOO = 10) (.58) The surface densities p, and J, are chosen so that à az Ar E f Px y dz = lx, y)= J plz, a)dz, (1.59) -Az -az and - Ar o Ar . . f Tx y di = Tila) -[ Tx,» ddz, (1.60) —Az —Az ' where Az is chosen so that all of the charge or current is within the range of integration, The relationship between p and p, is shown graphically in Figure 1.22. 1.3.2 Electromagnetic boundary conditions We now retum to our discussion of the behavior of the electromagnetic field at a material boundary, such as in Figure 1.17. In Figure 1.23 we have an enlarged 46 Basic theory of classical electromagnetism Fig. 1.25. Schemati drawing for explaining the surface equation of continuity for electric charge. the specialization of (1.41) to a surface: div À) = fim las É s.dae), (74) In the rectangular Cartesian coordinate system (x, y,2) the surface divergence can be evaluated using the Same arguments as used in Sectibn 1.2 for the regular divergence, For example, when the surface is the x-y plane with normal 2, and 4 is continuous and has continuous first partial derivatives, the surface divergence is da A. div, À, )= 53 + E =, (1.75) y tn the notation . - o =5 é. E (1.76) at es we have . divs Á(x,)) = Ve: Á(x,y). (117 With the definition for the surface divergence, (1.73) becomes " = a 8 ah D+ Ve d=-, (1:78) or % ' Sa Tn t Ve de= E, (1:19) a . where the subscript n signifies the component normal to the boundary surface. This is the surface equation of continuity for electric charge; it is a statement of conservation of chárge at a point on the surface and is illustrated in Figure 1.25. Charge can be taken away from the point P by the normal component of the volume 1.3 Surface densities, boundary conditions, and perfect conductors 47 Table LÁ. The electroniagnetic boundary conditionst General boundary Region: a perfect condition electric:conductor Ax(b-E)=0 AxB=0 ax -Ty)=d ixbh=ã AD -Dy)=a A-Dp=p, AtB-B)=0 A-B=0 + achem d=- o a htw da tp Note: tThe normal À points outward from region 1 into region 2. current in each region, — Jin and an. The surface current «; can also remove charge from this point if it has a net outward pointing compônent on the curve C surrounding Pí(ie., if V - j is positive). The net charge removed by these currents per unit area, per unit time must equal —3,/0t atP. The procedure used above can also be applied to-the integral forms of Gauss! electric law and Gauss" magnetic law, (1.3) and (1.4), to obtainbopndary conditions for the normal components of D and B, respectively: AD -Do=p 0. (1.80) Aà(B-B)=0, (1.81) or Dy — Din = ps o (1.82) Bu-Bn=0 0: (1.83) In words, these boundary conditions state that the norrhal component of D is con- tinuous on crossing a material boundary unless there is a surface charge density on the boundaiy; i in that case, it is discontinúous by the amount p;. The normal component of B is always continuous on crossing a material bouni These five boundary conditions [(1.65), (1.66), (1.78),:(1.80), and (1.81)] are sumimarized in Table 1.4. Recall that for time-varying electromagnetic fields, only three of the five Maxwell's equations in differential form are independent. Similar arguments can be used to show that only three of the boundary conditions, (1,65), (1.66), and (1.78), are independent (Problem 1.8). 1.3.3 The concept of a perfect conduitor Im our earlier discussion, we saw that the skin effect approximation holds for high- frequency oscillatory fields in good conductors. For this phenomenon, we must have o/ws >> 1 and the skin depth 8 = /Z/wpo must be small compared to all 48 « Basic theory of classical electromagnetism physical dimensions, The durrent in the conductor is then well approximated by a surface current density, and'the electromagnetic field is essentially zero everywhere in the conductor. Another way to view this approximation is to consider the limit as o > oo, that às, the case in which the conductor becomes a perfect conductor. Then o /ws — 00, 8 > 0, and the electromagnetic field is zero.in the conductor for all frequencies « > O, The perfect conductor is a model, an idealizatiqn that can be used whenever the skin effect approximation holds. The electromagnetic boundary conditions simplify when one of the two regions is a perfect conductor. For example, when region 1 is a perfect conductor (E = us O, B/=0,D = 0, and Hy'= 0), the boundary conditions (1.65), (1. 56), (1.78), (1.80), and (1.81) become Axê=0 (1.84) A x Ho = JJ (1.85) cê 2.3 : à. h+ Vs d=—E (1.86) a Do=ps . (1.87) à B=0 : (188) These conditions are summarized in Table 1.4, Note that the tangential electric field És, and the normal magnetic field 8», are always zero at the surface of a perfect conductor, 1.4 Energy of the electromagnetic field - Poynting's theorem The Lorentz force expression (1.12) provides the connection between electromag- netism and classical mechanics. Through this expression, concepts such as energy, linear momentum, and angular momentum, which are familiar in mechanics, can be associated with the electromagnetic field. In classical, nonrelativistic mechânics a particle of mass m, moving vita velocity 7 at position 7 in an inertial reference frame, has linear momentum [37] á dr - B=m=mi. = (189) Newton's (Isaac Newton; 1642-1727) second law of motion states that th time rate of change of the linear momentum of a particle is equal to the total force applied * to the particle: a dp Pe dt or, for a particle of constant mass, a di = =mã nom mã, (1.90) where à is the acceleration of the particle. 1.4 Energy of the electromagnetic field — Poynting's theorem 49 “The work done by the applied fórce on the particle when it moves through the displacement AF is defined to be aWw=Ê-aFr, (1.91) and the rate at which the work is done is the power: DO [BW O PA BE aa P= im, (47) = Jim, (F- 2) =f.i; (1.92) Wihen we introduce (1.90), the power can be expressed in terms of the kinesic energy of the particle X = or” - . t P=Êi= . (1.93) Thus, the rate at which work is done by the applied force — the power - is equal to the rate of increase in the kinetic energy of the particle. Now let's examine a charged particle moving in an electromagnetic field É, B in free space. The volume charge density in the particle is p, and its velocity is d. From (1.12) the force of electromagnetic origin on the particleis or, since J=pid, (1.94) P= ff sé+es Bu Cú The power (1.93) is then p=o ff ê+oixBiav = ff [oi Esos-cix BJav, where 7 was moved under the integral sign because the velocity is the game at all points in the particle. The second term in the integrand is zero: 5 - (ix B) = O;the magnetic field does no work on the charged particle, After substituting (1.94), we have . aa: p= ff E-dav= SE (1.96) We will assume that the electromagnetic field possesses energy and invoke the principle of conservation of energy. Equation (1.96) is then an expression for the rate at which energy is exchanged between the electromagnetic field and the mechanical motion of the charged particle. When P is positive, as in Figure 1.26a where É. 7 is positive throughout the particle, the kinetic energy of the particle is increasing; the ficld is instantancously supplying energy to the mechanical motion of the particle. s0 Basic theory df classical electromagnetism as E q. + 0 — = — — E do =—— po -— a, -— Fig. 126, Posiively charged pare moving in an electromagnete fel, ” P=dk/dt > O.bP<0. When 7 is negative, as in Figure 1.26b where É . J is negative throughout the particle, the kinetic energy of the particle is decreasing; the mechanical motion of the particle is instantaneously supplying energy to the field. The expression we obtained above is for the energy associated with the motion of a charged particle. What we would really like to have is an expression for the energy that applies to general volume distributions of charge and current (9, J). We can obtain such an expression by starting with the intégral from (1.96) applied toa general volume V and then substituting Maxwell's equations. First we will use the Ampêre-Maxwell law (1.51) to replace 7: JE: Jav= fik: vxdav- fi, E ay, Next, the general vector relation (Appendix B) AV xB)=B(VxÃ)-V(ÃxB) is applied to the term É (V x Hi) to give is: der = fl iocombmar= fp o era «Ne as and Faraday's law (1.50) is used to replace V x &. fl ss sav= -J, ear rã iv fox toa. am 14 Energy of the electromagnetic field — Poynting'sthéorem 51 Now we introduce the divergence thgorem or Gauss" theórem (Appendix B) [26]: fit -ÂdV = pr - ÀdS: (1.98) After applying this theorem to the last integral in (1.97), we have Hi ê. Jav+ fl (2.5 Ea Cave fhêxim-as=o (1.99) Itis customary to define the vector SE n)=EEo)xHçn. (1.100) Then (1.99) becomes hs savi fl (2.5 En. ave fjS.a3=0 (1.101) This relation is known as. Poynting' 's theorem, and the vector & is called the Poynting vector. This relation was first derived by John Henry Poynting (1852-1914) in 1884 [38], and, as is the case with many scientific discoveries, it was obtained independently, at about the same time, by another investigator, Oliver: Heaviside [89-41]. Before we examine the physical interpretation of this thegrem, we should say a few words-about the volume to which it applies. In the derivatión of (1.101), we used Maxwel]'s equations in differential form and the divergénoe theorem; this usage implies that the field and its first partial derivatives are continuous, which may not be the case if material boundaries are included within the volume. Of course, we would like'to apply the theorem to a large volume of space containing many subvolumes, each being a different material, such as a conductor, a dielectric, etc. Let's see how this can be done. We will only consider a volume V composed of two subvolumes V) and Vz, as in Figure 1.27, Our argument can easily be extended to a volume containing any number of subvolumes. The surface of Va is Sp, and the surface of Vj has an interior portion Sy; adjacent to S> plus an exterior portion $1e, which is also the surface of V.!? Poynting's theorem (1.101) can be applied to each of the vólumes Vi and Va separately, and the results can be added to give sos é cm (e 2098 +, (5 Ea Dart us Sua nto 1 The Poyntitig vector 5 and the differential area vector d5 should not be coifused, *2 In Figure 127 the sirface 1 is drawn slightly outside the boundary, and the surface 5) is drawn slightly inside the boundary. The fields and their first partial derivatíves are well behaved on these surfaces. 56 Basic theory of classical electromagnetism where ut, and itm are the electric and magnetic energy densities, respectively [42]: u= SetêR : (1.114) la um = >LIBP. = (LHS) 2u Ifthese same relations aré assumed to hold for a general time-varying electromag- , netic field, (1.111) becomes , Ii, (2.5 am Eav=? aU +Um)= where U = Ue + Um. We see that the second term in Poynting's theorem can be interpreted as the time rate of change of the electromagnetic energy stored within the volume V. “The third term in Poynting's theorem is the surface integral pas. A It is interpreted as the rate at which electromagnetic energy is leaving the volume V by passing through its surface S. The Poynting vector $, which appears in the integrand, has the units of energy per unit area per unit time (J/m?s) or power per unit area (W /m?). It can be regarded as the rate at which electromagnetic energy is passing through a unit areá whose normal is in the directioà of the vector É x FL. This interpretation proves particularly useful when dealing with the propagation of electromagnetic waves:(Chapters 2 and 3). The direction of the Poynting vector is then normal to the wavefront, and the magnitude of the Poynting vector is the power per unit area of the wavetront (see Figure 3.18). Now let's rewrite Poynting's theorem, substituting what we have karned from our discussion, = ff, 8-dav = ff l%fioav + (a dl serav 5 fl), pióav ) fp s-a5 QI (1.116) or (1118) The interpretation of this relation as a statement of conservation of energy within the vólume V is now clear; it is summarized in Figure 1.29. The rate at which the nonelectromagnetic soirces supply energy to the electromagnetic field must equal the rate at which energy is being transferred from the field as heat, plus the rate at which electromagnetic energy is being stored, plus the rate at which electromagnetic energy is leaving the volume by passing through its surfaces. 1.4 Energy of the electromagnetic field — Poynting's theorem 57 - dprmiendiitta, 3 E 4 So És o mIa EE Ee Sd BE mea O aa sé oa BB sis els + e a bo H & 1 E B 8 BE> als 3ê tã it Bis 8 dh B$85 FREE SPACE > SB 5 Es E GE es .28 ms osE. to gie = 5 £ = Sid = EiP pa I ES jo 385 l “ga Sis Ez aja 355 ei? mod VOLUME V WITH SURFACE: S. Fig. 129. Schematic drawing showing the interpreiatiom for the terms in Poynting's theorem. 58 . Basic theory of classical electromagnetism. When 8)/84 = O and 30/9t = 0, Equation (1.118) becomes Notice the similarity between this equation, which is a statement of conservation of energy, and the continuity Equation (1.5), which is'a statement of conservation of. charge. The energy density U corresponds to the charge density p, and the Poynting vector S, which represents a flow of energy, cprrespands to the current density 7; . which represents a flow of charge. Xt should be emphasized that Poynting's theorem (1.101) was obtained by sim- ply combining Maxwell's equations; hence, its validity rests on the validity of Maxwell's equations. The same thing cannot be said for a particular physical in- terpretation of the terms in the theorem. The physical interpretation of the theorem has been a controversial issue almost since its inception [43]. It would not suit our purposes to give a detailed account of this controversy; however, consideration of a few of the more important points will increase our understanding of the theorem. Any particular physical interpretation is, of course, tied to the physical makeup of the volume to which the theorem is applied. Recall that our interpretation is for a rigid isothermal volume composed of simple materials. We will not consider changes in the interpretation that result from changes in the properties of the ma- terials. Instead, we will concentrate on the ambiguities that exist in the particular results we have already presented, Equations (1.117) and (1.118). Letus accept that the surface integralin (1.117) represents the rate at which energy is leaving the volume V by passing through the surface S. Does it then necessarily follow that the Poynting vector S can be interpreted as the flow of electromagnetic energy at a point? That is, does S represent the rate at which energy is passing through a unit area whose normal is in the direction of $? The answer to this question is no, and we only need to consider the derivation of Poynting's theorem, namely Equation (1.97), to see this. A vector field 7 whose divergence is zero can be added to the Poynting vector without affecting the theorem. In particular, if we let an alternate Poynting vector S' be defined as S=847,.. “(Ilda where v.T=0, (1.119) then the last integral.in (1.97) is unchanged when &' is substituted fôr S = É x H. Which vector correctly describes the flow of electromagnetic energy at a point, S or any one of the myriad choices for $/? There is no conclusive answer to this question, although additional constraints can often be imposed that exclude certain choices for 5' [44-47], All we can really say is that in certain applications, particularly those involving wave propagation, the interpretation of the conventional Poynting vector S as representing the flow of electromagnetic energy at a point gives physically meaningful results. | , 14 Energy of the electromagnetic field — Poynting's theorem 59 Our conclusion is that we can apply Poynting's theorem'(1.101) to any electro- magnetic problem without reservation, because its validity rests on the validity of Maxwell's equations. However, when we use a particular physical interpretation for the terms in the theorem, for quantities that appear in the integrands such as S, ue, and um, wé are introducing an additional hypothesis whose usefulness will depend upon the particular problem being considered. The concepts associated with Poynting's theorem and the Poynting vector are . . nicely illustiated by the simple example shown in Figure 1.30a. Hére wehave a pair of batteries producing a steady current / in a long, straight wire. The straight wire has finite conductivity o, whereas the connecting wires are assumed to be perfectly conducting. We would like to know how the energy supplied:by the batteries is transported to the wire where it is dissipated as heat. This is à problem that was discussed by both Poynting and Heaviside [38, 40, 48]. Foo To simplify the analysis, the complex geometry of Figure 1:30a is replaced by the very idealized geometry of Figure 1.30b. The long, straight wire of radius a is surrounded by:a perfectly conducting cylindrical shell of inner radius b. A batteryis ” connectedto eachend of the structure. The batteries, which are shown schematically in Figure 1.30b, are assumed to be of radial design, so that the entire strmeture is rotationally symmetric about the z axis. This geometry and simple variations ofit have been discussed many times in the literature [6, 49, 50]. We want to determine the electromagnetic field and the Póynting vector in a region near the center of the wire (z = 0) away from both ends.: The interior of the wire is labeled 1, and the space between the conductors is labeled 2. The time-independent electric field is expressed in terms of the scalar potential , which is a solution to Laplace's partial differential equation (Pierre Simon de Laplace, 1749-1827): É=-vo, (1.120) where Vo =0. . (1.121) In the wire, we find (Problem 1.18) that SE : = Solé) : (1.122) a axa ; Bu dos ; Ee at (1.123) and, in the space between the conductors, LH z bm. = semezo(5) (5) So à saem ao 60 Basic theory of classical electromagnetism LONG STRAIGHT. a) PERFECTLY CONDUCTING CYLINDRICAL SHELL b) BATTERY Fig. 1.30, a) Batteries producing current 1 in a long, straight wire. b) Idealized geomeiry for studying the electromagnétic field of (a). . 1.4 Energy of the electromagnetic field — Poynting's theorem 61 The current density within the wire is then J=cê= La, (1.126) za and the charge density on its surface is Va a : . =1 fã =i AD Da =t(d: — É EEDp=a = ——— | -), p= (Ba Dora = (6 cos — À «SED poa atm (é) (1.127) where we have assumed £ = £o in the wire. From the current (1:126), the magnetic excitation is found to be > fo =>—[ É 1.128 = ma(2JA, (1.128) = 1 a " x = [28 1.129) h=a (5)s (1.129) The Poynting vector (1.100) is E in region 1 from (1.123) and (1.128) and in region 2 from (1.125) and (1,129): , a (PR PV. : s= mm) q.130 x R LM fo &= (estos mora (5) (5) (95) ao In Figures 1.31a and b the charge, current and electromagnetic field aré sketched for a region near the center of the wire.!3 The volume current density 7 is'in the axial direction and is uniform throughout the wire. The surface charge density ps is zero at the center of the wire (z = 0) and increases linearly with z; it is positive for z < O and negative for 2 > 0. The electric field É has only aú axial component within the wire. Outside the wire theie are axial and radial componerits to the electric field. The magnetic excitation H is in the azimuthal direction; it is zero on * thé axis of the wire, increases with increasing p until it reaches'a maximum at the surface of the wire, then decreases with increasing p. The Poynting vector S is sketched in Figure 1.31c. Outside the wire, the Poynting vector has an axial component S,,, which is due to £2, and Hg. This component “of the Poynting vector represents an axial flow of energy in the space between the conductors. Since the surface charge density p; produces £2, (1.127), itis essential for the axial flow of energy. é There is also a radial component $5, to the Poynting 3 tn Figures 1.31 and 1.32, the length of the arrow is used to indicate the relative magnitude of the, current density, electric field, and Poynting vector; the number of symbols indicates the relative magnitude of the charge density and magnetic excitation. * 4 In Poymting's analysis of the wire, he did not include the radial component of the electric field, Esp; therefore, he obtained no axial component &, to the Poynting vector. He concluded that all energy flow was normal to the conduetor [38]. This oversight was later pointed out by Heaviside [48]. 66 .“Basie theory of classical electromagnetism We wish to obtain a set of boundary conditions that ensures that a solution to Maxwell's equations within the volume V is unique. We will state qur results and then provide an argument [52, 53). Uniqueness Theorem For a volume V that contains only simple materials and for which the impressed currents Ji are specified, a solution to Maxwell's equations is uniquely specified for all times + > O by the initial values of the field (É and B attime t = 0) throughout V and the values of the tangential component of either Ê orB (à x E orhx B) over the boundary surfaces of V (S; and $.) for t > 0. The tangential component. of É cam be specified on a portion of the boundary surface and the tangential component of B on the remainder. Inother words, if a solution to Maxwell's equationt is obtained within the volume, and the solution satisfies all of the above conditions, it is the only solution. We begin our argument by assuming that there are two different solutions to Makwell's equations in the volume V for times 1 > O. We will call these two solutions É, B; and Ep, Bp. At time £ = O the field É, Bis specified everywhere in V, and both solutions a and b are equal to the specified field. Now we will consider the difference between the solutions: dE = É, — É», 88 = B. — By, etc, Since Maxwell's equations are linear in all field quantities, the difference field must be a solution to Maxwell's equations v x 08 = 205) ar vxsi= aj + MD), This can be seen by simply writing Maxwell's equations separately for &,, É, and És, Bp and then subtracting the equations for b from those for a. Maxwel!'s equations for the difference field can be combined as in the derivation of Poynting's theorem (Section 1.4) to obtain HI, sê. ads fl [18.2 285) +5H. 68) ay + fjos x6H)-fids =0. (1.133) Now we will assume. that there are ; only simple materials or perfect conductors, - within V, so the constitutive relations (1.9)-(1.11) apply. In particular, for the vol- ume density of current we have Ta Tear Tia = 084 Tia and ho = Jos Jo = cê, + Gi where the first term in each expression applies within a conduétor and the second term applies within aregion where there are impressed currents. Since the impressed 1.5 Eleciromagnetic boundary value problem — ufiqueness theorem 67 currents are specified, ja= aj = o(ê,- E) =o(8Ê). (1.134) After inserting (1.134) and the other constitutive relations, (1: ia3) becomes rota pa) + iaê x65)-hdS=0. É (1.135) “The integrand ofthe last integral in (1.135) is ' “ 0Êxa6)-A=5B-(Ax55)=-6Ê-(a x 85), (1136) where we have used the vector identity (Appendix B) A(Bx0)=C(AxB)=-B(Ãx 0), We will assume that both solutions satisfy the boundary conditions onsS stated above: À xEorh x B is specified on S for times + > 0. Thus À x 52 orA'x 86 is zero at every point on S; this makes (1.136) and thus the last integral in (1.135) zero: sa HI eltépav + À ii, alBpav) == Ml overam S=-[I, olêpavo “quam Here we inf: introduced the notation =5 If, elêpav + > SJ Iuápa, The term X is always a positive number or zero: : X>0, 1>0, (1.138) and from (1.137) the partial derivative of Xº with respect to time must be a negative number or zero: or De<0, +>6 (1.138b) “The initial conditions we have assumed att = O are É, = É, or dê =0, E, = By or 3B = 0, which make 4=0 1=0 (1.138c) These three 'statements (1:138a-c)' are satisfied only if X = O for > 0.16 This 18 “To see this, considêr (1.138b), which we can write as ax «mm [*tt80-20] . E w 8:50 at 68 Basic theory of classical eleciromagnetism means that 8É = 0, oB 0, and therefore E =Ê, B= B, for t > 0. Thus, there is only one solution, a unique solution, to Maxwell's equations within:V for the stated conditions. ln the above argument we have ignored an important fact: The electromagnetic field propagates with a finite velocity. This fact can often be used to reduce the amount of information needed to obtain a unique solution. In other words, our orig- inal statement for uniqueriess specifies information that is sufficient for obtaining a unique solution to Mpeg 's equations, but some of this information may not be necessary. This point is easily illustrated for a two-dimensional region of free space, the circular disc of radius p, on the x-y plane shown in Figure 1.34a.!? In two dimen- sions, our volume V with exterior surface S, becomes the surface S bounded by the curve C,. The electromagnetic field É, B on S is assumed to be zero for times ? < O. The semi-infinite cylinder in Figure 1.34a shows this surface at times 1 > O; that is, each cross section of the cylinder is the surface S at a different time. In the next chapter, we will show that the electromagnetic field propagates with the speed of light in free space, c. Now consider the point P(x = 0, y = 0) at time t=n < po/cin Figure 1.34b. Because of the finite velocity of propagation and causality, only the field at points within the right circular cone can influence the field at P.!8 The height of the cone is equal to the radius ofits base, cr, = p1. From this drawing we'see that the initial values of the field É and B at time t=0on the surface Sy (disc of radius.p1) determine the field at P. The initial values of the field over the remainder of-the surface S do not influence li field at -, This means XE+A) <A, and, with (1.1388), OSAUHADS AO, +>0. Now we will assume that X is continuous at 1 = O; then (1.1380) implies that £=0, 1=0+, and the above expression with : Os AA) AON=0, or * XMa)=0, The same argument can be applied'repeatedly to show that 2 = O for all > 0. ?” Here we present a simple physical argument for two spatial dimensions; a rigorous uniqueness proof for three spatial dimensions, which includes the finite velocity of, propagatio, is presented in References [54] and [55]. « !º For à causal system, only events that have occurred earlier in time can influence an event at the presenttime. If causality were not invoked, Figure 1.34b would include an additional inverted cone with apex at P and base att > 41. 1,5 Elecromagnetic boundary value problem - uniqueness theorem 69 e t=0— REGIONOF2-D| c a SPACE x e ct P=ch b) ' e | t=n-pgc — t=0 + o Fig. 1.34. Two-dimensional region in free space at various times. 7 Básic theory of classical electromagnetism Fig. 1.35. Schematic drawing for a radiator in infinite space, The surface $, encloses all sources. : Atthe later time t = fa > po/€ (Figure 1.340) the surface of the cone intersects the surface of the cylinder. Now the initial values of the field (t = 0) on the entire surface S and the values of the tangential component of either É or B on the boundary curve C, at times O <t < to — po/c (shaded portion of the cylindrical surface) determine the field at P, The values of the tangential components of É and B on C, for the times 22 — po/c <t < ta do not influence the field at P. Incertain applications, the region in which Maxwell's equations are to be solvedis infinitein extent; that is, in Figure 1.33 the external surface S, recedes to infinity. The boundary condition on this surface, the specification of the tangential component of É or B for t > O, then requires special consideration. This is best explained by discussing a simple example. Ê The example in Figure 1.35 is for a radiator (an antenna) in free space. The interior surface S; of our volume V coincides with the surface of the radiator, and the exterior surface 5, is a sphere of radius r.. The sphere of radius r, encloses the radiator, it contains all of the sources for the field. The electromagnetic field within V is zero for times + < O, after which the radiator is tumed on: Now the field is to be observed for times O < t < tmu. If we choose the radius 7, of the exterior surface S, so thatr. > ro + Ctmax, the physically relevant field É, B (one that propagates «with the speed of light and is causal) will be zero on this surface for the entire period of observation. As the exterior surface recedes to infinity, re —» oo, the period of observation becomes infinite, fmax — 00. We conclude that a unique, physically relevant solution to Maxwell's equations is obtained by specifying the tangential component of either É ór B on only the interior 1.6 Numerical solution of Maxwell's equations using finite differences a surface S;. On the exterior surface S, the tangential components of this field are zero due to the finite velocity of propagation for the electromágnetic field. 1.6 Numerical solution of Maxwell's equations using finite differences: An example Up to this point, we have described the basic elements of the electromagnetio boundary value problem: the partial differential equations (Maxwell's equations), the constitutiverelations, the boundary conditions, and thé. requirements for a unique solution. With these elements established, there arê various mathematical methods that can be used to obtain a solution for a particular problem, that is, for a certain geometrical configuration of conductors, dielectrics, etc, These methods roughly fall into two, categories: analytical methods and numerical methods. Analytical methods provide solutions that are closed-form expressions (i.e., for- mulas) for the electromagnetic field. These solutions can be examined to obtain Physical understanding of electromagnetic phenomena: For example, the behavior of the field at a large distante from a source can be found'by,evaluating the solution in the limit as the distance from the source becomes infiniite: (the radiated field, first discussed ir Section 3.5). A particular analytical method is generally applicable to only a few geometrical configurations. Consequently, there are a limited number of analytical solutions available for study, and these are for-rather simple geometries. Numerical methods use appropriate algorithms with a digital computer to obtain solutions to Maxwell's equations. The solutions are a set of numbers, such as numerical values for the electric field at points on a grid in space for a particular set of times. Such solutions are not always useful for obtaining general physical understanding of electromagnetic phenomena. Nevertheless, numerical methods have an advantage in that they can be used to solve problems involving complicated geometries thatare not solvable by standard analytical methods. Numerical methods ate often the only alternative when an accurate solution: is Tequired to a practical electromagnetics problem. In the remainder of this text, we will apply analytical methods to sólve a few basic electromagneticproblems, Our objective will be to obtaim a physical understanding Of electrornagnetic phenomena by examining the solutions. In this section, we will describe a particularly straightforward, numerical method based on approximating partial derivatives by finite-diffêrence quotients [56-61]. In the literature on electromagnetism, the method is often referred to as the finite- difference time-domain (FDTD) method. The approach, however, is not specific to electromagnetism, but is also used in other branches of physics, such as acoustics, and fluid mechanics. Our objective is simply to introduce numerical methods for solving electromagnetics problems. In this procéss we will see the solution of a practical electromagnetics problem from start to finish: the forinulation of a theo- Tetical model, a statement of the boundary value problem, the numerical solution, and the verification of the solution by comparison with experimental measurements. To gain a basic understanding of the method, we will begin with a simple, one- dimensional problem in which the electromagnetic field is assumed to depend only 7% Basic theory of classical electromagnetism ERA tir=5 o HE dis A o Ao 1F , vis=as AA o º 5 10 15 2 25 zfer Fig. 1.40. Numerical results showing the distribution of the electric field in space at various times, The pulse has not reached the right-hand boundary, to/t = 2.5. ofthe grid points. This procedure is often called “stepping-in-time” or “marching- intime," because the solution is developed incrementally in time starting with the initial values at t = 0, Numerical results calculated using this procedure are presented in Figure 1.40. The distribution of the electric field in space (Ey/E, versus z/ct) is shown for three times: t/r =5, 15, and 25. For this example £,/t = 2.5. The field is seen to be a Gaussian pulse in space. A simple calculation shows that the pulse is propagating fo the right with thé speed of light c. From these purely numerical results, weinfer | that the electric fieid has the following form in space and time: Elas t) = Ee Niue (14) and from similar numerical results for the magnetic field we find that Bde)=— dente, )=- Ipe icito-upie, (1.146) These are the equations for an electromagnetic plane wave propagating in free space, as we will show by analytical means in Section 2.1, For the times shówn in Figure 1.40, the pulse has not yet reached the right-hand " boundary at z = Zmax- Qur assumption (1.142) that the electric ficld is zero at this point for the-times of observation is consistent with these results. The pulse propagates as it would if the right-hand boundary were absent. - For the later times shown in Figure 1.41, the pulse has reached the boundary, where it is seen'to be reflected in inverted form. Our assumption that the electric ficld is always zero at z = zmax physically corresponds to the region z > Zu being à perfect conductor: The tangential component of the electric field, which is & in this case, is always zero at the surface of a perfect conductor. From these purely numerical results, we conclude that the electric field of a Plane wave (1.145) normally incident on a perfectly conducting half space is reflected in inverted form — a result that is easily confirmed by analytical means. . É 1,6 Numerical solution of Maxwell's equations using finite differences ” Fr urnas A o Br, à - trans AV af o[— «NV ur=ãas ar o Ss 10 15 20 25 = zter Fig. 1.41. Numerical results showing thê distribution of the electrie field in space ar various times. The pulse has reflected from the right-hand boundary, (9/7 = 2.5. We should say a few words about the size of the grid spacings Az and At. Of course, we have assumed that our method of solution is convergent; that is, the solution to the difference'equations, (1.143) and (1.144), tends' to the solution of the differential equations, (1.139) and (1.140), as Az and At tend to zero. Thus, we expect our results to differ negligibly from the actual solution when Az and At are chosen reasonably small, that is, small enough to resolve the spatial and temporal variations of the field. A mathematical argument shows that Az and At cannot be chosen indepen- dently if our method of solution is to be convergent [56-59]. In fact, for conver- gence the grid spacings must satisfy the domain of dependence condition (Courant- Friedrichs-Lewy condition), which, for a one-dimensional problem, is gar <1. : (1147 Az . : Below, we will offer a simple argument for this condition. * Consider'a point at which the field is to be determined, like the point marked A in Figure 1.42. Examination of the difference equations, (1.143) and (1.144), shôws * that the fleld at this point is determined solely by the values of the field at the points within the triangle shaded dark gray. The sides of this triangle havé slopes LAr/Az. The interior of the triangle is called the domain of dependence for the difference equations. We know from Maxwell's differential equations that the eleciromagnetic field propagates with a maximum speed equal to the speed of light c. Therefore, the field at any point within the triangle shaded light gray in Figure 1.42 can influence the field at point A. The sides of this triangle have slopes -1/€, and the interior of the triangle is called the domain of dependence for the differential equations. This domain of dependence for the difference equations must include the domain of dependence for.the differential equations. Otheryise, there would be points that could theoretically influence the result at A that would not be included in our 78 Basic theory of classical electromagnetism + + + + + SLOPE At/Az A . º . + + : SLOPE 1/c, º . To at * É Ê +; o. : Pol RR DOMAIN OF de DEPENDENCE - Fig. 1.42. Domains of dependente for the differential equations (triangle shaded light gray) and the difference equations (triangle shaded dark gray). : numerical scheme: So we must choose . At which is Equation (1.147). * The numerical inethod of solution we have described for one spatial dimension can be extended to two and three spatial dimensions. The mathematics are no more complicated; however, it is a bit more difficult to visualize thé spatial and temporal arrangement of the field components for the higher dimensions. For three spatial dimensions and the rectangular Cartesian coordihate system (x, y, 2), the grid in space (called the Yee lattice) is composed of cuboids with sides Ax, Ay, and Az. The six components of the electromagnetic field are evaluated at staggered positions in space as shown in Figure 1.43, and the finite-difference equations corresponding to the two Maxwell's equations, (1.50) and (1.51), in free space are [60, 61]: n 1 8 (aj qui 3) - " 1. 1 1 1 ct esgota d) semfilos(is+ 5441) 120) -afes(iseness) cer(rna+1)]] (1.148a) 1.6 Numerical solution of Maxwell's equations usingfnite differences 79 — sy 4 z Gir) Gob, eet) (i++) oDo Os La po q : (5,19 GL ” SB 8,8 x Fig. 1.43. Spatial grid (Ye latrice) for three-dimensional problém showing thé location of the field components about the point (i, j, k). : Poa : : mta Las D)- Edi esp fit sido +37 po 1 1, E 1 ted) rem Elo(isrts 1) (eua) foliar) -a(ieb0)]] (1.148) nim É casi + -E! LAN csrtt (i ++ 34) = 2 2 e A 1 1 1 cf (rparia) cenfila(im H nad pt))- Denis rss) mes ju afs(i+10+14) and 1 1(; gm (145 had 1 Aa ess ref esto 2510) so Basic theory of classical electromagnetism ceia; 237 2) Efer(s pit) (1.149) Lol —cB; prio). . (1.149) empre gi 1 + 1 Eslijak — *fi+- ef.» 43) femea fsy (+; cl DD) Ifatfo cB, (i put sf LItGk+s cem, jd k+ à) “o (11490) Here the conventional notation + Fi, jk) = FltAx, jAy,kAz,nAt) has been used. These finite-difference equations are solved in the same manner as for one dimen- sion: The equatioris are applied alternately to advance the solution in time. The grid spacings must satisfy the three-dimensional version of the domain of dependence condition (1.147): (1.150) 1.6 Numerical solution of Maxwell's equations using finite differences 8 FREE ANTENNA SPACE - COAXIAL TRANSMISSION LINE Fig. 1.44. Cylinárical monopole antenna fed through an image plane from a coaxial trans- mission line. ; which for Ax = Ay = Az is simply? cAt 1 Eid AT 343 The utility and accuracy of the finite-difference time-domain method are best, demonstrated by a practical example. Consider the cylindrical monopole antenna”. shown in Figure 1.44 [62]. The antenna is a vertical metal rod placed over a metal, image plane; it is fed through the image plane by a coaxial transmission line. This isan idealized theoretical model for a practical antenna; for example, the vertical rod might represent the radio antenna on an automobile, with the image plane representing the sheet metal (roof, fender, etc.) on which the antenna is mounted. A cross section of the antenna is shown in Figure 1.45. The image plane is assumed to be infinite in extent, and all conductors;are assuméd to be perfect. The volume of free space V in which the electromagnetic field is to be determined surrounds the antenna and extends into the coaxial line to the depth z = —£4. The boundary surface of this region is indicated by the dashed line in Figure 1.45. To-obtain a unique solution to Maxwell's equations within 'V for times O < 1 < ; tmax, We must specify Ê and É within V at time + = O. In addition À x É orfi x B: must be specified on the boundary surface of V for-all times O < + < max We will assume that the electromagnetic field (£ and B) is zero within V attime t = 0. On the cross section of the coaxial line at A (z = —£4), the tangential 2 When the grid spacings are equal, say all spacings equal Az, the domain of dependence condition for n-dimensionel space is simply . cara EA gd Ao ya which agrees with our results for one and three dimensions. 86 Basic theory of classical electromagnetism pulse; the electric field is then given by (1.151) with the incident voltage Vig) = Ve Cro, (1.155) The antenna is characterized by the time Ta = h/c; this is the time required for light to travel its length. The ratio of the characteristic time for the Gaussian pulse : 7 to the characteristic time for the antenna Ta is t/tg = 0.114.383 : Pigure 1.47 shows the surface charge density p, on the coaxial line and antenna as à function of the normalized position z, / h and the normalized time £/Ta. This is the surface charge density on the inner conductor of the coaxial line for — 1.0'< 2/h < O and on the antenna for 0 < z/h < 1.0. Each slice of this figure (with 1/Ta fixed) is similar to ône.of the graphs in Figure 1.40 for our one-dimensional problem. At the surface of'a perfect conductor, the normal component of the electric field is proportional to the surface charge density, so this is also a graph for the radial component of the electric field at the surface of the inner conductor and antenna. This diagram, called a “bounce diagram,” clearly shows that the pulse of electric field travels up the coaxial line until it reaches the aperture (2/h = 0,0,t/ta = 1.0; point A in Figure' 1.47). At this point, a portion of the pulse is reflected back into. the line and the remainder emerges on the antenna. The pulse is next reflected at the open end of the antenna (z/h = 1.0, t/t4 = 2.0; point B); then it travels down to the aperture (2/h = 0.0, t/ta = 3.0; point C) where it is partially reflected and enters the coaxial line. This process is repeated continuously, with the pulse alternately reflected from the open end of the antenna and from the coaxial aperture until there is no longer any charge on the antenna. The magnitude, of the electric field, |Ê|, in the space immediately surrounding the monopole antena is displayed on a gray scale in Figure 1.48 for three times. Graphs of the surface charge density on the antena are below each plot; these correspond to the slices marked a, b, and c in Figure 1,47. The spacing between the conductors of the coaxial line is expanded in these plots to clarify the presentation. In Figure 1.48a the pulse has been partially reflected at the aperture and is tray. eling out along thé antena. A spherical wavefront is centered on the aperture (W, in Figure 1.484); it terminates at the packets of charge on the antenna and on the image plane. À second spherical wavefront (Wz) is produced when this pulse is reflected at the open end of the antena. This spherical wavefront, centered on the open end of the antenna, is clearly shown in Figure 1.48b. The pulse, after reflection from the open end, travels down the antenna, eventually being partially refiected at the aperture and entering the coaxial line. À third spherical wavefront (W%), cen- tered at the apertire, is produced on this reflection (Figure 1.48c). The spherical wavefront War, which is the reflection of wavefront Wa at the image plane, is also clearly shown in this figure, For times beyond those shown in Figure 1.48, similar wavefronts are produced on each reflection of the pulse from the open end of the antena and the coaxial aperture, ? The pulse duration vias chosen short enough to resolve the reflections from different points on the antenna. 1.6 Numerical solution of Maxwell's equations using finite differences 87 Wo + War Hr Fig. 1.49. Electric field as seen on a spherical surface of large radius centered ou the monopole antena. Each graph shows the field at a fixed polar angle 8 as a function of the normalized time, t/ta: b/a = 2.30, h/a = 65.8, t/ta = 0.114. (After Maloney et al. [62], (O 1990 IEEE.) ; To complete the picture for the radiation, in Figure 1.49 we show the electric field as seen by a distant observer, one situated on a sphere of large radius centered on the antena. This is the radiated or far-zone field of the antenna, which we will discuss in more detail in Chapter 8. Each graph in this figure shows the electric field at a fixed polar angle 6 (measured from the axis of the antenna) as a function of the normalized time £/ta. The origin for the time, t/ta = O, and the amplitude of the field were selected to clarify the presentation. Notice that spherical wavefronts centered at the same point on the antenna are always separated by a time interval that is a multiple of 2ra, the round-trip transit time for the pulse on the antena. For example, wavefrónts Wj and Wa, which are centered on the drive point, are separated by-the time 2ra, as are wavefronts Wo and W4, which are centered on the open end. However, the relative times of arrival of the wavefront pairs, suchas Wi, W3 and Wa, Wa, change with the viewing angle 8. For example, at 6 =. 90º the wavefronts Wi, W>, Wa, and W4 are all separated, but atô =. 5º, Wand Wo are superposed, as are W; and Wy. The radiated field off the.end of the antenna (8 = 0º) is zero. Additional insight into the radiation from this antemna is given in Section 8.1. : : An experimental model was constructed for the cylindrical monopole antenna with the dimensions b/a = 2.30 and h/a = 32.8, This model was mounted on an aluminum' image plane (120 em x 155 cm) that was surrounded by absorbing 88 Basic theory of classical electromagnetism material to reduce reflections. The reflection coefficient in the coaxial line was measured at a number of frequencies (50 MHz < f < 18 GHz) and then used with a Fourier transform to óbtain the response of the antenna. toa Gaussianpulse (1.155). Ta Figure 1,50 the measured reflected voltage in the coaxial line (dois) is com. pared with results computed by the finite-difference time-domain method (solid line). The incident,voltage:is the Gaussian pulse (1.155) with X, = 1 V and t/Ta = 0,228. The agreement between the two sets of data is excellent. Note that the pealés in the reflected voltage are spaced by about the round tsip transit time for a pulse traveling at the speed of light along the antenna, t/Ta = 2.0, Figure 1.50b is a comparison of the measured (dots) and computed (solid line) electric fields on the image plane (9 = 90º) at the radial position p/h = 12.7. Agáin, the agreement is excellent. These results illustrate two important points: 1) that electromagnetic theory, as embodied in Makwell's equations, accurately predicts experimental observations and 2) that a particularly straightforward numerical method can be used with a digital computer to solve these equations. As the speed and memory of digital com- puters increase, the utility of such methods for solving practical electromagnetics problems will certainly increase, : 1.7 Harmonie time dependence and the Fourier transtorm Maxwell's equations as:written in Table 1.3 are for an electromagnetic field with general time dependence; that is, no restrictions are placed-on the temporal be- havior of quantities such as the electric field EF). In certain, applications, these equations can be Simplified by specifying a field with harmonio time dependence: A field for which all quantities are cosinusoidal in time, for example, (7,1) = $Avcos(ot + qr), where the parameter « is the angular frequency, The simplif- cation that results for hármonic time dependence is a consequence of the special properties of lincar systems. Thus, before we consider Maxwell's equations for harmonic time dependente, it will be helpful to review a few of the basic properties of linear systems [63]. “171 Linear systems The input to a system, which we will call ft), produces a unique output, which we will call g(t), as illustrated in Figure 1.514. The relationship between ft) and &(t) may be specified by'a set of equations; we will represent this felationship by the operator L: L[ro] = eq). — (156) We might think of /(%) as being the volume density of electrio current :/(5,1) and &() as being the resulting electromagnetic field E(F,t), B(F,1); the two -are related through Maxwel]'s equations. A linear system (Figure 1.51b) has the special property that if LAO] = e) 1.7 Harmonic time dependence and the Fourier transform 89 05 pe 4 — FD-TD RESULIS 1 04 * * MEASURED DATA] Los a é 02 o > a É o ê ã 0.0 -01 -02 o 06. 10 15 ao Dt . Era 15 = T—— 1] r — FDTD RESULTS | |] E * * MEASURED DATA | 10H 4 5 [ 1 n r 1 ã as q mor 1 8 r 4 E oo - a É 1 -0.5 1 E 14 19 É AN PA ao 15 : 20 b) ra i i for the cyliridrical monopole an- Fig. 1.50. Comparison of calculated and measured results y tenna excited by a 1 V Gaussian pulse: b/a = 2.30,h/a = 32.8, 7/ta = 0.228. a) Reflected voltage in the coaxial line. b) Electric field on image plane (9 = 90º) at p/ h = 12.7. (After Meloney et al, [62], ( 1990 IEEE.) so Basic theory of classical electromagnetism HO) — [e st) x. a) 9 h ' ,h() + az fa(r) gil) + az galo) DD) ; NO . 0 ' fe) — ea) o 5 ” ' 9 0 nto 1 a : ' Pig. 1.51. Diagrams illustrating à) genetal system, b) linear system, and c) time-invariant system. % , and LLh()] = 820), then “Marra fara), (1157) where ay and az 'are-arbitrary constants. This Property is called the “principle of superposition"” Examples of linear operations are addition, subtraction, differenti- ation, and integration. A time-invariant system is one for which Lft-r)]=e(t-t, (1.158) where 1, is an atbitrary constant. This property is illustrated by the drawings in, Figures 1.5la and 1.5lc, where a shift in the time at which the input stárts is seen: to cause a shift in the time at which the. output stárts, the shape of the output-is. unchanged by the shift. A system with this próperty is often called “stationary” or “non-time varying” A function that satisfies the relationship Lfo]=cre (the output is proportional to the input) is an eigenfunction of the operator. For a time-invariant, linear 'system (operator) the eigenfunctions are exponentials: Lele] = H(cw)eiot, (1.159) “1,7 Harmonic time dependence and the Fourier transform 9 The time independent factor H (co), which is a function of the parameter co, is called the “system function" Fora real system, a real input produces a real output. We will be dealing with real, time-invariant, linear systems. From our discussion, we see that for such systems the input? 6) = C(w) cos [wt + W(w)] = Re [F(c)e!e"] (1.160) with . É F(w) = Clajeivto) produces the qutput . e() = A(w) cos [wt + &(w)] = Re [G(we!e!] (1161) with ' Gu) = H(w)F(w) = A(w)elt, In words, this important result states that a cosinusoidal input to a real, time- inariant, linear system produces a cosinusoidal output. ** The following simple argument verifies (1.159) [64]. For a general system witivexponential input, we have fO=e, g)=Wea oe”, . Here we have faciored the exponential from the output, leaving the function W that depends on both r'and the parameter «», The system is time-invariant so * He-t)=eltcio, gs) =Wet — to, o)elet-to, This input also can be written as : 4 fe É to) = erteto got toe pg, where e=/oe cân be viewed as a coistant. Because the system is linear, the output for this input is . Bt) = e! gt) = e! WE, el o í ] = Wit, ajeteto, Equating the two expressions above for the output, we find that : E Wet — to 0) =W(t,0) am for arbitrary 4. Hence, W must be independent of t: : : ' Wit, 0) = Hu). 25 The complex input 1)= Fel = Re(Fel) + j Im(Feitt); produces the complex output : e)=HFeit =Re(HFe”) + jlm(HFel), vehere Re indicates the real part and Im the imaginary part of the complex number. The input can be considered asthe superposition of the two terms on the right-hand side of the equation for /(1). Because the system is linear, the output is the superposition of the outputs due to each of these two inputs. À real input produces a real output; hence, the component of the output Re(H Fe!) must * result from the component of the input Re( Fe"), 96 Basic theory of classical electromagnetism This result suggests that We should examine the integral Ii, [Fêe. 0). Tr, cofav (1.176) for fields with harmonic time dependence. Or substituting Mexwell's equations (1.169) and (1.170) into (1.176) and using the divergence theorem (1.98), we find that? REA Par E do fl (Gê br ip BJav les é 5a é . +fb(GE xa") ás=o . = qm This formula is known as lhe complex Poynting's theorem. When we introduce the complex Poynting vector RAÇA o)=5Ê€, 0) x TG, 0), (1.178) itbecomes : la ja . la a las o Ji. E Jar = so fi (8-5 -5Ê Pav fisas-o. “(IT From our discussion of time-average values, it is clear that Re [37,0] = (S(7, 1); (1.180) that is, the real part of thé complex Poynting vector is the time averáge of the Poynting vector (1.100). Notice that $.. is not a Phasor in the sense of j and É. It cannot be used as in (1.163) to obtain the time-varying Poynting vector, À physical interpretation of this theorem requires the specification of constitutive elations; we will assume simple materiáls (1.9)-(1.11): ' DF, 0) = cE(, q), (LI81) : ÉGo)= ae. o), (1.182) and , TE = TP o) + IC.) (1.183) with : I6,0)=0ÊG,0). . “ (1.184) Here we have separated the volume density of current into an impressed current J, and a conduction current J,; : *º These are the same mathematçal stepi that we used in obtaining Poynting's theorem for fields with general time dependencê (1.99), . º “1,7 Harmonic time dependence and the Fourier transform 97 With (1.181)-(1.184), the complex Poynting's theorem (1.179) becomes, Ms e aco | = so fi (Falêr SotdP)av + q) 8. asi (1.185) On equating the real and imaginary parts of this equation, we obtain the two equa- tions; - fi, Re (5E-J)av = !y, (Gitta av + fp reçõo vas (1.186) = fm (5ê ; Jia = o fi (qubt E anidt ar 80). d5, 1.187) + fp ( where we have indicated the Hermitian magnitude of a complex vector by lÃl= VÃ. A. (1.188) The first of these equations, (1.186), is an expression for the time-average power “within the volume V. To see this, we must identify each term in the equation as the time average of a quantity; from (1.106) Ii, (giro av = fl tia = (28) (1.189) and similarly from (1.108) II, E (58 ' JoJav = - fl 8 -Irav = pao, (1190) With these results and (1.180), (1.186) becomes DU = (28 5) .d5 1197 Emj-(ibegos am which states that the time-average power supplied by nonelectromagnetic sources to the electromagnetic field must equal the time-average power transferred from thé field as heat plus the time-average power leaving the volume by passing through its surface. The second equation, (1.187), concems the time-average energy stored in the elêctromagnetic field within the volume. From (1.112) and (1,113), we have Gera grama cm Basic theory of classical electromagnetism Ii, (GutdoJav == ff (SiiBejav =20. (1.193) Thus, (1.187) becomes o) — Um) = UM im(2e-1)av + fimo as), (194) which is an equation for the difference in the time-average energies stored in the electric and magnetic fields within the volume, When the right-hand side of this equation is zero, we have the interesting result We) = Um), os, for the total energy U = Ue + Um, (U) = UU) = Ulm). Half of the total time-average energy is stored in the electric field and half is stored in the magnetic field. 1.7.4 Uniqueness theorem Earlier we presented a uniqueness theorem for electromagnetic felds with gen- tal time dependence (Section 1.5). We will now develop a separate uniqueness theorem that applios specifically to fields with harmonic time dependence; it will involve the complex véctor phasors E(F, q), B(F, co) rather than the time-varying field EP, 1), B(F, 1). The procedure we will use is essentially the same as before, except that Maxwell's:equations and Poynting's theorem for general time depen- dence (Table 1.3) are replaced by their counterparts for harmonic time dependence (Table 1.5) [65]. As before, we will state our result and then present an argument. Uniqueness Theorem . For a volume V that contains only simple materials and for which the impressed currents J; are specified, a time-harmonic solution to Maxwell's equations is uniquely specified by the values ofthe tangential component f either É or É (à x É ori x 5) over the boundary surface of V (S; and 5, in Figure 1.33), The tangential component of É can be specified on a portion of the boundary surface and the tangential component of É on the remainder. Each tegion within V, with the exception of perfect conductors, is assumed to have fnite conductivity. The case for lossless materials (o = 0) is viewed as the case for lossy materials (o ;* 0) in the. limit as the loss goes to zero (o — 0). Tn other words, ifa solution to Maxwell's equations is obtained within the volume, and the solution satisfies all of the above conditions, it is the only solution. We will assume that there are two different solutions to Maxwell's equations de- seribed by the complex vector phasors Es, Bs and É», By. The difference between 1.7 Harmonic time dependence and the Fourier transform 99 these solutions, É = É, — E, 85 = B — By, 3 also a solution to Maxwell's equations; therefore, it can be used in the complex Poynting's theorem (1.185): ju Ii, (esp - SotBEJav =fil (Getér)av E |), (518 ' it)av = b Fusbt “axis = -p 1lip.ça xaBas q198) o s2u à a y We have assumed that both solutions arise from the same impressed sources and have the sametangential components for É or B on the surface S. Hence, 87 = O in the last integral on the left, and À x É = O or À. x 8º = O in the integral on the right, making (1.195) jo Looêp- 1jpp)av = | A pé dv. “(1.196) Hi Ro, 2H v AZ On separating the real and imaginary parts, we get fi, (GotEtJav =o, : (1.197) “II, (qeBr = SoltBeJav Foro *'0, Equation (1.197) can only be satisfied by [E] = Oor É, = És; then, from (1.198), itfollows that [58] = O or B, = B». Thus, there is only one solution, a unique solution, to Maxwell's equations within V for the stated conditions. For lossless materials (o = 0) our argument fails; we arê left with the inconclu- sive Equation (1.198). The case for lossless materials can be viewed as the case for lossy materials in the limit as the loss goes to zero. The physically relevant solution for the lossless.case (o = 0) is taken to be the physically relevant solution for the lossy case (o” ;* 0) in the limit as the conductivity goes to zero (o: — 0). This method is sométimes called the “principle of limiting absorption:30 As for fields with general time dependence, problems that involve regions of infinite extent Tequire special consideration. We will again restrict our discussion to the problem described in Figure 1.35: a source of finite extent ir infinite space. When the surrôunding material has loss (o 0), the Pphysically relevant solution to this problem is a ficld that decays with increasing distance from the source, For this solution, the integral over the surface $, in (1,195) goes to zero as re — 00; all of the energy supplied by the source is dissipated before it reaches the boundary and O. (1.198) *º For uniqueness theorems that do not require lossy materials see References [27] and (55). 100 Basic theory of classical electromagnetism surface at infinity 2! We conclude that a unique, physically relevant solution to Maxwell's equatioris is obtained by specifying the tangential component of either É or É on only thé interior surface S;. On the exterior surface 5, the tangential components of this field are zero due to the dissipation in the surrounding material. To obtain the physically relevant solution for the lossless case (o = 0), we take the limit of this solution as o > 0. seems disconcerting that the boundary conditions differ in the uniqueness theo- Tems for general time dependence (Section 1.5) and for harmonic time dependence: The former requires'specification of à x É or fi x B on the surface S as well as the Bt initial values of É and B throughout the volume V, whereas the latter requires only . specification of à.x. É or à x B on the surface S, This difference is explained by à simple physical argument, Consider the following situation: The field is initially zero within V, and at time é = O a source is tumed on, The source. gradually builds ap to à cosinusoidal oscillátion that continues for an indefinite time. For example, the source may be an imhprêssed current of the form : FX RH) cos(wr), = (1.199) where R(t) is a ramp function that goes from O at time £ = O to 1.0 aiter a time equal to several periods'(27r/(») of the cosinusoid. This is essentially the form that all osciliatory signals have in practice. The volume V is finite in extent and contains only Jossy materials (o * 0). Now the field is observed after a'very long period SÉ time; it is oscillating cosinusoidally everywhere within V; the transient field associated with tumihg on the source has long since died out due to the dissipation within V. The field has reached the steady state; it is harmonic in time, and it gontains no information about the initial conditions at time £ = 0, Since the initial values of the field do not affect the steady state solution, it is reasonable that they are not required in the uniqueness theorem for harmonic time dependence.?? *” The physically relevant solution behaves as an outward-going wave that decays exponentially with increasing radial distance re: ae JÊ tim) 0) acerte = eraregndirs noto | B withk= 8 — ja, where ea, [vi +o/0e7 — '] . b=u ly Fo/oi + | 22 A mathematical argument that supports tlis conclusion is presented in Reference [66]. ind 1.7 Harmonic time dependence and the Fourier transform 101 1.7.5 Fourier transform So far our discussion in this section has been confined to fields with harmonic time dependence. However, a field with general time dependence can be thought of asa linear superposition of fields that vary harmonically in time at different frequencies. The formal mechanism that describes this relationship is the. Fourier transformation (Jean Baptiste Joseph Fourier, 1768-1830): h os : . fe) = = f o Fel do, (1.200) where the weight associated with the harmônic function of frequency «is " x oo - & Fo)= f fee tear, : (1.201) Ea y Equation (1.201) is called the Fourier transform of the function (7); Equation (1.200) is called the Fourier inversion formula. The following shorthand notation will be used for a pair of functions that satisfy these relations: fe) + Fo). (1.202) We will not consider the conditions required for the existence of the transform; thêy are discussed in detail in References [63, 64] and [67]. We simply note that the transform alivays exists for a physically realizable function; f(t), such as one obtained from a measurement. Ítis easy to show that the transform of a real function (%) hãs Fo) = Eu), . (1.203) and that the transform of the n-th temporal derivative of a function is CÃO é (jo Fo) . (1.204) Now we will consider the Fourier transform of the electric field * ÊG.w)= f EG e iar. (1.205) -oo . Taking the curl of this equation and using the Maxwell's Equation (1.50), we have a o z Vx E(F,0w) =[ Vx EP ear og ora. o . = -/ [ão. 0] ecistar, (1.206) After introducing the Fourier transform of the magnetic field, é . BG o)= / Bç, periedo, (1.207) Es and employing (1.204), Equation (1.206) becomes : VxÊGo)=-joBGo). (1.208) 106 Basic theory of classical electromagnetism [85] G. S, Smith, “On the Skin Effect Approximation Am. J. Phys., Vol. 58, pp. 996-1,002, October 1990. [86] H.B. G. Casimir ând 1. Ubbink, “The Skin Effect” Philips Tech. Rev, Vol. 28, Part, No. 9, pp. 271-83; Part IL, No. 10, pp. 300-15; Part II, No. 12, pp. 366-81, 1967. [37] H. Goldstein, Classical Mechanics, 2nd Edition, Addison-Wesley, Reading, MA, 1980. - [38] 3. H. Poynting, “On the Transfer of Energy in the Electromagnetic Field” Philos. Trans. Roy. Soc. London, Vol, 175, Part II, pp. 343-61, 1884. [39] O. Heaviside, “The Induction of Currents in Cores,” The Electrician, Vol. 13, pp. 133-4, June 21, 1884. [40] —, “Electromagnétic Induction and Its Propagation” The Electriciar, Vol. 14, PP. 178-80, January 10, 1885; pp. 306-7, February 21, 1885. . [41] P.J. Nahin, Oliver Heaviside: Sage in Solitude, IEEE Press, New York, 1988. [42] 1. D. Jackson, Classical Electrodymamics, 2nd Edition, Wiley, New York, 1975. [43] M. Mason and W. Weaver, The Electromagnetic Field, University of Chicago Press, Chicago, 1929. [44] J. Slepian, “Energy and Energy Flow in the Electromagnetic Field,” . Appl. Phys., Vol. 13, pp. 512-8, August 1942. [45] W. H. Furry, “Examples of Momentum Distributions in the Electromagnetic Field in Matter,” Am. 4. Plhys., Vol. 37, pp. 621-36, June 1969. . [46] C.S. Lai, “Alternative Choice for the Energy Flow Vector of the Electromagnetic Field," Am J. Phys, Vol. 49, pp. 841-3, September. 1981. Comments on this paper: | P, Lorrain, Vol. 50, p. 492, June 1982; D, H, Kobe, Vol. 50, pp. 1,162-4, December 1982; P.C, Peters, Vol. 50, pp. 1,165-6, December 1982; R. H. Rome, Vol. 50, . pp. 1,166-8, Decêmber 1982. [47]: U. Backheus and K. Scháfer, “On the Uniqueness of the Vector for Energy Flow * Density in Electromagnetic Fields," Am. J. Phys., Vol. 54, pp. 279-80, March 1986. [48]' O. Heaviside, “Electromagnetic Induction and its Propagation, XXXV. The Transfer of Energy and its Application to Wires, Energy-Current” The Electrician, Vol. 18, pp- 211-3, January 14, 1887. [49] C. Schaefer, Einfilhrung in die theoretische Physik, Vol, II, Part 1, Walter de Gruyter, Berlin, 1932. [50] A. Marcus, “The Electric Field Associated with a Steady Current in Long Cylindrical Conductor," 4. J. Phys., Vol. 9, pp. 225-6, August 1941. [51] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics, Vol. 2, 4th Edition, Pergamon Press, New York, 1975. [52] 1. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. [53] L E. Tamm, Fundamentals ofthe Theory of Electricity, Mir Publishers, Moscow, USSR, 1979. First Russian Edition, 1929. [54] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume 11, Partial Differential Equations, Wiley, New York, 1962. . [55] D. 8, Jones, The Theory of Electromagnetism, Pergamon Press, Oxford, 1964, [56] R. Courant, K. Friedrichs, and H. Lewy, “Uber die partiellen Differenzengleichungen der mathematischen Physik,” Math. Ann., Vol. 100, pp 32-74, 1928. English translation, P. Fox, On the Partial Difference Equations of Mathematical Physics, Report NY0-7689, Courant Institute of Mathematical Sciences, New York University, Now York, September 1956; also IBM. Res. Dev, Vol. 11, pp. 215-34, March 1967. [57] P. Fox, “The Solution of Hyperbolic Partial Differentia! Equations by Difference Problems = 107 Methods)" in Mathematical Methods for Digital Computers, A Ralston and H. S. Wilf, editors, Ch. 16, Wiley, New York, 1964. : [58] E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. [59] W. E. Ames, Numerical Methods for Partial Differential Equations, 2nd Edition, Academic Press, New York, 1977. : [60] K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Iavolving Maxwel!'s Equations in Isotropic Media," JEEE Trans. Anténnas Propagat., Vol. AP-14, pp. 302-7, May 1966. Eae [61] A. Taflove, Computational Electrodynamics, the Finite-Difference Time-Domain Method, Anech House, Norwood, MA 1995. . [62] J. G. Maloney, G. S. Smith, and W. R. Scott, Jr, “Accurate Computation of the Radiation from Simple Antenas Using the Finite-Difference Time-Domain Method,” JEEE Trans. Antennas Propagat., Vol. AP-38, pp: 1,059-68, July 1990. [63] A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill, New York, 1962. Ê : [64] R.N. Bracéwell, The Fourier Transform and Its Applications, 2nd Edition, McGraw-Hill, New York, 1978. [65] R. F. Harringion, Time-Harmonic Electromagnetic Fields, MeGravi-Hill, New York, 1961. : - Lots [66] 1. Stakgold, Boundary Value Problems of Mathematical Physics, Volume , Macmillan, New York, 1968. : [67] E.C. Titchmacsh, Introduction to the Theory of Fourier Intégrals, 25d Edition, ”* " Oxford University Press, London, 1948. Republication, Chelsea, New York, 1986. [68] T. R. Carver and ]. Rajhel, “Direct “Literal” Demonstration pf the Éffect of a Displacement Current” Am. J. Phys., Vol. 42, pp. 246-9, March 1974. [69] D. F. Bartlett and T. R. Corle, “Measuring Maxwell's Displacement Current Inside a Capacitor,” Phys. Rey Lett., Vol. 55, pp. 59-62, July 1, 1985. [70] E. Kata, “Conceming the Number of Independent Variables of the Classical Electromagnetic Field,” Am. J, Phys., Vol. 33, pp. 306-12, April 1965. . [71 P.G.H. Sandars, “Magnetic Charge,” Contemporary Physics, Vol. 7, pp. 419-29, November 1966. “=-f92].D. R. Moorcroft, “Faraday's Law - Demonstration of a Teaser” Ar. J. Phys., Vol. 37, p. 221, February 1969; also, Vol. 38, pp. 376-7, March 1970. ““73]-R. EL Romer, "What do “Voltmeters” Measure?: Faraday's Law in à Multiply Comnected Region,” Am. J. Phys., Vol. 50, pp. 1,089-93, December 1982. “= [78] R.€. Peters, “The Role of Induced EMF's in Simple Circuits” Am.J. Phys., Vol: 52, pp. 208-11, March 1984, : [75] E. M. Purcell, Electricity and Magnetism, Berkeley Physics Course Vol. 2, 2nd Edition, McGraw-Hill, New York, 1985. : ' [76] M. A. Heald, “Electric Fields and Charges in Elementary Circuits,” Am. J. Phps., Vol. 52, pp. 522-6, June 1984. : [77 1. Van Bladel, Singular Electromagnetic Fields and Sources, Oxford University Press, Oxford, 1991. : Problems 1,1 Ampêre invented a device known as the astatic combination. for use in his experiments. This device is basically a detector for a spatially nonuniform 108 Basic theory of classical electromagnetism magnetic field. Its construction is shown in Figure P1.1, The wire forming the detector lies in the plane of the page and is bent to form two identical loops, L and R. The current ly is in opposite directions in the two loops. The detector is free to rotate about the vertical axis C—C', a) This device is called astatic (having no tendency toward a change in di- rection) because it is notaffected by the presence of the Earth's magnetic field. Assume that the Earth's magnetic field B, has the orientation shown in Figurê P1.2a and is uniform over the area of the detector. Show that the Earth's magnetic field does not cause the detector to rotate. b) The detector does rotate when it is placed in a magnetic field that is not uniform over its area. Show that the detector rotates when placed near the long, straight current-carrying wire with the orientation shown in Figure P1.2b.* . . e) Ampêre used this detector in some of his experiments based on the null principle, One of these experiments is shown in Figure P1.2c. The detector is placed midway between two parallel wires carrying the same current. The plane of the detector is normal to the plané cf the two wires, and one c Rr o =| eo Ra Fig. PL.1. Ampêre's astatic combination, Problems 109 DETECTOR INxZ PLANE a) DETECTOR * DETECTOR AF . def IT t | WIRE . LONG, . STRAIGHT | TREM: b) WIRE d SMALL BENDS s«d > - po ! la ã A 7) NENE : o o Pd Fig. PL.2. a) Astatic combination in Earth's magnetic field. b) Astatic combination near a long, straight, current-carrying wire. c) Ampêre's experiment. 1 ut Ho * Basie theory of classical electromagnetism Problems leg of the detector is parallel to the wires and in their plane, The wire on the left is straight, while the wire on the right has a series of small regular bends of size s, where s d. Ampêre found that the detector did not rotate for this arrangement. As stated by Maxwell [5], “This proves that the effect of the current running through a crooked Part of a wire The following simple experiment supports the inclusion of this term [68, 69]. The parallel plate capacitor shown in Figure P1.3 is connected to an alternat- ing current source. This produces the current T(t) = 1 sin(wt) in the wire connected tó the plates and the voltage V(t) = V, Cos(wt) across the plates. A voltmeter is connected to the terminals of a tightly wound, circular toroidal coil. The voltmeter measures the voltage Vn(t) = Vm cos(mr). o In position A, the coil surrounds the wire. The alternating current in the wire (J ) produces a magnetic field within the coil, and by Faraday's law, the time-varying magnetic field produces a voltage (electric field) at the terminals + Sfthe coil. n position B, the coil is inserted between the plates ofthe capacitor, and a voltage is again observed at the terminals of the coil, However, itis now N the alternating displacement current (80/01 ) that produces the time-varying » magnetic field within the coil and the resulting voltage across its terminais. ao Ó 2) Assumo that the capacitor is ideal: The electric feld exists only between a A] [PRA the plates (no fringing), and it is Uniform and normal to the plates. Let , “ the diameter. of the plates be d and their spacing s. The coil has radius b Fig. PL.3. Experiment for observing the “displacemient current: * 6 < d/2), Giroular cross section of area Sw = xa?, and N tums, The Soil is very tivin (a « b). Obtain an expression for Vy when the coil js . between the plates of the capacitor. approximately à : . + b) For the following practical values: de ns 1 (Qcoso ? + sing ô f “Tejatdr, . -o0) 3 Vo =100V s=10cm 4meor f=20kt b=35em Here we have assumed that the element is aligned with the 2 axis, and we have used the spherical coordinates shownin Figure 7.1b. — a “= 14em, Obtain the magnetic field 8, ofthe current element by applying ue my. i k Jay im integral form to the open surface S whose boundary is the E 9 “Urb ate needed on lhe coil to produce a voltage V = pd c da in Figure P1.4. Show that your resultis essentially e the Biot-Savart formula, which states that the magnetic field of a current 1.3 An infinitesimal current element is shown in Figure 7. la. Itis a current Z(1) - * elementis - a acting over the length AZ in the limit as AZ —»> 0, When the current varies . BE da seo xs, slowly with time, the electric field of the element, given in Table 7.1, is Cam ti6 Basic theory of classical electromagnetism Fig. P1.5. a) Models:for'a resistor and a voltmeter. b), c) measurements. E ' *19 "SHORT CIRCUIT! 5 o a (1.50) and (1.51). Use the rectangular Cartesian coordinate system (x,y, 2) and assume that the boundary surface is the x- ii s A=2asinFi -) plane with th = 2,asin Figure P1.6. Pl ith the unit normal Assume that the plane boundary in Figure P1.6is between a perfectconductor region 1, and free space, region 2. A small probe is used to ricasure the magnetic field in free space at the surface of the conductor, and itis found to be a simple function of position: a dx, y,0+,1) = Co(y8 — 49) cos(e). a) Use the electromagnetic boundary conditions to determine the surface ia of charge ps(x, y, t) and current J,(x, y, t) on the perfect con- luctor. b) What is the electric field Es(x, y, O, 1) in free i conntnn nt) space at the surface of the Problems n7 and d) Connections for várious * REGION 2 REGION 1 x Fig. PL6. Plane boundary with accompanying rectangular Cartesian coordinates. 1.10 The plane boundary in Figure P1.6 is between two simple materials with the constitutive parameters £1,01,/1 = io and E2,02,H2 = Ho. The electro- magnetic field is harmonic in time with frequency «w. At the surface, the phasor for the normal component of the electric field in region 1, Er(x, y, 0—, w),is known. Determine the phasor for the normal component of the electric field in region 2, E2:(x, y, 0+, w), and the surface density of charge os(x, y, tw) on the boundary. : a Li. Fora time harmonic field in a simple material (e, |4, o), show that the volume density of charge p must be zero and, as a result, V - E = O. [, 1.12 a) For a perfectly conducting body of volume V with surface 5, show that; the Lorentz force expression (1.12) becomes F= pras + x Bjs. b) Although the electromagnetic field on the surface of the conductor appears in this expression, it is not well defined. Just inside the surface, all of the components of both É and B are zero, while just outside the surface some of the components of £ and B are not zero. What value should be used in the expression for the force? The following argument shows that the correct value is the average of the fields just inside and just outside the surface [75]. The surface of the body is approximated locally by the planar geometry in Figure P1.7. The charge and current are assumed to be distributed in athin layer (-L < z < L) at the-surface, rather than ir an infinitesimal layer as for the perfect conductor. Within this layer, the volume densities of charge and current are functions of only the normal éoordinate z, and the current is in the y direction: p(z) and J(2) = Jy(2)5. To the left of the layer, the field is É, Bi, and fo the right É», B». For both regions, 1 and 2, assume thats = £, and 4 = Ho. Show that the electrostatic and magnetóstatie forces per unit area of the surface are = az +55] 1 us Basic theory of classical electromagnetism REGION 1 &,B1 REGION 2 &2,Br -L<:< (se TO=IMdS Fig. P1.7. Eletric charge and current distributed within a thin layer at the surface, and . = » La né dy = À, x Ea + bo] where a z a Es A a= Ff aee Jo f Foard 4 € Use the results from part band the boundary conditions at” Ethe surface of a perfect conductor to show that POLE/L a sas Fel fi(Li il as. 1.13 The constitutive relation between D and É is written as an operator L (1.156), LG, 0] = DG, 0. Which of the following relations are linear and which are time invariant? E, Do, aE(,+) de) =aêç,n+pDO,d a DP) =aedF + a 9 D(, 1) = atê, 1) Problems 119 1,14 1.16 117 118 De, )=aê(,t-t) a po - DE )= [ FÊ, — dr. =) Consider the time-harmonic electric field (1.165) EG)= D) AF, wo)cos [vt + di, wo)]1, ióepa . - where co, is used to emphasize that the frequency is a fixed parameter. Examine the relationship between the complex vector phasor for this field (1.166) EGu)= D) AF, wet ETA and the Fourier transform of the field (1.201) co Êc, w)= / EG net dr, a The following Fourier transform pair will be of use [63]: . elo + 275(o — wo. where 8 is the Dirac delta function (1.58), which is discussed in detail in Section 5:2. Derive the finite-difference equations (1.152), (1.153), and (1.154) for the electromagnetic field with the components £,, E, and By in-the circular cylindrical coordinate system (p, &, 2). Each of the ficld comiponents is inde- pendent of é. Use the spatial grid in Figure 1.46. a). Write a computer program to solve the one-dimentional, finite-difference equations (1.143) and (1.144). Checkyour. programiby obtaining theresults in Figures 1.40 and 1.41. u Violate the domain of dependence condition by letting c At/ Az be slightly * greater than one and make a graph as in Figure 1.º 40. What effect does this have on the solution? An ideal battery is connected to a resistor using perfectly conducting wire, as in Figure Pl.8a. A two-dimensional analogue of this circiit is shown in Fig- ure P1.8b. This is a simpler geometry to analyze because the electrômagnetic field is inyariant in the axial direction. Make a sketch showing the Poynting vector on.the cross section of this geometry. Your sketch should have the same form as Figure 1.31c. Assume that the magnetic field is udiform and in the axial direction inside the cylindrical tube, while it is insignificant outside the tube. Give a physical interpretation for what you observe [76]. Show that the scalar potential and the electric field for the'long, straight wire (Figure 1.30b), are given by Equations (1.122)-(1.125). Let the con- stitutive parameters for the wire be £o, io, and o. Hint: Assume that the Basic theory of classical electromagnetism PERFECTLY CONDUCTING WIRE RESISTOR. a) b) Fig. P1.8, a) Battery connected to a resistor. b) Two-dimensional aúalogue of a). solution to Láplace's equation (1.121) in cylindri inatesi à . ylindrical coordinates is separabl Pp, 2) = f(p)g(2). Solve for & in regions 1 and 2, and use the, praias of the geometry' and the electromagnetic boundary conditions to determine any unknowrt constants in the solutions. 1.19 Obtain the expression for the time avera; A ige Of a product of t | i quantities (1.175). º "o lime hamonie 1.20 We have examined boundary conditions for the cases where singlé layers of charge Or curtent are present on the interface between two materials, causing discontinuities in the normal component of D and the tangential component Problems 121 REGION? REGION 1 Fig. PL9. Double layers of charge and current at boundary. of. For some materials, the physical structure at the interface may be more complex and require a model containing more than one layer of charge or current. For example, when the interface is between a metal and a polar liquid of electrolytic solution, a “double layer” of charge may be present at the interface [77]. In this problem we will examine the boundary conditions that result for the simple case of a plane surface with a uniform double layer of charge or current. a) Double layer of charge: Let the plane surface separating regions 1 and 2 have the normal à = 2. A uniform layer of positive surface charge, ps, lies just above this surface at z = A /2, and à uniform layer of negative surface charge, —s, lies just below this surfacé at z = — A /2 (see Figure P1.9), The product 9; A = Cy. E Determine the electric field, É, in regions 1 and 2 and in the space between the layers of charge, —A./2 < z < A /2. Assume that the permit- tivity and permeability of all regions are those of free space. Show that this structure leads to the following boundary condition for the electrostatic potential-O: d — O =Cg/êo, where the potential at point b with respect to that at point a is Da ss =-[ E-dt, a Let C, remain constant in the limit as A — O and show that the volume density of charge for the structure can be expressed as 06(z) 38 where à is the Dirac deita function. 126 + Electromagnetic plane waves in free space: Polarized waves Equation (2.3) becomes = 1 ê Ea = pino, -— Q4R This equation shows that £, + is, at most, a constant independent of the three spatial variables and time. This constant is chosen to be zero, since we are only interested in solutions that vary in space and time, Thus, the electric field is transverse to the direction of propagation: Ês = Eat + Epa. 214) On substituting (2.14) into (2.1) and using (2.12), the following relations are obtained for the components of the electric and magnetic fields: ôBis pi 3-0 (215) Be 106 3 EE (2.15%) 9Bye (10 a ah ao (2150) After these three equations are integrated with respect to time, each equation con- tains a constant of integration representing a component of the field that is inde- pendent of time. Again, because we are only interested in solutions that vary in space and time, these constants are chosen to be zero, and (2.15a) through (2.15c) become Ba =0 (2.168) 1 Ba = Pb (2.16b) 1 By = doEn. (2.160) Ta vector notation, these three equations are simply a 1 ao Br = aê x Es. QI From (2.17) we ses that the magnetic field, as well as the electric field, is transverse to the direction of propagation. The magnetic field is also orthogonal to and simply related to the electric field. This is a TEM (transverse electromagnetic) plane wave. AI that remains to complete the description of a particular plane wave is to specify the function Z4 in Equation (2.112) or Equation (2.11b). As an example, we will consider the case where f, has only a y component, and this is a Gaussian function: Egte,t) = Fut — 2/0) = Ego Ult a À Bia 1) = —— Ee MUS ufito t c 2.1 General time dependerice 127 t=th=b+8/e Fig. 2.2, Propagation of a Gaussian pulse. Drawing shows spatial dependence of the field for the two times, to and tj. The spatial distribution for this field is shown for two instants of time in Fig- ure 2.2. This is a Gaussian pulse propagating in the direction of increasing 2 with a velocity equal to c. Recall that we encountered this electromagnetic field earlier in Section 1.6, where the analytical description given above was inferred from purely numerical results obtained by directly solving Maxwell's equations with a computer. . Our distussion to this point has been for a plane wave propagating in the +z dirêction. We will nów remove this restriction and surâmarize the properties of the plane wave by considering a wave propagating in the diregtión of the arbitrary unit vector É. This vector together with the unit vectors ú and Ô-are the basis vectors for the orthogonal, right-handed system (x, v, k) shown in Figure 2.3. The electromagnetic field is now EG) =Fu-E-zio j (2.189) - 1 =. Br, )= E x EF, - (2.18) with the transverse nature of the wave specified by the relations Rê=0 “(19% and : Ê.B=0. . (2.19b) Notice that it is no longer necessary to include the + sign to distinguish the two directions of propagation for the wave. The unit vector É includes this information; vhen the direetion'of É is reveised, the scalar product É . º automatically changes sign. Ata fixed time £ the electromagnetic field, (2.184) and (2.18b); has the same 128 Electromagnetic plane waves in free space: Polarized waves PLANE OF x Fig. 2.3. Geometiy for à plane wave propagating in a general direction. value at every point-P ón a plane for which É .F = rcosy = constant; this is indicated schematically in Figure 2,3. The magnetic excitatión f is o Ta bag É H=—B=-kxê; : 2.20) Ho &o 68:40) thus, the Poynting vector (1.100) for the plane wave becomes — E . S=ExH=IEPR, 221) bo where to = Volêo =316.7...8 (2.22) is the wave impedance of free space. The Poynting vector is uniform ovei any plane surface with normal É and represents the instantaneous rate at which ejectromag- netic energy is passing through a unit area of the plane (the power per unit area). 2.2 Harmonic time dependence: Monochromatic plane waves We will now consider a case of practical importance in 'which the source of the electromagnetic plane wavé varies harmonically in time with the frequency f (an- gular frequency « = 277 f) [1-3]. The components of A in (2.112) or(2.11b) are then cosinusoidal functions, and the electric field for the plane, wave propagating in the direction of the Ez axis (2.14) becomes Este, 1) = Exalt, DE A Eyale, 9 = Arcos [w(t 7 2/0) + dx]f + Apcos[wlt 2/0) + gy]. (2.23) 2.2 Harmonic time dependence: Monochromatic plane waves 129 Note that the amplitudes and thé phases of the field components £, and. are different, There is no required intorrelationship between these twa field compidnents for the plane wave. Thê wave number for propagation in free space is defined as ko =2m/ko = wfe, (2.24) where Ag is the wavelength in free space. The relative amplitde and the relative phase of the components £; and £, of the field are described by the angles y and à: tan(y) = Ay/Ax, Osy=<m/2 (2.25) ô=by— dr -m<ô<m. (226) After using (2.24)(2.26), the electric ficld (2.23) becomes Esta) = A:[ cos [sate, Aê +1anG) cos [sa(e, 1) + sb). em vit a o : so) = ot rh + de - (2:28) and the accompanying magnetic field is : Bdan= SE [cos [sstz, 9] = tanty)cos [sstz, ) + ]5). (2.29) Equations (2.27)-(2.29) completely describe a monochromatic plane wave propa- Eating in free space. The terin monochromatic indicates that the electromagnetic field oscillates at a single frequency; a wave of visible light would be a single (mono) color (chromatic). : “The electric field for a monochromatic plane wave propagating in the direction ” of the positive z axis is shown in Figure 2.4. In Figure 2.4a the spatial dependences of the two components of the field, E, and £,.,, are shown at a fixed time t,. The two components are scen to vary periodically along the z axis with a period equal to the wavelength Ap. The total electric field É, = E, % + Ej+9, shown in Fig- “ure 2.4b, rotates around the z axis, making one complete revolution per wavelength in z: For this example, the rotation of E, about the z axis has the same sense as a right-handed screw. « Ata fixed position z,, the electric field vector rotates in time with the angular frequency «, and the tip of the vector traces out a closed curve, the polarization ellipse, For the example in Figure 2.4b, the ellipse is traced outin à counterclockwise sense when viewed by an observer looking in the direction of propagation (ie,, ” looking in the direction of the positive z axis). The equation describing the polarization ellipse is obtained by combining the expressions for the field components £,. and E. to eliminate the function sa(z, t): ee = cos [su(z, )] Eyalz, t) A tantpã COS [setz, 6) + 8] = cos [su(z, BJ cos(s) — sin [sa(z, 1)] sin(8), 130 Electromágnétic plane waves in free space: Polarized waves Eeto Pol ELLIPSE RIGHT-HANDED SCREW SENSE IN SPACE Esto) LEFT-HANDED SENSE OF ») ROTATION IN TIME Fig. 2.4. Monochromatic plane wave. a) Spatial dependence for the components of the electric field, Er +, £y4, at the fixed time fo. b) Spatial dependence for the total field, É, at the fixed time 1, and the curve (polarization ellipse) traced out in time by.the electric ficld at the fixed position zo.- : 3 . 2.2 Harmonic time dependence: Monochromatic plane waves 131 or Eno) Estr) o fo [Esenflo As tan) — CA cos(ô) = RE — [582] | Ba), ; (2.30) After taking the square of (2.30) and rearranging terms, the équátion for the polar- ization ellipse results: Exelz 0)" Deo) Exales) Esales)] 1 [e ai Dra o a | rm] sito (2.31) The terminology used to describe the state of polarization for the electric field of a plane wave is a matter of convention. Here we will adopt the convention used by the Institute of Electrical and Electronics Engineers (IEEE) [4]. The sense in which the polarization ellipse is traced out is termed right-handed (left-handed) if an observer looking in the direction of propagation for the wave sees a clockwise (counterclockwise) rotation of the electric field vector as time-advances.! Thus, the sense of the polarization for the plane wave described in Figure. 2.4 is left-handed. Note that this is opposite to the right-handed screw sense for the rotation of the electric field vector in space (time fixed at t,). Iá:the optical literature, the-screw sense in space is sometimes used to describe the jtate of polarization. Thus, in the optical terminology the sense of the polarizationkfor the plare wave in Figure 2.4 would be right-handed. : The shapê and the orientation of the polarization ellipse and the sense in which the polarization ellipse is traced out are determined by. the angles y and 8. The size of the ellipse is determined by the amplitude of the electric field. Any quantity indicative of the amplitude, such as A,, can be used to set the size of the ellipse; this is evident in (2.31) where both field cômponents £,.+ and'£,+ are divided by Ay. When the polarization ellipse is circumscribed in a rectangle whose sides are parallel to the x and y axes (Figure 2.5a), the angle between the diagonal of the rectangle and the x axis is y. The effect of the angle 5 on the polarization ellipse is illustrated in Figure 2.6a, where polarization ellipses are shown for vatious values of 5 with the angle y fixed at tan(y) = 0.5 (y = 26.57º). Propagátion is in the direction of the positive z axis. For 5 = O we have lineár polarization; the field components E; and £, are in phase and the polarization éllipsé reduces to a straight line. For 0 <'8 < x the polarization ellipse is traced with a left-hânded sense; the component &, leads the component &, in time. When'ô = 7r.we again have linear polarization, and the polarization ellipse reduces to a straight line: For —y < 8 <Q the polarization elipse is traced with a right-handed sense; the component E, lags the component E, in time, fil For the special Cases tan(y) = 1 (A; = Ay) and d= m/2, 3/2, which are shown in Figure 2.6b, the polarization ellipse reduces to a circle. These two cases : A siinple procedure for remembering this terminology is to place the thumb &f the right (left) hand * “in the direction of propagation. If the curled fingers of the right (left) hand point in the direction in which the electric field vector is rotating, the sense of the polarization is righit- (left-) handed.  136 Electromagnetic plane waves in free space: Polatized waves and 2, secê(x) : = E “qu E secy) | 0 do (2a) tan*(x) The superscript T indicates the transpose of the matrix. The column vectors E, and E/; are related by the rotation matrix Re: En = R$E!, . (2.49) where ; [cost —sinz . The subscript £ indicates à left-handed rotation about the z axis. After substituting (2.49), Equation (2.44) becomes (EQTRÍQRE,=1, (2.51) which on comparison with“(2.45) shows that R7QRe = Q'. The inveise of the rotation matrix is equal to its transpose, Rj! = R7, and thus, Rr QR =". (2.52) Notice that this operation, transforms the real, symmetric matrix Q. into the real, diagonal matrix Q'. A transformation of this form is called a “similarity transfor- mation” . The matrix equation (2.52) is equivalent to four nonlinear equations involving the angles y, 8, x, and 7. These equations will be used to obtain the desired expressions for x and 7 in terms of y and 8. This can be accomplished by a “brute force” manipulation of (2.52), of it can be accomplished by a somewhat more elegant application of linear algebra in which the matrix Ry is determined in the similarity transformation (2.52) that-diagonalizes a general, real, symmetric matrix Q. We will outline the later approach [5]. If the column vectors Xy . X = V i=1,2 (2.53) Xy are the eigenvectors of the:miatrix Q, then OX =MXi ] “asa Here the À; are the eigenvalues of the matrix Q; that is, they are roots of the equation 1Q-»i=0, (2.55) , where | | indicates the determinant of the matrix and [ is the identity matrix. Now consider the matrix X whose column vectors are the eigenvectors X; of Q: Xu $] Km . (2.56, De; Xp (256) 2.3 The polarization elipse in the coordinate system of the principal axes 137 Equation (2.54) shows that for this matrix OX=XA; . (2.57) where the diagonal matrix A is “Ta O A-= [ a x . (2.58) From (2.57), we see that x xgx=A. x (2.59) A comparison of Equations (2.52) and (2.59) shows that the diagonal elements of the matrix ” must be the eigenvalues of the matrix Q, that is, Q' = A, and that the matrix Ry must be the matrix of the eigenvectors of Q, thatis, Re =X. “After applying the general reshlts presented above to our specific problem, we find that the eigenvalues and eigenvectors of the matrix Q are n= h 341 —'sinê(0) sin? (2y)] / 2sinZty) (2.60) and 1 X=C asgue | : (2.61) cos(8) where the —(+) sign in (2.60) is for i = 1(2) and the C; are atbitrary constants. Here À, is chosen to be the smallest eigenvalue, because 04 < 05, for —m/4 < x <m/4 - On equating the matrices Q' and A, we find that “sec!y)jsecê(y) = [1 = 1 — simê(o) sinê(2y)]| /2sint(y) [1+/1 sima siP(2y)]/2sin0y); 1-1 =sint6) sin(2y) tan(g) = | ESSE No 1+V1- sinZG)sini2y) After making the substitution sin(a:) = sin(ô) sin(2y), this equation becomes = A-cosçe) tan(x) = Trcos() — !n(x/2) and so it follows that a = 2x and sin(2x) = sin(8) sin(2y). (2.62) and r [sec?(x)/sec2(r)] ftanê (x) thus, 138 Electromagnetic plane waves in free space: Polarized waves 2.3 The polarization ellipse in the coordinate system of the principal axes 139 From the eigenvector X1 and the first column of the matrix Rg we have tan(r) = tan(y 1 — à1)/ cos(õ), or, on inserting A and rearranging, : 1 + cosZ(8) tan2(2y) — 1 E tati(r) = cos(ã) tan(2y) After making the substitution tan(8) = cos(8) tan(2y), we get >» 5 5 E) S b bo. se B&B s a as 1-cos(8) o = = 2). tan(r) ng) tan(8/2). It therefore follows that 8 = 2r and tan(27) = cos(8) tan(2y). (2.63) We now have the desired relationships for determining the angles x and from the angles y and 5: : sin(2x) = sjn(8) sin(27) ; (2.64) tan(2r) = cos(8) tan(27). (2.65) These results can also be used to express y and 8 in terins of x and r: = : . cos(27) = gos(2x) cos(2r) (2.66) tan(8) = tan(2x)/ sin(27). (2.67) When any one of these expressions is solved for a given arigle, an inverse trigono- y metric function occurs, and some care must be exercised to-ensure that the correct | values of these miultivalued functions are used. For example, ftom (2:64) 1 ' : ' x= ln [sinco) sinç2y)], (2.68) [o] z : and from (2.65) : à tan”) [ cos(õ) tan(2y)], v<=/4, lol <m/2 t= | itan”! [cos(8)tan(27)] +, v<m/4 ilol>m/2. (269) tan! [cos(8) tan(2y)] + 7/2, v>m/4 The principal values of the functions sin! and tan”! are assumed in these equations (ie, 7/2 <sin!() <7/2and -7/2 < tani() < m/2): Figure 2.7 is a chart for graphically converting between the angles y, 8 and x, t (Problem 2.6).? On this chart the curves of constant y are horizontal straight lines, and the curves of constant é are segments of ellipses, whereás the curves of constant x ate circles, and the curves of constant 7 are radial lines. The chart in Figure 2.7 is dyawn for left-handed states of polarization; for right-handed states the substitutions ? The use of graphical aids, such as this chart and the Poincaré sphere described in the next section, make it easier to determine the proper values for the inverse trigonometric functions, such às those in (2.68) anid (2.69). Fig. 2:7. Polarization chart for graphically converting between the angles y, 5 and x, 7. This chart is for left-handed polarization. For right-handed polarization, 5 > —5 and x > —x. 140 Electromagnetio plane waves in free space: Polarized waves S, Fig. 28. Coordinátes for the Poincaré sphere: 0 < y </2,-x <8 <m;-n/4<X< m/40<7<m. 8 > —5and x —> —y should be made. As an example, we can use the chart to find that, for y = 60º, é = 50º (this point is marked by a dot on the chart), he transformed angles are x = 21º, 7 = 66º; from Equations (2.68) arid (2.69), the precise values are x = 20.8º, 1 = 66.0º. 2.4 The Poincaré sphere and the Stokes parameters : In the preceding section, the state of polarization for the monochromatic plane wave was described in the two coordinate systems (x, y) and (x', y') by the parameters So y, é and So, x, t, respectively. We will now consider a valuable graphical rep- resentation due to Jules Henri Poincaré (1854-1912) for visualizing the state of polárization as destribed'by either set of parameters [6]. In Poincaré's representa- tion, shown in Figure 2.8, the state of polarization is indicated by the location of a point on the surface of a sphere — the Poincaré sphere. The radius of the sphere is equal to the time-average power per unit area So, and the spherical angular coordi- nates of the point are 27 'and 2x, the longitude and the latitude, respectively. It is easy to show from Equations (2.66) and (2.67) and Napier's rules for a right-angled spherical triangle that the angles 2y and 8 are as indicated in Figure 2.8 [7]. An application of the cosine rule for-sides to the triangle ABC yields cos(2y) = cos(2x) cos(2r), which is Equation (2.66). From the sine rule sin(6) = sin(B)sin(2x)/ sin(27), and from the cosine rule for angles cos(8) = sin(8) cos(2x): thus, tan(ô) = tan(2x)/ sin(27), which is Equation (2.67). 2.4 The Poincaré sphere and the Stokes parameters 141 RIGHT - HANDED Hindi CIRCULAR Fig. 2.9. Various states of polarization on the Poincaré sphere, There is a one-to-one correspondence between the states of polarization and the points on the Poincaré sphere. The various states of polarization:on the sphere are shownin Figure.2.9. On the equator the polarization is linear. States with left-handed polarization are on the upper hemisphere, and states with right-handed polarization are on the lower hemisphere. Left-handed circular polarization is at the north pole; and right-handed circular polarization is at the south pole. The Poincaré sphere provides a convenient graphical tool for visualizing changes in the state of polarization. For example, an optical system may contain several devices that alter the state of polarization of a light wave passing through the system. The state of polarization as. the wave passes through the system cân be graphed as a curve on the Poincaré sphere. Any alteration in the system is then visualized as a change in the curve on the Poincaré sphere. We will examine this application in more detail in the next section. The three rectangular coordinates (S1, S2, $3) ofa point (So, x» onthe Poincaré sphere in Figure 2.8 are S1 = So cos(2x) cos(27), e (2.708) Sa = So cos(2x) sin(27), (2.70b) and 83 = So sin(2). (2.700) The four parameters So, 1, S2, and S3 are the Stokes parameters (George Gabriel Stokes; 1819-1903) for the monochromatic plane wave [8]. Note that the four parameters are not independent, since SE =S/+ 82482. 271 146 Electromagnetic plane waves in free space: Polarized waves where —£ is the phase shift on transmission. Itis customary to normalize the Jones matrix by removing common phase factors such as exp(—jt): n=[10 2.82 "=10 0]: (2.82) When the transmission axis (x') of the linear polarizer is at the angle y, as in Figure 2.10b, the normalized Jones matrix, obtained by applying (2.79) to (2.82), is cost y cosysiny ts [es Vsiny sinZy ta8a) Noy consider a field linearly polarized in the direction of the x axis incident on an ideal linear polarizer with transmission axis at angle y. The Jones vector for the transmitted wave is; cos E'=TE' = Eie tt cosy | 5]. siny Thus, the ratio of the time-average powers per unit area for the transmitted and incident waves is : . Ep ( E! 7, fe (ESTE “This is a statement of the law of Malus (Étienne Louis Malus, 1775-181 2) for linear polarizers. : Retarders are optical elements used to change the state of polarization. The retarder works by dividing the incident field into two components, such as E! and El, and retarding the phase of one of these components relative to the other. When the two components of the wave are reunited to form the transmitted field, the state of polarization Is changed..We will consider an ideal linear retarder in.which the x axis is the fast axis and the y axis is thé slow axis, that is, the phase of the component E, is retarded relative to thê phase of the component E;. In.additioi, . we will assume that the retarder does not attenuate the wave. The Jones matrix for an ideal linear retarder with retardance A is then º 1.0 . peak . Ea a em or, in normalized form, esp q º a tr] (2.84) When the fast axis (x!) of the linear retarder is at the angle 4, as in Figure 2.10b, the normalized Jones matrix, obtained by applying (2.79) to (2.84), is T, = [CO(A/2)+ cos 2ysin(A/2) jsin2ysinç4/2) . ” jsin2y sinçA/2) cos(A /2) — j cos 2y sin(A /2) (2.85) 2.5 Optical elements for processing polarized light 147 A halfwave plate is a retarder with A = 77, and a quarter-wave plate is a retarder with A = 7/2-When the fast axis is along the x axis, the normalized Jones matrices for these wave plates are nlhem)= [g 8] (half-wave plate) (2.86) and ] TA = 7/9) = A [ õ á y 2] (quarter-wave plate). (2:87) There are many different physical constructions for linear polarizers and linear retarders; these are discussed in detail in the optical literature: Here we will only give a qualitative description of representative sheet-type linear polarizers and retarders, which have many commercial applications, The modem sheet-type polarizer was invented by Edwin Herbert Land and is referred to as Pólaroid sheet [11, 12]. A simple, commercial polarizer called an H-sheet polarizer is constructed from a polymeric material such as polyvinyl alcohol. A sheet of this material is heated and stretched unidirectionally, which causes the long-chain hiydrocarbon molecules to orient parallel to the stretched direction. The sheet is then stained with a solu- tion containing iodine. The iodine atoms form long strings:that lie parallel to the molecules ofthe polymer. Conduction electrons can move alông'the string of iodine atoms, and each string can be thought of crudely as a straight resistive conductor (wire) with its axis parallel to the stretched direction. To see how the sheet linearly polarizes light, consider a: wave Incident on the sheet, as in Figure 2.10a, with the electric field lying in the plane of the sheet. Let the field be decomposed into components parallel to and perpendicular to the stretched direction (y). The field component parallel to the strétched direction produces an axial electric current in each string of iodiné atoms. The currents dissipate power in these resistive conductors; thus, this component of the field is severely attefjuated as the wave passes through the sheet; The fielt component perpendicular to the stretched direction produces no significant axiál currents in the strings of iodine atoms, and it suffers little attenuation as the wave passes through the sheet. ConseguentIy, the wave that emerges from the sheet is linearly polarized with the electric field perpendicular to the stretched direction. Fon polarizers with this construction, the ratio |7)y/Tks| is typically in the range 5 x 1072 to 2x 1071 at optical wavelengths (4,000 À< Ao < 7,000 À); for an ideal polarizer, of course, this ratio is zero. The sheet-type polarizer is an example of a dichroic linear polarizer, à polarizer that works on the principle of differential absorption for two states of polarization of the incident wave. Clear sheets of stretched polymeric material are also used to coristruct linear re- tarders. Within the sheet, a wave with electric field parallel to the stretched direction experiences a higher effective permittivity, therefore a lower phase: velocity, than a wave with electric ficld perpendicular to the stretched direction. The thickness of the sheet is adjusted to obtain the desired relative phase retardance for waves with the two orientations of electric field. 148 Electromagnetic plane waves in free space: Polarized waves These two simple optical elements — the linear polarizer and the linear retarder = can be used to fabricate more complicated optical devices. As an example, we will describe the construction of an ideal circular polarizer. The polarizer has the following property: It produces either a left-handed or right-handed circularly . polarized transmitted field frôm an incident field of arbitrary polarization. Thus, the Jones matrix perfórms the following transformationi o Te Tol[A 1 E =c lr: Tola 45]! | (2.88) where the upper (lower) sign refers to left-handed (right-handed) circular polariza- tion. The complex constants À, B, and C are arbitrary, provided 2]C? < |AP-H|B| so that the polarizer does.not produce power. If we specify that the polarizer trans- mits a left-handed (right-handed) circularly polarized wave unattenuated and com- pletely absorbs a right-handed (lefi-handed) circularly polarized wave, then (2.88) can be used to write four equations in the four unknown transmission coefficients, Tax, Ty ---- After solving these équations, the normalized Jones matrix for the ideal left-handed (right-handed) circular polarizer becomes I[1 Fi T=> E : a [ “a (2.89) A circular polarizer with the Jones matrix (2.89) can be constructed from alinear polarizer sandwiched between two quarter-wave plates. The fast axes of the two quarter-wave plates are orthógonal, and the transmission axis of the linear polarizer makes an angle of 45º with the fast axis of either plate. The arrangemênt is shown in Figure 2.11. Note that the traúsmission axis of the linear polarizer is at the angle Y = 45º for a left-handed circular polarizer (LCP) and at the angle y = —45º for a right-handed circular polarizer (RCP). It is easy to verify that the arrangement shown in Figure 2.11 has the Jones matrix (2.89); we simply multiply the Jones matrices for the three elements: ecixt 0 Ji[i [eim q =ToTeTa = . > Fh= Tesla [ 0: eints as 7] 0 ecintt AMI Fj 2 [ &j. ] . The changes in the state of polarization for a wave passing through the ideal circular polarizer can be illustrated by use of the normalized Poincaré sphere. Let the arbitrary state of polarization'for the incident wave be represented by point A on the Poincaré sphere in Figure 2.12. On passing through the first quarter-wave plate, the relative phase angle ô between the two components (E, and E) of the wave is decreased by 1/2, while the relative amplitude, indicated by the angle y, remains fixed. This moves the state of polarization from point A to point B on the sphere, The linear polarizer, withtransmission axis at = 45º, moves the state of polarization on the sphere from point B to point C. The wave is now linearly polarized with its 2.5 Optical elements for processing polarized light 149 T SLOW INCIDENT WAVE AXIS A, WITH ARBITRARY / POLARIZATION QUARTER-WAVE PLATE +— FAST.AXIS TRANSMISSION AXIS LINEAR 45º T FORLCP POLARIZER =-4s é A TRANSMISSION AXIS FOR RCP QUARTER-WAVE PLATE 1— sLow Axis CIRCULARLY POLARIZED TRANSMITTED WAVE Fig*2.11, Three-element, ideal circular polarizer formed from a linear polarizer and two. quarter-wave plates. Angle of linear polarizer: y = 45º for left-handed circular polarizer, y = —45º for right-handed circular polarizer. axis at 45º (x = 0,7 =77/4,0r y = 7/4, 6 = 0). The second quarter-wave plate increases the relative phase angle 5 by 7/2, while the relative amplitude remains fixed at y = 7/4. This moves the state of polarization on the sphere from point C to the north pole, point D, where the wave is left-handed circularly polarized. We could have eliminated the first quarter-wave plate in the above construction and still have obtained a left-handed circularly polarized wave from an incident wave with arbitrary polarization. For this two-element polarizer, the states of polarization at points A and C on the Poincaré sphere in Figure 2.12 are connected by the dashed curve. The extra element (the first quarter-wave plate) in the three-element polarizer is added to make the polarizer pass unattenuated left-handed circularly polarized waves and completely absorb right-handed circularly polarized waves. The two- element polarizer does not have these properties. The normalized Jones matrices for the ideal elements we have discussed are summarized in Table 2.2. The eigenvalues and eigenvectors (A; and X,, i = 1,2) for the matrices are also given in the table. The eigenvectors are normalized so that XX* = 1. Recall from Section 2.3 that the eigenvectors and eigenvalues of 150 Electromagnetic plane waves in free space: Polarized waves Table 22. Normalized Jones matrices for ideal elements Jones matrix - - Eigenvalues Eigenvectors Element mo vd dp” X1 Xo Linear polarizer, 10 1 1 º transmission axis x oo 0 0 1 Left-handed afro 1 aq aq circular polarizer 2l; à o 2lj 2L-; Right-handed 115 1 ara 171 circular polarizer 2l-; 1 0 v2L-i 2lj Half-wave plate, jo j “mm º fast axis x 0 —j — 0 1 +) Quarter-wave plate, 1 [l+j 0 ] "2 no fast axis x AL o -j a-) oJli 2 Sam “ LEFT-HANDED : CIRCULAR y=n/4, FIXED 8,0+x2 Y. FIXED 5, 8,>8,-22 LINEAR +45º Fig. 2.12. The state of polarizatign on the normalized Poincaré sphere as a wave passes through the three-element, left-handed circular polarizer. The state of polarization at A is atbitrary, and the solid line traces the state of polarization as the wave passes through the various elements. Ê a matrix satisfy the relation TX =MX. - From this result we see that an eigenvector is the normalized Jones vector for a particular incident field and that for this incident field the state of polarization is not 2.5: Optical elements for processing polarized light * 151 changed when the wave passes through the optical element; that is, thg incident and transmitted fields have the same state of polarization. For example, for the linear polarizer with.transmiission axis along the x axis, the eigenvectors. are linearly polarized fields inthe x and y directions. The eigenvalue for the x-directed field is one, so this field passes unattenuated through the polarizer. The eigenvalue for the y-directed field is zero, so this field is completely absorbed by the polarizer. Similar arguments.apply to the left- and right-handed circular polarizers. For half- and quarter-wave plates whose fast axes lie along the x axis, the eigênvectors are linearly polarized fields in the x and y directions. The eigenvalues arg the relative . phase shifts for each of these fields on passing through the plate. In the above distussion, we have considered optical elements arrahged to form polarizers - devices that produce a wave with a known state of polatization from a wave with an arbitrary state of polarization. These same optical elements can be arranged to form devices for determining the state of polarization of'a wave.! We will illustrate this point with a simple polarimeter used to determirie the Stokes parameters of an arbitrarily polarized incident wave. The elements of the polarime. ter, shown in Figure 2.13, are a rotatable, ideal quarter-wave plate; a rotatable, ideal linear polarizer, referred to as an analyzer; and a detector. Thé fast axis of the quarter-wave plate and the transmission axis of the palarizer are, respectively, at the angles y, and Yp from the x axis. The normalized Jones matrix for the wave plate-analyzer combination is the product of the mátrices for the individual elements Ty = Tn2 Th! the elements in this matrix are Thxx = C08 Vp[ Cos Yp + j cos(Wp — 2y)]/ 2 Taxy = cos Wp[ sin rp — j sin(yp — 2y)]/2 Toys = sinpa[00s Vip + cost — 2y5)]/ VA Toy = sinbo[ sin vp — j sin(hp — 2y/)]/VD. 4 (2.90) The detector senses the time-average power per unit area' of thê transmitted wave (2.38): Solis bp) = (ENTE (2h, = (TED (Ta E (ho. 291) After some algebra, we find from (2.91) that tra ei Sib vp) = [MEL + EV) ato (Naa É + 1 + Ti? + 1a) 1 no + [MEL = 1858) /286 [Mas — [Toys Ê Tay Upa) + [RE ES ES to ]RECThes Ty + Tio Tapa) + [-Im( E E$)/60) ImCThss Thy + Ty Tre): (2.92) * Section 7.7 contains additional discussion of the use of optical elements to determine the state of * polarization for light. 156 Electromagnetic plane waves in free space: Polarized waves direction (x = 0,y = 0, 2). Locally, the field is seen to behave as a plane wave linearly polarized in thê direction £. : Consider the same dipole used as a receiving antena; this case is shown in Figure 2.15b. Let the field incident on the dipole be a plane electromagnetic wave propagating in the Z direction: Et) = El + Ei) = AL[£ + tan(yyetA ge Morto, (2.98) with the state of polarization determined by the angles y; and 8. The voltage produced at the open-circuited terminals of the receiving dipole can be shown to be Voc = —he?! Elo) = —h. ED É. - (299) From (2.99), we see that the dipole only responds to the component of the incident electric field that is parallel to its axis. Notice that there are two separate right- handed rectangular Cartesian coordinate systems in Figure 2.15b, The coordinates (x,y, 2) are used to describe the incident wave, and the coordinates (xº, y*, 2') are . used to describe the orientation of the antenna. 4 . A slightly more complicated antenna is shown in Figure 2.16a. Itis formed ftom two identical dipoles, one aligned with the x axis and the other aligned with the y axis. An additional device is connected to the terminals of the y-axis dipole, This device changes the amplitude:by the factor tan(y,) and the phase by the angle 5, for the current on transmissioh and for the open-circuit voltage on reception. We will not consider the construction of this device here, but an example is discussed in Problem 2.15. EA : : When this antenna is transmitting, a current source is connected in seriês with the dipoles, as shown in Figure 2.16b. The current at the terminal of the x-axis dipole is then. o, and that at the terminals of the y-axis dipole is 1, tan(y,) exp(j8,). The radiated electric field along the z axis (x.= 0,y = 0,2) is obtained by superimposing the fields for the two dipoles (2.9 . : : Titohelor, + tan(y)e it ye 1tor, (2.100) 2hoz With this antenna, any desired state of polarization for the field can be obtained by adjusting y and à,. For example, when tan(,) = 1 and é, = x/2 (-2r/2) the field is left-handed (right-handed) circularly polarized. Now consider this same antenna (y, and 6, fixed) as a receivirtg antenna placed in the path of an incident plane wave (2.98) propagating in the direction. The dipoles are connected in series as shown in Figure 2.16b, and the sum of their open-circuit voltages (2.99) is , Voc = —he[8"+ tân(ye lê: - Eito) = he AL [+ tan(y) tan(y Je! 48) stre=do, (2.101) Again, the coordinates (x, y, 2) are used for the incident wave and the coordinates Ei) $ Im Section 7,2 we will examine the plectrically short, linear antenna or dipole. The effective height for that antenna is discussed in Footnote 6 of Chapter 7. 2.6 Transmission and reception of polarized waves with antenas 157 v09: d c DIPOLES E AMPLITUDE:PHASE “lá A — CONTROL ES ———a rd q (e) f : RECEIVING a) TRANSMITTING / 2 AMPLITUDE-PHASE x(x) CONTROL, Ex - TRANSMITTING RECEIVING É o é d) h Fig. 2.16. a) Antenná formed from a pair of dipoles. b) Methods for connecting the antenna when transmitting and receiving. (x', y', 2) for the orientation of the antenna. We will examine the squared magnitude ofthis voltage. After introducing the time-average power per unitarea of the incident wave (2.38), gi = 14% P sec2(p) º Mo 158 Electromagnetic plane waves in free space: Polarized waves and multiplying and dividing by sec2(y;), we obtain Voe) = 2éo|he|? 50 sec) pr, (2.102) where the polarization mismatch factor py is — 1 + tanê(yo) tan? (no) — 2tan(ys) tan(yi) cos(B, + d) sec%y,) seci(y;) 62106) This factor determines how well the incident wave is received by the antenna. An examination of (2.103) (Problem 2.9) shows that-0 < p, < 1. The mismatch factor p, is a maximum(p, = 1) when the incident wave has the special state of polarization y = yr, & = ôr, such that Ww=Y. &= [ or Xr = Xn =m— te (2.104) The mismatch factor is zero (null) when the incident wave has the special state of polarization y; = ya: 6; = By, such that ' 8-7, 0O<g <r m=m/2=4m n=[E57 med 40" or o urm v<rf m=-d m=[CíNh vem, Here we used (2.68) and (2.69) to obtain x and 1 from y and 5, and we have been careful to keep the angles within the ranges we originally established/O < y < 7/2, -nx <ô<m,-n/i<x<n/4and0O<rt<y”. These are interesting results. They show that the states of polarization for the incidentfield ibat produce amaximumreception (y,, 8) and anullrgception (dh) for the receiving antenna are simply related to esa of polarization of the field that is radiated when the antenna is transmitting (y;, ,). This relationship is illustrated in Figure 2.17, where representative polarization ellipses for three fields (£, r, and n) are shown. For maximum reception the palarization ellipse of the incident field (r) has the same shape, orientation, and sense of rotation with respect to the direction of propagation (both aré left-handed in the figure) as the polarization ellipse of the (2.105) transmitted field (+). For “pull reception the polarizatior ellipse of the incident field, (n) has the same shape, an orthogonal orientation (the major axes are orthogónal), 7 The subscript é is used to indicate the state of polarization of the radiated Seld for the transmittirig antenna. The subscripts r and n are used to indicate the states of polarization of the incident feld for maximum reception and null reception, respectively, for the receiving antena. 2.6 Transmission and reception of polarized wáves with antenas, 159 WAVE FOR MAXIMUM RECEPTION INCIDENT WAVES => RECEPTION Fig. 2.17. Polarizationellipses for the feld ofthe transmitting anterina (1), the incfdent field for maximum repepio (7), and the incident feld for null reception (1). and an opposite sense of rotation (right-handed in the figurê) as the polarization ellipse of the field for maximum reception (r). The normalized Poincaré sphere is convenient for visualizing the states of po- larization for an antenna. The states of polarization for transmission (t) and for maximum reception (r) are shown on the Poincaré sphere in Figure 2.18a, and the states of polarization for maximum reception (r) and null reception (n) are shown on the Poincaré sphere in Figure 2.18b. Note that the points 7' and 1 are diametrically opposite on the Poincaré sphere. Thus, we see that an antenna that'has maximum reception for a horizontally (vertically) polarized incident field will have null recép- tion for a vertically (horizontally) polarized incident field. Similarly, if maximum reception is for a Ieft-handed (right-handed) circulariy polarized wave, there will be null reception for a right-handed (left-handed) circularly polarizêd wave. Consider an incident wave with an arbitrary state of polarization (y;, dj) repre- sented by the point i on the normalized Poincaré sphere in Figure 2. i8b. The points . 7 and í are joined by the great-circle arc of length s. We will now show that the polarization misinaich factor p, (2.103) is a function of the arc length 5, decreasing with increasing s. Thus, the polarization mismatch factor is the same for all of the states of polarization represented by points on a circle centered at r ôn the Poincaré sphere, This is illustrated in Figure 2. 18b. Note that as s increases and p, decreases, the circle moves away from the point r. Eventually, when s = 77 thg circle reduces to the point 2 and' p, = O. 160 Electromagnetic plane waves in free space: Polarized waves Pr CONSTANT ON CIRCLE o Sim Fig. 2.18. a) and b) States of polarization on the normalized Poincaré sphere: transmitting antenna (t), receiving antenna — maximum reception (r), receiving antenna — null reception (n), and general incident wave (1). c) Detail of spherical triangle, After introducing the angles y, and à, and using trigonometric identíties, (2.103) becomes p= 5h + cos(2y,) cos(2y;) + sin(2y,) sin(2y;) cos(ê; — 8,9]. (2.106) For the spherical triangle shown in' Figure 2.18c, we find from the cosine mile for sides that [7] cos(s) = cos(2y;) cos(2y;) + sin(2y,) sin(2yj) cos(B; — 8,). (2.107) A comparison of (2.106) with (2.107) shows that = E + cos(s)] = cos%(s/2), (2.108) proving our assertion that the polarization mismatch factor is a function of the arc length 5.8 " We have examined a very simple antenna, the pair of dipoles shown in Fig- ure 2.16a, for a specific-direction of propagation (x = 0, y = 0,2), and we have developed interesting relationships between the states of polarization for the fields transmitted and received by this antenna. These results are not restricted-to this $ Because we are dealing withia spheré ofunit radius, the arc length sis the saie as the angle subtended by the arc at the center of tie sphere, Some authors use this angle in the expression for p;. 2:7 Historical note: The experiments of Hertz 161 antenna or to a particular direction of propagation; they apply for any antena and for a general direction of propagation [13]. We will now summarize these results for a general antenna. The radiated field of a transmitting antenna has the state of polarization y;, ô in the direction determined by the angles 6, é. When the an- tenna is receiving an incident plane wave from the same direction 8, q, there will be maximum response, pr = 1, for the state of polarization of the incident field yr. 8, (2.104), and there will be a null response, p, = 0, for the state of polariza- tion 7a, 8n-(2.105). These two states of polarization (r and n) are at diametrically opposite points on the Poincaré sphere. For a general state of polarization y;, 8, of the incident field, the polarization mismatch factor is pr = C0S2(s/2), where 5: is the length of the great-circle arc joining the points that represent the states of . polarization y;, 8; and y,, 8, on the normalized Poincaré sphere. It follows that for two antennas (1 and 2) used in a communications link, their states of polarization must be related by (2.104) for maximum reception on the link GQe=miba=7-81,0<ô <m;in=-7—ên—r <ôn <0)Afew examples will illustrate this relationship. If antenna 1 transmits a linearly polarized wave with y = 7/4, 81 = O in the direction of antenna 2, then antenna 2 should trahsmit a lincarly polarized wave with ya = ya = 7/4, &o = 7 —8n = inthe direction of antenna 1: If antenna 1 transmits a left-handed circularly polarized wave (ya = 7/4, 811 = 7/2) in the direction of antenna 2; then antenna 2 should transmit a left-handed tircularly polarized wave (y2 = y1 = 7/4 82 =1—B1 = 7/2): in the direction of antenna 1. : 2.7 Historical note: The experiments of Hertz” Im this chapter, starting with only Maxwell's equations, we showed that the elec- tromagnetic field in free space can propagate as a plane wave with a velocity equal to the speed of light.The vectors for the electric and magnetic fields of the plane wave are transverse to the direction of propagation, and for the monochromatic Plane wave, the field can assume various states of polarization. One of the great and immediate achievements of Maxwell's theory was that it predicted the wave nature of the electromagnetic field, which we have described above, The theory showed that optics was a branch of electromagnetism and that - many of the phenomena that were observed for light waves should be present for electromagnetic waves of lower frequency. Maxwell's theory and its predictions were not accepted by all of his contemporaries, and indisputable experimental proof for the theory did.not come until somé fifteen years after the publication of Maxvwell's celebrated treatise (1873). This proof was provided by a series of inge- nious experiments performed by the German physicist Heinrich R. Hertz during the period 1887-1888. These experiments represent one of the greatest achievements in the development of electrodynamics, and we will now give a brief description - of a few of the more important experiments that deal with electromagnetic wave propagation in free space [14-18]. : 1 At the time of Hertz, there was no standard instrumentation for generating or detecting high-frequency electromagnetic $ignals,-and he had to develop suitable 166 Electromagnetic plane waves in free space: Polarized waves 13m e 4m A. go Na a : à ice SHEET a) DIPOLE ot = 2 3w/2 b» WALL 8m 6m /4m 2m o Fig. 2.22. a) Hertz's apparatuis for demonstrating the inteiference ofelectromagnetic waves. b) Standing wave formed when a plane wave is reflected from a perfectly conducting plane of infinite extent. c) Examplê of the interference patterns observed by Hertz. Hertz formed a slit using two large coplanar metal sheets, and with the transmit- ting and receiving antennás on opposite sides of the slit, he observed difiraction phenomena, such as we will discuss in Section 3.4. He constructed a large prism from hard pitch and showed that electromagnetic waves were refracted by the prism, just as light waves are tefracted by a glass prism. cá Hertz used a grating of parallel wires to demonstrate the polarization of electro- magnetic waves. The wires of the grating were 1 mm in diameter and spaced 3 cm 2.7 Historical note: The experiments of Hertz - 167 ZINC METAL PARABOLIC REFLECTOR DIPOLE 26cm — TRANSMITTING RECEIVING ANTENNA ANTENNA Fig. 2.23. Hertz's apparatus for observing electromagnetic waves of shortér wavelengih Mo =66em). apart. When the grating was placed between the transmítting and receiving anten- nas, as in Figure 2.24, the reception was a function of the oriêntation of the grating. With the wires of the grating parallel to the electric field of tie wave (Figure 2.244) there is no reception. In this case the incident electric field produces axial electrical currents in the wites. These currents radiate a field that nearly cancels the incident field in the forward direction (to the right of the grating in Figure 2.24a). The grating effectively reflects the incident wave back towatd the transmitting antenna. With the wires of the grating perpendicular to the electric field of the wave (Figure 2.24b) the field produces ho significant axial electric currents in the wires, and the grating has very little effect on the reception. Note the similarity between Hertz's grating of wires and thé-H-sheet polarizer discussed in Section 2.5. Both devices work on the samé principle, but at very different frequencies (radio versus optical frequencies). The only. difference is that the conducting wires of the grating reflect the incident wave, whereas the resistive atomic strings in-the polarizer absorb the incident wave. foi Perhaps the best statement of what Hertz's experiments accomplished is his.own conclusion [16]; * . Casting now a glance backwards we see that by the experiments above sketched the propa- Eation in time of a supposed action-at-a-distance is for the first time proved. This fact forms the philosophic result of the experiments; and, indeed, in a certain sense the: mostimportant Tesult. The proof includes a recognition of the fact that thé electric forces can disentengle themselves from material bodies, and can continue to subsist as conditions or changes in the state of space. The details of the experiments further prove that the particular manner in which the electric force is propagated exhibits the closest analogy with the propagation 168 Electromagnetic plane waves in free space: Polarized waves WRE GRATING RECEIVER | - EL. mede RECEIVER * 181181 b) GRATING Fig. 2.24. Wire grating for demonstrating the polarization of eleciromagnetic waves. a) AL- most complete blockage of wave by grating. b) Very little blockage of wave by grating. of light; indeed, that it corresponds almost completely to it, The hypothesis that light is an electrical phenomenon is thus made highly probable. To give a strict proof of this hypothesis would logically require experiments upon light itself. é References, [1] M. Bom and W. Wolf, Principles of Optics, 3rd Edition, Sec, L.4, Pergamon Press, New York, 1965. E [2] W. Swindell, editor, Polarized Light, Benchmark Papers in Optics/1, Halstead Press, New York, 1975. This reference contains papers of historical interest; included are [6, 8, 10, 11], and [12]. s [3) J.D. Kraus; Radio Astronomy, Ch. 4, McGraw-Hill, New York, 1966. [4] IEEE Standard Definitions of Terms for Radio Wave Propagation, The Institute of Electrical and Electronics Engineers, Inc., Std. 2111977, 1977. [5] A. J. Petirofrezzo,Matrices and Transformations, Ch. 4, Prentice-Hall, Englewood Cliffs, NI, 1966. Republication, Dover Publications, New York, 1978, [6] H. Poincaré, Théorie Mathématique de la Lumiêre, Vol. 2, Ch. 12, Gauthier-Villars, Paris, 1892, : : [7] W. Gellerr, H. Kustner, M. Hellwich, and H. Kastner, The VNR Concise Encyclopedia of Mathematics, Ch. 12; Van Nostrand Reinhold, New York, 1977. Problems 169 [8] G.G. Stokes, "On the Compositign and Resolution of Streams of Polarized Light from Different Sources) Trans. Cambridge. Philos. Soc., Vol. 9, pp. 399-416, 1852. Reprinted in Mathematical and Physical Papers, Volume II, pp. 233-58, Johnson Reprint Corp., New York, 1966. [9) WA. Shurcliff, Polarized Light: Production and Use, Harvard University Press, Cambridge, MA, 1966. : [O] R. C. Jones, "A New Calculus for the Treatment of Optical Systems,” Parts I through VIM, J. Opt. Soc. Am., HI, Vol.31, pp. 488-503, July 1941; IV, Vol. 32, pp. "48693, August 1942; V-VI, Vol 37, pp. 107-12, February 1947: VI, Vol, 38, “PP. 671-85, August 1948; VII, Vol. 46, pp. 126-31, February 1956, Part Eis “ conuthored with H. Hurwitz, . [11) E. H. Land and C, D. West, “Dichroism and Dichroie Polacizers,' in Colloid Chemistry, J. Alexander, editor, Vol. 6, pp. 160-90, Reinhold, New York, 1946. [12] E. H. Land, “Some Aspects of the Development of Sheet Polarizers," J, Op. Soc, Am, Vol. 41, pp. 957-63, December 1951. - [13] G. A. Déschamps, “Techniques for Handling Elliptically Polarized Waves with Special Reference to Antenas, Part II - Geometrical Representation of the Polarization of a Plane Electromagnetic Wave,” Proc. IRE, Vol. 39, pp: 540-4, May 1951. : “[14] H. Hertz, “Ucber electrodynamische Wellen im Luftraume und deren Eefiexion” Ann. Physik, Vol, 34, pp. 609-23, 1888. : [15] — “Ueber Strahlen electrischer Kraft” Arm. Physik, Vol, 36, pp. 764-83, 1889. pa = Electric Waves, Macmillan, London, 1893. Republication, Dover Publications, New York, 1962. - a 07 G. W. Pierce, Principles of Wireless Telegraphy, Ch. VII, McGraw-Hill, New York, 1910. - [18] ). A. Flemink, The Principles of Electric Wave Telegraphy, 3rd Edition, Longmans Green, London, 1916. [19] RW. P. King, “The Loop Antenna for Transmission and Receptiori in Antenna * Theory, R. E. Collin and F. J. Zucker, editors, Part I, Ch. LI, McGraw-Hill, New York, 1969. [20] 1. W. Evans, “The Birefringent Filter” J. Opt. Soc. Am., Vol. 39, pp. 229-42, March 1949. Problems 2.1 Consider the electric field of a plane wave with elliptical polarization: E(a) = [Açeite? + Ayelbo seita, Show that this field can be written as the sum of à right-handed circularly polarized wave and a left-handed circularly polarized wave. 2.2 Show that the expression for B(7) given in (2.33b) follows from.Maxwell's equations and the expression for E(F) given in (2.334). 2.3 Verify that the matrix product in Equation (2.44) is the equation for the polarization ellipse (2.42). . 170 2.6 27 Electromaghetic plane waves in free space: Polarized waves Two electromagnetic waves, one with frequency w and the other with fre- quency cw» (af q), propagate in the direction of the positive z axis. The electric field for each wave is Elo = Ai ( cos [at — ko(wn)z + Gui]? + tan(y) cos [mt — keo(wi)z + us + ap), i=1,2 a) Determine the real Poynting vectors &(g, 1) for the field É, alone, for the field É; alone, and for the combination E. = & + É. b) An instrument measures the intensity 7 = |(S(r, 1))| of the wave: Em 1f a (S€,0)= = / Sr, dr. The time 7 is very large compared to the period of either wave: T > 71 = 2r/w,T > = 2r/02,andT > T0/(T — 1). Show that . k=h+ bh, that is, the response of the instrument to the combination of fields is the same as the sum of the responses to the individual fields. c) If n plane-wave fields with different frequencies are combined, and the above inequalities are satisfied for all waves, will ES i=l Obtain the expressions for the Stokes parameters Si, Sa, and 55 in Equations (2.72b-4) from the corresponding expressions in Equations (2.70a-c). Hint: Use the relations (2.64)-(2.67) and the following trigonometric formulas: 9) = tan) sin(24) Tr) 1 1= tan?(y) cos(2y) = TER 2 tan(2y) = o cot(y) — tan(y) Consider the plane. that contains the equator of the normalized Poincaré sphere, the S1y-S2n'plane. Show that the polarization chart in Figure 2.7 is an orthographic projection of the lines of constant y, 6, x, and T from'the upper hemisphere of the Poincaré sphere onto this plane. For the orthograpihic pro- jection, a line perpendicular to the plane and passing through the point (y, 8) on the sphere locates the point (y, 8) on the plane. Obtain the equations that describe lines of constant y, 8, x, and 7 on this plane. A commercial, two-element, left-handed circular polarizer is formed from a sheet of linear polarizer and a sheet that is a quarter-wave plate. The construc- Problems im 28 29 tion is the same as that for the three-element polarizet in Figure 2.11, only the first quarter-wave plate, the one that is the closest to the source, is absent. a) Determine the Jones matrix for the circular polarizer and verify that the device produces a left-handed circularly polarized wave for an incident wave with general polarization. - Now assume that the. orientation of the polarizer is reversed, so that the incidént field first encounters the quarter-wave platé (that is, both elements Of the device are rotated through 180º about the fast axis, of the quarter- wave plate). Determine the Jones matrix for this device.; How does this device now affect the state of polarization of the incident wave? b Two linear polarizers are placed in series. The first polarizer has its transmis- sion axis atan angle of y = 45º with respect to the x axis, and the second polarizer has its transmission axis in the direction of the y axis. a) Determine the normalized Jones matrix (norimalized transmission matrix) for the combination. : b) The electric field of the incident light is linearly polarized in the direction of the x axis, with the normalized Jones vector o ad Show that the electric field of the transmitted light is linearly polarized in the direction of the y axis. Thus, this combination oflinear polarizers.has changed the linear polarization of the light from horizoital to vertical. Starting from Equation (2.103) obtain the values of y and ô, given in (2,104) and (2.105), for which p, is a maximum (pr = 1) and «minimum (p, = 0). Only consider the case for O < & < 7. a) The state of polarization of an incident plane wave:is to be determined using a single dipole receiving antenna. The dipole and wave have the orientation shown in Figure P2.1a. A device placed ai the terminals of the dipold'measures the modulus of the received voltagé | Vocl. The dipole is in the x"-y' plane, and it is rotated slowly about the 2! axis. Let y be the angle the dipole makes with the x axis. Obtain an expression for |Voe] as a function of the angle y.. i b) Assume that the incident wave has the state of polarization given by the following parameters: Ai = 1.0, y; = 33.21º, 8; = 40.90º. Also let lhe| = 1.0 m, Use the expression obtained in part ato make a polar graph with [Voe] the radial variable and y the angular variable, as indicated in Figure P2.1b. E ã £) Determine the angles x; and r; for the incident wave of part b. On your graph, draw the polarization ellipse for the incident wave so that it is circumscribed by the curve of part b. '