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Conductivity (� )

Relative Permeability (μ r )

Engineering Electromagnetics

EIGHTH EDITION

William H. Hayt, Jr.

Late Emeritus Professor Purdue University

John A. Buck

Georgia Institute of Technology

To Amanda and Olivia

A B O U T T H E A U T H O R S

William H. Hayt. Jr. (deceased) received his B.S. and M.S. degrees at Purdue Uni- versity and his Ph.D. from the University of Illinois. After spending four years in industry, Professor Hayt joined the faculty of Purdue University, where he served as professor and head of the School of Electrical Engineering, and as professor emeritus after retiring in 1986. Professor Hayt’s professional society memberships included Eta Kappa Nu, Tau Beta Pi, Sigma Xi, Sigma Delta Chi, Fellow of IEEE, ASEE, and NAEB. While at Purdue, he received numerous teaching awards, including the uni- versity’s Best Teacher Award. He is also listed in Purdue’s Book of Great Teachers, a permanent wall display in the Purdue Memorial Union, dedicated on April 23, 1999. The book bears the names of the inaugural group of 225 faculty members, past and present, who have devoted their lives to excellence in teaching and scholarship. They were chosen by their students and their peers as Purdue’s finest educators.

A native of Los Angeles, California, John A. Buck received his M.S. and Ph.D. degrees in Electrical Engineering from the University of California at Berkeley in 1977 and 1982, and his B.S. in Engineering from UCLA in 1975. In 1982, he joined the faculty of the School of Electrical and Computer Engineering at Georgia Tech, where he has remained for the past 28 years. His research areas and publications have centered within the fields of ultrafast switching, nonlinear optics, and optical fiber communications. He is the author of the graduate text Fundamentals of Optical Fibers (Wiley Interscience), which is now in its second edition. Awards include three institute teaching awards and the IEEE Third Millenium Medal. When not glued to his computer or confined to the lab, Dr. Buck enjoys music, hiking, and photography.

C O N T E N T S

6.5 Using Field Sketches to Estimate

Contents ix

Appendix A

Vector Analysis 553

A.1 General Curvilinear Coordinates 553

A.2 Divergence, Gradient, and Curl in General Curvilinear Coordinates 554

A.3 Vector Identities 556

Appendix B

Units 557

Appendix C

Material Constants 562

Appendix D

The Uniqueness Theorem 565

Appendix E

Origins of the Complex

Permittivity 567

Appendix F

Answers to Odd-Numbered

Problems 574

Index 580

P R E F A C E

It has been 52 years since the first edition of this book was published, then under the sole authorship of William H. Hayt, Jr. As I was five years old at that time, this would have meant little to me. But everything changed 15 years later when I used the second edition in a basic electromagnetics course as a college junior. I remember my sense of foreboding at the start of the course, being aware of friends’ horror stories. On first opening the book, however, I was pleasantly surprised by the friendly writing style and by the measured approach to the subject, which — at least for me — made it a very readable book, out of which I was able to learn with little help from my professor. I referred to it often while in graduate school, taught from the fourth and fifth editions as a faculty member, and then became coauthor for the sixth and seventh editions on the retirement (and subsequent untimely death) of Bill Hayt. The memories of my time as a beginner are clear, and I have tried to maintain the accessible style that I found so welcome then. Over the 50-year span, the subject matter has not changed, but emphases have. In the universities, the trend continues toward reducing electrical engineering core course allocations to electromagnetics. I have made efforts to streamline the presentation in this new edition to enable the student to get to Maxwell’s equations sooner, and I have added more advanced material. Many of the earlier chapters are now slightly shorter than their counterparts in the seventh edition. This has been done by economizing on the wording, shortening many sections, or by removing some entirely. In some cases, deleted topics have been converted to stand-alone articles and moved to the website, from which they can be downloaded. Major changes include the following: (1) The material on dielectrics, formerly in Chapter 6, has been moved to the end of Chapter 5. (2) The chapter on Poisson’s and Laplace’s equations has been eliminated, retaining only the one-dimensional treatment, which has been moved to the end of Chapter 6. The two-dimensional Laplace equation discussion and that of numerical methods have been moved to the website for the book. (3) The treatment on rectangular waveguides (Chapter 13) has been expanded, presenting the methodology of two-dimensional boundary value problems in that context. (4) The coverage of radiation and antennas has been greatly expanded and now forms the entire Chapter 14. Some 130 new problems have been added throughout. For some of these, I chose particularly good “classic” problems from the earliest editions. I have also adopted a new system in which the approximate level of difficulty is indicated beside each problem on a three-level scale. The lowest level is considered a fairly straightforward problem, requiring little work assuming the material is understood; a level 2 problem is conceptually more difficult, and/or may require more work to solve; a level 3 prob- lem is considered either difficult conceptually, or may require extra effort (including possibly the help of a computer) to solve.

x

xii Preface

Answers to the drill problems are given below each problem. Answers to odd- numbered end-of-chapter problems are found in Appendix F. A solutions manual and a set of PowerPoint slides, containing pertinent figures and equations, are avail- able to instructors. These, along with all other material mentioned previously, can be accessed on the website:

www.mhhe.com/haytbuck

I would like to acknowledge the valuable input of several people who helped to make this a better edition. Special thanks go to Glenn S. Smith (Georgia Tech), who reviewed the antennas chapter and provided many valuable comments and sug- gestions. Detailed suggestions and errata were provided by Clive Woods (Louisiana State University), Natalya Nikolova, and Don Davis (Georgia Tech). Accuracy checks on the new problems were carried out by Todd Kaiser (Montana State University) and Steve Weis (Texas Christian University). Other reviewers provided detailed com- ments and suggestions at the start of the project; many of the suggestions affected the outcome. They include:

Sheel Aditya – Nanyang Technological University, Singapore Yaqub M. Amani – SUNY Maritime College Rusnani Ariffin – Universiti Teknologi MARA Ezekiel Bahar – University of Nebraska Lincoln Stephen Blank – New York Institute of Technology Thierry Blu – The Chinese University of Hong Kong Jeff Chamberlain – Illinois College Yinchao Chen – University of South Carolina Vladimir Chigrinov – Hong Kong University of Science and Technology Robert Coleman – University of North Carolina Charlotte Wilbur N. Dale Ibrahim Elshafiey – King Saud University Wayne Grassel – Point Park University Essam E. Hassan – King Fahd University of Petroleum and Minerals David R. Jackson – University of Houston Karim Y. Kabalan – American University of Beirut Shahwan Victor Khoury, Professor Emeritus – Notre Dame University, Louaize-Zouk Mosbeh, Lebanon Choon S. Lee – Southern Methodist University Mojdeh J. Mardani – University of North Dakota Mohamed Mostafa Morsy – Southern Illinois University Carbondale Sima Noghanian – University of North Dakota W.D. Rawle – Calvin College G¨on¨ul Sayan – Middle East Technical University Fred H. Terry – Professor Emeritus, Christian Brothers University Denise Thorsen – University of Alaska Fairbanks Chi-Ling Wang – Feng-Chia University

Preface xiii

I also acknowledge the feedback and many comments from students, too numerous to name, including several who have contacted me from afar. I continue to be open and grateful for this feedback and can be reached at john.buck@ece.gatech.edu. Many suggestions were made that I considered constructive and actionable. I regret that not all could be incorporated because of time restrictions. Creating this book was a team effort, involving several outstanding people at McGraw-Hill. These include my publisher, Raghu Srinivasan, and sponsoring editor, Peter Massar, whose vision and encouragement were invaluable, Robin Reed, who deftly coordinated the production phase with excellent ideas and enthusiasm, and Darlene Schueller, who was my guide and conscience from the beginning, providing valuable insights, and jarring me into action when necessary. Typesetting was supervised by Vipra Fauzdar at Glyph International, who employed the best copy editor I ever had, Laura Bowman. Diana Fouts (Georgia Tech) applied her vast artistic skill to designing the cover, as she has done for the previous two editions. Finally, I am, as usual in these projects, grateful to a patient and supportive family, and particularly to my daughter, Amanda, who assisted in preparing the manuscript.

John A. Buck Marietta, Georgia December, 2010

On the cover: Radiated intensity patterns for a dipole antenna, showing the cases for which the wavelength is equal to the overall antenna length (red), two-thirds the antenna length (green), and one-half the antenna length (blue).

C H A P T E R 1

Vector Analysis

V

ector analysis is a mathematical subject that is better taught by mathematicians than by engineers. Most junior and senior engineering students have not had the time (or the inclination) to take a course in vector analysis, although it is likely that vector concepts and operations were introduced in the calculus sequence. These are covered in this chapter, and the time devoted to them now should depend on past exposure. The viewpoint here is that of the engineer or physicist and not that of the mathe- matician. Proofs are indicated rather than rigorously expounded, and physical inter- pretation is stressed. It is easier for engineers to take a more rigorous course in the mathematics department after they have been presented with a few physical pictures and applications. Vector analysis is a mathematical shorthand. It has some new symbols and some new rules, and it demands concentration and practice. The drill problems, first found at the end of Section 1.4, should be considered part of the text and should all be worked. They should not prove to be difficult if the material in the accompanying section of the text has been thoroughly understood. It takes a little longer to “read” the chapter this way, but the investment in time will produce a surprising interest. ■

1.1 SCALARS AND VECTORS The term scalar refers to a quantity whose value may be represented by a single (positive or negative) real number. The x , y , and z we use in basic algebra are scalars, and the quantities they represent are scalars. If we speak of a body falling a distance L in a time t , or the temperature T at any point in a bowl of soup whose coordinates are x , y , and z , then L , t , T , x , y , and z are all scalars. Other scalar quantities are mass, density, pressure (but not force), volume, volume resistivity, and voltage. A vector quantity has both a magnitude^1 and a direction in space. We are con- cerned with two- and three-dimensional spaces only, but vectors may be defined in

(^1) We adopt the convention that magnitude infers absolute value; the magnitude of any quantity is, therefore, always positive.

1

2 E N G I N E E R I N G E L E C T R O M A G N E T I C S

n -dimensional space in more advanced applications. Force, velocity, acceleration, and a straight line from the positive to the negative terminal of a storage battery are examples of vectors. Each quantity is characterized by both a magnitude and a direction. Our work will mainly concern scalar and vector field. A field (scalar or vector) may be defined mathematically as some function that connects an arbitrary origin to a general point in space. We usually associate some physical effect with a field, such as the force on a compass needle in the earth’s magnetic field, or the movement of smoke particles in the field defined by the vector velocity of air in some region of space. Note that the field concept invariably is related to a region. Some quantity is defined at every point in a region. Both scalar field and vector field exist. The temperature throughout the bowl of soup and the density at any point in the earth are examples of scalar fields. The gravitational and magnetic fields of the earth, the voltage gradient in a cable, and the temperature gradient in a soldering-iron tip are examples of vector fields. The value of a field varies in general with both position and time. In this book, as in most others using vector notation, vectors will be indicated by boldface type, for example, A. Scalars are printed in italic type, for example, A. When writing longhand, it is customary to draw a line or an arrow over a vector quantity to show its vector character. (C AUTION: This is the first pitfall. Sloppy notation, such as the omission of the line or arrow symbol for a vector, is the major cause of errors in vector analysis.)

1.2 VECTOR ALGEBRA With the definition of vectors and vector fields now established, we may proceed to define the rules of vector arithmetic, vector algebra, and (later) vector calculus. Some of the rules will be similar to those of scalar algebra, some will differ slightly, and some will be entirely new. To begin, the addition of vectors follows the parallelogram law. Figure 1.1 shows the sum of two vectors, A and B. It is easily seen that A + B = B + A , or that vector addition obeys the commutative law. Vector addition also obeys the associative law,

A + ( B + C ) = ( A + B ) + C

Note that when a vector is drawn as an arrow of finite length, its location is defined to be at the tail end of the arrow. Coplanar vectors are vectors lying in a common plane, such as those shown in Figure 1.1. Both lie in the plane of the paper and may be added by expressing each vector in terms of “horizontal” and “vertical” components and then adding the corresponding components. Vectors in three dimensions may likewise be added by expressing the vectors in terms of three components and adding the corresponding components. Examples of this process of addition will be given after vector components are discussed in Section 1.4.