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November 22, 2010 20:32 Hayt/Buck Page 2 hay80660 frontendsheet 2and3.pdf

Physical Constants

**Quantity Value
**

Electron charge *e *= (1*.*602 177 33 ± 0*.*000 000 46) × 10−19 C
Electron mass *m *= (9*.*109 389 7 ± 0*.*000 005 4) × 10−31 kg
Permittivity of free space *�*0 = 8*.*854 187 817 × 10−12 F/m
Permeability of free space *µ*0 = 4*π*10−7 H/m
Velocity of light *c *= 2*.*997 924 58 × 108 m/s

Dielectric Constant (*�*�*r *) and Loss Tangent (*�
*��*/�*�)

**Material **�** r **��

*/*�

Air 1.0005 Alcohol, ethyl 25 0.1 Aluminum oxide 8.8 0.000 6 Amber 2.7 0.002 Bakelite 4.74 0.022 Barium titanate 1200 0.013 Carbon dioxide 1.001 Ferrite (NiZn) 12.4 0.000 25 Germanium 16 Glass 4–7 0.002 Ice 4.2 0.05 Mica 5.4 0.000 6 Neoprene 6.6 0.011 Nylon 3.5 0.02 Paper 3 0.008 Plexiglas 3.45 0.03 Polyethylene 2.26 0.000 2 Polypropylene 2.25 0.000 3 Polystyrene 2.56 0.000 05 Porcelain (dry process) 6 0.014 Pyranol 4.4 0.000 5 Pyrex glass 4 0.000 6 Quartz (fused) 3.8 0.000 75 Rubber 2.5–3 0.002 Silica or SiO2 (fused) 3.8 0.000 75 Silicon 11.8 Snow 3.3 0.5 Sodium chloride 5.9 0.000 1 Soil (dry) 2.8 0.05 Steatite 5.8 0.003 Styrofoam 1.03 0.000 1 Teflon 2.1 0.000 3 Titanium dioxide 100 0.001 5 Water (distilled) 80 0.04 Water (sea) 4 Water (dehydrated) 1 0 Wood (dry) 1.5–4 0.01

November 22, 2010 20:32 Hayt/Buck Page 3 hay80660 frontendsheet 2and3.pdf

Conductivity (*σ *)

**Material ***σ*, **S/m Material ***σ*, **S/m
**

Silver 6*.*17 × 107 Nichrome 0*.*1 × 107
Copper 5*.*80 × 107 Graphite 7 × 104
Gold 4*.*10 × 107 Silicon 2300
Aluminum 3*.*82 × 107 Ferrite (typical) 100
Tungsten 1*.*82 × 107 Water (sea) 5
Zinc 1*.*67 × 107 Limestone 10−2
Brass 1*.*5 × 107 Clay 5 × 10−3
Nickel 1*.*45 × 107 Water (fresh) 10−3
Iron 1*.*03 × 107 Water (distilled) 10−4
Phosphor bronze 1 × 107 Soil (sandy) 10−5
Solder 0*.*7 × 107 Granite 10−6
Carbon steel 0*.*6 × 107 Marble 10−8
German silver 0*.*3 × 107 Bakelite 10−9
Manganin 0*.*227 × 107 Porcelain (dry process) 10−10
Constantan 0*.*226 × 107 Diamond 2 × 10−13
Germanium 0*.*22 × 107 Polystyrene 10−16
Stainless steel 0*.*11 × 107 Quartz 10−17

Relative Permeability (*µr *)

**Material ***µ r *

**Material**

*µ*Bismuth 0.999 998 6 Powdered iron 100 Paraffin 0.999 999 42 Machine steel 300 Wood 0.999 999 5 Ferrite (typical) 1000 Silver 0.999 999 81 Permalloy 45 2500 Aluminum 1.000 000 65 Transformer iron 3000 Beryllium 1.000 000 79 Silicon iron 3500 Nickel chloride 1.000 04 Iron (pure) 4000 Manganese sulfate 1.000 1 Mumetal 20 000 Nickel 50 Sendust 30 000 Cast iron 60 Supermalloy 100 000 Cobalt 60

**r**hay80660_frontendsheet_2and3_HR.indd 1 12/24/10 3:09 PM

November 22, 2010 20:32 Hayt/Buck Page 2 hay80660 frontendsheet 2and3.pdf

Physical Constants

**Quantity Value
**

Electron charge *e *= (1*.*602 177 33 ± 0*.*000 000 46) × 10−19 C
Electron mass *m *= (9*.*109 389 7 ± 0*.*000 005 4) × 10−31 kg
Permittivity of free space *�*0 = 8*.*854 187 817 × 10−12 F/m
Permeability of free space *µ*0 = 4*π*10−7 H/m
Velocity of light *c *= 2*.*997 924 58 × 108 m/s

Dielectric Constant (*�*�*r *) and Loss Tangent (*�
*��*/�*�)

**Material **�** r **��

*/*�

Air 1.0005 Alcohol, ethyl 25 0.1 Aluminum oxide 8.8 0.000 6 Amber 2.7 0.002 Bakelite 4.74 0.022 Barium titanate 1200 0.013 Carbon dioxide 1.001 Ferrite (NiZn) 12.4 0.000 25 Germanium 16 Glass 4–7 0.002 Ice 4.2 0.05 Mica 5.4 0.000 6 Neoprene 6.6 0.011 Nylon 3.5 0.02 Paper 3 0.008 Plexiglas 3.45 0.03 Polyethylene 2.26 0.000 2 Polypropylene 2.25 0.000 3 Polystyrene 2.56 0.000 05 Porcelain (dry process) 6 0.014 Pyranol 4.4 0.000 5 Pyrex glass 4 0.000 6 Quartz (fused) 3.8 0.000 75 Rubber 2.5–3 0.002 Silica or SiO2 (fused) 3.8 0.000 75 Silicon 11.8 Snow 3.3 0.5 Sodium chloride 5.9 0.000 1 Soil (dry) 2.8 0.05 Steatite 5.8 0.003 Styrofoam 1.03 0.000 1 Tefion 2.1 0.000 3 Titanium dioxide 100 0.001 5 Water (distilled) 80 0.04 Water (sea) 4 Water (dehydrated) 1 0 Wood (dry) 1.5–4 0.01

November 22, 2010 20:32 Hayt/Buck Page 3 hay80660 frontendsheet 2and3.pdf

Conductivity (*� *)

**Material *** *, **S/m Material *** *, **S/m
**

Silver 6*.*17 × 107 Nichrome 0*.*1 × 107
Copper 5*.*80 × 107 Graphite 7 × 104
Gold 4*.*10 × 107 Silicon 2300
Aluminum 3*.*82 × 107 Ferrite (typical) 100
Tungsten 1*.*82 × 107 Water (sea) 5
Zinc 1*.*67 × 107 Limestone 10−2
Brass 1*.*5 × 107 Clay 5 × 10−3
Nickel 1*.*45 × 107 Water (fresh) 10−3
Iron 1*.*03 × 107 Water (distilled) 10−4
Phosphor bronze 1 × 107 Soil (sandy) 10−5
Solder 0*.*7 × 107 Granite 10−6
Carbon steel 0*.*6 × 107 Marble 10−8
German silver 0*.*3 × 107 Bakelite 10−9
Manganin 0*.*227 × 107 Porcelain (dry process) 10−10
Constantan 0*.*226 × 107 Diamond 2 × 10−13
Germanium 0*.*22 × 107 Polystyrene 10−16
Stainless steel 0*.*11 × 107 Quartz 10−17

Relative Permeability (*µr *)

**Material ***µ r *

**Material**

*µ*Bismuth 0.999 998 6 Powdered iron 100 Paraffin 0.999 999 42 Machine steel 300 Wood 0.999 999 5 Ferrite (typical) 1000 Silver 0.999 999 81 Permalloy 45 2500 Aluminum 1.000 000 65 Transformer iron 3000 Beryllium 1.000 000 79 Silicon iron 3500 Nickel chloride 1.000 04 Iron (pure) 4000 Manganese sulfate 1.000 1 Mumetal 20 000 Nickel 50 Sendust 30 000 Cast iron 60 Supermalloy 100 000 Cobalt 60

**r**hay80660_frontendsheet_2and3_HR.indd 1 12/24/10 3:09 PM

**Engineering Electromagnetics
EIGHTH EDITION
**

**William H. Hayt, Jr.
***Late Emeritus Professor
Purdue University
*

**John A. Buck
***Georgia Institute of Technology*

ENGINEERING ELECTROMAGNETICS, EIGHTH EDITION

Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright C© 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions C© 2006, 2001, and 1989. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

Some ancillaries, including electronic and print components, may not be available to customers outside the United States.

This book is printed on acid-free paper.

1 2 3 4 5 6 7 8 9 0 DOC/DOC 1 0 9 8 7 6 5 4 3 2 1

ISBN 978-0-07-338066-7 MHID 0-07-338066-0

Vice President & Editor-in-Chief: *Marty Lange
*Vice President EDP/Central Publishing Services: *Kimberly Meriwether David
*Publisher: *Raghothaman Srinivasan
*Senior Sponsoring Editor: *Peter E. Massar
*Senior Marketing Manager: *Curt Reynolds
*Developmental Editor: *Darlene M. Schueller
*Project Manager: *Robin A. Reed
*Design Coordinator: *Brenda A. Rolwes
*Cover Design and Image: *Diana Fouts
*Buyer: *Kara Kudronowicz
*Media Project Manager: *Balaji Sundararaman
*Compositor: *Glyph International
*Typeface: 10*.*5*/*12 *Times Roman
*Printer: *R.R. Donnelley
*

All credits appearing on page or at the end of the book are considered to be an extension of the copyright page.

**Library of Congress Cataloging-in-Publication Data
**

Hayt, William Hart, 1920– Engineering electromagnetics / William H. Hayt, Jr., John A. Buck. — 8th ed.

p. cm. Includes bibliographical references and index. ISBN 978–0–07–338066–7 (alk. paper) 1. Electromagnetic theory. I. Buck, John A. II. Title. QC670.H39 2010 530.14′1—dc22 2010048332

www.mhhe.com

**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.

**B R I E F C O N T E N T S
**

Preface xii

**1 **Vector Analysis 1

**2 **Coulomb’s Law and Electric Field Intensity 26

**3 **Electric Flux Density, Gauss’s Law, and Divergence 48

**4 **Energy and Potential 75

**5 **Conductors and Dielectrics 109

**6 **Capacitance 143

**7 **The Steady Magnetic Field 180

**8 **Magnetic Forces, Materials, and Inductance 230

**9 **Time-Varying Fields and Maxwell’s Equations 277

**10 **Transmission Lines 301

**11 **The Uniform Plane Wave 367

**12 **Plane Wave Reflection and Dispersion 406

**13 **Guided Waves 453

**14 **Electromagnetic Radiation and Antennas 511

Appendix A Vector Analysis 553

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

**v**

**C O N T E N T S
**

Preface xii

Chapter **1
Vector Analysis **1

**1.1 **Scalars and Vectors 1
**1.2 **Vector Algebra 2
**1.3 **The Rectangular Coordinate System 3
**1.4 **Vector Components and Unit Vectors 5
**1.5 **The Vector Field 8
**1.6 **The Dot Product 9
**1.7 **The Cross Product 11
**1.8 **Other Coordinate Systems: Circular

Cylindrical Coordinates 13

**1.9 **The Spherical Coordinate System 18
References 22

Chapter 1 Problems 22

Chapter **2
Coulomb’s Law and Electric
Field Intensity **26

**2.1 **The Experimental Law of Coulomb 26
**2.2 **Electric Field Intensity 29
**2.3 **Field Arising from a Continuous Volume

Charge Distribution 33

**2.4 **Field of a Line Charge 35
**2.5 **Field of a Sheet of Charge 39
**2.6 **Streamlines and Sketches of Fields 41

References 44

Chapter 2 Problems 44

Chapter **3
Electric Flux Density, Gauss’s Law,
and Divergence **48

**3.1 **Electric Flux Density 48
**3.2 **Gauss’s Law 52
**3.3 **Application of Gauss’s Law: Some

Symmetrical Charge Distributions 56

**3.4 **Application of Gauss’s Law: Differential
Volume Element 61

**3.5 **Divergence and Maxwell’s First Equation 64
**3.6 **The Vector Operator ∇ and the Divergence

Theorem 67

References 70

Chapter 3 Problems 71

Chapter **4
Energy and Potential **75

**4.1 **Energy Expended in Moving a Point Charge in
an Electric Field 76

**4.2 **The Line Integral 77
**4.3 **Definition of Potential Difference

and Potential 82

**4.4 **The Potential Field of a Point Charge 84
**4.5 **The Potential Field of a System of Charges:

Conservative Property 86

**4.6 **Potential Gradient 90
**4.7 **The Electric Dipole 95
**4.8 **Energy Density in the Electrostatic

Field 100

References 104

Chapter 4 Problems 105

**vi**

Contents **vii
**

Chapter **5
Conductors and Dielectrics **109

**5.1 **Current and Current Density 110
**5.2 **Continuity of Current 111
**5.3 **Metallic Conductors 114
**5.4 **Conductor Properties and Boundary

Conditions 119

**5.5 **The Method of Images 124
**5.6 **Semiconductors 126
**5.7 **The Nature of Dielectric Materials 127
**5.8 **Boundary Conditions for Perfect

Dielectric Materials 133

References 137

Chapter 5 Problems 138

Chapter **6
Capacitance **143

**6.1 **Capacitance Defined 143
**6.2 **Parallel-Plate Capacitor 145
**6.3 **Several Capacitance Examples 147
**6.4 **Capacitance of a Two-Wire Line 150
**6.5 **Using Field Sketches to Estimate

Capacitance in Two-Dimensional Problems 154

**6.6 **Poisson’s and Laplace’s Equations 160
**6.7 **Examples of the Solution of Laplace’s

Equation 162

**6.8 **Example of the Solution of Poisson’s
Equation: the *p*-*n *Junction Capacitance 169

References 172

Chapter 6 Problems 173

Chapter **7
The Steady Magnetic Field **180

**7.1 **Biot-Savart Law 180
**7.2 **Ampère’s Circuital Law 188
**7.3 **Curl 195

**7.4 **Stokes’ Theorem 202
**7.5 **Magnetic Flux and Magnetic Flux

Density 207

**7.6 **The Scalar and Vector Magnetic
Potentials 210

**7.7 **Derivation of the Steady-Magnetic-Field
Laws 217

References 223

Chapter 7 Problems 223

Chapter **8
Magnetic Forces, Materials,
and Inductance **230

**8.1 **Force on a Moving Charge 230
**8.2 **Force on a Differential Current Element 232
**8.3 **Force between Differential Current

Elements 236

**8.4 **Force and Torque on a Closed Circuit 238
**8.5 **The Nature of Magnetic Materials 244
**8.6 **Magnetization and Permeability 247
**8.7 **Magnetic Boundary Conditions 252
**8.8 **The Magnetic Circuit 255
**8.9 **Potential Energy and Forces on Magnetic

Materials 261

**8.10 **Inductance and Mutual Inductance 263
References 270

Chapter 8 Problems 270

Chapter **9
Time-Varying Fields and Maxwell’s
Equations **277

**9.1 **Faraday’s Law 277
**9.2 **Displacement Current 284
**9.3 **Maxwell’s Equations in Point Form 288
**9.4 **Maxwell’s Equations in Integral Form 290
**9.5 **The Retarded Potentials 292

References 296

Chapter 9 Problems 296

**viii **Contents

Chapter **10
Transmission Lines **301

**10.1 **Physical Description of Transmission Line
Propagation 302

**10.2 **The Transmission Line Equations 304
**10.3 **Lossless Propagation 306
**10.4 **Lossless Propagation of Sinusoidal

Voltages 309

**10.5 **Complex Analysis of Sinusoidal Waves 311
**10.6 **Transmission Line Equations and Their

Solutions in Phasor Form 313

**10.7 **Low-Loss Propagation 315
**10.8 **Power Transmission and The Use of Decibels

in Loss Characterization 317

**10.9 **Wave Reflection at Discontinuities 320
**10.10 **Voltage Standing Wave Ratio 323
**10.11 **Transmission Lines of Finite Length 327
**10.12 **Some Transmission Line Examples 330
**10.13 **Graphical Methods: The Smith Chart 334
**10.14 **Transient Analysis 345

References 358

Chapter 10 Problems 358

Chapter **11
The Uniform Plane Wave **367

**11.1 **Wave Propagation in Free Space 367
**11.2 **Wave Propagation in Dielectrics 375
**11.3 **Poynting’s Theorem and Wave Power 384
**11.4 **Propagation in Good Conductors:

Skin Effect 387

**11.5 **Wave Polarization 394
References 401

Chapter 11 Problems 401

Chapter **12
Plane Wave Reflection and
Dispersion **406

**12.1 **Reflection of Uniform Plane Waves
at Normal Incidence 406

**12.2 **Standing Wave Ratio 413

**12.3 **Wave Reflection from Multiple
Interfaces 417

**12.4 **Plane Wave Propagation in General
Directions 425

**12.5 **Plane Wave Reflection at Oblique Incidence
Angles 428

**12.6 **Total Reflection and Total Transmission
of Obliquely Incident Waves 434

**12.7 **Wave Propagation in Dispersive Media 437
**12.8 **Pulse Broadening in Dispersive Media 443

References 447

Chapter 12 Problems 448

Chapter **13
Guided Waves **453

**13.1 **Transmission Line Fields and Primary
Constants 453

**13.2 **Basic Waveguide Operation 463
**13.3 **Plane Wave Analysis of the Parallel-Plate

Waveguide 467

**13.4 **Parallel-Plate Guide Analysis Using the Wave
Equation 476

**13.5 **Rectangular Waveguides 479
**13.6 **Planar Dielectric Waveguides 490
**13.7 **Optical Fiber 496

References 506

Chapter 13 Problems 506

Chapter **14
Electromagnetic Radiation
and Antennas **511

**14.1 **Basic Radiation Principles: The Hertzian
Dipole 511

**14.2 **Antenna Specifications 518
**14.3 **Magnetic Dipole 523
**14.4 **Thin Wire Antennas 525
**14.5 **Arrays of Two Elements 533
**14.6 **Uniform Linear Arrays 537
**14.7 **Antennas as Receivers 541

References 548

Chapter 14 Problems 548

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**

Preface **xi
**

As in the previous edition, the transmission lines chapter (10) is stand-alone, and can be read or covered in any part of a course, including the beginning. In it, transmission lines are treated entirely within the context of circuit theory; wave phenomena are introduced and used exclusively in the form of voltages and cur- rents. Inductance and capacitance concepts are treated as known parameters, and so there is no reliance on any other chapter. Field concepts and parameter com- putation in transmission lines appear in the early part of the waveguides chapter (13), where they play additional roles of helping to introduce waveguiding con- cepts. The chapters on electromagnetic waves, 11 and 12, retain their independence of transmission line theory in that one can progress from Chapter 9 directly to Chapter 11. By doing this, wave phenomena are introduced from first principles but within the context of the uniform plane wave. Chapter 11 refers to Chapter 10 in places where the latter may give additional perspective, along with a little more detail. Nevertheless, all necessary material to learn plane waves without previously studying transmission line waves is found in Chapter 11, should the student or instructor wish to proceed in that order.

The new chapter on antennas covers radiation concepts, building on the retarded potential discussion in Chapter 9. The discussion focuses on the dipole antenna, individually and in simple arrays. The last section covers elementary transmit-receive systems, again using the dipole as a vehicle.

The book is designed optimally for a two-semester course. As is evident, statics concepts are emphasized and occur first in the presentation, but again Chapter 10 (transmission lines) can be read first. In a single course that emphasizes dynamics, the transmission lines chapter can be covered initially as mentioned or at any point in the course. One way to cover the statics material more rapidly is by deemphasizing materials properties (assuming these are covered in other courses) and some of the advanced topics. This involves omitting Chapter 1 (assigned to be read as a review), and omitting Sections 2.5, 2.6, 4.7, 4.8, 5.5–5.7, 6.3, 6.4, 6.7, 7.6, 7.7, 8.5, 8.6, 8.8, 8.9, and 9.5.

A supplement to this edition is web-based material consisting of the afore- mentioned articles on special topics in addition to animated demonstrations and interactive programs developed by Natalya Nikolova of McMaster University and Vikram Jandhyala of the University of Washington. Their excellent contributions are geared to the text, and icons appear in the margins whenever an exercise that pertains to the narrative exists. In addition, quizzes are provided to aid in further study.

The theme of the text is the same as it has been since the first edition of 1958. An inductive approach is used that is consistent with the historical development. In it, the experimental laws are presented as individual concepts that are later unified in Maxwell’s equations. After the first chapter on vector analysis, additional math- ematical tools are introduced in the text on an as-needed basis. Throughout every edition, as well as this one, the primary goal has been to enable students to learn independently. Numerous examples, drill problems (usually having multiple parts), end-of-chapter problems, and material on the web site, are provided to facilitate this.

**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 Gönül 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).

**xiv **Preface

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1**C H A P T E R
Vector Analysis
**

**V **ector analysis is a mathematical subject that is better taught by mathematiciansthan by engineers. Most junior and senior engineering students have not hadthe 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 magnitude1 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 ENGINEERING ELECTROMAGNETICS
**

*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 *fields. *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 fields *and *vector fields *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. (CAUTION: 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.

**CHAPTER 1 **Vector Analysis **3
**

**Figure 1.1 **Two vectors may be added graphically either by drawing
both vectors from a common origin and completing the parallelogram or
by beginning the second vector from the head of the first and completing
the triangle; either method is easily extended to three or more vectors.

The rule for the subtraction of vectors follows easily from that for addition, for
we may always express **A**−**B **as **A**+(−**B**); the sign, or direction, of the second vector
is reversed, and this vector is then added to the first by the rule for vector addition.

Vectors may be multiplied by scalars. The magnitude of the vector changes, but its direction does not when the scalar is positive, although it reverses direction when multiplied by a negative scalar. Multiplication of a vector by a scalar also obeys the associative and distributive laws of algebra, leading to

(*r *+ *s*)(**A **+ **B**) = *r *(**A **+ **B**) + *s*(**A **+ **B**) = *r***A **+ *r***B **+ *s***A **+ *s***B
**Division of a vector by a scalar is merely multiplication by the reciprocal of that
scalar. The multiplication of a vector by a vector is discussed in Sections 1.6 and 1.7.
Two vectors are said to be equal if their difference is zero, or **A **= **B **if **A **− **B **= 0.

In our use of vector fields we shall always add and subtract vectors that are defined
at the same point. For example, the *total *magnetic field about a small horseshoe mag-
net will be shown to be the sum of the fields produced by the earth and the permanent
magnet; the total field at any point is the sum of the individual fields at that point.

If we are not considering a vector *field, *we may add or subtract vectors that are
not defined at the same point. For example, the sum of the gravitational force acting
on a 150 lb *f *(pound-force) man at the North Pole and that acting on a 175 lb *f *person
at the South Pole may be obtained by shifting each force vector to the South Pole
before addition. The result is a force of 25 lb *f *directed toward the center of the earth
at the South Pole; if we wanted to be difficult, we could just as well describe the force
as 25 lb *f *directed *away *from the center of the earth (or “upward”) at the North Pole.2

**1.3 THE RECTANGULAR
COORDINATE SYSTEM
**

To describe a vector accurately, some specific lengths, directions, angles, projections, or components must be given. There are three simple methods of doing this, and about eight or ten other methods that are useful in very special cases. We are going

2 Students have argued that the force might be described at the equator as being in a “northerly” direction. They are right, but enough is enough.

**4 ENGINEERING ELECTROMAGNETICS
**

to use only the three simple methods, and the simplest of these is the *rectangular*, or
*rectangular cartesian*, coordinate system.

In the rectangular coordinate system we set up three coordinate axes mutually
at right angles to each other and call them the *x, y*, and *z *axes. It is customary to
choose a *right-handed *coordinate system, in which a rotation (through the smaller
angle) of the *x *axis into the *y *axis would cause a right-handed screw to progress in
the direction of the *z *axis. If the right hand is used, then the thumb, forefinger, and
middle finger may be identified, respectively, as the *x, y*, and *z *axes. Figure 1.2*a
*shows a right-handed rectangular coordinate system.

A point is located by giving its *x, y*, and *z *coordinates. These are, respectively,
the distances from the origin to the intersection of perpendicular lines dropped from
the point to the *x, y*, and *z *axes. An alternative method of interpreting coordinate

**Figure 1.2 **(*a*) A right-handed rectangular coordinate system. If the curved fingers of the
right hand indicate the direction through which the *x *axis is turned into coincidence with the
*y *axis, the thumb shows the direction of the *z *axis. (*b*) The location of points *P*(1*, *2*, *3) and
*Q*(2*,*−2*, *1). (*c*) The differential volume element in rectangular coordinates; *dx*, *dy*, and *dz
*are, in general, independent differentials.

**CHAPTER 1 **Vector Analysis **5
**

values, which *must *be used in all other coordinate systems, is to consider the point as
being at the common intersection of three surfaces. These are the planes *x *= constant,
*y *= constant, and *z *= constant, where the constants are the coordinate values of the
point.

Figure 1.2*b *shows points *P *and *Q *whose coordinates are (1*, *2*, *3) and (2*, *−2*, *1),
respectively. Point *P *is therefore located at the common point of intersection of the
planes *x *= 1, *y *= 2, and *z *= 3, whereas point *Q *is located at the intersection of the
planes *x *= 2, *y *= −2, and *z *= 1.

As we encounter other coordinate systems in Sections 1.8 and 1.9, we expect points to be located at the common intersection of three surfaces, not necessarily planes, but still mutually perpendicular at the point of intersection.

If we visualize three planes intersecting at the general point *P *, whose coordinates
are *x, y*, and *z*, we may increase each coordinate value by a differential amount and
obtain three slightly displaced planes intersecting at point *P *′, whose coordinates are
*x *+ *dx *, *y *+ *dy*, and *z *+ *dz*. The six planes define a rectangular parallelepiped whose
volume is *dv *= *dxdydz*; the surfaces have differential areas *d S *of *dxdy*, *dydz*, and
*dzdx *. Finally, the distance *d L *from *P *to *P *′ is the diagonal of the parallelepiped and
has a length of

√
(*dx*)2 + (*dy*)2 + (*dz*)2. The volume element is shown in Figure 1.2*c*;

point *P *′ is indicated, but point *P *is located at the only invisible corner.
All this is familiar from trigonometry or solid geometry and as yet involves only

scalar quantities. We will describe vectors in terms of a coordinate system in the next section.

**1.4 VECTOR COMPONENTS
AND UNIT VECTORS
**

To describe a vector in the rectangular coordinate system, let us first consider a vector **r
**extending outward from the origin. A logical way to identify this vector is by giving
the three *component vectors*, lying along the three coordinate axes, whose vector sum
must be the given vector. If the component vectors of the vector **r **are **x**, **y**, and **z**,
then **r **= **x **+ **y **+ **z**. The component vectors are shown in Figure 1.3*a*. Instead of one
vector, we now have three, but this is a step forward because the three vectors are of
a very simple nature; each is always directed along one of the coordinate axes.

The component vectors have magnitudes that depend on the given vector (such
as **r**), but they each have a known and constant direction. This suggests the use of *unit
vectors *having unit magnitude by definition; these are parallel to the coordinate axes
and they point in the direction of increasing coordinate values. We reserve the symbol
**a **for a unit vector and identify its direction by an appropriate subscript. Thus **a***x , ***a***y *,
and **a***z *are the unit vectors in the rectangular coordinate system.3 They are directed
along the *x, y*, and *z *axes, respectively, as shown in Figure 1.3*b*.

If the component vector **y **happens to be two units in magnitude and directed
toward increasing values of *y*, we should then write **y **= 2**a***y *. A vector **r***P *pointing

3 The symbols **i***, ***j**, and **k **are also commonly used for the unit vectors in rectangular coordinates.

**6 ENGINEERING ELECTROMAGNETICS
**

**Figure 1.3 **(*a*) The component vectors x, y, and z of vector r. (*b*) The unit
vectors of the rectangular coordinate system have unit magnitude and are
directed toward increasing values of their respective variables. (*c*) The vector R*PQ
*is equal to the vector difference r*Q *− r*P .
*

from the origin to point *P*(1*, *2*, *3) is written **r***P *= **a***x *+ 2**a***y *+ 3**a***z *. The vector from
*P *to *Q *may be obtained by applying the rule of vector addition. This rule shows
that the vector from the origin to *P *plus the vector from *P *to *Q *is equal to the
vector from the origin to *Q*. The desired vector from *P*(1*, *2*, *3) to *Q*(2*, *−2*, *1) is
therefore

**R***P Q *= **r***Q *− **r***P *= (2 − 1)**a***x *+ (−2 − 2)**a***y *+ (1 − 3)**a***z
*= **a***x *− 4**a***y *− 2**a***z
*

The vectors **r***P , ***r***Q *, and **R***P Q *are shown in Figure 1.3*c*.
The last vector does not extend outward from the origin, as did the vector **r **we

initially considered. However, we have already learned that vectors having the same magnitude and pointing in the same direction are equal, so we see that to help our visualization processes we are at liberty to slide any vector over to the origin before

**CHAPTER 1 **Vector Analysis **7
**

determining its component vectors. Parallelism must, of course, be maintained during the sliding process.

If we are discussing a force vector **F**, or indeed any vector other than a
displacement-type vector such as **r**, the problem arises of providing suitable letters
for the three component vectors. It would not do to call them **x**, **y**, and **z**, for these
are displacements, or directed distances, and are measured in meters (abbreviated m)
or some other unit of length. The problem is most often avoided by using *component
scalars*, simply called *components*, *Fx , Fy *, and *Fz *. The components are the signed
magnitudes of the component vectors. We may then write **F **= *Fx ***a***x *+ *Fy***a***y *+ *Fz***a***z *.
The component vectors are *Fx ***a***x *, *Fy***a***y *, and *Fz***a***z .
*

Any vector **B **then may be described by **B **= *Bx ***a***x *+ *By***a***y *+ *Bz***a***z *. The magnitude
of **B **written |**B**| or simply *B*, is given by

|**B**| =
√

*B*2*x *+ *B*2*y *+ *B*2*z *(1)

Each of the three coordinate systems we discuss will have its three fundamental
and mutually perpendicular unit vectors that are used to resolve any vector into its
component vectors. Unit vectors are not limited to this application. It is helpful to
write a unit vector having a specified direction. This is easily done, for a unit vector
in a given direction is merely a vector in that direction divided by its magnitude. A
unit vector in the **r **direction is **r***/
*

√
*x*2 + *y*2 + *z*2, and a unit vector in the direction of

the vector **B **is

**a***B *= **B**√
*B*2*x *+ *B*2*y *+ *B*2*z
*

= **B**|**B**| (2)

**EXAMPLE 1.1
**

Specify the unit vector extending from the origin toward the point *G*(2*, *−2*, *−1).
** Solution. **We first construct the vector extending from the origin to point

*G*,

**G **= 2**a***x *− 2**a***y *− **a***z
*We continue by finding the magnitude of **G**,

|**G**| =
√

(2)2 + (−2)2 + (−1)2 = 3 and finally expressing the desired unit vector as the quotient,

**a***G *= **G**|**G**| =
2
3 **a***x *− 23 **a***y *− 13 **a***z *= 0*.*667**a***x *− 0*.*667**a***y *− 0*.*333**a***z
*

A special symbol is desirable for a unit vector so that its character is immediately
apparent. Symbols that have been used are **u***B, ***a***B, ***1B**, or even **b**. We will consistently
use the lowercase **a **with an appropriate subscript.