Engineering Electromagnetics - Physical Constants PDF

This page intentionally left blank  Physical Constants Quantity Value Conductivity (σ ) Electron charge...

This page intentionally left blank  Physical Constants Quantity Value Conductivity (σ ) Electron charge e = (1.602 177 33 ± 0.000 000 46) × 10−19 C Material σ, S/m Material σ, S/m Electron mass m = (9.109 389 7 ± 0.000 005 4) × 10−31 kg Silver 6.17 × 107 Nichrome 0.1 × 107 Permittivity of free space 0 = 8.854 187 817 × 10−12 F/m Copper 5.80 × 107 Graphite 7 × 104 Permeability of free space µ0 = 4π10−7 H/m Gold 4.10 × 107 Silicon 2300 Velocity of light c = 2.997 924 58 × 108 m/s Aluminum 3.82 × 107 Ferrite (typical) 100 Tungsten 1.82 × 107 Water (sea) 5 Dielectric Constant ( r ) and Loss Tangent ( / ) Zinc 1.67 × 107 Limestone 10−2 Material  r  / Brass 1.5 × 107 Clay 5 × 10−3 Air 1.0005 Nickel 1.45 × 107 Water (fresh) 10−3 Alcohol, ethyl 25 0.1 Iron 1.03 × 107 Water (distilled) 10−4 Aluminum oxide 8.8 0.000 6 Phosphor bronze 1 × 107 Soil (sandy) 10−5 Amber 2.7 0.002 Solder 0.7 × 107 Granite 10−6 Bakelite 4.74 0.022 Carbon steel 0.6 × 107 Marble 10−8 Barium titanate 1200 0.013 German silver 0.3 × 107 Bakelite 10−9 Carbon dioxide 1.001 Manganin 0.227 × 107 Porcelain (dry process) 10−10 Ferrite (NiZn) 12.4 0.000 25 Constantan 0.226 × 107 Diamond 2 × 10−13 Germanium 16 Germanium 0.22 × 107 Polystyrene 10−16 Glass 4–7 0.002 Ice 4.2 0.05 Stainless steel 0.11 × 107 Quartz 10−17 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 Relative Permeability (µr ) Quartz (fused) 3.8 0.000 75 Rubber 2.5–3 0.002 Material µr Material µr Silica or SiO2 (fused) 3.8 0.000 75 Bismuth 0.999 998 6 Powdered iron 100 Silicon 11.8 Paraffin 0.999 999 42 Machine steel 300 Snow 3.3 0.5 Wood 0.999 999 5 Ferrite (typical) 1000 Sodium chloride 5.9 0.000 1 Silver 0.999 999 81 Permalloy 45 2500 Soil (dry) 2.8 0.05 Aluminum 1.000 000 65 Transformer iron 3000 Steatite 5.8 0.003 Beryllium 1.000 000 79 Silicon iron 3500 Styrofoam 1.03 0.000 1 Nickel chloride 1.000 04 Iron (pure) 4000 Teflon 2.1 0.000 3 Manganese sulfate 1.000 1 Mumetal 20 000 Titanium dioxide 100 0.001 5 Nickel 50 Sendust 30 000 Water (distilled) 80 0.04 Cast iron 60 Supermalloy 100 000 Water (sea) 4 Cobalt 60 Water (dehydrated) 1 0 Wood (dry) 1.5–4 0.01 hay80660_frontendsheet_2and3_HR.indd November 22, 2010 20:32 1 Hayt/Buck Page 2 hay80660 frontendsheet 2and3.pdf 12/24/10 3:09 PM November 22, 2010 20:32 Hayt/Buck Page 3 hay80660 frontendsheet 2and3.pdf  Physical Constants Quantity Value Conductivity ( ) Electron charge e = (1.602 177 33 ± 0.000 000 46) × 10−19 C Material  , S/m Material  , S/m Electron mass m = (9.109 389 7 ± 0.000 005 4) × 10−31 kg Silver 6.17 × 107 Nichrome 0.1 × 107 Permittivity of free space 0 = 8.854 187 817 × 10−12 F/m Copper 5.80 × 107 Graphite 7 × 104 Permeability of free space µ0 = 4π10−7 H/m Gold 4.10 × 107 Silicon 2300 Velocity of light c = 2.997 924 58 × 108 m/s Aluminum 3.82 × 107 Ferrite (typical) 100 Tungsten 1.82 × 107 Water (sea) 5 Dielectric Constant ( r ) and Loss Tangent ( / ) Zinc 1.67 × 107 Limestone 10−2 Material  r  / Brass 1.5 × 107 Clay 5 × 10−3 Air 1.0005 Nickel 1.45 × 107 Water (fresh) 10−3 Alcohol, ethyl 25 0.1 Iron 1.03 × 107 Water (distilled) 10−4 Aluminum oxide 8.8 0.000 6 Phosphor bronze 1 × 107 Soil (sandy) 10−5 Amber 2.7 0.002 Solder 0.7 × 107 Granite 10−6 Bakelite 4.74 0.022 Carbon steel 0.6 × 107 Marble 10−8 Barium titanate 1200 0.013 German silver 0.3 × 107 Bakelite 10−9 Carbon dioxide 1.001 Manganin 0.227 × 107 Porcelain (dry process) 10−10 Ferrite (NiZn) 12.4 0.000 25 Constantan 0.226 × 107 Diamond 2 × 10−13 Germanium 16 Germanium 0.22 × 107 Polystyrene 10−16 Glass 4–7 0.002 Ice 4.2 0.05 Stainless steel 0.11 × 107 Quartz 10−17 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 Relative Permeability (µr ) Quartz (fused) 3.8 0.000 75 Rubber 2.5–3 0.002 Material µr Material µr Silica or SiO2 (fused) 3.8 0.000 75 Bismuth 0.999 998 6 Powdered iron 100 Silicon 11.8 Paraffin 0.999 999 42 Machine steel 300 Snow 3.3 0.5 Wood 0.999 999 5 Ferrite (typical) 1000 Sodium chloride 5.9 0.000 1 Silver 0.999 999 81 Permalloy 45 2500 Soil (dry) 2.8 0.05 Aluminum 1.000 000 65 Transformer iron 3000 Steatite 5.8 0.003 Beryllium 1.000 000 79 Silicon iron 3500 Styrofoam 1.03 0.000 1 Nickel chloride 1.000 04 Iron (pure) 4000 Tefion 2.1 0.000 3 Manganese sulfate 1.000 1 Mumetal 20 000 Titanium dioxide 100 0.001 5 Nickel 50 Sendust 30 000 Water (distilled) 80 0.04 Cast iron 60 Supermalloy 100 000 Water (sea) 4 Cobalt 60 Water (dehydrated) 1 0 Wood (dry) 1.5–4 0.01 hay80660_frontendsheet_2and3_HR.indd November 22, 2010 20:32 1 Hayt/Buck Page 2 hay80660 frontendsheet 2and3.pdf 12/24/10 3:09 PM November 22, 2010 20:32 Hayt/Buck Page 3 hay80660 frontendsheet 2and3.pdf Engineering Electromagnetics EIGH T H E D I T I O N 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 ABOUT THE AUTHORS 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.  BRIEF CONTENTS 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 CONTENTS Preface xii Chapter 3 Electric Flux Density, Gauss’s Law, and Divergence 48 Chapter 1 Vector Analysis 1 3.1 Electric Flux Density 48 3.2 Gauss’s Law 52 1.1 Scalars and Vectors 1 3.3 Application of Gauss’s Law: Some 1.2 Vector Algebra 2 Symmetrical Charge Distributions 56 1.3 The Rectangular Coordinate System 3 3.4 Application of Gauss’s Law: Differential 1.4 Vector Components and Unit Vectors 5 Volume Element 61 1.5 The Vector Field 8 3.5 Divergence and Maxwell’s First Equation 64 1.6 The Dot Product 9 3.6 The Vector Operator ∇ and the Divergence 1.7 The Cross Product 11 Theorem 67 1.8 Other Coordinate Systems: Circular References 70 Cylindrical Coordinates 13 Chapter 3 Problems 71 1.9 The Spherical Coordinate System 18 References 22 Chapter 4 Chapter 1 Problems 22 Energy and Potential 75 4.1 Energy Expended in Moving a Point Charge in an Electric Field 76 Chapter 2 4.2 The Line Integral 77 Coulomb’s Law and Electric 4.3 Definition of Potential Difference Field Intensity 26 and Potential 82 2.1 The Experimental Law of Coulomb 26 4.4 The Potential Field of a Point Charge 84 2.2 Electric Field Intensity 29 4.5 The Potential Field of a System of Charges: 2.3 Field Arising from a Continuous Volume Conservative Property 86 Charge Distribution 33 4.6 Potential Gradient 90 2.4 Field of a Line Charge 35 4.7 The Electric Dipole 95 2.5 Field of a Sheet of Charge 39 4.8 Energy Density in the Electrostatic 2.6 Streamlines and Sketches of Fields 41 Field 100 References 44 References 104 Chapter 2 Problems 44 Chapter 4 Problems 105 vi  Contents vii Chapter 5 7.4 Stokes’ Theorem 202 Conductors and Dielectrics 109 7.5 Magnetic Flux and Magnetic Flux Density 207 5.1 Current and Current Density 110 7.6 The Scalar and Vector Magnetic 5.2 Continuity of Current 111 Potentials 210 5.3 Metallic Conductors 114 7.7 Derivation of the Steady-Magnetic-Field 5.4 Conductor Properties and Boundary Laws 217 Conditions 119 References 223 5.5 The Method of Images 124 Chapter 7 Problems 223 5.6 Semiconductors 126 5.7 The Nature of Dielectric Materials 127 Chapter 8 5.8 Boundary Conditions for Perfect Magnetic Forces, Materials, Dielectric Materials 133 and Inductance 230 References 137 8.1 Force on a Moving Charge 230 Chapter 5 Problems 138 8.2 Force on a Differential Current Element 232 8.3 Force between Differential Current Chapter 6 Elements 236 Capacitance 143 8.4 Force and Torque on a Closed Circuit 238 6.1 Capacitance Defined 143 8.5 The Nature of Magnetic Materials 244 6.2 Parallel-Plate Capacitor 145 8.6 Magnetization and Permeability 247 6.3 Several Capacitance Examples 147 8.7 Magnetic Boundary Conditions 252 6.4 Capacitance of a Two-Wire Line 150 8.8 The Magnetic Circuit 255 6.5 Using Field Sketches to Estimate 8.9 Potential Energy and Forces on Magnetic Capacitance in Two-Dimensional Materials 261 Problems 154 8.10 Inductance and Mutual Inductance 263 6.6 Poisson’s and Laplace’s Equations 160 References 270 6.7 Examples of the Solution of Laplace’s Chapter 8 Problems 270 Equation 162 6.8 Example of the Solution of Poisson’s Chapter 9 Equation: the p-n Junction Capacitance 169 Time-Varying Fields and Maxwell’s References 172 Equations 277 Chapter 6 Problems 173 9.1 Faraday’s Law 277 9.2 Displacement Current 284 Chapter 7 9.3 Maxwell’s Equations in Point Form 288 The Steady Magnetic Field 180 9.4 Maxwell’s Equations in Integral Form 290 7.1 Biot-Savart Law 180 9.5 The Retarded Potentials 292 7.2 Ampère’s Circuital Law 188 References 296 7.3 Curl 195 Chapter 9 Problems 296 viii Contents Chapter 10 12.3 Wave Reflection from Multiple Transmission Lines 301 Interfaces 417 12.4 Plane Wave Propagation in General 10.1 Physical Description of Transmission Line Directions 425 Propagation 302 12.5 Plane Wave Reflection at Oblique Incidence 10.2 The Transmission Line Equations 304 Angles 428 10.3 Lossless Propagation 306 12.6 Total Reflection and Total Transmission 10.4 Lossless Propagation of Sinusoidal of Obliquely Incident Waves 434 Voltages 309 12.7 Wave Propagation in Dispersive Media 437 10.5 Complex Analysis of Sinusoidal Waves 311 12.8 Pulse Broadening in Dispersive Media 443 10.6 Transmission Line Equations and Their References 447 Solutions in Phasor Form 313 Chapter 12 Problems 448 10.7 Low-Loss Propagation 315 10.8 Power Transmission and The Use of Decibels Chapter 13 in Loss Characterization 317 Guided Waves 453 10.9 Wave Reflection at Discontinuities 320 13.1 Transmission Line Fields and Primary 10.10 Voltage Standing Wave Ratio 323 Constants 453 10.11 Transmission Lines of Finite Length 327 13.2 Basic Waveguide Operation 463 10.12 Some Transmission Line Examples 330 13.3 Plane Wave Analysis of the Parallel-Plate 10.13 Graphical Methods: The Smith Chart 334 Waveguide 467 10.14 Transient Analysis 345 13.4 Parallel-Plate Guide Analysis Using the Wave References 358 Equation 476 Chapter 10 Problems 358 13.5 Rectangular Waveguides 479 13.6 Planar Dielectric Waveguides 490 Chapter 11 13.7 Optical Fiber 496 The Uniform Plane Wave 367 References 506 11.1 Wave Propagation in Free Space 367 Chapter 13 Problems 506 11.2 Wave Propagation in Dielectrics 375 Chapter 14 11.3 Poynting’s Theorem and Wave Power 384 11.4 Propagation in Good Conductors: Electromagnetic Radiation Skin Effect 387 and Antennas 511 11.5 Wave Polarization 394 14.1 Basic Radiation Principles: The Hertzian References 401 Dipole 511 Chapter 11 Problems 401 14.2 Antenna Specifications 518 14.3 Magnetic Dipole 523 Chapter 12 14.4 Thin Wire Antennas 525 Plane Wave Reflection and 14.5 Arrays of Two Elements 533 Dispersion 406 14.6 Uniform Linear Arrays 537 12.1 Reflection of Uniform Plane Waves 14.7 Antennas as Receivers 541 at Normal Incidence 406 References 548 12.2 Standing Wave Ratio 413 Chapter 14 Problems 548  Contents ix Appendix A Appendix D Vector Analysis 553 The Uniqueness Theorem 565 A.1 General Curvilinear Coordinates 553 Appendix E A.2 Divergence, Gradient, and Curl Origins of the Complex in General Curvilinear Coordinates 554 Permittivity 567 A.3 Vector Identities 556 Appendix F Appendix B Answers to Odd-Numbered Units 557 Problems 574 Appendix C Material Constants 562 Index 580 PREFACE 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 [email protected]. 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 McGraw-Hill Digital Offerings Include: This text is available as an eBook at www.CourseSmart.com. At CourseSmart, stu- dents can take advantage of significant savings of the printed textbook, reduce their impact on the environment, and gain access to powerful web tools for learning. CourseSmart eBooks can be viewed online or downloaded to a computer. The eBooks allow students to do full text searches, add highlighting and notes, and share notes with classmates. Contact your McGraw-Hill sales representative or visit www.CourseSmart. com to learn more. Professors can benefit from McGraw-Hill’s Complete Online Solutions Manual Or- ganization System (COSMOS). COSMOS enables instructors to generate a limitless supply of problem material for assignment, as well as transfer and integrate their own problems into the software. For additional information, contact your McGraw-Hill sales representative. McGraw-Hill Create™ Craft your teaching resources to match the way you teach! With McGraw-Hill Create, www.mcgrawhillcreate.com, you can easily rearrange chapters, combine material from other content sources, and quickly upload content you have written like your course syllabus or teaching notes. Find the content you need in Create by searching through thousands of leading McGraw-Hill textbooks. Arrange your book to fit your teaching style. Create even allows you to personalize your book’s appearance by selecting the cover and adding your name, school, and course information. Order a Create book and you’ll receive a complimentary print review copy in 3–5 business days or a complimentary electronic review copy (eComp) via email in minutes. Go to www.mcgrawhillcreate.com today and register to experience how McGraw-Hill Create empowers you to teach your students your way. McGraw-Hill Higher Education and Blackboard® have teamed up. Blackboard, the web-based course management sys- tem, has partnered with McGraw-Hill to better allow students and faculty to use online materials and activities to complement face-to-face teaching. Blackboard features exciting social learning and teaching tools that foster more logical, visually impactful and active learning opportunities for stu- dents. You’ll transform your closed-door classrooms into communities where students remain connected to their educational experience 24 hours a day. This partnership allows you and your students access to McGraw-Hill’s Create right from within your Blackboard course—all with one single sign-on. McGraw- Hill and Blackboard can now offer you easy access to industry leading technology and content, whether your campus hosts it, or we do. Be sure to ask your local McGraw-Hill representative for details.  C H A P T E R 1 Vector Analysis ector analysis is a mathematical subject that is better taught by mathematicians V 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 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 + sA + sB 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 1.2a 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 1.2b 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 + d x, y + dy, and z + dz. The six planes define a rectangular parallelepiped whose volume is dv = d xd ydz; the surfaces have differential areas d S of d xd y, dydz, and dzd x. Finally,
the distance d L from P to P
is the diagonal of the parallelepiped and has a length of (d x)2 + (dy)2 + (dz)2. The volume element is shown in Figure 1.2c; 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 1.3a. 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 ax , a y , and az 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 1.3b. If the component vector y happens to be two units in magnitude and directed toward increasing values of y, we should then write y = 2a 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 P Q is equal to the vector difference r Q − r P. from the origin to point P(1, 2, 3) is written r P = ax + 2a y + 3az. 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)ax + (−2 − 2)a y + (1 − 3)az = ax − 4a y − 2az The vectors r P , r Q , and R P Q are shown in Figure 1.3c. 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 ax + Fy a y + Fz az. The component vectors are Fx ax , Fy a y , and Fz az. Any vector B then may be described by B = Bx ax + B y a y + Bz az. The magnitude of B written |B| or simply B, is given by  |B| = Bx2 + B y2 + Bz2 (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 B B aB =  = Bx2 + B y2 + Bz2 |B| (2) E X A M P L E 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 = 2ax − 2a y − az 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, G aG = = 23 ax − 23 a y − 13 az = 0.667ax − 0.667a y − 0.333az |G| 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. 8 ENGINEERING ELECTROMAGNETICS [NOTE: Throughout the text, drill problems appear following sections in which a new principle is introduced in order to allow students to test their understanding of the basic fact itself. The problems are useful in gaining familiarity with new terms and ideas and should all be worked. More general problems appear at the ends of the chapters. The answers to the drill problems are given in the same order as the parts of the problem.] D1.1. Given points M(−1, 2, 1), N (3, −3, 0), and P(−2, −3, −4), find: (a) R M N ; (b) R M N + R M P ; (c) |r M |; (d) a M P ; (e) |2r P − 3r N |. Ans. 4ax − 5a y − az ; 3ax − 10a y − 6az ; 2.45; −0.14ax − 0.7a y − 0.7az ; 15.56 1.5 THE VECTOR FIELD We have defined a vector field as a vector function of a position vector. In general, the magnitude and direction of the function will change as we move throughout the region, and the value of the vector function must be determined using the coordinate values of the point in question. Because we have considered only the rectangular coordinate system, we expect the vector to be a function of the variables x, y, and z. If we again represent the position vector as r, then a vector field G can be expressed in functional notation as G(r); a scalar field T is written as T (r). If we inspect the velocity of the water in the ocean in some region near the surface where tides and currents are important, we might decide to represent it by a velocity vector that is in any direction, even up or down. If the z axis is taken as upward, the x axis in a northerly direction, the y axis to the west, and the origin at the surface, we have a right-handed coordinate system and may write the velocity vector as v = v x ax + v y a y + v z az , or v(r) = v x (r)ax + v y (r)a y + v z (r)az ; each of the components v x , v y , and v z may be a function of the three variables x, y, and z. If we are in some portion of the Gulf Stream where the water is moving only to the north, then v y and v z are zero. Further simplifying assumptions might be made if the velocity falls off with depth and changes very slowly as we move north, south, east, or west. A suitable expression could be v = 2e z/100 ax. We have a velocity of 2 m/s (meters per second) at the surface and a velocity of 0.368 × 2, or 0.736 m/s, at a depth of 100 m (z = −100). The velocity continues to decrease with depth, while maintaining a constant direction. D1.2. A vector field S is expressed in rectangular coordinates as S = {125/ [(x − 1)2 + (y − 2)2 + (z + 1)2 ]}{(x − 1)ax + (y − 2)a y + (z + 1)az }. (a) Evaluate S at P(2, 4, 3). (b) Determine a unit vector that gives the direction of S at P. (c) Specify the surface f (x, y, z) on which |S| = 1. Ans.
5.95ax + 11.90a y + 23.8az ; 0.218ax + 0.436a y + 0.873az ; (x − 1)2 + (y − 2)2 + (z + 1)2 = 125  CHAPTER 1 Vector Analysis 9 1.6 THE DOT PRODUCT We now consider the first of two types of vector multiplication. The second type will be discussed in the following section. Given two vectors A and B, the dot product, or scalar product, is defined as the product of the magnitude of A, the magnitude of B, and the cosine of the smaller angle between them, A · B = |A| |B| cos θ AB (3) The dot appears between the two vectors and should be made heavy for emphasis. The dot, or scalar, product is a scalar, as one of the names implies, and it obeys the commutative law, A·B = B·A (4) for the sign of the angle does not affect the cosine term. The expression A · B is read “A dot B.” Perhaps the most common application of the dot product is in mechanics, where a constant force F applied over a straight displacement L does an amount of work F L cos θ , which is more easily written F · L. We might anticipate one of the results of Chapter 4 by pointing out that if the force varies along the path, integration is necessary to find the total work, and the result becomes  Work = F · dL Another example might be taken from magnetic fields. The total flux  crossing a surface of area S is given by B S if the magnetic flux density B is perpendicular to the surface and uniform over it. We define a vector surface S as having area for its magnitude and having a direction normal to the surface (avoiding for the moment the problem of which of the two possible normals to take). The flux crossing the surface is then B · S. This expression is valid for any direction of the uniform magnetic flux density. If the flux density is not constant over the surface, the total flux is  = B · dS. Integrals of this general form appear in Chapter 3 when we study electric flux density. Finding the angle between two vectors in three-dimensional space is often a job we would prefer to avoid, and for that reason the definition of the dot product is usually not used in its basic form. A more helpful result is obtained by considering two vectors whose rectangular components are given, such as A = A x ax + A y a y + A z az and B = Bx ax + B y a y + Bz az. The dot product also obeys the distributive law, and, therefore, A · B yields the sum of nine scalar terms, each involving the dot product of two unit vectors. Because the angle between two different unit vectors of the rectangular coordinate system is 90◦ , we then have a x · a y = a y · a x = a x · az = az · a x = a y · az = az · a y = 0 10 ENGINEERING ELECTROMAGNETICS Figure 1.4 (a) The scalar component of B in the direction of the unit vector a is B · a. (b) The vector component of B in the direction of the unit vector a is (B · a)a. The remaining three terms involve the dot product of a unit vector with itself, which is unity, giving finally A · B = A x B x + A y B y + A z Bz (5) which is an expression involving no angles. A vector dotted with itself yields the magnitude squared, or A · A = A2 = |A|2 (6) and any unit vector dotted with itself is unity, aA · aA = 1 One of the most important applications of the dot product is that of finding the component of a vector in a given direction. Referring to Figure 1.4a, we can obtain the component (scalar) of B in the direction specified by the unit vector a as B · a = |B| |a| cos θ Ba = |B| cos θ Ba The sign of the component is positive if 0 ≤ θ Ba ≤ 90◦ and negative whenever 90◦ ≤ θ Ba ≤ 180◦. To obtain the component vector of B in the direction of a, we multiply the component (scalar) by a, as illustrated by Figure 1.4b. For example, the component of B in the direction of ax is B · ax = Bx , and the component vector is Bx ax , or (B · ax )ax. Hence, the problem of finding the component of a vector in any direction becomes the problem of finding a unit vector in that direction, and that we can do. The geometrical term projection is also used with the dot product. Thus, B · a is the projection of B in the a direction. E X A M P L E 1.2 In order to illustrate these definitions and operations, consider the vector field G = yax − 2.5xa y + 3az and the point Q(4, 5, 2). We wish to find: G at Q; the scalar com- ponent of G at Q in the direction of a N = 13 (2ax + a y − 2az ); the vector component of G at Q in the direction of a N ; and finally, the angle θGa between G(r Q ) and a N.  CHAPTER 1 Vector Analysis 11 Solution. Substituting the coordinates of point Q into the expression for G, we have G(r Q ) = 5ax − 10a y + 3az Next we find the scalar component. Using the dot product, we have G · a N = (5ax − 10a y + 3az ) · 13 (2ax + a y − 2az ) = 13 (10 − 10 − 6) = −2 The vector component is obtained by multiplying the scalar component by the unit vector in the direction of a N, (G · a N )a N = −(2) 13 (2ax + a y − 2az ) = −1.333ax − 0.667a y + 1.333az The angle between G(r Q ) and a N is found from G · a N = |G| cos θGa √ −2 = 25 + 100 + 9 cos θGa and −2 θGa = cos−1 √ = 99.9◦ 134 D1.3. The three vertices of a triangle are located at A(6, −1, 2), B(−2, 3, −4), and C(−3, 1, 5). Find: (a) R AB ; (b) R AC ; (c) the angle θ B AC at vertex A; (d) the (vector) projection of R AB on R AC. Ans. −8ax + 4a y − 6az ; −9ax + 2a y + 3az ; 53.6◦ ; −5.94ax + 1.319a y + 1.979az 1.7 THE CROSS PRODUCT Given two vectors A and B, we now define the cross product, or vector product, of A and B, written with a cross between the two vectors as A × B and read “A cross B.” The cross product A × B is a vector; the magnitude of A × B is equal to the product of the magnitudes of A, B, and the sine of the smaller angle between A and B; the direction of A×B is perpendicular to the plane containing A and B and is along one of the two possible perpendiculars which is in the direction of advance of a right-handed screw as A is turned into B. This direction is illustrated in Figure 1.5. Remember that either vector may be moved about at will, maintaining its direction constant, until the two vectors have a “common origin.” This determines the plane containing both. However, in most of our applications we will be concerned with vectors defined at the same point. As an equation we can write A × B = a N |A| |B| sin θ AB (7) where an additional statement, such as that given above, is required to explain the direction of the unit vector a N. The subscript stands for “normal.” 12 ENGINEERING ELECTROMAGNETICS Figure 1.5 The direction of A × B is in the direction of advance of a right-handed screw as A is turned into B. Reversing the order of the vectors A and B results in a unit vector in the opposite direction, and we see that the cross product is not commutative, for B×A = −(A×B). If the definition of the cross product is applied to the unit vectors ax and a y , we find ax × a y = az , for each vector has unit magnitude, the two vectors are perpendicular, and the rotation of ax into a y indicates the positive z direction by the definition of a right-handed coordinate system. In a similar way, a y × az = ax and az × ax = a y. Note the alphabetic symmetry. As long as the three vectors ax , a y , and az are written in order (and assuming that ax follows az , like three elephants in a circle holding tails, so that we could also write a y , az , ax or az , ax , a y ), then the cross and equal sign may be placed in either of the two vacant spaces. As a matter of fact, it is now simpler to define a right-handed rectangular coordinate system by saying that ax × a y = az. A simple example of the use of the cross product may be taken from geometry or trigonometry. To find the area of a parallelogram, the product of the lengths of two adjacent sides is multiplied by the sine of the angle between them. Using vector notation for the two sides, we then may express the (scalar) area as the magnitude of A × B, or |A × B|. The cross product may be used to replace the right-hand rule familiar to all electrical engineers. Consider the force on a straight conductor of length L, where the direction assigned to L corresponds to the direction of the steady current I , and a uniform magnetic field of flux density B is present. Using vector notation, we may write the result neatly as F = I L × B. This relationship will be obtained later in Chapter 9. The evaluation of a cross product by means of its definition turns out to be more work than the evaluation of the dot product from its definition, for not only must we find the angle between the vectors, but we must also find an expression for the  CHAPTER 1 Vector Analysis 13 unit vector a N. This work may be avoided by using rectangular components for the two vectors A and B and expanding the cross product as a sum of nine simpler cross products, each involving two unit vectors, A × B = A x B x a x × a x + A x B y a x × a y + A x Bz a x × a z + A y B x a y × a x + A y B y a y × a y + A y Bz a y × a z + A z B x a z × a x + A z B y a z × a y + A z Bz a z × a z We have already found that ax × a y = az , a y × az = ax , and az × ax = a y. The three remaining terms are zero, for the cross product of any vector with itself is zero, since the included angle is zero. These results may be combined to give A × B = (A y Bz − A z B y )ax + (A z Bx − A x Bz )a y + (A x B y − A y Bx )az (8) or written as a determinant in a more easily remembered form,    a x a y az    A × B =  A x A y A z  (9)  B x B y Bz  Thus, if A = 2ax − 3a y + az and B = −4ax − 2a y + 5az , we have    a x a y az    A × B =  2 −3 1  −4 −2 5  = [(−3)(5) − (1(−2)]ax − [(2)(5) − (1)(−4)]a y + [(2)(−2) − (−3)(−4)]az = −13ax − 14a y − 16az D1.4. The three vertices of a triangle are located at A(6, −1, 2), B(−2, 3, −4), and C(−3, 1, 5). Find: (a) R AB × R AC ; (b) the area of the triangle; (c) a unit vector perpendicular to the plane in which the triangle is located. Ans. 24ax + 78a y + 20az ; 42.0; 0.286ax + 0.928a y + 0.238az 1.8 OTHER COORDINATE SYSTEMS: CIRCULAR CYLINDRICAL COORDINATES The rectangular coordinate system is generally the one in which students prefer to work every problem. This often means a lot more work, because many problems possess a type of symmetry that pleads for a more logical treatment. It is easier to do now, once and for all, the work required to become familiar with cylindrical and spherical coordinates, instead of applying an equal or greater effort to every problem involving cylindrical or spherical symmetry later. With this in mind, we will take a careful and unhurried look at cylindrical and spherical coordinates. 14 ENGINEERING ELECTROMAGNETICS The circular cylindrical coordinate system is the three-dimensional version of the polar coordinates of analytic geometry. In polar coordinates, a point is located in a plane by giving both its distance ρ from the origin and the angle φ between the line from the point to the origin and an arbitrary radial line, taken as φ = 0.4 In circular cylindrical coordinates, we also specify the distance z of the point from an arbitrary z = 0 reference plane that is perpendicular to the line ρ = 0. For simplicity, we usually refer to circular cylindrical coordinates simply as cylindrical coordinates. This will not cause any confusion in reading this book, but it is only fair to point out that there are such systems as elliptic cylindrical coordinates, hyperbolic cylindrical coordinates, parabolic cylindrical coordinates, and others. We no longer set up three axes as with rectangular coordinates, but we must instead consider any point as the intersection of three mutually perpendicular sur- faces. These surfaces are a circular cylinder (ρ = constant), a plane (φ = constant), and another plane (z = constant). This corresponds to the location of a point in a rectangular coordinate system by the intersection of three planes (x = constant, y = constant, and z = constant). The three surfaces of circular cylindrical coordinates are shown in Figure 1.6a. Note that three such surfaces may be passed through any point, unless it lies on the z axis, in which case one plane suffices. Three unit vectors must also be defined, but we may no longer direct them along the “coordinate axes,” for such axes exist only in rectangular coordinates. Instead, we take a broader view of the unit vectors in rectangular coordinates and realize that they are directed toward increasing coordinate values and are perpendicular to the surface on which that coordinate value is constant (i.e., the unit vector ax is normal to the plane x = constant and points toward larger values of x). In a corresponding way we may now define three unit vectors in cylindrical coordinates, aρ , aφ , and az. The unit vector aρ at a point P(ρ1 , φ1 , z 1 ) is directed radially outward, normal to the cylindrical surface ρ = ρ1. It lies in the planes φ = φ1 and z = z 1. The unit vector aφ is normal to the plane φ = φ1 , points in the direction of increasing φ, lies in the plane z = z 1 , and is tangent to the cylindrical surface ρ = ρ1. The unit vector az is the same as the unit vector az of the rectangular coordinate system. Figure 1.6b shows the three vectors in cylindrical coordinates. In rectangular coordinates, the unit vectors are not functions of the coordinates. Two of the unit vectors in cylindrical coordinates, aρ and aφ , however, do vary with the coordinate φ, as their directions change. In integration or differentiation with respect to φ, then, aρ and aφ must not be treated as constants. The unit vectors are again mutually perpendicular, for each is normal to one of the three mutually perpendicular surfaces, and we may define a right-handed cylindrical 4 The two variables of polar coordinates are commonly called r and θ. With three coordinates, however, it is more common to use ρ for the radius variable of cylindrical coordinates and r for the (different) radius variable of spherical coordinates. Also, the angle variable of cylindrical coordinates is customarily called φ because everyone uses θ for a different angle in spherical coordinates. The angle φ is common to both cylindrical and spherical coordinates. See?  CHAPTER 1 Vector Analysis 15 Figure 1.6 (a) The three mutually perpendicular surfaces of the circular cylindrical coordinate system. (b) The three unit vectors of the circular cylindrical coordinate system. (c) The differential volume unit in the circular cylindrical coordinate system; dρ, ρdφ, and dz are all elements of length. coordinate system as one in which aρ × aφ = az , or (for those who have flexible fingers) as one in which the thumb, forefinger, and middle finger point in the direction of increasing ρ, φ, and z, respectively. A differential volume element in cylindrical coordinates may be obtained by increasing ρ, φ, and z by the differential increments dρ, dφ, and dz. The two cylinders of radius ρ and ρ + dρ, the two radial planes at angles φ and φ + dφ, and the two “horizontal” planes at “elevations” z and z + dz now enclose a small volume, as shown in Figure 1.6c, having the shape of a truncated wedge. As the volume element becomes very small, its shape approaches that of a rectangular parallelepiped having sides of length dρ, ρdφ, and dz. Note that dρ and dz are dimensionally lengths, but dφ is not; ρdφ is the length. The surfaces have areas of ρ dρ dφ, dρ dz, and ρ dφ dz, and the volume becomes ρ dρ dφ dz. 16 ENGINEERING ELECTROMAGNETICS Figure 1.7 The relationship between the rectangular variables x, y, z and the cylindrical coordinate variables ρ, φ, z. There is no change in the variable z between the two systems. The variables of the rectangular and cylindrical coordinate systems are easily related to each other. Referring to Figure 1.7, we see that x = ρ cos φ y = ρ sin φ (10) z=z From the other viewpoint, we may express the cylindrical variables in terms of x, y, and z:
ρ= x 2 + y2 (ρ ≥ 0) y φ = tan−1 (11) x z=z We consider the variable ρ to be positive or zero, thus using only the positive sign for the radical in (11). The proper value of the angle φ is determined by inspecting the signs of x and y. Thus, if x = −3 and y = 4, we find that the point lies in the second quadrant so that ρ = 5 and φ = 126.9◦. For x = 3 and y = −4, we have φ = −53.1◦ or 306.9◦ , whichever is more convenient. Using (10) or (11), scalar functions given in one coordinate system are easily transformed into the other system. A vector function in one coordinate system, however, requires two steps in order to transform it to another coordinate system, because a different set of component  CHAPTER 1 Vector Analysis 17 vectors is generally required. That is, we may be given a rectangular vector A = A x a x + A y a y + A z az where each component is given as a function of x, y, and z, and we need a vector in cylindrical coordinates A = Aρ aρ + Aφ aφ + A z az where each component is given as a function of ρ, φ, and z. To find any desired component of a vector, we recall from the discussion of the dot product that a component in a desired direction may be obtained by taking the dot product of the vector and a unit vector in the desired direction. Hence, Aρ = A · aρ and Aφ = A · aφ Expanding these dot products, we have Aρ = (A x ax + A y a y + A z az ) · aρ = A x ax · aρ + A y a y · aρ (12) Aφ = (A x ax + A y a y + A z az ) · aφ = A x ax · aφ + A y a y · aφ (13) and A z = (A x ax + A y a y + A z az ) · az = A z az · az = A z (14) since az · aρ and az · aφ are zero. In order to complete the transformation of the components, it is necessary to know the dot products ax · aρ , a y · aρ , ax · aφ , and a y · aφ. Applying the definition of the dot product, we see that since we are concerned with unit vectors, the result is merely the cosine of the angle between the two unit vectors in question. Refer- ring to Figure 1.7 and thinking mightily, we identify the angle between ax and aρ as φ, and thus ax · aρ = cos φ, but the angle between a y and aρ is 90◦ − φ, and a y · aρ = cos (90◦ − φ) = sin φ. The remaining dot products of the unit vectors are found in a similar manner, and the results are tabulated as functions of φ in Table 1.1. Transforming vectors from rectangular to cylindrical coordinates or vice versa is therefore accomplished by using (10) or (11) to change variables, and by using the dot products of the unit vectors given in Table 1.1 to change components. The two steps may be taken in either order. Table 1.1 Dot products of unit vectors in cylindrical and rectangular coordinate systems aρ aφ az ax · cos φ − sin φ 0 ay · sin φ cos φ 0 az · 0 0 1 18 ENGINEERING ELECTROMAGNETICS E X A M P L E 1.3 Transform the vector B = yax − xa y + zaz into cylindrical coordinates. Solution. The new components are Bρ = B · aρ = y(ax · aρ ) − x(a y · aρ ) = y cos φ − x sin φ = ρ sin φ cos φ − ρ cos φ sin φ = 0 Bφ = B · aφ = y(ax · aφ ) − x(a y · aφ ) = −y sin φ − x cos φ = −ρ sin2 φ − ρ cos2 φ = −ρ Thus, B = −ρaφ + zaz D1.5. (a) Give the rectangular coordinates of the point C(ρ = 4.4, φ = −115◦ , z = 2). (b) Give the cylindrical coordinates of the point D(x = −3.1, y = 2.6, z = −3). (c) Specify the distance from C to D. Ans. C(x = −1.860, y = −3.99, z = 2); D(ρ = 4.05, φ = 140.0◦ , z = −3); 8.36 D1.6. Transform to cylindrical coordinates: (a) F = 10ax −8a y +6az at point P(10, −8, 6); (b) G = (2x + y)ax − (y − 4x)a y at point Q(ρ, φ, z). (c) Give the rectangular components of the vector H = 20aρ − 10aφ + 3az at P(x = 5, y = 2, z = −1). Ans. 12.81aρ + 6az ; (2ρ cos2 φ − ρ sin2 φ + 5ρ sin φ cos φ)aρ + (4ρ cos2 φ − ρ sin2 φ − 3ρ sin φ cos φ)aφ ; Hx = 22.3, Hy = −1.857, Hz = 3 1.9 THE SPHERICAL COORDINATE SYSTEM We have no two-dimensional coordinate system to help us understand the three- dimensional spherical coordinate system, as we have for the circular cylindrical coordinate system. In certain respects we can draw on our knowledge of the latitude- and-longitude system of locating a place on the surface of the earth, but usually we consider only points on the surface and not those below or above ground. Let us start by building a spherical coordinate system on the three rectangular axes (Figure 1.8a). We first define the distance from the origin to any point as r. The surface r = constant is a sphere. The second coordinate is an angle θ between the z axis and the line drawn from the origin to the point in question. The surface θ = constant is a cone, and the two surfaces, cone and sphere, are everywhere perpendicular along their inter- section, which is a circle of radius r sin θ. The coordinate θ corresponds to latitude,  CHAPTER 1 Vector Analysis 19 Figure 1.8 (a) The three spherical coordinates. (b) The three mutually perpendicular surfaces of the spherical coordinate system. (c) The three unit vectors of spherical coordinates: ar × aθ = aφ. (d) The differential volume element in the spherical coordinate system. except that latitude is measured from the equator and θ is measured from the “North Pole.” The third coordinate φ is also an angle and is exactly the same as the angle φ of cylindrical coordinates. It is the angle between the x axis and the projection in the z = 0 plane of the line drawn from the origin to the point. It corresponds to the angle of longitude, but the angle φ increases to the “east.” The surface φ = constant is a plane passing through the θ = 0 line (or the z axis). We again consider any point as the intersection of three mutually perpendicular surfaces—a sphere, a cone, and a plane—each oriented in the manner just described. The three surfaces are shown in Figure 1.8b. Three unit vectors may again be defined at any point. Each unit vector is per- pendicular to one of the three mutually perpendicular surfaces and oriented in that 20 ENGINEERING ELECTROMAGNETICS direction in which the coordinate increases. The unit vector ar is directed radially outward, normal to the sphere r = constant, and lies in the cone θ = constant and the plane φ = constant. The unit vector aθ is normal to the conical surface, lies in the plane, and is tangent to the sphere. It is directed along a line of “longitude” and points “south.” The third unit vector aφ is the same as in cylindrical coordinates, being normal to the plane and tangent to both the cone and the sphere. It is directed to the “east.” The three unit vectors are shown in Figure 1.8c. They are, of course, mutually per- pendicular, and a right-handed coordinate system is defined by causing ar × aθ = aφ. Our system is right-handed, as an inspection of Figure 1.8c will show, on application of the definition of the cross product. The right-hand rule identifies the thumb, fore- finger, and middle finger with the direction of increasing r , θ , and φ, respectively. (Note that the identification in cylindrical coordinates was with ρ, φ, and z, and in rectangular coordinates with x, y, and z.) A differential volume element may be con- structed in spherical coordinates by increasing r , θ , and φ by dr , dθ, and dφ, as shown in Figure 1.8d. The distance between the two spherical surfaces of radius r and r + dr is dr ; the distance between the two cones having generating angles of θ and θ + dθ is r dθ; and the distance between the two radial planes at angles φ and φ + dφ is found to be r sin θ dφ, after a few moments of trigonometric thought. The surfaces have areas of r dr dθ , r sin θ dr dφ, and r 2 sin θ dθ dφ, and the volume is r 2 sin θ dr dθ dφ. The transformation of scalars from the rectangula