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This document provides an overview of practical applications covered in a textbook on electromagnetics. It lists over 50 applications in various fields such as electrostatics, magnetostatics, electromagnetic waves, transmission lines, waveguides, and antennas. These applications include examples such as electrostatic separation of solids, the xerographic copying machine, magnetic resonance imaging, optical fibers, radar, and antennas. The document also provides physical constants and their standard symbols that are relevant to electromagnetics.

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PRACTICAL APPLICATIONS Some of the real-life applications covered in this book are listed in order of appearance. • Applications of electrostatics (Section 4.1) • Electrostatic separation of solids (Example 4.3) • Electrostatic discharge (ESD) (Section 4.11) • Electrostatic shielding (Section 5.9B) • High dielectric constant materials (Section 5.10) • Graphene (Section 5.11) NEW • Electrohydrodynamic pump (Example 6.1) • Xerographic copying machine (Example 6.2) • Parallel-plate capacitor, coaxial capacitor, and spherical capacitor (Section 6.5) • RF MEMS (Section 6.8) (Chapter 12 opener) NEW • Ink-jet printer (Problem 6.52) • Microstrip lines (Sections 6.7, 11.8, and 14.6) • Applications of magnetostatics (Section 7.1) • Coaxial transmission line (Section 7.4C) • Lightning (Section 7.9) • Polywells (Section 7.10) NEW • Magnetic resonant imaging (MRI) (Chapter 8 opener) • Magnetic focusing of a beam of electrons (Example 8.2, Figure 8.2) • Velocity filter for charged particles (Example 8.3, Figure 8.3) • Inductance of common elements (Table 8.3) • Electromagnet (Example 8.16) • Magnetic levitation (Section 8.12) • Hall effect (Section 8.13) NEW • Direct current machine (Section 9.3B) • Memristor (Section 9.8) NEW • Optical nanocircuits (Section 9.9) NEW • Homopolar generator disk (Problem 9.14) • Microwaves (Section 10.11) • Radar (Sections 10.11 and 13.9) • 60 GHz technology (Section 10.12) NEW • Bioelectromagnetics (Chapter 11 opener) • Coaxial, two-line, and planar lines (Figure 11.1, Section 11.2) • Quarter-wave transformer (Section 11.6A) • Data cables (Section 11.8B) • Metamaterials (Section 11.9) NEW • Microwave imaging (Section 11.10) NEW • Optical fiber (Section 12.9) • Cloaking and invisibility (Section 12.10) NEW • Smart antenna (Chapter 13 opener) • Typical antennas (Section 13.1, Figure 13.2) • Electromagnetic interference and compatibility (Section 13.10) • Grounding and filtering (Section 13.10) 00_Sadiku_FRONTCOVER.indd 2 16/11/17 2:09 PM
• Textile antennas and sensors (Section 13.11) NEW • RFID (Section 13.12) NEW • Commercial EM software—FEKO (Section 14.7) NEW • COMSOL Multiphysics (Section 14.8) NEW • CST Microwave Studio (Section 14.9) NEW Approximate Best Experimental Value for Problem Quantity (Units) Symbol Value* Work Permittivity of free space (F/m) eo 8.854  1012 1029 36p Permeability of free space (H/m) mo 4p  107 12.6  107 Intrinsic impedance of free space (V) ho 376.6 120p Speed of light in vacuum (m/s) c 2.998  108 3  108 Electron charge (C) e 1.6022  1019 1.6  1019 Electron mass (kg) me 9.1093  1031 9.1  1031 Proton mass (kg) mp 1.6726  1027 1.67  1027 Neutron mass (kg) mn 1.6749  1027 1.67  1027 Boltzmann constant (J/K) k 1.38065  1023 1.38  1023 Avogadro number (/kg-mole) N 6.0221  1023 6  1023 Planck constant (J  s) h 6.626  1034 6.62  1034 Acceleration due to gravity (m/s2 ) g 9.80665 9.8 Universal constant of gravitation G 6.673  1011 6.66  1011 N (m/kg)2 Electron-volt (J) eV 1.602176  1019 1.6  1019    * Values recommended by CODATA (Committee on Data for Science and Technology, Paris). PHYSICAL CONSTANTS 00_Sadiku_FRONTCOVER.indd 3 16/11/17 2:09 PM
ELEMENTS OF ELECTROMAGNETICS 00_Sadiku_FM.indd 1 16/11/17 3:36 PM
THE OXFORD SERIES IN ELECTRICAL AND COMPUTER ENGINEERING Adel S. Sedra, Series Editor Allen and Holberg, CMOS Analog Circuit Design, 3rd edition Boncelet, Probability, Statistics, and Random Signals Bobrow, Elementary Linear Circuit Analysis, 2nd edition Bobrow, Fundamentals of Electrical Engineering, 2nd edition Campbell, Fabrication Engineering at the Micro- and Nanoscale, 4th edition Chen, Digital Signal Processing Chen, Linear System Theory and Design, 4th edition Chen, Signals and Systems, 3rd edition Comer, Digital Logic and State Machine Design, 3rd edition Comer, Microprocessor-Based System Design Cooper and McGillem, Probabilistic Methods of Signal and System Analysis, 3rd edition Dimitrijev, Principles of Semiconductor Device, 2nd edition Dimitrijev, Understanding Semiconductor Devices Fortney, Principles of Electronics: Analog Digital Franco, Electric Circuits Fundamentals Ghausi, Electronic Devices and Circuits: Discrete and Integrated Guru and Hiziroğlu, Electric Machinery and Transformers, 3rd edition Houts, Signal Analysis in Linear Systems Jones, Introduction to Optical Fiber Communication Systems Krein, Elements of Power Electronics, 2nd edition Kuo, Digital Control Systems, 3rd edition Lathi and Green, Linear Systems and Signals, 3rd edition Lathi and Ding, Modern Digital and Analog Communication Systems, 5th edition Lathi, Signal Processing and Linear Systems Martin, Digital Integrated Circuit Design Miner, Lines and Electromagnetic Fields for Engineers Mitra, Signals and Systems Parhami, Computer Architecture Parhami, Computer Arithmetic, 2nd edition Roberts and Sedra, SPICE, 2nd edition Roberts, Taenzler, and Burns, An Introduction to Mixed-Signal IC Test and Measurement, 2nd edition Roulston, An Introduction to the Physics of Semiconductor Devices Sadiku, Elements of Electromagnetics, 7th edition Santina, Stubberud, and Hostetter, Digital Control System Design, 2nd edition Sarma, Introduction to Electrical Engineering Schaumann, Xiao, and Van Valkenburg, Design of Analog Filters, 3rd edition Schwarz and Oldham, Electrical Engineering: An Introduction, 2nd edition Sedra and Smith, Microelectronic Circuits, 7th edition Stefani, Shahian, Savant, and Hostetter, Design of Feedback Control Systems, 4th edition Tsividis, Operation and Modeling of the MOS Transistor, 3rd edition Van Valkenburg, Analog Filter Design Warner and Grung, Semiconductor Device Electronics Wolovich, Automatic Control Systems Yariv and Yeh, Photonics: Optical Electronics in Modern Communications, 6th edition Żak, Systems and Control 00_Sadiku_FM.indd 2 16/11/17 3:36 PM
ELEMENTS OF ELECTROMAGNETICS MATTHEW N. O. SADIKU Prairie View AM University New York • Oxford OXFORD UNIVERSITY PRESS SEVENTH EDITION 00_Sadiku_FM.indd 3 16/11/17 3:36 PM
Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and certain other countries. Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America. © 2018, 2015, 2010, 2007, 2000 by Oxford University Press © 1994, 1989 by Holt, Rinehart, Winston, Inc For titles covered by Section 112 of the US Higher Education Opportunity Act, please visit www.oup.com/us/he for the latest information about pricing and alternate formats. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by license, or under terms agreed with the appropriate reproduction rights organization. Inquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not circulate this work in any other form and you must impose this same condition on any acquirer. Library of Congress Cataloging-in-Publication Data Names: Sadiku, Matthew N. O., author. Title: Elements of Electromagnetics / Matthew N.O. Sadiku, Prairie View AM University. Description: Seventh edition. | New York, NY, United States of America : Oxford University Press, [2018] | Series: The Oxford series in electrical and computer engineering Identifiers: LCCN 2017046497 | ISBN 9780190698614 (hardcover) Subjects: LCSH: Electromagnetism. Classification: LCC QC760 .S23 2018 | DDC 537—dc23 LC record available at https://lccn.loc.gov/2017046497 9 8 7 6 5 4 3 2 1 Printed by LSC Communications, United States of America 00_Sadiku_FM.indd 4 16/11/17 3:36 PM
To my wife, Kikelomo 00_Sadiku_FM.indd 5 16/11/17 3:36 PM
vi BRIEF TABLE OF CONTENTS MATH ASSESSMENT  MA-1 Chapter 1   Vector Algebra  3 Chapter 2   Coordinate Systems and Transformation  31 Chapter 3   Vector Calculus  59 Chapter 4   Electrostatic Fields  111 Chapter 5    Electric Fields in Material Space   177 Chapter 6   Electrostatic Boundary-Value Problems  225 Chapter 7   Magnetostatic Fields  297 Chapter 8    Magnetic Forces, Materials, and Devices   349 Chapter 9   Maxwell’s Equations  421 Chapter 10  Electromagnetic Wave Propagation  473 Chapter 11  Transmission Lines  553 Chapter 12  Waveguides  633 Chapter 13  Antennas  691 Chapter 14  Numerical Methods  757 Appendix A    Mathematical Formulas  835 Appendix B    Material Constants  845 Appendix C   MATLAB  847 Appendix D The Complete Smith Chart   860 Appendix E    Answers to Odd-Numbered Problems   861 Index  889 00_Sadiku_FM.indd 6 16/11/17 3:36 PM
vii BRIEF TABLE OF CONTENTS  vi PREFACE xiii A NOTE TO THE STUDENT   xvii ABOUT THE AUTHOR   xviii MATH ASSESSMENT   MA-1 † Indicates sections that may be skipped, explained briefly, or assigned as homework if the text is covered in one semester. PART 1:   VECTOR ANALYSIS 1 VECTOR ALGEBRA   3 1.1 Introduction  3   † 1.2 A Preview of the Book   4 1.3 Scalars and Vectors  4 1.4 Unit Vector   5 1.5 Vector Addition and Subtraction   6 1.6 Position and Distance Vectors   7 1.7 Vector Multiplication   11 1.8 Components of a Vector   16 Summary  23 Review Questions  24 Problems  25 2 COORDINATE SYSTEMS AND TRANSFORMATION    31 2.1 Introduction   31 2.2 Cartesian Coordinates (x, y, z)  32 2.3 Circular Cylindrical Coordinates (r, f, z)  32 2.4 Spherical Coordinates (r, u, f)  35 2.5 Constant-Coordinate Surfaces   44 Summary  51 Review Questions  52 Problems  54 3 VECTOR CALCULUS   59 3.1 Introduction   59 3.2 Differential Length, Area, and Volume   59 3.3 Line, Surface, and Volume Integrals   66 CONTENTS 00_Sadiku_FM.indd 7 16/11/17 3:36 PM
3.4 Del Operator   69 3.5 Gradient of a Scalar   71 3.6 Divergence of a Vector and Divergence Theorem   75 3.7 Curl of a Vector and Stokes’s Theorem   82 3.8 Laplacian of a Scalar   90 † 3.9 Classification of Vector Fields   92 Summary  97 Review Questions  98 Problems  100 PART 2:  ELECTROSTATICS 4 ELECTROSTATIC FIELDS   111 4.1 Introduction  111 4.2 Coulomb’s Law and Field Intensity   112 4.3 Electric Fields due to Continuous Charge Distributions   119 4.4 Electric Flux Density   130 4.5 Gauss’s Law—Maxwell’s Equation   132 4.6 Applications of Gauss’s Law   134 4.7 Electric Potential   141 4.8 Relationship between E and V—Maxwell’s Equation   147 4.9 An Electric Dipole and Flux Lines   150 4.10 Energy Density in Electrostatic Fields   154 † 4.11 Application Note—Electrostatic Discharge   159 Summary  164 Review Questions  167 Problems  168 5 ELECTRIC FIELDS IN MATERIAL SPACE    177 5.1 Introduction  177 5.2 Properties of Materials   177 5.3 Convection and Conduction Currents   178 5.4 Conductors  181 5.5 Polarization in Dielectrics   187 5.6 Dielectric Constant and Strength   190 † 5.7 Linear, Isotropic, and Homogeneous Dielectrics   191 5.8 Continuity Equation and Relaxation Time   196 5.9 Boundary Conditions   198 † 5.10 Application Note— Materials with High Dielectric Constant   207 5.11 Application Note—Graphene  208 † 5.12 Application Note—Piezoelectrics  210 Summary  214 Review Questions  215 Problems  217 viii CONTENTS 00_Sadiku_FM.indd 8 16/11/17 3:36 PM
6 ELECTROSTATIC BOUNDARY-VALUE PROBLEMS    225 6.1 Introduction  225 6.2 Poisson’s and Laplace’s Equations   225 † 6.3 Uniqueness Theorem   227 6.4 General Procedures for Solving Poisson’s or Laplace’s Equation   228 6.5 Resistance and Capacitance   249 6.6 Method of Images   266 † 6.7 Application Note—Capacitance of Microstrip Lines   272 6.8 Application Note—RF MEMS   275 † 6.9 Application Note—Supercapacitors   276 Summary  280 Review Questions  281 Problems  282 PART 3:  MAGNETOSTATICS 7 MAGNETOSTATIC FIELDS   297 7.1 Introduction   297 7.2 Biot–Savart’s Law   298 7.3 Ampère’s Circuit Law—Maxwell’s Equation   309 7.4 Applications of Ampère’s Law   309 7.5 Magnetic Flux Density—Maxwell’s Equation   317 7.6 Maxwell’s Equations for Static Fields   319 7.7 Magnetic Scalar and Vector Potentials   320 † 7.8 Derivation of Biot–Savart’s Law and Ampère’s Law   326 † 7.9 Application Note—Lightning   328 7.10 Application Note—Polywells   329 Summary  333 Review Questions  335 Problems  338 8 MAGNETIC FORCES, MATERIALS, AND DEVICES    349 8.1 Introduction  349 8.2 Forces due to Magnetic Fields   349 8.3 Magnetic Torque and Moment   361 8.4 A Magnetic Dipole   363 8.5 Magnetization in Materials   368 † 8.6 Classification of Materials   372 8.7 Magnetic Boundary Conditions   376 8.8 Inductors and Inductances   381 8.9 Magnetic Energy   384 † 8.10 Magnetic Circuits   392 † 8.11 Force on Magnetic Materials   394 CONTENTS ix 00_Sadiku_FM.indd 9 16/11/17 3:36 PM
8.12 Application Note—Magnetic Levitation   399 † 8.13 Application Note—SQUIDs   401 Summary  405 Review Questions  407 Problems  409 PART 4:   WAVES AND ­ APPLICATIONS 9 MAXWELL’S EQUATIONS   421 9.1 Introduction  421 9.2 Faraday’s Law   422 9.3 Transformer and Motional Electromotive Forces   424 9.4 Displacement Current   433 9.5 Maxwell’s Equations in Final Forms   436 † 9.6 Time-Varying Potentials   439 9.7 Time-Harmonic Fields   441 † 9.8 Application Note—Memristor 454 † 9.9 Application Note—Optical Nanocircuits  455 † 9.10 Application Note—Wireless Power Transfer and Qi Standard  457 Summary  460 Review Questions  461 Problems  463 10 ELECTROMAGNETIC WAVE PROPAGATION    473 10.1 Introduction  473 † 10.2 Waves in General   474 10.3 Wave Propagation in Lossy Dielectrics   480 10.4 Plane Waves in Lossless Dielectrics   487 10.5 Plane Waves in Free Space   487 10.6 Plane Waves in Good Conductors   489 10.7 Wave Polarization   498 10.8 Power and the Poynting Vector   502 10.9 Reflection of a Plane Wave at Normal Incidence   506 † 10.10 Reflection of a Plane Wave at Oblique Incidence   517 † 10.11 Application Note—Microwaves   529 10.12 Application Note—60 GHz Technology   534 Summary  537 Review Questions  538 Problems  540 11 TRANSMISSION LINES   553 11.1 Introduction   553 11.2 Transmission Line Parameters   554 11.3 Transmission Line Equations   557 x CONTENTS 00_Sadiku_FM.indd 10 16/11/17 3:36 PM
11.4 Input Impedance, Standing Wave Ratio, and Power   564 11.5 The Smith Chart   572 11.6 Some Applications of Transmission Lines   585 † 11.7 Transients on Transmission Lines   592 † 11.8 Application Note—Microstrip Lines and Characterization of Data Cables   604 11.9 Application Note—Metamaterials   612 † 11.10 Application Note—Microwave Imaging   613 Summary  617 Review Questions  618 Problems  621 12 WAVEGUIDES   633 12.1 Introduction  633 12.2 Rectangular Waveguides   634 12.3 Transverse Magnetic Modes   638 12.4 Transverse Electric Modes   643 12.5 Wave Propagation in the Guide   654 12.6 Power Transmission and Attenuation   656 † 12.7 Waveguide Current and Mode Excitation   660 12.8 Waveguide Resonators   666 † 12.9 Application Note—Optical Fiber   672 † 12.10 Application Note—Cloaking and Invisibility   678 Summary  680 Review Questions  682 Problems  683 13 ANTENNAS   691 13.1 Introduction   691 13.2 Hertzian Dipole   693 13.3 Half-Wave Dipole Antenna   697 13.4 Quarter-Wave Monopole Antenna   701 13.5 Small-Loop Antenna   702 13.6 Antenna Characteristics   707 13.7 Antenna Arrays   715 † 13.8   Effective Area and the Friis Equation   725 † 13.9   The Radar Equation   728 † 13.10 Application Note—Electromagnetic Interference and Compatibility  732 † 13.11 Application Note—Textile Antennas and Sensors   737 † 13.12 Application Note—Fractal Antennas   739 13.13 Application Note—RFID   742 Summary  745 Review Questions  746 Problems  747 CONTENTS xi 00_Sadiku_FM.indd 11 16/11/17 3:36 PM
14 NUMERICAL METHODS   757 14.1 Introduction  757 † 14.2 Field Plotting   758 14.3 The Finite Difference Method   766 14.4 The Moment Method   779 14.5 The Finite Element Method   791 † 14.6 Application Note—Microstrip Lines   810 Summary  820 Review Questions  820 Problems  822 APPENDIX A Mathematical Formulas   835 APPENDIX B Material Constants   845 APPENDIX C MATLAB  847 APPENDIX D The Complete Smith Chart   860 APPENDIX E Answers to Odd-Numbered Problems   861 INDEX  889 xii CONTENTS 00_Sadiku_FM.indd 12 16/11/17 3:36 PM
xiii This new edition is intended to provide an introduction to engineering electromagnetics (EM) at the junior or senior level. Although the new edition improves on the previous edi- tions, the core of the subject of EM has not changed. The fundamental objective of the first edition has been retained: to present EM concepts in a clearer and more interesting manner than other texts. This objective is achieved in the following ways: 1. To avoid complicating matters by covering EM and mathematical concepts simultane­ ously, vector analysis is covered at the beginning of the text and applied gradually. This approach avoids breaking in repeatedly with more background on vector analysis, thereby creating discontinuity in the flow of thought. It also separates mathematical theorems from physical concepts and makes it easier for the student to grasp the generality of those theorems. Vector analysis is the backbone of the mathematical formulation of EM problems. 2. Each chapter opens either with a historical profile of some electromagnetic pioneers or with a discussion of a modern topic related to the chapter. The chapter starts with a brief introduction that serves as a guide to the whole chapter and also links the chapter to the rest of the book. The introduction helps the students see the need for the chapter and how it relates to the previous chapter. Key points are emphasized to draw the reader’s attention. A brief summary of the major concepts is discussed toward the end of the chapter. 3. To ensure that students clearly get the gist of the matter, key terms are defined and highlighted. Important formulas are boxed to help students identify essential formulas. 4. Each chapter includes a reasonable amount of solved examples. Since the examples are part of the text, they are clearly explained without asking the reader to fill in missing steps. In writing out the solution, we aim for clarity rather than efficiency. Thoroughly worked out examples give students confidence to solve problems themselves and to learn to apply concepts, which is an integral part of engineering education. Each illustrative example is followed by a problem in the form of a Practice Exercise, with the answer provided. 5. At the end of each chapter are ten review questions in the form of multiple-choice ­ objective items. Open-ended questions, although they are intended to be thought-provoking, are ignored by most students. Objective review questions with answers immediately following them provide encouragement for students to do the problems and gain immediate feedback. A large number of problems are provided and are presented in the same order as the material in the main text. Approximately 20 to 25 percent of the problems in this edition have been replaced. Problems of intermediate difficulty are identified by a single asterisk; the most diffi- cult problems are marked with a double asterisk. Enough problems are provided to allow the instructor to choose some as examples and assign some as homework problems. Answers to odd-numbered problems are provided in Appendix E. 6. Since most practical applications involve time-varying fields, six chapters are devoted to such fields. However, static fields are given proper emphasis because they are special cases of dynamic fields. Ignorance of electrostatics is no longer acceptable PREFACE 00_Sadiku_FM.indd 13 16/11/17 3:36 PM
because there are large industries, such as copier and computer peripheral manufactur- ing, that rely on a clear understanding of electrostatics. 7. The last section in each chapter is devoted to applications of the concepts covered in the chapter. This helps students see how concepts apply to real-life situations. 8. The last chapter covers numerical methods with practical applications and MATLAB programs. This chapter is of paramount importance because most practical prob- lems are only solvable using numerical techniques. Since MATLAB is used throughout the book, an introduction to MATLAB is provided in Appendix C. 9. Over 130 illustrative examples and 300 figures are given in the text. Some additional learning aids such as basic mathematical formulas and identities are included in Appendix A. Another guide is a special note to students, which follows this preface. NEW TO THE SIXTH EDITION •  Five new Application Notes designed to explain the real-world connections between the concepts discussed in the text. •  A revised Math Assessment test, for instructors to gauge their students’ mathematical knowledge and preparedness for the course. •  New and updated end-of-chapter problems. Solutions to the end-of-chapter problems and the Math Assessment, as well as PowerPoint slides of all figures in the text, can be found at the Oxford University Press Ancillary Resource Center. Students and professors can view Application Notes from previous editions of the text on the book’s companion website www.oup.com/us/sadiku. Although this book is intended to be self-explanatory and useful for self-instruction, the personal contact that is always needed in teaching is not forgotten. The actual choice of course topics, as well as emphasis, depends on the preference of the individual instruc- tor. For example, an instructor who feels that too much space is devoted to vector anal- ysis or static fields may skip some of the materials; however, the students may use them as reference. Also, having covered Chapters 1 to 3, it is possible to explore Chapters 9 to 14. Instructors who disagree with the vector-calculus-first approach may proceed with Chapters 1 and 2, then skip to Chapter 4, and refer to Chapter 3 as needed. Enough mate- rial is covered for two-semester courses. If the text is to be covered in one semester, cover- ing Chapters 1 to 9 is recommended; some sections may be skipped, explained briefly, or assigned as homework. Sections marked with the dagger sign (†) may be in this category. ACKNOWLEDGMENTS I thank Dr. Sudarshan Nelatury of Penn State University for providing the new Application Notes and the Math Assessment test. It would not be possible to prepare this edition without the efforts of Executive Editor Dan Kaveney, Associate Editor Christine Mahon, Assistant Editor Megan Carlson, Marketing Manager David Jurman, Marketing Assistant Colleen Rowe, Production Editor Claudia Dukeshire, and Designer Michele Laseau at Oxford University Press, as well as Susan Brown and Betty Pessagno. xiv PREFACE 00_Sadiku_FM.indd 14 16/11/17 3:36 PM
I thank the reviewers who provided helpful feedback for this edition: Mohammadreza (Reza) Barzegaran Lamar University Sudarshan Nelatury Penn State Erie Sharif M. A. Bhuiyan Tuskegee University Sima Noghanian University of North Dakota Muhammad Dawood New Mexico State University Vladimir Rakov University of Florida Robert Gauthier Carleton University Lisa Shatz Suffolk University Jesmin Khan Tuskegee University Kyle Sundqvist Texas AM University Edwin Marengo Northeastern University Lili H. Tabrizi California State University, Los Angeles Perambur S. Neelakanta Florida Atlantic University I also offer thanks to those who reviewed previous editions of the text: Yinchao Chen University of South Carolina Douglas T. Petkie Wright State University Satinderpaul Singh Devgan Tennessee State University James E. Richie Marquette University Dentcho Angelov Genov Louisiana Tech University Elena Semouchkina Michigan Technological University Scott Grenquist Wentworth Institute of Technology Barry Spielman Washington University Xiaomin Jin Cal Poly State University, San Luis Obispo Murat Tanik University of Alabama–Birmingham Jaeyoun Kim Iowa State University Erdem Topsakal Mississippi State University Caicheng Lu University of Kentucky Charles R. Westgate Sr. SUNY–Binghamton Perambur S. Neelakantaswamy Florida Atlantic University Weldon J. Wilson University of Central Oklahoma Kurt E. Oughstun University of Vermont Yan Zhang University of Oklahoma I am grateful to Dr. Kendall Harris, dean of the College of Engineering at Prairie View AM University, and Dr. Pamela Obiomon, head of the Department of Electrical and PREFACE xv 00_Sadiku_FM.indd 15 16/11/17 3:36 PM
xvi PREFACE Computer Engineering, for their constant support. I would like to express my gratitude to Dr. Vichate Ungvichian, at Florida Atlantic University, for pointing out some errors. I acknowledge Emmanuel Shadare for help with the figures. A well-deserved expression of appreciation goes to my wife and our children for their constant support and prayer. I owe special thanks for those professors and students who have used earlier edi- tions of the book. Please keep sending those errors directly to the publisher or to me at sadiku@ieee.org. —Matthew N.O. Sadiku Prairie View, Texas 00_Sadiku_FM.indd 16 16/11/17 3:36 PM
A NOTE TO THE STUDENT Electromagnetic theory is generally regarded by students as one of the most difficult cours- es in physics or the electrical engineering curriculum. But this conception may be proved wrong if you take some precautions. From experience, the following ideas are provided to help you perform to the best of your ability with the aid of this textbook: 1. Pay particular attention to Part 1 on vector analysis, the mathematical tool for this course. Without a clear understanding of this section, you may have problems with the rest of the book. 2. Do not attempt to memorize too many formulas. Memorize only the basic ones, which are usually boxed, and try to derive others from these. Try to understand how formu- las are related. There is nothing like a general formula for solving all problems. Each for- mula has limitations owing to the assumptions made in obtaining it. Be aware of those as- sumptions and use the formula accordingly. 3. Try to identify the key words or terms in a given definition or law. Knowing the meaning of these key words is essential for proper application of the definition or law. 4. Attempt to solve as many problems as you can. Practice is the best way to gain skill. The best way to understand the formulas and assimilate the material is by solving problems. It is recommended that you solve at least the problems in the Practice Exercise immediately following each illustrative example. Sketch a diagram illustrating the problem before ­ attempting to solve it mathematically. Sketching the diagram not only makes the problem easier to solve, but also helps you understand the problem by simplifying and organizing your thinking process. Note that unless otherwise stated, all distances are in meters. For example (2, 1, 5) actually means (2 m, 1 m, 5 m). You may use MATLAB to do number crunching and plotting. A brief introduction to MATLAB is provided in Appendix C. A list of the powers of 10 and Greek letters commonly used throughout this text is provided in the tables located on the inside cover. Important formulas in calculus, vectors, and complex analysis are provided in Appendix A. Answers to odd-numbered problems are in Appendix E. xvii xvii 00_Sadiku_FM.indd 17 16/11/17 3:36 PM
ABOUT THE AUTHOR Matthew N. O. Sadiku received his BSc degree in 1978 from Ahmadu Bello University, Zaria, Nigeria, and his MSc and PhD degrees from Tennessee Technological University, Cookeville, Tennessee, in 1982 and 1984, respectively. From 1984 to 1988, he was an assis- tant professor at Florida Atlantic University, Boca Raton, Florida, where he did graduate work in computer science. From 1988 to 2000, he was at Temple University, Philadelphia, Pennsylvania, where he became a full professor. From 2000 to 2002, he was with Lucent/ Avaya, Holmdel, New Jersey, as a system engineer and with Boeing Satellite Systems, Los Angeles, California, as a senior scientist. He is currently a professor of electrical and com- puter engineering at Prairie View AM University, Prairie View, Texas. He is the author of over 370 professional papers and over 70 books, including Elements of Electromagnetics (Oxford University Press, 7th ed., 2018), Fundamentals of Electric Circuits (McGraw-Hill, 6th ed., 2017, with C. Alexander), Computational Electromagnetics with MATLAB (CRC, 4th ed., 2018), Metropolitan Area Networks (CRC Press, 1995), and Principles of Modern Communication Systems (Cambridge University Press, 2017, with S. O. Agbo). In addition to the engineering books, he has written Christian books including Secrets of Successful Marriages, How to Discover God’s Will for Your Life, and commentaries on all the books of the New Testament Bible. Some of his books have been translated into French, Korean, Chinese (and Chinese Long Form in Taiwan), Italian, Portuguese, and Spanish. He was the recipient of the 2000 McGraw-Hill/Jacob Millman Award for out- standing contributions in the field of electrical engineering. He was also the reci­ pient of Regents Professor award for 2012–2013 by the Texas AM University System. He is a registered professional engineer and a fellow of the Institute of Electrical and Electronics Engineers (IEEE) “for contributions to computational electromagnetics and engineering education.” He was the IEEE Region 2 Student Activities Committee Chairman. He was an associate editor for IEEE Transactions on Education. He is also a member of the Association for Computing Machinery (ACM) and the American Society of Engineering Education (ASEE). His current research interests are in the areas of computational electromagnetics, computer networks, and engineering education. His works can be found in his autobiography, My Life and Work (Trafford Publishing, 2017) or on his website, www.matthewsadiku.com. He currently resides with his wife Kikelomo in Hockley, Texas. He can be reached via email at sadiku@ieee.org. xviii 00_Sadiku_FM.indd 18 16/11/17 3:36 PM
MA-1 MATH ASSESSMENT 1.1 Let u be the angle between the vectors A and B. What can be said about u if (i) |A +B| , |A 2 B|, (ii) |A 1 B| 5 |A 2 B|, (iii) |A 1 B| . |A 2 B|? 1.2 Two sides of a parallelogram ABCD denoted as p = 5ax and q = 3ax + 4ay are shown in Figure MA-1 Let the diagonals intersect at O and make an angle a. Find the coordinates of O and the magnitude of a. Based on the value of a, what can we call ABCD? B O α A p = 5ax C D q = 3ax + 4ay FIGURE MA-1 Parallelogram ABCD. 1.3  What is the distance R between the two points A(3, 5, 1) and B(5, 7, 2)? Also find its reciprocal, 1 R . 1.4  What is the distance vector RAB from A(3, 7, 1) to B(8, 19, 2) and a unit vector aAB in the direction of RAB? 1.5  Find the interval of values x takes so that a unit vector u satisfies |(x 2 2)u| , |3u|. 1.6  There are four charges in space at four points A, B, C, and D, each 1 m from every other. You are asked to make a selection of coordinates for these charges. How do you place them in space and select their coordinates? There is no unique way. 1.7  A man driving a car starts at point O, drives in the following pattern 15 km northeast to point A, 20 km southwest to point B, 25 km north to C, 10 km southeast to D, 15 km west to E, and stops. How far is he from his starting point, and in what direction? 00_Sadiku_FM.indd 1 16/11/17 3:36 PM
1.8  A unit vector an makes angles , , and  with the x-, y-, and z-axes, respectively. Express an in the rectangular coordinate system. Also express a nonunit vector OP of length  parallel to an. 1.9 Three vectors p, q, and r sum to a zero vector and have the magnitude of 10, 11, and 15, respectively. Determine the value of p ? q 1 q ? r 1 r ? p. 1.10  An experiment revealed that the point Q(x, y, z) is 4 m from P(2, 1, 4) and that the vector QP makes 45.5225, 59.4003, and 60 with the x-, y-, and z-axes, respectively. Determine the location of Q. 1.11  In a certain frame of reference with x-, y-, and z-axes, imagine the first octant to be a room with a door. Suppose that the height of the door is h and its width is . The top-right corner P of the door when it is shut has the rectangular coordinates (, 0, h). Now if the door is turned by angle , so we can enter the room, what are the coordinates of P? What is the length of its diagonal r 5 OP in terms of  and z? Suppose the vector OP makes an angle  with the z-axis; express  and h in terms of r and . 1.12  Consider two vectors p 5 OP and q=OQ in Figure MA-2. Express the vector GR in terms of p and q. Assume that /ORQ 5 90°. MA-2 MATH ASSESSMENT P R Q O FIGURE MA-2 Orthogonal projection of one vector over another. 1.13  Consider the equations of two planes: 3x 2 2y 2 z 5 8 2x 1 y 1 4z 5 3 Let them intersect along the straight line . Obtain the coordinates of the points where meets the xy– and the yz–planes. Also determine the angle between and the xz-plane. 1.14 Given two vectors p 5 ax 1 ay and q 5 ay 1 az of equal length, find a third vector r such that it has the same length and the angle between any two of them is 608. 1.15 Given A 5 2xy ax 1 3zy ay 1 5z az and B 5 sinx ax 1 2y ay 1 5y az, find (i)   A, (ii)  3 A, (iii)    3 A, and (iv)   (A 3 B). 00_Sadiku_FM.indd 2 16/11/17 3:36 PM
2.1  A triangular plate of base b 5 5 and height h 5 4 shown in Figure MA-3 is uni- formly charged with a uniform surface charge density rs 5 10 C/m2 . You are to cut a rectangular piece so that maximum amount of charge is taken out. What should be the dimensions x and y of the rectangle? What is the magnitude of the charge extracted out? 2.2  Consider two fixed points F1(2c, 0) and F2(c, 0) in the xy-plane. Show that the locus of a point P(x, y) that satisfies the constraint that the sum PF1 +PF2 remains constant and is equal to 2a is an ellipse. The equipotential loci due to a uniform line charge of length 2c are family of ellipses in the plane containing the charge. This problem helps in proving it. 2.3  Show that the ordinary angle subtended by a closed curve lying in a plane at a point P is 2 radians if P is enclosed by the curve and zero if not. MATH ASSESSMENT MA-3 x y M M′ O A θ F FIGURE MA-3 A rectangular piece cut out from a triangular plate. 2.4  Show that the solid angle subtended by a closed surface at a point P is 4 steradians if P is enclosed by the closed surface and zero if not. 2.5  The electrostatic potential V(r) is known to obey the equation V(r) = 2V (2r) with the boundary condition V(5) 5 3 volts. Determine V(15). 2.6  Evaluate the indefinite integrals (i) 3cosec u du and (ii) 3sec u du. Ignore the arbi- trary constant. 2.7  A liquid drop is in the form of an ellipsoid x2 a2 1 y2 b2 1 z2 c2 5 1 shown in Figure MA-4 and is filled with a charge of nonuniform density ry 5 x2 C/m3 . Find the total charge in the drop. 2.8  Two families of curves are said to be orthogonal to each other if they intersect at 908. Given a family y2 5 cx3 , find the equation for orthogonal trajectories and plot three to four members of each on the same graph. 2.9  Consider a vector given by E 5 (4xy 1 z)ax 1 2x2 ay 1 xaz. Find the line integral from A(3, 7, 1) to B(8, 9, 2) by (i)evaluatingthelineintegralVAB 52 3 B A Edlalong     the line joining A to B and (ii) evaluating e23 C A E # dl 2 3 D C E # dl 2 3 B D E # dlf, where the stopovers C and D are C(8, 7, 1) and D(8, 9, 1). 00_Sadiku_FM.indd 3 16/11/17 3:36 PM
2.10  Find the trigonometric Fourier series of a function f(x) 5 x 1 x2 defined over the interval 2p , x , p. 2.11  In a certain electrostatic system, there are found an infinite set of image point charges. The field intensity at a point may be written as E 5 A a ` m51 a ` n5m (21)(n21) n2 Simplify the double summation. Hint: Integrating the following series term by term and substituting x 5 1 helps in finding the result. 1 1 1 x 5 12x 1 x2 2 x3 1 ... 1 2 ... 2.12  Solve the differential equation d2 V(x) dx2 5 k V(x) subject to the boundary conditions dV dx ` x50 5 0 and V(0) 5 0. Assume that k is a constant. 3.1  The location of a moving charge is given by the time-varying radius vector r 5 2 cos t ax 1 2 sin t ay + 3taz. Describe the trajectory of motion. Find the velocity and acceleration vectors at any instant t. In particular, indicate their directions at the specific instants t 5 0 and t 5 p/2. Find their magnitudes at any instant. 3.2  The magnetic field strength H(z) at a point on the z-axis shown in Figure MA-5 is proportional to the sum of cosine of angles and is given by H 5 k(cos u1 1 cos u2). Find H(0). Also show that if a ≪ ,, H (6,) 1 2 H(0). This helps in finding the magnetic field along the axis of a long solenoid. a x –c y b S –a τ –b c z FIGURE MA-4 A non uniformly charged liquid drop. MA-4 MATH ASSESSMENT 00_Sadiku_FM.indd 4 16/11/17 3:36 PM
3.3 Suppose it is suggested that B 5 r (k 3 r) is the magnetic flux density vector, where k is a constant vector and r 5 rar verify if it is solenoidal. MATH ASSESSMENT MA-5 3.4  Evaluate the line integral I C (x 1 y)dx 1 (x 2 y)dy x 2 1 y 2 where C is the circle x2 1 y2 5 a2 of constant radius a. 3.5 Evaluate the line integral I C xdy 2 ydx x2 1 y2 where C is a closed curve (i) encircling the origin n times, (ii) not enclosing the origin. 4.1 Show that    3 A 5 0. 4.2 Show that  3  5 0. 4.3  Given that the imaginary unit is j 5 21 and that x 5 jj , could the value of x be real? If so, is it unique? Can x have one value in the interval (100, 120)? 4.4 Show that   A 3 B 5 B   3 A 2 A   3 B. 4.5 Use De Moivre's theorem to prove that cos 3 5 cos3  2 3 cos  sin2 : 4.6 Determine j. 4.7 Determine j using the Euler formula. 4.8  Find the phasors for the following field quantities: (a) Ex(z, t) 5 Eo cos (t 2 z 1 ) (V/m) (b) Ey(z, t) 5 100e23z cos (t 2 5z 1 /4) (V/m) (c) Hx(z, t) 5 Ho cos (t 1 z) (A/m) (d) Hy(z, t) 5 120e25z cos (t 1 z 1 h) (A/m) 4.9  Find the instantaneous time domain sinusoidal functions corresponding to the following phasors: (a) Ex(z) 5 Eoejz (V/m) (b) Ey(z) 5 100e23z e2j5z (V/m) (c) Is(z) 5 5 1 j4 (A) (d) Vs(z) 5 j10ej/3 (V) 4.10  Write the phasor expression ~ I for the following current using a cosine reference. (a) i(t) 5 Io cos (t 2 /6) (b) i(t) 5 Io sin (t 1 /3) z z a 0 θ1 θ2 2 a FIGURE MA-5 Toward finding magnetic field along the axis of a solenoid. 00_Sadiku_FM.indd 5 16/11/17 3:36 PM
4.11  In a certain resonant cavity, the resonant modes are described by a triplet of ­nonnegative integers m, n, and p. Find possible solutions under the inequality ­constraints, mn 1 np 1 pm 2 0 13 16 # m2 4 1 n2 9 1 p2 # 5 4 4.12  A voltage source V (t) 5 100 cos (6109 t 2 45) (V) is connected to a series RLC circuit, as shown in Figure MA-6. Given R 5 10M, C 5 100pF, and L 5 1H, use phasor notation to find the following: (a) i(t) (b) Vc (t), the voltage across the capacitor 4.13  (i) Show that the locus of the points P(x, y) obeying the equation x2 1 y2 1 2gx 1 2fy 1 c 5 0 represents a circle. (ii) Express the coordinates of the center and the radius. Use the following equations of circles to find the centers and radii. x2 1 y2 1 8x 2 4y 1 11 5 0 x2 1 y2 2 10x 2 6y 1 9 5 0 225x2 1 225y2 1 90x 2 300y 1 28 5 0 MA-6 MATH ASSESSMENT FIGURE MA-6 A series RLC circuit for Problem 4.12. + + – Vs Vc R jωL 1 jωC – i(t) 4.14 Recall the vector identity  3 A   3 A 1  3 A, where  is a scalar function and A is a vector point function. Suppose A 5 Azaz, where Az 5 e2jkr r and k is a ­constant. Simplify  3 A. 4.15  Between two points A and B on the brink of a circular water pond, a transmission line has to be run. It costs twice the money per meter length to install the cable through the water compared to installation on the edge. One might take the cable (a) completely around the arc on the surrounding land or (b) straight through in the water or (c) partly on the arc and for the remaining, straight in the water. (i) What path costs the maximum money? (ii) Suggest an arrangement that minimizes the cost. With some numerical values, plot the cost function. 4.16  Show the following series expansion assuming |x|  1: 1 11 2 x22 5 1 1 2x 1 3x2 1 4x3 1 . . . 00_Sadiku_FM.indd 6 16/11/17 3:36 PM
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PA R T 1 VECTOR ANALYSIS 01_Sadiku_Ch01.indd 1 01/11/17 1:14 PM
CODES OF ETHICS Engineering is a profession that makes significant contributions to the economic and social ­ well-being of people all over the world. As members of this important profession, engineers are expected to exhibit the highest standards of honesty and integrity. Unfortu­ nately, the engineering curriculum is so crowded that there is no room for a course on ethics in most schools. Although there are over 850 codes of ethics for different professions all over the world, the code of ethics of the Institute of Electrical and Electronics Engineers (IEEE) is presented here to give students a flavor of the importance of ethics in engineer­ ing professions. We, the members of the IEEE, in recognition of the importance of our technologies in affecting the quality of life throughout the world, and in accepting a personal obligation to our profession, its members and the communities we serve, do hereby commit ourselves to the highest ethical and professional conduct and agree: 1.  to accept responsibility in making engineering decisions consistent with the safety, health, and welfare of the public, and to disclose promptly factors that might endanger the public or the environment; 2.  to avoid real or perceived conflicts of interest whenever possible, and to disclose them to affected parties when they do exist; 3. to be honest and realistic in stating claims or estimates based on available data; 4. to reject bribery in all its forms; 5.  to improve the understanding of technology, its appropriate application, and po­ tential consequences; 6.  to maintain and improve our technical competence and to undertake technologi­ cal tasks for others only if qualified by training or experience, or after full disclo­ sure of pertinent limitations; 7.  to seek, accept, and offer honest criticism of technical work, to acknowledge and correct errors, and to credit properly the contributions of others; 8.  to treat fairly all persons regardless of such factors as race, religion, gender, ­ dis­ ability, age, or national origin; 9.  to avoid injuring others, their property, reputation, or employment by false or ma­ licious action; 10.  to assist colleagues and co-workers in their professional development and to ­ sup­ port them in following this code of ethics. —Courtesy of IEEE 01_Sadiku_Ch01.indd 2 01/11/17 1:14 PM
3 C H A P T E R 3 1.1 INTRODUCTION Electromagnetics (EM) may be regarded as the study of the interactions between electric charges at rest and in motion. It entails the analysis, synthesis, physical interpretation, and application of electric and magnetic fields. Electromagnetics (EM) is a branch of physics or electrical engineering in which electric and magnetic phenomena are studied. EM principles find applications in various allied disciplines such as microwaves, antennas, electric machines, satellite communications, bioelectromagnetics, plasmas, nuclear research, fiber optics, electromagnetic interference and compatibility, electromechanical energy conver- sion, radar meteorology, and remote sensing.1,2 In physical medicine, for example, EM power, in the form either of shortwaves or microwaves, is used to heat deep tissues and to stimulate certain physiological responses in order to relieve certain pathological conditions. EM fields are used in induction heaters for melting, forging, annealing, surface hardening, and soldering operations. Dielectric heating equipment uses shortwaves to join or seal thin sheets of plastic materials. EM energy offers many new and exciting possibilities in agriculture. It is used, for example, to change vegetable taste by reducing acidity. EM devices include transformers, electric relays, radio/TV, telephones, electric motors, transmission lines, waveguides, antennas, optical fibers, radars, and lasers. The design of these devices requires thorough knowledge of the laws and principles of EM. 1 For numerous applications of electrostatics, see J. M. Crowley, Fundamentals of Applied Electrostatics. New York: John Wiley Sons, 1986. 2 For other areas of applications of EM, see, for example, D. Teplitz, ed., Electromagnetism: Paths to Research. New York: Plenum Press, 1982. VECTOR ALGEBRA Books are the quietest and most constant friends; they are the most accessible and wisest of counselors, and most patient of teachers. —CHARLES W. ELLIOT 1 01_Sadiku_Ch01.indd 3 01/11/17 1:14 PM
4 CHAPTER 1 VECTOR ALGEBRA The subject of electromagnetic phenomena in this book can be summarized in Maxwell’s equations: = # D 5 rv (1.1) = # B 5 0 (1.2) = 3 E 5 2 'B 't (1.3) = 3 H 5 J 1 'D 't (1.4) where = 5 the vector differential operator D 5 the electric flux density B 5 the magnetic flux density E 5 the electric field intensity H 5 the magnetic field intensity v 5 the volume charge density J 5 the current density Maxwell based these equations on previously known results, both experimental and theore­ tical. A quick look at these equations shows that we shall be dealing with vector quantities. It is consequently logical that we spend some time in Part 1 examining the mathematical tools required for this course. The derivation of eqs. (1.1) to (1.4) for time-invariant conditions and the physical significance of the quantities D, B, E, H, J, and v will be our aim in Parts 2 and 3. In Part 4, we shall reexamine the equations for time-varying situations and apply them in our study of practical EM devices such as transmission lines, waveguides, antennas, fiber optics, and radar systems. † 1.2 A PREVIEW OF THE BOOK 1.3 SCALARS AND VECTORS Vector analysis is a mathematical tool with which electromagnetic concepts are most con- veniently expressed and best comprehended. We must learn its rules and techniques before we can confidently apply it. Since most students taking this course have little exposure to vector analysis, considerable attention is given to it in this and the next two chapters.3 This chapter introduces the basic concepts of vector algebra in Cartesian coordinates only. The next chapter builds on this and extends to other coordinate systems. A quantity can be either a scalar or a vector. A scalar is a quantity that is completely specified by its magnitude. † Indicates sections that may be skipped, explained briefly, or assigned as homework if the text is covered in one semester. 3 The reader who feels no need for review of vector algebra can skip to the next chapter. 01_Sadiku_Ch01.indd 4 01/11/17 1:14 PM
1.4 Unit Vector 5 A scalar is a quantity that has only magnitude. Quantities such as time, mass, distance, temperature, entropy, electric potential, and popu- lation are scalars. A vector has not only magnitude, but direction in space. A vector is a quantity that is described by both magnitude and direction. Vector quantities include velocity, force, momentum, acceleration displacement, and electric field intensity. Another class of physical quantities is called tensors, of which scalars and vectors are special cases. For most of the time, we shall be concerned with scalars and vectors.4 To distinguish between a scalar and a vector it is customary to represent a vector by a letter with an arrow on top of it, such as A and B , or by a letter in boldface type such as A and B. A scalar is represented simply by a letter—for example, A, B, U, and V. EM theory is essentially a study of some particular fields. A field is a function that specifies a particular quantity everywhere in a region. A field may indicate variation of a quantity throughout space and perhaps with time. If the quantity is scalar (or vector), the field is said to be a scalar (or vector) field. Examples of scalar fields are temperature distribution in a building, sound intensity in a theater, electric potential in a region, and refractive index of a stratified medium. The gravitational force on a body in space and the velocity of raindrops in the atmosphere are examples of vector fields. 1.4 UNIT VECTOR A vector A has both magnitude and direction. The magnitude of A is a scalar written as A or 0 A 0 . A unit vector aA along A is defined as a vector whose magnitude is unity (i.e., 1) and its direction is along A; that is, aA 5 A 0 A 0 5 A A (1.5) Note that 0 aA 0 5 1. Thus we may write A as A 5 AaA (1.6) which completely specifies A in terms of its magnitude A and its direction aA. A vector A in Cartesian (or rectangular) coordinates may be represented as 1Ax, Ay, Az 2    or    Axax 1 Ayay 1 Azaz (1.7) 4 For an elementary treatment of tensors, see, for example, A. I. Borisenko and I. E. Tarapor, Vector and Tensor Analysis with Applications. New York: Dover, 1979. 01_Sadiku_Ch01.indd 5 01/11/17 1:14 PM
6 CHAPTER 1 VECTOR ALGEBRA where Ax, Ay, and Az are called the components of A in the x-, y-, and z-directions, respec- tively; ax, ay, and az are unit vectors in the x-, y-, and z-directions, respectively. For example, ax is a dimensionless vector of magnitude one in the direction of the increase of the x-axis. The unit vectors ax, ay, and az are illustrated in Figure 1.1(a), and the components of A along the coordinate axes are shown in Figure 1.1(b). The magnitude of vector A is given by A 5 Ax 2 1 Ay 2 1 Az 2 (1.8) and the unit vector along A is given by aA 5 Axax 1 Ayay 1 Azaz Ax 2 1 Ay 2 1 Az 2 (1.9) FIGURE 1.1 (a) Unit vectors ax, ay, and az, (b) components of A along ax, ay, and az. 1.5 VECTOR ADDITION AND SUBTRACTION Two vectors A and B can be added together to give another vector C; that is, C 5 A 1 B (1.10) The vector addition is carried out component by component. Thus, if A 5 1Ax, Ay, Az) and B 5 1Bx, By, Bz). C 5 1Ax 1 Bx 2ax 1 1Ay 1 By 2ay 1 1Az 1 Bz 2az (1.11) Vector subtraction is similarly carried out as D 5 A 2 B 5 A 1 12B2 5 1Ax 2 Bx 2ax 1 1Ay 2 By 2ay 1 1Az 2 Bz 2az (1.12) 01_Sadiku_Ch01.indd 6 01/11/17 1:14 PM
1.6 Position and Distance Vectors 7 Graphically, vector addition and subtraction are obtained by either the parallelogram rule or the head-to-tail rule as portrayed in Figures 1.2 and 1.3, respectively. The three basic laws of algebra obeyed by any given vectors A, B, and C are summa- rized as follows: Law Addition Multiplication Commutative A 1 B 5 B 1 A kA 5 Ak Associative A 1 1B 1 C2 5 1A 1 B2 1 C k(,A) 5 (k,)A Distributive k1A 1 B2 5 kA 1 kB where k and , are scalars. Multiplication of a vector with another vector will be discussed in Section 1.7. FIGURE 1.3 Vector subtraction D 5 A 2 B: (a) parallelogram rule, (b) ­ head-to-tail rule. FIGURE 1.2 Vector addition C 5 A 1 B: (a) parallelogram rule, (b) head-to-tail rule. 1.6 POSITION AND DISTANCE VECTORS A point P in Cartesian coordinates may be represented by (x, y, z). The position vector rP (or radius vector) of point P is defined as the directed dis- tance from the origin O to P; that is, rP 5 OP 5 xax 1 yay 1 zaz (1.13) 01_Sadiku_Ch01.indd 7 01/11/17 1:14 PM
8 CHAPTER 1 VECTOR ALGEBRA The position vector of point P is useful in defining its position in space. Point (3, 4, 5), for example, and its position vector 3ax 1 4ay 1 5az are shown in Figure 1.4. The distance vector is the displacement from one point to another. If two points P and Q are given by (xP, yP, zP) and (xQ, yQ, zQ), the distance vector (or separation vector) is the displacement from P to Q as shown in Figure 1.5; that is, rPQ 5 rQ 2 rP 5 1xQ 2 xP 2ax 1 1yQ 2 yP 2ay 1 1zQ 2 zP 2az (1.14) The difference between a point P and a vector A should be noted. Though both P and A may be represented in the same manner as (x, y, z) and (Ax, Ay, Az), respectively, the point P is not a vector; only its position vector rP is a vector. Vector A may depend on point P, however. For example, if A 5 2xyax 1 y2 ay 2 xz2 az and P is 12, 21, 42, then A at P would be 24ax 1 ay 2 32az. A vector field is said to be constant or uniform if it does not depend on space variables x, y, and z. For example, vector B 5 3ax 2 2ay 1 10az is a uniform vector while vector A 5 2xyax 1 y2 ay 2 xz2 az is not uniform because B is the same everywhere, whereas A varies from point to point. O FIGURE 1.4 Illustration of position vector rP 5 3ax 1 4ay 5 5az. FIGURE 1.5 Distance vector rPQ. EXAMPLE 1.1 If A 5 10ax 2 4ay 1 6az and B 5 2ax 1 ay, find (a) the component of A along ay, (b) the magnitude of 3A 2 B, (c) a unit vector along A 1 2B. 01_Sadiku_Ch01.indd 8 01/11/17 1:14 PM
1.6 Position and Distance Vectors 9 Solution: (a) The component of A along ay is Ay 5 24. (b) 3A 2 B 5 3110, 24, 62 2 12, 1, 02 5 130, 212, 182 2 12, 1, 02 5 128, 213, 182 Hence, 0 3A 2 B 0 5 282 1 121322 1 11822 5 1277 5 35.74 (c) Let C 5 A 1 2B 5 110, 24, 62 1 14, 2, 02 5 114, 22, 62. A unit vector along C is ac 5 C 0 C 0 5 114, 22, 62 142 1 12222 1 62 or ac 5 0.9113ax 2 0.1302ay 1 0.3906az Note that 0 ac 0 5 1 as expected. PRACTICE EXERCISE 1.1 Given vectors A 5 ax 1 3az and B 5 5ax 1 2ay 2 6az, determine (a) uA 1 Bu (b) 5A 2 B (c) The component of A along ay (d) A unit vector parallel to 3A 1 B Answer: (a) 7, (b) (0, 22, 21), (c) 0, (d) 6(0.9117, 0.2279, 0.3419). Points P and Q are located at (0, 2, 4) and 123, 1, 52. Calculate (a) The position of vector rP (b) The distance vector from P to Q (c) The distance between P and Q (d) A vector parallel to PQ with magnitude of 10 EXAMPLE 1.2 01_Sadiku_Ch01.indd 9 01/11/17 1:14 PM
10 CHAPTER 1 VECTOR ALGEBRA Solution: (a) rP 5 0ax 1 2ay 1 4az 5 2ay 1 4az (b)  rPQ 5 rQ 2 rP 5 123, 1, 52 2 10, 2, 42 5 123, 21, 12 or rPQ 5 23ax 2 ay 1 az (c) Since rPQ is the distance vector from P to Q, the distance between P and Q is the mag- nitude of this vector; that is, d 5 0 rPQ 0 5 9 1 1 1 1 5 3.317 Alternatively: d 5 1xQ 2 xP 22 1 1yQ 2 yP 22 1 1zQ 2 zP 22 5 9 1 1 1 1 5 3.317 (d) Let the required vector be A, then A 5 AaA where A 5 10 is the magnitude of A. Since A is parallel to PQ, it must have the same unit vector as rPQ or rQP. Hence, aA 5 6 rPQ 0 rPQ 0 5 6 123, 21, 12 3.317 and A 5 6 10123, 21, 12 3.317 5 6129.045ax 2 3.015ay 1 3.015az 2 PRACTICE EXERCISE 1.2 Given points P(1, 23, 5), Q(2, 4, 6), and R(0, 3, 8), find (a) the position vectors of P and R, (b) the distance vector rQR, (c) the distance between Q and R. Answer: (a) ax 2 3ay 1 5az, 3ax 1 8az, (b) 22ax 2 ay 1 2az, (c) 3. A river flows southeast at 10 km/hr and a boat floats upon it with its bow pointed in the direction of travel. A man walks upon the deck at 2 km/hr in a direction to the right and perpendicular to the direction of the boat’s movement. Find the velocity of the man with respect to the earth. Solution: Consider Figure 1.6 as illustrating the problem. The velocity of the boat is 5 7.071ax 2 7.071ay km/hr EXAMPLE 1.3 01_Sadiku_Ch01.indd 10 01/11/17 1:14 PM ub 5 101cos 45° ax 2 sin 45° ay 2
1.7 Vector Multiplication 11 PRACTICE EXERCISE 1.3 An airplane has a ground speed of 350 km/hr in the direction due west. If there is a wind blowing northwest at 40 km/hr, calculate the true air speed and heading of the airplane. Answer: 379.3 km/hr, 4.275° north of west. FIGURE 1.6 For Example 1.3. 1.7 VECTOR MULTIPLICATION When two vectors A and B are multiplied, the result is either a scalar or a vector depending on how they are multiplied. Thus there are two types of vector multiplication: 1. Scalar (or dot) product: A # B 2. Vector (or cross) product: A 3 B 01_Sadiku_Ch01.indd 11 01/11/17 1:14 PM t uab 5 um 1 ub 5 5.657ax 2 8.485ay 0 uab 0 5 10.2l256.3° i ii that is, 10.2 km/hr at 56.3° south of east. The velocity of the man with respect to the boat (relative velocity) is um 5 212 cos 45° ax 2 sin 45° ay 2 5 21.414ax 2 1.414ay km/hr Thus the absolute velocity of the man is
12 CHAPTER 1 VECTOR ALGEBRA Multiplication of three vectors A, B, and C can result in either: 3. Scalar triple product: A # 1B 3 C2 or 4. Vector triple product: A 3 1B 3 C2 A. Dot Product The dot product of two vectors A and B, written as A ? B, is defined geometrically as the product of the magnitudes of A and B and the cosine of the smaller angle between them when they are drawn tail to tail. Thus, A # B 5 AB cos uAB (1.15) where uAB is the smaller angle between A and B. The result of A # B is called either the scalar product because it is scalar or the dot product due to the dot sign. If A 5 1Ax, Ay, Az 2 and B 5 1Bx, By, Bz), then A # B 5 AxBx 1 AyBy 1 AzBz (1.16) which is obtained by multiplying A and B component by component. Two vectors A and B are said to be orthogonal (or perpendicular) with each other if A # B 5 0. Note that dot product obeys the following: (i) Commutative law: A # B 5 B # A (1.17) (ii) Distributive law: A # 1B 1 C2 5 A # B 1 A # C (1.18) (iii) A # A 5 0 A 0 2 5 A2 (1.19) Also note that ax # ay 5 ay # az 5 az # ax 5 0 (1.20a) ax # ax 5 ay # ay 5 az # az 5 1 (1.20b) It is easy to prove the identities in eqs. (1.17) to (1.20) by applying eq. (1.15) or (1.16). If A # B 5 0, the two vectors A and B are orthogonal or perpendicular. 01_Sadiku_Ch01.indd 12 01/11/17 1:14 PM
1.7 Vector Multiplication 13 B. Cross Product The cross product of two vectors A and B, written as A 3 B, is a vector quantity whose magnitude is the area of the parallelogram formed by A and B (see Figure 1.7) and is in the direction of advance of a right-handed screw as A is turned into B. Thus, A 3 B 5 AB sin uABan (1.21) where an is a unit vector normal to the plane containing A and B. The direction of an is taken as the direction of the right thumb when the fingers of the right hand rotate from A to B as shown in Figure 1.8(a). Alternatively, the direction of an is taken as that of the advance of a right-handed screw as A is turned into B as shown in Figure 1.8(b). The vector multiplication of eq. (1.21) is called cross product owing to the cross sign; it is also called vector product because the result is a vector. If A 5 1Ax, Ay, Az) and B 5 1Bx, By, Bz), then A 3 B 5 3 ax ay az Ax Ay Az Bx By Bz 3 (1.22a) 5 1AyBz 2 AzBy 2ax 1 1AzBx 2 AxBz 2ay 1 1AxBy 2 AyBx 2az (1.22b) which is obtained by “crossing” terms in cyclic permutation, hence the name “cross product.” A A B B FIGURE 1.7 The cross product of A and B is a vector with magnitude equal to the area of the parallelogram and direction as indicated. 01_Sadiku_Ch01.indd 13 01/11/17 1:14 PM
14 CHAPTER 1 VECTOR ALGEBRA Note that the cross product has the following basic properties: (i) It is not commutative: A 3 B 2 B 3 A (1.23a) It is anticommutative: A 3 B 5 2B 3 A (1.23b) (ii) It is not associative: A 3 1B 3 C2 2 1A 3 B2 3 C (1.24) (iii) It is distributive: A 3 1B 1 C2 5 A 3 B 1 A 3 C (1.25) (iv) Scaling: kA 3 B 5 A 3 kB 5 k1A 3 B2 (1.26) (v) A 3 A 5 0 (1.27) Also note that ax 3 ay 5 az ay 3 az 5 ax (1.28) az 3 ax 5 ay which are obtained in cyclic permutation and illustrated in Figure 1.9. The identities in eqs. (1.23) to (1.28) are easily verified by using eq. (1.21) or (1.22). It should be noted that in obtaining an, we have used the right-hand or right-handed-screw rule because we want to FIGURE 1.8 Direction of A 3 B and an using (a) the right-hand rule and (b) the right-handed-screw rule. 01_Sadiku_Ch01.indd 14 01/11/17 1:14 PM
1.7 Vector Multiplication 15 be consistent with our coordinate system illustrated in Figure 1.1, which is right-handed. A right-handed coordinate system is one in which the right-hand rule is satisfied: that is, ax 3 ay 5 az is obeyed. In a left-handed system, we follow the left-hand or left-handed screw rule and ax 3 ay 5 2az is satisfied. Throughout this book, we shall stick to right- handed coordinate systems. Just as multiplication of two vectors gives a scalar or vector result, multiplication of three vectors A, B, and C gives a scalar or vector result, depending on how the vectors are multiplied. Thus we have a scalar or vector triple product. C. Scalar Triple Product Given three vectors A, B, and C, we define the scalar triple product as A # 1B 3 C2 5 B # 1C 3 A2 5 C # 1A 3 B2 (1.29) obtained in cyclic permutation. If A 5 1Ax, Ay, Az), B 5 1Bx, By, Bz), and C 5 1Cx, Cy, Cz), then A # 1B 3 C2 is the volume of a parallelepiped having A, B, and C as edges and is easily obtained by finding the determinant of the 3 3 3 matrix formed by A, B, and C; that is, A # 1B 3 C2 5 3 Ax Ay Az Bx By Bz Cx Cy Cz 3 (1.30) Since the result of this vector multiplication is scalar, eq. (1.29) or (1.30) is called the scalar triple product. D. Vector Triple Product For vectors A, B, and C, we define the vector triple product as A 3 1B 3 C2 5 B1A # C2 2 C1A # B2 (1.31) FIGURE 1.9 Cross product using cyclic permutation. (a) Moving clockwise leads to positive results. (b) Moving counterclockwise leads to negative results. 01_Sadiku_Ch01.indd 15 01/11/17 1:14 PM
16 CHAPTER 1 VECTOR ALGEBRA which may be remembered as the “bac-cab” rule. It should be noted that 1A # B2C 2 A1B # C2 (1.32) but 1A # B2C 5 C1A # B2 (1.33) 1.8 COMPONENTS OF A VECTOR A direct application of scalar product is its use in determining the projection (or compo- nent) of a vector in a given direction. The projection can be scalar or vector. Given a vector A, we define the scalar component AB of A along vector B as [see Figure 1.10(a)] AB 5 A cos uAB 5 0 A 0 0 aB 0 cos uAB or AB 5 A # aB (1.34) The vector component AB of A along B is simply the scalar component in eq. (1.34) multi- plied by a unit vector along B; that is, AB 5 ABaB 5 1A # aB 2aB (1.35) Both the scalar and vector components of A are illustrated in Figure 1.10. Notice from Figure 1.10(b) that the vector can be resolved into two orthogonal components: one ­ component AB par- allel to B, another 1A 2 AB 2 perpendicular to B. In fact, our Cartesian representation of a vector is essentially resolving the vector into three mutually orthogonal components as in Figure 1.1(b). We have considered addition, subtraction, and multiplication of vectors. However, divi- sion of vectors A/B has not been considered because it is undefined except when A and B are parallel so that A 5 kB, where k is a constant. Differentiation and integration of vectors will be considered in Chapter 3. FIGURE 1.10 Components of A along B: (a) scalar component AB, (b) vector component AB. 01_Sadiku_Ch01.indd 16 01/11/17 1:14 PM
1.8 Components of a Vector 17 Given vectors A 5 3ax 1 4ay 1 az and B 5 2ay 2 5az, find the angle between A and B. Solution: The angle uAB can be found by using either dot product or cross product. A # B 5 13, 4, 12 # 10, 2, 252 5 0 1 8 2 5 5 3 0 A 0 5 32 1 42 1 12 5 26 0 B 0 5 02 1 22 1 12522 5 29 cos uAB 5 A # B 0 A 0 0 B 0 5 3 1262 1292 5 0.1092 uAB 5 cos21 Alternatively: A 3 B 5 3 ax ay az 3 4 1 0 2 25 3 5 1220 2 22ax 1 10 1 152ay 1 16 2 02az 5 1222, 15, 62 0 A 3 B 0 5 122222 1 152 1 62 5 745 sin uAB 5 0 A 3 B 0 0 A 0 0 B 0 5 745 1262 1292 5 0.994 uAB 5 sin21 0.994 5 83.73° PRACTICE EXERCISE 1.4 If A 5 ax 1 3az and B 5 5ax 1 2ay 2 6az, find uAB. Answer: 120.6°. Three field quantities are given by P 5 2ax 2 az Q 5 2ax 2 ay 1 2az R 5 2ax 2 3ay 1 az Determine (a) 1P 1 Q2 3 1P 2 Q2 (b) Q # R 3 P (c) P # Q 3 R EXAMPLE 1.4 EXAMPLE 1.5 01_Sadiku_Ch01.indd 17 01/11/17 1:14 PM 0.1092 5 83.73°
18 CHAPTER 1 VECTOR ALGEBRA (d) sin uQR (e) P 3 1Q 3 R2 (f) A unit vector perpendicular to both Q and R (g) The component of P along Q Solution: (a) 1P 1 Q2 3 1P 2 Q2 5 P 3 1P 2 Q2 1 Q 3 1P 2 Q2 5 P 3 P 2 P 3 Q 1 Q 3 P 2 Q 3 Q 5 0 1 Q 3 P 1 Q 3 P 2 0 5 2Q 3 P 5 2 3 ax ay az 2 21 2 2 0 21 3 5 211 2 02 ax 1 214 1 22 ay 1 210 1 22 az 5 2ax 1 12ay 1 4az (b) The only way Q # R 3 P makes sense is Q # 1R 3 P2 5 12, 21, 22 # 3 ax ay az 2 23 1 2 0 21 3 5 12, 21, 22 # 13, 4, 62 5 6 2 4 1 12 5 14 Alternatively: Q # 1R 3 P2 5 2 21 2 2 23 1 2 0 21 01_Sadiku_Ch01.indd 18 01/11/17 1:14 PM 3 3 To find the determinant of a 3 3 3 matrix, we repeat the first two rows and cross multiply; when the cross multiplication is from right to left, the result should be negated as shown diagrammatically here. This technique of finding a determinant applies only to a 3 3 3 matrix. Hence, Q # 1R 3 P2 5 5 2 21 2 2 23 1 2 0 21 2 21 2 2 23 1 5 5 16 1 0 2 2 1 12 2 0 2 2 5 14 as obtained before.
1.8 Components of a Vector 19 (c) From eq. (1.29) P # 1Q 3 R2 5 Q # 1R 3 P2 5 14 or P # 1Q 3 R2 5 12, 0, 212 # 15, 2, 242 5 10 1 0 1 4 5 14 (d) sin uQR 5 0 Q 3 R 0 0 Q 0 0 R 0 5 15, 2, 242 0 0 12, 21, 22 0 0 12, 23, 12 0 5 45 314 5 5 14 5 0.5976 (e) P 3 1Q 3 R2 5 12, 0, 212 3 15, 2, 242 5 12, 3, 42 Alternatively, using the bac-cab rule, P 3 1Q 3 R2 5 Q1P # R2 2 R1P # Q2 5 12, 21, 22 14 1 0 2 12 2 12, 23, 12 14 1 0 2 22 5 12, 3, 42 (f) A unit vector perpendicular to both Q and R is given by a 5 6Q 3 R 0 Q 3 R 0 5 615, 2, 242 45 5 610.745, 0.298, 20.5962 Note that 0 a 0 5 1, a # Q 5 0 5 a # R. Any of these can be used to check a. (g) The component of P along Q is PQ 5 0 P 0 cos uPQaQ 5 1P # aQ 2aQ 5 1P # Q2Q 0 Q 0 2 5 14 1 0 2 22 12, 21, 22 14 1 1 1 42 5 2 9 12, 21, 22 5 0.4444ax 2 0.2222ay 1 0.4444a 01_Sadiku_Ch01.indd 19 01/11/17 1:14 PM
20 CHAPTER 1 VECTOR ALGEBRA PRACTICE EXERCISE 1.5 Let E 5 3ay 1 4az and F 5 4ax 2 10ay 1 5az. (a) Find the component of E along F. (b) Determine a unit vector perpendicular to both E and F. Answer: (a) (20.2837, 0.7092, 20.3546), (b) 6(0.9398, 0.2734, 20.205). Derive the cosine formula a2 5 b2 1 c2 2 2bc cos A and the sine formula sin A a 5 sin B b 5 sin C c using dot product and cross product, respectively. Solution: Consider a triangle as shown in Figure 1.11. From the figure, we notice that a 1 b 1 c 5 0 that is, b 1 c 5 2a Hence, a2 5 a # a 5 1b 1 c2 # 1b 1 c2 5 b # b 1 c # c 1 2b # c a2 5 b2 1 c2 2 2bc cos A where (p 2 A) is the angle between b and c. The area of a triangle is half of the product of its height and base. Hence, ` 1 2 a 3 b` 5 ` 1 2 b 3 c` 5 ` 1 2 c 3 a` ab sin C 5 bc sin A 5 ca sin B Dividing through by abc gives sin A a 5 sin B b 5 sin C c EXAMPLE 1.6 01_Sadiku_Ch01.indd 20 01/11/17 1:14 PM
1.8 Components of a Vector 21 EXAMPLE 1.7 FIGURE 1.11 For Example 1.6. PRACTICE EXERCISE 1.6 Show that vectors a 5 (4, 0, 21), b 5 (1, 3, 4), and c 5 (25, 23, 23) form the sides of a triangle. Is this a right angle triangle? Calculate the area of the triangle. Answer: Yes, 10.5. Show that points P1 15, 2, 242, P2 11, 1, 22, and P3 123, 0, 82 all lie on a straight line. Determine the shortest distance between the line and point P4 13, 21, 02. Solution: The distance vector rP1P2 is given by rP1P2 5 rP2 2 rP1 5 11, 1, 22 2 15, 2, 242 5 124, 21, 62 Similarly, rP1P3 5 rP3 2 rP1 5 123, 0, 82 2 15, 2, 242 5 128, 22, 122 rP1P4 5 rP4 2 rP1 5 13, 21, 02 2 15, 2, 242 5 122, 23, 42 rP1P2 3 rP1P3 5 3 ax ay az 24 21 6 28 22 12 3 5 10, 0, 02 showing that the angle between rP1P2 and rP1P3 is zero 1sin u 5 02. This implies that P1, P2, and P3 lie on a straight line. Alternatively, the vector equation of the straight line is easily determined from Figure 1.12(a). For any point P on the line joining P1 and P2 rP1P 5 lrP1P2 where λ is a constant. Hence the position vector rP of the point P must satisfy rP 2 rP1 5 l1rP2 2 rP1 2 01_Sadiku_Ch01.indd 21 01/11/17 1:14 PM
22 CHAPTER 1 VECTOR ALGEBRA that is, rP 5 rP1 1 l1rP2 2 rP1 2 5 15, 2, 242 2 l14, 1, 262 rP 5 15 2 4l, 2 2 l, 24 1 6l2 This is the vector equation of the straight line joining P1 and P2. If P3 is on this line, the position vector of P3 must satisfy the equation; r3 does satisfy the equation when l 5 2. The shortest distance between the line and point P4 13, 21, 02 is the perpendicular distance from the point to the line. From Figure 1.12(b), it is clear that d 5 rP1P4 sin u 5 0 rP1P4 3 aP1P2 0 5 0 122, 23, 42 3 124, 21, 62 0 0 124, 21, 62 0 5 312 53 5 2.426 Any point on the line may be used as a reference point. Thus, instead of using P1 as a reference point, we could use P3. If jP4P3 P2 5 , then d 5 0 rP3P4 0 sin ur 5 0 rP3P4 3 aP3P2 0 PRACTICE EXERCISE 1.7 If P1 is (1, 2, 23) and P2 is (24, 0, 5), find (a) The distance P1P2 (b) The vector equation of the line P1P2 (c) The shortest distance between the line P1P2 and point P3 (7, 21, 2) Answer: (a) 9.644, (b) (1 2 5l)ax 1 2(1 2 l)ay 1 (8l 2 3)az, (c) 8.2. FIGURE 1.12 For Example 1.7. 01_Sadiku_Ch01.indd 22 20/11/17 2:38 PM
Summary 23 1. A field is a function that specifies a quantity in space. For example, A(x, y, z) is a vector field, whereas V(x, y, z) is a scalar field. 2. AvectorAisuniquelyspecifiedbyitsmagnitudeandaunitvectoralongit,thatis,A 5 AaA. 3. Multiplying two vectors A and B results in either a scalar A # B 5 AB cos uAB or a vector A 3 B 5 AB sin uAB an. Multiplying three vectors A, B, and C yields a scalar A # 1B 3 C2 or a vector A 3 1B 3 C2. 4. The scalar projection (or component) of vector A onto B is AB 5 A # aB, whereas vector projection of A onto B is AB 5 ABaB. 5. The MATLAB commands dot(A,B) and cross(A,B) are used for dot and cross products, respectively. % This script allows the user to input two vectors and % then compute their dot product, cross product, sum, % and difference clear vA = input(‵Enter vector A in the format [x y z]... n ‵); if isempty(vA); vA = [0 0 0]; end % if the input is % entered incorrectly set the vector to 0 vB = input(‵Enter vector B in the format [x y z]... n ‵); if isempty(vB); vB = [0 0 0]; end disp(‵Magnitude of A:’) disp(norm(vA)) % norm finds the magnitude of a % multi-dimensional vector disp(‵Magnitude of B:’) disp(norm(vB)) disp(‵Unit vector in direction of A:’) disp(vA/norm(vA)) % unit vector is the vector % divided by its magnitude disp(‵Unit vector in direction of B:’) disp(vB/norm(vB)) disp(‵Sum A+B:’) disp(vA+vB) disp(‵Difference A-B:’) disp(vA-vB) disp(‵Dot product (A . B):’) disp(dot(vA,vB)) % dot takes the dot product of vectors disp(‵Cross product (A x B):’) disp(cross(vA,vB)) % cross takes cross product of vectors MATLAB 1.1 SUMMARY 01_Sadiku_Ch01.indd 23 16/11/17 3:00 PM
24 CHAPTER 1 VECTOR ALGEBRA 1.1 Tell which of the following quantities is not a vector: (a) force, (b) momentum, (c) accelera- tion, (d) work, (e) weight. 1.2 Which of the following is not a scalar field? (a) Displacement of a mosquito in space (b) Light intensity in a drawing room (c) Temperature distribution in your classroom (d) Atmospheric pressure in a given region (e) Humidity of a city 1.3 Of the rectangular coordinate systems shown in Figure 1.13, which are not right handed? 1.4 Which of these is correct? (a) A 3 A 5 0 A 0 2 (d) ax # ay 5 az (b) A 3 B 1 B 3 A 5 0 (e) ak 5 ax 2 ay, where ak is a unit vector (c) A # B # C 5 B # C # A 1.5 Which of the following identities is not valid? (a) a1b 1 c2 5 ab 1 bc (d) c # 1a 3 b2 5 2b # 1a 3 c2 (b) a 3 1b 1 c2 5 a 3 b 1 a 3 c (e) aA # aB 5 cos uAB (c) a # b 5 b # a 1.6 Which of the following statements are meaningless? (a) A # B 1 2A 5 0 (c) A1A 1 B2 1 2 5 0 (b) A # B 1 5 5 2A (d) A # A 1 B # B 5 0 1.7 Let F 5 2ax 2 6ay 1 10az and G 5 ax 1 Gyay 1 5az. If F and G have the same unit vector, Gy is (a) 6 (c) 0 (b) 23 (d) 6 1.8 Given that A 5 ax 1 aay 1 az and B 5 aax 1 ay 1 az, if A and B are normal to each other, αis (a) 22 (d) 1 (b) 21/2 (e) 2 (c) 0 1.9 The component of 6ax 1 2ay 2 3az along 3ax 2 4ay is (a) 212ax 2 9ay 2 3az (d) 2 (b) 30ax 2 40ay (e) 10 (c) 10/7 REVIEW QUESTIONS 01_Sadiku_Ch01.indd 24 16/11/17 1:22 PM
Problems 25 1.10 Given A 5 26ax 1 3ay 1 2az, the projection of A along ay is (a) 212 (d) 7 (b) 24 (e) 12 (c) 3 Answers: 1.1d, 1.2a, 1.3b,e, 1.4b, 1.5a, 1.6a,b,c, 1.7b, 1.8b, 1.9d, 1.10c. Section 1.4—Unit Vector 1.1 Determine the unit vector along the direction OP, where O is the origin and P is point (4, 25, 1). 1.2 Points A(4, 26, 2), B(22, 0, 3), and C(10, 1, 27) form a triangle. Show that rAB 1 rBC 1 rCA = 0. Sections 1.5–1.7—Vector Addition, Subtraction, and Multiplication 1.3 If A 5 4ax 2 2ay 1 6az and B 5 12ax 1 18ay 2 8az, determine: (a) A 2 3B (b) 12A 1 5B2/|B| (c) ax 3 A (d) 1B 3 ax 2 # ay 1.4 Let vectors A 5 10ax 2 6ay 1 8az and B 5 ax 1 2az. Find: (a) A  B, (b) A 3 B, (c) 2A – 3B. FIGURE 1.13 For Review Question 1.3. PROBLEMS 01_Sadiku_Ch01.indd 25 16/11/17 1:22 PM
26 CHAPTER 1 VECTOR ALGEBRA 1.5 Let A 5 22ax 1 5ay 1 az, B 5 ax 1 3az, and C 5 4ax 26ay 1 10az. (a) Determine A 2 B 1 C (b) Find A  (B 3 C) (c) Calculate the angle between A and B 1.6 Let A 5 ax 2 az, B 5 ax 1 ay 1 az, C 5 ay 1 2az, find: (a) A # 1B 3 C2 (b) 1A 3 B2 # C (c) A 3 1B 3 C2 (d) 1A 3 B2 3 C 1.7 Given that the position vectors of points T and S are 4ax 1 6ay 2 az and 10ax 1 12ay 1 8az, respectively, find: (a) the coordinates of T and S, (b) the distance vector from T to S, (c) the distance between T and S. 1.8 Let A 5 4ax 1 2ay 2 az and B 5 aax 1 bay 1 3az (a) If A and B are parallel, find a and b (b) If A and B are perpendicular, find a and b 1.9 Let A 5 10ax 1 5ay 2 2az. Find: (a) A 3 ay, (b) A  az, (c) the angle between A and az. 1.10 (a) Show that 1A # B22 1 |A 3 B|2 5 1AB22 (b) Show that ax 5 ay 3 az ax # ay 3 az , ay 5 az 3 ax ax # ay 3 az , az 5 ax 3 ay ax # ay 3 az 1.11 Given that P 5 2ax 2 ay 2 2az Q 5 4ax 1 3ay 1 2az R 5 2ax 1 ay 1 2az find: (a) 0 P 1 Q 2 R 0 , (b) P # Q 3 R, (c) Q 3 P # R, (d) 1P 3 Q2 # 1Q 3 R2, (e) 1P 3 Q2 3 1Q 3 R2, (f) cos uPR, (g) sin uPQ. 1.12 If A 5 4ax 2 6ay 1 az and B 5 2ax 1 5az , find: (a) A  B + 2|B|2 (b) a unit vector perpendicular to both A and B 01_Sadiku_Ch01.indd 26 16/11/17 1:22 PM
Problems 27 1.13 Determine the dot product, cross product, and angle between P 5 2ax 2 6ay 1 5az    and    Q 5 3ay 1 az 1.14 Prove that vectors P 5 2ax 1 4ay 2 6az and Q 5 5ax 1 2ay 2 3az are orthogonal ­ vectors. 1.15 Simplify the following expressions: (a) A 3 1A 3 B2 (b) A 3 3A 3 1A 3 B2 4 1.16 A right angle triangle has its corners located at P1(5, 23, 1), P2(1, 22, 4), and P3(3, 3, 5). (a) Which corner is a right angle? (b) Calculate the area of the triangle. 1.17 Points P, Q, and R are located at 121, 4, 82, 12, 21, 32, and 121, 2, 32, respectively. Determine (a) the distance between P and Q, (b) the distance vector from P to R, (c) the angle between QP and QR, (d) the area of triangle PQR, (e) the perimeter of triangle PQR. 1.18 Two points P12, 4, 212 and Q(12, 16, 9) form a straight line. Calculate the time taken for a sonar signal traveling at 300 m/s to get from the origin to the midpoint of PQ. 1.19 Find the area of the parallelogram formed by the vectors D 5 4ax 1 ay 1 5az and E 5 2ax 1 2ay 1 3az. *1.20 (a) Prove that P 5 cos u1ax 1 sin u1ay and Q 5 cos u2ax 1 sin u2ay are unit vectors in the xy-plane, respectively, making angles u1 and u2 with the x-axis. (b)  By means of dot product, obtain the formula for cos1u2 2 u1 2. By similarly formu- lating P and Q, obtain the formula for cos1u2 1 u1 2. (c) If u is the angle between P and Q, find 1 2 0 P 2 Q 0 in terms of u. 1.21 Consider a rigid body rotating with a constant angular velocity v radians per second about a fixed axis through O as in Figure 1.14. Let r be the distance vector from O to P, the position of a particle in the body. The magnitude of the velocity u of the body at P is 0 u 0 5 d 0 v 0 5 0 r 0 sin u 0 v 0 or u 5 v 3 r. If the rigid body is rotating at 3 rad/s about an axis parallel to ax 2 2ay 1 2az and passing through point 12, 23, 12, determine the velocity of the body at (1, 3, 4). 1.22 A cube of side 1 m has one corner placed at the origin. Determine the angle between the diagonals of the cube. 1.23 Given vectors T 5 2ax 2 6ay 1 3az and S 5 ax 1 2ay 1 az, find (a) the scalar projection of T on S, (b) the vector projection of S on T, (c) the smaller angle between T and S. *Single asterisks indicate problems of intermediate difficulty. 01_Sadiku_Ch01.indd 27 16/11/17 1:22 PM
28 CHAPTER 1 VECTOR ALGEBRA FIGURE 1.14 For Problem 1.21. Section 1.8—Components of a Vector 1.24 Given two vectors A and B, show that the vector component of A perpendicular to B is C 5 A 2 A # B B # B B 1.25 Let A 5 20ax 1 15ay 2 10az and B 5 ax 1 ay. Find: (a) A  B, (b) A 3 B, (c) the compo- nent of A along B. 1.26 Figure 1.15 shows that A makes specific angles with respect to each axis. For A 5 2ax 2 4ay 1 6az, find the direction angles a, b, and g. 1.27 If H 5 2xyax 2 1x 1 z2ay 1 z2 az, find: (a) A unit vector parallel to H at P11, 3, 222 (b) The equation of the surface on which 0 H 0 5 10 1.28 Let P 5 2ax 2 4ay 1 az and Q 5 ax 1 2ay. Find R which has magnitude 4 and is perpen- dicular to both P and Q. 1.29 Let G 5 x2 ax 2 yay 1 2zaz and H 5 yzax 1 3ay 2 xzaz. At point (1, 22, 3), (a) calculate the magnitude of G and H, (b) determine G  H, (c) find the angle between G and H. 1.30 A vector field is given by H 5 10yz2 ax 2 8xyzay 1 12y2 az (a) Evaluate H at P(21, 2, 4) (b) Find the component of H along ax 2 ay at P. 01_Sadiku_Ch01.indd 28 16/11/17 2:22 PM
1.31 E and F are vector fields given by E 5 2xax 1 ay 1 yzaz and F 5 xyax 2 y2 ay1 xyzaz. Determine: (a) 0 E 0 at (1, 2, 3) (b) The component of E along F at (1, 2, 3) (c) A vector perpendicular to both E and F at 10, 1, 232 whose magnitude is unity 1.32 Given two vector fields D 5 yzax 1 xzay 1 xyaz  and  E 5 5xyax 1 6(x2 1 3)ay 1 8z2 az (a) Evaluate C 5 D 1 E at point P(21, 2, 4). (b) Find the angle C makes with the x-axis at P. z x y A α β γ FIGURE 1.15 For Problem 1.26. Problems 29 01_Sadiku_Ch01.indd 29 16/11/17 2:22 PM
The Accreditation Board for Engineering and Technology (ABET) establishes eleven criteria for accrediting engineering, technology, and computer science programs. The criteria are as follows: A. Ability to apply mathematics science and engineering principles B. Ability to design and conduct experiments and interpret data C. Ability to design a system, component, or process to meet desired needs D. Ability to function on multidisciplinary teams E. Ability to identify, formulate, and solve engineering problems F. Ability to understand professional and ethical responsibility G. Ability to communicate effectively H. Ability to understand the impact of engineering solutions in a global context I. Ability to recognize the need for and to engage in lifelong learning J. Ability to know of contemporary issues K. Ability to use the techniques, skills, and modern engineering tools necessary for engineering practice Criterion A applies directly to electromagnetics. As students, you are expected to study math- ematics, science, and engineering with the purpose of being able to apply that knowledge to the solution of engineering problems. The skill needed here is the ability to apply the fundamentals of EM in solving a problem. The best approach is to attempt as many problems as you can. This will help you to understand how to use formulas and assimilate the material. Keep nearly all your basic mathematics, science, and engineering textbooks. You may need to consult them from time to time. ENHANCING YOUR SKILLS AND CAREER 02_Sadiku_Ch02.indd 30 01/11/17 1:50 PM
31 C H A P T E R 31 2.1 INTRODUCTION In general, the physical quantities we shall be dealing with in EM are functions of space and time. In order to describe the spatial variations of the quantities, we must be able to ­ define all points uniquely in space in a suitable manner. This requires using an appropriate coordinate system. A point or vector can be represented in any curvilinear coordinate system, which may be orthogonal or nonorthogonal. An orthogonal system is one in which the coordinate surfaces are mutually perpendicular. Nonorthogonal systems are hard to work with, and they are of little or no practical use. Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the cir- cular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal.1 A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem. A hard problem in one coordinate system may turn out to be easy in another system. In this text, we shall restrict ourselves to the three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical. Although we have considered the Cartesian system in Chapter 1, we shall consider it in detail in this chapter. We should bear in mind that the concepts covered in Chapter 1 and demonstrated in Cartesian coordinates are equally applicable to other systems of coordinates. For example, the procedure for find- ing the dot or cross product of two vectors in a cylindrical system is the same as that used in the Cartesian system in Chapter 1. COORDINATE SYSTEMS AND TRANSFORMATION History teaches us that man learns nothing from history. —HEGEL 2 1 For an introductory treatment of these coordinate systems, see M. R. Spiegel and J. Liu, Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 2nd ed., 1999, pp. 126–130. 02_Sadiku_Ch02.indd 31 01/11/17 1:50 PM
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eBook Elements of Electromagnetics, 7e Matthew Sadiku.pdf

  • 1.
  • 2. Get Complete eBook Download Link below for instant download https://browsegrades.net/documents/2 86751/ebook-payment-link-for-instant- download-after-payment
  • 3. PRACTICAL APPLICATIONS Some of the real-life applications covered in this book are listed in order of appearance. • Applications of electrostatics (Section 4.1) • Electrostatic separation of solids (Example 4.3) • Electrostatic discharge (ESD) (Section 4.11) • Electrostatic shielding (Section 5.9B) • High dielectric constant materials (Section 5.10) • Graphene (Section 5.11) NEW • Electrohydrodynamic pump (Example 6.1) • Xerographic copying machine (Example 6.2) • Parallel-plate capacitor, coaxial capacitor, and spherical capacitor (Section 6.5) • RF MEMS (Section 6.8) (Chapter 12 opener) NEW • Ink-jet printer (Problem 6.52) • Microstrip lines (Sections 6.7, 11.8, and 14.6) • Applications of magnetostatics (Section 7.1) • Coaxial transmission line (Section 7.4C) • Lightning (Section 7.9) • Polywells (Section 7.10) NEW • Magnetic resonant imaging (MRI) (Chapter 8 opener) • Magnetic focusing of a beam of electrons (Example 8.2, Figure 8.2) • Velocity filter for charged particles (Example 8.3, Figure 8.3) • Inductance of common elements (Table 8.3) • Electromagnet (Example 8.16) • Magnetic levitation (Section 8.12) • Hall effect (Section 8.13) NEW • Direct current machine (Section 9.3B) • Memristor (Section 9.8) NEW • Optical nanocircuits (Section 9.9) NEW • Homopolar generator disk (Problem 9.14) • Microwaves (Section 10.11) • Radar (Sections 10.11 and 13.9) • 60 GHz technology (Section 10.12) NEW • Bioelectromagnetics (Chapter 11 opener) • Coaxial, two-line, and planar lines (Figure 11.1, Section 11.2) • Quarter-wave transformer (Section 11.6A) • Data cables (Section 11.8B) • Metamaterials (Section 11.9) NEW • Microwave imaging (Section 11.10) NEW • Optical fiber (Section 12.9) • Cloaking and invisibility (Section 12.10) NEW • Smart antenna (Chapter 13 opener) • Typical antennas (Section 13.1, Figure 13.2) • Electromagnetic interference and compatibility (Section 13.10) • Grounding and filtering (Section 13.10) 00_Sadiku_FRONTCOVER.indd 2 16/11/17 2:09 PM
  • 4. • Textile antennas and sensors (Section 13.11) NEW • RFID (Section 13.12) NEW • Commercial EM software—FEKO (Section 14.7) NEW • COMSOL Multiphysics (Section 14.8) NEW • CST Microwave Studio (Section 14.9) NEW Approximate Best Experimental Value for Problem Quantity (Units) Symbol Value* Work Permittivity of free space (F/m) eo 8.854  1012 1029 36p Permeability of free space (H/m) mo 4p  107 12.6  107 Intrinsic impedance of free space (V) ho 376.6 120p Speed of light in vacuum (m/s) c 2.998  108 3  108 Electron charge (C) e 1.6022  1019 1.6  1019 Electron mass (kg) me 9.1093  1031 9.1  1031 Proton mass (kg) mp 1.6726  1027 1.67  1027 Neutron mass (kg) mn 1.6749  1027 1.67  1027 Boltzmann constant (J/K) k 1.38065  1023 1.38  1023 Avogadro number (/kg-mole) N 6.0221  1023 6  1023 Planck constant (J  s) h 6.626  1034 6.62  1034 Acceleration due to gravity (m/s2 ) g 9.80665 9.8 Universal constant of gravitation G 6.673  1011 6.66  1011 N (m/kg)2 Electron-volt (J) eV 1.602176  1019 1.6  1019    * Values recommended by CODATA (Committee on Data for Science and Technology, Paris). PHYSICAL CONSTANTS 00_Sadiku_FRONTCOVER.indd 3 16/11/17 2:09 PM
  • 6. THE OXFORD SERIES IN ELECTRICAL AND COMPUTER ENGINEERING Adel S. Sedra, Series Editor Allen and Holberg, CMOS Analog Circuit Design, 3rd edition Boncelet, Probability, Statistics, and Random Signals Bobrow, Elementary Linear Circuit Analysis, 2nd edition Bobrow, Fundamentals of Electrical Engineering, 2nd edition Campbell, Fabrication Engineering at the Micro- and Nanoscale, 4th edition Chen, Digital Signal Processing Chen, Linear System Theory and Design, 4th edition Chen, Signals and Systems, 3rd edition Comer, Digital Logic and State Machine Design, 3rd edition Comer, Microprocessor-Based System Design Cooper and McGillem, Probabilistic Methods of Signal and System Analysis, 3rd edition Dimitrijev, Principles of Semiconductor Device, 2nd edition Dimitrijev, Understanding Semiconductor Devices Fortney, Principles of Electronics: Analog Digital Franco, Electric Circuits Fundamentals Ghausi, Electronic Devices and Circuits: Discrete and Integrated Guru and Hiziroğlu, Electric Machinery and Transformers, 3rd edition Houts, Signal Analysis in Linear Systems Jones, Introduction to Optical Fiber Communication Systems Krein, Elements of Power Electronics, 2nd edition Kuo, Digital Control Systems, 3rd edition Lathi and Green, Linear Systems and Signals, 3rd edition Lathi and Ding, Modern Digital and Analog Communication Systems, 5th edition Lathi, Signal Processing and Linear Systems Martin, Digital Integrated Circuit Design Miner, Lines and Electromagnetic Fields for Engineers Mitra, Signals and Systems Parhami, Computer Architecture Parhami, Computer Arithmetic, 2nd edition Roberts and Sedra, SPICE, 2nd edition Roberts, Taenzler, and Burns, An Introduction to Mixed-Signal IC Test and Measurement, 2nd edition Roulston, An Introduction to the Physics of Semiconductor Devices Sadiku, Elements of Electromagnetics, 7th edition Santina, Stubberud, and Hostetter, Digital Control System Design, 2nd edition Sarma, Introduction to Electrical Engineering Schaumann, Xiao, and Van Valkenburg, Design of Analog Filters, 3rd edition Schwarz and Oldham, Electrical Engineering: An Introduction, 2nd edition Sedra and Smith, Microelectronic Circuits, 7th edition Stefani, Shahian, Savant, and Hostetter, Design of Feedback Control Systems, 4th edition Tsividis, Operation and Modeling of the MOS Transistor, 3rd edition Van Valkenburg, Analog Filter Design Warner and Grung, Semiconductor Device Electronics Wolovich, Automatic Control Systems Yariv and Yeh, Photonics: Optical Electronics in Modern Communications, 6th edition Żak, Systems and Control 00_Sadiku_FM.indd 2 16/11/17 3:36 PM
  • 7. ELEMENTS OF ELECTROMAGNETICS MATTHEW N. O. SADIKU Prairie View AM University New York • Oxford OXFORD UNIVERSITY PRESS SEVENTH EDITION 00_Sadiku_FM.indd 3 16/11/17 3:36 PM
  • 8. Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and certain other countries. Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America. © 2018, 2015, 2010, 2007, 2000 by Oxford University Press © 1994, 1989 by Holt, Rinehart, Winston, Inc For titles covered by Section 112 of the US Higher Education Opportunity Act, please visit www.oup.com/us/he for the latest information about pricing and alternate formats. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by license, or under terms agreed with the appropriate reproduction rights organization. Inquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not circulate this work in any other form and you must impose this same condition on any acquirer. Library of Congress Cataloging-in-Publication Data Names: Sadiku, Matthew N. O., author. Title: Elements of Electromagnetics / Matthew N.O. Sadiku, Prairie View AM University. Description: Seventh edition. | New York, NY, United States of America : Oxford University Press, [2018] | Series: The Oxford series in electrical and computer engineering Identifiers: LCCN 2017046497 | ISBN 9780190698614 (hardcover) Subjects: LCSH: Electromagnetism. Classification: LCC QC760 .S23 2018 | DDC 537—dc23 LC record available at https://lccn.loc.gov/2017046497 9 8 7 6 5 4 3 2 1 Printed by LSC Communications, United States of America 00_Sadiku_FM.indd 4 16/11/17 3:36 PM
  • 9. To my wife, Kikelomo 00_Sadiku_FM.indd 5 16/11/17 3:36 PM
  • 10. vi BRIEF TABLE OF CONTENTS MATH ASSESSMENT  MA-1 Chapter 1   Vector Algebra  3 Chapter 2   Coordinate Systems and Transformation  31 Chapter 3   Vector Calculus  59 Chapter 4   Electrostatic Fields  111 Chapter 5    Electric Fields in Material Space   177 Chapter 6   Electrostatic Boundary-Value Problems  225 Chapter 7   Magnetostatic Fields  297 Chapter 8    Magnetic Forces, Materials, and Devices   349 Chapter 9   Maxwell’s Equations  421 Chapter 10  Electromagnetic Wave Propagation  473 Chapter 11  Transmission Lines  553 Chapter 12  Waveguides  633 Chapter 13  Antennas  691 Chapter 14  Numerical Methods  757 Appendix A    Mathematical Formulas  835 Appendix B    Material Constants  845 Appendix C   MATLAB  847 Appendix D The Complete Smith Chart   860 Appendix E    Answers to Odd-Numbered Problems   861 Index  889 00_Sadiku_FM.indd 6 16/11/17 3:36 PM
  • 11. vii BRIEF TABLE OF CONTENTS  vi PREFACE xiii A NOTE TO THE STUDENT   xvii ABOUT THE AUTHOR   xviii MATH ASSESSMENT   MA-1 † Indicates sections that may be skipped, explained briefly, or assigned as homework if the text is covered in one semester. PART 1:   VECTOR ANALYSIS 1 VECTOR ALGEBRA   3 1.1 Introduction  3   † 1.2 A Preview of the Book   4 1.3 Scalars and Vectors  4 1.4 Unit Vector   5 1.5 Vector Addition and Subtraction   6 1.6 Position and Distance Vectors   7 1.7 Vector Multiplication   11 1.8 Components of a Vector   16 Summary  23 Review Questions  24 Problems  25 2 COORDINATE SYSTEMS AND TRANSFORMATION    31 2.1 Introduction   31 2.2 Cartesian Coordinates (x, y, z)  32 2.3 Circular Cylindrical Coordinates (r, f, z)  32 2.4 Spherical Coordinates (r, u, f)  35 2.5 Constant-Coordinate Surfaces   44 Summary  51 Review Questions  52 Problems  54 3 VECTOR CALCULUS   59 3.1 Introduction   59 3.2 Differential Length, Area, and Volume   59 3.3 Line, Surface, and Volume Integrals   66 CONTENTS 00_Sadiku_FM.indd 7 16/11/17 3:36 PM
  • 12. 3.4 Del Operator   69 3.5 Gradient of a Scalar   71 3.6 Divergence of a Vector and Divergence Theorem   75 3.7 Curl of a Vector and Stokes’s Theorem   82 3.8 Laplacian of a Scalar   90 † 3.9 Classification of Vector Fields   92 Summary  97 Review Questions  98 Problems  100 PART 2:  ELECTROSTATICS 4 ELECTROSTATIC FIELDS   111 4.1 Introduction  111 4.2 Coulomb’s Law and Field Intensity   112 4.3 Electric Fields due to Continuous Charge Distributions   119 4.4 Electric Flux Density   130 4.5 Gauss’s Law—Maxwell’s Equation   132 4.6 Applications of Gauss’s Law   134 4.7 Electric Potential   141 4.8 Relationship between E and V—Maxwell’s Equation   147 4.9 An Electric Dipole and Flux Lines   150 4.10 Energy Density in Electrostatic Fields   154 † 4.11 Application Note—Electrostatic Discharge   159 Summary  164 Review Questions  167 Problems  168 5 ELECTRIC FIELDS IN MATERIAL SPACE    177 5.1 Introduction  177 5.2 Properties of Materials   177 5.3 Convection and Conduction Currents   178 5.4 Conductors  181 5.5 Polarization in Dielectrics   187 5.6 Dielectric Constant and Strength   190 † 5.7 Linear, Isotropic, and Homogeneous Dielectrics   191 5.8 Continuity Equation and Relaxation Time   196 5.9 Boundary Conditions   198 † 5.10 Application Note— Materials with High Dielectric Constant   207 5.11 Application Note—Graphene  208 † 5.12 Application Note—Piezoelectrics  210 Summary  214 Review Questions  215 Problems  217 viii CONTENTS 00_Sadiku_FM.indd 8 16/11/17 3:36 PM
  • 13. 6 ELECTROSTATIC BOUNDARY-VALUE PROBLEMS    225 6.1 Introduction  225 6.2 Poisson’s and Laplace’s Equations   225 † 6.3 Uniqueness Theorem   227 6.4 General Procedures for Solving Poisson’s or Laplace’s Equation   228 6.5 Resistance and Capacitance   249 6.6 Method of Images   266 † 6.7 Application Note—Capacitance of Microstrip Lines   272 6.8 Application Note—RF MEMS   275 † 6.9 Application Note—Supercapacitors   276 Summary  280 Review Questions  281 Problems  282 PART 3:  MAGNETOSTATICS 7 MAGNETOSTATIC FIELDS   297 7.1 Introduction   297 7.2 Biot–Savart’s Law   298 7.3 Ampère’s Circuit Law—Maxwell’s Equation   309 7.4 Applications of Ampère’s Law   309 7.5 Magnetic Flux Density—Maxwell’s Equation   317 7.6 Maxwell’s Equations for Static Fields   319 7.7 Magnetic Scalar and Vector Potentials   320 † 7.8 Derivation of Biot–Savart’s Law and Ampère’s Law   326 † 7.9 Application Note—Lightning   328 7.10 Application Note—Polywells   329 Summary  333 Review Questions  335 Problems  338 8 MAGNETIC FORCES, MATERIALS, AND DEVICES    349 8.1 Introduction  349 8.2 Forces due to Magnetic Fields   349 8.3 Magnetic Torque and Moment   361 8.4 A Magnetic Dipole   363 8.5 Magnetization in Materials   368 † 8.6 Classification of Materials   372 8.7 Magnetic Boundary Conditions   376 8.8 Inductors and Inductances   381 8.9 Magnetic Energy   384 † 8.10 Magnetic Circuits   392 † 8.11 Force on Magnetic Materials   394 CONTENTS ix 00_Sadiku_FM.indd 9 16/11/17 3:36 PM
  • 14. 8.12 Application Note—Magnetic Levitation   399 † 8.13 Application Note—SQUIDs   401 Summary  405 Review Questions  407 Problems  409 PART 4:   WAVES AND ­ APPLICATIONS 9 MAXWELL’S EQUATIONS   421 9.1 Introduction  421 9.2 Faraday’s Law   422 9.3 Transformer and Motional Electromotive Forces   424 9.4 Displacement Current   433 9.5 Maxwell’s Equations in Final Forms   436 † 9.6 Time-Varying Potentials   439 9.7 Time-Harmonic Fields   441 † 9.8 Application Note—Memristor 454 † 9.9 Application Note—Optical Nanocircuits  455 † 9.10 Application Note—Wireless Power Transfer and Qi Standard  457 Summary  460 Review Questions  461 Problems  463 10 ELECTROMAGNETIC WAVE PROPAGATION    473 10.1 Introduction  473 † 10.2 Waves in General   474 10.3 Wave Propagation in Lossy Dielectrics   480 10.4 Plane Waves in Lossless Dielectrics   487 10.5 Plane Waves in Free Space   487 10.6 Plane Waves in Good Conductors   489 10.7 Wave Polarization   498 10.8 Power and the Poynting Vector   502 10.9 Reflection of a Plane Wave at Normal Incidence   506 † 10.10 Reflection of a Plane Wave at Oblique Incidence   517 † 10.11 Application Note—Microwaves   529 10.12 Application Note—60 GHz Technology   534 Summary  537 Review Questions  538 Problems  540 11 TRANSMISSION LINES   553 11.1 Introduction   553 11.2 Transmission Line Parameters   554 11.3 Transmission Line Equations   557 x CONTENTS 00_Sadiku_FM.indd 10 16/11/17 3:36 PM
  • 15. 11.4 Input Impedance, Standing Wave Ratio, and Power   564 11.5 The Smith Chart   572 11.6 Some Applications of Transmission Lines   585 † 11.7 Transients on Transmission Lines   592 † 11.8 Application Note—Microstrip Lines and Characterization of Data Cables   604 11.9 Application Note—Metamaterials   612 † 11.10 Application Note—Microwave Imaging   613 Summary  617 Review Questions  618 Problems  621 12 WAVEGUIDES   633 12.1 Introduction  633 12.2 Rectangular Waveguides   634 12.3 Transverse Magnetic Modes   638 12.4 Transverse Electric Modes   643 12.5 Wave Propagation in the Guide   654 12.6 Power Transmission and Attenuation   656 † 12.7 Waveguide Current and Mode Excitation   660 12.8 Waveguide Resonators   666 † 12.9 Application Note—Optical Fiber   672 † 12.10 Application Note—Cloaking and Invisibility   678 Summary  680 Review Questions  682 Problems  683 13 ANTENNAS   691 13.1 Introduction   691 13.2 Hertzian Dipole   693 13.3 Half-Wave Dipole Antenna   697 13.4 Quarter-Wave Monopole Antenna   701 13.5 Small-Loop Antenna   702 13.6 Antenna Characteristics   707 13.7 Antenna Arrays   715 † 13.8   Effective Area and the Friis Equation   725 † 13.9   The Radar Equation   728 † 13.10 Application Note—Electromagnetic Interference and Compatibility  732 † 13.11 Application Note—Textile Antennas and Sensors   737 † 13.12 Application Note—Fractal Antennas   739 13.13 Application Note—RFID   742 Summary  745 Review Questions  746 Problems  747 CONTENTS xi 00_Sadiku_FM.indd 11 16/11/17 3:36 PM
  • 16. 14 NUMERICAL METHODS   757 14.1 Introduction  757 † 14.2 Field Plotting   758 14.3 The Finite Difference Method   766 14.4 The Moment Method   779 14.5 The Finite Element Method   791 † 14.6 Application Note—Microstrip Lines   810 Summary  820 Review Questions  820 Problems  822 APPENDIX A Mathematical Formulas   835 APPENDIX B Material Constants   845 APPENDIX C MATLAB  847 APPENDIX D The Complete Smith Chart   860 APPENDIX E Answers to Odd-Numbered Problems   861 INDEX  889 xii CONTENTS 00_Sadiku_FM.indd 12 16/11/17 3:36 PM
  • 17. xiii This new edition is intended to provide an introduction to engineering electromagnetics (EM) at the junior or senior level. Although the new edition improves on the previous edi- tions, the core of the subject of EM has not changed. The fundamental objective of the first edition has been retained: to present EM concepts in a clearer and more interesting manner than other texts. This objective is achieved in the following ways: 1. To avoid complicating matters by covering EM and mathematical concepts simultane­ ously, vector analysis is covered at the beginning of the text and applied gradually. This approach avoids breaking in repeatedly with more background on vector analysis, thereby creating discontinuity in the flow of thought. It also separates mathematical theorems from physical concepts and makes it easier for the student to grasp the generality of those theorems. Vector analysis is the backbone of the mathematical formulation of EM problems. 2. Each chapter opens either with a historical profile of some electromagnetic pioneers or with a discussion of a modern topic related to the chapter. The chapter starts with a brief introduction that serves as a guide to the whole chapter and also links the chapter to the rest of the book. The introduction helps the students see the need for the chapter and how it relates to the previous chapter. Key points are emphasized to draw the reader’s attention. A brief summary of the major concepts is discussed toward the end of the chapter. 3. To ensure that students clearly get the gist of the matter, key terms are defined and highlighted. Important formulas are boxed to help students identify essential formulas. 4. Each chapter includes a reasonable amount of solved examples. Since the examples are part of the text, they are clearly explained without asking the reader to fill in missing steps. In writing out the solution, we aim for clarity rather than efficiency. Thoroughly worked out examples give students confidence to solve problems themselves and to learn to apply concepts, which is an integral part of engineering education. Each illustrative example is followed by a problem in the form of a Practice Exercise, with the answer provided. 5. At the end of each chapter are ten review questions in the form of multiple-choice ­ objective items. Open-ended questions, although they are intended to be thought-provoking, are ignored by most students. Objective review questions with answers immediately following them provide encouragement for students to do the problems and gain immediate feedback. A large number of problems are provided and are presented in the same order as the material in the main text. Approximately 20 to 25 percent of the problems in this edition have been replaced. Problems of intermediate difficulty are identified by a single asterisk; the most diffi- cult problems are marked with a double asterisk. Enough problems are provided to allow the instructor to choose some as examples and assign some as homework problems. Answers to odd-numbered problems are provided in Appendix E. 6. Since most practical applications involve time-varying fields, six chapters are devoted to such fields. However, static fields are given proper emphasis because they are special cases of dynamic fields. Ignorance of electrostatics is no longer acceptable PREFACE 00_Sadiku_FM.indd 13 16/11/17 3:36 PM
  • 18. because there are large industries, such as copier and computer peripheral manufactur- ing, that rely on a clear understanding of electrostatics. 7. The last section in each chapter is devoted to applications of the concepts covered in the chapter. This helps students see how concepts apply to real-life situations. 8. The last chapter covers numerical methods with practical applications and MATLAB programs. This chapter is of paramount importance because most practical prob- lems are only solvable using numerical techniques. Since MATLAB is used throughout the book, an introduction to MATLAB is provided in Appendix C. 9. Over 130 illustrative examples and 300 figures are given in the text. Some additional learning aids such as basic mathematical formulas and identities are included in Appendix A. Another guide is a special note to students, which follows this preface. NEW TO THE SIXTH EDITION •  Five new Application Notes designed to explain the real-world connections between the concepts discussed in the text. •  A revised Math Assessment test, for instructors to gauge their students’ mathematical knowledge and preparedness for the course. •  New and updated end-of-chapter problems. Solutions to the end-of-chapter problems and the Math Assessment, as well as PowerPoint slides of all figures in the text, can be found at the Oxford University Press Ancillary Resource Center. Students and professors can view Application Notes from previous editions of the text on the book’s companion website www.oup.com/us/sadiku. Although this book is intended to be self-explanatory and useful for self-instruction, the personal contact that is always needed in teaching is not forgotten. The actual choice of course topics, as well as emphasis, depends on the preference of the individual instruc- tor. For example, an instructor who feels that too much space is devoted to vector anal- ysis or static fields may skip some of the materials; however, the students may use them as reference. Also, having covered Chapters 1 to 3, it is possible to explore Chapters 9 to 14. Instructors who disagree with the vector-calculus-first approach may proceed with Chapters 1 and 2, then skip to Chapter 4, and refer to Chapter 3 as needed. Enough mate- rial is covered for two-semester courses. If the text is to be covered in one semester, cover- ing Chapters 1 to 9 is recommended; some sections may be skipped, explained briefly, or assigned as homework. Sections marked with the dagger sign (†) may be in this category. ACKNOWLEDGMENTS I thank Dr. Sudarshan Nelatury of Penn State University for providing the new Application Notes and the Math Assessment test. It would not be possible to prepare this edition without the efforts of Executive Editor Dan Kaveney, Associate Editor Christine Mahon, Assistant Editor Megan Carlson, Marketing Manager David Jurman, Marketing Assistant Colleen Rowe, Production Editor Claudia Dukeshire, and Designer Michele Laseau at Oxford University Press, as well as Susan Brown and Betty Pessagno. xiv PREFACE 00_Sadiku_FM.indd 14 16/11/17 3:36 PM
  • 19. I thank the reviewers who provided helpful feedback for this edition: Mohammadreza (Reza) Barzegaran Lamar University Sudarshan Nelatury Penn State Erie Sharif M. A. Bhuiyan Tuskegee University Sima Noghanian University of North Dakota Muhammad Dawood New Mexico State University Vladimir Rakov University of Florida Robert Gauthier Carleton University Lisa Shatz Suffolk University Jesmin Khan Tuskegee University Kyle Sundqvist Texas AM University Edwin Marengo Northeastern University Lili H. Tabrizi California State University, Los Angeles Perambur S. Neelakanta Florida Atlantic University I also offer thanks to those who reviewed previous editions of the text: Yinchao Chen University of South Carolina Douglas T. Petkie Wright State University Satinderpaul Singh Devgan Tennessee State University James E. Richie Marquette University Dentcho Angelov Genov Louisiana Tech University Elena Semouchkina Michigan Technological University Scott Grenquist Wentworth Institute of Technology Barry Spielman Washington University Xiaomin Jin Cal Poly State University, San Luis Obispo Murat Tanik University of Alabama–Birmingham Jaeyoun Kim Iowa State University Erdem Topsakal Mississippi State University Caicheng Lu University of Kentucky Charles R. Westgate Sr. SUNY–Binghamton Perambur S. Neelakantaswamy Florida Atlantic University Weldon J. Wilson University of Central Oklahoma Kurt E. Oughstun University of Vermont Yan Zhang University of Oklahoma I am grateful to Dr. Kendall Harris, dean of the College of Engineering at Prairie View AM University, and Dr. Pamela Obiomon, head of the Department of Electrical and PREFACE xv 00_Sadiku_FM.indd 15 16/11/17 3:36 PM
  • 20. xvi PREFACE Computer Engineering, for their constant support. I would like to express my gratitude to Dr. Vichate Ungvichian, at Florida Atlantic University, for pointing out some errors. I acknowledge Emmanuel Shadare for help with the figures. A well-deserved expression of appreciation goes to my wife and our children for their constant support and prayer. I owe special thanks for those professors and students who have used earlier edi- tions of the book. Please keep sending those errors directly to the publisher or to me at sadiku@ieee.org. —Matthew N.O. Sadiku Prairie View, Texas 00_Sadiku_FM.indd 16 16/11/17 3:36 PM
  • 21. A NOTE TO THE STUDENT Electromagnetic theory is generally regarded by students as one of the most difficult cours- es in physics or the electrical engineering curriculum. But this conception may be proved wrong if you take some precautions. From experience, the following ideas are provided to help you perform to the best of your ability with the aid of this textbook: 1. Pay particular attention to Part 1 on vector analysis, the mathematical tool for this course. Without a clear understanding of this section, you may have problems with the rest of the book. 2. Do not attempt to memorize too many formulas. Memorize only the basic ones, which are usually boxed, and try to derive others from these. Try to understand how formu- las are related. There is nothing like a general formula for solving all problems. Each for- mula has limitations owing to the assumptions made in obtaining it. Be aware of those as- sumptions and use the formula accordingly. 3. Try to identify the key words or terms in a given definition or law. Knowing the meaning of these key words is essential for proper application of the definition or law. 4. Attempt to solve as many problems as you can. Practice is the best way to gain skill. The best way to understand the formulas and assimilate the material is by solving problems. It is recommended that you solve at least the problems in the Practice Exercise immediately following each illustrative example. Sketch a diagram illustrating the problem before ­ attempting to solve it mathematically. Sketching the diagram not only makes the problem easier to solve, but also helps you understand the problem by simplifying and organizing your thinking process. Note that unless otherwise stated, all distances are in meters. For example (2, 1, 5) actually means (2 m, 1 m, 5 m). You may use MATLAB to do number crunching and plotting. A brief introduction to MATLAB is provided in Appendix C. A list of the powers of 10 and Greek letters commonly used throughout this text is provided in the tables located on the inside cover. Important formulas in calculus, vectors, and complex analysis are provided in Appendix A. Answers to odd-numbered problems are in Appendix E. xvii xvii 00_Sadiku_FM.indd 17 16/11/17 3:36 PM
  • 22. ABOUT THE AUTHOR Matthew N. O. Sadiku received his BSc degree in 1978 from Ahmadu Bello University, Zaria, Nigeria, and his MSc and PhD degrees from Tennessee Technological University, Cookeville, Tennessee, in 1982 and 1984, respectively. From 1984 to 1988, he was an assis- tant professor at Florida Atlantic University, Boca Raton, Florida, where he did graduate work in computer science. From 1988 to 2000, he was at Temple University, Philadelphia, Pennsylvania, where he became a full professor. From 2000 to 2002, he was with Lucent/ Avaya, Holmdel, New Jersey, as a system engineer and with Boeing Satellite Systems, Los Angeles, California, as a senior scientist. He is currently a professor of electrical and com- puter engineering at Prairie View AM University, Prairie View, Texas. He is the author of over 370 professional papers and over 70 books, including Elements of Electromagnetics (Oxford University Press, 7th ed., 2018), Fundamentals of Electric Circuits (McGraw-Hill, 6th ed., 2017, with C. Alexander), Computational Electromagnetics with MATLAB (CRC, 4th ed., 2018), Metropolitan Area Networks (CRC Press, 1995), and Principles of Modern Communication Systems (Cambridge University Press, 2017, with S. O. Agbo). In addition to the engineering books, he has written Christian books including Secrets of Successful Marriages, How to Discover God’s Will for Your Life, and commentaries on all the books of the New Testament Bible. Some of his books have been translated into French, Korean, Chinese (and Chinese Long Form in Taiwan), Italian, Portuguese, and Spanish. He was the recipient of the 2000 McGraw-Hill/Jacob Millman Award for out- standing contributions in the field of electrical engineering. He was also the reci­ pient of Regents Professor award for 2012–2013 by the Texas AM University System. He is a registered professional engineer and a fellow of the Institute of Electrical and Electronics Engineers (IEEE) “for contributions to computational electromagnetics and engineering education.” He was the IEEE Region 2 Student Activities Committee Chairman. He was an associate editor for IEEE Transactions on Education. He is also a member of the Association for Computing Machinery (ACM) and the American Society of Engineering Education (ASEE). His current research interests are in the areas of computational electromagnetics, computer networks, and engineering education. His works can be found in his autobiography, My Life and Work (Trafford Publishing, 2017) or on his website, www.matthewsadiku.com. He currently resides with his wife Kikelomo in Hockley, Texas. He can be reached via email at sadiku@ieee.org. xviii 00_Sadiku_FM.indd 18 16/11/17 3:36 PM
  • 23. MA-1 MATH ASSESSMENT 1.1 Let u be the angle between the vectors A and B. What can be said about u if (i) |A +B| , |A 2 B|, (ii) |A 1 B| 5 |A 2 B|, (iii) |A 1 B| . |A 2 B|? 1.2 Two sides of a parallelogram ABCD denoted as p = 5ax and q = 3ax + 4ay are shown in Figure MA-1 Let the diagonals intersect at O and make an angle a. Find the coordinates of O and the magnitude of a. Based on the value of a, what can we call ABCD? B O α A p = 5ax C D q = 3ax + 4ay FIGURE MA-1 Parallelogram ABCD. 1.3  What is the distance R between the two points A(3, 5, 1) and B(5, 7, 2)? Also find its reciprocal, 1 R . 1.4  What is the distance vector RAB from A(3, 7, 1) to B(8, 19, 2) and a unit vector aAB in the direction of RAB? 1.5  Find the interval of values x takes so that a unit vector u satisfies |(x 2 2)u| , |3u|. 1.6  There are four charges in space at four points A, B, C, and D, each 1 m from every other. You are asked to make a selection of coordinates for these charges. How do you place them in space and select their coordinates? There is no unique way. 1.7  A man driving a car starts at point O, drives in the following pattern 15 km northeast to point A, 20 km southwest to point B, 25 km north to C, 10 km southeast to D, 15 km west to E, and stops. How far is he from his starting point, and in what direction? 00_Sadiku_FM.indd 1 16/11/17 3:36 PM
  • 24. 1.8  A unit vector an makes angles , , and  with the x-, y-, and z-axes, respectively. Express an in the rectangular coordinate system. Also express a nonunit vector OP of length  parallel to an. 1.9 Three vectors p, q, and r sum to a zero vector and have the magnitude of 10, 11, and 15, respectively. Determine the value of p ? q 1 q ? r 1 r ? p. 1.10  An experiment revealed that the point Q(x, y, z) is 4 m from P(2, 1, 4) and that the vector QP makes 45.5225, 59.4003, and 60 with the x-, y-, and z-axes, respectively. Determine the location of Q. 1.11  In a certain frame of reference with x-, y-, and z-axes, imagine the first octant to be a room with a door. Suppose that the height of the door is h and its width is . The top-right corner P of the door when it is shut has the rectangular coordinates (, 0, h). Now if the door is turned by angle , so we can enter the room, what are the coordinates of P? What is the length of its diagonal r 5 OP in terms of  and z? Suppose the vector OP makes an angle  with the z-axis; express  and h in terms of r and . 1.12  Consider two vectors p 5 OP and q=OQ in Figure MA-2. Express the vector GR in terms of p and q. Assume that /ORQ 5 90°. MA-2 MATH ASSESSMENT P R Q O FIGURE MA-2 Orthogonal projection of one vector over another. 1.13  Consider the equations of two planes: 3x 2 2y 2 z 5 8 2x 1 y 1 4z 5 3 Let them intersect along the straight line . Obtain the coordinates of the points where meets the xy– and the yz–planes. Also determine the angle between and the xz-plane. 1.14 Given two vectors p 5 ax 1 ay and q 5 ay 1 az of equal length, find a third vector r such that it has the same length and the angle between any two of them is 608. 1.15 Given A 5 2xy ax 1 3zy ay 1 5z az and B 5 sinx ax 1 2y ay 1 5y az, find (i)   A, (ii)  3 A, (iii)    3 A, and (iv)   (A 3 B). 00_Sadiku_FM.indd 2 16/11/17 3:36 PM
  • 25. 2.1  A triangular plate of base b 5 5 and height h 5 4 shown in Figure MA-3 is uni- formly charged with a uniform surface charge density rs 5 10 C/m2 . You are to cut a rectangular piece so that maximum amount of charge is taken out. What should be the dimensions x and y of the rectangle? What is the magnitude of the charge extracted out? 2.2  Consider two fixed points F1(2c, 0) and F2(c, 0) in the xy-plane. Show that the locus of a point P(x, y) that satisfies the constraint that the sum PF1 +PF2 remains constant and is equal to 2a is an ellipse. The equipotential loci due to a uniform line charge of length 2c are family of ellipses in the plane containing the charge. This problem helps in proving it. 2.3  Show that the ordinary angle subtended by a closed curve lying in a plane at a point P is 2 radians if P is enclosed by the curve and zero if not. MATH ASSESSMENT MA-3 x y M M′ O A θ F FIGURE MA-3 A rectangular piece cut out from a triangular plate. 2.4  Show that the solid angle subtended by a closed surface at a point P is 4 steradians if P is enclosed by the closed surface and zero if not. 2.5  The electrostatic potential V(r) is known to obey the equation V(r) = 2V (2r) with the boundary condition V(5) 5 3 volts. Determine V(15). 2.6  Evaluate the indefinite integrals (i) 3cosec u du and (ii) 3sec u du. Ignore the arbi- trary constant. 2.7  A liquid drop is in the form of an ellipsoid x2 a2 1 y2 b2 1 z2 c2 5 1 shown in Figure MA-4 and is filled with a charge of nonuniform density ry 5 x2 C/m3 . Find the total charge in the drop. 2.8  Two families of curves are said to be orthogonal to each other if they intersect at 908. Given a family y2 5 cx3 , find the equation for orthogonal trajectories and plot three to four members of each on the same graph. 2.9  Consider a vector given by E 5 (4xy 1 z)ax 1 2x2 ay 1 xaz. Find the line integral from A(3, 7, 1) to B(8, 9, 2) by (i)evaluatingthelineintegralVAB 52 3 B A Edlalong     the line joining A to B and (ii) evaluating e23 C A E # dl 2 3 D C E # dl 2 3 B D E # dlf, where the stopovers C and D are C(8, 7, 1) and D(8, 9, 1). 00_Sadiku_FM.indd 3 16/11/17 3:36 PM
  • 26. 2.10  Find the trigonometric Fourier series of a function f(x) 5 x 1 x2 defined over the interval 2p , x , p. 2.11  In a certain electrostatic system, there are found an infinite set of image point charges. The field intensity at a point may be written as E 5 A a ` m51 a ` n5m (21)(n21) n2 Simplify the double summation. Hint: Integrating the following series term by term and substituting x 5 1 helps in finding the result. 1 1 1 x 5 12x 1 x2 2 x3 1 ... 1 2 ... 2.12  Solve the differential equation d2 V(x) dx2 5 k V(x) subject to the boundary conditions dV dx ` x50 5 0 and V(0) 5 0. Assume that k is a constant. 3.1  The location of a moving charge is given by the time-varying radius vector r 5 2 cos t ax 1 2 sin t ay + 3taz. Describe the trajectory of motion. Find the velocity and acceleration vectors at any instant t. In particular, indicate their directions at the specific instants t 5 0 and t 5 p/2. Find their magnitudes at any instant. 3.2  The magnetic field strength H(z) at a point on the z-axis shown in Figure MA-5 is proportional to the sum of cosine of angles and is given by H 5 k(cos u1 1 cos u2). Find H(0). Also show that if a ≪ ,, H (6,) 1 2 H(0). This helps in finding the magnetic field along the axis of a long solenoid. a x –c y b S –a τ –b c z FIGURE MA-4 A non uniformly charged liquid drop. MA-4 MATH ASSESSMENT 00_Sadiku_FM.indd 4 16/11/17 3:36 PM
  • 27. 3.3 Suppose it is suggested that B 5 r (k 3 r) is the magnetic flux density vector, where k is a constant vector and r 5 rar verify if it is solenoidal. MATH ASSESSMENT MA-5 3.4  Evaluate the line integral I C (x 1 y)dx 1 (x 2 y)dy x 2 1 y 2 where C is the circle x2 1 y2 5 a2 of constant radius a. 3.5 Evaluate the line integral I C xdy 2 ydx x2 1 y2 where C is a closed curve (i) encircling the origin n times, (ii) not enclosing the origin. 4.1 Show that    3 A 5 0. 4.2 Show that  3  5 0. 4.3  Given that the imaginary unit is j 5 21 and that x 5 jj , could the value of x be real? If so, is it unique? Can x have one value in the interval (100, 120)? 4.4 Show that   A 3 B 5 B   3 A 2 A   3 B. 4.5 Use De Moivre's theorem to prove that cos 3 5 cos3  2 3 cos  sin2 : 4.6 Determine j. 4.7 Determine j using the Euler formula. 4.8  Find the phasors for the following field quantities: (a) Ex(z, t) 5 Eo cos (t 2 z 1 ) (V/m) (b) Ey(z, t) 5 100e23z cos (t 2 5z 1 /4) (V/m) (c) Hx(z, t) 5 Ho cos (t 1 z) (A/m) (d) Hy(z, t) 5 120e25z cos (t 1 z 1 h) (A/m) 4.9  Find the instantaneous time domain sinusoidal functions corresponding to the following phasors: (a) Ex(z) 5 Eoejz (V/m) (b) Ey(z) 5 100e23z e2j5z (V/m) (c) Is(z) 5 5 1 j4 (A) (d) Vs(z) 5 j10ej/3 (V) 4.10  Write the phasor expression ~ I for the following current using a cosine reference. (a) i(t) 5 Io cos (t 2 /6) (b) i(t) 5 Io sin (t 1 /3) z z a 0 θ1 θ2 2 a FIGURE MA-5 Toward finding magnetic field along the axis of a solenoid. 00_Sadiku_FM.indd 5 16/11/17 3:36 PM
  • 28. 4.11  In a certain resonant cavity, the resonant modes are described by a triplet of ­nonnegative integers m, n, and p. Find possible solutions under the inequality ­constraints, mn 1 np 1 pm 2 0 13 16 # m2 4 1 n2 9 1 p2 # 5 4 4.12  A voltage source V (t) 5 100 cos (6109 t 2 45) (V) is connected to a series RLC circuit, as shown in Figure MA-6. Given R 5 10M, C 5 100pF, and L 5 1H, use phasor notation to find the following: (a) i(t) (b) Vc (t), the voltage across the capacitor 4.13  (i) Show that the locus of the points P(x, y) obeying the equation x2 1 y2 1 2gx 1 2fy 1 c 5 0 represents a circle. (ii) Express the coordinates of the center and the radius. Use the following equations of circles to find the centers and radii. x2 1 y2 1 8x 2 4y 1 11 5 0 x2 1 y2 2 10x 2 6y 1 9 5 0 225x2 1 225y2 1 90x 2 300y 1 28 5 0 MA-6 MATH ASSESSMENT FIGURE MA-6 A series RLC circuit for Problem 4.12. + + – Vs Vc R jωL 1 jωC – i(t) 4.14 Recall the vector identity  3 A   3 A 1  3 A, where  is a scalar function and A is a vector point function. Suppose A 5 Azaz, where Az 5 e2jkr r and k is a ­constant. Simplify  3 A. 4.15  Between two points A and B on the brink of a circular water pond, a transmission line has to be run. It costs twice the money per meter length to install the cable through the water compared to installation on the edge. One might take the cable (a) completely around the arc on the surrounding land or (b) straight through in the water or (c) partly on the arc and for the remaining, straight in the water. (i) What path costs the maximum money? (ii) Suggest an arrangement that minimizes the cost. With some numerical values, plot the cost function. 4.16  Show the following series expansion assuming |x|  1: 1 11 2 x22 5 1 1 2x 1 3x2 1 4x3 1 . . . 00_Sadiku_FM.indd 6 16/11/17 3:36 PM
  • 29. Get Complete eBook Download Link below for instant download https://browsegrades.net/documents/2 86751/ebook-payment-link-for-instant- download-after-payment
  • 30. PA R T 1 VECTOR ANALYSIS 01_Sadiku_Ch01.indd 1 01/11/17 1:14 PM
  • 31. CODES OF ETHICS Engineering is a profession that makes significant contributions to the economic and social ­ well-being of people all over the world. As members of this important profession, engineers are expected to exhibit the highest standards of honesty and integrity. Unfortu­ nately, the engineering curriculum is so crowded that there is no room for a course on ethics in most schools. Although there are over 850 codes of ethics for different professions all over the world, the code of ethics of the Institute of Electrical and Electronics Engineers (IEEE) is presented here to give students a flavor of the importance of ethics in engineer­ ing professions. We, the members of the IEEE, in recognition of the importance of our technologies in affecting the quality of life throughout the world, and in accepting a personal obligation to our profession, its members and the communities we serve, do hereby commit ourselves to the highest ethical and professional conduct and agree: 1.  to accept responsibility in making engineering decisions consistent with the safety, health, and welfare of the public, and to disclose promptly factors that might endanger the public or the environment; 2.  to avoid real or perceived conflicts of interest whenever possible, and to disclose them to affected parties when they do exist; 3. to be honest and realistic in stating claims or estimates based on available data; 4. to reject bribery in all its forms; 5.  to improve the understanding of technology, its appropriate application, and po­ tential consequences; 6.  to maintain and improve our technical competence and to undertake technologi­ cal tasks for others only if qualified by training or experience, or after full disclo­ sure of pertinent limitations; 7.  to seek, accept, and offer honest criticism of technical work, to acknowledge and correct errors, and to credit properly the contributions of others; 8.  to treat fairly all persons regardless of such factors as race, religion, gender, ­ dis­ ability, age, or national origin; 9.  to avoid injuring others, their property, reputation, or employment by false or ma­ licious action; 10.  to assist colleagues and co-workers in their professional development and to ­ sup­ port them in following this code of ethics. —Courtesy of IEEE 01_Sadiku_Ch01.indd 2 01/11/17 1:14 PM
  • 32. 3 C H A P T E R 3 1.1 INTRODUCTION Electromagnetics (EM) may be regarded as the study of the interactions between electric charges at rest and in motion. It entails the analysis, synthesis, physical interpretation, and application of electric and magnetic fields. Electromagnetics (EM) is a branch of physics or electrical engineering in which electric and magnetic phenomena are studied. EM principles find applications in various allied disciplines such as microwaves, antennas, electric machines, satellite communications, bioelectromagnetics, plasmas, nuclear research, fiber optics, electromagnetic interference and compatibility, electromechanical energy conver- sion, radar meteorology, and remote sensing.1,2 In physical medicine, for example, EM power, in the form either of shortwaves or microwaves, is used to heat deep tissues and to stimulate certain physiological responses in order to relieve certain pathological conditions. EM fields are used in induction heaters for melting, forging, annealing, surface hardening, and soldering operations. Dielectric heating equipment uses shortwaves to join or seal thin sheets of plastic materials. EM energy offers many new and exciting possibilities in agriculture. It is used, for example, to change vegetable taste by reducing acidity. EM devices include transformers, electric relays, radio/TV, telephones, electric motors, transmission lines, waveguides, antennas, optical fibers, radars, and lasers. The design of these devices requires thorough knowledge of the laws and principles of EM. 1 For numerous applications of electrostatics, see J. M. Crowley, Fundamentals of Applied Electrostatics. New York: John Wiley Sons, 1986. 2 For other areas of applications of EM, see, for example, D. Teplitz, ed., Electromagnetism: Paths to Research. New York: Plenum Press, 1982. VECTOR ALGEBRA Books are the quietest and most constant friends; they are the most accessible and wisest of counselors, and most patient of teachers. —CHARLES W. ELLIOT 1 01_Sadiku_Ch01.indd 3 01/11/17 1:14 PM
  • 33. 4 CHAPTER 1 VECTOR ALGEBRA The subject of electromagnetic phenomena in this book can be summarized in Maxwell’s equations: = # D 5 rv (1.1) = # B 5 0 (1.2) = 3 E 5 2 'B 't (1.3) = 3 H 5 J 1 'D 't (1.4) where = 5 the vector differential operator D 5 the electric flux density B 5 the magnetic flux density E 5 the electric field intensity H 5 the magnetic field intensity v 5 the volume charge density J 5 the current density Maxwell based these equations on previously known results, both experimental and theore­ tical. A quick look at these equations shows that we shall be dealing with vector quantities. It is consequently logical that we spend some time in Part 1 examining the mathematical tools required for this course. The derivation of eqs. (1.1) to (1.4) for time-invariant conditions and the physical significance of the quantities D, B, E, H, J, and v will be our aim in Parts 2 and 3. In Part 4, we shall reexamine the equations for time-varying situations and apply them in our study of practical EM devices such as transmission lines, waveguides, antennas, fiber optics, and radar systems. † 1.2 A PREVIEW OF THE BOOK 1.3 SCALARS AND VECTORS Vector analysis is a mathematical tool with which electromagnetic concepts are most con- veniently expressed and best comprehended. We must learn its rules and techniques before we can confidently apply it. Since most students taking this course have little exposure to vector analysis, considerable attention is given to it in this and the next two chapters.3 This chapter introduces the basic concepts of vector algebra in Cartesian coordinates only. The next chapter builds on this and extends to other coordinate systems. A quantity can be either a scalar or a vector. A scalar is a quantity that is completely specified by its magnitude. † Indicates sections that may be skipped, explained briefly, or assigned as homework if the text is covered in one semester. 3 The reader who feels no need for review of vector algebra can skip to the next chapter. 01_Sadiku_Ch01.indd 4 01/11/17 1:14 PM
  • 34. 1.4 Unit Vector 5 A scalar is a quantity that has only magnitude. Quantities such as time, mass, distance, temperature, entropy, electric potential, and popu- lation are scalars. A vector has not only magnitude, but direction in space. A vector is a quantity that is described by both magnitude and direction. Vector quantities include velocity, force, momentum, acceleration displacement, and electric field intensity. Another class of physical quantities is called tensors, of which scalars and vectors are special cases. For most of the time, we shall be concerned with scalars and vectors.4 To distinguish between a scalar and a vector it is customary to represent a vector by a letter with an arrow on top of it, such as A and B , or by a letter in boldface type such as A and B. A scalar is represented simply by a letter—for example, A, B, U, and V. EM theory is essentially a study of some particular fields. A field is a function that specifies a particular quantity everywhere in a region. A field may indicate variation of a quantity throughout space and perhaps with time. If the quantity is scalar (or vector), the field is said to be a scalar (or vector) field. Examples of scalar fields are temperature distribution in a building, sound intensity in a theater, electric potential in a region, and refractive index of a stratified medium. The gravitational force on a body in space and the velocity of raindrops in the atmosphere are examples of vector fields. 1.4 UNIT VECTOR A vector A has both magnitude and direction. The magnitude of A is a scalar written as A or 0 A 0 . A unit vector aA along A is defined as a vector whose magnitude is unity (i.e., 1) and its direction is along A; that is, aA 5 A 0 A 0 5 A A (1.5) Note that 0 aA 0 5 1. Thus we may write A as A 5 AaA (1.6) which completely specifies A in terms of its magnitude A and its direction aA. A vector A in Cartesian (or rectangular) coordinates may be represented as 1Ax, Ay, Az 2    or    Axax 1 Ayay 1 Azaz (1.7) 4 For an elementary treatment of tensors, see, for example, A. I. Borisenko and I. E. Tarapor, Vector and Tensor Analysis with Applications. New York: Dover, 1979. 01_Sadiku_Ch01.indd 5 01/11/17 1:14 PM
  • 35. 6 CHAPTER 1 VECTOR ALGEBRA where Ax, Ay, and Az are called the components of A in the x-, y-, and z-directions, respec- tively; ax, ay, and az are unit vectors in the x-, y-, and z-directions, respectively. For example, ax is a dimensionless vector of magnitude one in the direction of the increase of the x-axis. The unit vectors ax, ay, and az are illustrated in Figure 1.1(a), and the components of A along the coordinate axes are shown in Figure 1.1(b). The magnitude of vector A is given by A 5 Ax 2 1 Ay 2 1 Az 2 (1.8) and the unit vector along A is given by aA 5 Axax 1 Ayay 1 Azaz Ax 2 1 Ay 2 1 Az 2 (1.9) FIGURE 1.1 (a) Unit vectors ax, ay, and az, (b) components of A along ax, ay, and az. 1.5 VECTOR ADDITION AND SUBTRACTION Two vectors A and B can be added together to give another vector C; that is, C 5 A 1 B (1.10) The vector addition is carried out component by component. Thus, if A 5 1Ax, Ay, Az) and B 5 1Bx, By, Bz). C 5 1Ax 1 Bx 2ax 1 1Ay 1 By 2ay 1 1Az 1 Bz 2az (1.11) Vector subtraction is similarly carried out as D 5 A 2 B 5 A 1 12B2 5 1Ax 2 Bx 2ax 1 1Ay 2 By 2ay 1 1Az 2 Bz 2az (1.12) 01_Sadiku_Ch01.indd 6 01/11/17 1:14 PM
  • 36. 1.6 Position and Distance Vectors 7 Graphically, vector addition and subtraction are obtained by either the parallelogram rule or the head-to-tail rule as portrayed in Figures 1.2 and 1.3, respectively. The three basic laws of algebra obeyed by any given vectors A, B, and C are summa- rized as follows: Law Addition Multiplication Commutative A 1 B 5 B 1 A kA 5 Ak Associative A 1 1B 1 C2 5 1A 1 B2 1 C k(,A) 5 (k,)A Distributive k1A 1 B2 5 kA 1 kB where k and , are scalars. Multiplication of a vector with another vector will be discussed in Section 1.7. FIGURE 1.3 Vector subtraction D 5 A 2 B: (a) parallelogram rule, (b) ­ head-to-tail rule. FIGURE 1.2 Vector addition C 5 A 1 B: (a) parallelogram rule, (b) head-to-tail rule. 1.6 POSITION AND DISTANCE VECTORS A point P in Cartesian coordinates may be represented by (x, y, z). The position vector rP (or radius vector) of point P is defined as the directed dis- tance from the origin O to P; that is, rP 5 OP 5 xax 1 yay 1 zaz (1.13) 01_Sadiku_Ch01.indd 7 01/11/17 1:14 PM
  • 37. 8 CHAPTER 1 VECTOR ALGEBRA The position vector of point P is useful in defining its position in space. Point (3, 4, 5), for example, and its position vector 3ax 1 4ay 1 5az are shown in Figure 1.4. The distance vector is the displacement from one point to another. If two points P and Q are given by (xP, yP, zP) and (xQ, yQ, zQ), the distance vector (or separation vector) is the displacement from P to Q as shown in Figure 1.5; that is, rPQ 5 rQ 2 rP 5 1xQ 2 xP 2ax 1 1yQ 2 yP 2ay 1 1zQ 2 zP 2az (1.14) The difference between a point P and a vector A should be noted. Though both P and A may be represented in the same manner as (x, y, z) and (Ax, Ay, Az), respectively, the point P is not a vector; only its position vector rP is a vector. Vector A may depend on point P, however. For example, if A 5 2xyax 1 y2 ay 2 xz2 az and P is 12, 21, 42, then A at P would be 24ax 1 ay 2 32az. A vector field is said to be constant or uniform if it does not depend on space variables x, y, and z. For example, vector B 5 3ax 2 2ay 1 10az is a uniform vector while vector A 5 2xyax 1 y2 ay 2 xz2 az is not uniform because B is the same everywhere, whereas A varies from point to point. O FIGURE 1.4 Illustration of position vector rP 5 3ax 1 4ay 5 5az. FIGURE 1.5 Distance vector rPQ. EXAMPLE 1.1 If A 5 10ax 2 4ay 1 6az and B 5 2ax 1 ay, find (a) the component of A along ay, (b) the magnitude of 3A 2 B, (c) a unit vector along A 1 2B. 01_Sadiku_Ch01.indd 8 01/11/17 1:14 PM
  • 38. 1.6 Position and Distance Vectors 9 Solution: (a) The component of A along ay is Ay 5 24. (b) 3A 2 B 5 3110, 24, 62 2 12, 1, 02 5 130, 212, 182 2 12, 1, 02 5 128, 213, 182 Hence, 0 3A 2 B 0 5 282 1 121322 1 11822 5 1277 5 35.74 (c) Let C 5 A 1 2B 5 110, 24, 62 1 14, 2, 02 5 114, 22, 62. A unit vector along C is ac 5 C 0 C 0 5 114, 22, 62 142 1 12222 1 62 or ac 5 0.9113ax 2 0.1302ay 1 0.3906az Note that 0 ac 0 5 1 as expected. PRACTICE EXERCISE 1.1 Given vectors A 5 ax 1 3az and B 5 5ax 1 2ay 2 6az, determine (a) uA 1 Bu (b) 5A 2 B (c) The component of A along ay (d) A unit vector parallel to 3A 1 B Answer: (a) 7, (b) (0, 22, 21), (c) 0, (d) 6(0.9117, 0.2279, 0.3419). Points P and Q are located at (0, 2, 4) and 123, 1, 52. Calculate (a) The position of vector rP (b) The distance vector from P to Q (c) The distance between P and Q (d) A vector parallel to PQ with magnitude of 10 EXAMPLE 1.2 01_Sadiku_Ch01.indd 9 01/11/17 1:14 PM
  • 39. 10 CHAPTER 1 VECTOR ALGEBRA Solution: (a) rP 5 0ax 1 2ay 1 4az 5 2ay 1 4az (b)  rPQ 5 rQ 2 rP 5 123, 1, 52 2 10, 2, 42 5 123, 21, 12 or rPQ 5 23ax 2 ay 1 az (c) Since rPQ is the distance vector from P to Q, the distance between P and Q is the mag- nitude of this vector; that is, d 5 0 rPQ 0 5 9 1 1 1 1 5 3.317 Alternatively: d 5 1xQ 2 xP 22 1 1yQ 2 yP 22 1 1zQ 2 zP 22 5 9 1 1 1 1 5 3.317 (d) Let the required vector be A, then A 5 AaA where A 5 10 is the magnitude of A. Since A is parallel to PQ, it must have the same unit vector as rPQ or rQP. Hence, aA 5 6 rPQ 0 rPQ 0 5 6 123, 21, 12 3.317 and A 5 6 10123, 21, 12 3.317 5 6129.045ax 2 3.015ay 1 3.015az 2 PRACTICE EXERCISE 1.2 Given points P(1, 23, 5), Q(2, 4, 6), and R(0, 3, 8), find (a) the position vectors of P and R, (b) the distance vector rQR, (c) the distance between Q and R. Answer: (a) ax 2 3ay 1 5az, 3ax 1 8az, (b) 22ax 2 ay 1 2az, (c) 3. A river flows southeast at 10 km/hr and a boat floats upon it with its bow pointed in the direction of travel. A man walks upon the deck at 2 km/hr in a direction to the right and perpendicular to the direction of the boat’s movement. Find the velocity of the man with respect to the earth. Solution: Consider Figure 1.6 as illustrating the problem. The velocity of the boat is 5 7.071ax 2 7.071ay km/hr EXAMPLE 1.3 01_Sadiku_Ch01.indd 10 01/11/17 1:14 PM ub 5 101cos 45° ax 2 sin 45° ay 2
  • 40. 1.7 Vector Multiplication 11 PRACTICE EXERCISE 1.3 An airplane has a ground speed of 350 km/hr in the direction due west. If there is a wind blowing northwest at 40 km/hr, calculate the true air speed and heading of the airplane. Answer: 379.3 km/hr, 4.275° north of west. FIGURE 1.6 For Example 1.3. 1.7 VECTOR MULTIPLICATION When two vectors A and B are multiplied, the result is either a scalar or a vector depending on how they are multiplied. Thus there are two types of vector multiplication: 1. Scalar (or dot) product: A # B 2. Vector (or cross) product: A 3 B 01_Sadiku_Ch01.indd 11 01/11/17 1:14 PM t uab 5 um 1 ub 5 5.657ax 2 8.485ay 0 uab 0 5 10.2l256.3° i ii that is, 10.2 km/hr at 56.3° south of east. The velocity of the man with respect to the boat (relative velocity) is um 5 212 cos 45° ax 2 sin 45° ay 2 5 21.414ax 2 1.414ay km/hr Thus the absolute velocity of the man is
  • 41. 12 CHAPTER 1 VECTOR ALGEBRA Multiplication of three vectors A, B, and C can result in either: 3. Scalar triple product: A # 1B 3 C2 or 4. Vector triple product: A 3 1B 3 C2 A. Dot Product The dot product of two vectors A and B, written as A ? B, is defined geometrically as the product of the magnitudes of A and B and the cosine of the smaller angle between them when they are drawn tail to tail. Thus, A # B 5 AB cos uAB (1.15) where uAB is the smaller angle between A and B. The result of A # B is called either the scalar product because it is scalar or the dot product due to the dot sign. If A 5 1Ax, Ay, Az 2 and B 5 1Bx, By, Bz), then A # B 5 AxBx 1 AyBy 1 AzBz (1.16) which is obtained by multiplying A and B component by component. Two vectors A and B are said to be orthogonal (or perpendicular) with each other if A # B 5 0. Note that dot product obeys the following: (i) Commutative law: A # B 5 B # A (1.17) (ii) Distributive law: A # 1B 1 C2 5 A # B 1 A # C (1.18) (iii) A # A 5 0 A 0 2 5 A2 (1.19) Also note that ax # ay 5 ay # az 5 az # ax 5 0 (1.20a) ax # ax 5 ay # ay 5 az # az 5 1 (1.20b) It is easy to prove the identities in eqs. (1.17) to (1.20) by applying eq. (1.15) or (1.16). If A # B 5 0, the two vectors A and B are orthogonal or perpendicular. 01_Sadiku_Ch01.indd 12 01/11/17 1:14 PM
  • 42. 1.7 Vector Multiplication 13 B. Cross Product The cross product of two vectors A and B, written as A 3 B, is a vector quantity whose magnitude is the area of the parallelogram formed by A and B (see Figure 1.7) and is in the direction of advance of a right-handed screw as A is turned into B. Thus, A 3 B 5 AB sin uABan (1.21) where an is a unit vector normal to the plane containing A and B. The direction of an is taken as the direction of the right thumb when the fingers of the right hand rotate from A to B as shown in Figure 1.8(a). Alternatively, the direction of an is taken as that of the advance of a right-handed screw as A is turned into B as shown in Figure 1.8(b). The vector multiplication of eq. (1.21) is called cross product owing to the cross sign; it is also called vector product because the result is a vector. If A 5 1Ax, Ay, Az) and B 5 1Bx, By, Bz), then A 3 B 5 3 ax ay az Ax Ay Az Bx By Bz 3 (1.22a) 5 1AyBz 2 AzBy 2ax 1 1AzBx 2 AxBz 2ay 1 1AxBy 2 AyBx 2az (1.22b) which is obtained by “crossing” terms in cyclic permutation, hence the name “cross product.” A A B B FIGURE 1.7 The cross product of A and B is a vector with magnitude equal to the area of the parallelogram and direction as indicated. 01_Sadiku_Ch01.indd 13 01/11/17 1:14 PM
  • 43. 14 CHAPTER 1 VECTOR ALGEBRA Note that the cross product has the following basic properties: (i) It is not commutative: A 3 B 2 B 3 A (1.23a) It is anticommutative: A 3 B 5 2B 3 A (1.23b) (ii) It is not associative: A 3 1B 3 C2 2 1A 3 B2 3 C (1.24) (iii) It is distributive: A 3 1B 1 C2 5 A 3 B 1 A 3 C (1.25) (iv) Scaling: kA 3 B 5 A 3 kB 5 k1A 3 B2 (1.26) (v) A 3 A 5 0 (1.27) Also note that ax 3 ay 5 az ay 3 az 5 ax (1.28) az 3 ax 5 ay which are obtained in cyclic permutation and illustrated in Figure 1.9. The identities in eqs. (1.23) to (1.28) are easily verified by using eq. (1.21) or (1.22). It should be noted that in obtaining an, we have used the right-hand or right-handed-screw rule because we want to FIGURE 1.8 Direction of A 3 B and an using (a) the right-hand rule and (b) the right-handed-screw rule. 01_Sadiku_Ch01.indd 14 01/11/17 1:14 PM
  • 44. 1.7 Vector Multiplication 15 be consistent with our coordinate system illustrated in Figure 1.1, which is right-handed. A right-handed coordinate system is one in which the right-hand rule is satisfied: that is, ax 3 ay 5 az is obeyed. In a left-handed system, we follow the left-hand or left-handed screw rule and ax 3 ay 5 2az is satisfied. Throughout this book, we shall stick to right- handed coordinate systems. Just as multiplication of two vectors gives a scalar or vector result, multiplication of three vectors A, B, and C gives a scalar or vector result, depending on how the vectors are multiplied. Thus we have a scalar or vector triple product. C. Scalar Triple Product Given three vectors A, B, and C, we define the scalar triple product as A # 1B 3 C2 5 B # 1C 3 A2 5 C # 1A 3 B2 (1.29) obtained in cyclic permutation. If A 5 1Ax, Ay, Az), B 5 1Bx, By, Bz), and C 5 1Cx, Cy, Cz), then A # 1B 3 C2 is the volume of a parallelepiped having A, B, and C as edges and is easily obtained by finding the determinant of the 3 3 3 matrix formed by A, B, and C; that is, A # 1B 3 C2 5 3 Ax Ay Az Bx By Bz Cx Cy Cz 3 (1.30) Since the result of this vector multiplication is scalar, eq. (1.29) or (1.30) is called the scalar triple product. D. Vector Triple Product For vectors A, B, and C, we define the vector triple product as A 3 1B 3 C2 5 B1A # C2 2 C1A # B2 (1.31) FIGURE 1.9 Cross product using cyclic permutation. (a) Moving clockwise leads to positive results. (b) Moving counterclockwise leads to negative results. 01_Sadiku_Ch01.indd 15 01/11/17 1:14 PM
  • 45. 16 CHAPTER 1 VECTOR ALGEBRA which may be remembered as the “bac-cab” rule. It should be noted that 1A # B2C 2 A1B # C2 (1.32) but 1A # B2C 5 C1A # B2 (1.33) 1.8 COMPONENTS OF A VECTOR A direct application of scalar product is its use in determining the projection (or compo- nent) of a vector in a given direction. The projection can be scalar or vector. Given a vector A, we define the scalar component AB of A along vector B as [see Figure 1.10(a)] AB 5 A cos uAB 5 0 A 0 0 aB 0 cos uAB or AB 5 A # aB (1.34) The vector component AB of A along B is simply the scalar component in eq. (1.34) multi- plied by a unit vector along B; that is, AB 5 ABaB 5 1A # aB 2aB (1.35) Both the scalar and vector components of A are illustrated in Figure 1.10. Notice from Figure 1.10(b) that the vector can be resolved into two orthogonal components: one ­ component AB par- allel to B, another 1A 2 AB 2 perpendicular to B. In fact, our Cartesian representation of a vector is essentially resolving the vector into three mutually orthogonal components as in Figure 1.1(b). We have considered addition, subtraction, and multiplication of vectors. However, divi- sion of vectors A/B has not been considered because it is undefined except when A and B are parallel so that A 5 kB, where k is a constant. Differentiation and integration of vectors will be considered in Chapter 3. FIGURE 1.10 Components of A along B: (a) scalar component AB, (b) vector component AB. 01_Sadiku_Ch01.indd 16 01/11/17 1:14 PM
  • 46. 1.8 Components of a Vector 17 Given vectors A 5 3ax 1 4ay 1 az and B 5 2ay 2 5az, find the angle between A and B. Solution: The angle uAB can be found by using either dot product or cross product. A # B 5 13, 4, 12 # 10, 2, 252 5 0 1 8 2 5 5 3 0 A 0 5 32 1 42 1 12 5 26 0 B 0 5 02 1 22 1 12522 5 29 cos uAB 5 A # B 0 A 0 0 B 0 5 3 1262 1292 5 0.1092 uAB 5 cos21 Alternatively: A 3 B 5 3 ax ay az 3 4 1 0 2 25 3 5 1220 2 22ax 1 10 1 152ay 1 16 2 02az 5 1222, 15, 62 0 A 3 B 0 5 122222 1 152 1 62 5 745 sin uAB 5 0 A 3 B 0 0 A 0 0 B 0 5 745 1262 1292 5 0.994 uAB 5 sin21 0.994 5 83.73° PRACTICE EXERCISE 1.4 If A 5 ax 1 3az and B 5 5ax 1 2ay 2 6az, find uAB. Answer: 120.6°. Three field quantities are given by P 5 2ax 2 az Q 5 2ax 2 ay 1 2az R 5 2ax 2 3ay 1 az Determine (a) 1P 1 Q2 3 1P 2 Q2 (b) Q # R 3 P (c) P # Q 3 R EXAMPLE 1.4 EXAMPLE 1.5 01_Sadiku_Ch01.indd 17 01/11/17 1:14 PM 0.1092 5 83.73°
  • 47. 18 CHAPTER 1 VECTOR ALGEBRA (d) sin uQR (e) P 3 1Q 3 R2 (f) A unit vector perpendicular to both Q and R (g) The component of P along Q Solution: (a) 1P 1 Q2 3 1P 2 Q2 5 P 3 1P 2 Q2 1 Q 3 1P 2 Q2 5 P 3 P 2 P 3 Q 1 Q 3 P 2 Q 3 Q 5 0 1 Q 3 P 1 Q 3 P 2 0 5 2Q 3 P 5 2 3 ax ay az 2 21 2 2 0 21 3 5 211 2 02 ax 1 214 1 22 ay 1 210 1 22 az 5 2ax 1 12ay 1 4az (b) The only way Q # R 3 P makes sense is Q # 1R 3 P2 5 12, 21, 22 # 3 ax ay az 2 23 1 2 0 21 3 5 12, 21, 22 # 13, 4, 62 5 6 2 4 1 12 5 14 Alternatively: Q # 1R 3 P2 5 2 21 2 2 23 1 2 0 21 01_Sadiku_Ch01.indd 18 01/11/17 1:14 PM 3 3 To find the determinant of a 3 3 3 matrix, we repeat the first two rows and cross multiply; when the cross multiplication is from right to left, the result should be negated as shown diagrammatically here. This technique of finding a determinant applies only to a 3 3 3 matrix. Hence, Q # 1R 3 P2 5 5 2 21 2 2 23 1 2 0 21 2 21 2 2 23 1 5 5 16 1 0 2 2 1 12 2 0 2 2 5 14 as obtained before.
  • 48. 1.8 Components of a Vector 19 (c) From eq. (1.29) P # 1Q 3 R2 5 Q # 1R 3 P2 5 14 or P # 1Q 3 R2 5 12, 0, 212 # 15, 2, 242 5 10 1 0 1 4 5 14 (d) sin uQR 5 0 Q 3 R 0 0 Q 0 0 R 0 5 15, 2, 242 0 0 12, 21, 22 0 0 12, 23, 12 0 5 45 314 5 5 14 5 0.5976 (e) P 3 1Q 3 R2 5 12, 0, 212 3 15, 2, 242 5 12, 3, 42 Alternatively, using the bac-cab rule, P 3 1Q 3 R2 5 Q1P # R2 2 R1P # Q2 5 12, 21, 22 14 1 0 2 12 2 12, 23, 12 14 1 0 2 22 5 12, 3, 42 (f) A unit vector perpendicular to both Q and R is given by a 5 6Q 3 R 0 Q 3 R 0 5 615, 2, 242 45 5 610.745, 0.298, 20.5962 Note that 0 a 0 5 1, a # Q 5 0 5 a # R. Any of these can be used to check a. (g) The component of P along Q is PQ 5 0 P 0 cos uPQaQ 5 1P # aQ 2aQ 5 1P # Q2Q 0 Q 0 2 5 14 1 0 2 22 12, 21, 22 14 1 1 1 42 5 2 9 12, 21, 22 5 0.4444ax 2 0.2222ay 1 0.4444a 01_Sadiku_Ch01.indd 19 01/11/17 1:14 PM
  • 49. 20 CHAPTER 1 VECTOR ALGEBRA PRACTICE EXERCISE 1.5 Let E 5 3ay 1 4az and F 5 4ax 2 10ay 1 5az. (a) Find the component of E along F. (b) Determine a unit vector perpendicular to both E and F. Answer: (a) (20.2837, 0.7092, 20.3546), (b) 6(0.9398, 0.2734, 20.205). Derive the cosine formula a2 5 b2 1 c2 2 2bc cos A and the sine formula sin A a 5 sin B b 5 sin C c using dot product and cross product, respectively. Solution: Consider a triangle as shown in Figure 1.11. From the figure, we notice that a 1 b 1 c 5 0 that is, b 1 c 5 2a Hence, a2 5 a # a 5 1b 1 c2 # 1b 1 c2 5 b # b 1 c # c 1 2b # c a2 5 b2 1 c2 2 2bc cos A where (p 2 A) is the angle between b and c. The area of a triangle is half of the product of its height and base. Hence, ` 1 2 a 3 b` 5 ` 1 2 b 3 c` 5 ` 1 2 c 3 a` ab sin C 5 bc sin A 5 ca sin B Dividing through by abc gives sin A a 5 sin B b 5 sin C c EXAMPLE 1.6 01_Sadiku_Ch01.indd 20 01/11/17 1:14 PM
  • 50. 1.8 Components of a Vector 21 EXAMPLE 1.7 FIGURE 1.11 For Example 1.6. PRACTICE EXERCISE 1.6 Show that vectors a 5 (4, 0, 21), b 5 (1, 3, 4), and c 5 (25, 23, 23) form the sides of a triangle. Is this a right angle triangle? Calculate the area of the triangle. Answer: Yes, 10.5. Show that points P1 15, 2, 242, P2 11, 1, 22, and P3 123, 0, 82 all lie on a straight line. Determine the shortest distance between the line and point P4 13, 21, 02. Solution: The distance vector rP1P2 is given by rP1P2 5 rP2 2 rP1 5 11, 1, 22 2 15, 2, 242 5 124, 21, 62 Similarly, rP1P3 5 rP3 2 rP1 5 123, 0, 82 2 15, 2, 242 5 128, 22, 122 rP1P4 5 rP4 2 rP1 5 13, 21, 02 2 15, 2, 242 5 122, 23, 42 rP1P2 3 rP1P3 5 3 ax ay az 24 21 6 28 22 12 3 5 10, 0, 02 showing that the angle between rP1P2 and rP1P3 is zero 1sin u 5 02. This implies that P1, P2, and P3 lie on a straight line. Alternatively, the vector equation of the straight line is easily determined from Figure 1.12(a). For any point P on the line joining P1 and P2 rP1P 5 lrP1P2 where λ is a constant. Hence the position vector rP of the point P must satisfy rP 2 rP1 5 l1rP2 2 rP1 2 01_Sadiku_Ch01.indd 21 01/11/17 1:14 PM
  • 51. 22 CHAPTER 1 VECTOR ALGEBRA that is, rP 5 rP1 1 l1rP2 2 rP1 2 5 15, 2, 242 2 l14, 1, 262 rP 5 15 2 4l, 2 2 l, 24 1 6l2 This is the vector equation of the straight line joining P1 and P2. If P3 is on this line, the position vector of P3 must satisfy the equation; r3 does satisfy the equation when l 5 2. The shortest distance between the line and point P4 13, 21, 02 is the perpendicular distance from the point to the line. From Figure 1.12(b), it is clear that d 5 rP1P4 sin u 5 0 rP1P4 3 aP1P2 0 5 0 122, 23, 42 3 124, 21, 62 0 0 124, 21, 62 0 5 312 53 5 2.426 Any point on the line may be used as a reference point. Thus, instead of using P1 as a reference point, we could use P3. If jP4P3 P2 5 , then d 5 0 rP3P4 0 sin ur 5 0 rP3P4 3 aP3P2 0 PRACTICE EXERCISE 1.7 If P1 is (1, 2, 23) and P2 is (24, 0, 5), find (a) The distance P1P2 (b) The vector equation of the line P1P2 (c) The shortest distance between the line P1P2 and point P3 (7, 21, 2) Answer: (a) 9.644, (b) (1 2 5l)ax 1 2(1 2 l)ay 1 (8l 2 3)az, (c) 8.2. FIGURE 1.12 For Example 1.7. 01_Sadiku_Ch01.indd 22 20/11/17 2:38 PM
  • 52. Summary 23 1. A field is a function that specifies a quantity in space. For example, A(x, y, z) is a vector field, whereas V(x, y, z) is a scalar field. 2. AvectorAisuniquelyspecifiedbyitsmagnitudeandaunitvectoralongit,thatis,A 5 AaA. 3. Multiplying two vectors A and B results in either a scalar A # B 5 AB cos uAB or a vector A 3 B 5 AB sin uAB an. Multiplying three vectors A, B, and C yields a scalar A # 1B 3 C2 or a vector A 3 1B 3 C2. 4. The scalar projection (or component) of vector A onto B is AB 5 A # aB, whereas vector projection of A onto B is AB 5 ABaB. 5. The MATLAB commands dot(A,B) and cross(A,B) are used for dot and cross products, respectively. % This script allows the user to input two vectors and % then compute their dot product, cross product, sum, % and difference clear vA = input(‵Enter vector A in the format [x y z]... n ‵); if isempty(vA); vA = [0 0 0]; end % if the input is % entered incorrectly set the vector to 0 vB = input(‵Enter vector B in the format [x y z]... n ‵); if isempty(vB); vB = [0 0 0]; end disp(‵Magnitude of A:’) disp(norm(vA)) % norm finds the magnitude of a % multi-dimensional vector disp(‵Magnitude of B:’) disp(norm(vB)) disp(‵Unit vector in direction of A:’) disp(vA/norm(vA)) % unit vector is the vector % divided by its magnitude disp(‵Unit vector in direction of B:’) disp(vB/norm(vB)) disp(‵Sum A+B:’) disp(vA+vB) disp(‵Difference A-B:’) disp(vA-vB) disp(‵Dot product (A . B):’) disp(dot(vA,vB)) % dot takes the dot product of vectors disp(‵Cross product (A x B):’) disp(cross(vA,vB)) % cross takes cross product of vectors MATLAB 1.1 SUMMARY 01_Sadiku_Ch01.indd 23 16/11/17 3:00 PM
  • 53. 24 CHAPTER 1 VECTOR ALGEBRA 1.1 Tell which of the following quantities is not a vector: (a) force, (b) momentum, (c) accelera- tion, (d) work, (e) weight. 1.2 Which of the following is not a scalar field? (a) Displacement of a mosquito in space (b) Light intensity in a drawing room (c) Temperature distribution in your classroom (d) Atmospheric pressure in a given region (e) Humidity of a city 1.3 Of the rectangular coordinate systems shown in Figure 1.13, which are not right handed? 1.4 Which of these is correct? (a) A 3 A 5 0 A 0 2 (d) ax # ay 5 az (b) A 3 B 1 B 3 A 5 0 (e) ak 5 ax 2 ay, where ak is a unit vector (c) A # B # C 5 B # C # A 1.5 Which of the following identities is not valid? (a) a1b 1 c2 5 ab 1 bc (d) c # 1a 3 b2 5 2b # 1a 3 c2 (b) a 3 1b 1 c2 5 a 3 b 1 a 3 c (e) aA # aB 5 cos uAB (c) a # b 5 b # a 1.6 Which of the following statements are meaningless? (a) A # B 1 2A 5 0 (c) A1A 1 B2 1 2 5 0 (b) A # B 1 5 5 2A (d) A # A 1 B # B 5 0 1.7 Let F 5 2ax 2 6ay 1 10az and G 5 ax 1 Gyay 1 5az. If F and G have the same unit vector, Gy is (a) 6 (c) 0 (b) 23 (d) 6 1.8 Given that A 5 ax 1 aay 1 az and B 5 aax 1 ay 1 az, if A and B are normal to each other, αis (a) 22 (d) 1 (b) 21/2 (e) 2 (c) 0 1.9 The component of 6ax 1 2ay 2 3az along 3ax 2 4ay is (a) 212ax 2 9ay 2 3az (d) 2 (b) 30ax 2 40ay (e) 10 (c) 10/7 REVIEW QUESTIONS 01_Sadiku_Ch01.indd 24 16/11/17 1:22 PM
  • 54. Problems 25 1.10 Given A 5 26ax 1 3ay 1 2az, the projection of A along ay is (a) 212 (d) 7 (b) 24 (e) 12 (c) 3 Answers: 1.1d, 1.2a, 1.3b,e, 1.4b, 1.5a, 1.6a,b,c, 1.7b, 1.8b, 1.9d, 1.10c. Section 1.4—Unit Vector 1.1 Determine the unit vector along the direction OP, where O is the origin and P is point (4, 25, 1). 1.2 Points A(4, 26, 2), B(22, 0, 3), and C(10, 1, 27) form a triangle. Show that rAB 1 rBC 1 rCA = 0. Sections 1.5–1.7—Vector Addition, Subtraction, and Multiplication 1.3 If A 5 4ax 2 2ay 1 6az and B 5 12ax 1 18ay 2 8az, determine: (a) A 2 3B (b) 12A 1 5B2/|B| (c) ax 3 A (d) 1B 3 ax 2 # ay 1.4 Let vectors A 5 10ax 2 6ay 1 8az and B 5 ax 1 2az. Find: (a) A  B, (b) A 3 B, (c) 2A – 3B. FIGURE 1.13 For Review Question 1.3. PROBLEMS 01_Sadiku_Ch01.indd 25 16/11/17 1:22 PM
  • 55. 26 CHAPTER 1 VECTOR ALGEBRA 1.5 Let A 5 22ax 1 5ay 1 az, B 5 ax 1 3az, and C 5 4ax 26ay 1 10az. (a) Determine A 2 B 1 C (b) Find A  (B 3 C) (c) Calculate the angle between A and B 1.6 Let A 5 ax 2 az, B 5 ax 1 ay 1 az, C 5 ay 1 2az, find: (a) A # 1B 3 C2 (b) 1A 3 B2 # C (c) A 3 1B 3 C2 (d) 1A 3 B2 3 C 1.7 Given that the position vectors of points T and S are 4ax 1 6ay 2 az and 10ax 1 12ay 1 8az, respectively, find: (a) the coordinates of T and S, (b) the distance vector from T to S, (c) the distance between T and S. 1.8 Let A 5 4ax 1 2ay 2 az and B 5 aax 1 bay 1 3az (a) If A and B are parallel, find a and b (b) If A and B are perpendicular, find a and b 1.9 Let A 5 10ax 1 5ay 2 2az. Find: (a) A 3 ay, (b) A  az, (c) the angle between A and az. 1.10 (a) Show that 1A # B22 1 |A 3 B|2 5 1AB22 (b) Show that ax 5 ay 3 az ax # ay 3 az , ay 5 az 3 ax ax # ay 3 az , az 5 ax 3 ay ax # ay 3 az 1.11 Given that P 5 2ax 2 ay 2 2az Q 5 4ax 1 3ay 1 2az R 5 2ax 1 ay 1 2az find: (a) 0 P 1 Q 2 R 0 , (b) P # Q 3 R, (c) Q 3 P # R, (d) 1P 3 Q2 # 1Q 3 R2, (e) 1P 3 Q2 3 1Q 3 R2, (f) cos uPR, (g) sin uPQ. 1.12 If A 5 4ax 2 6ay 1 az and B 5 2ax 1 5az , find: (a) A  B + 2|B|2 (b) a unit vector perpendicular to both A and B 01_Sadiku_Ch01.indd 26 16/11/17 1:22 PM
  • 56. Problems 27 1.13 Determine the dot product, cross product, and angle between P 5 2ax 2 6ay 1 5az    and    Q 5 3ay 1 az 1.14 Prove that vectors P 5 2ax 1 4ay 2 6az and Q 5 5ax 1 2ay 2 3az are orthogonal ­ vectors. 1.15 Simplify the following expressions: (a) A 3 1A 3 B2 (b) A 3 3A 3 1A 3 B2 4 1.16 A right angle triangle has its corners located at P1(5, 23, 1), P2(1, 22, 4), and P3(3, 3, 5). (a) Which corner is a right angle? (b) Calculate the area of the triangle. 1.17 Points P, Q, and R are located at 121, 4, 82, 12, 21, 32, and 121, 2, 32, respectively. Determine (a) the distance between P and Q, (b) the distance vector from P to R, (c) the angle between QP and QR, (d) the area of triangle PQR, (e) the perimeter of triangle PQR. 1.18 Two points P12, 4, 212 and Q(12, 16, 9) form a straight line. Calculate the time taken for a sonar signal traveling at 300 m/s to get from the origin to the midpoint of PQ. 1.19 Find the area of the parallelogram formed by the vectors D 5 4ax 1 ay 1 5az and E 5 2ax 1 2ay 1 3az. *1.20 (a) Prove that P 5 cos u1ax 1 sin u1ay and Q 5 cos u2ax 1 sin u2ay are unit vectors in the xy-plane, respectively, making angles u1 and u2 with the x-axis. (b)  By means of dot product, obtain the formula for cos1u2 2 u1 2. By similarly formu- lating P and Q, obtain the formula for cos1u2 1 u1 2. (c) If u is the angle between P and Q, find 1 2 0 P 2 Q 0 in terms of u. 1.21 Consider a rigid body rotating with a constant angular velocity v radians per second about a fixed axis through O as in Figure 1.14. Let r be the distance vector from O to P, the position of a particle in the body. The magnitude of the velocity u of the body at P is 0 u 0 5 d 0 v 0 5 0 r 0 sin u 0 v 0 or u 5 v 3 r. If the rigid body is rotating at 3 rad/s about an axis parallel to ax 2 2ay 1 2az and passing through point 12, 23, 12, determine the velocity of the body at (1, 3, 4). 1.22 A cube of side 1 m has one corner placed at the origin. Determine the angle between the diagonals of the cube. 1.23 Given vectors T 5 2ax 2 6ay 1 3az and S 5 ax 1 2ay 1 az, find (a) the scalar projection of T on S, (b) the vector projection of S on T, (c) the smaller angle between T and S. *Single asterisks indicate problems of intermediate difficulty. 01_Sadiku_Ch01.indd 27 16/11/17 1:22 PM
  • 57. 28 CHAPTER 1 VECTOR ALGEBRA FIGURE 1.14 For Problem 1.21. Section 1.8—Components of a Vector 1.24 Given two vectors A and B, show that the vector component of A perpendicular to B is C 5 A 2 A # B B # B B 1.25 Let A 5 20ax 1 15ay 2 10az and B 5 ax 1 ay. Find: (a) A  B, (b) A 3 B, (c) the compo- nent of A along B. 1.26 Figure 1.15 shows that A makes specific angles with respect to each axis. For A 5 2ax 2 4ay 1 6az, find the direction angles a, b, and g. 1.27 If H 5 2xyax 2 1x 1 z2ay 1 z2 az, find: (a) A unit vector parallel to H at P11, 3, 222 (b) The equation of the surface on which 0 H 0 5 10 1.28 Let P 5 2ax 2 4ay 1 az and Q 5 ax 1 2ay. Find R which has magnitude 4 and is perpen- dicular to both P and Q. 1.29 Let G 5 x2 ax 2 yay 1 2zaz and H 5 yzax 1 3ay 2 xzaz. At point (1, 22, 3), (a) calculate the magnitude of G and H, (b) determine G  H, (c) find the angle between G and H. 1.30 A vector field is given by H 5 10yz2 ax 2 8xyzay 1 12y2 az (a) Evaluate H at P(21, 2, 4) (b) Find the component of H along ax 2 ay at P. 01_Sadiku_Ch01.indd 28 16/11/17 2:22 PM
  • 58. 1.31 E and F are vector fields given by E 5 2xax 1 ay 1 yzaz and F 5 xyax 2 y2 ay1 xyzaz. Determine: (a) 0 E 0 at (1, 2, 3) (b) The component of E along F at (1, 2, 3) (c) A vector perpendicular to both E and F at 10, 1, 232 whose magnitude is unity 1.32 Given two vector fields D 5 yzax 1 xzay 1 xyaz  and  E 5 5xyax 1 6(x2 1 3)ay 1 8z2 az (a) Evaluate C 5 D 1 E at point P(21, 2, 4). (b) Find the angle C makes with the x-axis at P. z x y A α β γ FIGURE 1.15 For Problem 1.26. Problems 29 01_Sadiku_Ch01.indd 29 16/11/17 2:22 PM
  • 59. The Accreditation Board for Engineering and Technology (ABET) establishes eleven criteria for accrediting engineering, technology, and computer science programs. The criteria are as follows: A. Ability to apply mathematics science and engineering principles B. Ability to design and conduct experiments and interpret data C. Ability to design a system, component, or process to meet desired needs D. Ability to function on multidisciplinary teams E. Ability to identify, formulate, and solve engineering problems F. Ability to understand professional and ethical responsibility G. Ability to communicate effectively H. Ability to understand the impact of engineering solutions in a global context I. Ability to recognize the need for and to engage in lifelong learning J. Ability to know of contemporary issues K. Ability to use the techniques, skills, and modern engineering tools necessary for engineering practice Criterion A applies directly to electromagnetics. As students, you are expected to study math- ematics, science, and engineering with the purpose of being able to apply that knowledge to the solution of engineering problems. The skill needed here is the ability to apply the fundamentals of EM in solving a problem. The best approach is to attempt as many problems as you can. This will help you to understand how to use formulas and assimilate the material. Keep nearly all your basic mathematics, science, and engineering textbooks. You may need to consult them from time to time. ENHANCING YOUR SKILLS AND CAREER 02_Sadiku_Ch02.indd 30 01/11/17 1:50 PM
  • 60. 31 C H A P T E R 31 2.1 INTRODUCTION In general, the physical quantities we shall be dealing with in EM are functions of space and time. In order to describe the spatial variations of the quantities, we must be able to ­ define all points uniquely in space in a suitable manner. This requires using an appropriate coordinate system. A point or vector can be represented in any curvilinear coordinate system, which may be orthogonal or nonorthogonal. An orthogonal system is one in which the coordinate surfaces are mutually perpendicular. Nonorthogonal systems are hard to work with, and they are of little or no practical use. Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the cir- cular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal.1 A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem. A hard problem in one coordinate system may turn out to be easy in another system. In this text, we shall restrict ourselves to the three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical. Although we have considered the Cartesian system in Chapter 1, we shall consider it in detail in this chapter. We should bear in mind that the concepts covered in Chapter 1 and demonstrated in Cartesian coordinates are equally applicable to other systems of coordinates. For example, the procedure for find- ing the dot or cross product of two vectors in a cylindrical system is the same as that used in the Cartesian system in Chapter 1. COORDINATE SYSTEMS AND TRANSFORMATION History teaches us that man learns nothing from history. —HEGEL 2 1 For an introductory treatment of these coordinate systems, see M. R. Spiegel and J. Liu, Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 2nd ed., 1999, pp. 126–130. 02_Sadiku_Ch02.indd 31 01/11/17 1:50 PM
  • 61. Get Complete eBook Download Link below for instant download https://browsegrades.net/documents/2 86751/ebook-payment-link-for-instant- download-after-payment