Table Of ContentElectromagnetics
Fourth Edition
Joseph A. Edminister
Professor Emeritus of Electrical Engineering
The University of Akron
Mahmood Nahvi, PhD
Professor Emeritus of Electrical Engineering
California Polytechnic State University
Schaum’s Outline Series
New York Chicago San Francisco Athens London Madrid
Mexico City Milan New Delhi Singapore Sydney Toronto
JOSEPH A. EDMINISTER is currently Director of Corporate Relations for the College of Engineering at Cornell University. In 1984, he
held an IEEE Congressional Fellowship in the offi ce of Congressman Dennis E. Eckart (D-OH). He received BEE, MSE, and JD degrees
from the University of Akron. He served as professor of electrical engineering, acting department head of electrical engineering, assistant
dean and acting dean of engineering, all at the University of Akron. He is an attorney in the state of Ohio and a registered patent attorney.
He taught electric circuit analysis and electromagnetic theory throughout his academic career. He is a Professor Emeritus of Electrical
Engineering from the University of Akron.
MAHMOOD NAHVI is Professor Emeritus of Electrical Engineering at California Polytechnic State University in San Luis Obispo,
California. He earned his BSc, MSc, and PhD, all in electrical engineering, and has 50 years of teaching and research in this fi eld. Dr.
Nahvi’s areas of special interest and expertise include network theory, control theory, communications engineering, signal processing,
neural networks, adaptive control and learning in synthetic and living systems, communication and control in the central nervous system,
and engineering education. In the area of engineering education, he has developed computer modules for electric circuits, signals, and
systems which improve the teaching and learning of the fundamentals of electrical engineering.
Copyright © 2014, 2011, 1993, 1979 by McGraw-Hill Education. All rights reserved. Except as permitted under the United States
Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or
retrieval system, without the prior written permission of the publisher.
ISBN: 978-0-07-183148-2
MHID: 0-07-183148-7
The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-183147-5,
MHID: 0-07-183147-9.
eBook conversion by codeMantra
Version 1.0
All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked
name, we use names in an editorial fashion only, and to the benefi t of the trademark owner, with no intention of infringement of the
trademark. Where such designations appear in this book, they have been printed with initial caps.
McGraw-Hill Education eBooks are available at special quantity discounts to use as premiums and sales promotions or for use in corporate
training programs. To contact a representative, please visit the Contact Us page at www.mhprofessional.com.
Trademarks: McGraw-Hill Education, the McGraw-Hill Education logo, Schaum’s, and related trade dress are trademarks or registered
trademarks of McGraw-Hill Education and/or its affi liates in the United States and other countries, and may not be used without written
permission. All other trademarks are the property of their respective owners. McGraw-Hill Education is not associated with any product or
vendor mentioned in this book.
TERMS OF USE
This is a copyrighted work and McGraw-Hill Education and its licensors reserve all rights in and to the work. Use of this work is subject
to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not
decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell,
publish or sublicense the work or any part of it without McGraw-Hill Education’s prior consent. You may use the work for your own
noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail
to comply with these terms.
THE WORK IS PROVIDED “AS IS.” McGRAW-HILL EDUCATION AND ITS LICENSORS MAKE NO GUARANTEES OR
WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM
USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA
HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT
NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill
Education and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its
operation will be uninterrupted or error free. Neither McGraw-Hill Education nor its licensors shall be liable to you or anyone else for any
inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill Education has no
responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill Education and/or
its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability
to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim
or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.
Preface
The third edition ofSchaum’s Outline of Electromagneticsoffers several new features which make it a more pow-
erful tool for students and practitioners of electromagnetic field theory. The book is designed for use as a textbook
for a first course in electromagnetics or as a supplement to other standard textbooks, as well as a reference and
an aid to professionals. Chapter 1, which is a new chapter, presents an overview of the subject including funda-
mental theories, new examples and problems (from static fields through Maxwell’s equations), wave propagation,
and transmission lines. Chapters 5, 10, and 13 are changed greatly and reorganized. Mathematical tools such as
the gradient, divergence, curl, and Laplacian are presented in the modified Chapter 5. The magnetic field and
boundary conditions are now organized and presented in a single Chapter 10. Similarly, time-varying fields and
Maxwell’s equations are presented in a single Chapter 13. Transmission lines are discussed in Chapter 15. This
chapter can, however, be used independently from other chapters if the program of study would recommend it.
The basic approach of the previous editions has been retained. As in other Schaum’s Outlines, the empha-
sis is on how to solve problems and learning through examples. Each chapter includes statements of pertinent
definitions, simplified outlines of the principles, and theoretical foundations needed to understand the subject,
interleaved with illustrative examples. Each chapter then contains an ample set of problems with detailed solu-
tions and another set of problems with answers. The study of electromagnetics requires the use of rather advanced
mathematics, specifically vector analysis in Cartesian, cylindrical and spherical coordinates. Throughout the
book, the mathematical treatment has been kept as simple as possible and an abstract approach has been avoided.
Concrete examples are liberally used and numerous graphs and sketches are given. We have found in many
years of teaching that the solution of most problems begins with a carefully drawn and labeled sketch.
This book is dedicated to our students from whom we have learned to teach well. Contributions of Messrs
M. L. Kult and K. F. Lee to material on transmission lines, waveguides, and antennas are acknowledged. Finally,
we wish to thank our wives Nina Edminister and Zahra Nahvi for their continuing support.
JOSEPHA. EDMINISTER
MAHMOODNAHVI
iii
Contents
CHAPTER 1 The Subject of Electromagnetics 1
1.1Historical Background 1.2Objectives of the Chapter 1.3Electric Charge
1.4Units 1.5Vectors 1.6Electrical Force, Field, Flux, and Potential 1.7Magnetic
Force, Field, Flux, and Potential 1.8Electromagnetic Induction 1.9Mathematical
Operators and Identities 1.10Maxwell’s Equations 1.11Electromagnetic Waves
1.12Trajectory of a Sinusoidal Motion in Two Dimensions 1.13Wave Polarization
1.14Electromagnetic Spectrum 1.15Transmission Lines
CHAPTER 2 Vector Analysis 31
2.1Introduction 2.2Vector Notation 2.3Vector Functions 2.4Vector Algebra
2.5Coordinate Systems 2.6Differential Volume, Surface, and Line Elements
CHAPTER 3 Electric Field 44
3.1Introduction 3.2Coulomb’s Law in Vector Form 3.3Superposition
3.4Electric Field Intensity 3.5Charge Distributions 3.6Standard Charge
Configurations
CHAPTER 4 Electric Flux 63
4.1Net Charge in a Region 4.2Electric Flux and Flux Density 4.3Gauss’s Law
4.4Relation between Flux Density and Electric Field Intensity 4.5Special
Gaussian Surfaces
CHAPTER 5 Gradient,Divergence,Curl,and Laplacian 78
5.1Introduction 5.2Gradient 5.3The Del Operator 5.4The Del Operator and
Gradient 5.5Divergence 5.6Expressions for Divergence in Coordinate Systems
5.7The Del Operator and Divergence 5.8Divergence of D 5.9The Divergence
Theorem 5.10Curl 5.11Laplacian 5.12Summary of Vector Operations
CHAPTER 6 Electrostatics:Work,Energy,and Potential 97
6.1Work Done in Moving a Point Charge 6.2Conservative Property of the
Electrostatic Field 6.3Electric Potential between Two Points 6.4Potential of a
Point Charge 6.5Potential of a Charge Distribution 6.6Relationship between E
and V 6.7 Energy in Static Electric Fields
iv
Contents v
CHAPTER 7 Electric Current 113
7.1Introduction 7.2Charges in Motion 7.3Convection Current Density J
7.4Conduction Current Density J 7.5Concductivity σ7.6Current I
7.7Resistance R 7.8Current Sheet Density K 7.9Continuity of Current
7.10Conductor-Dielectric Boundary Conditions
CHAPTER 8 Capacitance and Dielectric Materials 131
8.1Polarization Pand Relative Permittivity (cid:2) 8.2Capacitance 8.3Multiple-
r
Dielectric Capacitors 8.4Energy Stored in a Capacitor 8.5Fixed-Voltage Dand E
8.6Fixed-Charge Dand E 8.7Boundary Conditions at the Interface of Two
Dielectrics 8.8Method of Images
CHAPTER 9 Laplace’s Equation 151
9.1Introduction 9.2Poisson’s Equation and Laplace’s Equation 9.3Explicit Forms
of Laplace’s Equation 9.4Uniqueness Theorem 9.5Mean Value and Maximum
Value Theorems 9.6Cartesian Solution in One Variable 9.7Cartesian Product
Solution 9.8Cylindrical Product Solution 9.9Spherical Product Solution
CHAPTER 10 Magnetic Filed and Boundary Conditions 172
10.1Introduction 10.2Biot-Savart Law 10.3Ampere’s Law 10.4Relationship of
Jand H 10.5Magnetic Flux Density B 10.6Boundary Relations for Magnetic
Fields 10.7Current Sheet at the Boundary 10.8Summary of Boundary Conditions
10.9Vector Magnetic Potential A 10.10Stokes’Theorem
CHAPTER 11 Forces and Torques in Magnetic Fields 193
11.1Magnetic Force on Particles 11.2Electric and Magnetic Fields Combined
11.3Magnetic Force on a Current Element 11.4Work and Power 11.5Torque
11.6Magnetic Moment of a Planar Coil
CHAPTER 12 Inductance and Magnetic Circuits 209
12.1Inductance 12.2Standard Conductor Configurations 12.3Faraday’s
Law and Self-Inductance 12.4Internal Inductance 12.5Mutual Inductance
12.6Magnetic Circuits 12.7The B-H Curve 12.8Ampere’s Law for Magnetic
Circuits 12.9Cores with Air Gaps 12.10Multiple Coils 12.11Parallel
Magnetic Circuits
CHAPTER 13 Time-Varying Fields and Maxwell’s Equations 233
13.1Introduction 13.2Maxwell’s Equations for Static Fields 13.3Faraday’s
Law and Lenz’s Law 13.4Conductors’Motion in Time-Independent Fields
13.5Conductors’Motion in Time-Dependent Fields 13.6Displacement Current
13.7Ratio of J to J 13.8Maxwell’s Equations for Time-Varying Fields
C D
CHAPTER 14 Electromagnetic Waves 251
14.1Introduction 14.2Wave Equations 14.3Solutions in Cartesian Coordinates
14.4Plane Waves 14.5Solutions for Partially Conducting Media 14.6Solutions
for Perfect Dielectrics 14.7Solutions for Good Conductors; Skin Depth
vi Contents
14.8Interface Conditions at Normal Incidence 14.9Oblique Incidence and
Snell’s Laws 14.10Perpendicular Polarization 14.11Parallel Polarization
14.12Standing Waves 14.13Power and the Poynting Vector
CHAPTER 15 Transmission Lines 273
15.1Introduction 15.2Distributed Parameters 15.3Incremental Models
15.4Transmission Line Equation 15.5Sinusoidal Steady-State Excitation
15.6Sinusoidal Steady-State in Lossless Lines 15.7The Smith Chart
15.8Impedance Matching 15.9Single-Stub Matching 15.10Double-Stub
Matching 15.11Impedance Measurement 15.12Transients in Lossless Lines
CHAPTER 16 Waveguides 311
16.1Introduction 16.2Transverse and Axial Fields 16.3TE and TM Modes;
Wave Impedances 16.4Determination of the Axial Fields 16.5Mode Cutoff
Frequencies 16.6Dominant Mode 16.7Power Transmitted in a Lossless
Waveguide 16.8Power Dissipation in a Lossy Waveguide
CHAPTER 17 Antennas 330
17.1Introduction 17.2Current Source and the Eand HFields 17.3Electric
(Hertzian) Dipole Antenna 17.4Antenna Parameters 17.5Small Circular-Loop
Antenna 17.6Finite-Length Dipole 17.7Monopole Antenna 17.8Self- and
Mutual Impedances 17.9The Receiving Antenna 17.10Linear Arrays
17.11Reflectors
APPENDIX 349
INDEX 350
CHAPTER 1
The Subject of Electromagnetics
1.1 Historical Background
Electric and magnetic phenomena have been known to mankind since early times. The amber effect is an exam-
ple of an electrical phenomenon: a piece of amber rubbed against the sleeve becomes electrified, acquiring a force
field which attracts light objects such as chaff and paper. Rubbing one’s woolen jacket on the hair of one’s head
elicits sparks which can be seen in the dark. Lightning between clouds (or between clouds and the earth) is
another example of familiar electrical phenomena. The woolen jacket and the clouds are electrified, acquiring
a force field which leads to sparks. Examples of familiar magnetic phenomena are natural or magnetized min-
eral stones that attract metals such as iron. The magical magnetic force, it is said, had even kept some objects
in temples floating in the air.
The scientific and quantitative exploration of electric and magnetic phenomena started in the seventeenth
and eighteenth centuries (Gilbert, 1600, Guericke, 1660, Dufay, 1733, Franklin, 1752, Galvani, 1771,
Cavendish, 1775, Coulomb, 1785, Volta, 1800). Forces between stationary electric charges were explained by
Coulomb’s law. Electrostatics and magnetostatics (fields which do not change with time) were formulated and
modeled mathematically. The study of the interrelationship between electric and magnetic fields and their time-
varying behavior progressed in the nineteenth century (Oersted, 1820 and 1826, Ampere, 1820, Faraday, 1831,
Henry, 1831, Maxwell, 1856 and 1873, Hertz, 1893)1. Oersted observed that an electric current produces a
magnetic field. Faraday verified that a time-varying magnetic field produces an electric field (emf). Henry
constructed electromagnets and discovered self-inductance. Maxwell, by introducing the concept of the dis-
placement current, developed a mathematical foundation for electromagnetic fields and waves currently known
as Maxwell’s equations. Hertz verified, experimentally, propagation of electromagnetic waves predicted by
Maxwell’s equations. Despite their simplicity, Maxwell’s equations are comprehensive in that they account
for all classical electromagnetic phenomena, from static fields to electromagnetic induction and wave propa-
gation. Since publication of Maxwell’s historical manuscript in 1873 more advances have been made in the
field, culminating in what is presently known as classical electromagnetics (EM). Currently, the important
applications of EM are in radiation and propagation of electromagnetic waves in free space, by transmission
lines, waveguides, fiber optics, and other methods. The power of these applications far surpasses any alleged
historical magical powers of healing patients or suspending objects in the air.
In order to study the subject of electromagnetics, one may start with electrostatic and magnetostatic fields,
continue with time-varying fields and Maxwell’s equations, and move on to electromagnetic wave propagation
and radiation. Alternatively, one may start with Maxwell’s equations. This book uses the first approach, starting
with the Coulomb force law between two charges. Vector algebra and vector calculus are introduced early and
as needed throughout the book.
1For some historical timelines see the references at the end of this chapter.
1
2 CHAPTER 1 The Subject of Electromagnetics
1.2 Objectives of the Chapter
This chapter is intended to provide a brief glance (and be easily understood by an undergraduate student in the
sciences and engineering) of some basic concepts and methods of the subject of electromagnetics. The objec-
tive is to familiarize the reader with the subject and let him or her know what to expect from it. The chapter can
also serve as a short summary of the main tools and techniques used throughout the book. Detailed treatments
of the concepts are provided throughout the rest of the book.
1.3 Electric Charge
The source of the force field associated with an electrified object (such as the amber rubbed against the sleeve) is
a quantity called electric charge which we will denote by Qor q. The unit of electric charge is the coulomb, shown
by the letter C (see the next section for a definition). Electric charges are of two types, labeled positive and neg-
ative. Charges of the same type repel while those of the opposite type attract each other. At the atomic level we
recognize two types of charged particles of equal numbers in the natural state: electrons and protons. An electron
has a negative charge of 1.60219 (cid:3)10(cid:4)9C (sometimes shown by the letter e) and a proton has a positive charge
of precisely the same amount as that of an electron. The choice of negative and positive labels for electric charges
on electrons and protons is accidental and rooted in history. The electric charge on an electron is the smallest
amount one may find. This quantization of charge, however, is not of interest in classical electromagnetics and
will not be discussed. Instead we will have charges as a continuous quantity concentrated at a point or distributed
on a line, a surface, or in a volume, with the charge density normally denoted by the letter ρ.
It is much easier to remove electrons from their host atoms than protons. If some electrons leave a piece of
matter which is electrically neutral, then that matter becomes positively charged. To takes our first example
again, electrons are transferred from cloth to amber when they are rubbed together. The amber then accumulates
a negative charge which becomes the source of an electric field. Some numerical properties of electrons are
given in Table 1-1.
TABLE1-1 Some Numerical Properties of Electrons
Electric charge (cid:4)1.60219 (cid:3)10(cid:4)19C
Resting mass 9.10939 (cid:3)10(cid:4)31kg
Charge to mass ratio 1.75 (cid:3)1011C/kg
Order of radius 3.8 (cid:3)10(cid:4)15m
Number of electrons per 1 C 6.24 (cid:3)1018
1.4 Units
In electromagnetics we use the International System of Units, abbreviated SI from the French le Système inter-
national d’unités (also called the rationalized MKS system). The SI system has seven basic units for seven basic
quantities. Three units come from the MKS mechanical system (the meter, the kilogram, and the second). The
fourth unit is the amperefor electric current. One ampereis the amount of constant current in each of two infi-
nitely long parallel conductors with negligible diameters separated by one meter with a resulting force between
them of 2 (cid:3)10(cid:4)7newtons per meter. These basic units are summarized in Table 1-2.
TABLE1-2 Four Basic Units in the SI System
QUANTITY SYMBOL SI UNIT ABBREVIATION
Length L, (cid:2) Meter m
Mass M, m Kilogram kg
Time T, t Second s
Current I, i Ampere A
CHAPTER 1 The Subject of Electromagnetics 3
The other three basic quantities and their corresponding SI units are the temperature in degrees kelvin (K),
the luminous intensity in candelas (cd), and the amount of a substance in moles (mol). These are not of interest
to us. Units for all other quantities of interest are derived from the four basic units of length, mass, time,and
currentusing electromechanical formulae. For example, the unit of electric charge is found from its relationship
with current and time to be q(cid:5)(cid:2) i dt. Thus, one coulomb is the amount of charge passed by one ampere in one
second, 1 C(cid:5)1 A(cid:3)s. The derived units are shown in Table 1-3.
TABLE1-3 Additional Units in the SI System Derived from the Basic Units
QUANTITY SYMBOL SI UNIT ABBREVIATION
Force F, ƒ Newton N
Energy, work W, w Joule J
Power P, p Watt W
Electric charge Q, q Coulomb C
Electric field E, e Volt/meter V/m
Electric potential V, v Volt V
Displacement D Coulomb/meter2 C/m2
Resistance R Ohm Ω
Conductance G Siemens S
Capacitance C Farad F
Inductance L Henry H
Magnetic field intensity H Ampere/meter A/m
Magnetic flux φ Weber Wb
Magnetic flux density B Tesla T
Frequency ƒ Hertz Hz
Magnetic flux density Bis sometimes measured in gauss, where 104gauss (cid:5)1 tesla. The decimal multiples and
submultiples of SI units will be used whenever possible. The symbols given in Table 1-4 are prefixed to the unit
symbols of Tables 1-2 and 1-3.
TABLE1-4 Decimal Multiples and Submultiples of Units in the SI System
PREFIX FACTOR SYMBOL
Atto 10(cid:4)18 a
Femto 10(cid:4)15 f
Pico 10(cid:4)12 p
Nano 10(cid:4)9 n
Micro 10(cid:4)6 μ
Mili 10(cid:4)3 m
Centi 10(cid:4)2 c
Deci 10(cid:4)1 d
Kilo 103 k
Mega 106 M
Giga 109 G
Tera 1012 T
Peta 1015 P
Exa 1018 E
