Schaum's Outlines Electromagnetics 4th Edition pdf pdf

  Electromagnetics 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 of Schaum’s Outline of Electromagnetics offers 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.

  J OSEPH

  A. E DMINISTER M AHMOOD N AHVI

  Contents

  CHAPTER 1 The Subject of Electromagnetics

  1

  1.1 Historical Background 1.2 Objectives of the Chapter 1.3 Electric Charge

  1.4 Units 1.5 Vectors 1.6 Electrical Force, Field, Flux, and Potential 1.7 Magnetic

  Force, Field, Flux, and Potential 1.8 Electromagnetic Induction 1.9 Mathematical Operators and Identities 1.10 Maxwell’s Equations 1.11 Electromagnetic Waves

  1.12 Trajectory of a Sinusoidal Motion in Two Dimensions 1.13 Wave Polarization

  1.14 Electromagnetic Spectrum 1.15 Transmission Lines

  CHAPTER 2 Vector Analysis

  31

  2.1 Introduction 2.2 Vector Notation 2.3 Vector Functions 2.4 Vector Algebra

  2.5 Coordinate Systems 2.6 Differential Volume, Surface, and Line Elements

  CHAPTER 3 Electric Field

  44

  

3.1 Introduction 3.2 Coulomb’s Law in Vector Form 3.3 Superposition

  3.4 Electric Field Intensity 3.5 Charge Distributions 3.6 Standard Charge

  Configurations

  CHAPTER 4 Electric Flux

  63

  4.1 Net Charge in a Region 4.2 Electric Flux and Flux Density 4.3 Gauss’s Law

  4.4 Relation between Flux Density and Electric Field Intensity 4.5 Special

  Gaussian Surfaces

  CHAPTER 5 Gradient, Divergence, Curl, and Laplacian

  78

  5.1 Introduction 5.2 Gradient 5.3 The Del Operator 5.4 The Del Operator and

  Gradient 5.5 Divergence 5.6 Expressions for Divergence in Coordinate Systems

  5.7 The Del Operator and Divergence 5.8 Divergence of D 5.9 The Divergence

  Theorem 5.10 Curl 5.11 Laplacian 5.12 Summary of Vector Operations

  CHAPTER 6 Electrostatics: Work, Energy, and Potential

  97

  6.1 Work Done in Moving a Point Charge 6.2 Conservative Property of the

  Electrostatic Field 6.3 Electric Potential between Two Points 6.4 Potential of a Point Charge 6.5 Potential of a Charge Distribution 6.6 Relationship between E and V 6.7 Energy in Static Electric Fields

  v Contents CHAPTER 7 Electric Current

  113

  7.1 Introduction 7.2 Charges in Motion 7.3 Convection Current Density J

  7.4 Conduction Current Density J 7.5 Concductivity

  7.6 Current I σ

  

7.7 Resistance R 7.8 Current Sheet Density K 7.9 Continuity of Current

  7.10 Conductor-Dielectric Boundary Conditions

  

CHAPTER 8 Capacitance and Dielectric Materials 131

  8.1 Polarization P and Relative Permittivity

  8.2 Capacitance 8.3 Multiple- ⑀ r

  Dielectric Capacitors 8.4 Energy Stored in a Capacitor 8.5 Fixed-Voltage D and E

  8.6 Fixed-Charge D and E 8.7 Boundary Conditions at the Interface of Two

  Dielectrics 8.8 Method of Images

  CHAPTER 9 Laplace’s Equation 151

  9.1 Introduction 9.2 Poisson’s Equation and Laplace’s Equation 9.3 Explicit Forms

  of Laplace’s Equation 9.4 Uniqueness Theorem 9.5 Mean Value and Maximum Value Theorems 9.6 Cartesian Solution in One Variable 9.7 Cartesian Product Solution 9.8 Cylindrical Product Solution 9.9 Spherical Product Solution

  

CHAPTER 10 Magnetic Filed and Boundary Conditions 172

  10.1 Introduction 10.2 Biot-Savart Law 10.3 Ampere’s Law 10.4 Relationship of J and H 10.5 Magnetic Flux Density B 10.6 Boundary Relations for Magnetic

  Fields 10.7 Current Sheet at the Boundary 10.8 Summary of Boundary Conditions

  10.9 Vector Magnetic Potential A 10.10 Stokes’ Theorem

  

CHAPTER 11 Forces and Torques in Magnetic Fields 193

  11.1 Magnetic Force on Particles 11.2 Electric and Magnetic Fields Combined

  11.3 Magnetic Force on a Current Element 11.4 Work and Power 11.5 Torque

  11.6 Magnetic Moment of a Planar Coil

  

CHAPTER 12 Inductance and Magnetic Circuits 209

  12.1 Inductance 12.2 Standard Conductor Configurations 12.3 Faraday’s

  Law and Self-Inductance 12.4 Internal Inductance 12.5 Mutual Inductance

  12.6 Magnetic Circuits 12.7 The B-H Curve 12.8 Ampere’s Law for Magnetic

  Circuits 12.9 Cores with Air Gaps 12.10 Multiple Coils 12.11 Parallel Magnetic Circuits

  

CHAPTER 13 Time-Varying Fields and Maxwell’s Equations 233

  13.1 Introduction 13.2 Maxwell’s Equations for Static Fields 13.3 Faraday’s

  Law and Lenz’s Law 13.4 Conductors’ Motion in Time-Independent Fields

  13.5 Conductors’ Motion in Time-Dependent Fields 13.6 Displacement Current

  13.7 Ratio of J to J

  13.8 Maxwell’s Equations for Time-Varying Fields C D

  CHAPTER 14 Electromagnetic Waves 251

  14.1 Introduction 14.2 Wave Equations 14.3 Solutions in Cartesian Coordinates

  14.4 Plane Waves 14.5 Solutions for Partially Conducting Media 14.6 Solutions

  for Perfect Dielectrics 14.7 Solutions for Good Conductors; Skin Depth

  vi Contents

  14.8 Interface Conditions at Normal Incidence 14.9 Oblique Incidence and

  Snell’s Laws 14.10 Perpendicular Polarization 14.11 Parallel Polarization

  14.12 Standing Waves 14.13 Power and the Poynting Vector

  273

  15.1 Introduction 15.2 Distributed Parameters 15.3 Incremental Models

  15.4 Transmission Line Equation 15.5 Sinusoidal Steady-State Excitation

  

15.6 Sinusoidal Steady-State in Lossless Lines 15.7 The Smith Chart

  15.8 Impedance Matching 15.9 Single-Stub Matching 15.10 Double-Stub

  Matching 15.11 Impedance Measurement 15.12 Transients in Lossless Lines

  311

16.1 Introduction 16.2 Transverse and Axial Fields 16.3 TE and TM Modes;

  Wave Impedances 16.4 Determination of the Axial Fields 16.5 Mode Cutoff Frequencies 16.6 Dominant Mode 16.7 Power Transmitted in a Lossless Waveguide 16.8 Power Dissipation in a Lossy Waveguide

  330

  17.1 Introduction 17.2 Current Source and the E and H Fields 17.3 Electric

  (Hertzian) Dipole Antenna 17.4 Antenna Parameters 17.5 Small Circular-Loop Antenna 17.6 Finite-Length Dipole 17.7 Monopole Antenna 17.8 Self- and Mutual Impedances 17.9 The Receiving Antenna 17.10 Linear Arrays

  17.11 Reflectors APPENDIX

  349

  INDEX 350

  C H A P T E R 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,

1 Henry, 1831, Maxwell, 1856 and 1873, Hertz, 1893) . 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.

1 For some historical timelines see the references at the end of this chapter.

  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 Q or 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

  ⫺9

  has a negative charge of 1.60219 ⫻ 10 C (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.

  TABLE 1-1 Some Numerical Properties of Electrons ⫺19

  Electric charge ⫺1.60219 ⫻ 10 C

  ⫺31

  Resting mass 9.10939 ⫻ 10 kg

  11 Charge to mass ratio 1.75 ⫻ 10 C/kg ⫺15

  Order of radius 3.8 ⫻ 10 m

  18 Number of electrons per 1 C 6.24 ⫻ 10

  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 ampere for electric current. One ampere is 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

  ⫺7 them of 2 ⫻ 10 newtons per meter. These basic units are summarized in Table 1-2.

  

TABLE 1-2 Four Basic Units in the SI System

QUANTITY SYMBOL SI UNIT ABBREVIATION

  Length L, ℓ Meter m Mass M, m Kilogram kg Time T, t Second s Current I, i Ampere A

CHAPTER 1 The Subject of Electromagnetics

  Giga

  

⫺3

  m Centi

  10

  

⫺2

  c Deci

  10

  

⫺1

  d Kilo

  10

  3

  k Mega

  10

  6 M

  10

  Mili

  9 G

  Tera

  10

  12 T

  Peta

  10

  15 P

  Exa

  10

  18 E

  Magnetic flux density B is sometimes measured in gauss, where 10

  4

  gauss ⫽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.

  10

  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

  

current using electromechanical formulae. For example, the unit of electric charge is found from its relationship

  Weber Wb Magnetic flux density B Tesla T Frequency ƒ Hertz Hz

  with current and time to be q ⫽

  冮 i dt. Thus, one coulomb is the amount of charge passed by one ampere in one second, 1 C ⫽ 1 A ⫻ s. The derived units are shown in Table 1-3.

  3 TABLE 1-3 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/meter

  2 C/m

  2 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

  

TABLE 1-4 Decimal Multiples and Submultiples of Units in the SI System

PREFIX FACTOR SYMBOL

  10

  Atto

  10

  

⫺18

  a Femto

  10

  

⫺15

  f Pico

  10

  

⫺12

  p Nano

  10

  

⫺9

  n Micro

  

⫺6

µ

  4

CHAPTER 1 The Subject of Electromagnetics

  1.5 Vectors

  In electromagnetics we use vectors to facilitate our calculations and explanations. A vector is a quantity specified by its magnitude and direction. Forces and force fields are examples of quantities expressed by vectors. To distin- guish vectors from scalar quantities, bold-face letters are used for the former. A vector whose magnitude is 1 is called a unit vector. To represent vectors in the Cartesian coordinate space, we employ three basic unit vectors: a , a , and

  x y a

  in the x, y, and z directions, respectively. For example, a vector connecting the origin to point A at

  z

  2

  2

  2

  ⫺ a ⫹ 3a ⫹ ⫹3 ⫽ (x ⫽ 2, y ⫽ ⫺1, z ⫽ 3) is shown by A ⫽ 2a . Its magnitude is  A ⫽ A ⫽ 兹苵苵苵苵 2 (⫺1) 苵苵苵苵 苵苵苵苵 兹苵苵 14.

  x y z

  Its direction is given by the unit vector

  A

  ⫽ ⫺ ⫹ 0 5345 . a 0 2673 . a 0 8018 . a

  x y z

  14 Three basic vector operations are addition and subtraction, A ⫾ B ⫽ (A ⫾ B )a ⫹ (A ⫾ B )a ⫹ (A ⫾ B )a ,

  

x x x y y y z z z

  dot product, A · B ⫽ AB cos , where is the smaller angle between A and B,

  θ θ cross product, A ⫻ B ⫽ AB sin a , where a is the unit vector normal to the plane parallel to A and B. θ n n

  The dot product results in a scalar quantity; hence it is also called the scalar product. It can easily be shown that A · B ⫽ A B ⫹ A B ⫹ A B . The cross product results in a vector quantity; hence, it is also called the

  x x y y z z

  vector product. A ⫻ B is normal to both A and B and follows the right-hand rule: With the fingers of the right hand rotating from A to B through angle , the thumb points in the direction of A ⫻ B. It can easily

  θ

  be shown that

  a ⫻ a ⫽ a  x y z

  

  

a ⫻ a ⫽ a A ⫻ ⫽ B ( A B ⫺ A B ) a ⫹ ( A B ⫺ A B ) a ⫹ ( A B ⫺ A B ) a

  ⇒  y z z yy x z x x z y x x y x z

  y z x

   ⫻ ⫽

  a a a z x y

  

  1.6 Electrical Force, Field, Flux, and Potential

Electrical Force. There is a force between two point charges. The force is directly proportional to the charge

  magnitudes and inversely proportional to the square of their separation distance. The direction of the force is along the line joining the two charges. For point charges of like sign the force is one of repulsion, while for unlike charges the force is attractive. The magnitude of the force is given by

  

Q Q

  

1

  2

  ⫽

  F

  2

  4 d

  

π⑀

  This is Coulomb’s law, which was developed from work with small charged bodies (spheres) and a delicate tor- sion balance (Coulomb 1785). Rationalized SI units are used. The force is in newtons (N), the distance is in meters (m), and the charge is in units called the coulomb (C). The SI system is rationalized by the factor 4 , intro-

  π

  duced in Coulomb’s law in order that it not appear later in Maxwell’s equation. is the permittivity of the medium

  ⑀

  2

  2

  with the unit C /(N . m ) or, equivalently, farads per meter (F/m). For free space or vacuum,

  9 ⫺

  10

  

12

⫺

  ⫽ ⫽ ⫻ ⑀ ⑀ 8 854 .

10 F /m ⬇ F /m

  36

  π

  For media other than free space, ⫽ , where is the relative permittivity or dielectric constant. Free space

  ⑀ ⑀ ⑀ ⑀ r r

  is to be assumed in all problems and examples as well as the approximate value for , unless there is a statement

  ⑀ to the contrary.

  5

CHAPTER 1 The Subject of Electromagnetics

  Q1 F F Q2

(a)

  

Q1 Q2

F F

d

(b)

  Q Q

  1 2 Fig. 1-1 Coulomb’s law F ⫽ . In (a) Q and Q are unlike charges, and in (b) they are of the same type.

  1

  2

  2

  4 π⑀ d

  ⫺10

  EXAMPLE 1. Two electrons in free space are separated by 1 A° (1 Angstrom ⫽ 10 m). We want to find and compare Coulomb’s electrostatic force and Newton’s gravitational force between them. Since the distance

  ⫺10 ⫺15

  between the two electrons, 10 m, is much more than their radii, 艐 3.8 ⫻ 10 m, they can be considered point charges and masses. Coulomb’s electrostatic force between them is

  2

  19 ⫺

  ⫻ 1 60219 .

  10 Q Q

  1 2  

  9 ⫺ F ⫽ ⫽ 23 1 ⫻

  e

  2

  10 N ⬇ .

  

9

⫺

  ⑀ 4 d

  10

  20 π ⫺

  ⫻ ⫻

  4

  1

  π

  4

  π

  2

  ⫽ GM Newton’s gravitational force between two masses M and M separated by a distance d is F M /d , where

  1 2 g

  1

  2 ⫺11

F is in newtons, M and M are in kg, d is in meters, and G, the gravitational constant, is G ⫽ 6.674 ⫻ 10 . With

g

  1

  2 ⫺31

  the resting mass of an electron being 9.10939 ⫻ 10 kg, the gravitational force between them is

  2

  31 ⫺

  ⫻ 9 10939 .

  1

  2 11 ⫺

  10 M M

  52 −   F ⫽ G ⫽ 6 674 . ⫻

  10 ⬇ . ⫻ 55 4

  10 N

  g

  2

  20

  2 ⫺ d

  10 Therefore, the electrical force between two electrons is approximately 42 orders of magnitude stronger than their gravitational force.

  Superposition Property.

  The presence of a third charge doesn’t change the mutual force between the other two charges, but rather adds (vectorially) its own force contribution. This is called the linear superposition prop- erty. It helps us define a vector quantity called the field intensity and use it in order to find the force on a charge at any point in the field.

  

Electric Field. The force field associated with a charge configuration is called an electric field. It is a vector

  field specified by a quantity called the electric field intensity, shown by vector E. The electric field intensity at a given point is the force on a positive unit charge, called the test charge, placed at that point. The intensity of the electric field due to a point charge Q at a distance d is a vector directed away from the point charge (if Q is positive) or toward it (if Q is negative). Its magnitude is

  Q

  ⫽

  E

  2

  4 d

  

π⑀

  In vector notation

  

Q

E ⫽ a

  2

  4 d

  

π⑀

  where a is the unit vector directed from the point charge to the test point. The unit of the electric field inten- sity is V/m. The superposition property of electric field is used to find the field due to any spatial charge configuration.

  6

CHAPTER 1 The Subject of Electromagnetics

  EXAMPLE 2.

  The electric field intensity at 10 cm away from a point charge of 0.1 C in vacuum is

  µ

  7 ⫺

  10 ⫽ ⫽

  E a

  90 a kV/m

  r r

  9 ⫺

   

  10

  

2

⫺

  4

  10

  π

  36  

  π

  where a is the radial unit vector with the charge at the center. At the distance of 1 m its magnitude is reduced to

  r

  900 V/m. If the medium is a dielectric with relative permittivity ⫽ 100 (such as titanium dioxide), the above

  ⑀ r field intensities are reduced to 900 and 9 V/m, respectively.

  

Electric Flux. An electric field is completely specified by its intensity vector. However, to help better explain

  some phenomena, we also define a scalar field called the electric flux. Electric flux is considered a quantity, albeit an imaginary one, which originates on the positive charge, moves along a stream of directional lines (called flux lines), and terminates on the negative charge, or at infinity if there are no other charges in the field. Electrical forces are thus experienced when an electric charge encounters lines of electric flux. This is analogous to the example of fluid flow where the flux originates from a source and terminates at a sink or dissipates into the environment. In this case, a vector field such as velocity defines the flux density, from which one can deter- mine the amount of fluid flow passing through a surface. Faraday envisioned the concept of electric flux, shown by Ψ, to explain how a positive charge induces an equal but negative charge on a shell which encloses it. His experimental setup consisted of an inner shell enclosed by an outer sphere. Unlike flux, which is a scalar field, its density is a vector field.

  Electric Flux Density. Flux density D in an electric field is defined by D ⫽ ⑀

  E. In the SI system, the unit of

  2

  electric flux is the coulomb (C) and that of flux density is C/m . The flux passing through a differential surface element ds is the dot product D · ds, which numerically is equal to the product of the differential surface ele- ment’s area, with the magnitude of the component of the flux density perpendicular to it.

  Gauss’s Law. The total flux out of a closed surface is equal to the net charge enclosed within the surface.

  EXAMPLE 3.

  Flux density through a spherical surface with radius d enclosing a point charge Q is

  Q

D ⫽ E ⫽ a

  ⑀

  2

  4 d

  π

  where a is the radial unit vector from the point charge to the test point on the sphere. The total flux coming out of the sphere is

  Q

  2

  2

  ⌿ ⫽ 4 ⫻ ⫽ 4 ⫻ ⫽

  d D d Q π π

  2

  4 d

  π

Electric Potential. The work done to move a unit charge from point B to point A in an electric field is called

  the potential of point A with reference to point B and shown by V . It is equal to the line integral

  AB A

  ⫽⫺

  

V E · l

d AB

  

冮

B

  The value of the integral depends only on the electric field and the two end-points. It is independent of the path traversed by the charge as long as all attempted paths share the same initial and destination points. The value of the integral over a closed path is, therefore, zero. This is a property of a conservative field such as E. If the reference point B is moved to infinity, the integral defines a scalar field called the potential field. The unit of the potential is the volt (V), the work needed to move one C of charge a distance of 1 m along a field of E ⫽ 1 V/m.

  The electric potential is introduced as a line integral of the electric field. It can also be found from charge

  7

CHAPTER 1 The Subject of Electromagnetics

  EXAMPLE 4. There exists a static electric field in the atmosphere directed downward which depends on weather conditions and decreases with height. Assuming that its intensity near the ground is about 150 V/m and remains the same within the tropospheric height, find the electric potential at a height of 333 m with respect to the ground.

  V ⫽ 150 V/m ⫻ 333 V/m 艐 50kV.

1.7 Magnetic Force, Field, Flux, and Potential

  

Magnetic Force. Permanent magnets, be they natural, such as a lodestone (Gilbert, 1600 AD), or manufactured,

  such as one bought from a store, establish a force field in their vicinity which exerts a force on some metallic objects. The force here is called the magnetic force, and the field is called a magnetic field. The source of the mag- netic field is the motion of electric charges within the atomic structure of such permanent magnets. Moving elec- tric charges (such as an electric current) also produce a magnetic field which may be detected in the same way as that of a permanent magnet. Hold a compass needle close to a wire carrying a DC current and the needle will align itself, at a right angle with the current. Change the current direction in the wire and the needle will change its direction. This experiment, performed by Oersted in 1820, indicates that the electric current produces a mag- netic field in its surroundings which exerts a force on the compass needle. Replace the compass with a solenoid carrying a DC current and the solenoid will align itself, just like the compass, in a direction perpendicular to the current. With a DC current, such generated magnetic fields are static in nature. In a similar experiment, a wire suspended along the direction of a magnetic field and carrying a sinusoidal current will vibrate at the frequency of the current. (This effect was used in the early days of EKG monitoring. Passage of electrical heart pulses through a wire which was suspended in the field of a permanent magnet causes it to deflect and vibrate, with a pen recording the viberations and hence the electrical activity of the heart.) These observations indicate that the magnetic field exerts a force on the compass, another magnet, or a current-carrying wire or solenoid. They specif- ically show that a current-carrying wire generates a magnetic field in its neighborhood which exerts a force on another current-carrying wire placed there.

  Force Between Two Wires.

  Two infinitely long parallel wires carrying currents I and I and separated by dis-

  1

  2

  tance d experience a mutual force. They are pushed apart (when currents are in the same direction) or pulled together (when currents are in opposite directions). The magnitude of the magnetic force between the two wires in free space is given by the following:

  I I µ 1 2

  F ⫽

  2

  d

π

  ⫺7

  where ⫽ 4 ⫻ 10 (H /m) is the permeability of free space. The force is in newtons (N), the distance is in

  µ π meters (m), and the currents are in amperes (A).

  

Magnetic Field Strength. The magnetic field strength is a vector quantity, specified by a magnitude and a

direction. In this book we work with magnetic fields whose sources are electric currents and moving charges.

  The magnitude of the differential field strength of a small element of conducting wire dl carrying a current I is

  I dl sin θ d H ⫽

  2

  4 R

  π

  where R is the distance from the current element to the test point at which dH is being measured and is the angle

  θ

  between the current element and the line connecting it to the test point. The direction of the field is normal to the plane made of the current element and the connecting line, following the right-hand rule: With the right thumb pointing along the direction of current, the fingers point in the direction of the field. This is called the Biot-Savart

  law. In vector notation it is given by I d l ⫻ a R d H ⫽

  2

4 R

  π

R

  8

CHAPTER 1 The Subject of Electromagnetics

  Z dH R dI

  Fig. 1-2 Differential magnetic field d H at the distance R due to a differential current element d l The unit of magnetic field strength is A /m. The superposition property is used to find the magnetic field due to any current configuration.

  

Magnetic Field Strength of a Long Wire. By using the superposition property, we can integrate the above