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Engineering Electromagnetics

McGraw-Hill Series in Electrical and Computer Engineering

SENIOR CONSULTING EDITOR

Stephen W. Director, University of Michigan, Ann Arbor

Circuits and Systems Communications and Signal Processing Computer Engineering Control Theory and Robotics Electromagnetics Electronics and VLSI Circuits Introductory Power Antennas, Microwaves, and Radar

Previous Consulting Editors

Ronald N. Bracewell, Colin Cherry, James F. Gibbons, Willis W. Harman,

Hubert Heffner, Edward W. Herold, John G. Linvill, Simon Ramo, Ronald A.

Rohrer, Anthony E. Siegman, Charles Susskind, Frederick E. Terman, John G.

Truxal, Ernst Weber, and John R. Whinnery

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

To find Appendix E, please visit the expanded book website: www.mhhe.com/engcs/electrical/haytbuck

P R E F A C E

Over the years, I have developed a familiarity with this book in its various

editions, having learned from it, referred to it, and taught from it. The second edition was used in my first electromagnetics course as a junior during the early '70's. Its simple and easy-to-read style convinced me that this material could be learned, and it helped to confirm my latent belief at the time that my specialty would lie in this direction. Later, it was not surprising to see my own students coming to me with heavily-marked copies, asking for help on the drill problems, and taking a more active interest in the subject than I usually observed. So, when approached to be the new co-author, and asked what I would do to change the book, my initial feeling wasÐnothing. Further reflection brought to mind earlier wishes for more material on waves and transmission lines. As a result, Chapters 1 to 10 are original, while 11 to 14 have been revised, and contain new material. A conversation with Bill Hayt at the project's beginning promised the start of what I thought would be a good working relationship. The rapport was immediate. His declining health prevented his active participation, but we seemed to be in general agreement on the approach to a revision. Although I barely knew him, his death, occurring a short time later, deeply affected me in the sense that someone that I greatly respected was gone, along with the promise of a good friendship. My approach to the revision has been as if he were still here. In the front of my mind was the wish to write and incorporate the new material in a manner that he would have approved, and which would have been consistent with the original objectives and theme of the text. Much more could have been done, but at the risk of losing the book's identity and possibly its appeal. Before their deaths, Bill Hayt and Jack Kemmerly completed an entirely new set of drill problems and end-of-chapter problems for the existing material at that time, up to and including the transmission lines chapter. These have been incorporated, along with my own problems that pertain to the new topics. The other revisions are summarized as follows: The original chapter on plane waves has now become two. The first (Chapter 11) is concerned with the development of the uniform plane wave and the treatment wave propagation in various media. These include lossy materials, where propagation and loss are now modeled in a general way using the complex permittivity. Conductive media are presented as special cases, as are materials that exhibit electronic or molecular resonances. A new appendix provides background on resonant media. A new section on wave polarization is also included. Chapter 12 deals with wave reflection at single and multiple interfaces, and at oblique incidence angles. An additional section on dispersive media has been added, which introduces the concepts of group velo- city and group dispersion. The effect of pulse broadening arising from group dispersion is treated at an elementary level. Chapter 13 is essentially the old transmission lines chapter, but with a new section on transients. Chapter 14 is intended as an introduction to waveguides and antennas, in which the underlying

xi

Qaboos University, Juri Silmberg, Ryerson Polytechnic University and Robert M. Weikle II, University of Virginia. My editors at McGraw-Hill, Catherine Fields, Michelle Flomenhoft, and Betsy Jones, provided excellent expertise and supportÐparticularly Michelle, who was almost in daily contact, and provided immediate and knowledgeable answers to all questions and concerns. My see- mingly odd conception of the cover illustration was brought into reality through the graphics talents of Ms Diana Fouts at Georgia Tech. Finally, much is owed to my wife and daughters for putting up with a part-time husband and father for many a weekend.

John A. Buck Atlanta, 2000

PREFACE xiii

C O N T E N T S

• Chapter 1 Vector Analysis Preface xi
• Chapter 2 Coulomb's Law and Electric Field Intensity
• Chapter 3 Electric Flux Density, Gauss' Law, and Divergence
• Chapter 4 Energyand Potential
• Chapter 5 Conductors, Dielectrics, and Capacitance
• Chapter 6 Experimental Mapping Methods
• Chapter 7 Poisson's and Laplace's Equations
• Chapter 8 The SteadyMagnetic Field
• Chapter 9 Magnetic Forces, Materials, and Inductance
• Chapter 10 Time-Varying Fields and Maxwell's Equations
• Chapter 11 The Uniform Plane Wave
• Chapter 12 Plane Waves at Boundaries and in Dispersive Media
• Chapter 13 Transmission Lines
• Chapter 14 Waveguide and Antenna Fundamentals - Appendix A Vector Analysis - Appendix B Units - Appendix C Material Constants - Appendix D Origins of the Complex Permittivity - Index Appendix E Answers to Selected Problems
• Chapter 1 Vector Analysis Preface xi
  • 1.1. Scalars and Vectors
  • 1.2. Vector Algebra
  • 1.3. The Cartesian Coordinate System
  • 1.4. Vector Components and Unit Vectors
  • 1.5. The Vector Field
  • 1.6. The Dot Product
  • 1.7. The Cross Product
    • Coordinates 1.8. Other Coordinate Systems: Circular Cylindrical
  • 1.9. The Spherical Coordinate System
• Chapter 2 Coulomb's Law and Electric Field Intensity
  • 2.1. The Experimental Law of Coulomb
  • 2.2. Electric Field Intensity
  • 2.3. Field Due to a Continuous Volume Charge Distribution
  • 2.4. Field of a Line Charge
  • 2.5. Field of a Sheet Charge
  • 2.6. Streamlines and Sketches of Fields
• Chapter 3 Electric Flux Density, Gauss' Law, and Divergence
  • 3.1. Electric Flux Density
  • 3.2. Gauss' Law
    • Distributions 3.3. Applications of Gauss' Law: Some Symmetrical Charge
  • 3.4. Application of Gauss' Law: Differential Volume Element
  • 3.5. Divergence
  • 3.6. Maxwell's First Equation (Electrostatics)
  • 3.7. The Vector Operator r and the Divergence Theorem
• Chapter 4 Energyand Potential - Electric Field 4.1. Energy and Potential in a Moving Point Charge in an
  • 4.2. The Line Integral
  • 4.3. De®nition of Potential Difference and Potential
  • 4.4. The Potential Field of a Point Charge
    • Property 4.5. The Potential Field of a System of Charges: Conservative
  • 4.6. Potential Gradient
  • 4.7. The Dipole
  • 4.8. Energy Density in the Electric Field
• Chapter 5 Conductors, Dielectrics, and Capacitance
  • 5.1. Current and Current Density
  • 5.2. Continuity of Current
  • 5.3. Metallic Conductors
  • 5.4. Conductor Properties and Boundary Conditions
  • 5.5. The Method of Images
  • 5.6. Semiconductors
  • 5.7. The Nature of Dielectric Materials
  • 5.8. Boundary Conditions for Perfect Dielectric Materials
  • 5.9. Capacitance
  • 5.10. Several Capacitance Examples
  • 5.11. Capacitance of a Two-Wire Line
• Chapter 6 Experimental Mapping Methods
  • 6.1. Curvilinear Squares
  • 6.2. The Iteration Method
  • 6.3. Current Analogies
  • 6.4. Physical Models
• Chapter 7 Poisson's and Laplace's Equations
  • 7.1 Poisson's and Laplace's Equations
  • 7.2. Uniqueness Theorem
  • 7.3. Examples of the Solution of Laplace's Equation
  • 7.4. Example of the Solution of Poisson's Equation
  • 7.5. Product Solution of Laplace's Equation
• Chapter 8 The SteadyMagnetic Field
  • 8.1. Biot-Savart Law
  • 8.2. Ampere's Circuital Law
  • 8.3. Curl
  • 8.4. Stokes' Theorem
  • 8.5. Magnetic Flux and Magnetic Flux Density
  • 8.6. The Scalar and Vector Magnetic Potentials
  • 8.7. Derivation of the Steady-Magnetic-Field Laws
  • Chapter 9 Magnetic Forces, Materials and Inductance
    • 9.1. Force on a Moving Charge
    • 9.2. Force on a Differential Current Element
    • 9.3. Force Between Differential Current Elements
    • 9.4. Force and Torque on a Closed Circuit
    • 9.5. The Nature of Magnetic Materials
    • 9.6. Magnetization and Permeability
    • 9.7. Magnetic Boundary Conditions
    • 9.8. The Magnetic Circuit
    • 9.9. Potential Energy and Forces on Magnetic Materials
    • 9.10. Inductance and Mutual Inductance
• Chapter 10 Time-Varying Fields and Maxwell's Equations - 10.1. Faraday's Law - 10.2. Displacement Current - 10.3. Maxwell's Equations in Point Form - 10.4. Maxwell's Equations in Integral Form - 10.5. The Retarded Potentials
• Chapter 11 The Uniform Plane Wave - 11.1. Wave Propagation in Free Space - 11.2. Wave Propagation in Dielectrics - 11.3. The Poynting Vector and Power Considerations - 11.4. Propagation in Good Conductors: Skin Effect - 11.5. Wave Polarization
• Chapter 12 Plane Waves at Boundaries and in Dispersive Media - 12.1. Re¯ection of Uniform Plane Waves at Normal Incidence - 12.2. Standing Wave Ratio - 12.3. Wave Re¯ection from Multiple Interfaces - 12.4. Plane Wave Propagation in General Directions - 12.5. Plane Wave Re¯ection at Oblique Incidence Angles - 12.6. Wave Propagation in Dispersive Media
• Chapter 13 Transmission Lines - 13.1. The Transmission-Line Equations - 13.2. Transmission-Line Parameters - 13.3. Some Transmission-Line Examples - 13.4. Graphical Methods - 13.5. Several Practical Problems - 13.6. Transients on Transmission Lines

Chapter 14 Waveguide and Antenna Fundamentals 484

14.1. Basic Waveguide Operation 485 14.2. Plane Wave Analysis of the Parallel-Plate Waveguide 488 14.3. Parallel-Plate Guide Analysis Using the Wave Equation 497 14.4. Rectangular Waveguides 501 14.5. Dielectric Waveguides 506 14.6. Basic Antenna Principles 514

Appendix A Vector Analysis 529

Appendix B Units 534

Appendix C Material Constants 540

Appendix D Origins of the Complex Permittivity 544

Appendix E Answers to Selected Problems

Index 551

To find Appendix E, please visit the expanded website: www.mhhe.com/engcs/electrical/haytbuck

x CONTENTS

worked. They should not prove to be difficult if the material in the accompany- ing section of the text has been thoroughly understood. It take a little longer to ``read'' the chapter this way, but the investment in time will produce a surprising interest.

1.1 SCALARS AND VECTORS

The term scalar refers to a quantity whose value may be represented by a single (positive or negative) real number. The x; y, and z we used in basic algebra are scalars, and the quantities they represent are scalars. If we speak of a body falling a distance L in a time t, or the temperature T at any point in a bowl of soup whose coordinates are x; y, and z, then L; t; T; x; y, and z are all scalars. Other scalar quantities are mass, density, pressure (but not force), volume, and volume resistivity. Voltage is also a scalar quantity, although the complex representation of a sinusoidal voltage, an artificial procedure, produces a complex scalar, or phasor, which requires two real numbers for its representation, such as amplitude and phase angle, or real part and imaginary part. A vector quantity has both a magnitude^1 and a direction in space. We shall be concerned with two- and three-dimensional spaces only, but vectors may be defined in n-dimensional space in more advanced applications. Force, velocity, acceleration, and a straight line from the positive to the negative terminal of a storage battery are examples of vectors. Each quantity is characterized by both a magnitude and a direction. We shall be mostly concerned with scalar and vector fields. A field (scalar or vector) may be defined mathematically as some function of that vector which connects an arbitrary origin to a general point in space. We usually find it possible to associate some physical effect with a field, such as the force on a compass needle in the earth's magnetic field, or the movement of smoke particles in the field defined by the vector velocity of air in some region of space. Note that the field concept invariably is related to a region. Some quantity is defined at every point in a region. Both scalar fields and vector fields exist. The temperature throughout the bowl of soup and the density at any point in the earth are examples of scalar fields. The gravitational and magnetic fields of the earth, the voltage gradient in a cable, and the temperature gradient in a soldering- iron tip are examples of vector fields. The value of a field varies in general with both position and time. In this book, as in most others using vector notation, vectors will be indi- cated by boldface type, for example, A. Scalars are printed in italic type, for example, A. When writing longhand or using a typewriter, it is customary to drawa line or an arrowover a vector quantity to showits vector character. (C AUTION : This is the first pitfall. Sloppy notation, such as the omission of the line or arrowsymbol for a vector, is the major cause of errors in vector analysis.)

2 ENGINEERING ELECTROMAGNETICS

(^1) We adopt the convention that magnitude'' infersabsolute value''; the magnitude of any quantity is therefore always positive.

VECTOR ANALYSIS 3

1.2 VECTOR ALGEBRA

With the definitions of vectors and vector fields nowaccomplished, we may proceed to define the rules of vector arithmetic, vector algebra, and (later) of vector calculus. Some of the rules will be similar to those of scalar algebra, some will differ slightly, and some will be entirely new and strange. This is to be expected, for a vector represents more information than does a scalar, and the multiplication of two vectors, for example, will be more involved than the multi- plication of two scalars. The rules are those of a branch of mathematics which is firmly established. Everyone plays by the same rules,'' and we, of course, are merely going to look at and interpret these rules. However, it is enlightening to consider ourselves pioneers in the field. We are making our own rules, and we can make any rules we wish. The only requirement is that the rules be self-consistent. Of course, it would be nice if the rules agreed with those of scalar algebra where possible, and it would be even nicer if the rules enabled us to solve a few practical problems. One should not fall into the trap ofalgebra worship'' and believe that the rules of college algebra were delivered unto man at the Creation. These rules are merely self-consistent and extremely useful. There are other less familiar alge- bras, however, with very different rules. In Boolean algebra the product AB can be only unity or zero. Vector algebra has its own set of rules, and we must be constantly on guard against the mental forces exerted by the more familiar rules or scalar algebra. Vectorial addition follows the parallelogram law, and this is easily, if inac- curately, accomplished graphically. Fig. 1.1 shows the sum of two vectors, A and B. It is easily seen that A ‡ B ˆ B ‡ A, or that vector addition obeys the com- mutative law. Vector addition also obeys the associative law,

A ‡ B ‡ C† ˆ A ‡ B† ‡ C

Note that when a vector is drawn as an arrow of finite length, its location is defined to be at the tail end of the arrow. Coplanar vectors, or vectors lying in a common plane, such as those shown in Fig. 1.1, which both lie in the plane of the paper, may also be added by expressing each vector in terms of horizontal'' andvertical'' components and adding the corresponding components. Vectors in three dimensions may likewise be added by expressing the vec- tors in terms of three components and adding the corresponding components. Examples of this process of addition will be given after vector components are discussed in Sec. 1.4. The rule for the subtraction of vectors follows easily from that for addition, for we may always express A  B as A ‡ B†; the sign, or direction, of the second vector is reversed, and this vector is then added to the first by the rule for vector addition. Vectors may be multiplied by scalars. The magnitude of the vector changes, but its direction does not when the scalar is positive, although it reverses direc-

In the cartesian coordinate system we set up three coordinate axes mutually at right angles to each other, and call them the x; y, and z axes. It is customary to choose a right-handed coordinate system, in which a rotation (through the smal- ler angle) of the x axis into the y axis would cause a right-handed screw to progress in the direction of the z axis. If the right hand is used, then the thumb, forefinger, and middle finger may then be identified, respectively, as the x; y, and z axes. Fig. 1.2a shows a right-handed cartesian coordinate system. A point is located by giving its x; y, and z coordinates. These are, respec- tively, the distances from the origin to the intersection of a perpendicular dropped from the point to the x; y, and z axes. An alternative method of inter- preting coordinate values, and a method corresponding to that which must be used in all other coordinate systems, is to consider the point as being at the

VECTOR ANALYSIS 5

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

common intersection of three surfaces, the planes x ˆ constant, y ˆ constant, and z ˆ constant, the constants being the coordinate values of the point. Fig. 1.2b shows the points P and Q whose coordinates are 1 ; 2 ; 3 † and

2 ;  2 ; 1 †, respectively. Point P is therefore located at the common point of intersection of the planes x ˆ 1, y ˆ 2, and z ˆ 3, while point Q is located at the intersection of the planes x ˆ 2, y ˆ 2, z ˆ 1. As we encounter other coordinate systems in Secs. 1.8 and 1.9, we should expect points to be located at the common intersection of three surfaces, not necessarily planes, but still mutually perpendicular at the point of intersection. If we visualize three planes intersecting at the general point P, whose coor- dinates are x; y, and z, we may increase each coordinate value by a differential amount and obtain three slightly displaced planes intersecting at point P 0 , whose coordinates are x ‡ dx, y ‡ dy, and z ‡ dz. The six planes define a rectangular parallelepiped whose volume is dv ˆ dxdydz; the surfaces have differential areas dS of dxdy, dydz, and dzdx. Finally, the distance dL from P to P 0 is the diagonal of the parallelepiped and has a length of

dx†^2 ‡ dy†^2 ‡ dz†^2

q

. The volume element is shown in Fig. 1.2c; point P 0 is indicated, but point P is located at the only invisible corner. All this is familiar from trigonometry or solid geometry and as yet involves only scalar quantities. We shall begin to describe vectors in terms of a coordinate system in the next section.

1.4 VECTOR COMPONENTS AND UNIT

VECTORS

To describe a vector in the cartesian coordinate system, let us first consider a vector r extending outward from the origin. A logical way to identify this vector is by giving the three component vectors, lying along the three coordinate axes, whose vector sum must be the given vector. If the component vectors of the vector r are x, y, and z, then r ˆ x ‡ y ‡ z. The component vectors are shown in Fig. 1.3a. Instead of one vector, we now have three, but this is a step forward, because the three vectors are of a very simple nature; each is always directed along one of the coordinate axes. In other words, the component vectors have magnitudes which depend on the given vector (such as r above), but they each have a known and constant direction. This suggests the use of unit vectors having unit magnitude, by defini- tion, and directed along the coordinate axes in the direction of the increasing coordinate values. We shall reserve the symbol a for a unit vector and identify the direction of the unit vector by an appropriate subscript. Thus ax, ay, and az are the unit vectors in the cartesian coordinate system. 3 They are directed along the x; y, and z axes, respectively, as shown in Fig. 1.3b.

6 ENGINEERING ELECTROMAGNETICS

(^3) The symbols i; j, and k are also commonly used for the unit vectors in cartesian coordinates.

This last vector does not extend outward from the origin, as did the vector r we initially considered. However, we have already learned that vectors having the same magnitude and pointing in the same direction are equal, so we see that to help our visualization processes we are at liberty to slide any vector over to the origin before determining its component vectors. Parallelism must, of course, be maintained during the sliding process. If we are discussing a force vector F, or indeed any vector other than a displacement-type vector such as r, the problem arises of providing suitable letters for the three component vectors. It would not do to call them x; y, and z, for these are displacements, or directed distances, and are measured in meters (abbreviated m) or some other unit of length. The problem is most often avoided by using component scalars, simply called components, Fx; Fy, and Fz. The com- ponents are the signed magnitudes of the component vectors. We may then write F ˆ Fxax ‡ Fyay ‡ Fzaz. The component vectors are F (^) xax, Fyay, and Fzaz: Any vector B then may be described by B ˆ B (^) xax ‡ B (^) yay ‡ B (^) zaz. The mag- nitude of B written jBj or simply B, is given by

jBj ˆ

B^2 x ‡ B^2 y ‡ B^2 z

q

1 †

Each of the three coordinate systems we discuss will have its three funda- mental and mutually perpendicular unit vectors which are used to resolve any vector into its component vectors. However, unit vectors are not limited to this application. It is often helpful to be able to write a unit vector having a specified direction. This is simply done, for a unit vector in a given direction is merely a vector in that direction divided by its magnitude. A unit vector in the r direction is r=

x^2 ‡ y^2 ‡ z^2

p , and a unit vector in the direction of the vector B is

aB ˆ

B

B^2 x ‡ B^2 y ‡ B^2 z

q ˆ

B

jBj

h

Example 1.

Specify the unit vector extending from the origin toward the point G 2 ;  2 ;  1 †.

Solution. We first construct the vector extending from the origin to point G, G ˆ 2 ax  2 ay  az We continue by finding the magnitude of G,

jGj ˆ



2 †^2 ‡  2 †^2 ‡  1 †^2

q ˆ 3

8 ENGINEERING ELECTROMAGNETICS

and finally expressing the desired unit vector as the quotient,

aG ˆ G jGj ˆ 23 ax  23 ay  13 az ˆ 0 : 667 ax  0 : 667 ay  0 : 333 az

A special identifying symbol is desirable for a unit vector so that its character is immediately apparent. Symbols which have been used are uB; aB; (^1) B, or even b. We shall consistently use the lowercase a with an appropriate subscript. [N OTE : Throughout the text, drill problems appear following sections in which a newprinciple is introduced in order to allowstudents to test their understanding of the basic fact itself. The problems are useful in gaining familiarization with new terms and ideas and should all be worked. More general problems appear at the ends of the chapters. The answers to the drill problems are given in the same order as the parts of the problem.]

\ D1.1. Given points M  1 ; 2 ; 1 †, N 3 ;  3 ; 0 †, and P  2 ;  3 ;  4 †, find: (a) RMN ; (b) RMN ‡ RMP; (c) jrM j; (d) aMP; (e) j 2 rP  3 rN j:

Ans. 4ax  5 ay  az; 3ax  10 ay  6 az; 2.45;  0 : 1400 ax  0 : 700 ay  0 : 700 az; 15.

1.5 THE VECTOR FIELD

We have already defined a vector field as a vector function of a position vector. In general, the magnitude and direction of the function will change as we move throughout the region, and the value of the vector function must be determined using the coordinate values of the point in question. Since we have considered only the cartesian coordinate system, we should expect the vector to be a func- tion of the variables x; y, and z: If we again represent the position vector as r, then a vector field G can be expressed in functional notation as G r†; a scalar field T is written as T r†. If we inspect the velocity of the water in the ocean in some region near the surface where tides and currents are important, we might decide to represent it by a velocity vector which is in any direction, even up or down. If the z axis is taken as upward, the x axis in a northerly direction, the y axis to the west, and the origin at the surface, we have a right-handed coordinate system and may write the velocity vector as v ˆ vxax ‡ v (^) yay ‡ v (^) zaz, or v r† ˆ v (^) x r†ax ‡ vy r†ay ‡ vz r†az; each of the components vx; v (^) y, and v (^) z may be a function of the three variables x; y, and z. If the problem is simplified by assuming that we are in some portion of the Gulf Stream where the water is moving only to the north, then vy, and v (^) z are zero. Further simplifying assumptions might be made if the velocity falls off with depth and changes very slowly as we move north, south, east, or west. A suitable expression could be v ˆ 2 ez=^100 ax. We have a velocity of 2 m/s (meters per second) at the surface and a velocity of 0: 368  2, or 0.736 m/s, at a depth of 100 m z ˆ  100 †, and the velocity continues to decrease with depth; in this example the vector velocity has a constant direction. While the example given above is fairly simple and only a rough approx- imation to a physical situation, a more exact expression would be correspond-

VECTOR ANALYSIS 9