Electromagnetism: A Comprehensive Guide to Electric and Magnetic Fields, Lecture notes of Engineering

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Engineering Electromagnetics

NINTH EDITION

William H. Hayt, Jr.

Late Emeritus Professor Purdue University

John A. Buck

Georgia Institute of Technology

A B O U T T H E A U T H O R S

William H. Hayt. Jr. (deceased) received his B.S. and M.S. degrees at Purdue Uni- versity and his Ph.D. from the University of Illinois. After spending four years in industry, Professor Hayt joined the faculty of Purdue University, where he served as professor and head of the School of Electrical Engineering, and as professor emeritus after retiring in 1986. Professor Hayt’s professional society memberships included Eta Kappa Nu, Tau Beta Pi, Sigma Xi, Sigma Delta Chi, Fellow of IEEE, ASEE, and NAEB. While at Purdue, he received numerous teaching awards, including the uni- versity’s Best Teacher Award. He is also listed in Purdue’s Book of Great Teachers, a permanent wall display in the Purdue Memorial Union, dedicated on April 23, 1999. The book bears the names of the inaugural group of 225 faculty members, past and present, who have devoted their lives to excellence in teaching and scholarship. They were chosen by their students and their peers as Purdue’s finest educators.

A native of Los Angeles, California, John A. Buck received his B.S. in Engineering from UCLA in 1975, and the M.S. and Ph.D. degrees in Electrical Engineering from the University of California at Berkeley in 1977 and 1982. In 1982, he joined the faculty of the School of Electrical and Computer Engineering at Georgia Tech, and is now Professor Emeritus. His research areas and publications have centered within the fields of ultrafast switching, nonlinear optics, and optical fiber communications. He is the author of the graduate text Fundamentals of Optical Fibers (Wiley Interscience), which is in its second edition. Dr. Buck is the recipient of four institute teaching awards and the IEEE Third Millenium Medal.

vi

C O N T E N T S

Contents ix

Appendix A

Vector Analysis 557

A.1 General Curvilinear Coordinates 557

A.2 Divergence, Gradient, and Curl in General Curvilinear Coordinates 558

A.3 Vector Identities 560

Appendix B

Units 561

Appendix C

Material Constants 566

Appendix D

The Uniqueness Theorem 569

Appendix E

Origins of the Complex

Permittivity 571

Appendix F

Answers to Odd-Numbered

Problems 578

Index 584

x

P R E F A C E

The printing of this book occurs one year short of 60 years since its first edition, which was at that time under the sole authorship of William H. Hayt, Jr. In a sense, I grew up with the book, having used the second edition in a basic electromagnetics course as a college junior. The reputation of the subject matter precedes itself. The prospect of taking the first course in electromagnetics was then, as now, a matter of dread to many if not most. One professor of mine at Berkeley put it succinctly through the rather negative observation that electromagetics is “a test of your ability to bend your mind”. But on entering the course and first opening the book, I was surprised and relieved to find the friendly writing style and the measured approach to the subject. This for me made it a very readable book, out of which I was able to learn with little help from my instructor. I referred to the book often while in grad- uate school, taught from the fourth and fifth editions as a faculty member, and then became coauthor for the sixth edition on the retirement (and subsequent untimely death) of Bill Hayt. To this day, the memories of my time as a beginner are vivid, and in preparing the sixth and subsequent editions, I have tried to maintain the accessible style that I found so encouraging and useful then. Over the 60-year span, the subject matter has not changed, but emphases have. In universities, the trend continues toward reducing electrical engineering core course allocations to electromagnetics. This is a matter of economy, rather than any belief in diminished relevance. Quite the contrary: A knowledge of electromagnetic field the- ory is in the present day more important than ever for the electrical engineer. Exam- ples that demonstrate this include the continuing expansion of high-speed wireless and optical fiber communication. Additionally, the need continues for ever-smaller and denser microcircuitry, in which a command of field theory is essential for suc- cessful designs. The more traditional applications of electrical power generation and distribution remain as important as ever. I have made efforts to further improve the presentation in this new edition. Most changes occur in the earlier chapters, in which much of the wording has been short- ened, and several explanations were improved. Additional introductory material has been added in several places to provide perspective. In addition, all chapters are now subsectioned, to improve the organization and to make topics easier to locate. Some 100 new end-of-chapter problems have been added throughout, all of which replaced older problems that I considered well-worn. For some of these, I chose par- ticularly good “classic” problems from the earliest editions. I have retained the previous system in which the approximate level of difficulty is indicated beside each problem on a three-level scale. The lowest level is considered a fairly straightforward problem, requiring little work assuming the material is understood; a level 2 problem is con- ceptually more difficult, and/or may require more work to solve; a level 3 problem is

xii Preface

Answers to the drill problems are given below each problem. Answers to odd-numbered end-of-chapter problems are found in Appendix F. A solutions man- ual and a set of PowerPoint slides, containing pertinent figures and equations, are available to instructors. These, along with all other material mentioned previously, can be accessed on the website:

www.mhhe.com/haytbuck

I would like to acknowledge the valuable input of several people who helped to make this a better edition. They include:

Gerald Whitman – New Jersey Institute of Technology Andrew F. Peterson – Georgia Institute of Technology M. Chris Wernicki, Ph.D. – NYIT David Baumann – Lake Superior State University Jesmin Khan – Tuskegee University Dr. S. Hossein Mousavinezhad – Idaho State University Kiyun Han – Wichita State University Anand Gopinath – University of Minnesota Donald M. Keller – Point Park University Argyrios VAronides – University of Scranton Otsebele Nare – Hampton University Robert Wayne Scharstein – University of Alabama Virgil Thomason – University of Tennessee at Chattanooga Gregory M. Wilkins, Ph.D. – Morgan State University Mark A. Jerabek – West Virginia University James Richie – Marquette University Dean Johnson – Western Michigan David A. Rogers – North Dakota State University Tomasz Petelenz – University of Utah Surendra Singh – The University of Tulsa Tom Vandervelde – Tufts John Zwart – Dordt College Taan ElAli – Embry-Riddle Aeronautical University R. Clive Woods – Louisiana State University Jack Adams – Merrimack College

I also acknowledge the feedback and many comments from students, too numerous to name, including several who have contacted me from afar. I continue to be open and grateful for this feedback and can be reached at john.buck@ece.gatech.edu. Many suggestions were made that I considered constructive and actionable. I regret that not all could be incorporated because of time restrictions. Creating this book was a team effort, involving several outstanding people at McGraw-Hill. These include my editors, Raghu Srinivasan and Tomm Scaife, whose vision and encouragement

Preface xiii

were invaluable. Jenilynn McAtee and Lora Neyens deftly coordinated the production phase with excellent ideas and enthusiasm, and Tina Bower, who was my guide and conscience from the beginning, providing valuable insights, and jarring me into ac- tion when necessary. I am, as usual in these projects, grateful to a patient and sup- portive family.

John A. Buck Marietta, Georgia May, 2017

2 E N G I N E E R I N G E L E C T R O M A G N E T I C S

n -dimensional space in more advanced applications. Force, velocity, acceleration, and a straight line from the positive to the negative terminal of a storage battery are examples of vectors. Each quantity is characterized by both a magnitude and a direction. Our work will mainly concern scalar and vector fields. A field (scalar or vector) may be defined mathematically as some function that connects an arbitrary origin to a general point in space. We usually associate some physical effect with a field, such as the force on a compass needle in the earth’s magnetic field, or the movement of smoke particles in the field defined by the vector velocity of air in some region of space. Note that the field concept invariably is related to a region. Some quantity is defined at every point in a region. Both scalar fields and vector fields exist. The tem- perature and the density at any point in the earth are examples of scalar fields. The gravitational and magnetic fields of the earth, the voltage gradient in a cable, and the temperature gradient in a soldering-iron tip are examples of vector fields. The value of a field varies in general with both position and time. In this book, as in most others using vector notation, vectors will be indicated by boldface type, for example, A. Scalars are printed in italic type, for example, A. When writing longhand, it is customary to draw a line or an arrow over a vector quan- tity to show its vector character. (CAUTION: This is the first pitfall. Sloppy notation, such as the omission of the line or arrow symbol for a vector, is the major cause of errors in vector analysis.)

1.2 VECTOR ALGEBRA In this section, the rules of vector arithmetic, vector algebra, and (later) vector calcu- lus are defined. Some of the rules will be similar to those of scalar algebra, some will differ slightly, and some will be entirely new.

1.2.1 Addition and Subtraction

The addition of vectors follows the parallelogram law. Figure 1.1 shows the sum of two vectors, A and B. It is easily seen that A + B = B + A , or that vector addition obeys the commutative law. Vector addition also obeys the associative law,

A + ( B + C ) = ( A + B ) + C

Note that when a vector is drawn as an arrow of finite length, its location is de- fined to be at the tail end of the arrow. Coplanar vectors are vectors lying in a common plane, such as those shown in Figure 1.1. Both lie in the plane of the paper and may be added by expressing each vector in terms of “horizontal” and “vertical” components and then adding the cor- responding components. Vectors in three dimensions may likewise be added by expressing the vectors in terms of three components and adding the corresponding components. Examples of this process of addition will be given after vector components are discussed in Section 1.4.

C H A P T E R 1 Vector Analysis 3

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 vec- tor is reversed, and this vector is then added to the first by the rule for vector addition.

1.2.2 Multiplication and Division

Vectors may be multiplied by scalars. The magnitude of the vector changes, but its direction does not when the scalar is positive, although it reverses direction when multiplied by a negative scalar. Multiplication of a vector by a scalar also obeys the associative and distributive laws of algebra, leading to

( r + s ) ( A + B ) = r ( A + B ) + s ( A + B ) = r A + r B + s A + s B

Division of a vector by a scalar is merely multiplication by the reciprocal of that scalar. The multiplication of a vector by a vector is discussed in Sections 1.6 and 1.7. Two vectors are said to be equal if their difference is zero, or A = B if A − B = 0. In our use of vector fields we always add and subtract vectors that are defined at the same point. For example, the total magnetic field about a small horseshoe magnet will be shown to be the sum of the fields produced by the earth and the permanent magnet; the total field at any point is the sum of the individual fields at that point.

1.3 THE RECTANGULAR COORDINATE SYSTEM

To describe a vector accurately, some specific lengths, directions, angles, projec- tions, or components must be given. There are three simple coordinate systems by which this is done, and about eight or ten other systems that are useful in very special cases. We are going to use only the three simple systems, the simplest of which is the rectangular , or rectangular cartesian , coordinate system.

1.3.1 Right-Handed Coordinate Systems

In the rectangular coordinate system we set up three coordinate axes mutually at right angles to each other and call them the x, y , and z axes. It is customary to choose a right-handed coordinate system, in which a rotation (through the smaller angle) of the x axis into the y axis would cause a right-handed screw to progress in the direction of the z axis. If the right hand is used, then the thumb, forefinger, and

B

A + B A + B A

B

A

Figure 1.1 Two vectors may be added graphically either by drawing both vectors from a common origin and completing the parallelogram or by beginning the second vector from the head of the first and completing the triangle; either method is easily extended to three or more vectors.

C H A P T E R 1 Vector Analysis 5

planes x = 1, y = 2, and z = 3, whereas point Q is located at the intersection of the planes x = 2, y = −2, and z = 1. In other coordinate systems, as discussed in Sections 1.8 and 1.9, we expect points to be located at the common intersection of three surfaces, not necessarily planes, but still mutually perpendicular at the point of intersection. If we visualize three planes intersecting at the general point P , whose coordinates are x, y , and z , we may increase each coordinate value by a differential amount and obtain three slightly displaced planes intersecting at point P ′, whose coordinates are x + dx , y + dy , and z + dz. The six planes define a rectangular parallelepiped whose volume is dv = dxdydz ; the surfaces have differential areas dS of dxdy , dydz , and dzdx. Finally, the distance dL from P to P ′ is the diagonal of the parallelepiped and has a length of √

---

( dx ) 2 + ( dy ) 2 + ( dz ) 2. The volume element is shown in Figure 1.2 c ; point P ′ is indicated, but point P is located at the only invisible corner. All this is familiar from trigonometry or solid geometry and as yet involves only sca- lar quantities. We will describe vectors in terms of a coordinate system in the next section.

1.4 VECTOR COMPONENTS

AND UNIT VECTORS

To describe a vector in the rectangular coordinate system, first consider a vector r extending outward from the origin. A logical way to identify this vector is by giving the three component vectors , lying along the three coordinate axes, whose vector sum must be the given vector. If the component vectors of the vector r are x , y , and z , then r = x + y + z. The component vectors are shown in Figure 1.3 a. Instead of one vector, we now have three, but this is a step forward because the three vectors are of a very simple nature; each is always directed along one of the coordinate axes. The component vectors in Figure 1.3 have magnitudes that depend on the given vector (such as r ), but they each have a known and constant direction. This suggests the use of unit vectors having unit magnitude by definition; these are parallel to the coordinate axes and they point in the direction of increasing coordinate values. We reserve the symbol a for a unit vector and identify its direction by an appropriate sub- script. Thus a x , a y , and a z are the unit vectors in the rectangular coordinate system. 2 They are directed along the x , y , and z axes, respectively, as shown in Figure 1.3 b. If the component vector y happens to be two units in magnitude and directed toward increasing values of y , we then write y = 2 a y. A vector r P pointing from the origin to point P (1, 2, 3) is written r P = a x + 2 a y + 3 a z. The vector from P to Q is obtained by applying the rule of vector addition. This rule shows that the vector from the origin to P plus the vector from P to Q is equal to the vector from the origin to Q. The desired vector from P (1, 2, 3) to Q (2, −2, 1) is therefore

R PQ = r Q − r P = (2 − 1) a x + (− 2 − 2) a y + (1 − 3) a z = a x − 4 a y − 2 a z

The vectors r P , r Q , and R PQ are shown in Figure 1.3 c.

(^2) The symbols i , j , and k are also commonly used for the unit vectors in rectangular coordinates.

6 E N G I N E E R I N G E L E C T R O M A G N E T I C S

The last vector does not extend outward from the origin, as did the vector r we initially considered. However, we have already learned that vectors having the same magnitude and pointing in the same direction are equal, so we see that to help our visualization processes we are at liberty to slide any vector over to the origin before determining its component vectors. Parallelism must, of course, be maintained dur- ing the sliding process. In discussing a force vector F , or any vector other than a displacement-type vector such as r , the problem arises of providing suitable letters for the three compo- nent 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 , F (^) x , Fy , and Fz. The components are the signed magnitudes of the com- ponent vectors. We may then write F = F (^) x a x + F (^) y a y + F (^) z a z. The component vectors are Fx a x , Fy a y , and Fz a z.

Figure 1.3 (a) The component vectors x , y , and z of vector r. (b) The unit vectors of the rectangular coordinate system have unit magnitude and are directed toward increasing val- ues of their respective variables. (c) The vector R PQ is equal to the vector difference r Q − r P.

z z

z

y (^) y

y

x

x

x

( a ) ( b )

( c )

Q (2, –2, 1)

P (1, 2, 3)

r P

a y a x

a z

r y

z

x r = x + y + z

r Q

R PQ

8 E N G I N E E R I N G E L E C T R O M A G N E T I C S

1.5 THE VECTOR FIELD We have defined a vector field as a vector function of a position vector. In general, the magnitude and direction of the function will change as we move throughout the region, and the value of the vector function must be determined using the coordinate values of the point in question. In the rectangular coordinate system, the vector will be a function of the variables x, y , and z. Again, representing the position vector as r , a vector field G can be expressed in functional notation as G ( r ); a scalar field T is written as T ( r ). If we inspect the velocity of the water in the ocean in some region near the surface where tides and currents are important, we might decide to represent it by a velocity vector that is in any direction, even up or down. If the z axis is taken as upward, the x axis in a northerly direction, the y axis to the west, and the origin at the surface, we have a right-handed coordinate system and may write the velocity vector as v = vx a x + vy a y + vz a z , or v ( r ) = vx ( r ) a x + vy ( r ) a y + vz ( r ) a z ; each of the components vx, vy , and vz may be a function of the three variables x, y , and z. If we are in some portion of the Gulf Stream where the water is moving only to the north, then vy and vz 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 = 2 ez^ /100 a x. We have a velocity of 2 m/s (meters per second) at the surface and a velocity of 0.368 × 2, or 0.736 m/s, at a depth of 100 m ( z = −100). The velocity con- tinues to decrease with depth while maintaining a constant direction.

1.6 THE DOT PRODUCT The dot product (or scalar product) is used to multiply a given vector field by the compo- nent of another field that is parallel to the first. This gives the same result when the roles of the fields are reversed. In that sense, the dot product is a projection operation, which can be used to obtain the magnitude of a given field in a specific direction in space.

1.6.1 Geometric Definition

Given two vectors A and B , the dot product is geometrically defined as the product of the magnitude of A , the magnitude of B , and the cosine of the smaller angle between them, thus projecting one field onto the other:

A · B = | A || B | cos θAB (^) (3)

D1.2. A vector field S is expressed in rectangular coordinates as S = {125/[( x − 1)^2 + ( y − 2)^2 + ( z + 1)^2 ]}{( x − 1) a x + ( y − 2) a y + ( z + 1) a z }. ( a ) Evaluate S at P (2, 4, 3). ( b ) Determine a unit vector that gives the direction of S at P. ( c ) Specify the surface f ( x, y, z ) on which | S | = 1.

Ans. ( a ) 5.95 a x + 11.90 a y + 23.8 a z ; ( b ) 0.218 a x + 0.436 a y + 0.873 a z ; ( c ) √

---

( x – 1)^2 + ( y – 2)^2 + ( z + 1)^2 = 125

C H A P T E R 1 Vector Analysis 9

The dot appears between the two vectors and should be made heavy for emphasis. The dot, or scalar, product is a scalar, as one of the names implies, and it obeys the commutative law,

A · B = B · A (4)

for the sign of the angle does not affect the cosine term. The expression A · B is read “ A dot B .” A common application of the dot product is in mechanics, where a constant force F applied over a straight displacement L does an amount of work FL cos θ , which is more easily written F · L. If the force varies along the path, integration is necessary to find the total work (as is taken up in Chapter 4), and the result becomes

Work = ∫ F · d L

Another example occurs in magnetic fields. The total flux Φ crossing a surface of area S is given by BS if the magnetic flux density B is perpendicular to the surface and uniform over it. We define a vector surface S as having area for its magnitude and having a direction normal to the surface (avoiding for the moment the problem of which of the two possible normals to take). The flux crossing the surface is then B · S. This expression is valid for any direction of the uniform magnetic flux density.

If the flux density is not constant over the surface, the total flux is Φ = ∫ B · d S. Integrals of this general form appear in Chapter 3 in the context of electric flux density.

1.6.2 Operational Definition

Finding the angle between two vectors in three-dimensional space is often a job we would prefer to avoid, and for that reason the definition of the dot product is usually not used in its basic form. A more helpful result is obtained by considering two vectors whose rectangular components are given, such as A = A (^) x a x + Ay a y + A (^) z a z and B = B (^) x a x + B (^) y a y + B (^) z a z. The dot product also obeys the distributive law, and, therefore, A · B yields the sum of nine scalar terms, each involving the dot product of two unit vectors. Because the an- gle between two different unit vectors of the rectangular coordinate system is 90°, we then have

a x · a y = a y · a x = a x · a z = a z · a x = a y · a z = a z · a y = 0

The remaining three terms involve the dot product of a unit vector with itself, which is unity, giving finally the operational definition:

A · B = Ax Bx + Ay By + Az Bz (5)

which is an expression involving no angles.