Engineering Electromagnetic Chapter 4 problems

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chapter 4 problem (10 problems) can you solve them?

attached the book if you wanna review the equations

Please don't do nothing similar from any website or solution manual attached.

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| ▲ ▲ Engineering Electromagnetics | e-Text Main Menu | Textbook Table of Contents | 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 | e-Text Main Menu | Textbook Table of Contents | Engineering Electromagnetics SIXTH EDITION William H. Hayt, Jr. Late Emeritus Professor Purdue University John A. Buck Georgia Institute of Technology Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogotá Caracas Lisbon London Madrid Mexico City Milan New Delhi Seoul Singapore Sydney Taipei Toronto | ▲ ▲ Boston | e-Text Main Menu | Textbook Table of Contents | BRIEF CONTENTS Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Preface xi Vector Analysis Coulomb's Law and Electric Field Intensity Electric Flux Density, Gauss' Law, and Divergence Energy and Potential Conductors, Dielectrics, and Capacitance Experimental Mapping Methods Poisson's and Laplace's Equations The Steady Magnetic Field Magnetic Forces, Materials, and Inductance Time-Varying Fields and Maxwell's Equations The Uniform Plane Wave Plane Waves at Boundaries and in Dispersive Media Transmission Lines Waveguide and Antenna Fundamentals 1 Appendix Appendix Appendix Appendix Appendix Index 27 53 83 119 169 195 224 274 322 348 387 435 484 A Vector Analysis 529 B Units 534 C Material Constants 540 D Origins of the Complex Permittivity E Answers to Selected Problems 544  551  To find Appendix E, please visit the expanded book website: www.mhhe.com/engcs/electrical/haytbuck | ▲ ▲ v | e-Text Main Menu | Textbook Table of Contents | PREFACE 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 velocity 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 | e-Text Main Menu | Textbook Table of Contents | PREFACE physical concepts are emphasized. The waveguide sections are all new, but the antennas treatment is that of the previous editions. The approach taken in the new material, as was true in the original work, is to emphasize physical understanding and problem-solving skills. I have also moved the work more in the direction of communications-oriented material, as this seemed a logical way in which the book could evolve, given the material that was already there. The perspective has been broadened by an expanded emphasis toward optics concepts and applications, which are presented along with the more traditional lower-frequency discussions. This again seemed to be a logical step, as the importance of optics and optical communications has increased significantly since the earlier editions were published. The theme of the text has not changed since the first edition of 1958. An inductive approach is used that is consistent with the historical development. In it, the experimental laws are presented as individual concepts that are later unified in Maxwell's equations. Apart from the first chapter on vector analysis, the mathematical tools are introduced in the text on an as-needed basis. Throughout every edition, as well as this one, the primary goal has been to enable students to learn independently. Numerous examples, drill problems (usually having multiple parts), and end-of-chapter problems are provided to facilitate this. Answers to the drill problems are given below each problem. Answers to selected end-of-chapter problems can be found on the internet at www.mhhe.com/engcs/electrical/haytbuck. A solutions manual is also available. The book contains more than enough material for a one-semester course. As is evident, statics concepts are emphasized and occur first in the presentation. In a course that places more emphasis on dynamics, the later chapters can be reached earlier by omitting some or all of the material in Chapters 6 and 7, as well as the later sections of Chapter 8. The transmission line treatment (Chapter 13) relies heavily on the plane wave development in Chapters 11 and 12. A more streamlined presentation of plane waves, leading to an earlier arrival at transmission lines, can be accomplished by omitting sections 11.5, 12.5, and 12.6. Chapter 14 is intended as an ``advanced topics'' chapter, in which the development of waveguide and antenna concepts occurs through the application of the methods learned in earlier chapters, thus helping to solidify that knowledge. It may also serve as a bridge between the basic course and more advanced courses that follow it. I am deeply indebted to several people who provided much-needed feedback and assistance on the work. Glenn S. Smith, Georgia Tech, reviewed parts of the manuscript and had many suggestions on the content and the philosophy of the revision. Several outside reviewers pointed out errors and had excellent suggestions for improving the presentation, most of which, within time limitations, were taken. These include Madeleine Andrawis, South Dakota State University, M. Yousif El-Ibiary, University of Oklahoma, Joel T. Johnson, Ohio State University, David Kelley, Pennsylvania State University, Sharad R. Laxpati, University of Illinois at Chicago, Masoud Mostafavi, San Jose State University, Vladimir A. Rakov, University of Florida, Hussain Al-Rizzo, Sultan | ▲ ▲ xii | e-Text Main Menu | Textbook Table of Contents | PREFACE 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 seemingly 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 | e-Text Main Menu | Textbook Table of Contents | xiii CONTENTS Chapter 1 Chapter 2 Chapter 3 Chapter 4 Preface xi Vector Analysis 1 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. Scalars and Vectors Vector Algebra The Cartesian Coordinate System Vector Components and Unit Vectors The Vector Field The Dot Product The Cross Product Other Coordinate Systems: Circular Cylindrical Coordinates 1.9. The Spherical Coordinate System 2 3 4 6 9 10 13 Coulomb's Law and Electric Field Intensity 27 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 28 31 36 38 44 46 15 20 The Experimental Law of Coulomb Electric Field Intensity Field Due to a Continuous Volume Charge Distribution Field of a Line Charge Field of a Sheet Charge Streamlines and Sketches of Fields Electric Flux Density, Gauss' Law, and Divergence 53 3.1. Electric Flux Density 3.2. Gauss' Law 3.3. Applications of Gauss' Law: Some Symmetrical Charge Distributions 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 54 57 Energy and Potential 83 4.1. Energy and Potential in a Moving Point Charge in an Electric Field 4.2. The Line Integral 4.3. De®nition of Potential Difference and Potential 4.4. The Potential Field of a Point Charge 84 85 91 93 62 67 70 73 74 | ▲ ▲ vii | e-Text Main Menu | Textbook Table of Contents | CONTENTS Chapter 5 Chapter 6 Chapter 7 Chapter 8 | 4.5. The Potential Field of a System of Charges: Conservative Property 4.6. Potential Gradient 4.7. The Dipole 4.8. Energy Density in the Electric Field 95 99 106 110 Conductors, Dielectrics, and Capacitance 119 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 120 122 124 129 134 136 138 144 150 154 157 Experimental Mapping Methods 169 6.1. 6.2. 6.3. 6.4. 170 176 183 186 Curvilinear Squares The Iteration Method Current Analogies Physical Models Poisson's and Laplace's Equations 195 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 196 198 200 207 211 The Steady Magnetic Field 224 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 225 232 239 246 251 254 261 ▲ ▲ viii Biot-Savart Law Ampere's Circuital Law Curl Stokes' Theorem Magnetic Flux and Magnetic Flux Density The Scalar and Vector Magnetic Potentials Derivation of the Steady-Magnetic-Field Laws | e-Text Main Menu | Textbook Table of Contents | CONTENTS Chapter 9 Chapter 10 Chapter 11 Chapter 12 | 274 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 275 276 280 283 288 292 297 299 306 308 Time-Varying Fields and Maxwell's Equations 322 10.1. 10.2. 10.3. 10.4. 10.5. 323 329 334 336 338 Faraday's Law Displacement Current Maxwell's Equations in Point Form Maxwell's Equations in Integral Form The Retarded Potentials The Uniform Plane Wave 348 11.1. 11.2. 11.3. 11.4. 11.5. 348 356 365 369 376 Wave Propagation in Free Space Wave Propagation in Dielectrics The Poynting Vector and Power Considerations Propagation in Good Conductors: Skin Effect Wave Polarization Plane Waves at Boundaries and in Dispersive Media 387 12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 388 395 400 408 411 421 Re¯ection of Uniform Plane Waves at Normal Incidence Standing Wave Ratio Wave Re¯ection from Multiple Interfaces Plane Wave Propagation in General Directions Plane Wave Re¯ection at Oblique Incidence Angles Wave Propagation in Dispersive Media Transmission Lines 435 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 436 442 448 452 460 463 ▲ ▲ Chapter 13 Magnetic Forces, Materials and Inductance | The Transmission-Line Equations Transmission-Line Parameters Some Transmission-Line Examples Graphical Methods Several Practical Problems Transients on Transmission Lines e-Text Main Menu | Textbook Table of Contents | ix CONTENTS Chapter 14 Appendix A Appendix B Appendix C Appendix D Appendix E Waveguide and Antenna Fundamentals 484 14.1. 14.2. 14.3. 14.4. 14.5. 14.6. 485 488 497 501 506 514 Basic Waveguide Operation Plane Wave Analysis of the Parallel-Plate Waveguide Parallel-Plate Guide Analysis Using the Wave Equation Rectangular Waveguides Dielectric Waveguides Basic Antenna Principles Vector Analysis Units Material Constants Origins of the Complex Permittivity Answers to Selected Problems 529 Index 551 534 540 544  To find Appendix E, please visit the expanded website: www.mhhe.com/engcs/electrical/haytbuck | ▲ ▲ x | e-Text Main Menu | Textbook Table of Contents | CHAPTER 1 VECTOR ANALYSIS Vector analysis is a mathematical subject which is much better taught by mathematicians than by engineers. Most junior and senior engineering students, however, have not had the time (or perhaps the inclination) to take a course in vector analysis, although it is likely that many elementary vector concepts and operations were introduced in the calculus sequence. These fundamental concepts and operations are covered in this chapter, and the time devoted to them now should depend on past exposure. The viewpoint here is also that of the engineer or physicist and not that of the mathematician in that proofs are indicated rather than rigorously expounded and the physical interpretation is stressed. It is easier for engineers to take a more rigorous and complete course in the mathematics department after they have been presented with a few physical pictures and applications. It is possible to study electricity and magnetism without the use of vector analysis, and some engineering students may have done so in a previous electrical engineering or basic physics course. Carrying this elementary work a bit further, however, soon leads to line-filling equations often composed of terms which all look about the same. A quick glance at one of these long equations discloses little of the physical nature of the equation and may even lead to slighting an old friend. Vector analysis is a mathematical shorthand. It has some new symbols, some new rules, and a pitfall here and there like most new fields, and it demands concentration, attention, and practice. The drill problems, first met at the end of Sec. 1.4, should be considered an integral part of the text and should all be | ▲ ▲ 1 | e-Text Main Menu | Textbook Table of Contents | ENGINEERING ELECTROMAGNETICS worked. They should not prove to be difficult if the material in the accompanying 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 magnitude1 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 solderingiron 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 or using a typewriter, it is customary to draw a line or an arrow over a vector quantity 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 We adopt the convention that ``magnitude'' infers ``absolute value''; the magnitude of any quantity is therefore always positive. | ▲ ▲ 2 ...
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Tutor Answer

Ace_Tutor
School: University of Maryland

I have the completed answer for the first 7 problems (the last 3 problems (#37, #45 and #48) are not founded in chapter 4)

Problem 8:
With given charge Q1 , the electric field E at Q2 due to Q1 by definition is

E=

Q1

4 0 r 2

r

And the differential of length dl for integration when finding the work can be converted from
rectangular coordinates to spherical coordinates as follow

dl = drr + rd + r sin  d
Thus, their product or the integrand is

E  dl =

Q1

4 0 r 2

 E  dl =

(

r  drr + rd + r sin  d

Q1

4 0 r

2

( )

Q1

dr r  r +

4 0 r

2

)

( )

rd r   +

Q1

4 0 r 2

( )

r sin  d r  

 Q1

 Q1

 Q1

 E  dl = 
dr  (1) + 
rd  ( 0 ) + 
r sin  d  ( 0 )
2
2
2
 4 0 r

 4 0 r

 4 0 r

Q1
 E  dl =
dr
4 0 r 2
a) Hence, the work done WB →C in carrying the charge Q2 from B to C while the 2 last variables
are held constant is calculated by
rA

rA

rB

rB

WB →C = −Q2  E  dl = −Q2 

Q1

4 0 r 2

dr

rA

 WB →C


Q 
= −Q2   − 1 
 4 0 r  rB

 WB →C =

Q1Q2
QQ
− 1 2
4 0 rA 4 0 rB

And the unit for this answer is joule ( J ) .
b) If we write the electric field E in terms of spherical coordinates, it follows that

E=

Q1

4 0 r 2

r = Er r + E  + E 

Q1

Hence, according to the notation, we have Er =

4 0 r 2

, E = 0 and E = 0 .

Similarly, the work done WC → D in carrying the charge Q2 from C ( rA , B , B ) to D ( rA , A , B )
while r and  are held constant is calculated by
A

WC → D = −Q2  E d
B

A

A

B

B

 WC → D = −Q2  0  d = −Q2  0
 WC → D = 0
And the unit for this answer is joule ( J ) .
c) By the result in part b, we have E = 0 . Finally, the work done in carrying the charge Q2 from

D to A while the 2 first variables are held constant is calculated by
A

WD → A = −Q2  E d
B

A

A

B

B

 WD → A = −Q2  0  d = −Q2  0
 WD → A = 0
And the unit for this answer is joule ( J ) .
Problem 12:
a) The coor...

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Anonymous
Excellent job

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