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It is a commonly held view, and one often vehemently expressed by students-particularly recent survivors of a course on the subject-that electromagnetics is a difficult, complicated, mysterious discipline. It requires mastery of abstruse mathematical techniques, they say; it entails juggling a bewildering variety of equations and laws and rules, they decide. Even intense study has left them with only a superficial grasp of the concepts. Few see the beauty of electromagnetics; not many appreciate the simplicity and economy of its fundamental laws. A minority of its practitioners realize its wide-ranging utility, the breadth and scope of its applications. Only a minority master it enough to be able to use its principles to understand or predict the capabilities and limitations of the engineering systems they need to analyze or design.
Instructors of the subject are in a better position to assess the grandeur and importance of electromagnetics but are plagued by other demons as they plan the presentation of the subject. There is never enough time to cover all the aspects of electromagnetics that they deem essential, indispensable, and obligatory. They are forced to make coldhearted choices of their topics and suffer the agony of having to discard this or that favorite theme. Some find that they must turn their course into lectures on applied mathematics; some concentrate on mastery of electrostatics and magnetostatics, leaving little time for the radically different phenomena of dynamics. Agreement on what topics must be included and which may be discarded, what areas are to be emphasized and which to be glossed over, is indeed rare.
The present work will not lay these matters to rest but offers an approach and a pedagogic philosophy aimed at making a respectable contribution to the debate over an appropriate syllabus for the subject. It is intended as a textbook for a one-semester first course in electromagnetics for engineers and physicists. It is an outgrowth of decades of experience in teaching the subject to juniors in electrical engineering at the School of Engineering and Applied Science of Columbia University. The students come to the course with preparation in calculus, including a smattering of vector analysis, and with some exposure to notions of electricity and magnetism in a physics course, usually with emphasis on electrostatics and magnetostatics. Several elective courses follow this one, but at Columbia this is the only required course in electromagnetics in a crowded undergraduate curriculum in electrical engineering.
What approach underlies this work and how is it different? The pedagogic preferences expressed in this text include the following.
1. Time-varying fields. Time variation is central and paramount virtually from the start. The emphasis is on distributed systems, featuring action at a distance and after a delay, so that time development and dynamics are deeply involved and inescapable. A modern curriculum should avoid the classic textbook's concentration on statics. Few practicing engineers hang pith balls and rub combs to transfer charges. Even the engineers or designers who appear to deal with purely static systems usually intend to have them process time-varying signals. Those practitioners who design devices and systems based on only electrostatics and magnetostatics principles, or on software that deals only with statics, remain uncertain, if not actually ignorant, of the dynamic response or bandwidth of their systems. They are likely to be left behind, wondering why their designs misbehave or fail at today's required speeds or data rates.
2. Simple mathematics. The mathematics is kept simple, at the level of integral calculus. The text avoids approaching the subject from the standpoint of solving partial differential equations and boundary value problems that invoke esoteric mathematical functions and apply to only a few types of geometry. Useful as they undoubtedly are, Bessel and Hankel functions, Legendre polynomials, elliptic integrals, and the like are likely to overwhelm students at the level of this text, keep them from appreciating the underlying phenomena, and leave them helpless when confronted with some unfamiliar geometry. The special functions of mathematical analysis should be taught, but at a later stage. There are enough important concepts to learn without adding these special functions into the mix, at this elementary level.
Instead of handling differential equations, the present text makes extensive use of Maxwell's equations in integral form. Note that this does not mean we are dealing with integral equations, any more than asking to find the derivative of a given function is the same as solving a differential equation. We just need integrals of familiar functions; the integrals need not involve functions more complicated than powers or logarithms of the variable. The use of equations in integral form allows us to deal with any geometry, at least approximately. As an added bonus, the integral form applies, unchanged, to moving media as well as to stationary ones.
3. Quasistatics for high frequencies. The text approaches high-frequency phenomena by way of quasistatic analysis. This furnishes a more user-friendly, gentle transition from the more intuitive low-frequency circuit theory results to those of faster circuitry. Quasistatic analysis imparts insights difficult to gain from formal solutions to differential equations that feature advanced, special functions of mathematical analysis. Does one readily extract insights into the skin effect in a cylindrical wire by expressing the fields in terms of Kelvin functions or Bessel functions with complex arguments? Quasistatics emphasizes deviations from the results of circuit theory that are due to interactions between the electric and the magnetic fields in the configuration. It can be applied to geometries that defy analytic solutions. It is particularly helpful in readily furnishing approximate answers in complicated cases. Most important, it provides the engineer with estimates of the frequency range of applicability of possible designs.
4.Transmission lines paramount. The main focus of the work is on wave propagation, exemplified particularly by the behavior and response of transmission lines, both transient and steady state. Such structures are realistic and practical signal-transmission systems. Transmission lines illustrate time delay, reflections, standing waves, matching procedures, the effects of a mismatch, measurement techniques, and power transfer. The simplicity and far-ranging applicability of harmonic plane waves are also stressed, however. The two are carefully integrated to support and complement their concepts and applications.
5.Electric and magnetic fields on a par. Electric and magnetic fields are treated as twins, with virtually equal importance. Too many textbooks and courses stress the electric field and introduce the magnetic field as an afterthought or oddity. While this can be sensible in the static case, the two types of fields are inextricably entwined in the time-varying case. We seek to treat them on an equal footing and emphasize how each affects and actually creates and maintains the other.
In this connection, it is interesting that modern electronic circuit design has emphasized capacitive elements in integrated circuits; these are based on the behavior of electric fields. More recently, the design of inductive elements, based on magnetic field behavior, has regained importance, driven particularly by the exigencies of wireless communication circuits and their tuning elements.
6.Avoid potentials! We studiously and deliberately avoid introducing potentials and the notion that the electric field is the gradient of a scalar potential. This radical attitude and apparent heresy calls for some explanation.
We contend that the student who hasn't learned or has forgotten the notions that electric fields are conservative and that conductors are equipotentials is more ready to tackle time-varying situations than is the rival who must unlearn and abandon these incorrect assertions when dealing with dynamic fields. Avoiding the potentials is, in fact, a consistent element of the approach that emphasizes dynamics over statics.
Potentials are enormously helpful when dealing with purely electrostatic or magnetostatic effects, but they become a major obstacle to making the transition to the time-varying case, when we need to unlearn almost all about the scalar potential. For example, students who have been taught that all points in a circuit that are connected by conductors must be at the same voltage typically have a difficult time with the fact that the time-varying voltages at various points of a circuit or structure may be, and usually are, different even when they are connected by perfect conductors.
That we avoid introducing potentials is not to say that we don't deal with voltage; we do of course, but as an electromotive force (emf), not as a potential. The distinction is vital: Static potentials are conservative (zero closed-loop integral of the field) but emf (voltage) is not. The nonzero voltage around a closed curve is crucial to time-varying fields and to circuits with energy sources in them; it is incompatible with fields derived from a scalar potential. Our experience has been that it is easier to impart an understanding of dynamic field effects when the scalar potential has not been inculcated prior to the need to modify or abandon it and the limitations it imposes. If we insist on deriving the electric field from a scalar potential, we find that it is only part of the field and we need to add the time derivative of the vector potential. The new potential fields are further removed from the physical interaction, and this complicates the situation to an extent that may make the potentials more of a pedagogic liability than an asset. We choose to postpone discussion of potentials to the end, when we look back on the entire development and review the overall principles of electromagnetics.
7. Interpretations stressed. As a matter of habit, the examples in the text do not end when the answer has been found. Those answers are extensively discussed and interpreted, a practice that should be made routine. Students should be encouraged not merely to extract a number or formula from the equations but to look at the answer, examine it from the standpoint of physical plausibility and logic, and especially to interpret the result. This book features careful derivations and emphasizes interpretations.
It is also often helpful to draw analogies among different results; we have deliberately emphasized such analogies and similarities of seemingly disparate aspects of the subject, such as the electric and the magnetic versions of certain effects.
8. Realistic figures. We have felt strongly that figures should never be misleading, and we have kept them realistic whenever it may matter. Curves have been properly calculated and plotted, not merely sketched. In some cases, perspective views are the only renderings that can avoid confusion, and these have been carefully computed and presented as realistically as possible.
9.Answers to all problems. Answers to all problems are provided at the back of the book. Obviously, judicious use of the answers can be of great help to the student; indiscriminate abuse of the answer key is just as obviously harmful to the learning process. Answers are most often sought in formulaic or symbolic form; such forms are usually much more informative than a numerical answer that applies to only a specific instance. In the guise of a formula, the answer provides information on how the result varies with the parameters of the problem. Nevertheless, we do ask for a numerical value when appreciation of typical magnitudes is important.
For the instructor, a Solutions Manual has been prepared; it provides complete solutions to every problem.
10. Some topics treated in problems. We have sought to make the problems instructive rather than mere drill. A variety of problems has been provided, some easy, some more challenging, some illustrative, some to fill in gaps in the text, some for practice, some particularly instructive, some to pique the student's interest.
In a few cases where we have omitted formal demonstrations of certain assertions, guidance as to how to prove them has been relegated to the problems. A few topics that might seem to be missing from the text may be found among the sets of problems. Instructors may prefer to lecture on some of these topics, rather than leave them as homework assignments. The Solutions Manual can help guide such lectures.
11. So many equations, so little time? Although the level of the mathematics used has been kept at that of elementary integral calculus, there are occasions when the onslaught of equations may seem like that of a blizzard. There are several reasons for it when the mathematical development becomes intense.
Show all steps.
The primary reason is that we prefer to err on the side of showing all the steps in developing a result; our experience is that most students appreciate seeing the intermediate steps between the starting point and the final equation.
Another cause is that we prefer, whenever practical, to show where a result comes from, rather than merely assert its truth. One example for which this leads to an intensive development is the discussion of the parallel-wire transmission line. This development leads to an important engineering design equation for the characteristic impedance of transmission lines, specifically the parallel-wire line, in terms of the geometry and electrical constitution of the structure. This is something an engineer should know how to obtain and use, for whatever configuration may be under design. Most textbooks don't even attempt to calculate this; they may simply furnish a ready-made formula for the characteristic impedance, descendent from heaven or from the manufacturer of the transmission line-but then who will be hired by the manufacturer to design that line, if engineers are not taught how such formulas are developed?
Yet another instance is represented by the development of the results for oblique incidence of plane waves. In many textbooks, the derivation is confined to Snell's laws and the polarizations are merely quoted. In this text, we use the concept of impedance to unify, simplify, and develop the full set of equations. As a result, many high-frequency or optical system design methods reduce to special cases of impedance matching or transforming techniques.
Types of boundary conditions.
Another section that may seem overly elaborate is that on boundary conditions. Our experience has been that this subject often causes confusion. For most applications, the boundary conditions are comfortably simple; for other cases, they can become baffling. We have divided the presentation of how to select and apply boundary conditions into ''ordinary'' and ''extraordinary'' types to clarify this circumstance.
Corrections to circuit theory.
The material on quasistatic analysis is extensive, though not intensive, and may be controversial for that reason alone. The aim is to present a systematic, tabular approach to the adjustments and corrections that must be made when a design based on statics is subjected to time-varying signals. Pushing electromagnetic designs to higher and higher frequencies is currently an important aim of engineers, particularly for communications applications. If an instructor feels the material on quasistatics is superfluous or too extensive, it can be skipped or truncated and still leave a coherent treatment of electromagnetics. However, this topic has much that is of great conceptual value, especially to unveil, correct, and extend certain circuit-theory results. It also serves as a transition to wave phenomena, an approach that avoids handing students the wave solution from on high and merely asking or allowing them to verify that it satisfies the equations.
A few words on some technical aspects of the text are in order. We have used the MKSA subset of the SI (Système Internationale) system of units; we do not want confusion over units to cloud the meaning of the concepts. Examples have been set off from the rest of the text by a vertical line; within the example, the end of the question has been marked with a symbol <<>>, to separate it from the start of the solution. Each chapter ends with a summary or review of the material covered.
Special efforts by some and special patience on the part of others have been indispensable to the production of this textbook. Particular recognition and thanks are due to Dr. Perry Malouf, who graciously undertook to attempt and to criticize the problem sets. The book was years in the making, but there were peaks of sometimes frantic activity that deprived family members of the attention they deserved. I take this opportunity to recognize and applaud their patience, forbearance, and assistance.
Diament, Paul : Columbia University in the City of New York
1. Gauss's Law, Surface Integrals, and Electric Fields.
Motivation. Field Concept. Definition Of Field. Electric Field. Effect Of The Medium. Flux Of Vector Field. Gauss's Law. Determination Of Field From Flux. Point Charge. Coulomb's Law. Superposition Of Fields. Summary. Problems.
2. Ampère's Law, Line Integrals, and Magnetic Fields.
Magnetic Field. Effect Of The Medium. Magnetomotive Force. Ampère's Law. Determination Of Field From Current. Current And Current Density. Conservation Of Charge. Magnetic Field Of Semi-Infinite Filament. Current Element. Superposition Of Fields. Biot-Savart Law. Current Element As Dipole Flow Source. Summary. Problems.
3. Emf, Field Dynamics, and Maxwell's Equations.
Electromotive Force. Ohm's Law. Electric Power Density. Faraday-Maxwell Law. How Ampère's Law Fails. Ampère-Maxwell Law. Types Of Current. Magnetic Field Of A Moving Charge. Gauss's Magnetic Law. Boundary Conditions. Extraordinary Boundary Conditions. Summary Of Boundary Conditions And Effects Of Conductivity. Superconductors, Plasmas, And Space Charge. Summary. Problems.
4. Maxwell's Equations and Quasistatic Analysis.
Field Equations. Maxwell's Equations. Quasistatic Fields. Surprises For Circuit Analysts. Systematic Quasistatic Analysis. Normal Modes. Cylindrical Cavity. Fields In A Wire. Microstrip Lines. Summary. Problems.
5. Transmission Lines, Time Delay, and Wave Propagation.
Transmission Line Parameters. Transmission Line Equations. Quasistatic Analysis Of Transmission Line. Summation Of Series. Interpretation: Matched Line. Interpretation Waves. Mismatched Line. Incident And Reflected Waves. Summary. Problems.
6. Steady-State Wave Transmission and Plane Waves.
Sinusoidal Steady State. Complex Exponentials. Sinusoidal Waves. Transmission Lines In The Steady State. Power Transfer. Plane Waves. Wave Designators. Sinusoidal Plane Waves. Polarization. Linear And Circular Polarization. Optically Active Media. Incident And Reflected Plane Waves. Source Of Plane Waves. Load Impedances For Plane Waves. Summary. Problems.
7. Impedance Matching Techniques and Oblique Waves.
Standing Waves. Impedance Measurement And Standing Wave Ratio. Impedance Matching With A Tuning Stub. Smith Chart. Reflection At A Mirror. Antireflection Coating. Oblique Incidence. Oblique Waves As Nonuniform Transverse Waves. Snell's Laws. Transverse Electric And Transverse Magnetic Polarizations. Brewster's Angle. Failure Of Snell's Law. Reactive Impedances. Total Internal Reflection. Significance For Dielectric Waveguides. Reflection At Grazing Incidence. Summary. Problems.
8. Poynting Theorems and Lossy Transmission Lines.
Poynting Theorem. Interpretation. Coaxial Line. Complex Poynting Theorem. Interpretation. General Transmission Lines. Solution To General Transmission Line Equations. Interpretation. Plane Waves In Conductors. Summary. Problems.
9. Waveguiding and Radiating Structures.
Parallel-Plate Transmission Line. Waveguides. Separation Of Variables. TE And TM Modes. Interpretation Of Waveguide Modes. Wave Dispersion And Group Velocity. Antennas And Radiation. Antenna Terminology. Summary. Problems.
10. Review, Epilogue, and Appendix.
Review Of Concepts. Vistas Ahead. Appendix. Problems.
Answers to Problems.
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