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    Introduction to Electromagnetic Theory
    Introduction to Electromagnetic Theory
    Introduction to Electromagnetic Theory
    Ebook911 pages7 hours

    Introduction to Electromagnetic Theory

    By George E. Owen

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    A direct, stimulating approach to electromagnetic theory, this text employs matrices and matrix methods for the simple development of broad theorems. The author uses vector representation throughout the book, with numerous applications of Poisson’s equation and the Laplace equation (the latter occurring in both electronics and magnetic media). Contents include the electrostatics of point charges, distributions of charge, conductors and dielectrics, currents and circuits, and the Lorentz force and the magnetic field. Additional topics comprise the magnetic field of steady currents, induced electric fields, magnetic media, the Maxwell equations, radiation, and time-varying current circuits.
    Geared toward advanced undergraduate and first-year graduate students, this text features a large selection of problems. It also contains useful appendixes on vector analysis, matrices, elliptic functions, partial differential equations, Fourier series, and conformal transformations. 228 illustrations by the author. Appendixes. Problems. Index.

    LanguageEnglish
    PublisherDover Publications
    Release dateJan 23, 2013
    ISBN9780486174440
    Introduction to Electromagnetic Theory

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      Introduction to Electromagnetic Theory - George E. Owen

      Illustrations by the Author

      Bibliographical Note

      This Dover edition, first published in 2003, is an unabridged republication of the work originally published by Allyn and Bacon, Inc., Boston, in 1963.

      Library of Congress Cataloging-in-Publication Data

      Owen, George E. (George Ernest), 1922

      Introduction to electromagnetic theory / George E. Owen ; [illustrations by the author].

      p. cm.

      Originally published: Boston: Allyn and Bacon, 1963.

      Includes index.

      9780486174440

      1. Electricity. 2. Electromagnetism. I. Title.

      QC523.09 2003

      537-dc21

      2003043801

      Manufactured in the United States of America

      Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

      To my son o lum Stephen

      Preface

      Although the topics contained in the subject of electricity and magnetism are standard, this text will present some fresh viewpoints and some original material.

      This manuscript was developed over a period of ten years from my course for undergraduates and first-year graduate students. The results of this course were first presented in a set of bound notes in 1956. The kind reception of these notes led me to revise and rewrite them until the text presented here was a reality.

      Among the variations from the standard treatment is a development of the electric displacement field, incorporating contributions from quadrupole distributions and higher moments. Also, I have found that the use of the Dirac delta function enables the intermediate student to gain a much greater insight into the relation between point configurations and continuous distributions. In fact, by introducing the delta function with the development of the point charge potentials and fields, I have found that students are able to gain physical interpretations of both the delta function and later the Green’s function. The gross features of these functions are grasped more readily in some respects than the solutions of partial differential equations which have become a standard part of a course at this level.

      As far as possible, exact solutions are presented along with various far field and near field approximations. In particular, the fields of circular configurations are discussed in detail. This necessitates the introduction of the complete elliptic functions. To allow for gaps in the student’s background these functions are described briefly in an appendix.

      Seven appendices have been included as the final portion of this book to allow the reader a brief review of elementary differential geometry, matrices, elliptic functions, partial differential equations, Fourier series, conformal transformations, and physical constants. Perhaps the most obvious departure from the standard treatment lies in the use of matrices and matrix notation. I would compare the usual tendency to avoid matrix notation to an attitude prevalent in the thirties which neglected the more convenient vector notation. We now accept vector methods as an absolute necessity.

      Matrix methods make the presentation more powerful and often simpler. Whenever the discussion could benefit, the matrix representation has been employed.

      To note further some recent additions, the vector harmonics are developed and employed to describe the radiation field.

      By and large the text portion alone would appear to be highly formal and to require a fair mathematical background. The book must be used in conjunction with the appendices. With the appendices the discussions are complete, and the reader should not find the level too advanced.

      It was my intention to leave many of the applications and so-called physical discussions as problems. As a result the problems are essential to the work. For instance, applications of the formalism to such items as Brewster’s law, total reflection, etc., appear in a systematic fashion in the problems.

      Because the scope of this text was intended for both undergraduate and first-year graduate students, courses at the intermediate level may well omit the more advanced portions of the book. The ordering of topics is appropriate for such deletions.

      I wish to thank my colleagues Professor Feldman, Professor Fulton, Professor Kerr, Professor Pevsner, Professor Madansky and Professor Franco Rasetti for their comments and suggestions.

      This has been an enjoyable endeavor, and I trust that the reader will gain some new insights from this effort.

      Finally, my deepest regards are extended to Mrs. Dencie Kent, who typed this manuscript so many times.

      George E. Owen

      Table of Contents

      Title Page

      Copyright Page

      Dedication

      Preface

      Notation

      I. Preliminaries

      II. The Electrostatics of Point Charges

      III. Distributions of Charge

      IV. Conductors and Dielectrics

      V. Currents and Circuits

      VI. The Lorentz Force and the Magnetic Field

      VII. The Magnetic Field of Steady Currents

      VIII. Induced Electric Fields

      IX. Magnetic Media

      X. The Maxwell Equations

      XI. Radiation

      XII. Time- Varying Current Circuits

      Appendix

      Problems

      Index

      Notation

      in the illustrations will be represented by bold face letters such as A in the text.

      Matrices will be represented by open face letters such as S in the text. The elements of a matrix are shown as symbols having two subscripts.

      I. Preliminaries

      Force, kinematics, and mass form the foundation of mechanics. The proper definition of the mathematical objects called force and mass is involved and in some instances is controversial. Because the theory of electricity and magnetism utilizes the concept of forces and displacements of charged bodies, it is essential that some elements of mechanics be incorporated in this development. Under the assumption that the definitions of force and mass have been established, our major interest in this section will be to review the definitions of work, potential energy, and the conservative force. Appendix A contains a brief review of vector analysis using generalized coordinates; hereafter the vector and coordinate notation can be referred to in that Appendix.

      A. WORK

      Assume that a point particle, m, moves along a path labeled C. If at every point on the path the particle is acted upon by a force F(x,y,z), then the work to move the particle from point A to point B is DEFINED as

      The force F(x,y,z) is a vector point function,

      F(x,y,z) = Fx(x,y,z)i + Fy(x,y,z)j + Fz(x,y,z)k,

      and

      dr = dx i + dy j + dz k.

      Considering only those vectors which are independent of time,

      In generalized orthogonal coordinates ξm,

      To evaluate the line integral an equation of constraint (representing the specified path) must be utilized. The space curve connecting A and B can be written

      f(x) = g(y) = h(z).

      Actually this equation of constraint can be considered to be three equations giving the projections of the path or space curve on the three mutually orthogonal planes xy, yz, and zx. By substituting the equations of constraint into WA→B, the integral is evaluated. In other words, to evaluate WA→B we reduce each element of the integrand to a function of the single variable corresponding to the associated differential. If f(x) = g(y) = h(z) can be solved to give

      y = φ(x) and z = ζ(x),

      then for instance

      Fx(x,y,z) dx = Fx(x,φ(x),ζ(x)) dx.

      To illustrate this consider the following example. Let

      F = xy i + zj + k,

      (where the appropriate units are implied) and let the path be represented by

      . Now evaluating,

      The force vector must have a constant before each component carrying the appropriate units; such as newtons/m² for the first component. Then the units of W will be newton meters.

      B. THE CONSERVATIVE FORCE

      Under certain conditions the LINE INTEGRAL is independent of the path of integration, depending only upon the coordinates of the end points. If F · dr is an EXACT differential then it is equal to the total differential of some scalar point function V(x,y,z).

      F · dr = Fx(x,y,z) dx + Fy(x,y,z) dy + Fz(x,y,z) dz = −dV(x,y,z)

      or in more general terms

      The components of the vector F under such circumstances are directly related to the derivatives of the scalar function V, because

      F ··dr = −dV = Fx(x,y,z) dx + Fy(x,y,z) dy + Fz(x,y,z) dz

      then

      and

      This statement of course relates the vector F and the gradient of V:

      −dV = −grad V · dr = F ··dr

      whence

      F = −grad V,

      which is a statement of the exact differential relationship.

      In generalized coordinates

      If we now examine the integral under these conditions, we find

      This expansion demonstrates that the integral is independent of the particular path chosen between A and B.

      The NEGATIVE sign assigned to the relation between F and grad V is a convention. As we shall observe, this choice of sign defines the total energy of a system as the sum of the kinetic and potential energies. The opposite sign convention would require that the total energy to be the difference of these two constituents.

      Forces which have the properties just derived are called CONSERVATIVE. A conservative force must satisfy two conditions.

      The vector point function representing the force must be generated from a scalar point function by

      F(ξ1,ξ2,ξ3) = −grad V(ξ1,ξ2,ξ3);

      The force F must not be an explicit function of time,

      F ≠ F(ξ1,ξ2,ξ3;t).

      If these two conditions are satisfied, the conservative property is concisely stated by the zero-value closed path integral

      The time independence of F is implied if we assume that the integration over the closed path is performed in a nonzero time interval. Finally we should remark that the vanishing of the curl of F (see Appendix A) is a sufficient condition for the closed path integral to be zero (assuming that curl F is defined at every point on S). Stokes’ theorem provides a proof of this statement. If

      curl F = 0

      then

      The curl of F is a measure of the circulation of F.

      One can demonstrate further that when

      then

      C. CONSERVATION OF ENERGY

      For simplicity of development consider the motion of a point mass m in a force field F = −grad V. By Newton’s second law,

      where F is the force acting upon a point mass moving with a velocity v = dr/dt. The mechanical momentum¹ is designated by

      The first quadrature of the second law is obtained by taking the scalar product of drwith the equation in question ²:

      .

      The power dissipated is defined as

      If F is conservative, the power equation reduces to the equation for the conservation of energy: when

      F = −grad V,

      then

      thus under these conditions

      and

      T + V = E = a constant of the motion.

      E = T + V is called the total energy of the system.

      When the conservative force was defined we stipulated that F be independent of time. The consequences of having a time-dependent potential demonstrate vividly the reasons for this restriction. If we allow a force field F to be generated from a time-dependent scalar point function U(x,y,z;t), then

      F(x,y,z;t) = −grad U(x,y,z;t)

      and

      When U is an explicit function of t we find an extra term in the expansion of the total derivative,

      Therefore

      and

      The total energy E under such circumstances is not constant in time.

      II. The Electrostatics of Point Charges

      A. DEFINITION OF CHARGE

      In the natural sciences it is often difficult to prepare the definitions of the terms which are to be introduced. Strangely enough, sometimes the very basis of a science (and by this we mean the initial definitions) is a region of controversy. In the field of analytical dynamics one must proceed quite carefully in the manner of introducing the definitions of mass and force. In all cases we postulate that observers are endowed with spatial and temporal intuition. This is to say, we assume that observers can detect and measure the displacement of material bodies. In addition we assume that they can detect displacement in time; in particular, that they are endowed with the ability to discern, of two events a and b, whether event a occurs before or after event b.

      Such assumptions remove the concepts of displacement, velocity, and acceleration from the realm of philosophical argument. However, even when we have established our right to space and time measurement, still greater obstacles can be encountered in attempting to define force and mass. There is a danger in this instance that two undefined quantities will be defined one in terms of the other.

      The main objective is to establish a one-to-one correspondence between our observations of natural phenomena and the representation of these observations in terms of certain mathematical objects. There are several ways in which the correspondence of the objects in the mathematical representation and the observations can be presented. From a practical viewpoint anyone of these approaches is sufficient for our purposes. A standard method consists of postulating the basic equations and observables of electromagnetic theory. One then proceeds to demonstrate that the postulated quantities are consistent with all natural observations. The operational definition represents a different approach. All quantities such as mass, charge, etc., are defined in terms of a set of experimental observations. In the end quantities have reality in terms of these recipes for definition.

      In order to introduce a definition of charge we will assume that displacement, time, force, and mass are quantities already defined. A body is said to be charged or to possess charge when it exhibits forces characteristic of a charging process. We now must explain what is meant by a charging process.

      A primitive charging process is an operation such as rubbing glass or amber with fur, silk, or other materials, which cause the bodies involved to exhibit forces attractive or repulsive. These forces cannot be attributed to gravitational forces.

      A more sophisticated charging operation involves the use of a battery. For instance a simple battery can be constructed by placing a Zn or Mg rod in water. A number of ions separate from the metallic terminal and leave it charged or in a state of electrification. External bodies can now be charged by bringing them into contact with the terminal, and after charging will show the forces characteristic of this process.

      The so-called forces characteristic of the charging process can be detected in a number of ways. If several bodies such as pithballs, suspended by insulating strings from a common point, are charged, when brought together they will assume positions of equilibrium consistent with the electrical forces. It is possible to take one charged body and use it to place (induce) charge on an electroscope. The displacement of the leaves of the instrument then indicates the presence of charge.

      There are but two kinds of electrical charge, positive and negative. Verification of this statement lies in experimental evidence, that

      the mutual attraction of three or more

      charged bodies has never been observed.

      Experimental evidence also indicates that a lower limit of electrical charge exists. This discrete lower limit is the magnitude of the electron or proton charge.

      The unit of charge which will be used in this work is the coulomb. The standard coulomb can be specified from Coulomb’s law, from the faraday, or indirectly from the standard ampere.

      Present day practice utilizes the standard ampere as a means of measuring the standard coulomb. This technique is by far the most accurate, uniform, and convenient. The unit ampere corresponds to the rate of flow of one coulomb per second across a specified boundary. Because the ampere and coulomb are used as standards, the units to be employed throughout the discussions to follow will be the rationalized MKS units. This procedure also has become standard. Since lengthy discussions of units tend to distract rather than enlighten, the comparison of units will be confined to a table in Appendix G at the termination of this study.

      B. COULOMB’S LAW

      The forces characteristic of charged bodies have been alluded to previously. By and large the forces set up between extended charged bodies can have a relatively complicated dependence upon the laboratory coordinates. Fortunately the force fields set up by extended charged bodies can be understood by a linear superposition of the fields of all of the infinitesimal volumes of charge which make up the body in question. There is no a priori reason why the total field should be obtainable by linear superposition; however, experimental observation demonstrates the reliability of this principle.

      Because the fields can be constructed essentially by superposing fields of point charges, we shall begin with a study of the forces between point charges.

      A true point charge of course may not be realized in an actual experiment. The forces between point charges, however, can be indicated in the limit, if a series of force experiments are performed with a set of charged bodies, a smaller body being used in each succeeding experiment. Alternatively, the forces can be determined by use of a torsion balance (page 11). By charging the two bodies employed to different degrees one can establish quite quickly that the magnitude of the force F12 between two very small charged bodies 1 and 2 is proportional to the PRODUCT of the charges,

      |F12| ∝ q1q2.

      Next one can demonstrate that the force between charged bodies is a function of the distance between them. In fact, in the limit as the dimensions of the bodies become very much smaller than their separation r12, the force depends upon the inverse square of the separation:

      Further experiments would indicate that the force between two point charges q1 and q2 has the direction of the line of centers connecting the two points:

      where

      r1 = the position vector of q1

      r2 = the position vector of q2.

      The constant of proportionality in rationalized mks units is determined from the velocity of light and the magnetic permeability of free space. The reasons for these dependences will be seen in later sections (see Chapters X and XI). The constant of proportionality is

      k = 10⁹ meters/farad,

      where k, written in terms of the PERMITTIVITY of free space ε0, is

      ε0, the permittivity of free space, is 8.85(10)−12 farad/meter. Thus

      or

      In these units we should find a force of 9(10)⁹ newtons between two point charges one meter apart, each having a charge of one coulomb.

      In electrostatic units (esu), the unit of charge is defined as the charge on each of two point bodies one centimeter apart which produces a mutual force of interaction of one dyne. Thus the constant of proportionality in the esu system is dimensionless and equal to unity.

      The relation between ε0, the permittivity of free space, µ0, the magnetic permeability of free space, and c, the velocity of light in a vacuum, is

      c is a measured quantity equal to 2.998(10)⁸ meters/sec while µ0 is taken arbitrarily as

      µ0 = 4π(10)−7 henry/meter.

      The relation connecting ε0, µ0, and c comes from the equations governing the propagation of electromagnetic waves in free space (see Chapters X and XI). This relation fixes ε0, once c and µ0 are determined : ε0 = 1/c²,µ0.

      Under the conditions considered for the two point charges, the force exerted by q2 on q1, F12, and that which q1 exerts upon q2, F21, are equal in magnitude and opposite in direction:

      F21 = −F12.

      Thus Newton’s third law is satisfied. This relation has been implied already in the vector form of Coulomb’s law; by interchanging the indices in the equation for F12, we find

      C. THE ELECTRIC FIELD INTENSITY

      1. THE PRINCIPLE OF LINEAR SUPERPOSITION

      Experimentally it has been verified that the total force Ft on a test charge Qt in the presence of N charges qj is given by the vector sum of all of the separate Coulomb interactions, Ftj:

      or

      2. THE ELECTRIC FIELD INTENSITY

      Consider a region of space which contains N point source charges qj. In general we must consider that the N source charges occupy certain positions of equilibrium. This equilibrium is the final result of the electrostatic forces between the source charges coupled with certain mechanical constraints which must be specified for each particular situation. For instance, in Chapter IV we will find that when charges are placed on the surface of a conductor they are constrained to remain on the surface. This situation can be represented as a geometric constraint.

      Our purpose is now to obtain a measurement of the net force field set up by the N point charges, at any point in space. To accomplish this we introduce a test point charge Qt and measure the force exerted on it by the N source charges.

      Caution Must be Observed!

      If the magnitude of the test charge Qt is too large we may obtain a measurable rearrangement of the source charges. In other words, the act of making the measurement tends to alter the initial situation we wish to measure. Our desire therefore is to introduce the test charge without influencing the positions of the source charges. This of course is impossible.

      There is a technique, however, which will allow us to gauge the original force field in the limit as the magnitude of the test charge approaches zero. A series of experiments can be performed with a continual decrease in magnitude of Qt. In the limit as Qt goes to zero we will obtain the magnitude and direction of the original force field of the N source charges.

      To illustrate this we show an example plot of the magnitude of the net force Ft as a function of Qt. It is quite clear that when Qt approaches zero the force on Qt goes to-zero. On the other hand, the ratio of Ft to Qt has a nonzero value in the limit.

      To represent the field of the source charges we define the ELECTRIC FIELD INTENSITY as the net force per unit charge exerted on the test charge Qt as the magnitude of Qt approaches zero:

      This vector field is characteristic of the N is measured as the slope of the |Ft| vs. Qt curve in the linear region near Qt = 0.

      We are now in a position to compute the electric field intensity for our N source charges. If Qt is at a point P, defined by the position vector rp, then

      and

      The field of a single point charge q situated at a point designated by rp is therefore

      is conservative, we will develop the energy stored in a charge ensemble and then investigate the electrical properties of special configurations of point charges.

      D. THE ELECTROSTATIC POTENTIAL

      is a conservative vector field. This can be demonstrated in various ways. A series of path integrations would show that the work done in moving a test charge from a point a to a point b is independent of the path taken between a and b:

      (x,y,z) can be generated from a scalar point function V(x,y,z) by

      p = − gradp Vp.

      Examine the field of a single point charge q:

      We assert that

      If the reader is in doubt about the details of this operation he should expand |rp − r|−1 in terms of its components and take the gradient term by term.

      Notice that the gradient represents variations in the coordinates of the point P, at which the field is measured, wherefore the derivatives are taken with respect to the variables xp/, yp, and zp.

      = −grad V, the field can be recognized as a conservative field.

      Going back to the more general problem of N source charges, the principle of superposition implies that

      Using the result for the single charge, we find that

      p is

      and

      p = −gradp Vp.

      is conservative, the integration is independent of the path connecting a and b:

      The difference (Vb − Va) is called the potential difference, and we have shown that

      The negative of the potential difference between two points is equal to the positive work done per unit test charge.

      The potential difference is positive when the work is contributed by the external mechanism which moves the test charge. The electrical energy increases.

      The potential difference is negative if the electrical system of N charges dissipates energy in moving Qt. This energy is absorbed by the external mechanism controlling Qt.

      In the case of N source charges the potential difference is

      assuming of course that the constant (which sets the zero of potential) is the same in both Vb and Va and cancels in the difference.

      When the source consists of a single point charge q at the origin (r = 0), then

      Because the functional form of V goes as 1 /r, the work required to bring a test charge from infinity to a point rp depends upon the coordinates specified by rp. Thus it is convenient to take the zero of potential at infinity. Then the constant appearing in V(xp,yp,zp) is ZERO, and

      For N source charges, the potential at P is then

      p from the integral form can become quite complex and tedious, and the development of the scalar potential function is usually much easier. Therefore the usual procedure to obtain the vector field will consist of first solving for the potential function, after which the vector field can be obtained quite readily by taking the gradient of the scalar function.

      E. THE TOTAL ENERGY STORED IN A SYSTEM OF POINT CHARGES

      The work required to bring a point charge q1 from ∞ to a point designated by r1 in a field-free region is ZERO. However, if a second charge q2 is brought from ∞ to a point r2, the region is no longer field-free, and WORK is done by the agent bringing up the second charge. As a result, electrostatic energy (positive or negative depending upon the relative signs of q1 and q2) is stored in the system of two charges.

      The symbol for the total energy stored in the electrostatic system will be U. U is equal to the negative of the work done by the field, or to the positive work done by an external agency:

      The term grad2 implies a variation in the coordinates of charge q2.

      This equation has been written in such a manner that the storage is considered to be concentrated in q2. Actually in a system composed of two charges the energy can be considered to be shared between the two.

      Let us call V21 the

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