Sec. 0.1 0.1 PREFACE

Preface

1

The text is aimed at an audience that has seen Maxwells equations in integral or dierential form (second-term Freshman Physics) and had some exposure to integral theorems and dierential operators (second term Freshman Calculus). The rst two chapters and supporting problems and appendices are a review of this material. In Chap. 3, a simple and physically appealing argument is presented to show that Maxwells equations predict the time evolution of a eld, produced by free charges, given the initial charge densities and velocities, and electric and magnetic elds. This is a form of the uniqueness theorem that is established more rigorously later. As part of this development, it is shown that a eld is completely specied by its divergence and its curl throughout all of space, a proof that explains the general form of Maxwells equations. With this background, Maxwells equations are simplied into their electro quasistatic (EQS) and magnetoquasistatic (MQS) forms. The stage is set for taking a structured approach that gives a physical overview while developing the mathe matical skills needed for the solution of engineering problems. The text builds on and reinforces an understanding of analog circuits. The elds are never static. Their dynamics are often illustrated with step and sinusoidal steady state responses in systems where the spatial dependence has been encapsu lated in time-dependent coecients (of solutions to partial dierential equations) satisfying ordinary dierential equations. However, the connection with analog cir cuits goes well beyond the same approach to solving dierential equations as used in circuit theory. The approximations inherent in the development of circuit theory from Maxwells equations are brought out very explicitly, so that the student ap preciates under what conditions the assumptions implicit in circuit theory cease to be applicable. To appreciate the organization of material in this text, it may be helpful to make a more subtle connection with electrical analog circuits. We think of circuit theory as being analogous to eld theory. In this analogy, our development begins with capacitors charges and their associated elds, equipotentials used to repre sent perfect conductors. It continues with resistors steady conduction to represent losses. Then these elements are combined to represent charge relaxation, i.e. RC systems dynamics (Chaps. 4-7). Because EQS elds are not necessarily static, the student can appreciate R-C type dynamics, where the distribution of free charge is determined by the continuum analog of R-C systems. Using the same approach, we then take up the continuum generalization of L-R systems (Chaps. 810). As before, we rst are given the source (the current density) and nd the magnetic eld. Then we consider perfectly conducting systems and once again take the boundary value point of view. With the addition of nite conductivity to this continuum analog of systems of inductors, we arrive at the dynamics of systems that are L-R-like in the circuit analogy. Based on an appreciation of the connection between sources and elds aorded by these quasistatic developments, it is natural to use the study of electric and magnetic energy storage and dissipation as an entree into electrodynamics (Chap. 11). Central to electrodynamics are electromagnetic waves in loss-free media (Chaps. 1214). In this limit, the circuit analog is a system of distributed dierential induc-

2

Chapter 0

tors and capacitors, an L-C system. Following the same pattern used for EQS and MQS systems, elds are rst found for given sources antennae and arrays. The boundary value point of view then brings in microwave and optical waveguides and transmission lines. We conclude with the electrodynamics of lossy material, the generalization of L-R-C systems (Chaps. 1415). Drawing on what has been learned for EQS, MQS, and electrodynamic systems, for example, on the physical signicance of the dominant characteristic times, we form a perspective as to how electromagnetic elds are exploited in practical systems. In the circuit analogy, these characteristic times are RC, L/R, and 1/ LC. One benet of the eld theory point of view is that it shows the inuence of physical scale and conguration on the dynamics represented by these times. The circuit analogy gives a hint as to why it is so often possible to view the world as either EQS or MQS. The time 1/ is the geometric LC mean of RC and L/R. Either RC or L/R is smaller than 1/ LC, but not both. For large R, RC dynamics comes rst as the frequency is raised (EQS), followed by electrodynamics. For small R, L/R dynamics comes rst (MQS), again followed by electrodynamics. Implicit is the enormous dierence between what is meant by a perfect conductor in systems appropriately modeled as EQS and MQS. This organization of the material is intended to bring the student to the realization that electric, magnetic, and electromagnetic devices and systems can be broken into parts, often described by one or another limiting form of Maxwells equations. Recognition of these limits is part of the art and science of modeling, of making the simplications necessary to make the device or system amenable to analytic treatment or computer analysis and of eectively using appropriate simplications of the laws to guide in the process of invention. With the EQS approximation comes the opportunity to treat such devices as transistors, electrostatic precipitators, and electrostatic sensors and actuators, while relays, motors, and magnetic recording media are examples of MQS systems. Transmission lines, antenna arrays, and dielectric waveguides (i.e., optical bers) are examples where the full, dynamic Maxwells equations must be used. In connection with examples, about 40 demonstrations are described in this text. These are designed to make the mathematical results take on physical mean ing. Based upon relatively simple congurations and arrangements of equipment, they incorporate no more complexity then required to make a direct connection between what has been derived and what is observed. Their purpose is to help the student observe physically what has been described symbolically. Often coming with a plot of the theoretical predictions that can be compared to data taken in the classroom, they give the opportunity to test the range of validity of the theory and to promulgate a quantitative approach to dealing with the physical world. More detailed consideration of the demonstrations can be the basis for special projects, often bringing in computer modeling. For the student having only the text as a resource, the descriptions of the experiments stand on their own as a connection between the abstractions and the physical reality. For those fortunate enough to have some of the demonstrations used in the classroom, they serve as documenta tion of what was done. All too often, students fail to prot from demonstrations because conventional note taking fails to do justice to the presentation. The demonstrations included in the text are of physical phenomena more than of practical applications. To ll out the classroom experience, to provide the

Sec. 0.1

Preface

3

engineering motivation, applications should also be exemplied. In the subject as we teach it, and as a practical matter, these are more of the nature of show and tell than of working demonstrations, often reecting the current experience and interests of the instructor and usually involving more complexity than appropriate for more than a qualitative treatment. The text provides a natural frame of reference for developing numerical ap proaches to the details of geometry and nonlinearity, beginning with the method of moments as the superposition integral approach to boundary value problems and culminating in energy methods as a basis for the nite element approach. Profes sor J. L. Kirtley and Dr. S. D. Umans are currently spearheading our eorts to expose the student to the muscle provided by the computer for making practical use of eld theory while helping the student gain physical insight. Work stations, nite element packages, and the like make it possible to take detailed account of geometric eects in routine engineering design. However, no matter how advanced the computer packages available to the student may become in the future, it will remain essential that a student comprehend the physical phenomena at work with the aid of special cases. This is the reason for the emphasis of the text on simple ge ometries to provide physical insight into the processes at work when elds interact with media. The mathematics of Maxwells equations leads the student to a good understanding of the gradient, divergence, and curl operators. This mathematical con versance will help the student enter other areas such as uid and solid mechanics, heat and mass transfer, and quantum mechanics that also use the language of clas sical elds. So that the material serves this larger purpose, there is an emphasis on source-eld relations, on scalar and vector potentials to represent the irrotational and solenoidal parts of elds, and on that understanding of boundary conditions that accounts for nite system size and nite time rates of change. Maxwells equations form an intellectual edice that is unsurpassed by any other discipline of physics. Very few equations encompass such a gamut of physical phenomena. Conceived before the introduction of relativity Maxwells equations not only survived the formulation of relativity, but were instrumental in shaping it. Because they are linear in the elds, the replacement of the eld vectors by operators is all that is required to make them quantum theoretically correct; thus, they also survived the introduction of quantum theory. The introduction of magnetizable materials deviates from the usual treatment in that we use paired magnetic charges, magnetic dipoles, as the source of magneti zation. The often-used alternative is circulating Amp`rian currents. The magnetic e charge approach is based on the Chu formulation of electrodynamics. Chu exploited the symmetry of the equations obtained in this way to facilitate the study of mag netism by analogy with polarization. As the years went by, it was unavoidable that this approach would be criticized, because the dipole moment of the electron, the main source of ferromagnetism, is associated with the spin of the electron, i.e., seems to be more appropriately pictured by circulating currents. Tellegen in particular, of Tellegen-theorem fame, took issue with this ap proach. Whereas he conceded that a choice between two approaches that give iden tical answers is a matter of taste, he gave a derivation of the force on a current loop (the Amp`rian model of a magnetic dipole) and showed that it gave a dierent e answer from that on a magnetic dipole. The dierence was small, the correction term was relativistic in nature; thus, it would have been dicult to detect the

4

Chapter 0

eect in macroscopic measurements. It occurred only in the presence of a timevarying electric eld. Yet this criticism, if valid, would have made the treatment of magnetization in terms of magnetic dipoles highly suspect. The resolution of this issue followed a careful investigation of the force exerted on a current loop on one hand, and a magnetic dipole on the other. It turned out that Tellegens analysis, in postulating a constant circulating current around the loop, was in error. A time-varying electric eld causes changes in the circulating current that, when taken into account, causes an additional force that cancels the critical term. Both models of a magnetic dipole yield the same force expression. The diculty in the analysis arose because the current loop contains moving parts, i.e., a circulating current, and therefore requires the use of relativistic corrections in the rest-frame of the loop. Hence, the current loop model is inherently much harder to analyze than the magnetic chargedipole model. The resolution of the force paradox also helped clear up the question of the symmetry of the energy momentum tensor. At about the same time as this work was in progress, Shockley and James at Stanford independently raised related questions that led to a lively exchange between them and Coleman and Van Vleck at Harvard. Shockley used the term hidden momentum for contributions to the momentum of the electromagnetic eld in the presence of magnetizable materials. Coleman and Van Vleck showed that a proper formulation based on the Dirac equation (i.e., a relativistic description) automatically includes such terms. With all this theoretical work behind us, we are comfortable with the use of the magnetic charge dipole model for the source of magnetization. The student is not introduced to the intricacies of the issue, although brief mention is made of them in the text. As part of curriculum development over a period about equal in time to the age of a typical student studying this material (the authors began their collaboration in 1968) this text ts into an evolution of eld theory with its origins in the Radiation Lab days during and following World War II. Quasistatics, promulgated in texts by Professors Richard B. Adler, L.J. Chu, and Robert M. Fano, is a major theme in this text as well. However, the notion has been broadened and made more rigorous and useful by recognizing that electromagnetic phenomena that are quasistatic, in the sense that electromagnetic wave phenomena can be ignored, can nevertheless be rate dependent. As used in this text, a quasistatic regime includes dynamical phenomena with characteristic times longer than those associated with electromagnetic waves. (A model in which no time-rate processes are included is termed quasistationary for distinction.) In recognition of the lineage of our text, it is dedicated to Professors R. B. Adler, L. J. Chu and R. M. Fano. Professor Adler, as well as Professors J. Moses, G. L. Wilson, and L. D. Smullin, who headed the department during the period of development, have been a source of intellectual, moral, and nancial support. Our inspiration has also come from colleagues in teaching faculty and teaching assistants, and those students who provided insight concerning the many evolutions of the notes. The teaching of Professor Alan J. Grodzinsky, whose latterday lectures have been a mainstay for the course, is reected in the text itself. A partial list of others who contributed to the curriculum development includes Professors J. A. Kong, J. H. Lang, T. P. Orlando, R. E. Parker, D. H. Staelin, and M. Zahn (who helped with a nal reading of the text). With macros written by Ms. Amy Hendrickson, the text was Text by Ms. Cindy Kopf, who managed to make the nal publication process a pleasure for the authors.

1 MAXWELLS INTEGRAL LAWS IN FREE SPACE

1.0 INTRODUCTION Practical, intellectual, and cultural reasons motivate the study of electricity and magnetism. The operation of electrical systems designed to perform certain engineering tasks depends, at least in part, on electrical, electromechanical, or electrochemical phenomena. The electrical aspects of these applications are described by Maxwells equations. As a description of the temporal evolution of electromagnetic elds in three-dimensional space, these same equations form a concise summary of a wider range of phenomena than can be found in any other discipline. Maxwells equations are an intellectual achievement that should be familiar to every student of physical phenomena. As part of the theory of elds that includes continuum mechanics, quantum mechanics, heat and mass transfer, and many other disciplines, our subject develops the mathematical language and methods that are the basis for these other areas. For those who have an interest in electromechanical energy conversion, transmission systems at power or radio frequencies, waveguides at microwave or optical frequencies, antennas, or plasmas, there is little need to argue the necessity for becoming expert in dealing with electromagnetic elds. There are others who may require encouragement. For example, circuit designers may be satised with circuit theory, the laws of which are stated in terms of voltages and currents and in terms of the relations imposed upon the voltages and currents by the circuit elements. However, these laws break down at high frequencies, and this cannot be understood without electromagnetic eld theory. The limitations of circuit models come into play as the frequency is raised so high that the propagation time of electromagnetic elds becomes comparable to a period, with the result that inductors behave as capacitors and vice versa. Other limitations are associated with loss phenomena. As the frequency is raised, resistors and transistors are limited by capacitive eects, and transducers and transformers by eddy currents. 1

2

Maxwells Integral Laws in Free Space

Chapter 1

Anyone concerned with developing circuit models for physical systems requires a eld theory background to justify approximations and to derive the values of the circuit parameters. Thus, the bioengineer concerned with electrocardiography or neurophysiology must resort to eld theory in establishing a meaningful connection between the physical reality and models, when these are stated in terms of circuit elements. Similarly, even if a control theorist makes use of a lumped parameter model, its justication hinges on a continuum theory, whether electromagnetic, mechanical, or thermal in nature. Computer hardware may seem to be another application not dependent on electromagnetic eld theory. The software interface through which the computer is often seen makes it seem unrelated to our subject. Although the hardware is generally represented in terms of circuits, the practical realization of a computer designed to carry out logic operations is limited by electromagnetic laws. For example, the signal originating at one point in a computer cannot reach another point within a time less than that required for a signal, propagating at the speed of light, to traverse the interconnecting wires. That circuit models have remained useful as computation speeds have increased is a tribute to the solid state technology that has made it possible to decrease the size of the fundamental circuit elements. Sooner or later, the fundamental limitations imposed by the electromagnetic elds dene the computation speed frontier of computer technology, whether it be caused by electromagnetic wave delays or electrical power dissipation. Overview of Subject. As illustrated diagrammatically in Fig. 1.0.1, we start with Maxwells equations written in integral form. This chapter begins with a denition of the elds in terms of forces and sources followed by a review of each of the integral laws. Interwoven with the development are examples intended to develop the methods for surface and volume integrals used in stating the laws. The examples are also intended to attach at least one physical situation to each of the laws. Our objective in the chapters that follow is to make these laws useful, not only in modeling engineering systems but in dealing with practical systems in a qualitative fashion (as an inventor often does). The integral laws are directly useful for (a) dealing with elds in this qualitative way, (b) nding elds in simple congurations having a great deal of symmetry, and (c) relating elds to their sources. Chapter 2 develops a dierential description from the integral laws. By following the examples and some of the homework associated with each of the sections, a minimum background in the mathematical theorems and operators is developed. The dierential operators and associated integral theorems are brought in as needed. Thus, the divergence and curl operators, along with the theorems of Gauss and Stokes, are developed in Chap. 2, while the gradient operator and integral theorem are naturally derived in Chap. 4. Static elds are often the rst topic in developing an understanding of phenomena predicted by Maxwells equations. Fields are not measurable, let alone of practical interest, unless they are dynamic. As developed here, elds are never truly static. The subject of quasistatics, begun in Chap. 3, is central to the approach we will use to understand the implications of Maxwells equations. A mature understanding of these equations is achieved when one has learned how to neglect complications that are inconsequential. The electroquasistatic (EQS) and magne-

Sec. 1.0

Introduction

3

4

Maxwells Integral Laws in Free Space

Chapter 1

Fig. 1.0.1 Outline of Subject. The three columns, respectively for electroquasistatics, magnetoquasistatics and electrodynamics, show parallels in development.

toquasistatic (MQS) approximations are justied if time rates of change are slow enough (frequencies are low enough) so that time delays due to the propagation of electromagnetic waves are unimportant. The examples considered in Chap. 3 give some notion as to which of the two approximations is appropriate in a given situation. A full appreciation for the quasistatic approximations will come into view as the EQS and MQS developments are drawn together in Chaps. 11 through 15. Although capacitors and inductors are examples in the electroquasistatic and magnetoquasistatic categories, respectively, it is not true that quasistatic systems can be generally modeled by frequency-independent circuit elements. Highfrequency models for transistors are correctly based on the EQS approximation. Electromagnetic wave delays in the transistors are not consequential. Nevertheless, dynamic eects are important and the EQS approximation can contain the nite time for charge migration. Models for eddy current shields or heaters are correctly based on the MQS approximation. Again, the delay time of an electromagnetic wave is unimportant while the all-important diusion time of the magnetic eld

Sec. 1.0

Introduction

5

is represented by the MQS laws. Space charge waves on an electron beam or spin waves in a saturated magnetizable material are often described by EQS and MQS laws, respectively, even though frequencies of interest are in the GHz range. The parallel developments of EQS (Chaps. 47) and MQS systems (Chaps. 8 10) is emphasized by the rst page of Fig. 1.0.1. For each topic in the EQS column to the left there is an analogous one at the same level in the MQS column. Although the eld concepts and mathematical techniques used in dealing with EQS and MQS systems are often similar, a comparative study reveals as many contrasts as direct analogies. There is a two-way interplay between the electric and magnetic studies. Not only are results from the EQS developments applied in the description of MQS systems, but the examination of MQS situations leads to a greater appreciation for the EQS laws. At the tops of the EQS and the MQS columns, the rst page of Fig. 1.0.1, general (contrasting) attributes of the electric and magnetic elds are identied. The developments then lead from situations where the eld sources are prescribed to where they are to be determined. Thus, EQS electric elds are rst found from prescribed distributions of charge, while MQS magnetic elds are determined given the currents. The development of the EQS eld solution is a direct investment in the subsequent MQS derivation. It is then recognized that in many practical situations, these sources are induced in materials and must therefore be found as part of the eld solution. In the rst of these situations, induced sources are on the boundaries of conductors having a suciently high electrical conductivity to be modeled as perfectly conducting. For the EQS systems, these sources are surface charges, while for the MQS, they are surface currents. In either case, elds must satisfy boundary conditions, and the EQS study provides not only mathematical techniques but even partial dierential equations directly applicable to MQS problems. Polarization and magnetization account for eld sources that can be prescribed (electrets and permanent magnets) or induced by the elds themselves. In the Chu formulation used here, there is a complete analogy between the way in which polarization and magnetization are represented. Thus, there is a direct transfer of ideas from Chap. 6 to Chap. 9. The parallel quasistatic studies culminate in Chaps. 7 and 10 in an examination of loss phenomena. Here we learn that very dierent answers must be given to the question When is a conductor perfect? for EQS on one hand, and MQS on the other. In Chap. 11, many of the concepts developed previously are put to work through the consideration of the ow of power, storage of energy, and production of electromagnetic forces. From this chapter on, Maxwells equations are used without approximation. Thus, the EQS and MQS approximations are seen to represent systems in which either the electric or the magnetic energy storage dominates respectively. In Chaps. 12 through 14, the focus is on electromagnetic waves. The development is a natural extension of the approach taken in the EQS and MQS columns. This is emphasized by the outline represented on the right page of Fig. 1.0.1. The topics of Chaps. 12 and 13 parallel those of the EQS and MQS columns on the previous page. Potentials used to represent electrodynamic elds are a natural generalization of those used for the EQS and MQS systems. As for the quasistatic elds, the elds of given sources are considered rst. An immediate practical application is therefore the description of radiation elds of antennas.

6

Maxwells Integral Laws in Free Space

Chapter 1

The boundary value point of view, introduced for EQS systems in Chap. 5 and for MQS systems in Chap. 8, is the basic theme of Chap. 13. Practical examples include simple transmission lines and waveguides. An understanding of transmission line dynamics, the subject of Chap. 14, is necessary in dealing with the conventional ideal lines that model most high-frequency systems. They are also shown to provide useful models for representing quasistatic dynamical processes. To make practical use of Maxwells equations, it is necessary to master the art of making approximations. Based on the electromagnetic properties and dimensions of a system and on the time scales (frequencies) of importance, how can a physical system be broken into electromagnetic subsystems, each described by its dominant physical processes? It is with this goal in mind that the EQS and MQS approximations are introduced in Chap. 3, and to this end that Chap. 15 gives an overview of electromagnetic elds.

1.1 THE LORENTZ LAW IN FREE SPACE There are two points of view for formulating a theory of electrodynamics. The older one views the forces of attraction or repulsion between two charges or currents as the result of action at a distance. Coulombs law of electrostatics and the corresponding law of magnetostatics were rst stated in this fashion. Faraday[1] introduced a new approach in which he envisioned the space between interacting charges to be lled with elds, by which the space is activated in a certain sense; forces between two interacting charges are then transferred, in Faradays view, from volume element to volume element in the space between the interacting bodies until nally they are transferred from one charge to the other. The advantage of Faradays approach was that it brought to bear on the electromagnetic problem the then well-developed theory of continuum mechanics. The culmination of this point of view was Maxwells formulation[2] of the equations named after him. From Faradays point of view, electric and magnetic elds are dened at a point r even when there is no charge present there. The elds are dened in terms of the force that would be exerted on a test charge q if it were introduced at r moving at a velocity v at the time of interest. It is found experimentally that such a force would be composed of two parts, one that is independent of v, and the other proportional to v and orthogonal to it. The force is summarized in terms of the electric eld intensity E and magnetic ux density o H by the Lorentz force law. (For a review of vector operations, see Appendix 1.) f = q(E + v o H) (1)

The superposition of electric and magnetic force contributions to (1) is illustrated in Fig. 1.1.1. Included in the gure is a reminder of the right-hand rule used to determine the direction of the cross-product of v and o H. In general, E and H are not uniform, but rather are functions of position r and time t: E = E(r, t) and o H = o H(r, t). In addition to the units of length, mass, and time associated with mechanics, a unit of charge is required by the theory of electrodynamics. This unit is the

Sec. 1.1

The Lorentz Law in Free Space

7

Fig. 1.1.1 Lorentz force f in geometric relation to the electric and magnetic eld intensities, E and H, and the charge velocity v: (a) electric force, (b) magnetic force, and (c) total force.

coulomb. The Lorentz force law, (1), then serves to dene the units of E and of o H. 2 newton kilogram meter/(second) units of E = = (2) coulomb coulomb units of o H = kilogram newton = coulomb meter/second coulomb second (3)

We can only establish the units of the magnetic ux density o H from the force law and cannot argue until Sec. 1.4 that the derived units of H are ampere/meter and hence of o are henry/meter. In much of electrodynamics, the predominant concern is not with mechanics but with electric and magnetic elds in their own right. Therefore, it is inconvenient to use the unit of mass when checking the units of quantities. It proves useful to introduce a new name for the unit of electric eld intensity the unit of volt/meter. In the summary of variables given in Table 1.8.2 at the end of the chapter, the fundamental units are SI, while the derived units exploit the fact that the unit of mass, kilogram = volt-coulomb-second2 /meter2 and also that a coulomb/second = ampere. Dimensional checking of equations is guaranteed if the basic units are used, but may often be accomplished using the derived units. The latter communicate the physical nature of the variable and the natural symmetry of the electric and magnetic variables.Example 1.1.1. Electron Motion in Vacuum in a Uniform Static Electric Field

In vacuum, the motion of a charged particle is limited only by its own inertia. In the uniform electric eld illustrated in Fig. 1.1.2, there is no magnetic eld, and an electron starts out from the plane x = 0 with an initial velocity vi . The imposed electric eld is E = ix Ex , where ix is the unit vector in the x direction and Ex is a given constant. The trajectory is to be determined here and used to exemplify the charge and current density in Example 1.2.1.

8

Maxwells Integral Laws in Free Space

Chapter 1

Fig. 1.1.2 An electron, subject to the uniform electric eld intensity Ex , has the position x , shown as a function of time for positive and negative elds.

With m dened as the electron mass, Newtons law combines with the Lorentz law to describe the motion. m d2 x = f = eEx dt2 (4)

The electron position x is shown in Fig. 1.1.2. The charge of the electron is customarily denoted by e (e = 1.6 1019 coulomb) where e is positive, thus necessitating an explicit minus sign in (4). By integrating twice, we get x = 1 e Ex t2 + c1 t + c2 2m (5)

where c1 and c2 are integration constants. If we assume that the electron is at x = 0 and has velocity vi when t = ti , it follows that these constants are c1 = v i + e Ex t i ; m c2 = vi ti 1 e Ex t2 i 2m (6)

Thus, the electron position and velocity are given as a function of time by x = 1 e Ex (t ti )2 + vi (t ti ) 2m (7) (8)

dx e = Ex (t ti ) + vi dt m

With x dened as upward and Ex > 0, the motion of an electron in an electric eld is analogous to the free fall of a mass in a gravitational eld, as illustrated by Fig. 1.1.2. With Ex < 0, and the initial velocity also positive, the velocity is a monotonically increasing function of time, as also illustrated by Fig. 1.1.2. Example 1.1.2. Electron Motion in Vacuum in a Uniform Static Magnetic Field

The magnetic contribution to the Lorentz force is perpendicular to both the particle velocity and the imposed eld. We illustrate this fact by considering the trajectory

Sec. 1.1

The Lorentz Law in Free Space

9

Fig. 1.1.3 (a) In a uniform magnetic ux density o Ho and with no initial velocity in the y direction, an electron has a circular orbit. (b) With an initial velocity in the y direction, the orbit is helical.

resulting from an initial velocity viz along the z axis. With a uniform constant magnetic ux density o H existing along the y axis, the force is f = e(v o H) (9)

The cross-product of two vectors is perpendicular to the two vector factors, so the acceleration of the electron, caused by the magnetic eld, is always perpendicular to its velocity. Therefore, a magnetic eld alone cannot change the magnitude of the electron velocity (and hence the kinetic energy of the electron) but can change only the direction of the velocity. Because the magnetic eld is uniform, because the velocity and the rate of change of the velocity lie in a plane perpendicular to the magnetic eld, and, nally, because the magnitude of v does not change, we nd that the acceleration has a constant magnitude and is orthogonal to both the velocity and the magnetic eld. The electron moves in a circle so that the centrifugal force counterbalances the magnetic force. Figure 1.1.3a illustrates the motion. The radius of the circle is determined by equating the centrifugal force and radial Lorentz force eo |v|Ho = which leads to r= mv 2 r (10)

m |v| e o Ho

(11)

The foregoing problem can be modied to account for any arbitrary initial angle between the velocity and the magnetic eld. The vector equation of motion (really three equations in the three unknowns x , y , z ) m d d2 = e o H dt2 dt (12)

is linear in and so solutions can be superimposed to satisfy initial conditions that , include not only a velocity viz but one in the y direction as well, viy . Motion in the same direction as the magnetic eld does not give rise to an additional force. Thus,

10

Maxwells Integral Laws in Free Space

Chapter 1

the y component of (12) is zero on the right. An integration then shows that the y directed velocity remains constant at its initial value, viy . This uniform motion can be added to that already obtained to see that the electron follows a helical path, as shown in Fig. 1.1.3b. It is interesting to note that the angular frequency of rotation of the electron around the eld is independent of the speed of the electron and depends only upon the magnetic ux density, o Ho . Indeed, from (11) we nd e v c = o H o r m (13)

For a ux density of 1 volt-second/meter (or 1 tesla), the cyclotron frequency is fc = c /2 = 28 GHz. (For an electron, e = 1.6021019 coulomb and m = 9.1061031 kg.) With an initial velocity in the z direction of 3 107 m/s, the radius of gyration in the ux density o H = 1 tesla is r = viz /c = 1.7 104 m.

1.2 CHARGE AND CURRENT DENSITIES In Maxwells day, it was not known that charges are not innitely divisible but occur in elementary units of 1.6 1019 coulomb, the charge of an electron. Hence, Maxwells macroscopic theory deals with continuous charge distributions. This is an adequate description for elds of engineering interest that are produced by aggregates of large numbers of elementary charges. These aggregates produce charge distributions that are described conveniently in terms of a charge per unit volume, a charge density . Pick an incremental volume and determine the net charge within. Then (r, t) net charge in V V (1)

is the charge density at the position r when the time is t. The units of are coulomb/meter3 . The volume V is chosen small as compared to the dimensions of the system of interest, but large enough so as to contain many elementary charges. The charge density is treated as a continuous function of position. The graininess of the charge distribution is ignored in such a macroscopic treatment. Fundamentally, current is charge transport and connotes the time rate of change of charge. Current density is a directed current per unit area and hence measured in (coulomb/second)/meter2 . A charge density moving at a velocity v implies a rate of charge transport per unit area, a current density J, given by J = v (2)

One way to envision this relation is shown in Fig. 1.2.1, where a charge density having velocity v traverses a dierential area a. The area element has a unit normal n, so that a dierential area vector can be dened as a = na. The charge that passes during a dierential time t is equal to the total charge contained in the volume v adt. Therefore, d(q) = v adt (3)

Sec. 1.2

Charge and Current Densities

11

Fig. 1.2.1

Current density J passing through surface having a normal n.

Fig. 1.2.2 Charge injected at the lower boundary is accelerated upward by an electric eld. Vertical distributions of (a) eld intensity, (b) velocity and (c) charge density.

Divided by dt, we expect (3) to take the form J a, so it follows that the current density is related to the charge density by (2). The velocity v is the velocity of the charge. Just how the charge is set into motion depends on the physical situation. The charge might be suspended in or on an insulating material which is itself in motion. In that case, the velocity would also be that of the material. More likely, it is the result of applying an electric eld to a conductor, as considered in Chap. 7. For charged particles moving in vacuum, it might result from motions represented by the laws of Newton and Lorentz, as illustrated in the examples in Sec.1.1. This is the case in the following example.Example 1.2.1. Charge and Current Densities in a Vacuum Diode

Consider the charge and current densities for electrons being emitted with initial velocity v from a cathode in the plane x = 0, as shown in Fig. 1.2.2a.1 Electrons are continuously injected. As in Example 1.1.1, where the motions of the individual electrons are considered, the electric eld is assumed to be uniform. In the next section, it is recognized that charge is the source of the electric eld. Here it is assumed that the charge used to impose the uniform eld is much greater than the space charge associated with the electrons. This is justied in the limit of a low electron current. Any one of the electrons has a position and velocity given by (1.1.7) and (1.1.8). If each is injected with the same initial velocity, the charge and current densities in any given plane x = constant would be expected to be independent of time. Moreover, the current passing any x-plane should be the same as that passing any other such plane. That is, in the steady state, the current density is independent1 Here we picture the eld variables E , v , and as though they were positive. For electrons, x x < 0, and to make vx > 0, we must have Ex < 0.

12

Maxwells Integral Laws in Free Spaceof not only time but x as well. Thus, it is possible to write (x)vx (x) = Jo

Chapter 1

(4)

where Jo is a given current density. The following steps illustrate how this condition of current continuity makes it possible to shift from a description of the particle motions described with time as the independent variable to one in which coordinates (x, y, z) (or for short r) are the independent coordinates. The relation between time and position for the electron described by (1.1.7) takes the form of a quadratic in (t ti ) 1 e Ex (t ti )2 vi (t ti ) + x = 0 2m (5)

This can be solved to give the elapsed time for a particle to reach the position x . Note that of the two possible solutions to (5), the one selected satises the condition that when t = ti , x = 0. t ti = vi e 2 v i 2 m E x x e E m x

(6)

With the benet of this expression, the velocity given by (1.1.8) is written as dx = dt2 vi

2e E x x m

(7)

Now we make a shift in viewpoint. On the left in (7) is the velocity vx of the particle that is at the location x = x. Substitution of variables then gives vx =2 vi 2

e Ex x m

(8)

so that x becomes the independent variable used to express the dependent variable vx . It follows from this expression and (4) that the charge density = Jo = vx Jo2 vi

2e E x m x

(9)

is also expressed as a function of x. In the plots shown in Fig. 1.2.2, it is assumed that Ex < 0, so that the electrons have velocities that increase monotonically with x. As should be expected, the charge density decreases with x because as they speed up, the electrons thin out to keep the current density constant.

1.3 GAUSS INTEGRAL LAW OF ELECTRIC FIELD INTENSITY The Lorentz force law of Sec. 1.1 expresses the eect of electromagnetic elds on a moving charge. The remaining sections in this chapter are concerned with the reaction of the moving charges upon the electromagnetic elds. The rst of

Sec. 1.3

Gauss Integral Law

13

Fig. 1.3.1

General surface S enclosing volume V .

Maxwells equations to be considered, Gauss law, describes how the electric eld intensity is related to its source. The net charge within an arbitrary volume V that is enclosed by a surface S is related to the net electric ux through that surface byoE S

da =V

dv

(1)

With the surface normal dened as directed outward, the volume is shown in Fig. 1.3.1. Here the permittivity of free space, o = 8.854 1012 farad/meter, is an empirical constant needed to express Maxwells equations in SI units. On the right in (1) is the net charge enclosed by the surface S. On the left is the summation over this same closed surface of the dierential contributions of ux o E da. The quantity o E is called the electric displacement ux density and, [from (1)], has the units of coulomb/meter2 . Out of any region containing net charge, there must be a net displacement ux. The following example illustrates the mechanics of carrying out the volume and surface integrations.Example 1.3.1. Electric Field Due to Spherically Symmetric Charge Distribution

Given the charge and current distributions, the integral laws fully determine the electric and magnetic elds. However, they are not directly useful unless there is a great deal of symmetry. An example is the distribution of charge density (r) =r o R ; 0;

rR

(2)

in the spherical coordinate system of Fig. 1.3.2. Here o and R are given constants. An argument based on the spherical symmetry shows that the only possible component of E is radial. E = ir Er (r) (3)

Indeed, suppose that in addition to this r component the eld possesses a component. At a given point, the components of E then appear as shown in Fig. 1.3.2b. Rotation of the system about the axis shown results in a component of E in some new direction perpendicular to r. However, the rotation leaves the source of that eld, the charge distribution, unaltered. It follows that E must be zero. A similar argument shows that E also is zero.

14

Maxwells Integral Laws in Free Space

Chapter 1

Fig. 1.3.2 (a) Spherically symmetric charge distribution, showing radial dependence of charge density and associated radial electric eld intensity. (b) Axis of rotation for demonstration that the components of E transverse to the radial coordinate are zero.

The incremental volume element is dv = (dr)(rd)(r sin d) and it follows that for a spherical volume having arbitrary radius r,r 2

(4)

dv =V

0 0 0 R 2 0 0 0

o r (r sin d)(r d)dr = R o r (r sin d)(r d)dr = R

o 4 r ; R o R3 ;

r a on the z axis.

56

Electroquasistatic Fields: The Superposition Integral Point of View

Chapter 4

Fig. P4.5.9

4.5.7

A strip of charge lying in the xz plane between x = b and x = b extends to in the z direction. On this strip the surface charge density is s = o (d b) (d x) (a)

where d > b. Show that at the location (x, y) = (d, 0), the potential is (d, 0) = o (d b){[ln(d b)]2 [ln(d + b)]2 } 4 o (b)

4.5.8

A pair of charge strips lying in the xz plane and running from z = + to z = are each of width 2d with their left and right edges, respectively, located on the z axis. The one between the z axis and (x, y) = (2d, 0) has a uniform surface charge density o , while the one between (x, y) = (2d, 0) and the z axis has s = o . (Note that the symmetry makes the plane x = 0 one of zero potential.) What must be the value of o if the potential at the center of the right strip, where (x, y) = (d, 0), is to be V ?

4.5.9 Distributions of line charge can be approximated by piecing together uniformly charged segments. Especially if a computer is to be used to carry out the integration by summing over the elds due to the linear elements of line charge, this provides a convenient basis for calculating the electric potential for a given line distribution of charge. In the following, you determine the potential at an arbitrary observer coordinate r due to a line charge that is uniformly distributed between the points r + b and r + c, as shown in Fig. P4.5.9a. The segment over which this charge (of line charge density l ) is distributed is denoted by the vector a, as shown in the gure. Viewed in the plane in which the position vectors a, b, and c lie, a coordinate denoting the position along the line charge is as shown in Fig. P4.5.9b. The origin of this coordinate is at the position on the line segment collinear with a that is nearest to the observer position r.

Sec. 4.5

Problems

57

(a) Argue that in terms of , the base and tip of the a vector are as designated in Fig. P4.5.9b along the axis. (b) Show that the superposition integral for the potential due to the segment of line charge at r isba/|a|

=ca/|a|

l d 4 o |r r | |b a|2 |a|2

(a)

where |r r | = 2 + (b)

(c) Finally, show that the potential isba |a| ca |a|

ln = 4 o

+ +

ba 2 |a| ca 2 |a|

+ +

|ba|2 |a|2

(c)|ba|2 |a|2

(d) A straight segment of line charge has the uniform density o between the points (x, y, z) = (0, 0, d) and (x, y, z) = (d, d, d). Using (c), show that the potential (x, y, z) is = 2d x y + 2[(d x)2 + (d y)2 + (d z)2 ] o ln 4 o x y + 2[x2 + y 2 + (d z)2 ] (d)

4.5.10 Given the charge distribution, (r), the potential follows from (3). This expression has the disadvantage that to nd E, derivatives of must be taken. Thus, it is not enough to know at one location if E is to be determined. Start with (3) and show that a superposition integral for the electric eld intensity is E= 1 4 o (r )ir r dv |r r |2 (a)

V

where ir r is a unit vector directed from the source coordinate r to the observer coordinate r. (Hint: Remember that when the gradient of is taken to obtain E, the derivatives are with respect to the observer coordinates with the source coordinates held xed.) A similar derivation is given in Sec. 8.2, where an expression for the magnetic eld intensity H is obtained from a superposition integral for the vector potential A. 4.5.11 For a better understanding of the concepts underlying the derivation of the superposition integral for Poissons equation, consider a hypothetical situation where a somewhat dierent equation is to be solved. The charge

58

Electroquasistatic Fields: The Superposition Integral Point of View

Chapter 4

density is assumed in part to be a predetermined density s(x, y, z), and in part to be induced at a given point (x, y, z) in proportion to the potential itself at that same point. That is, =so 2

(a)

(a) Show that the expression to be satised by is then not Poissons equation but rather2

2 =

so

(b)

where s(x, y, z) now plays the role of . (b) The rst step in the derivation of the superposition integral is to nd the response to a point source at the origin, dened such thatR R0

lim

s4r2 dr = Q

(c)

0

Because the situation is then spherically symmetric, the desired response to this point source must be a function of r only. Thus, for this response, (b) becomes 1 2 s r 2 = 2 r r r o Show that for r = 0, a solution is =A er r (e) (d)

and use (c) to show that A = Q/4 o . (c) What is the superposition integral for ? 4.5.12 Because there is a jump in potential across a dipole layer, given by (31), there is an innite electric eld within the layer. (a) With n dened as the unit normal to the interface, argue that this internal electric eld is Eint = o s n (a)

(b) In deriving the continuity condition on E, (1.6.12), using (4.1.1), it was assumed that E was nite everywhere, even within the interface. With a dipole layer, this assumption cannot be made. For example, suppose that a nonuniform dipole layer s (x) is in the plane y = 0. Show that there is a jump in tangential electric eld, Ex , given bya b Ex Ex = o

s x

(b)

Sec. 4.6

Problems

59

Fig. P4.6.1

4.6 Electroquasistatic Fields in the Presence of Perfect Conductors 4.6.1 A charge distribution is represented by a line charge between z = c and z = b along the z axis, as shown in Fig. P4.6.1a. Between these points, the line charge density is given by l = o (a z) (a c) (a)

and so it has the distribution shown in Fig. P4.6.1b. It varies linearly from the value o where z = c to o (a b)/(a c) where z = b. The only other charges in the system are at innity, where the potential is dened as being zero. An equipotential surface for this charge distribution passes through the point z = a on the z axis. [This is the same a as appears in (a).] If this equipotential surface is replaced by a perfectly conducting electrode, show that the capacitance of the electrode relative to innity is C = 2 o (2a c b) 4.6.2 (b)

Charges at innity are used to impose a uniform eld E = Eo iz on a region of free space. In addition to the charges that produce this eld, there are positive and negative charges, of magnitude q, at z = +d/2 and z = d/2, respectively, as shown in Fig. P4.6.2. Spherical coordinates (r, , ) are dened in the gure. (a) The potential, radial coordinate and charge are normalized such that = ; Eo d r= r ; d q= q 4 o Eo d2 (a)

Show that the normalized electric potential can be written as = r cos + q r2 + 1 r cos 41/2

r2 +

1 + r cos 4

1/2

}

(b)

60

Electroquasistatic Fields: The Superposition Integral Point of View

Chapter 4

Fig. P4.6.2

(b) There is an equipotential surface = 0 that encloses these two charges. Thus, if a perfectly conducting object having a surface taking the shape of this = 0 surface is placed in the initially uniform electric eld, the result of part (a) is a solution to the boundary value problem representing the potential, and hence electric eld, around the object. The following establishes the shape of the object. Use (b) to nd an implicit expression for the radius r at which the surface intersects the z axis. Use a graphical solution to show that there will always be such an intersection with r > d/2. For q = 2, nd this radius to two-place accuracy. (c) Make a plot of the surface = 0 in a = constant plane. One way to do this is to use a programmable calculator to evaluate given r and . It is then straightforward to pick a and iterate on r to nd the location of the surface of zero potential. Make q = 2. (d) We expect E to be largest at the poles of the object. Thus, it is in these regions that we expect electrical breakdown to rst occur. In terms of E o and with q = 2, what is the electric eld at the north pole of the object? (e) In terms of Eo and d, what is the total charge on the northern half of the object. [Hint: A numerical calculation is not required.] 4.6.3 For the disk of charge shown in Fig. 4.5.3, there is an equipotential surface that passes through the point z = d on the z axis and encloses the disk. Show that if this surface is replaced by a perfectly conducting electrode, the capacitance of this electrode relative to innity is 2R2 o C= ( R2 + d2 d) (a)

4.6.4

The purpose of this problem is to get an estimate of the capacitance of, and the elds surrounding, the two conducting spheres of radius R shown in Fig. P4.6.4, with the centers separated by a distance h. We construct

Sec. 4.6

Problems

61

Fig. P4.6.4

an approximate eld solution for the eld produced by charges Q on the two spheres, as follows: (a) First we place the charges at the centers of the spheres. If R h, the two equipotentials surrounding the charges at r1 R and r2 R are almost spherical. If we assume that they are spherical, what is the potential dierence between the two spherical conductors? Where does the maximum eld occur and how big is it? (b) We can obtain a better solution by noting that a spherical equipotential coincident with the top sphere is produced by a set of three charges. These are the charge Q at z = h/2 and the two charges inside the top sphere properly positioned according to (33) of appropriate magnitude and total charge +Q. Next, we replace the charge Q by two charges, just like we did for the charge +Q. The net eld is now due to four charges. Find the potential dierence and capacitance for the new eld conguration and compare with the previous result. Do you notice that you have obtained higher-order terms in R/h? You are in the process of obtaining a rapidly convergent series in powers of R/h. 4.6.5 This is a continuation of Prob. 4.5.4. The line distribution of charge given there is the only charge in the region 0 x. However, the y z plane is now a perfectly conducting surface, so that the electric eld is normal to the plane x = 0. (a) Determine the potential in the half-space 0 x. (b) For the potential found in part (a), what is the equation for the equipotential surface passing through the point (x, y, z) = (a/2, 0, 0)? (c) For the remainder of this problem, assume that d = 4a. Make a sketch of this equipotential surface as it intersects the plane z = 0. In doing this, it is convenient to normalize x and y to a by dening = x/a and

62

Electroquasistatic Fields: The Superposition Integral Point of View

Chapter 4

= y/a. A good way to make the plot is then to compute the potential using a programmable calculator. By iteration, you can quickly zero in on points of the desired potential. It is sucient to show that in addition to the point of part (a), your curve passes through three well-dened points that suggest its being a closed surface. (d) Suppose that this closed surface having potential V is actually a metallic (perfect) conductor. Sketch the lines of electric eld intensity in the region between the electrode and the ground plane. (e) The capacitance of the electrode relative to the ground plane is dened as C = q/V , where q is the total charge on the surface of the electrode having potential V . For the electrode of part (c), what is C? 4.7 Method of Images 4.7.1 A point charge Q is located on the z axis a distance d above a perfect conductor in the plane z = 0. (a) Show that above the plane is = Q 4 o 1 + (z d)2 ]1/2

[x2

+

y2

1 2 2 + (z + d)2 ]1/2 [x + y

(a)

(b) Show that the equation for the equipotential surface = V passing through the point z = a < d is [x2 + y 2 + (z d)2 ]1/2 [x2 + y 2 + (z + d)2 ]1/2 2a = 2 d a2

(b)

(c) Use intuitive arguments to show that this surface encloses the point charge. In terms of a, d, and o , show that the capacitance relative to the ground plane of an electrode having the shape of this surface is C= 2 o (d2 a2 ) a (c)

4.7.2

A positive uniform line charge is along the z axis at the center of a perfectly conducting cylinder of square cross-section in the x y plane. (a) Give the location and sign of the image line charges. (b) Sketch the equipotentials and E lines in the x y plane.

Sec. 4.7

Problems

63

Fig. P4.7.3

4.7.3

When a bird perches on a dc high-voltage power line and then ies away, it does so carrying a net charge. (a) Why? (b) For the purpose of measuring this net charge Q carried by the bird, we have the apparatus pictured in Fig. P4.7.3. Flush with the ground, a strip electrode having width w and length l is mounted so that it is insulated from ground. The resistance, R, connecting the electrode to ground is small enough so that the potential of the electrode (like that of the surrounding ground) can be approximated as zero. The bird ies in the x direction at a height h above the ground with a velocity U . Thus, its position is taken as y = h and x = U t. (c) Given that the bird has own at an altitude sucient to make it appear as a point charge, what is the potential distribution? (d) Determine the surface charge density on the ground plane at y = 0. (e) At a given instant, what is the net charge, q, on the electrode? (Assume that the width w is small compared to h so that in an integration over the electrode surface, the integration in the z direction is simply a multiplication by w.) (f) Sketch the time dependence of the electrode charge. (g) The current through the resistor is dq/dt. Find an expression for the voltage, v, that would be measured across the resistance, R, and sketch its time dependence.

4.7.4 Uniform line charge densities +l and l run parallel to the z axis at x = a, y = 0 and x = b, y = 0, respectively. There are no other charges in the half-space 0 < x. The y z plane where x = 0 is composed of nely segmented electrodes. By connecting a voltage source to each segment, the potential in the x = 0 plane can be made whatever we want. Show that the potential distribution you would impose on these electrodes to insure that there is no normal component of E in the x = 0 plane, Ex (0, y, z), is (0, y, z) = l (a2 + y 2 ) ln 2 2 o (b + y 2 ) (a)

64

Electroquasistatic Fields: The Superposition Integral Point of View

Chapter 4

Fig. P4.7.5

4.7.5

The two-dimensional system shown in cross-section in Fig. P4.7.5 consists of a uniform line charge at x = d, y = d that extends to innity in the z directions. The charge per unit length in the z direction is the constant . Metal electrodes extend to innity in the x = 0 and y = 0 planes. These electrodes are grounded so that the potential in these planes is zero. (a) Determine the electric potential in the region x > 0, y > 0. (b) An equipotential surface passes through the line x = a, y = a(a < d). This surface is replaced by a metal electrode having the same shape. In terms of the given constants a, d, and o , what is the capacitance per unit length in the z direction of this electrode relative to the ground planes?

4.7.6 The disk of charge shown in Fig. 4.5.3 is located at z = s rather than z = 0. The plane z = 0 consists of a perfectly conducting ground plane. (a) Show that for 0 < z, the electric potential along the z axis is given by o = R2 + (z s)2 |z s| 2 o (a) R2 + (z + s)2 |z + s| (b) Show that the capacitance relative to the ground plane of an electrode having the shape of the equipotential surface passing through the point z = d < s on the z axis and enclosing the disk of charge is C= 2R2 R2 + (d s)2 o

R2 + (d + s)2 + 2d

(b)

4.7.7

The disk of charge shown in Fig. P4.7.7 has radius R and height h above a perfectly conducting plane. It has a surface charge density s = o r/R. A perfectly conducting electrode has the shape of an equipotential surface

Sec. 4.8

Problems

65

Fig. P4.7.7

that passes through the point z = a < h on the z axis and encloses the disk. What is the capacitance of this electrode relative to the plane z = 0? 4.7.8 A straight segment of line charge has the uniform density o between the points (x, y, z) = (0, 0, d) and (x, y, z) = (d, d, d). There is a perfectly conducting material in the plane z = 0. Determine the potential for z 0. [See part (d) of Prob. 4.5.9.]

4.8 Charge Simulation Approach to Boundary Value Problems 4.8.1 For the six-segment approximation to the elds of the parallel plate capacitor in Example 4.8.1, determine the respective strip charge densities in terms of the voltage V and dimensions of the system. What is the approximate capacitance?

5 ELECTROQUASISTATIC FIELDS FROM THE BOUNDARY VALUE POINT OF VIEW5.0 INTRODUCTION The electroquasistatic laws were discussed in Chap. 4. The electric eld intensity E is irrotational and represented by the negative gradient of the electric potential. E= (1) Gauss law is then satised if the electric potential is related to the charge density by Poissons equation 2 = (2)o

In charge-free regions of space, obeys Laplaces equation, (2), with = 0. The last part of Chap. 4 was devoted to an opportunistic approach to nding boundary value solutions. An exception was the numerical scheme described in Sec. 4.8 that led to the solution of a boundary value problem using the sourcesuperposition approach. In this chapter, a more direct attack is made on solving boundary value problems without necessarily resorting to numerical methods. It is one that will be used extensively not only as eects of polarization and conduction are added to the EQS laws, but in dealing with MQS systems as well. Once again, there is an analogy useful for those familiar with the description of linear circuit dynamics in terms of ordinary dierential equations. With time as the independent variable, the response to a drive that is turned on when t = 0 can be determined in two ways. The rst represents the response as a superposition of impulse responses. The resulting convolution integral represents the response for all time, before and after t = 0 and even when t = 0. This is the analogue of the point of view taken in the rst part of Chap. 4. The second approach represents the history of the dynamics prior to when t = 0 in terms of initial conditions. With the understanding that interest is conned to times subsequent to t = 0, the response is then divided into particular 1

2

Electroquasistatic Fields from the Boundary Value Point of View

Chapter 5

and homogeneous parts. The particular solution to the dierential equation representing the circuit is not unique, but insures that at each instant in the temporal range of interest, the dierential equation is satised. This particular solution need not satisfy the initial conditions. In this chapter, the drive is the charge density, and the particular potential response guarantees that Poissons equation, (2), is satised everywhere in the spatial region of interest. In the circuit analogue, the homogeneous solution is used to satisfy the initial conditions. In the eld problem, the homogeneous solution is used to satisfy boundary conditions. In a circuit, the homogeneous solution can be thought of as the response to drives that occurred prior to when t = 0 (outside the temporal range of interest). In the determination of the potential distribution, the homogeneous response is one predicted by Laplaces equation, (2), with = 0, and can be regarded either as caused by ctitious charges residing outside the region of interest or as caused by the surface charges induced on the boundaries. The development of these ideas in Secs. 5.15.3 is self-contained and does not depend on a familiarity with circuit theory. However, for those familiar with the solution of ordinary dierential equations, it is satisfying to see that the approaches used here for dealing with partial dierential equations are a natural extension of those used for ordinary dierential equations. Although it can often be found more simply by other methods, a particular solution always follows from the superposition integral. The main thrust of this chapter is therefore toward a determination of homogeneous solutions, of nding solutions to Laplaces equation. Many practical congurations have boundaries that are described by setting one of the coordinate variables in a three-dimensional coordinate system equal to a constant. For example, a box having rectangular crosssections has walls described by setting one Cartesian coordinate equal to a constant to describe the boundary. Similarly, the boundaries of a circular cylinder are naturally described in cylindrical coordinates. So it is that there is great interest in having solutions to Laplaces equation that naturally t these congurations. With many examples interwoven into the discussion, much of this chapter is devoted to cataloging these solutions. The results are used in this chapter for describing EQS elds in free space. However, as eects of polarization and conduction are added to the EQS purview, and as MQS systems with magnetization and conduction are considered, the homogeneous solutions to Laplaces equation established in this chapter will be a continual resource. A review of Chap. 4 will identify many solutions to Laplaces equation. As long as the eld source is outside the region of interest, the resulting potential obeys Laplaces equation. What is dierent about the solutions established in this chapter? A hint comes from the numerical procedure used in Sec. 4.8 to satisfy arbitrary boundary conditions. There, a superposition of N solutions to Laplaces equation was used to satisfy conditions at N points on the boundaries. Unfortunately, to determine the amplitudes of these N solutions, N equations had to be solved for N unknowns. The solutions to Laplaces equation found in this chapter can also be used as the terms in an innite series that is made to satisfy arbitrary boundary conditions. But what is dierent about the terms in this series is their orthogonality. This property of the solutions makes it possible to explicitly determine the individual amplitudes in the series. The notion of the orthogonality of functions may already

Sec. 5.1

Particular and Homogeneous Solutions

3

Fig. 5.1.1 Volume of interest in which there can be a distribution of charge density. To illustrate bounding surfaces on which potential is constrained, n isolated surfaces and one enclosing surface are shown.

be familiar through an exposure to Fourier analysis. In any case, the fundamental ideas involved are introduced in Sec. 5.5.

5.1 PARTICULAR AND HOMOGENEOUS SOLUTIONS TO POISSONS AND LAPLACES EQUATIONS Suppose we want to analyze an electroquasistatic situation as shown in Fig. 5.1.1. A charge distribution (r) is specied in the part of space of interest, designated by the volume V . This region is bounded by perfect conductors of specied shape and location. Known potentials are applied to these conductors and the enclosing surface, which may be at innity. In the space between the conductors, the potential function obeys Poissons equation, (5.0.2). A particular solution of this equation within the prescribed volume V is given by the superposition integral, (4.5.3). p (r) =V

(r )dv 4 o |r r |

(1)

This potential obeys Poissons equation at each point within the volume V . Since we do not evaluate this equation outside the volume V , the integration over the sources called for in (1) need include no sources other than those within the volume V . This makes it clear that the particular solution is not unique, because the addition to the potential made by integrating over arbitrary charges outside the volume V will only give rise to a potential, the Laplacian derivative of which is zero within the volume V . Is (1) the complete solution? Because it is not unique, the answer must be, surely not. Further, it is clear that no information as to the position and shape of the conductors is built into this solution. Hence, the electric eld obtained as the negative gradient of the potential p of (1) will, in general, possess a nite tangential component on the surfaces of the electrodes. On the other hand, the conductors

4

Electroquasistatic Fields from the Boundary Value Point of View

Chapter 5

have surface charge distributions which adjust themselves so as to cause the net electric eld on the surfaces of the conductors to have vanishing tangential electric eld components. The distribution of these surface charges is not known at the outset and hence cannot be included in the integral (1). A way out of this dilemma is as follows: The potential distribution we seek within the space not occupied by the conductors is the result of two charge distributions. First is the prescribed volume charge distribution leading to the potential function p , and second is the charge distributed on the conductor surfaces. The potential function produced by the surface charges must obey the source-free Poissons equation in the space V of interest. Let us denote this solution to the homogeneous form of Poissons equation by the potential function h . Then, in the volume V, h must satisfy Laplaces equation.2

h = 0

(2)

The superposition principle then makes it possible to write the total potential as = p + h (3)

The problem of nding the complete eld distribution now reduces to that of nding a solution such that the net potential of (3) has the prescribed potentials vi on the surfaces Si . Now p is known and can be evaluated on the surface Si . Evaluation of (3) on Si gives vi = p (Si ) + h (Si ) so that the homogeneous solution is prescribed on the boundaries Si . h (Si ) = vi p (Si ) (5) (4)

Hence, the determination of an electroquasistatic eld with prescribed potentials on the boundaries is reduced to nding the solution to Laplaces equation, (2), that satises the boundary condition given by (5). The approach which has been formalized in this section is another point of view applicable to the boundary value problems in the last part of Chap. 4. Certainly, the abstract view of the boundary value situation provided by Fig. 5.1.1 is not dierent from that of Fig. 4.6.1. In Example 4.6.4, the eld shown in Fig. 4.6.8 is determined for a point charge adjacent to an equipotential charge-neutral spherical electrode. In the volume V of interest outside the electrode, the volume charge distribution is singular, the point charge q. The potential given by (4.6.35), in fact, takes the form of (3). The particular solution can be taken as the rst term, the potential of a point charge. The second and third terms, which are equivalent to the potentials caused by the ctitious charges within the sphere, can be taken as the homogeneous solution. Superposition to Satisfy Boundary Conditions. In the following sections, superposition will often be used in another way to satisfy boundary conditions.

Sec. 5.2

Uniqueness of Solutions

5

Suppose that there is no charge density in the volume V , and again the potentials on each of the n surfaces Sj are vj . Then2

=0

(6) (7)

= vj

on Sj , j = 1, . . . n

The solution is broken into a superposition of solutions j that meet the required condition on the j-th surface but are zero on all of the others.n

=j=1

j

(8)

j

vj 0

on Sj on S1 . . . Sj1 , Sj+1 . . . Sn

(9)

Each term is a solution to Laplaces equation, (6), so the sum is as well.2

j = 0

(10)

In Sec. 5.5, a method is developed for satisfying arbitrary boundary conditions on one of four surfaces enclosing a volume of interest. Capacitance Matrix. Suppose that in the n electrode system the net charge on the i-th electrode is to be found. In view of (8), the integral of E da over the surface Si enclosing this electrode then givesn

qi = Si

o

da = Si

o j=1

j da

(11)

Because of the linearity of Laplaces equation, the potential j is proportional to the voltage exciting that potential, vj . It follows that (11) can be written in terms of capacitance parameters that are independent of the excitations. That is, (11) becomesn

qi =j=1

Cij vj

(12)

where the capacitance coecients are Cij = Si o

j da

vj

(13)

The charge on the i-th electrode is a linear superposition of the contributions of all n voltages. The coecient multiplying its own voltage, Cii , is called the selfcapacitance, while the others, Cij , i = j, are the mutual capacitances.

6

Electroquasistatic Fields from the Boundary Value Point of View

Chapter 5

Fig. 5.2.1 Field line originating on one part of bounding surface and terminating on another after passing through the point ro .

5.2 UNIQUENESS OF SOLUTIONS TO POISSONS EQUATION We shall show in this section that a potential distribution obeying Poissons equation is completely specied within a volume V if the potential is specied over the surfaces bounding that volume. Such a uniqueness theorem is useful for two reasons: (a) It tells us that if we have found such a solution to Poissons equation, whether by mathematical analysis or physical insight, then we have found the only solution; and (b) it tells us what boundary conditions are appropriate to uniquely specify a solution. If there is no charge present in the volume of interest, then the theorem states the uniqueness of solutions to Laplaces equation. Following the method reductio ad absurdum, we assume that the solution is not unique that two solutions, a and b , exist, satisfying the same boundary conditions and then show that this is impossible. The presumably dierent solutions a and b must satisfy Poissons equation with the same charge distribution and must satisfy the same boundary conditions. 2 a = ; a = i on Si (1)o 2

b =

o

;

b = i

on

Si

(2)

It follows that with d dened as the dierence in the two potentials, d = a b ,2

d

( d ) = 0;

d = 0 on

Si

(3)

A simple argument now shows that the only way d can both satisfy Laplaces equation and be zero on all of the bounding surfaces is for it to be zero. First, it is argued that d cannot possess a maximum or minimum at any point within V . With the help of Fig. 5.2.1, visualize the negative of the gradient of d , a eld line, as it passes through some point ro . Because the eld is solenoidal (divergence free), such a eld line cannot start or stop within V (Sec. 2.7). Further, the eld denes a potential (4.1.4). Hence, as one proceeds along the eld line in the direction of the negative gradient, the potential has to decrease until the eld line reaches one of the surfaces Si bounding V . Similarly, in the opposite direction, the potential has to increase until another one of the surfaces is reached. Accordingly, all maximum and minimum values of d (r) have to be located on the surfaces.

Sec. 5.3

Continuity Conditions

7

The dierence potential at any interior point cannot assume a value larger than or smaller than the largest or smallest value of the potential on the surfaces. But the surfaces are themselves at zero potential. It follows that the dierence potential is zero everywhere in V and that a = b . Therefore, only one solution exists to the boundary value problem stated with (1).

5.3 CONTINUITY CONDITIONS At the surfaces of metal conductors, charge densities accumulate that are only a few atomic distances thick. In describing their elds, the details of the distribution within this thin layer are often not of interest. Thus, the charge is represented by a surface charge density (1.3.11) and the surface supporting the charge treated as a surface of discontinuity. In such cases, it is often convenient to divide a volume in which the eld is to be determined into regions separated by the surfaces of discontinuity, and to use piece-wise continuous functions to represent the elds. Continuity conditions are then needed to connect eld solutions in two regions separated by the discontinuity. These conditions are implied by the dierential equations that apply throughout the region. They assure that the elds are consistent with the basic laws, even in passing through the discontinuity. Each of the four Maxwells equations implies a continuity condition. Because of the singular nature of the source distribution, these laws are used in integral form to relate the elds to either side of the surface of discontinuity. With the vector n dened as the unit normal to the surface of discontinuity and pointing from region (b) to region (a), the continuity conditions were summarized in Table 1.8.3. In the EQS approximation, the laws of primary interest are Faradays law without the magnetic induction and Gauss law, the rst two equations of Chap. 4. Thus, the corresponding EQS continuity conditions are n [Ea Eb ] = 0 n ( oE a oE b

(1) (2)

) = s

Because the magnetic induction makes no contribution to Faradays continuity condition in any case, these conditions are the same as for the general electrodynamic laws. As a reminder, the contour enclosing the integration surface over which Faradays law was integrated (Sec. 1.6) to obtain (1) is shown in Fig. 5.3.1a. The integration volume used to obtain (2) from Gauss law (Sec. 1.3) is similarly shown in Fig. 5.3.1b. What are the continuity conditions on the electric potential? The potential is continuous across a surface of discontinuity even if that surface carries a surface charge density. This will be the case when the E eld is nite (a dipole layer containing an innite eld causes a jump of potential), because then the line integral of the electric eld from one side of the surface to the other side is zero, the pathlength being innitely small. a b = 0 (3) To determine the jump condition representing Gauss law through the surface of discontinuity, it was integrated (Sec. 1.3) over the volume shown intersecting the

8

Electroquasistatic Fields from the Boundary Value Point of View

Chapter 5

Fig. 5.3.1 (a) Dierential contour intersecting surface supporting surface charge density. (b) Dierential volume enclosing surface charge on surface having normal n.

surface in Fig. 5.3.1b. The resulting continuity condition, (2), is written in terms of the potential by recognizing that in the EQS approximation, E = . n [( )a ( )b ] = so

(4)

At a surface of discontinuity that carries a surface charge density, the normal derivative of the potential is discontinuous. The continuity conditions become boundary conditions if they are made to represent physical constraints that go beyond those already implied by the laws that prevail in the volume. A familiar example is one where the surface is that of an electrode constrained in its potential. Then the continuity condition (3) requires that the potential in the volume adjacent to the electrode be the given potential of the electrode. This statement cannot be justied without invoking information about the physical nature of the electrode (that it is innitely conducting, for example) that is not represented in the volume laws and hence is not intrinsic to the continuity conditions.

5.4 SOLUTIONS TO LAPLACES EQUATION IN CARTESIAN COORDINATES Having investigated some general properties of solutions to Poissons equation, it is now appropriate to study specic methods of solution to Laplaces equation subject to boundary conditions. Exemplied by this and the next section are three standard steps often used in representing EQS elds. First, Laplaces equation is set up in the coordinate system in which the boundary surfaces are coordinate surfaces. Then, the partial dierential equation is reduced to a set of ordinary dierential equations by separation of variables. In this way, an innite set of solutions is generated. Finally, the boundary conditions are satised by superimposing the solutions found by separation of variables. In this section, solutions are derived that are natural if boundary conditions are stated along coordinate surfaces of a Cartesian coordinate system. It is assumed that the elds depend on only two coordinates, x and y, so that Laplaces equation

Sec. 5.4 is (Table I)

Solutions to Laplaces Equation 2 2 + =0 x2 y 2

9

(1)

This is a partial dierential equation in two independent variables. One timehonored method of mathematics is to reduce a new problem to a problem previously solved. Here the process of nding solutions to the partial dierential equation is reduced to one of nding solutions to ordinary dierential equations. This is accomplished by the method of separation of variables. It consists of assuming solutions with the special space dependence (x, y) = X(x)Y (y) (2)

In (2), X is assumed to be a function of x alone and Y is a function of y alone. If need be, a general space dependence is then recovered by superposition of these special solutions. Substitution of (2) into (1) and division by then gives 1 d2 Y (y) 1 d2 X(x) = X(x) dx2 Y (y) dy 2 (3)

Total derivative symbols are used because the respective functions X and Y are by denition only functions of x and y. In (3) we now have on the left-hand side a function of x alone, on the righthand side a function of y alone. The equation can be satised independent of x and y only if each of these expressions is constant. We denote this separation constant by k 2 , and it follows that d2 X = k 2 X (4) dx2 and d2 Y = k2 Y (5) dy 2 These equations have the solutions X cos kx Y cosh ky If k = 0, the solutions degenerate into X constant Y constant or or x y (8) (9) or or sin kx sinh ky (6) (7)

The product solutions, (2), are summarized in the rst four rows of Table 5.4.1. Those in the right-hand column are simply those of the middle column with the roles of x and y interchanged. Generally, we will leave the prime o the k in writing these solutions. Exponentials are also solutions to (7). These, sometimes more convenient, solutions are summarized in the last four rows of the table.

10

Electroquasistatic Fields from the Boundary Value Point of View

Chapter 5

The solutions summarized in this table can be used to gain insight into the nature of EQS elds. A good investment is therefore made if they are now visualized. The elds represented by the potentials in the left-hand column of Table 5.4.1 are all familiar. Those that are linear in x and y represent uniform elds, in the x and y directions, respectively. The potential xy is familiar from Fig. 4.1.3. We will use similar conventions to represent the potentials of the second column, but it is helpful to have in mind the three-dimensional portrayal exemplied for the potential xy in Fig. 4.1.4. In the more complicated eld maps to follow, the sketch is visualized as a contour map of the potential with peaks of positive potential and valleys of negative potential. On the top and left peripheries of Fig. 5.4.1 are sketched the functions cos kx and cosh ky, respectively, the product of which is the rst of the potentials in the middle column of Table 5.4.1. If we start out from the origin in either the +y or y directions (north or south), we climb a potential hill. If we instead proceed in the +x or x directions (east or west), we move downhill. An easterly path begun on the potential hill to the north of the origin corresponds to a decrease in the cos kx factor. To follow a path of equal elevation, the cosh ky factor must increase, and this implies that the path must turn northward. A good starting point in making these eld sketches is the identication of the contours of zero potential. In the plot of the second potential in the middle column of Table 5.4.1, shown in Fig. 5.4.2, these are the y axis and the lines kx = +/2, +3/2, etc. The dependence on y is now odd rather than even, as it was for the plot of Fig. 5.4.1. Thus, the origin is now on the side of a potential hill that slopes downward from north to south. The solutions in the third and fourth rows of the second column possess the same eld patterns as those just discussed provided those patterns are respectively shifted in the x direction. In the last four rows of Table 5.4.1 are four additional possible solutions which are linear combinations of the previous four in that column. Because these decay exponentially in either the +y or y directions, they are useful for representing solutions in problems where an innite half-space is considered. The solutions in Table 5.4.1 are nonsingular throughout the entire xy plane. This means that Laplaces equation is obeyed everywhere within the nite x y plane, and hence the eld lines are continuous; they do not appear or disappear. The sketches show that the elds become stronger and stronger as one proceeds in the positive and negative y directions. The lines of electric eld originate on positive charges and terminate on negative charges at y . Thus, for the plots shown in Figs. 5.4.1 and 5.4.2, the charge distributions at innity must consist of alternating distributions of positive and negative charges of innite amplitude. Two nal observations serve to further develop an appreciation for the nature of solutions to Laplaces equation. First, the third dimension can be used to represent the potential in the manner of Fig. 4.1.4, so that the potential surface has the shape of a membrane stretched from boundaries that are elevated in proportion to their potentials. Laplaces equation, (1), requires that the sum of quantities that reect the curvatures in the x and y directions vanish. If the second derivative of a function is positive, it is curved upward; and if it is negative, it is curved downward. If the curvature is positive in the x direction, it must be negative in the y direction. Thus, at the origin in Fig. 5.4.1, the potential is cupped downward for excursions in the

Sec. 5.5

Modal Expansion

11

Fig. 5.4.1 Equipotentials for = cos(kx) cosh(ky) and eld lines. As an aid to visualizing the potential, the separate factors cos(kx) and cosh(ky) are, respectively, displayed at the top and to the left.

x direction, and so it must be cupped upward for variations in the y direction. A similar deduction must apply at every point in the x y plane. Second, because the k that appears in the periodic functions of the second column in Table 5.4.1 is the same as that in the exponential and hyperbolic functions, it is clear that the more rapid the periodic variation, the more rapid is the decay or apparent growth.

5.5 MODAL EXPANSION TO SATISFY BOUNDARY CONDITIONS Each of the solutions obtained in the preceding section by separation of variables could be produced by an appropriate potential applied to pairs of parallel surfaces

12

Electroquasistatic Fields from the Boundary Value Point of View

Chapter 5

Fig. 5.4.2 Equipotentials for = cos(kx) sinh(ky) and eld lines. As an aid to visualizing the potential, the separate factors cos(kx) and sinh(ky) are, respectively, displayed at the top and to the left.

in the planes x = constant and y = constant. Consider, for example, the fourth solution in the column k 2 0 of Table 5.4.1, which with a constant multiplier is = A sin kx sinh ky (1)

This solution has = 0 in the plane y = 0 and in the planes x = n/k, where n is an integer. Suppose that we set k = n/a so that = 0 in the plane y = a as well. Then at y = b, the potential of (1) (x, b) = A sinh n n b sin x a a (2)

Sec. 5.5

Modal ExpansionTABLE 5.4.1 TWO-DIMENSIONAL CARTESIAN SOLUTIONS OF LAPLACES EQUATION

13

k=0 Constant y x xy

k2 0 cos kx cosh ky cos kx sinh ky sin kx cosh ky sin kx sinh ky cos kx eky

k2 0 (k jk ) cosh k x cos k y cosh k x sin k y sinh k x cos k y sinh k x sin k y ekx x

cos k y cos k y sin k y sin k y

cos kx eky sin kx eky

ek e

k x x

sin kx eky

ek

Fig. 5.5.1 Two of the innite number of potential functions having the form of (1) that will t the boundary conditions = 0 at y = 0 and at x = 0 and x = a.

has a sinusoidal dependence on x. If a potential of the form of (2) were applied along the surface at y = b, and the surfaces at x = 0, x = a, and y = 0 were held at zero potential (by, say, planar conductors held at zero potential), then the potential, (1), would exist within the space 0 < x < a, 0 < y < b. Segmented electrodes having each segment constrained to the appropriate potential could be used to approximate the distribution at y = b. The potential and eld plots for n = 1 and n = 2 are given in Fig. 5.5.1. Note that the theorem of Sec. 5.2 insures

14

Electroquasistatic Fields from the Boundary Value Point of View

Chapter 5

Fig. 5.5.2 Cross-section of zero-potential rectangular slot with an electrode having the potential v inserted at the top.

that the specied potential is unique. But what can be done to describe the eld if the wall potentials are not constrained to t neatly the solution obtained by separation of variables? For example, suppose that the elds are desired in the same region of rectangular cross-section, but with an electrode at y = b constrained to have a potential v