Sec. 0.1 Preface 1 0.1 PREFACE 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 electroquasistatic (EQS) and magnetoquasistatic (MQS) forms. The stage is set for taking a structured approach that gives a physical overview while developing the mathematical 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 encapsulated in time-dependent coecients (of solutions to partial dierential equations) satisfying ordinary dierential equations. However, the connection with analog circuits 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 appreciates 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 represent 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 asto why it is so often possible to view the world as either EQS or MQS. The time 1/ LC is the geometric 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 meaning. 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 documentation 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 approaches 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. Professor 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 geometries 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 under-standing of the gradient, divergence, and curl operators. This mathematical conversance 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 classical 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 magnetization. The often-used alternative is circulating Amp`erian currents. The magnetic 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 magnetism 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 approach. Whereas he conceded that a choice between two approaches that give identical answers is a matter of taste, he gave a derivation of the force on a current loop (the Amp`erian model of a magnetic dipole) and showed that it gave a dierent 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 time-varying 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 curriculumdevelopment 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. 1MAXWELLSINTEGRAL LAWSIN FREE SPACE1.0 INTRODUCTIONPractical, intellectual, and cultural reasons motivate the study of electricity andmagnetism. The operation of electrical systems designed to perform certain engi-neering tasks depends, at least in part, on electrical, electromechanical, or electro-chemical phenomena. The electrical aspects of these applications are described byMaxwells equations. As a description of the temporal evolution of electromagneticelds in three-dimensional space, these same equations form a concise summary ofa wider range of phenomena than can be found in any other discipline. Maxwellsequations are an intellectual achievement that should be familiar to every studentof physical phenomena. As part of the theory of elds that includes continuum me-chanics, quantum mechanics, heat and mass transfer, and many other disciplines,our subject develops the mathematical language and methods that are the basis forthese other areas.For those who have an interest in electromechanical energy conversion, trans-mission systems at power or radio frequencies, waveguides at microwave or opticalfrequencies, antennas, or plasmas, there is little need to argue the necessity forbecoming expert in dealing with electromagnetic elds. There are others who mayrequire encouragement. For example, circuit designers may be satised with circuittheory, the laws of which are stated in terms of voltages and currents and in termsof the relations imposed upon the voltages and currents by the circuit elements.However, these laws break down at high frequencies, and this cannot be understoodwithout electromagnetic eld theory. The limitations of circuit models come intoplay as the frequency is raised so high that the propagation time of electromagneticelds becomes comparable to a period, with the result that inductors behave ascapacitors and vice versa. Other limitations are associated with loss phenom-ena. As the frequency is raised, resistors and transistors are limited by capacitiveeects, and transducers and transformers by eddy currents.12 Maxwells Integral Laws in Free Space Chapter 1Anyone concerned with developing circuit models for physical systems requiresa eld theory background to justify approximations and to derive the values of thecircuit parameters. Thus, the bioengineer concerned with electrocardiography orneurophysiology must resort to eld theory in establishing a meaningful connectionbetween the physical reality and models, when these are stated in terms of circuitelements. Similarly, even if a control theorist makes use of a lumped parametermodel, its justication hinges on a continuum theory, whether electromagnetic,mechanical, or thermal in nature.Computer hardware may seem to be another application not dependent onelectromagnetic eld theory. The software interface through which the computeris often seen makes it seem unrelated to our subject. Although the hardware isgenerally represented in terms of circuits, the practical realization of a computerdesigned to carry out logic operations is limited by electromagnetic laws. For exam-ple, the signal originating at one point in a computer cannot reach another pointwithin 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 ascomputation speeds have increased is a tribute to the solid state technology thathas made it possible to decrease the size of the fundamental circuit elements. Sooneror later, the fundamental limitations imposed by the electromagnetic elds denethe computation speed frontier of computer technology, whether it be caused byelectromagnetic wave delays or electrical power dissipation.Overview of Subject. As illustrated diagrammatically in Fig. 1.0.1, westart with Maxwells equations written in integral form. This chapter begins witha denition of the elds in terms of forces and sources followed by a review ofeach of the integral laws. Interwoven with the development are examples intendedto 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 eachof 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 systemsin a qualitative fashion (as an inventor often does). The integral laws are directlyuseful for (a) dealing with elds in this qualitative way, (b) nding elds in simplecongurations having a great deal of symmetry, and (c) relating elds to theirsources.Chapter 2 develops a dierential description from the integral laws. By follow-ing 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 andStokes, are developed in Chap. 2, while the gradient operator and integral theoremare naturally derived in Chap. 4.Static elds are often the rst topic in developing an understanding of phe-nomena predicted by Maxwells equations. Fields are not measurable, let aloneof practical interest, unless they are dynamic. As developed here, elds are nevertruly static. The subject of quasistatics, begun in Chap. 3, is central to the approachwe will use to understand the implications of Maxwells equations. A mature un-derstanding of these equations is achieved when one has learned how to neglectcomplications that are inconsequential. The electroquasistatic (EQS) and magne-Sec. 1.0 Introduction 34 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.0.1 Outline of Subject. The three columns, respectively for electro-quasistatics, magnetoquasistatics and electrodynamics, show parallels in de-velopment.toquasistatic (MQS) approximations are justied if time rates of change are slowenough (frequencies are low enough) so that time delays due to the propagation ofelectromagnetic waves are unimportant. The examples considered in Chap. 3 givesome notion as to which of the two approximations is appropriate in a given situa-tion. A full appreciation for the quasistatic approximations will come into view asthe EQS and MQS developments are drawn together in Chaps. 11 through 15.Although capacitors and inductors are examples in the electroquasistaticand magnetoquasistatic categories, respectively, it is not true that quasistatic sys-tems can be generally modeled by frequency-independent circuit elements. High-frequency 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 nitetime for charge migration. Models for eddy current shields or heaters are correctlybased on the MQS approximation. Again, the delay time of an electromagneticwave is unimportant while the all-important diusion time of the magnetic eldSec. 1.0 Introduction 5is represented by the MQS laws. Space charge waves on an electron beam or spinwaves in a saturated magnetizable material are often described by EQS and MQSlaws, respectively, even though frequencies of interest are in the GHz range.The parallel developments of EQS (Chaps. 47) and MQS systems (Chaps. 810) is emphasized by the rst page of Fig. 1.0.1. For each topic in the EQS columnto the left there is an analogous one at the same level in the MQS column. Althoughthe eld concepts and mathematical techniques used in dealing with EQS and MQSsystems are often similar, a comparative study reveals as many contrasts as directanalogies. 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 MQSsystems, but the examination of MQS situations leads to a greater appreciation forthe 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 prescribedto where they are to be determined. Thus, EQS electric elds are rst found fromprescribed distributions of charge, while MQS magnetic elds are determined giventhe currents. The development of the EQS eld solution is a direct investment in thesubsequent 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 theeld solution. In the rst of these situations, induced sources are on the boundariesof conductors having a suciently high electrical conductivity to be modeled asperfectly conducting. For the EQS systems, these sources are surface charges,while for the MQS, they are surface currents. In either case, elds must satisfyboundary conditions, and the EQS study provides not only mathematical techniquesbut even partial dierential equations directly applicable to MQS problems.Polarization and magnetization account for eld sources that can be pre-scribed (electrets and permanent magnets) or induced by the elds themselves.In the Chu formulation used here, there is a complete analogy between the wayin which polarization and magnetization are represented. Thus, there is a directtransfer of ideas from Chap. 6 to Chap. 9.The parallel quasistatic studies culminate in Chaps. 7 and 10 in an examina-tion of loss phenomena. Here we learn that very dierent answers must be given tothe question When is a conductor perfect? for EQS on one hand, and MQS onthe other.In Chap. 11, many of the concepts developed previously are put to workthrough the consideration of the ow of power, storage of energy, and productionof electromagnetic forces. From this chapter on, Maxwells equations are used with-out approximation. Thus, the EQS and MQS approximations are seen to representsystems in which either the electric or the magnetic energy storage dominates re-spectively.In Chaps. 12 through 14, the focus is on electromagnetic waves. The develop-ment 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. Thetopics of Chaps. 12 and 13 parallel those of the EQS and MQS columns on theprevious page. Potentials used to represent electrodynamic elds are a natural gen-eralization 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 applicationis therefore the description of radiation elds of antennas.6 Maxwells Integral Laws in Free Space Chapter 1The 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. Practicalexamples include simple transmission lines and waveguides. An understanding oftransmission line dynamics, the subject of Chap. 14, is necessary in dealing with theconventional ideal lines that model most high-frequency systems. They are alsoshown to provide useful models for representing quasistatic dynamical processes.To make practical use of Maxwells equations, it is necessary to master theart of making approximations. Based on the electromagnetic properties and dimen-sions of a system and on the time scales (frequencies) of importance, how can aphysical system be broken into electromagnetic subsystems, each described by itsdominant physical processes? It is with this goal in mind that the EQS and MQSapproximations are introduced in Chap. 3, and to this end that Chap. 15 gives anoverview of electromagnetic elds.1.1 THE LORENTZ LAW IN FREE SPACEThere are two points of view for formulating a theory of electrodynamics. The olderone views the forces of attraction or repulsion between two charges or currents as theresult of action at a distance. Coulombs law of electrostatics and the correspondinglaw of magnetostatics were rst stated in this fashion. Faraday[1]introduced a newapproach in which he envisioned the space between interacting charges to be lledwith elds, by which the space is activated in a certain sense; forces between twointeracting charges are then transferred, in Faradays view, from volume elementto volume element in the space between the interacting bodies until nally theyare transferred from one charge to the other. The advantage of Faradays approachwas that it brought to bear on the electromagnetic problem the then well-developedtheory of continuum mechanics. The culmination of this point of view was Maxwellsformulation[2]of the equations named after him.From Faradays point of view, electric and magnetic elds are dened at apoint r even when there is no charge present there. The elds are dened in termsof the force that would be exerted on a test charge q if it were introduced at rmoving at a velocity v at the time of interest. It is found experimentally that sucha force would be composed of two parts, one that is independent of v, and the otherproportional to v and orthogonal to it. The force is summarized in terms of theelectric eld intensity E and magnetic ux density oH by the Lorentz force law.(For a review of vector operations, see Appendix 1.)f = q(E+v oH) (1)The superposition of electric and magnetic force contributions to (1) is illus-trated in Fig. 1.1.1. Included in the gure is a reminder of the right-hand rule usedto determine the direction of the cross-product of v and oH. In general, E and Hare not uniform, but rather are functions of position r and time t: E = E(r, t) andoH = oH(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 theSec. 1.1 The Lorentz Law in Free Space 7Fig. 1.1.1 Lorentz force f in geometric relation to the electric and magneticeld 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 ofoH.units of E = newtoncoulomb = kilogram meter/(second)2coulomb (2)units of oH = newtoncoulomb meter/second = kilogramcoulomb second (3)We can only establish the units of the magnetic ux density oHfrom the forcelaw and cannot argue until Sec. 1.4 that the derived units of H are ampere/meterand hence of o are henry/meter.In much of electrodynamics, the predominant concern is not with mechanicsbut with electric and magnetic elds in their own right. Therefore, it is inconvenientto use the unit of mass when checking the units of quantities. It proves useful tointroduce 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, thefundamental units are SI, while the derived units exploit the fact that the unit ofmass, kilogram = volt-coulomb-second2/meter2and 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 communicatethe physical nature of the variable and the natural symmetry of the electric andmagnetic variables.Example 1.1.1. Electron Motion in Vacuum in a Uniform StaticElectric FieldIn vacuum, the motion of a charged particle is limited only by its own inertia. Inthe uniform electric eld illustrated in Fig. 1.1.2, there is no magnetic eld, and anelectron starts out from the plane x = 0 with an initial velocity vi.The imposed electric eld is E = ixEx, where ix is the unit vector in the xdirection and Ex is a given constant. The trajectory is to be determined here andused to exemplify the charge and current density in Example 1.2.1.8 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.1.2 An electron, subject to the uniform electric eld intensityEx, has the position x, shown as a function of time for positive andnegative elds.With m dened as the electron mass, Newtons law combines with the Lorentzlaw to describe the motion.md2xdt2 = f = eEx (4)The electron position x is shown in Fig. 1.1.2. The charge of the electron is custom-arily denoted by e (e = 1.6 1019coulomb) where e is positive, thus necessitatingan explicit minus sign in (4).By integrating twice, we getx = 12emExt2+ c1t + c2 (5)where c1 and c2 are integration constants. If we assume that the electron is at x = 0and has velocity vi when t = ti, it follows that these constants arec1 = vi + emExti; c2 = viti 12emExt2i (6)Thus, the electron position and velocity are given as a function of time byx = 12emEx(t ti)2+ vi(t ti) (7)dxdt = emEx(t ti) + vi (8)With x dened as upward and Ex > 0, the motion of an electron in an electriceld is analogous to the free fall of a mass in a gravitational eld, as illustratedby Fig. 1.1.2. With Ex < 0, and the initial velocity also positive, the velocity is amonotonically increasing function of time, as also illustrated by Fig. 1.1.2.Example 1.1.2. Electron Motion in Vacuum in a Uniform StaticMagnetic FieldThe magnetic contribution to the Lorentz force is perpendicular to both the particlevelocity and the imposed eld. We illustrate this fact by considering the trajectorySec. 1.1 The Lorentz Law in Free Space 9Fig. 1.1.3 (a) In a uniform magnetic ux density oHo and with noinitial 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 constantmagnetic ux density oH existing along the y axis, the force isf = e(v oH) (9)The cross-product of two vectors is perpendicular to the two vector factors, so theacceleration of the electron, caused by the magnetic eld, is always perpendicularto its velocity. Therefore, a magnetic eld alone cannot change the magnitude ofthe electron velocity (and hence the kinetic energy of the electron) but can changeonly the direction of the velocity. Because the magnetic eld is uniform, because thevelocity and the rate of change of the velocity lie in a plane perpendicular to themagnetic eld, and, nally, because the magnitude of v does not change, we nd thatthe acceleration has a constant magnitude and is orthogonal to both the velocityand the magnetic eld. The electron moves in a circle so that the centrifugal forcecounterbalances the magnetic force. Figure 1.1.3a illustrates the motion. The radiusof the circle is determined by equating the centrifugal force and radial Lorentz forceeo|v|Ho = mv2r (10)which leads tor = me|v|oHo(11)The foregoing problem can be modied to account for any arbitrary initial anglebetween the velocity and the magnetic eld. The vector equation of motion (reallythree equations in the three unknowns x, y, z)md2dt2 = e

ddt oH

(12)is linear in , and so solutions can be superimposed to satisfy initial conditions thatinclude not only a velocity viz but one in the y direction as well, viy. Motion in thesame direction as the magnetic eld does not give rise to an additional force. Thus,10 Maxwells Integral Laws in Free Space Chapter 1the y component of (12) is zero on the right. An integration then shows that the ydirected velocity remains constant at its initial value, viy. This uniform motion canbe added to that already obtained to see that the electron follows a helical path, asshown in Fig. 1.1.3b.It is interesting to note that the angular frequency of rotation of the electronaround the eld is independent of the speed of the electron and depends only uponthe magnetic ux density, oHo. Indeed, from (11) we ndvr c = emoHo (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 = 1.6021019coulomb and m = 9.1061031kg.) With an initial velocity in the z direction of 3 107m/s, the radius of gyrationin the ux density oH = 1 tesla is r = viz/c = 1.7 104m.1.2 CHARGE AND CURRENT DENSITIESIn Maxwells day, it was not known that charges are not innitely divisible butoccur in elementary units of 1.6 1019coulomb, the charge of an electron. Hence,Maxwells macroscopic theory deals with continuous charge distributions. This isan adequate description for elds of engineering interest that are produced by ag-gregates of large numbers of elementary charges. These aggregates produce chargedistributions 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 VV (1)is the charge density at the position r when the time is t. The units of arecoulomb/meter3. The volume V is chosen small as compared to the dimensions ofthe 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 graini-ness of the charge distribution is ignored in such a macroscopic treatment.Fundamentally, current is charge transport and connotes the time rate ofchange of charge. Current density is a directed current per unit area and hencemeasured in (coulomb/second)/meter2. A charge density moving at a velocity vimplies a rate of charge transport per unit area, a current density J, given byJ = 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 unitnormal n, so that a dierential area vector can be dened as a = na. The chargethat passes during a dierential time t is equal to the total charge contained inthe volume v adt. Therefore,d(q) = v adt (3)Sec. 1.2 Charge and Current Densities 11Fig. 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 up-ward 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 currentdensity is related to the charge density by (2).The velocity v is the velocity of the charge. Just how the charge is set intomotion depends on the physical situation. The charge might be suspended in or onan insulating material which is itself in motion. In that case, the velocity wouldalso be that of the material. More likely, it is the result of applying an electric eldto 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, asillustrated 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 DiodeConsider the charge and current densities for electrons being emitted with initialvelocity v from a cathode in the plane x = 0, as shown in Fig. 1.2.2a.1Electrons are continuously injected. As in Example 1.1.1, where the motions of theindividual electrons are considered, the electric eld is assumed to be uniform. In thenext section, it is recognized that charge is the source of the electric eld. Here it isassumed that the charge used to impose the uniform eld is much greater than thespace charge associated with the electrons. This is justied in the limit of a lowelectron 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 currentdensities in any given plane x = constant would be expected to be independent oftime. Moreover, the current passing any x-plane should be the same as that passingany other such plane. That is, in the steady state, the current density is independent1Here we picture the eld variables Ex, vx, and as though they were positive. For electrons, < 0, and to make vx > 0, we must have Ex < 0.12 Maxwells Integral Laws in Free Space Chapter 1of not only time but x as well. Thus, it is possible to write(x)vx(x) = Jo (4)where Jo is a given current density.The following steps illustrate how this condition of current continuity makesit possible to shift from a description of the particle motions described with time asthe independent variable to one in which coordinates (x, y, z) (or for short r) are theindependent coordinates. The relation between time and position for the electrondescribed by (1.1.7) takes the form of a quadratic in (t ti)12emEx(t ti)2vi(t ti) + x = 0 (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 conditionthat when t = ti, x = 0.t ti =vi

v2i 2 emExxemEx(6)With the benet of this expression, the velocity given by (1.1.8) is written asdxdt =

v2i 2emExx (7)Now we make a shift in viewpoint. On the left in (7) is the velocity vx of theparticle that is at the location x = x. Substitution of variables then givesvx =

v2i 2 emExx (8)so that x becomes the independent variable used to express the dependent variablevx. It follows from this expression and (4) that the charge density = Jovx= Jo

v2i 2emExx(9)is also expressed as a function of x. In the plots shown in Fig. 1.2.2, it is assumedthat Ex < 0, so that the electrons have velocities that increase monotonically withx. As should be expected, the charge density decreases with x because as they speedup, the electrons thin out to keep the current density constant.1.3 GAUSS INTEGRAL LAW OF ELECTRIC FIELD INTENSITYThe Lorentz force law of Sec. 1.1 expresses the eect of electromagnetic eldson a moving charge. The remaining sections in this chapter are concerned withthe reaction of the moving charges upon the electromagnetic elds. The rst ofSec. 1.3 Gauss Integral Law 13Fig. 1.3.1 General surface S enclosing volume V .Maxwells equations to be considered, Gauss law, describes how the electric eldintensity is related to its source. The net charge within an arbitrary volume V thatis enclosed by a surface S is related to the net electric ux through that surface by

S

oE da =

Vdv(1)With the surface normal dened as directed outward, the volume is shown inFig. 1.3.1. Here the permittivity of free space, o = 8.854 1012farad/meter, is anempirical constant needed to express Maxwells equations in SI units. On the rightin (1) is the net charge enclosed by the surface S. On the left is the summationover this same closed surface of the dierential contributions of ux oE da. Thequantity oE is called the electric displacement ux density and, [from (1)], has theunits of coulomb/meter2. Out of any region containing net charge, there must be anet displacement ux.The following example illustrates the mechanics of carrying out the volumeand surface integrations.Example 1.3.1. Electric Field Due to Spherically Symmetric ChargeDistributionGiven the charge and current distributions, the integral laws fully determine theelectric and magnetic elds. However, they are not directly useful unless there is agreat deal of symmetry. An example is the distribution of charge density(r) =

orR; r < R0; r > R (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 com-ponent of E is radial.E = irEr(r) (3)Indeed, suppose that in addition to this r component the eld possesses a com-ponent. 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 somenew direction perpendicular to r. However, the rotation leaves the source of thateld, the charge distribution, unaltered. It follows that E must be zero. A similarargument shows that E also is zero.14 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.3.2 (a) Spherically symmetric charge distribution, showing ra-dial dependence of charge density and associated radial electric eldintensity. (b) Axis of rotation for demonstration that the componentsof E transverse to the radial coordinate are zero.The incremental volume element isdv = (dr)(rd)(r sin d) (4)and it follows that for a spherical volume having arbitrary radius r,

Vdv =

r0

0

20

or

R

(r

sin d)(r

d)dr

= oR r4; r < R

R0

0

20

or

R

(r

sin d)(r

d)dr

= oR3; R < r(5)To evaluate the left-hand side of (1), note thatn = ir; da = ir(rd)(r sin d) (6)Thus, for the spherical surface at the arbitrary radius r,

S

oE da =

0

20

oEr(r sin d)(rd) = oEr4r2(7)With the volume and surface integrals evaluated in (5) and (7), Gauss law, (l),shows that

oEr4r2= oR r4Er = or24oR; r < R (8a)

oEr4r2= oR3Er = oR34or2; R < r (8b)Inside the spherical charged region, the radial electric eld increases with the squareof the radius because even though the associated surface increases like the squareSec. 1.3 Gauss Integral Law 15Fig. 1.3.3 Singular charge distributions: (a) point charge, (b) line charge,(c) surface charge.Fig. 1.3.4 Filamentary volume element having cross-section da used to de-ne line charge density.of the radius, the enclosed charge increases even more rapidly. Figure 1.3.2 illus-trates this dependence, as well as the exterior eld decay. Outside, the surface areacontinues to increase in proportion to r2, but the enclosed charge remains constant.Singular Charge Distributions. Examples of singular functions from circuittheory are impulse and step functions. Because there is only the one independentvariable, namely time, circuit theory is concerned with only one dimension. Inthree-dimensional eld theory, there are three spatial analogues of the temporalimpulse function. These are point, line, and surface distributions of , as illustratedin Fig. 1.3.3. Like the temporal impulse function of circuit theory, these singulardistributions are dened in terms of integrals.A point charge is the limit of an innite charge density occupying zero volume.With q dened as the net charge,q = limV 0

Vdv (9)the point charge can be pictured as a small charge-lled region, the outside of whichis charge free. An example is given in Fig. 1.3.2 in the limit where the volume 4R3/3goes to zero, while q = oR3remains nite.A line charge density represents a two-dimensional singularity in charge den-sity. It is the mathematical abstraction representing a thin charge lament. In termsof the lamentary volume shown in Fig. 1.3.4, the line charge per unit length l(the line charge density) is dened as the limit where the cross-sectional area of thevolume goes to zero, goes to innity, but the integral16 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.3.5 Volume element having thickness h used to dene surface chargedensity.Fig. 1.3.6 Point charge q at origin of spherical coordinate system.l = limA0

Ada (10)remains nite. In general, l is a function of position along the curve.The one-dimensional singularity in charge density is represented by the surfacecharge density. The charge density is very large in the vicinity of a surface. Thus,as a function of a coordinate perpendicular to that surface, the charge density isa one-dimensional impulse function. To dene the surface charge density, mount apillbox as shown in Fig. 1.3.5 so that its top and bottom surfaces are on the twosides of the surface. The surface charge density is then dened as the limits = limh0

+h2h2d (11)where the coordinate is picked parallel to the direction of the normal to thesurface, n. In general, the surface charge density s is a function of position in thesurface.Illustration. Field of a Point ChargeA point charge q is located at the origin in Fig. 1.3.6. There are no other charges.By the same arguments as used in Example 1.3.1, the spherical symmetry of thecharge distribution requires that the electric eld be radial and be independent of and . Evaluation of the surface integral in Gauss integral law, (1), amounts tomultiplying oEr by the surface area. Because all of the charge is concentrated atthe origin, the volume integral gives q, regardless of radial position of the surface S.Thus,4r2

oEr = q E = q4or2ir (12)Sec. 1.3 Gauss Integral Law 17Fig. 1.3.7 Uniform line charge distributed from innity to + in-nity along z axis. Rotation by 180 degrees about axis shown leads toconclusion that electric eld is radial.is the electric eld associated with a point charge q.Illustration. The Field Associated with Straight Uniform Line ChargeA uniform line charge is distributed along the z axis from z = to z = +, asshown in Fig. 1.3.7. For an observer at the radius r, translation of the line sourcein the z direction and rotation of the source about the z axis (in the direction)results in the same charge distribution, so the electric eld must only depend onr. Moreover, E can only have a radial component. To see this, suppose that therewere a z component of E. Then a 180 degree rotation of the system about an axisperpendicular to and passing through the z axis must reverse this eld. However,the rotation leaves the charge distribution unchanged. The contradiction is resolvedonly if Ez = 0. The same rotation makes it clear that E must be zero.This time, Gauss integral law is applied using for S the surface of a rightcircular cylinder coaxial with the z axis and of arbitrary radius r. Contributionsfrom the ends are zero because there the surface normal is perpendicular to E.With the cylinder taken as having length l, the surface integration amounts to amultiplication of oEr by the surface area 2rl while, the volume integral gives llregardless of the radius r. Thus, (1) becomes2rloEr = ll E = l2orir (13)for the eld of an innitely long uniform line charge having density l.Example 1.3.2. The Field of a Pair of Equal and Opposite InnitePlanar Charge DensitiesConsider the eld produced by a surface charge density +o occupying all the xyplane at z = s/2 and an opposite surface charge density o at z = s/2.First, the eld must be z directed. Indeed there cannot be a component ofE transverse to the z axis, because rotation of the system around the z axis leavesthe same source distribution while rotating that component of E. Hence, no suchcomponent exists.18 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.3.8 Sheets of surface charge and volume of integration withupper surface at arbitrary position x. With eld Eo due to externalcharges equal to zero, the distribution of electric eld is the discontinu-ous function shown at right.Because the source distribution is independent of x and y, Ez is independent ofthese coordinates. The z dependence is now established by means of Gauss integrallaw, (1). The volume of integration, shown in Fig. 1.3.8, has cross-sectional area Ain the x y plane. Its lower surface is located at an arbitrary xed location belowthe lower surface charge distribution, while its upper surface is in the plane denotedby z. For now, we take Ez as being Eo on the lower surface. There is no contributionto the surface integral from the side walls because these have normals perpendicularto E. It follows that Gauss law, (1), becomesA(oEzoEo) = 0; < z < s2 Ez = EoA(oEzoEo) = Ao; s2 < z < s2 Ez = o

o+ EoA(oEzoEo) = 0; s2 < z < Ez = Eo(14)That is, with the upper surface below the lower charge sheet, no charge is enclosedby the surface of integration, and Ez is the constant Eo. With the upper surfaceof integration between the charge sheets, Ez is Eo minus o/o. Finally, with theupper integration surface above the upper charge sheet, Ez returns to its value ofEo. The external electric eld Eo must be created by charges at z = +, much asthe eld between the charge sheets is created by the given surface charges. Thus,if these charges at innity are absent, Eo = 0, and the distribution of Ez is asshown to the right in Fig. 1.3.8.Illustration. Coulombs Force Law for Point ChargesIt is worthwhile to see that for charges at rest, Gauss integral law and the Lorentzforce law give the familiar action at a distance force law. The force on a charge qis given by the Lorentz law, (1.1.1), and if the electric eld is caused by a secondcharge at the origin in Fig. 1.3.9, thenf = qE = q1q24or2ir (15)Coulombs famous statement that the force exerted by one charge on another isproportional to the product of their charges, acts along a line passing through eachSec. 1.3 Gauss Integral Law 19Fig. 1.3.9 Coulomb force induced on charge q2 due to eld from q1.Fig. 1.3.10 Like-charged particles on ends of thread are pushed apartby the Coulomb force.charge, and is inversely proportional to the square of the distance between them, isnow demonstrated.Demonstration 1.3.1. Coulombs Force LawThe charge resulting on the surface of adhesive tape as it is pulled from a dispenseris a common nuisance. As the tape is brought toward a piece of paper, the forceof attraction that makes the paper jump is an aggravating reminder that there arecharges on the tape. Just how much charge there is on the tape can be approximatelydetermined by means of the simple experiment shown in Fig. 1.3.10.Two pieces of freshly pulled tape about 7 cm long are folded up into balls andstuck on the ends of a thread having a total length of about 20 cm. The middle ofthe thread is then tied up so that the charged balls of tape are suspended free toswing. (By electrostatic standards, our ngers are conductors, so the tape should bemanipulated chopstick fashion by means of plastic rods or the like.) It is then easyto measure approximately l and r, as dened in the gure. The force of repulsionthat separates the balls of tape is presumably predicted by (15). In Fig. 1.3.10,the vertical component of the tension in the thread must balance the gravitationalforce Mg (where g is the gravitational acceleration and M is the mass). It followsthat the horizontal component of the thread tension balances the Coulomb force ofrepulsion.q24or2 = Mg(r/2)l q =

Mgr32ol (16)As an example, tape balls having an area of A = 14 cm2, (7 cm length of 2 cmwide tape) weighing 0.1 mg and dangling at a length l = 20 cm result in a distanceof separation r = 3 cm. It follows from (16) (with all quantities expressed in SIunits) that q = 2.7 109coulomb. Thus, the average surface charge density isq/A = 1.9106coulomb/meter or 1.21013electronic charges per square meter. If20 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.3.11 Pillbox-shaped incremental volume used to deduce the jumpcondition implied by Gauss integral law.these charges were in a square array with spacing s between charges, then s = e/s2,and it follows that the approximate distance between the individual charge in thetape surface is 0.3m. This length is at the limit of an optical microscope and mayseem small. However, it is about 1000 times larger than a typical atomic dimension.2Gauss Continuity Condition. Each of the integral laws summarized in thischapter implies a relationship between eld variables evaluated on either side of asurface. These conditions are necessary for dealing with surface singularities in theeld sources. Example 1.3.2 illustrates the jump in the normal component of E thataccompanies a surface charge.A surface that supports surface charge is pictured in Fig. 1.3.11, as havinga unit normal vector directed from region (b) to region (a). The volume to whichGauss integral law is applied has the pillbox shape shown, with endfaces of areaA on opposite sides of the surface. These are assumed to be small enough so thatover the area of interest the surface can be treated as plane. The height h of thepillbox is very small so that the cylindrical sideface of the pillbox has an area muchsmaller than A.Now, let h approach zero in such a way that the two sides of the pillbox remainon opposite sides of the surface. The volume integral of the charge density, on theright in (1), gives As. This follows from the denition of the surface charge density,(11). The electric eld is assumed to be nite throughout the region of the surface.Hence, as the area of the sideface shrinks to zero, so also does the contribution ofthe sideface to the surface integral. Thus, the displacement ux through the closedsurface consists only of the contributions from the top and bottom surfaces. Appliedto the pillbox, Gauss integral law requires thatn (oEaoEb) = s (17)where the area A has been canceled from both sides of the equation.The contribution from the endface on side (b) comes with a minus sign becauseon that surface, n is opposite in direction to the surface element da.Note that the eld found in Example 1.3.2 satises this continuity conditionat z = s/2 and z = s/2.2An alternative way to charge a particle, perhaps of low density plastic, is to place it in thecorona discharge around the tip of a pin placed at high voltage. The charging mechanism at workin this case is discussed in Chapter 7 (Example 7.7.2).Sec. 1.4 Amp`eres Integral Law 21Fig. 1.4.1 Surface S is enclosed by contour C having positive direction de-termined by the right-hand rule. With the ngers in the direction of ds, thethumb passes through the surface in the direction of positive da.1.4 AMP`ERES INTEGRAL LAWThe law relating the magnetic eld intensity H to its source, the current density J,is

CH ds =

SJ da + ddt

S

oE da(1)Note that by contrast with the integral statement of Gauss law, (1.3.1), thesurface integral symbols on the right do not have circles. This means that theintegrations are over open surfaces, having edges denoted by the contour C. Such asurface S enclosed by a contour C is shown in Fig. 1.4.1. In words, Amp`eres integrallaw as given by (1) requires that the line integral (circulation) of the magnetic eldintensity H around a closed contour is equal to the net current passing through thesurface spanning the contour plus the time rate of change of the net displacementux density oE through the surface (the displacement current).The direction of positive da is determined by the right-hand rule, as alsoillustrated in Fig. 1.4.1. With the ngers of the right-hand in the direction of ds,the thumb has the direction of da. Alternatively, with the right hand thumb in thedirection of ds, the ngers will be in the positive direction of da.In Amp`eres law, H appears without o. This law therefore establishes thebasic units of H as coulomb/(meter-second). In Sec. 1.1, the units of the ux den-sity oH are dened by the Lorentz force, so the second empirical constant, thepermeability of free space, is o = 4 107henry/m (henry = volt sec/amp).Example 1.4.1. Magnetic Field Due to Axisymmetric CurrentA constant current in the z direction within the circular cylindrical region of radiusR, shown in Fig. 1.4.2, extends from innity to + innity along the z axis and isrepresented by the densityJ =

Jo

rR

; r < R0; r > R (2)22 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.4.2 Axially symmetric current distribution and associated ra-dial distribution of azimuthal magnetic eld intensity. Contour C is usedto determine azimuthal H, while C

is used to show that the z-directedeld must be uniform.where Jo and R are given constants. The associated magnetic eld intensity hasonly an azimuthal component.H = Hi (3)To see that there can be no r component of this eld, observe that rotationof the source around the radial axis, as shown in Fig. 1.4.2, reverses the source(the current is then in the z direction) and hence must reverse the eld. But anr component of the eld does not reverse under such a rotation and hence must bezero. The H and Hz components are not ruled out by this argument. However, ifthey exist, they must not depend upon the and z coordinates, because rotation ofthe source around the z axis and translation of the source along the z axis does notchange the source and hence does not change the eld.The current is independent of time and so we assume that the elds are aswell. Hence, the last term in (1), the displacement current, is zero. The law is thenused with S, a surface having its enclosing contour C at the arbitrary radius r, asshown in Fig. 1.4.2. Then the area and line elements areda = rddriz; ds = ird (4)and the right-hand side of (1) becomes

SJ da =

20

r0 JorRrddr = Jor323R ; r < R

20

R0 JorRrddr = JoR223 ; R < r(5)Integration on the left-hand side amounts to a multiplication of the independentH by the length of C.

CH ds =

20Hrd = H2r (6)Sec. 1.4 Amp`eres Integral Law 23Fig. 1.4.3 (a) Line current enclosed by volume having cross-sectional areaA. (b) Surface current density enclosed by contour having thickness h.These last two expressions are used to evaluate (1) and obtain2rH = Jor323R H = Jor23R ; r < R2rH = JoR223 H = JoR23r ; r < R (7)Thus, the azimuthal magnetic eld intensity has the radial distribution shown inFig. 1.4.2.The z component of H is, at most, uniform. This can be seen by applying theintegral law to the contour C

, also shown in Fig. 1.4.2. Integration on the top andbottom legs gives zero because Hr = 0. Thus, to make the contributions due to Hzon the vertical legs cancel, it is necessary that Hz be independent of radius. Such auniform eld must be caused by sources at innity and is therefore set equal to zeroif such sources are not postulated in the statement of the problem.Singular Current Distributions. The rst of two singular forms of the currentdensity shown in Fig. 1.4.3a is the line current. Formally, it is the limit of an innitecurrent density distributed over an innitesimal area.i = lim|J|A0

AJ da (8)With i a constant over the length of the line, a thin wire carrying a current iconjures up the correct notion of the line current. However, in general, the currenti may depend on the position along the line if it varies with time as in an antenna.The second singularity, the surface current density, is the limit of a verylarge current density J distributed over a very thin layer adjacent to a surface. InFig. 1.4.3b, the current is in a direction parallel to the surface. If the layer extendsbetween = h/2 and = +h/2, the surface current density K is dened asK = lim|J|h0 h2h2Jd (9)24 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.4.4 Uniform line current with contours for determining H. Axis ofrotation is used to deduce that radial component of eld must be zero.By denition, K is a vector tangential to the surface that has units of am-pere/meter.Illustration. H eld Produced by a Uniform Line CurrentA uniform line current of magnitude i extends from innity to + innity alongthe z axis, as shown in Fig. 1.4.4. The symmetry arguments of Example 1.4.1 showthat the only component of H is azimuthal. Application of Amp`eres integral law,(1), to the contour of Fig. 1.4.4 having arbitrary radius r gives a line integral thatis simply the product of H and the circumference 2r and a surface integral thatis simply i, regardless of the radius.2rH = i H = i2r (10)This expression makes it especially clear that the units of Hare ampere/meter.Demonstration 1.4.1. Magnetic Field of a Line CurrentAt 60 Hz, the displacement current contribution to the magnetic eld of the exper-iment shown in Fig. 1.4.5 is negligible. So long as the eld probe is within a distancer from the wire that is small compared to the distance to the ends of the wire orto the return wires below, the magnetic eld intensity is predicted quantitativelyby (10). The curve shown is typical of demonstration measurements illustrating theradial dependence. Because the Hall-eect probe fundamentally exploits the Lorentzforce law, it measures the ux density oH. A common unit for ux density is theGauss. For conversion of units, 10,000 gauss = 1 tesla, where the tesla is the SI unit.Illustration. Uniform Axial Surface CurrentAt the radius R from the z axis, there is a uniform z directed surface currentdensity Ko that extends from - innity to + innity in the z direction. The sym-metry arguments of Example 1.4.1 show that the resulting magnetic eld intensitySec. 1.4 Amp`eres Integral Law 25Fig. 1.4.5 Demonstration of peak magnetic ux density induced by linecurrent of 6 ampere (peak).Fig. 1.4.6 Uniform current density Ko is z directed in circular cylin-drical shell at r = R. Radially discontinuous azimuthal eld shown isdetermined using the contour at arbitrary radius r.is azimuthal. To determine that eld, Amp`eres integral law is applied to a contourhaving the arbitrary radius r, shown in Fig. 1.4.6. As in the previous illustration,the line integral is the product of the circumference and H. The surface integralgives nothing if r < R, but gives 2R times the surface current density if r > R.Thus,2rH =

0; r < R2RKo; r > R H =

0; r < RKoRr ; r > R (11)Thus, the distribution of H is the discontinuous function shown in Fig. 1.4.6. Theeld tangential to the surface current undergoes a jump that is equal in magnitude26 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.4.7 Amp`eres integral law is applied to surface S

enclosed by a rect-angular contour that intersects a surface S carrying the current density K. Interms of the unit normal to S, n, the resulting continuity condition is given by(16).to the surface current density.Amp`eres Continuity Condition. A surface current density in a surface Scauses a discontinuity of the magnetic eld intensity. This is illustrated in Fig. 1.4.6.To obtain a general relation between elds evaluated to either side of S, a rectan-gular surface of integration is mounted so that it intersects S as shown in Fig. 1.4.7.The normal to S is in the plane of the surface of integration. The length l of therectangle is assumed small enough so that the surface of integration can be consid-ered plane over this length. The width w of the rectangle is assumed to be muchsmaller than l . It is further convenient to introduce, in addition to the normal nto S, the mutually orthogonal unit vectors is and in as shown.Now apply the integral form of Amp`eres law, (1), to the rectangular surfaceof area lw. For the right-hand side we obtain

S

J da +

S

t

oE da K inl (12)Only J gives a contribution, and then only if there is an innite current densityover the zero thickness of S, as required by the denition of the surface currentdensity, (9). The time rate of change of a nite displacement ux density integratedover zero area gives zero, and hence there is no contribution from the second term.The left-hand side of Amp`eres law, (1), is a contour integral following therectangle. Because w has been assumed to be very small compared with l, and His assumed nite, no contribution is made by the two short sides of the rectangle.Hence,l is (HaHb) = K inl (13)From Fig. 1.4.7, note thatis = inn (14)Sec. 1.5 Charge Conservation in Integral 27The cross and dot can be interchanged in this scalar triple product without aectingthe result (Appendix 1), so introduction of (14) into (13) givesin n (HaHb) = in K (15)Finally, note that the vector in is arbitrary so long as it lies in the surface S. Sinceit multiplies vectors tangential to the surface, it can be omitted.n (HaHb) = K (16)There is a jump in the tangential magnetic eld intensity as one passes througha surface current. Note that (16) gives a prediction consistent with what was foundfor the illustration in Fig. 1.4.6.1.5 CHARGE CONSERVATION IN INTEGRAL FORMEmbedded in the laws of Gauss and Amp`ere is a relationship that must existbetween the charge and current densities. To see this, rst apply Amp`eres law toa closed surface, such as sketched in Fig. 1.5.1. If the contour C is regarded asthedrawstring and S as the bag, then this limit is one in which the string isdrawn tight so that the contour shrinks to zero. Thus, the open surface integrals of(1.4.1) become closed, while the contour integral vanishes.

SJ da + ddt

S

oE da = 0 (1)But now, in view of Gauss law, the surface integral of the electric displacementcan be replaced by the total charge enclosed. That is, (1.3.1) is used to write (1) as

SJ da + ddt

Vdv = 0(2)This is the law of conservation of charge. If there is a net current out of thevolume shown in Fig. 1.5.2, (2) requires that the net charge enclosed be decreasingwith time.Charge conservation, as expressed by (2), was a compelling reason for Maxwellto add the electric displacement term to Amp`eres law. Without the displacementcurrent density, Amp`eres law would be inconsistent with charge conservation. Thatis, if the second term in (1) would be absent, then so would the second term in (2). Ifthe displacement current term is dropped in Amp`eres law, then net current cannotenter, or leave, a volume.The conservation of charge is consistent with the intuitive picture of the rela-tionship between charge and current developed in Example 1.2.1.Example 1.5.1. Continuity of Convection Current28 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.5.1 Contour C enclosing an open surface can be thought of as thedrawstring of a bag that can be closed to create a closed surface.Fig. 1.5.2 Current density leaves a volume V and hence the net charge mustdecrease.Fig. 1.5.3 In steady state, charge conservation requires that the cur-rent density entering through the x = 0 plane be the same as thatleaving through the plane at x = x.The steady state current of electrons accelerated through vacuum by a uniformelectric eld is described in Example 1.2.1 by assuming that in any plane x = con-stant the current density is the same. That this must be true is now seen formally byapplying the charge conservation integral theorem to the volume shown in Fig. 1.5.3.Here the lower surface is in the injection plane x = 0, where the current density isknown to be Jo. The upper surface is at the arbitrary level denoted by x. Becausethe steady state prevails, the time derivative in (2) is zero. The remaining surfaceintegral has contributions only from the top and bottom surfaces. Evaluation ofthese, with the recognition that the area element on the top surface is (ixdydz)while it is (ixdydz) on the bottom surface, makes it clear thatAJxAJo = 0 vx = Jo (3)This same relation was used in Example 1.2.1, (1.2.4), as the basis for convertingfrom a particle point of view to the one used here, where (x, y, z) are independentof t.Example 1.5.2. Current Density and Time-Varying ChargeSec. 1.5 Charge Conservation in Integral 29Fig. 1.5.4 With the given axially symmetric charge distribution pos-itive and decreasing with time (/t < 0), the radial current densityis positive, as shown.With the charge density a given function of time with an axially symmetric spatialdistribution, (2) can be used to deduce the current density. In this example, thecharge density is = o(t)er/a(4)and can be pictured as shown in Fig. 1.5.4. The function of time o is given, as isthe dimension a.As the rst step in nding J, we evaluate the volume integral in (2) for acircular cylinder of radius r having z as its axis and length l in the z direction.

Vdv =

l0

20

r0oeradr(rd)dz= 2la2

1 era

1 + ra

o(5)The axial symmetry demands that J is in the radial direction and indepen-dent of and z. Thus, the evaluation of the surface integral in (2) amounts to amultiplication of Jr by the area 2rl, and that equation becomes2rlJr + 2la2

1 era

1 + ra

dodt = 0 (6)Finally, this expression can be solved for Jr.Jr = a2r

era

1 + ra

1

dodt (7)Under the assumption that the charge density is positive and decreasing, sothat do/dt < 0, the radial distribution of Jr is shown at an instant in time in30 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.5.5 When a charge q is introduced into an essentially groundedmetal sphere, a charge q is induced on its inner surface. The inte-gral form of charge conservation, applied to the surface S, shows thati = dq/dt. The net excursion of the integrated signal is then a directmeasurement of q.Fig. 1.5.4. In this case, the radial current density is positive at any radius r becausethe net charge within that radius, given by (5), is decreasing with time.The integral form of charge conservation provides the link between the currentcarried by a wire and the charge. Thus, if we can measure a current, this law providesthe basis for measuring the net charge. The following demonstration illustrates itsuse.Demonstration 1.5.1. Measurement of ChargeIn Demonstration 1.3.1, the net charge is deduced from mechanical measurementsand Coulombs force law. Here that same charge is deduced electrically. The ballcarrying the charge is stuck to the end of a thin plastic rod, as in Fig. 1.5.5. Theobjective is to measure this charge, q, without removing it from the ball.We know from the discussion of Gauss law in Sec. 1.3 that this charge is thesource of an electric eld. In general, this eld terminates on charges of oppositesign. Thus, the net charge that terminates the eld originating from q is equalin magnitude and opposite in sign to q. Measurement of this image charge istantamount to measuring q.How can we design a metal electrode so that we are guaranteed that all ofthe lines of E originating from q will be terminated on its surface? It would seemthat the electrode should essentially surround q. Thus, in the experiment shown inFig. 1.5.5, the charge is transported to the interior of a metal sphere through a holein its top. This sphere is grounded through a resistance R and also surrounded bya grounded shield. This resistance is made low enough so that there is essentiallyno electric eld in the region between the spherical electrode, and the surroundingshield. As a result, there is negligible charge on the outside of the electrode and thenet charge on the spherical electrode is just that inside, namely q.Now consider the application of (2) to the surface S shown in Fig. 1.5.5. Thesurface completely encloses the spherical electrode while excluding the charge q atits center. On the outside, it cuts through the wire connecting the electrode tothe resistance R. Thus, the volume integral in (2) gives the net charge q, whileSec. 1.6 Faradays Integral Law 31contributions to the surface integral only come from where S cuts through the wire.By denition, the integral of J da over the cross-section of the wire gives the currenti (amps). Thus, (2) becomes simplyi + d(q)dt = 0 i = dqdt (8)This current is the result of having pushed the charge through the hole to aposition where all the eld lines terminated on the spherical electrode.3Although small, the current through the resistor results in a voltage.v iR = Rdqdt (9)The integrating circuit is introduced into the experiment in Fig. 1.5.5 so that theoscilloscope directly displays the charge. With this circuit goes a gain A such thatvo = A

vdt = ARq (10)Then, the voltage vo to which the trace on the scope rises as the charge is insertedthrough the hole reects the charge q. This measurement of q corroborates that ofDemonstration 1.3.1.In retrospect, because S and V are arbitrary in the integral laws, the experi-ment need not be carried out using an electrode and shield that are spherical. Thesecould just as well have the shape of boxes.Charge Conservation Continuity Condition. The continuity condition asso-ciated with charge conservation can be derived by applying the integral law to thesame pillbox-shaped volume used to derive Gauss continuity condition, (1.3.17). Itcan also be found by simply recognizing the similarity between the integral laws ofGauss and charge conservation. To make this similarity clear, rewrite (2) puttingthe time derivative under the integral. In doing so, d/dt must again be replaced by/t, because the time derivative now operates on , a function of t and r.

SJ da +

VtdV = 0 (11)Comparison of (11) with Gauss integral law, (1.3.1), shows the similarity. The roleof oE in Gauss law is played by J, while that of is taken by /t. Hence,by analogy with the continuity condition for Gauss law, (1.3.17), the continuitycondition for charge conservation is3Note that if we were to introduce the charged ball without having the spherical electrodeessentially grounded through the resistance R, charge conservation (again applied to the surfaceS) would require that the electrode retain charge neutrality. This would mean that there wouldbe a charge q on the outside of the electrode and hence a eld between the electrode and thesurrounding shield. With the charge at the center and the shield concentric with the electrode,this outside eld would be the same as in the absence of the electrode, namely the eld of a pointcharge, (1.3.12).32 Maxwells Integral Laws in Free Space Chapter 1Fig. 1.6.1 Integration line for denition of electromotive force.n (JaJb) + st = 0(12)Implicit in this condition is the assumption that J is nite. Thus, the conditiondoes not include the possibility of a surface current.1.6 FARADAYS INTEGRAL LAWThe laws of Gauss and Amp`ere relate elds to sources. The statement of chargeconservation implied by these two laws relates these sources. Thus, the previousthree sections either relate elds to their sources or interrelate the sources. In thisand the next section, integral laws are introduced that do not involve the chargeand current densities.Faradays integral law states that the circulation of E around a contour Cis determined by the time rate of change of the magnetic ux linking the surfaceenclosed by that contour (the magnetic induction).

CE ds = ddt

SoH da(1)As in Amp`eres integral law and Fig. 1.4.1, the right-hand rule relates ds andda.The electromotive force, or EMF, between points (a) and (b) along the pathP shown in Fig. 1.6.1 is dened asEab =

(b)(a)E ds (2)We will accept this denition for now and look forward to a careful development ofthe circumstances under which the EMF is measured as a voltage in Chaps. 4 and10.Electric Field Intensity with No Circulation. First, suppose that the timerate of change of the magnetic ux is negligible, so that the electric eld is essentiallySec. 1.6 Faradays Integral Law 33Fig. 1.6.2 Uniform electric eld intensity Eo, between plane paralleluniform distributions of surface charge density, has no circulation aboutcontours C1 and C2.free of circulation. This means that no matter what closed contour C is chosen, theline integral of E must vanish.

CE ds = 0 (3)We will nd that this condition prevails in electroquasistatic systems and that allof the elds in Sec. 1.3 satisfy this requirement.Illustration. A Field Having No CirculationA static eld between plane parallel sheets of uniform charge density has no circu-lation. Such a eld, E = Eoix, exists in the region 0 < y < s between the sheets ofsurface charge density shown in Fig. 1.6.2. The most convenient contour for testingthis claim is denoted C1 in Fig. 1.6.2.Along path 1, E ds = Eody, and integration from y = 0 to y = s gives sEo forthe EMF of point (a) relative to point (b). Note that the EMF between the planeparallel surfaces in Fig. 1.6.2 is the same regardless of where the points (a) and (b)are located in the respective surfaces.On segments 2 and 4, E is orthogonal to ds, so there is no contribution tothe line integral on these two sections. Because ds has a direction opposite to E onsegment 3, the line integral is the integral from y = 0 to y = s of E ds = Eody.The result of this integration is sEo, so the contributions from segments 1 and 3cancel, and the circulation around the closed contour is indeed zero.4In this planar geometry, a eld that has only a y component cannot be afunction of x without incurring a circulation. This is evident from carrying out thisintegration for such a eld on the rectangular contour C1. Contributions to paths 1and 3 cancel only if E is independent of x.Example 1.6.1. Contour IntegrationTo gain some appreciation for what it means to require of E that it have no circu-lation, no matter what contour is chosen, consider the somewhat more complicatedcontour C2 in the uniform eld region of Fig. 1.6.2. Here, C2 is composed of the4In setting up the line integral on a contour such as 3, which has a direction opposite to thatin which the coordinate increases, it is tempting to double-account for the direction of ds not onlybe recognizing that ds = iydy, but by integrating from y = s to y = 0 as well.34 Maxwells Integral Laws in Free Space Chapter 1semicircle (5) and the straight segment (6). On the latter, E is perpendicular to dsand so there is no contribution there to the circulation.

CE ds =

5E ds +

6E ds =

5E ds (4)On segment 5, the vector dierential ds is rst written in terms of the unit vectori, and that vector is in turn written (with the help of the vector decompositionshown in the gure) in terms of the Cartesian unit vectors.ds = iRd; i = iy cos ix sin (5)It follows that on the segment 5 of contour C2E ds = Eo cos Rd (6)and integration gives

CE ds =

0Eo cos Rd = [EoRsin ]0 = 0 (7)So for contour C2, the circulation of E is also zero.When the electromotive force between two points is path independent, we callit the voltage between the two points. For a eld having no circulation, the EMFmust be independent of path. This we will recognize formally in Chap. 4.Electric Field Intensity with Circulation. The second limiting situation,typical of the magnetoquasistatic systems to be considered, is primarily concernedwith the circulation of E, and hence with the part of the electric eld generated bythe time-varying magnetic ux density. The remarkable fact is that Faradays lawholds for any contour, whether in free space or in a material. Often, however, thecontour of interest coincides with a conducting wire, which comprises a coil thatlinks a magnetic ux density.Illustration. Terminal EMF of a CoilA coil with one turn is shown in Fig. 1.6.3. Contour (1) is inside the wire, while(2) joins the terminals along a dened path. With these contours constituting C,Faradays integral law as given by (1) determines the terminal electromotive force.If the electrical resistance of the wire can be regarded as zero, in the sense thatthe electric eld intensity inside the wire is negligible, the contour integral reducesto an integration from (b) to (a).5In view of the denition of the EMF, (2), thisintegration gives the negative of the EMF. Thus, Faradays law gives the terminalEMF asEab = ddtf; f

SoH da (8)5With the objectives here limited to attaching an intuitive meaning to Faradays law, wewill give careful attention to the conditions required for this terminal relation to hold in Chaps.8, 9, and 10.Sec. 1.6 Faradays Integral Law 35Fig. 1.6.3 Line segment (1) through a perfectly conducting wire and(2) joining the terminals (a) and (b) form closed contour.Fig. 1.6.4 Demonstration of voltmeter reading induced at terminals ofa coil in accordance with Faradays law. To plot data on graph, normalizevoltage to Vo as dened with (11). Because I is the peak current, v isthe peak voltage.where f, the total ux of magnetic eld linking the coil, is dened as the uxlinkage. Note that Faradays law makes it possible to measure oH electrically (asnow demonstrated).Demonstration 1.6.1. Voltmeter Reading Induced by Magnetic InductionThe rectangular coil shown in Fig. 1.6.4 is used to measure the magnetic eldintensity associated with current in a wire. Thus, the arrangement and eld are thesame as in Demonstration 1.4.1. The height and length of the coil are h and l asshown, and because the coil has N turns, i