Chapter 1 Introductory concepts 1.1 Notation, conventions, and symbology Any book that covers a broad range of topics will likely harbor some problems with notation and symbology. This results from having the same symbol used in di?erent areas to represent di?erent quantities, and also from having too many quantities to represent. Rather than invent new symbols, we choose to stay close to the standards and warn the reader about any symbol used to represent more than one distinct quantity. The basic nature of a physical quantity is indicated by typeface or by the use of a diacritical mark. Scalars are shown in ordinary typeface: q,Phi1, for example. Vectors are shown in boldface: E,Π. Dyadics are shown in boldface with an overbar: ˉepsilon1, ˉ A. Frequency dependent quantities are indicated by a tilde, whereas time dependent quan- tities are written without additional indication; thus we write ? E(r,ω)and E(r,t). (Some quantities, such as impedance, are used in the frequency domain to interrelate Fourier spectra; although these quantities are frequency dependent they are seldom written in the time domain, and hence we do not attach tildes to their symbols.) We often combine diacritical marks:for example, ? ˉepsilon1 denotes a frequency domain dyadic. We distinguish carefully between phasor and frequency domain quantities. The variable ω is used for the frequency variable of the Fourier spectrum, while ˇω is used to indicate the constant frequency of a time harmonic signal. We thus further separate the notion of a phasor ?eld from a frequency domain ?eld by using a check to indicate a phasor ?eld: ˇ E(r). However, there is often a simple relationship between the two, such as ˇ E = ? E( ˇω). We designate the ?eld and source point position vectors by r and r prime , respectively, and the corresponding relative displacement or distance vector by R: R = r ? r prime . A hat designates a vector as a unit vector (e.g., ?x). The sets of coordinate variables in rectangular, cylindrical, and spherical coordinates are denoted by (x, y, z), (ρ,φ, z), (r,θ,φ), respectively. (In the spherical system φ is the azimuthal angle and θ is the polar angle.) We freely use the “del” operator notation ? for gradient, curl, divergence, Laplacian, and so on. The SI (MKS) system of units is employed throughout the book. 1.2 The ?eld concept of electromagnetics Introductory treatments of electromagnetics often stress the role of the ?eld in force transmission:the individual ?elds E and B are de?ned via the mechanical force on a small test charge. This is certainly acceptable, but does not tell the whole story. We might, for example, be left with the impression that the EM ?eld always arises from an interaction between charged objects. Often coupled with this is the notion that the ?eld concept is meant merely as an aid to the calculation of force, a kind of notational convenience not placed on the same physical footing as force itself. In fact, ?elds are more than useful — they are fundamental. Before discussing electromagnetic ?elds in more detail, let us attempt to gain a better perspective on the ?eld concept and its role in modern physical theory. Fields play a central role in any attempt to describe physical reality. They are as real as the physical substances we ascribe to everyday experience. In the words of Einstein [63], “It seems impossible to give an obvious qualitative criterion for distinguishing between matter and ?eld or charge and ?eld.” We must therefore put ?elds and particles of matter on the same footing:both carry energy and momentum, and both interact with the observable world. 1.2.1 Historical perspective Early nineteenth century physical thought was dominated by the action at a distance concept, formulated by Newton more than 100 years earlier in his immensely successful theory of gravitation. In this view the in?uence of individual bodies extends across space, instantaneously a?ects other bodies, and remains completely una?ected by the presence of an intervening medium. Such an idea was revolutionary; until thenactionbycontact,in which objects are thought to a?ect each other through physical contact or by contact with the intervening medium, seemed the obvious and only means for mechanical interaction. Priestly’s experiments in 1766 and Coulomb’s torsion-bar experiments in 1785 seemed to indicate that the force between two electrically charged objects behaves in strict analogy with gravitation:both forces obey inverse square laws and act along a line joining the objects. Oersted, Ampere, Biot, and Savart soon showed that the magnetic force on segments of current-carrying wires also obeys an inverse square law. The experiments of Faraday in the 1830s placed doubt on whether action at a distance really describes electric and magnetic phenomena. When a material (such as a dielec- tric) is placed between two charged objects, the force of interaction decreases; thus, the intervening medium does play a role in conveying the force from one object to the other. To explain this, Faraday visualized “lines of force” extending from one charged object to another. The manner in which these lines were thought to interact with materials they intercepted along their path was crucial in understanding the forces on the objects. This also held for magnetic e?ects. Of particular importance was the number of lines passing through a certain area (the ?ux), which was thought to determine the amplitude of the e?ect observed in Faraday’s experiments on electromagnetic induction. Faraday’s ideas presented a new world view:electromagnetic phenomena occur in the region surrounding charged bodies, and can be described in terms of the laws governing the “?eld” of his lines of force. Analogies were made to the stresses and strains in material objects, and it appeared that Faraday’s force lines created equivalent electromagnetic stresses and strains in media surrounding charged objects. His law of induction was formulated not in terms of positions of bodies, but in terms of lines of magnetic force. Inspired by Faraday’s ideas, Gauss restated Coulomb’s law in terms of ?ux lines, and Maxwell extended the idea to time changing ?elds through his concept of displacement current. In the 1860s Maxwell created what Einstein called “the most important invention since Newton’s time”— a set of equations describing an entirely ?eld-based theory of electromagnetism. These equations do not model the forces acting between bodies, as do Newton’s law of gravitation and Coulomb’s law, but rather describe only the dynamic, time-evolving structure of the electromagnetic ?eld. Thus bodies are not seen to inter- act with each other, but rather with the (very real) electromagnetic ?eld they create, an interaction described by a supplementary equation (the Lorentz force law). To bet- ter understand the interactions in terms of mechanical concepts, Maxwell also assigned properties of stress and energy to the ?eld. Using constructs that we now call the electric and magnetic ?elds and potentials, Maxwell synthesized all known electromagnetic laws and presented them as a system of di?erential and algebraic equations. By the end of the nineteenth century, Hertz had devised equations involving only the electric and magnetic ?elds, and had derived the laws of circuit theory (Ohm’s law and Kircho?’s laws) from the ?eld expressions. His experiments with high-frequency ?elds veri?ed Maxwell’s predictions of the existence of electromagnetic waves propagating at ?nite velocity, and helped solidify the link between electromagnetism and optics. But one problem remained:if the electromagnetic ?elds propagated by stresses and strains on a medium, how could they propagate through a vacuum? A substance called the luminiferous aether, long thought to support the trans- verse waves of light, was put to the task of carrying the vibrations of the electromagnetic ?eld as well. However, the pivotal experiments of Michelson and Morely showed that the aether was ?ctitious, and the physical existence of the ?eld was ?rmly established. The essence of the ?eld concept can be conveyed through a simple thought experiment. Consider two stationary charged particles in free space. Since the charges are stationary, we know that (1) another force is present to balance the Coulomb force between the charges, and (2) the momentum and kinetic energy of the system are zero. Now suppose one charge is quickly moved and returned to rest at its original position. Action at a distance would require the second charge to react immediately (Newton’s third law), but by Hertz’s experiments it does not. There appears to be no change in energy of the system:both particles are again at rest in their original positions. However, after a time (given by the distance between the charges divided by the speed of light) we ?nd that the second charge does experience a change in electrical force and begins to move away from its state of equilibrium. But by doing so it has gained net kinetic energy and momentum, and the energy and momentum of the system seem larger than at the start. This can only be reconciled through ?eld theory. If we regard the ?eld as a physical entity, then the nonzero work required to initiate the motion of the ?rst charge and return it to its initial state can be seen as increasing the energy of the ?eld. A disturbance propagates at ?nite speed and, upon reaching the second charge, transfers energy into kinetic energy of the charge. Upon its acceleration this charge also sends out a wave of ?eld disturbance, carrying energy with it, eventually reaching the ?rst charge and creating a second reaction. At any given time, the net energy and momentum of the system, composed of both the bodies and the ?eld, remain constant. We thus come to regard the electromagnetic ?eld as a true physical entity:an entity capable of carrying energy and momentum. 1.2.2 Formalization of ?eld theory Before we can invoke physical laws, we must ?nd a way to describe the state of the system we intend to study. We generally begin by identifying a set of state variables that can depict the physical nature of the system. In a mechanical theory such as Newton’s law of gravitation, the state of a system of point masses is expressed in terms of the instantaneous positions and momenta of the individual particles. Hence 6N state variables are needed to describe the state of a system of N particles, each particle having three position coordinates and three momentum components. The time evolution of the system state is determined by a supplementary force function (e.g., gravitational attraction), the initial state (initial conditions), and Newton’s second law F = dP/dt. Descriptions using ?nite sets of state variables are appropriate for action-at-a-distance interpretations of physical laws such as Newton’s law of gravitation or the interaction of charged particles. If Coulomb’s law were taken as the force law in a mechanical description of electromagnetics, the state of a system of particles could be described completely in terms of their positions, momenta, and charges. Of course, charged particle interaction is not this simple. An attempt to augment Coulomb’s force law with Ampere’s force law would not account for kinetic energy loss via radiation. Hence we abandon 1 the mechanical viewpoint in favor of the ?eld viewpoint, selecting a di?erent set of state variables. The essence of ?eld theory is to regard electromagnetic phenomena as a?ecting all of space. We shall ?nd that we can describe the ?eld in terms of the four vector quantities E, D, B, and H. Because these ?elds exist by de?nition at each point in space and each time t, a ?nite set of state variables cannot describe the system. Here then is an important distinction between ?eld theories and mechanical theories: the state of a ?eld at any instant can only be described by an in?nite number of state variables. Mathematically we describe ?elds in terms of functions of continuous variables; however, we must be careful not to confuse all quantities described as “?elds” with those ?elds innate to a scienti?c ?eld theory. For instance, we may refer to a temperature “?eld” in the sense that we can describe temperature as a function of space and time. However, we do not mean by this that temperature obeys a set of physical laws analogous to those obeyed by the electromagnetic ?eld. What special character, then, can we ascribe to the electromagnetic ?eld that has meaning beyond that given by its mathematical implications? In this book, E, D, B, and H are integral parts of a ?eld-theory description of electromagnetics. In any ?eld theory we need two types of ?elds:a mediating ?eld generated by a source, and a ?eld describing the source itself. In free-space electromagnetics the mediating ?eld consists of E and B, while the source ?eld is the distribution of charge or current. An important consideration is that the source ?eld must be independent of the mediating ?eld that it “sources.” Additionally, ?elds are generally regarded as unobservable:they can only be measured indirectly through interactions with observable quantities. We need a link to mechanics to observe E and B:we might measure the change in kinetic energy of a particle as it interacts with the ?eld through the Lorentz force. The Lorentz force becomes the force function in the mechanical interaction that uniquely determines the (observable) mechanical state of the particle. A ?eld is associated with a set of ?eldequations and a set of constitutiverelations. The ?eld equations describe, through partial derivative operations, both the spatial distribu- tion and temporal evolution of the ?eld. The constitutive relations describe the e?ect 1 Attempts have been made to formulate electromagnetic theory purely in action-at-a-distance terms, but this viewpoint has not been generally adopted [69]. of the supporting medium on the ?elds and are dependent upon the physical state of the medium. The state may include macroscopic e?ects, such as mechanical stress and thermodynamic temperature, as well as the microscopic, quantum-mechanical properties of matter. The value of the ?eld at any position and time in a bounded region V is then determined uniquely by specifying the sources within V , the initial state of the ?elds within V , and the value of the ?eld or ?nitely many of its derivatives on the surface bounding V .If the boundary surface also de?nes a surface of discontinuity between adjacent regions of di?ering physical characteristics, or across discontinuous sources, then jump conditions may be used to relate the ?elds on either side of the surface. The variety of forms of ?eld equations is restricted by many physical principles in- cluding reference-frame invariance, conservation, causality, symmetry, and simplicity. Causality prevents the ?eld at time t = 0 from being in?uenced by events occurring at subsequent times t > 0. Of course, we prefer that a ?eld equation be mathematically robust and well-posed to permit solutions that are unique and stable. Many of these ideas are well illustrated by a consideration of electrostatics. We can describe the electrostatic ?eld through a mediating scalar ?eld Phi1(x, y, z) known as the electrostatic potential. The spatial distribution of the ?eld is governed by Poisson’s equation ? 2 Phi1 ?x 2 + ? 2 Phi1 ?y 2 + ? 2 Phi1 ?z 2 =? ρ epsilon1 0 ,θ where ρ = ρ(x, y, z) is the source charge density. No temporal derivatives appear, and the spatial derivatives determine the spatial behavior of the ?eld. The function ρ represents the spatially-averaged distribution of charge that acts as the source term for the ?eld Phi1. Note that ρ incorporates no information about Phi1. To uniquely specify the ?eld at any point, we must still specify its behavior over a boundary surface. We could, for instance, specify Phi1 on ?ve of the six faces of a cube and the normal derivative ?Phi1/?n on the remaining face. Finally, we cannot directly observe the static potential ?eld, but we can observe its interaction with a particle. We relate the static potential ?eld theory to the realm of mechanics via the electrostatic force F = qE acting on a particle of charge q. In future chapters we shall present a classical ?eld theory for macroscopic electromag- netics. In that case the mediating ?eld quantities are E, D, B, and H, and the source ?eld is the current density J. 1.3 The sources of the electromagnetic ?eld Electric charge is an intriguing natural entity. Human awareness of charge and its e?ects dates back to at least 600 BC, when the Greek philosopher Thales of Miletus observed that rubbing a piece of amber could enable the amber to attract bits of straw. Although charging by friction is probably still the most common and familiar manifes- tation of electric charge, systematic experimentation has revealed much more about the behavior of charge and its role in the physical universe. There are two kinds of charge, to which Benjamin Franklin assigned the respective names positive and negative. Franklin observed that charges of opposite kind attract and charges of the same kind repel. He also found that an increase in one kind of charge is accompanied by an increase in the other, and so ?rst described the principle of charge conservation. Twentieth century physics has added dramatically to the understanding of charge: 1. Electric charge is a fundamental property of matter, as is mass or dimension. 2. Charge is quantized:there exists a smallest quantity ( quantum) of charge that can be associated with matter. No smaller amount has been observed, and larger amounts always occur in integral multiples of this quantity. 3. The charge quantum is associated with the smallest subatomic particles, and these particles interact through electrical forces. In fact, matter is organized and arranged through electrical interactions; for example, our perception of physical contact is merely the macroscopic manifestation of countless charges in our ?ngertips pushing against charges in the things we touch. 4. Electric charge is an invariant:the value of charge on a particle does not depend on the speed of the particle. In contrast, the mass of a particle increases with speed. 5. Charge acts as the source of an electromagnetic ?eld; the ?eld is an entity that can carry energy and momentum away from the charge via propagating waves. We begin our investigation of the properties of the electromagnetic ?eld with a detailed examination of its source. 1.3.1 Macroscopic electromagnetics We are interested primarily in those electromagnetic e?ects that can be predicted by classical techniques using continuous sources (charge and current densities). Although macroscopic electromagnetics is limited in scope, it is useful in many situations en- countered by engineers. These include, for example, the determination of currents and voltages in lumped circuits, torques exerted by electrical machines, and ?elds radiated by antennas. Macroscopic predictions can fall short in cases where quantum e?ects are im- portant:e.g., with devices such as tunnel diodes. Even so, quantum mechanics can often be coupled with classical electromagnetics to determine the macroscopic electromagnetic properties of important materials. Electric charge is not of a continuous nature. The quantization of atomic charge — ±e for electrons and protons, ±e/3 and ±2e/3 for quarks — is one of the most precisely established principles in physics (veri?ed to 1 part in 10 21 ). The value of e itself is known to great accuracy: e = 1.60217733 × 10 ?19 Coulombs (C). However, the discrete nature of charge is not easily incorporated into everyday engineer- ing concerns. The strange world of the individual charge — characterized by particle spin, molecular moments, and thermal vibrations — is well described only by quantum theory. There is little hope that we can learn to describe electrical machines using such concepts. Must we therefore retreat to the macroscopic idea and ignore the discretization of charge completely? A viable alternative is to use atomic theories of matter to estimate the useful scope of macroscopic electromagnetics. Remember, we are completely free to postulate a theory of nature whose scope may be limited. Like continuum mechanics, which treats distributions of matter as if they were continuous, macroscopic electromagnetics is regarded as valid because it is veri?ed by experiment over a certain range of conditions. This applicability range generally corresponds to dimensions on a laboratory scale, implying a very wide range of validity for engineers. Macroscopic e?ects as averaged microscopic e?ects. Macroscopic electromag- netics can hold in a world of discrete charges because applications usually occur over physical scales that include vast numbers of charges. Common devices, generally much larger than individual particles, “average” the rapidly varying ?elds that exist in the spaces between charges, and this allows us to view a source as a continuous “smear” of charge. To determine the range of scales over which the macroscopic viewpoint is valid, we must compare averaged values of microscopic ?elds to the macroscopic ?elds we mea- sure in the lab. But if the e?ects of the individual charges are describable only in terms of quantum notions, this task will be daunting at best. A simple compromise, which produces useful results, is to extend the macroscopic theory right down to the micro- scopic level and regard discrete charges as “point” entities that produce electromagnetic ?elds according to Maxwell’s equations. Then, in terms of scales much larger than the classical radius of an electron (≈ 10 ?14 m), the expected rapid ?uctuations of the ?elds in the spaces between charges is predicted. Finally, we ask:over what spatial scale must we average the e?ects of the ?elds and the sources in order to obtain agreement with the macroscopic equations? In the spatial averaging approach a convenient weighting function f (r) is chosen, and is normalized so that integraltext f (r)dV = 1. An example is the Gaussian distribution f (r) = (πa 2 ) ?3/2 e ?r 2 /a 2 , where a is the approximate radial extent of averaging. The spatial average of a micro- scopic quantity F(r,t) is given by 〈F(r,t)〉= integraldisplay F(r ? r prime ,t) f (r prime )dV prime . (1.1) The scale of validity of the macroscopic model can be found by determining the averaging radius a that produces good agreement between the averaged microscopic ?elds and the macroscopic ?elds. The macroscopic volume charge density. At this point we do not distinguish between the “free” charge that is unattached to a molecular structure and the charge found near the surface of a conductor. Nor do we consider the dipole nature of polarizable materials or the microscopic motion associated with molecular magnetic moment or the magnetic moment of free charge. For the consideration of free-space electromagnetics, we assume charge exhibits either three degrees of freedom (volume charge), two degrees of freedom (surface charge), or one degree of freedom (line charge). In typical matter, the microscopic ?elds vary spatially over dimensions of 10 ?10 m or less, and temporally over periods (determined by atomic motion) of 10 ?13 s or less. At the surface of a material such as a good conductor where charge often concentrates, averaging with a radius on the order of 10 ?10 m may be required to resolve the rapid variation in the distribution of individual charged particles. However, within a solid or liquid material, or within a free-charge distribution characteristic of a dense gas or an electron beam, a radius of 10 ?8 m proves useful, containing typically 10 6 particles. A di?use gas, on the other hand, may have a particle density so low that the averaging radius takes on laboratory dimensions, and in such a case the microscopic theory must be employed even at macroscopic dimensions. Once the averaging radius has been determined, the value of the charge density may be found via (1.1). The volume density of charge for an assortment of point sources can be written in terms of the three-dimensional Dirac delta as ρ o (r,t) = summationdisplay i q i δ(r ? r i (t)), where r i (t) is the position of the charge q i at time t. Substitution into (1.1) gives ρ(r,t) =〈ρ o (r,t)〉= summationdisplay i q i f (r ? r i (t)) (1.2) as the averaged charge density appropriate for use in a macroscopic ?eld theory. Because the oscillations of the atomic particles are statistically uncorrelated over the distances used in spatial averaging, the time variations of microscopic ?elds are not present in the macroscopic ?elds and temporal averaging is unnecessary. In (1.2) the time dependence of the spatially-averaged charge density is due entirely to bulk motion of the charge aggregate (macroscopic charge motion). With the de?nition of macroscopic charge density given by (1.2), we can determine the total charge Q(t) in any macroscopic volume region V using Q(t) = integraldisplay V ρ(r,t)dV. (1.3) We have Q(t) = summationdisplay i q i integraldisplay V f (r ? r i (t))dV = summationdisplay r i (t)∈V q i . Here we ignore the small discrepancy produced by charges lying within distance a of the boundary of V . It is common to employ a box B having volume Delta1V : braceleftbigg f (r) = 1/Delta1V, r ∈ B, 0, r /∈ B. In this case ρ(r,t) = 1 Delta1V summationdisplay r?r i (t)∈B q i . The size of B is chosen with the same considerations as to atomic scale as was the averaging radius a. Discontinuities at the edges of the box introduce some di?culties concerning charges that move in and out of the box because of molecular motion. The macroscopic volume current density. Electric charge in motion is referred to as electric current. Charge motion can be associated with external forces and with microscopic ?uctuations in position. Assuming charge q i has velocity v i (t) = dr i (t)/dt, the charge aggregate has volume current density J o (r,t) = summationdisplay i q i v i (t)δ(r ? r i (t)). Spatial averaging gives the macroscopic volume current density J(r,t) =〈J o (r,t)〉= summationdisplay i q i v i (t) f (r ? r i (t)). (1.4) Figure 1.1:Intersection of the averaging function of a point charge with a surface S,as the charge crosses S with velocity v:(a) at some time t = t 1 , and (b) at t = t 2 > t 1 . The averaging function is represented by a sphere of radius a. Spatial averaging at time t eliminates currents associated with microscopic motions that are uncorrelated at the scale of the averaging radius (again, we do not consider the magnetic moments of particles). The assumption of a su?ciently large averaging radius leads to J(r,t) = ρ(r,t)v(r,t). (1.5) The total ?ux I(t) of current through a surface S is given by I(t) = integraldisplay S J(r,t)· ?n dS where ?n is the unit normal to S. Hence, using (4), we have I(t) = summationdisplay i q i d dt (r i (t)· ?n) integraldisplay S f (r ? r i (t))dS if ?n stays approximately constant over the extent of the averaging function and S is not in motion. We see that the integral e?ectively intersects S with the averaging function sur- roundingeachmovingpointcharge (Figure1.1).Thetimederivativeof r i · ?n represents the velocity at which the averaging function is “carried across” the surface. Electric current takes a variety of forms, each described by the relation J = ρv. Isolated charged particles (positive and negative) and charged insulated bodies moving through space comprise convection currents. Negatively-charged electrons moving through the positive background lattice within a conductor comprise a conductioncurrent. Empirical evidence suggests that conduction currents are also described by the relation J = σE known as Ohm’slaw. A third type of current, called electrolyticcurrent, results from the ?ow of positive or negative ions through a ?uid. 1.3.2 Impressed vs. secondary sources In addition to the simple classi?cation given above we may classify currents as primary or secondary, depending on the action that sets the charge in motion. It is helpful to separate primary or “impressed” sources, which are independent of the ?elds they source, from secondary sources which result from interactions between the sourced ?elds and the medium in which the ?elds exist. Most familiar is the conduc- tion current set up in a conducting medium by an externally applied electric ?eld. The impressed source concept is particularly important in circuit theory, where independent voltage sources are modeled as providing primary voltage excitations that are indepen- dent of applied load. In this way they di?er from the secondary or “dependent” sources that react to the e?ect produced by the application of primary sources. In applied electromagnetics the primary source may be so distant that return e?ects resulting from local interaction of its impressed ?elds can be ignored. Other examples of primary sources include the applied voltage at the input of an antenna, the current on a probe inserted into a waveguide, and the currents producing a power-line ?eld in which a biological body is immersed. 1.3.3 Surface and line source densities Because they are spatially averaged e?ects, macroscopic sources and the ?elds they source cannot have true spatial discontinuities. However, it is often convenient to work with sources in one or two dimensions. Surface and line source densities are idealizations of actual, continuous macroscopic densities. The entity we describe as a surface charge is a continuous volume charge distributed in a thin layer across some surface S. If the thickness of the layer is small compared to laboratory dimensions, it is useful to assign to each point r on the surface a quantity describing the amount of charge contained within a cylinder oriented normal to the surface and having in?nitesimal cross section dS. We call this quantity the surface charge density ρ s (r,t), and write the volume charge density as ρ(r,w,t) = ρ s (r,t) f (w,Delta1), where w is distance from S in the normal direction and Delta1 in some way parameterizes the “thickness” of the charge layer at r. The continuous density function f (x,Delta1)satis?es integraldisplay ∞ ?∞ f (x,Delta1)dx = 1 and lim Delta1→0 f (x,Delta1)= δ(x). For instance, we might have f (x,Delta1)= e ?x 2 /Delta1 2 Delta1 √ π . (1.6) With this de?nition the total charge contained in a cylinder normal to the surface at r and having cross-sectional area dS is dQ(t) = integraldisplay ∞ ?∞ [ρ s (r,t)dS] f (w,Delta1)dw = ρ s (r,t)dS, and the total charge contained within any cylinder oriented normal to S is Q(t) = integraldisplay S ρ s (r,t)dS. (1.7) We may describe a line charge as a thin “tube” of volume charge distributed along some contour Gamma1. The amount of charge contained between two planes normal to the contour and separated by a distance dl is described by the line charge density ρ l (r,t). The volume charge density associated with the contour is then ρ(r,ρ,t) = ρ l (r,t) f s (ρ,Delta1), where ρ is the radial distance from the contour in the plane normal to Gamma1 and f s (ρ,Delta1) is a density function with the properties integraldisplay ∞ 0 f s (ρ,Delta1)2πρ dρ = 1 and lim Delta1→0 f s (ρ,Delta1) = δ(ρ) 2πρ . For example, we might have f s (ρ,Delta1) = e ?ρ 2 /Delta1 2 πDelta1 2 . (1.8) Then the total charge contained between planes separated by a distance dl is dQ(t) = integraldisplay ∞ 0 [ρ l (r,t)dl] f s (ρ,Delta1)2πρ dρ = ρ l (r,t)dl and the total charge contained between planes placed at the ends of a contour Gamma1 is Q(t) = integraldisplay Gamma1 ρ l (r,t)dl. (1.9) We may de?ne surface and line currents similarly. A surface current is merely a volume current con?ned to the vicinity of a surface S. The volume current density may be represented using a surface current density function J s (r,t), de?ned at each point r on the surface so that J(r,w,t) = J s (r,t) f (w,Delta1). Here f (w,Delta1) is some appropriate density function such as (1.6), and the surface current vector obeys ?n·J s = 0 where ?n is normal to S. The total current ?owing through a strip of width dl arranged perpendicular to S at r is dI(t) = integraldisplay ∞ ?∞ [J s (r,t)· ?n l (r)dl] f (w,Delta1)dw = J s (r,t)· ?n l (r)dl where ?n l is normal to the strip at r (and thus also tangential to S at r). The total current passing through a strip intersecting with S along a contour Gamma1 is thus I(t) = integraldisplay Gamma1 J s (r,t)· ?n l (r)dl. We may describe a line current as a thin “tube” of volume current distributed about some contour Gamma1 and ?owing parallel to it. The amount of current passing through a plane normal to the contour is described by the linecurrentdensity J l (r,t). The volume current density associated with the contour may be written as J(r,ρ,t) = ?u(r)J l (r,t) f s (ρ,Delta1), where ?u is a unit vector along Gamma1, ρ is the radial distance from the contour in the plane normal to Gamma1, and f s (ρ,Delta1) is a density function such as (1.8). The total current passing through any plane normal to Gamma1 at r is I(t) = integraldisplay ∞ 0 [J l (r,t)?u(r)· ?u(r)] f s (ρ,Delta1)2πρ dρ = J l (r,t). It is often convenient to employ singular models for continuous source densities. For instance, it is mathematically simpler to regard a surface charge as residing only in the surface S than to regard it as being distributed about the surface. Of course, the source is then discontinuous since it is zero everywhere outside the surface. We may obtain a representation of such a charge distribution by letting the thickness parameter Delta1 in the density functions recede to zero, thus concentrating the source into a plane or a line. We describe the limit of the density function in terms of the δ-function. For instance, the volume charge distribution for a surface charge located about the xy-plane is ρ(x, y, z,t) = ρ s (x, y,t) f (z,Delta1). As Delta1 → 0 we have ρ(x, y, z,t) = ρ s (x, y,t) lim Delta1→0 f (z,Delta1)= ρ s (x, y,t)δ(z). It is a simple matter to represent singular source densities in this way as long as the surface or line is easily parameterized in terms of constant values of coordinate variables. However, care must be taken to represent the δ-function properly. For instance, the density of charge on the surface of a cone at θ = θ 0 may be described using the distance normal to this surface, which is given by rθ ? rθ 0 : ρ(r,θ,φ,t) = ρ s (r,φ,t)δ(r[θ ?θ 0 ]). Using the property δ(ax) = δ(x)/a, we can also write this as ρ(r,θ,φ,t) = ρ s (r,φ,t) δ(θ ?θ 0 ) r . 1.3.4 Charge conservation There are four fundamental conservation laws in physics:conservation of energy, mo- mentum, angular momentum, and charge. These laws are said to be absolute; they have never been observed to fail. In that sense they are true empirical laws of physics. However, in modern physics the fundamental conservation laws have come to represent more than just observed facts. Each law is now associated with a fundamental symme- try of the universe; conversely, each known symmetry is associated with a conservation principle. For example, energy conservation can be shown to arise from the observation that the universe is symmetric with respect to time; the laws of physics do not depend on choice of time origin t = 0. Similarly, momentum conservation arises from the obser- vation that the laws of physics are invariant under translation, while angular momentum conservation arises from invariance under rotation. The law of conservation of charge also arises from a symmetry principle. But instead of being spatial or temporal in character, it is related to the invariance of electrostatic potential. Experiments show that there is no absolute potential, only potential di?erence. The laws of nature are invariant with respect to what we choose as the “reference” potential. This in turn is related to the invariance of Maxwell’s equations under gauge transforms; the values of the electric and magnetic ?elds do not depend on which gauge transformation we use to relate the scalar potential Phi1 to the vector potential A. We may state the conservation of charge as follows: The net charge in any closed system remains constant with time. This does not mean that individual charges cannot be created or destroyed, only that the total charge in any isolated system must remain constant. Thus it is possible for a positron with charge e to annihilate an electron with charge ?e without changing the net charge of the system. Only if a system is not closed can its net charge be altered; since moving charge constitutes current, we can say that the total charge within a system depends on the current passing through the surface enclosing the system. This is the essence of the continuity equation. To derive this important result we consider a closed system within which the charge remains constant, and apply the Reynolds transport theorem (see § A.2). The continuity equation. Consider a region of space occupied by a distribution of charge whose velocity is given by the vector ?eld v. We surround a portion of charge by a surface S and let S deform as necessary to “follow” the charge as it moves. Since S always contains precisely the same charged particles, we have an isolated system for which the time rate of change of total charge must vanish. An expression for the time rate of change is given by the Reynolds transport theorem (A.66); we have 2 DQ Dt = D Dt integraldisplay V(t) ρ dV = integraldisplay V(t) ?ρ ?t dV + contintegraldisplay S(t) ρv · dS = 0. The “D/Dt” notation indicates that the volume region V(t) moves with its enclosed particles. Since ρv represents current density, we can write integraldisplay V(t) ?ρ(r,t) ?t dV + contintegraldisplay S(t) J(r,t)· dS = 0. (1.10) In this large-scaleform of the continuity equation, the partial derivative term describes the time rate of change of the charge density for a ?xed spatial position r. At any time t, the time rate of change of charge density integrated over a volume is exactly compensated by the total current exiting through the surrounding surface. We can obtain the continuity equation in point form by applying the divergence the- orem to the second term of (1.10) to get integraldisplay V(t) bracketleftbigg ?ρ(r,t) ?t +?·J(r,t) bracketrightbigg dV = 0. Since V(t) is arbitrary we can set the integrand to zero to obtain ?ρ(r,t) ?t +?·J(r,t) = 0. (1.11) 2 Note that in Appendix A we use the symbol u to represent the velocity of a material and v to represent the velocity of an arti?cial surface. This expression involves the time derivative of ρ with r ?xed. We can also ?nd an expression in terms of the material derivative by using the transport equation (A.67). Enforcing conservation of charge by setting that expression to zero, we have Dρ(r,t) Dt +ρ(r,t)?·v(r,t) = 0. (1.12) Here Dρ/Dt is the time rate of change of the charge density experienced by an observer moving with the current. We can state the large-scale form of the continuity equation in terms of a stationary volume. Integrating (1.11) over a stationary volume region V and using the divergence theorem, we ?nd that integraldisplay V ?ρ(r,t) ?t dV =? contintegraldisplay S J(r,t)· dS. Since V is not changing with time we have dQ(t) dt = d dt integraldisplay V ρ(r,t)dV =? contintegraldisplay S J(r,t)· dS. (1.13) Hence any increase of total charge within V must be produced by current entering V through S. Use of the continuity equation. As an example, suppose that in a bounded region of space we have ρ(r,t) = ρ 0 re ?βt . We wish to ?nd J and v, and to verify both versions of the continuity equation in point form. The spherical symmetry of ρ requires that J = ?rJ r . Application of (1.13) over a sphere of radius a gives 4π d dt integraldisplay a 0 ρ 0 re ?βt r 2 dr =?4π J r (a)a 2 . Hence J = ?rβρ 0 r 2 4 e ?βt and therefore ?·J = 1 r 2 ? ?r (r 2 J r ) = βρ 0 re ?βt . The velocity is v = J ρ = ?rβ r 4 , and we have ?·v = 3β/4. To verify the continuity equations, we compute the time derivatives ?ρ ?t =?βρ 0 re ?βt , Dρ Dt = ?ρ ?t + v ·?ρ =?βρ 0 re ?βt + parenleftBig ?rβ r 4 parenrightBig · parenleftbig ?rρ 0 e ?βt parenrightbig =? 3 4 βρ 0 re ?βt . Note that the charge density decreases with time less rapidly for a moving observer than for a stationary one (3/4 as fast):the moving observer is following the charge outward, and ρ ∝ r. Now we can check the continuity equations. First we see Dρ Dt +ρ?·v =? 3 4 βρ 0 re ?βt +(ρ 0 re ?βt ) parenleftbigg 3 4 β parenrightbigg = 0, as required for a moving observer; second we see ?ρ ?t +?·J =?βρ 0 re ?βt +βρ 0 e ?βt = 0, as required for a stationary observer. The continuity equation in fewer dimensions. The continuity equation can also be used to relate current and charge on a surface or along a line. By conservation of charge we can write d dt integraldisplay S ρ s (r,t)dS =? contintegraldisplay Gamma1 J s (r,t)· ?m dl (1.14) where ?m is the vector normal to the curve Gamma1 and tangential to the surface S. By the surface divergence theorem (B.20), the corresponding point form is ?ρ s (r,t) ?t +? s · J s (r,t) = 0. (1.15) Here ? s · J s is the surface divergence of the vector ?eld J s . For instance, in rectangular coordinates in the z = 0 plane we have ? s · J s = ? J sx ?x + ? J sy ?y . In cylindrical coordinates on the cylinder ρ = a, we would have ? s · J s = 1 a ? J sφ ?φ + ? J sz ?z . A detailed description of vector operations on a surface may be found in Tai [190], while many identities may be found in Van Bladel [202]. TheequationofcontinuityforalineiseasilyestablishedbyreferencetoFigure1.2. Here the net charge exiting the surface during time Delta1t is given by Delta1t[I(u 2 ,t)? I(u 1 ,t)]. Thus, the rate of net increase of charge within the system is dQ(t) dt = d dt integraldisplay ρ l (r,t)dl =?[I(u 2 ,t)? I(u 1 ,t)]. (1.16) The corresponding point form is found by letting the length of the curve approach zero: ?I(l,t) ?l + ?ρ l (l,t) ?t = 0, (1.17) where l is arc length along the curve. As an example, suppose the line current on a circular loop antenna is approximately I(φ,t) = I 0 cos parenleftBig ωa c φ parenrightBig cosωt, Figure 1.2:Linear form of the continuityequation. where a is the radius of the loop, ω is the frequency of operation, and c is the speed of light. We wish to ?nd the line charge density on the loop. Since l = aφ, we can write I(l,t) = I 0 cos parenleftbigg ωl c parenrightbigg cosωt. Thus ?I(l,t) ?l =?I 0 ω c sin parenleftbigg ωl c parenrightbigg cosωt =? ?ρ l (l,t) ?t . Integrating with respect to time and ignoring any constant (static) charge, we have ρ(l,t) = I 0 c sin parenleftbigg ωl c parenrightbigg sinωt or ρ(φ,t) = I 0 c sin parenleftBig ωa c φ parenrightBig sinωt. Note that we could have used the chain rule ?I(φ,t) ?l = ?I(φ,t) ?φ ?φ ?l and ?φ ?l = bracketleftbigg ?l ?φ bracketrightbigg ?1 = 1 a to calculate the spatial derivative. We can apply the volume density continuity equation (1.11) directly to surface and line distributions written in singular notation. For the loop of the previous example, we write the volume current density corresponding to the line current as J(r,t) = ? φδ(ρ? a)δ(z)I(φ,t). Substitution into (1.11) then gives ?·[ ? φδ(ρ? a)δ(z)I(φ,t)] =? ?ρ(r,t) ?t . The divergence formula for cylindrical coordinates gives δ(ρ? a)δ(z) ?I(φ,t) ρ?φ =? ?ρ(r,t) ?t . Next we substitute for I(φ,t) to get ? I 0 ρ ωa c sin parenleftBig ωa c φ parenrightBig δ(ρ? a)δ(z)cosωt =? ?ρ(r,t) ?t . Finally, integrating with respect to time and ignoring any constant term, we have ρ(r,t) = I 0 c δ(ρ? a)δ(z)sin parenleftBig ωa c φ parenrightBig sinωt, where we have set ρ = a because of the presence of the factor δ(ρ? a). 1.3.5 Magnetic charge We take for granted that electric ?elds are produced by electric charges, whether stationary or in motion. The smallest element of electric charge is the electric monopole: a single discretely charged particle from which the electric ?eld diverges. In contrast, experiments show that magnetic ?elds are created only by currents or by time changing electric ?elds; hence, magnetic ?elds have moving electric charge as their source. The elemental source of magnetic ?eld is the magnetic dipole, representing a tiny loop of electric current (or a spinning electric particle). The observation made in 1269 by Pierre De Maricourt, that even the smallest magnet has two poles, still holds today. In a world ?lled with symmetry at the fundamental level, we ?nd it hard to understand why there should not be a source from which the magnetic ?eld diverges. We would call such a source magnetic charge, and the most fundamental quantity of magnetic charge would be exhibited by amagneticmonopole. In 1931 Paul Dirac invigorated the search for magnetic monopoles by making the ?rst strong theoretical argument for their existence. Dirac showed that the existence of magnetic monopoles would imply the quantization of electric charge, and would thus provide an explanation for one of the great puzzles of science. Since that time magnetic monopoles have become important players in the “Grand Uni?ed Theories” of modern physics, and in cosmological theories of the origin of the universe. If magnetic monopoles are ever found to exist, there will be both positive and negatively charged particles whose motions will constitute currents. We can de?ne a macroscopic magnetic charge density ρ m and current density J m exactly as we did with electric charge, and use conservation of magnetic charge to provide a continuity equation: ?·J m (r,t)+ ?ρ m (r,t) ?t = 0. (1.18) With these new sources Maxwell’s equations become appealingly symmetric. Despite uncertainties about the existence and physical nature of magnetic monopoles, magnetic charge and current have become an integral part of electromagnetic theory. We often use the concept of ?ctitious magnetic sources to make Maxwell’s equations symmetric, and then derive various equivalence theorems for use in the solution of important problems. Thus we can put the idea of magnetic sources to use regardless of whether these sources actually exist. 1.4 Problems 1.1 Write the volume charge density for a singular surface charge located on the sphere r = r 0 , entirely in terms of spherical coordinates. Find the total charge on the sphere. 1.2 Repeat Problem 1.1 for a charged half plane φ = φ 0 . 1.3 Write the volume charge density for a singular surface charge located on the cylin- der ρ = ρ 0 , entirely in terms of cylindrical coordinates. Find the total charge on the cylinder. 1.4 Repeat Problem 1.3 for a charged half plane φ = φ 0 .