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2Quasistatic Electromagnetic Fields May one plow with an ox and an ass together? The like of you may write everything and prove everything in quaternions, but in the transition period the bilingual method may help to explain the more perfect. James Clerk Maxwell, in a letter to P. G. Tait. 2Quasistatic Electromagnetic Fields The purpose of this chapter is to articulate the notion of a quasistatic elec- tromagnetic system, and develop the topological aspects of the boundary value problems encountered in the analysis of quasistatic systems. The topological ap- proach gives a new perspective on variational formulations which form the basis of finite element analysis. 2A. The Quasistatic Limit Of Maxwell's Equations Maxwell's Equations. Let S be a surface with boundary, V a volume in R3, and note that @ denotes the boundary operator. The integral versions of Maxwell's equations are as follows: d (2{1) E dl = B dS (Faraday's Law) · −dt · Z@S ZS (2{2) B dS = 0 (Gauss' Law for magnetic charge) 0 · Z@V d (2{3) H dl = D dS + J dS (Amp`ere's Law) 0 · dt 0 · 0 · Z@S ZS ZS (2{4) D dS = ρ dV (Gauss' Law) · Z@V ZV where E = Electric field intensity vector; B = Magnetic field flux density vector; H = Magnetic field intensity vector; D = Electric field flux density vector; 49 50 2. QUASISTATIC ELECTROMAGNETIC FIELDS and the current and charge sources are described by J = Electric current flux density vector; ρ = Electric charge density. If we let S0 = @V in Amp`ere's Law (2{3) and remember that @@ = 0, then the field vector can be eliminated between Amp`ere's and Gauss' law, (2{3) and (2{4) to reveal a statement of charge conservation: d 0 = ρ dV + J dS: dt · ZV Z@V This shows that conservation of charge is implicit in Maxwell's equations. When the surfaces and volumes S, S0, V , V 0 are stationary with respect to the inertial reference frame of the field vectors, one can use the standard theorems of vector calculus to rewrite Maxwell's equations as follows: @B (2{5) curl E = ; − @t (2{6) div B = 0; @D (2{7) curl H = + J; @t (2{8) div D = ρ, @ρ (2{9) div J + = 0: @t Equations (2{5){(2{8) are the differential versions of Maxwell's equations. Equa- tion (2{9) is the differential version of the conservation of charge and can be obtained independently by taking the divergence of Equation (2{7) and substi- tuting in (2{9), remembering that div curl = 0. The differential versions of Maxwell's equations are much less general than the integral laws for three main reasons: (1) They need to be modified for use in problems involving noninertial reference frames. Differential forms are an essential tool for this modification. (2) The differential laws do not contain \global topological" information which can be deduced from the integral laws. The machinery of cohomology groups is the remedy for this problem. (3) The differential laws assume differentiability of the field vectors with respect to spatial coordinates and time. However, there can be discontinuities in fields across medium interfaces, so it is not possible to deduce the proper interface conditions from the differential laws. In order to get around the third obstacle, we must go back to the integral laws and derive so-called interface conditions which give relations for the fields across material or media interfaces (Figure 2.1). Here we simply state the interface 2A. THE QUASISTATIC LIMIT OF MAXWELL'S EQUATIONS 51 £ ¡ b b b b ¤ ( ¢ ) ( ) b ( ¥ ) σs a £ ¡ a a a a ¤ ( ¢ ) ( ) Figure 2.1. conditions: (2{10) n (Ea Eb) = 0; × − (2{11) n (Ba Bb) = 0; · − (2{12) n (Ha Hb) = K; × − (2{13) n (Da Db) = σ ; · − s where the superscripts refer to limiting values of the field at on the interface when the interface is approached from the side of the indicated medium. The source vectors do not really exist in nature but rather represent a way of modeling current distribution in the limit of zero skin depth (that is, when ! = 2πf or σ 0). This approximation will be discussed soon. From a limiting case! 1of the in!tegral laws one has a statement of conservation of surface charge: @σ n (J a J b) + div K + S = 0; · − S @t where divS is the divergence operator in the interface. In summary, one can say that the integral version of Maxwell's equations is equivalent to three distinct pieces of information: (1) the differential version in regions where constitutive laws are continuous, and the field vectors are continuous and have the appropriate degree of differentiability; (2) interface conditions where constitutive laws are discontinuous; (3) global topological information which is lost when problems are specified on subsets of Euclidean space | these are the lumped parameters of circuit theory. Constitutive Laws. In general, Maxwell's equations are insufficient for de- termining the electromagnetic field since there are six independent equations in twelve unknowns, namely the components of E, B, D, and H. The first step to closing this gap is to introduce constitutive laws. For stationary media, these 52 2. QUASISTATIC ELECTROMAGNETIC FIELDS typically take the form D = "0E + P ; B = µ0(H + M) and, if J is not fixed, but related to E by Ohm's law, J = σ(E + Ei): The new variables are " = 8:854 10−12farad/meter (permittivity of free space); 0 × µ = 4π 10−7henry/meter (permeability of free space); 0 × σ = conductivity; P = polarization; M = magnetization; Ei = external impressed field due to forces of chemical origin. Generally the functional relationships P = P (E; r); M = M(H; r) are assumed, though sometimes it makes sense to have M = M(B; r). There are several ways to characterize media: (1) Lossless: @Pi=@Ej = @Pj =@Ei, @Mi=@Hj = @Mj =@Hi when referred to Cartesian coordinates. (2) Homogeneous: P = P (E), M = M(H). 2 2 (3) Isotropic: P = "0χe( E ; r)E, M = χm( H ; r)H where χe and χm are called the electric andj magneticj susceptibilities,j j respectively. (4) Linear: P is a linear function of E, and M is a linear function of H. Occasionally the constitutive laws are replaced by linear differential operators: 2 di P = " χ E: 0 i dti i=0 X This leads into the topic of dispersion relations in optics and other interesting phenomena which we will sidestep. Remark on the concept of energy. The condition for lossless media is equiva- lent to saying that the integrals E1 H1 D(E) dE; B(H) dH E · H · Z 0 Z 0 are independent of path connecting initial and final states (at each point in physical space). Often integration by parts or some monotonicity assumption then shows that the values of the integrals D(E1) B(H1) we = E(D) dD; wm = H(B) dB D E · B H · Z ( 0) Z ( 0) 2A. THE QUASISTATIC LIMIT OF MAXWELL'S EQUATIONS 53 representing \electric and magnetic energy densities" are defined independent of how final states are achieved. When this is so, we define total energies by: We = we dV; Wm = wm dV ZV ZV so that dW @D dW @B e = E dV; m = H dV dt · @t dt · @t ZV ZV Parameters Characterizing Linear Isotropic Media. For linear media the constitutive relations are simply D = "E (" = "0(1 + χe)); B = µH (µ = µ0(1 + χm)); J = σE (Ohm's Law). It is often necessary to take a Fourier transform with respect to time in which case the frequency parameter ! plays an important role in approximations. Thus the four parameters, !; σ; µ, " describe the behavior of the constitutive relations at each point in space. From the point of view of wave propagation, it is usually more convenient to introduce a different set of parameters: 1 v = (wave speed) pµε µ η = (wave impedance in lossless media, i.e. σ = 0) jσ=0 " r" τ = (dielectric relaxation time) e σ 2 δ = (skin depth) !σµ r !µ R = (surface resistivity) S 2σ r (Note that c = 1=p"0µ0 is the speed of light in vacuum.) Although this set of pa- rameters is unmotivated at this point, their interpretation and role in simplifying solutions will become clear when we consider electromagnetic waves. The Standard Potentials for Maxwell's Equations. In this section we present the familiar potentials for the electromagnetic field as an efficient way to end up with \fewer equations in fewer unknowns." Recall Maxwell's equations in space-time R3 R: × @D (2{14) curl H = J + ; @t (2{15) div D = ρ, @B (2{16) curl E = ; − @t (2{17) div B = 0: 54 2. QUASISTATIC ELECTROMAGNETIC FIELDS It appears that we have eight scalar equations in twelve unknowns, the twelve unknowns being the components of E; B; D, and H. However, constitutive laws provide six more equations so that we should think of the above as eight equations in six unknowns. The standard potentials enable us to solve equations (2{16) and (2{17) explicitly so that we end up with four equations in four unknowns. Since Equation (2{17) is valid throughout R3 and the second cohomology group of R3 is trivial, we have (2{18) B = curl A for some vector potential A. Equations (2{16) and (2{18) can be combined to give @A curl E + = 0: @t Using the fact that this equation is valid throughout R3, one can conclude that for a scalar potential φ @A E + = grad φ @t − or @A (2{19) E = grad φ.
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