hich fo h E 2j sin k,h The scattered field has the form of (5.7)but must be odd. Thus A+=-A-and the total field for y h E2( 2JA(kx, a)sin kyy -ou lo()2j sin k, Setting E, =0 at z=d and solving for A+ we find that the total field for this case is +j△ , (. y, 0)=2/[ jk ly-hl 2 i sin k、he jkr sin k, d Adding the fields for the two cases we find that +j△ oulo(o) -jkyIy-hl y dk (5.8) which is a superposition of impressed and scattered fields 5.2 Solenoidal-lamellar decomposition We now discuss the decomposition of a general vector field into a lamellar comp aving zero curl and a solenoidal component having zero divergence. This is known as a Helmholtz decomposition. If V is any vector field then we wish to write where Vs and Vi are the solenoidal and lamellar components of v. Formulas expressing these components in terms of V are obtained as follows. We first write Vs in terms of a vector potential”Aas v=V×A. This is possible by virtue of(B 49). Similarly, we write VI in terms of a"scalar potential v=Vφ ②2001
which for y > h is E˜ i z(x, y,ω) = −ωµ˜ ˜I0(ω) 2 2π ∞+ j −∞+ j 2 j sin kyh 2ky e− jky y e− jkx x dkx . The scattered field has the form of (5.7) but must be odd. Thus A+ = −A− and the total field for y > h is E˜ z(x, y,ω) = 1 2π ∞+ j −∞+ j 2 j A+(kx ,ω)sin ky y − ωµ˜ ˜I0(ω) 2 2 j sin kyh 2ky e− jky y e− jkx x dkx . Setting E˜ z = 0 at z = d and solving for A+ we find that the total field for this case is E˜ z(x, y,ω) = −ωµ˜ ˜I0(ω) 2 2π ∞+ j −∞+ j e− jky |y−h| − e− jky |y+h| 2ky − − 2 j sin kyh 2ky e− jkyd sin kyd sin ky y e− jkx x dkx . Adding the fields for the two cases we find that E˜ z(x, y,ω) = −ωµ˜ ˜I0(ω) 2π ∞+ j −∞+ j e− jky |y−h| 2ky e− jkx x dkx + +ωµ˜ ˜I0(ω) 2π ∞+ j −∞+ j cos kyh cos ky y cos kyd + j sin kyh sin ky y sin kyd e− jkyd 2ky e− jkx x dkx , (5.8) which is a superposition of impressed and scattered fields. 5.2 Solenoidal–lamellar decomposition We nowdiscuss the decomposition of a general vector field into a lamellar component having zero curl and a solenoidal component having zero divergence. This is known as a Helmholtz decomposition. If V is any vector field then we wish to write V = Vs + Vl, (5.9) where Vs and Vl are the solenoidal and lamellar components of V. Formulas expressing these components in terms of V are obtained as follows. We first write Vs in terms of a “vector potential” A as Vs =∇× A. (5.10) This is possible by virtue of (B.49). Similarly, we write Vl in terms of a “scalar potential” φ as Vl = ∇φ. (5.11)
To obtain a formula for VI we take the divergence of(5. 9 )and use(.11)to get v=vvt=V.Vφ=Vφ The result may be regarded as Poissons equation for the unknown This equation is solved in Chapter 3. By (3.61)we have V.V(r) 4丌R where R= r-r, and we have V. V(r v1(r)=-V Similarly, a formula for Vs can be obtained by taking the curl of (5.9)to get VxV=VXV Substituting(. 10)we have xv=V×(V×A)=V(·A)-V2A We may choose any value we wish for V. A, since this does not alter Vs=VxA (We discuss such "gauge transformations"in greater detail later in this chapter. )With V.A=O we obtain VxV=V-A This is Poissons equation for each rectangular component of A; therefore V'xV(r) 4丌R and we have V' x V(r) V,(r=VX Summing the results we obtain the Helmholtz decomposition V=V+V dV′+V 4T R (5.13) Identification of the electromagnetic potentials. Let us write the electromagnetic fields as a general superposition of solenoidal and lamellar components E=V×AE+VφE (5.14) B=V×AB+Vφ (5.15) One possible form of the potentials AE, AB, E, and B appears in(5.13).However, because e and B are related by Maxwells equations, the potentials should be related to the sources. We can determine the explicit relationship by substituting(5. 14) and (5.15) ②2001
To obtain a formula for Vl we take the divergence of (5.9) and use (5.11) to get ∇ · V =∇· Vl =∇·∇φ = ∇2 φ. The result, ∇2 φ =∇· V, may be regarded as Poisson’s equation for the unknown φ. This equation is solved in Chapter 3. By (3.61) we have φ(r) = − V ∇ · V(r ) 4π R dV , where R = |r − r |, and we have Vl(r) = −∇ V ∇ · V(r ) 4π R dV . (5.12) Similarly, a formula for Vs can be obtained by taking the curl of (5.9) to get ∇ × V =∇× Vs. Substituting (5.10) we have ∇ × V =∇× (∇ × A) = ∇(∇ · A) − ∇2 A. We may choose any value we wish for ∇ · A, since this does not alter Vs =∇× A. (We discuss such “gauge transformations” in greater detail later in this chapter.) With ∇ · A = 0 we obtain −∇ × V = ∇2 A. This is Poisson’s equation for each rectangular component of A; therefore A(r) = V ∇ × V(r ) 4π R dV , and we have Vs(r) =∇× V ∇ × V(r ) 4π R dV . Summing the results we obtain the Helmholtz decomposition V = Vl + Vs = −∇ V ∇ · V(r ) 4π R dV +∇× V ∇ × V(r ) 4π R dV . (5.13) Identification of the electromagnetic potentials. Let us write the electromagnetic fields as a general superposition of solenoidal and lamellar components: E =∇× AE + ∇φE , (5.14) B =∇× AB + ∇φB. (5.15) One possible form of the potentials AE , AB, φE , and φB appears in (5.13). However, because E and B are related by Maxwell’s equations, the potentials should be related to the sources. We can determine the explicit relationship by substituting (5.14) and (5.15)
into Ampere's and Faradays laws. It is most convenient to analyze the relationship Ising superposition of the cases for which Jm =0 and J=0. With Jm=0 Faraday's law V×E (5.16) Since v xe is solenoidal that B=0, which is equivalent to the auxiliary Maxwell equation V.B= 0. Now, substitution of (5. 14)and(5.15)into(5.16)gives Using V×(VφE)=0 and combining the terms we get da V×v×AE+ dAg + ve Substitution into(. 14)gives +VφE+V] Combining the two gradient functions together, we see that we can write both e and B in terms of two potentials dA E= Vφe, B=V×A (518) where the negative sign on the gradient term is introduced by convention. Gauge transformations and the Coulomb gauge. We pay a price for the simplicity of using only two potentials to represent E and B. While V x ae is definitely solenoidal, Ae itself may not be: because of this (5.17) may not be a decomposition into solenoidal and lamellar components. However, a corollary of the Helmholtz theorem states that a vector field is uniquely specified only when both its curl and divergence are specified. Here there is an ambiguity in the representation of E and B; we may remove this ambiguity and define Ae uniquely by requiring that Then Ae is solenoidal and the decomposition (5.17) is solenoidal-lamellar. This require- ment on Ae is called the Coulomb gauge. The ambiguity implied by the non-uniqueness of V. Ae can also be expressed by the observation that a transformation of the type A→A+Vr, (5.20) (5.21) ②2001
into Ampere’s and Faraday’s laws. It is most convenient to analyze the relationships using superposition of the cases for which Jm = 0 and J = 0. With Jm = 0 Faraday’s lawis ∇ × E = −∂B ∂t . (5.16) Since ∇ × E is solenoidal, B must be solenoidal and thus ∇φB = 0. This implies that φB = 0, which is equivalent to the auxiliary Maxwell equation ∇ · B = 0. Now , substitution of (5.14) and (5.15) into (5.16) gives ∇ × [∇ × AE + ∇φE ] = − ∂ ∂t [∇ × AB] . Using ∇ × (∇φE ) = 0 and combining the terms we get ∇ × ∇ × AE + ∂AB ∂t = 0, hence ∇ × AE = −∂AB ∂t + ∇ξ. Substitution into (5.14) gives E = −∂AB ∂t + [∇φE + ∇ξ ] . Combining the two gradient functions together, we see that we can write both E and B in terms of two potentials: E = −∂Ae ∂t − ∇φe, (5.17) B =∇× Ae, (5.18) where the negative sign on the gradient term is introduced by convention. Gauge transformations and the Coulomb gauge. We pay a price for the simplicity of using only two potentials to represent E and B. While ∇ × Ae is definitely solenoidal, Ae itself may not be: because of this (5.17) may not be a decomposition into solenoidal and lamellar components. However, a corollary of the Helmholtz theorem states that a vector field is uniquely specified only when both its curl and divergence are specified. Here there is an ambiguity in the representation of E and B; we may remove this ambiguity and define Ae uniquely by requiring that ∇ · Ae = 0. (5.19) Then Ae is solenoidal and the decomposition (5.17) is solenoidal–lamellar. This requirement on Ae is called the Coulomb gauge. The ambiguity implied by the non-uniqueness of ∇ · Ae can also be expressed by the observation that a transformation of the type Ae → Ae + ∇, (5.20) φe → φe − ∂ ∂t , (5.21)
leaves the expressions (5.17)and (5. 18)unchanged. This is called a gauge transformation, and the choice of a certain r alters the specification of v. Ae. Thus we may begin with the Coulomb gauge as our baseline, and allow any alteration of Ae according to(5. 20) vA by v Vr= v2r Once V. Ae is specified, the relationship between the potentials and the current J can be found by substitution of (5.17) and(5. 18) into Ampere's law. At this point e assume media that are linear, homogeneous, isotropic, and described by the time- invariant parameters A, e, and o. Writing J=J+oE we have V×(V×A)=J (5.22) Taking the divergence of both sides of (5. 22 )we get 0=V·J Then, by substitution from the continuity equation and use of (5. 19)along with VVoe= V-pe we obtain (p'+eVe)=-ov2oe For a lossless medium this reduces to Vφe=-p/∈ (5.24) pe r, 0= p(r,) av (5.25) 4丌∈R We can obtain an equation for Ae by expanding the left-hand side of (5. 22) to get v(v- A)-v2A=uJ'-op--ouVp a-A at 2-kea vae VA J+σμ一+aμVφe+∈V中e under the Coulomb gauge. For lossless media this becomes vA-H∈a2=-H+∈aV中 (5.27) Observe that the left-hand side of (5.27) is solenoidal(since the Laplacian term came from the curl-curl, and V. Ae=0), while the right-hand side contains a general vector field J and a lamellar term. We might expect the Voe term to cancel the lamellar portion of J, and this does happen 91. By(5.12) and the continuity equation we can write the lamellar component of the current as i(,)=-p/VJ(r,t) J -v/pr,t) Thus(5.27)becomes (5.28) ②2001
leaves the expressions (5.17) and (5.18) unchanged. This is called a gauge transformation, and the choice of a certain alters the specification of ∇ · Ae. Thus we may begin with the Coulomb gauge as our baseline, and allowany alteration of Ae according to (5.20) as long as we augment ∇ · Ae by ∇·∇ = ∇2. Once ∇ · Ae is specified, the relationship between the potentials and the current J can be found by substitution of (5.17) and (5.18) into Ampere’s law. At this point we assume media that are linear, homogeneous, isotropic, and described by the timeinvariant parameters µ, , and σ. Writing J = Ji + σE we have 1 µ ∇ × (∇ × Ae) = Ji − σ ∂Ae ∂t − σ∇φe − ∂2Ae ∂t 2 − ∂ ∂t ∇φe. (5.22) Taking the divergence of both sides of (5.22) we get 0 =∇· Ji − σ ∂ ∂t ∇ · A − σ∇·∇φe − ∂2 ∂t 2 ∇ · Ae − ∂ ∂t ∇·∇φe. (5.23) Then, by substitution from the continuity equation and use of (5.19) along with ∇·∇φe = ∇2φe we obtain ∂ ∂t ρi + ∇2 φe = −σ∇2 φe. For a lossless medium this reduces to ∇2 φe = −ρi / (5.24) and we have φe(r, t) = V ρi (r , t) 4πR dV . (5.25) We can obtain an equation for Ae by expanding the left-hand side of (5.22) to get ∇ (∇ · Ae) − ∇2 Ae = µJi − σµ ∂Ae ∂t − σµ∇φe − µ ∂2Ae ∂t 2 − µ ∂ ∂t ∇φe, (5.26) hence ∇2 Ae − µ ∂2Ae ∂t 2 = −µJi + σµ ∂Ae ∂t + σµ∇φe + µ ∂ ∂t ∇φe under the Coulomb gauge. For lossless media this becomes ∇2 Ae − µ ∂2Ae ∂t 2 = −µJi + µ ∂ ∂t ∇φe. (5.27) Observe that the left-hand side of (5.27) is solenoidal (since the Laplacian term came from the curl-curl, and ∇ · Ae = 0), while the right-hand side contains a general vector field Ji and a lamellar term. We might expect the ∇φe term to cancel the lamellar portion of Ji , and this does happen [91]. By (5.12) and the continuity equation we can write the lamellar component of the current as Ji l(r, t) = −∇ V ∇ · Ji (r , t) 4π R dV = ∂ ∂t ∇ V ρi (r , t) 4π R dV = ∂ ∂t ∇φe. Thus (5.27) becomes ∇2 Ae − µ ∂2Ae ∂t 2 = −µJi s. (5.28)
Therefore the vector potential Ae, which describes the solenoidal portion of both E and B, is found from just the solenoidal portion of the current. On the other hand, the scalar potential, which describes the lamellar portion of E, is found from p' which arises from V- J, the lamellar portion of the current From the perspective of field computation, we see that the introduction of potential functions has reoriented the solution process from dealing with two coupled first-order partial differential equations(Maxwell's equations), to two uncoupled second-order equa- tions(the potential equations(5. 24)and(5.28). The decoupling of the equations is often worth the added complexity of dealing with potentials, and, in fact, is the solution tech nique of choice in such areas as radiation and guided waves. It is worth pausing for a moment to examine the form of these equations. We see that the scalar potential obeys Poisson's equation with the solution(5.25), while the vector potential obeys the wave equation. As a wave, the vector potential must propagate aw from the source with finite velocity. However, the solution for the scalar potential (5.25)shows no such behavior. In fact, any change to the charge distribution instantaneously permeates all of space. This apparent violation of Einsteins postulate shows that we must be careful hen interpreting the physical meaning of the potentials. Once the computations (5.17) and(5. 18)are undertaken, we find that both e and B behave as waves, and thus propa gate at finite velocity. Mathematically, the conundrum can be resolved by realizing that individually the solenoidal and lamellar components of current must occupy all of space even if their sum, the actual current J, is localized [911 The Lorentz gauge. a different choice of gauge condition can allow both the vector and scalar potentials to act as waves. In this case e may be written as a sum of two terms: one purely solenoidal, and the other a superposition of lamellar and solenoidal Let us examine the effect of choosing the lorentz gauge (5.29) Substituting this expression into(5.26)we find that the gradient terms cancel, giving (5.30) a2A (531) and(5.23)become For lossy media we have obtained a second-order differential equation for Ae, but e must be found through the somewhat cumbersome relation(5.29). For lossless media e coupled Maxwell equations have been decoupled into two second-order equations, one involving Ae and one involving e. Both(5.31)and(5.32) are wave equations, with J as the source for Ae and p' as the source for e. Thus the expected finite-velocity wave nature of the electromagnetic fields is also manifested in each of the potential functions The drawback is that, even though we can still use(5.17)and(5.18), the expression for E is no longer a decomposition into solenoidal and lamellar components. Nevertheless, the choice of the Lorentz gauge is very popular in the study of radiated and guided waves ②2001
Therefore the vector potential Ae, which describes the solenoidal portion of both E and B, is found from just the solenoidal portion of the current. On the other hand, the scalar potential, which describes the lamellar portion of E, is found from ρi which arises from ∇ · Ji , the lamellar portion of the current. From the perspective of field computation, we see that the introduction of potential functions has reoriented the solution process from dealing with two coupled first-order partial differential equations (Maxwell’s equations), to two uncoupled second-order equations (the potential equations (5.24) and (5.28)). The decoupling of the equations is often worth the added complexity of dealing with potentials, and, in fact, is the solution technique of choice in such areas as radiation and guided waves. It is worth pausing for a moment to examine the form of these equations. We see that the scalar potential obeys Poisson’s equation with the solution (5.25), while the vector potential obeys the wave equation. As a wave, the vector potential must propagate away from the source with finite velocity. However, the solution for the scalar potential (5.25) shows no such behavior. In fact, any change to the charge distribution instantaneously permeates all of space. This apparent violation of Einstein’s postulate shows that we must be careful when interpreting the physical meaning of the potentials. Once the computations (5.17) and (5.18) are undertaken, we find that both E and B behave as waves, and thus propagate at finite velocity. Mathematically, the conundrum can be resolved by realizing that individually the solenoidal and lamellar components of current must occupy all of space, even if their sum, the actual current Ji , is localized [91]. The Lorentz gauge. A different choice of gauge condition can allowboth the vector and scalar potentials to act as waves. In this case E may be written as a sum of two terms: one purely solenoidal, and the other a superposition of lamellar and solenoidal parts. Let us examine the effect of choosing the Lorentz gauge ∇ · Ae = −µ ∂φe ∂t − µσφe. (5.29) Substituting this expression into (5.26) we find that the gradient terms cancel, giving ∇2 Ae − µσ ∂Ae ∂t − µ ∂2Ae ∂t 2 = −µJi . (5.30) For lossless media ∇2 Ae − µ ∂2Ae ∂t 2 = −µJi , (5.31) and (5.23) becomes ∇2 φe − µ ∂2φe ∂t 2 = −ρi . (5.32) For lossy media we have obtained a second-order differential equation for Ae, but φe must be found through the somewhat cumbersome relation (5.29). For lossless media the coupled Maxwell equations have been decoupled into two second-order equations, one involving Ae and one involving φe. Both (5.31) and (5.32) are wave equations, with Ji as the source for Ae and ρi as the source for φe. Thus the expected finite-velocity wave nature of the electromagnetic fields is also manifested in each of the potential functions. The drawback is that, even though we can still use (5.17) and (5.18), the expression for E is no longer a decomposition into solenoidal and lamellar components. Nevertheless, the choice of the Lorentz gauge is very popular in the study of radiated and guided waves.