Figure 3.6: Geometry for establishing the singular property of V-(1/R) f(r)(di holds for any continuous function f(r). By direct differentiation we have v2()=0forr'≠r hence the second part of (3.59)is established. This also shows that if r e V then the domain of integration in(3. 59) can be restricted to a sphere of arbitrarily small radius e centered at r(Figure 3.6). The result we seek is found in the limit as 8-0. Thus we are interested in computing f(rv/I )dv=im/f(r)四2/ dv Since f is continuous atr=r, we have by the mean value theorem r()ay-==/=() The integral over Ve can be computed using V(1/R)=V.V(1/R)and the divergence n’.V′ d s where Se bounds Ve. Noting that f= -R, using(57), and writing the integral spherical coordinates(E, 0, )centered at the point r, we have R f(r)v dv= f(r)li 一R(2) 8- sin 6 de do=-4rf(r Hence the first part of (3.59) is also established. The Greens function for unbounded space. In view of (3.58), one solution 4r-r ②2001 by CRC Press LLC
Figure 3.6: Geometry for establishing the singular property of ∇2(1/R). by showing that V f (r )∇2 1 R dV = −4π f (r), r ∈ V, 0, r ∈/ V, (3.59) holds for any continuous function f (r). By direct differentiation we have ∇2 1 R = 0 for r = r, hence the second part of (3.59)is established. This also shows that if r ∈ V then the domain of integration in (3.59)can be restricted to a sphere of arbitrarily small radius ε centered at r (Figure 3.6). The result we seek is found in the limit as ε → 0. Thus we are interested in computing V f (r )∇2 1 R dV = lim ε→0 Vε f (r )∇2 1 R dV . Since f is continuous at r = r, we have by the mean value theorem V f (r )∇2 1 R dV = f (r) lim ε→0 Vε ∇2 1 R dV . The integral over Vε can be computed using ∇2 (1/R) = ∇ · ∇ (1/R) and the divergence theorem: Vε ∇2 1 R dV = Sε nˆ · ∇ 1 R d S , where Sε bounds Vε. Noting that nˆ = −Rˆ , using (57), and writing the integral in spherical coordinates (ε,θ,φ)centered at the point r, we have V f (r )∇2 1 R dV = f (r) lim ε→0 2π 0 π 0 −Rˆ · Rˆ ε2 ε2 sin θ dθ dφ = −4π f (r). Hence the first part of (3.59)is also established. The Green’s function for unbounded space. In view of (3.58), one solution to (3.52)is G(r|r ) = 1 4π|r − r | . (3.60)
This simple Greens function is generally used to find the potential produced by charge in unbounded space. Here N=0(no internal surfaces) and SB-00. Thus p(r) a(r) ap(r)= G(rr) dv+ lim aG(rr p(r) an' G(rr) We have seen that the Greens function varies inversely with distance from the source, and thus expect that, as a superposition of point-source potentials, p(r) will also vary inversely with distance from a source of finite extent as that distance becomes large with respect to the size of the source. The normal derivatives then vary inversely with distance squared. Thus, each term in the surface integrand will vary inversely with distance cubed while the surface area itself varies with distance squared. The result is that the surface integral vanishes as the surface recedes to infinity, giving Φ(r)=G(r|r) By 3.60)we then have p(r) (361) where the integration is performed over all of space. Since points at infinity are a convenient reference for the absolute potential Later we shall need to know the amount of work required to move a charge o from infinity to a point P located at r. If a potential field is produced by charge located in unbounded space, moving an additional charge into position requires the work W2=-QE.d=QΦ(r)-Φ()=QΦr (3.62) Coulombs law. We can obtain E from(61)by direct differentiation. We have E(r) 4丌∈ Ir-rdv.I 4r6/p(r)v/1 r-r E(r) 46o(r (3.63) by (3.57). So Coulomb's law follows from the two fundamental postulates of electrostatics (35)and(3.6) Greens function for unbounded space: two dimensions. We define the two- dimensional Green's function as the potential at a point r= p+iz produced by a z-directed line source of constant density located at r= p. Perhaps the simplest way to compute this is to first find e produced by a line source on the z-axis. By (3.63)we have r) TE()F-r ②2001 by CRC Press LLC
This simple Green’s function is generally used to find the potential produced by charge in unbounded space. Here N = 0 (no internal surfaces)and SB → ∞. Thus (r) = V G(r|r ) ρ(r ) dV + lim SB→∞ SB (r ) ∂G(r|r ) ∂n − G(r|r ) ∂(r ) ∂n d S . We have seen that the Green’s function varies inversely with distance from the source, and thus expect that, as a superposition of point-source potentials, (r) will also vary inversely with distance from a source of finite extent as that distance becomes large with respect to the size of the source. The normal derivatives then vary inversely with distance squared. Thus, each term in the surface integrand will vary inversely with distance cubed, while the surface area itself varies with distance squared. The result is that the surface integral vanishes as the surface recedes to infinity, giving (r) = V G(r|r ) ρ(r ) dV . By (3.60)we then have (r) = 1 4π V ρ(r ) |r − r | dV (3.61) where the integration is performed over all of space. Since lim r→∞ (r) = 0, points at infinity are a convenient reference for the absolute potential. Later we shall need to know the amount of work required to move a charge Q from infinity to a point P located at r. If a potential field is produced by charge located in unbounded space, moving an additional charge into position requires the work W21 = −Q P ∞ E · dl = Q[(r) − (∞)] = Q(r). (3.62) Coulomb’s law. We can obtain E from (61)by direct differentiation. We have E(r) = − 1 4π ∇ V ρ(r ) |r − r | dV = − 1 4π V ρ(r )∇ 1 |r − r | dV , hence E(r) = 1 4π V ρ(r ) r − r |r − r | 3 dV (3.63) by (3.57). So Coulomb’s law follows from the two fundamental postulates of electrostatics (3.5)and (3.6). Green’s function for unbounded space: two dimensions. We define the twodimensional Green’s function as the potential at a point r = ρ + zˆz produced by a z-directed line source of constant density located at r = ρ . Perhaps the simplest way to compute this is to first find E produced by a line source on the z-axis. By (3.63)we have E(r) = 1 4π ρl(z ) r − r |r − r | 3 dl .
Then, since r=iz+pp, r= iz, and dr=dz, we have E(p) Pp+i(z-2 02]32 Carrying out the integration we find that e has only a p-component which varies only with p E(p) The absolute potential referred to a radius po can be found by computing the line integral of e from p to po Φ(p) We may choose any reference point po except po =0 or po= oo. This choice is equivalent to the addition of an arbitrary constant, hence we can also write C (365) The potential for a general two-dimensional charge distribution in unbounded space is y superposition Φ(p)= Pr(p) G(plp)ds where the Greens function is the potential of a unit line source located at p: G(plp) Here Sr denotes the transverse (xy) plane, and Pr denotes the two-dimensional charge distribution(C/m )within that plane We note that the potential field (3.66)of a two-dimensional source decreases logarith- mically with distance. Only the potential produced by a source of finite extent decreases inversely with distance Dirichlet and Neumann Greens functions. The unbounded space Greens func tion may be inconvenient for expressing the potential in a region having internal surfaces In fact, (3.56)shows that to use this function we would be forced to specify both p and its normal derivative over all surfaces. This, of course, would exceed the actual requirements for uniqueness Many functions can satisfy (3.52). For instance, G(rr) (3.68) satisfies (3.52) if r: V. Evaluation of (3.55) with the Greens function (3.68) repro duces the general formulation (3.56) aplacian of the second term in(3.68)is identically zero in V. In fact, we can add any function to the free-space Green's function provided that the additional term obeys Laplace 's equation within V F(rr) F(rr)=0 ②2001 by CRC Press LLC
Then, since r = zˆz + ρˆ ρ, r = zˆz , and dl = dz , we have E(ρ) = ρl 4π ∞ −∞ ρˆ ρ + zˆ(z − z ) ρ2 + (z − z )2 3/2 dz . Carrying out the integration we find that E has only a ρ-component which varies only with ρ: E(ρ) = ρˆ ρl 2π ρ . (3.64) The absolute potential referred to a radius ρ0 can be found by computing the line integral of E from ρ to ρ0: (ρ) = − ρl 2π ρ ρ0 dρ ρ = ρl 2π lnρ0 ρ . We may choose any reference point ρ0 except ρ0 = 0 or ρ0 = ∞. This choice is equivalent to the addition of an arbitrary constant, hence we can also write (ρ) = ρl 2π ln 1 ρ + C. (3.65) The potential for a general two-dimensional charge distribution in unbounded space is by superposition (ρ) = ST ρT (ρ ) G(ρ|ρ ) d S , (3.66) where the Green’s function is the potential of a unit line source located at ρ : G(ρ|ρ ) = 1 2π ln ρ0 |ρ − ρ | . (3.67) Here ST denotes the transverse (xy)plane, and ρT denotes the two-dimensional charge distribution (C/m2)within that plane. We note that the potential field (3.66)of a two-dimensional source decreases logarithmically with distance. Only the potential produced by a source of finite extent decreases inversely with distance. Dirichlet and Neumann Green’s functions. The unbounded space Green’s function may be inconvenient for expressing the potential in a region having internal surfaces. In fact, (3.56)shows that to use this function we would be forced to specify both and its normal derivative over all surfaces. This, of course, would exceed the actual requirements for uniqueness. Many functions can satisfy (3.52). For instance, G(r|r ) = A |r − r | + B |r − ri| (3.68) satisfies (3.52)if ri ∈/ V. Evaluation of (3.55)with the Green’s function (3.68)reproduces the general formulation (3.56)since the Laplacian of the second term in (3.68)is identically zero in V. In fact, we can add any function to the free-space Green’s function, provided that the additional term obeys Laplace’s equation within V: G(r|r ) = A |r − r | + F(r|r ), ∇2 F(r|r ) = 0. (3.69)
a good choice for G(rr) will minimize the effort required to evaluate p(r). Examining (3.56)we notice two possibilities. If we demand that then the surface integral terms in (3.56) involving ap/an will vanish. The Green's function satisfying (3.70)is known as the Dirichlet Green's function. Let us designate it by Gp and use reciprocity to write(3.70)as GD(rr)=0 for all r∈S. The resulting specialization of (3.56) p(r) agp(rr) p(r) Gp(rr) V+∮(r) dGp(rr q(r) ds (371) equires the specification of (but not its normal derivative) over the boundary surfaces In case SB and Sn surround and are adjacent to perfect conductors, the Dirichlet bound ary condition has an important physical meaning. The corresponding Greens function is the potential at point r produced by a point source at r in the presence of the conductors when the conductors are grounded -i.e, held at zero potential. Then we must specify the actual constant potentials on the conductors to determine p everywhere within V using(3.71). The additional term F(rlr) in(3.69)accounts for the potential produced by surface charges on the grounded conductors By analogy with(3. 70) it is tempting to try to define another electrostatic Green's ag(rr) 0 for all r'∈S. (372) But this choice is not permissible if V is a finite-sized region. Let us integrate(3.54)over V and employ the divergence theorem and the sifting property to get dS"=-1: in conjunction with this, equation (3.72)would imply the false statement 0=-1. Sup- ose instead that we introduce a Greens function according to =-- for all r'∈S (3.74) There A is the total area of S. This choice avoids a contradiction in (3. 73); it does not nullify any terms in(3.56), but does reduce the surface integral terms involving p to constants. Taken together, these terms all comprise a single additive constant on the ight-hand side: although the corresponding potential p(r) is thereby determined only to within this additive constant, the value of e(r)=-vo(r) will be unaffected. By reciprocity we can rewrite(3. 74)as aGn(rr) for all r∈S. (375) ②2001 by CRC Press LLC
A good choice for G(r|r ) will minimize the effort required to evaluate (r). Examining (3.56)we notice two possibilities. If we demand that G(r|r ) = 0 for all r ∈ S (3.70) then the surface integral terms in (3.56)involving ∂/∂n will vanish. The Green’s function satisfying (3.70)is known as the Dirichlet Green’s function. Let us designate it by GD and use reciprocity to write (3.70)as GD(r|r ) = 0 for all r ∈ S. The resulting specialization of (3.56), (r) = V GD(r|r ) ρ(r ) dV + SB (r ) ∂GD(r|r ) ∂n d S + + N n=1 Sn (r ) ∂GD(r|r ) ∂n d S , (3.71) requires the specification of (but not its normal derivative)over the boundary surfaces. In case SB and Sn surround and are adjacent to perfect conductors, the Dirichlet boundary condition has an important physical meaning. The corresponding Green’s function is the potential at point r produced by a point source at r in the presence of the conductors when the conductors are grounded — i.e., held at zero potential. Then we must specify the actual constant potentials on the conductors to determine everywhere within V using (3.71). The additional term F(r|r ) in (3.69)accounts for the potential produced by surface charges on the grounded conductors. By analogy with (3.70)it is tempting to try to define another electrostatic Green’s function according to ∂G(r|r ) ∂n = 0 for all r ∈ S. (3.72) But this choice is not permissible if V is a finite-sized region. Let us integrate (3.54)over V and employ the divergence theorem and the sifting property to get S ∂G(r|r ) ∂n d S = −1; (3.73) in conjunction with this, equation (3.72)would imply the false statement 0 = −1. Suppose instead that we introduce a Green’s function according to ∂G(r|r ) ∂n = − 1 A for all r ∈ S. (3.74) where A is the total area of S. This choice avoids a contradiction in (3.73); it does not nullify any terms in (3.56), but does reduce the surface integral terms involving to constants. Taken together, these terms all comprise a single additive constant on the right-hand side; although the corresponding potential (r) is thereby determined only to within this additive constant, the value of E(r) = −∇(r) will be unaffected. By reciprocity we can rewrite (3.74)as ∂GN (r|r ) ∂n = − 1 A for all r ∈ S. (3.75)
The Greens function GN so defined is known as the Neumann Green's function. Observe that if V is not finite-sized then A->oo and according to(3. 74)the choice(3.72 )becomes allowable Finding the Green's function that obeys one of the boundary conditions for a given geometry is often a difficult task. Nevertheless, certain canonical geometries make the Greens function approach straightforward and simple. Such is the case in image theory when a charge is located near a simple conducting body such as a ground screen or a sphere. In these cases the function F(rr) consists of a single correction term as (3.68). We shall consider these simple cases in examples to follow Reciprocity of the static Green's function. It remains to show that for any of the Greens functions introduced above. The unbounded-space Greens function is reciprocal by inspection; Ir-r,l is unaffected by interchanging r and r. However, we can give a more general treatment covering this case as well as the dirichlet and Neumann cases. We begin with v2G(r)=-8(r-r) In Green's second identity let p(r)=G(r]ra) y(r)=G(rrb) where ra and rh are arbitrary points, and integrate over the unprimed coordinates. We [G(rlra)VG(r/rb)-G(rrb)VG(rlraldv Gra C(m/r、aGra) aG(r]rb) If G is the unbounded-space Greens function, the surface integral must vanish since SB-00. It must also vanish under Dirichlet or Neumann boundary conditions. Since V-G(rra)=-8(r-ra) VG(rrb)=-s(r-rb ve have [G(rra)8(r-rb)-Grlrbs(r-raldV=0, hence G(rblra)=g(ralrb) by the sifting property. by the arbitrariness of ra and rb, reciprocity is established Electrostatic shielding. The Dirichlet Green's function can be used to explain trostatic shielding. We consider a closed, grounded, conducting shell with charge outside but not inside(Figure 3.7). By(3. 71)the potential at points inside the shell is d(r)=∮中(r) agp(rr ②2001 by CRC Press LLC
The Green’s function GN so defined is known as the Neumann Green’s function. Observe that if V is not finite-sized then A → ∞ and according to (3.74)the choice (3.72)becomes allowable. Finding the Green’s function that obeys one of the boundary conditions for a given geometry is often a difficult task. Nevertheless, certain canonical geometries make the Green’s function approach straightforward and simple. Such is the case in image theory, when a charge is located near a simple conducting body such as a ground screen or a sphere. In these cases the function F(r|r ) consists of a single correction term as in (3.68). We shall consider these simple cases in examples to follow. Reciprocity of the static Green’s function. It remains to show that G(r|r ) = G(r |r) for any of the Green’s functions introduced above. The unbounded-space Green’s function is reciprocal by inspection; |r − r | is unaffected by interchanging r and r . However, we can give a more general treatment covering this case as well as the Dirichlet and Neumann cases. We begin with ∇2G(r|r ) = −δ(r − r ). In Green’s second identity let φ(r) = G(r|ra), ψ(r) = G(r|rb), where ra and rb are arbitrary points, and integrate over the unprimed coordinates. We have V [G(r|ra)∇2G(r|rb) − G(r|rb)∇2G(r|ra)] dV = − S G(r|ra) ∂G(r|rb) ∂n − G(r|rb) ∂G(r|ra) ∂n d S. If G is the unbounded-space Green’s function, the surface integral must vanish since SB → ∞. It must also vanish under Dirichlet or Neumann boundary conditions. Since ∇2G(r|ra) = −δ(r − ra), ∇2G(r|rb) = −δ(r − rb), we have V [G(r|ra)δ(r − rb) − G(r|rb)δ(r − ra)] dV = 0, hence G(rb|ra) = G(ra|rb) by the sifting property. By the arbitrariness of ra and rb, reciprocity is established. Electrostatic shielding. The Dirichlet Green’s function can be used to explain electrostatic shielding. We consider a closed, grounded, conducting shell with charge outside but not inside (Figure 3.7). By (3.71) the potential at points inside the shell is (r) = SB (r ) ∂GD(r|r ) ∂n d S ,