Page 145 - Electromagnetics

The Green’s function G N 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
2

∇ G(r|r ) =−δ(r − r ).

In Green’s second identity let
φ(r) = G(r|r a ), ψ(r) = G(r|r b ),
where r a and r b are arbitrary points, and integrate over the unprimed coordinates. We
have

2 2
[G(r|r a )∇ G(r|r b ) − G(r|r b )∇ G(r|r a )] dV =
V

∂G(r|r b ) ∂G(r|r a )
− G(r|r a ) − G(r|r b ) dS.
∂n ∂n
S
If G is the unbounded-space Green’s function, the surface integral must vanish since
S B →∞. It must also vanish under Dirichlet or Neumann boundary conditions. Since
2
2
∇ G(r|r a ) =−δ(r − r a ), ∇ G(r|r b ) =−δ(r − r b ),
we have

[G(r|r a )δ(r − r b ) − G(r|r b )δ(r − r a )] dV = 0,
V
hence
G(r b |r a ) = G(r a |r b )

by the sifting property. By the arbitrariness of r a and r b , reciprocity is established.

Electrostatic shielding. The Dirichlet Green’s function can be used to explain elec-
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

∂G D (r|r )

(r) = (r ) dS ,
∂n
S B
© 2001 by CRC Press LLC