Page 139 - Electromagnetics

Figure 3.5: Computation of potential from known sources and values on bounding sur-
faces.

plates of a capacitor, which may be charged and on which the potential is assumed
to be known. To solve for (r) within V we must know the potential produced by a

point source. This potential, called the Green’s function, is denoted G(r|r ); it has two

arguments because it satisfies Poisson’s equation at r when the source is located at r :
2

∇ G(r|r ) =−δ(r − r ). (3.52)
Later we shall demonstrate that in all cases of interest to us the Green’s function is
symmetric in its arguments:

G(r |r) = G(r|r ). (3.53)
This property of G is known as reciprocity.
Our development rests on the mathematical result (B.30)known as Green’s second
identity. We can derive this by subtracting the identities

∇· (φ∇ψ) = φ∇· (∇ψ) + (∇φ) · (∇ψ),
∇· (ψ∇φ) = ψ∇· (∇φ) + (∇ψ) · (∇φ),
to obtain

2
2
∇· (φ∇ψ − ψ∇φ) = φ∇ ψ − ψ∇ φ.
Integrating this over a volume region V with respect to the dummy variable r and using

the divergence theorem, we obtain

2 2
[φ(r )∇ ψ(r ) − ψ(r )∇ φ(r )] dV =− [φ(r )∇ ψ(r ) − ψ(r )∇ φ(r )] · dS .
V S
The negative sign on the right-hand side occurs because ˆ n is an inward normal to V .
Finally, since ∂ψ(r )/∂n = ˆ n ·∇ ψ(r ),wehave

∂ψ(r ) ∂φ(r )
2 2
[φ(r )∇ ψ(r ) − ψ(r )∇ φ(r )] dV =− φ(r ) − ψ(r ) dS
V S ∂n ∂n
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