Page 141 - Electromagnetics

2
Figure 3.6: Geometry for establishing the singular property of ∇ (1/R).

by showing that

−4π f (r), r ∈ V,
1
2
f (r )∇ dV = (3.59)
V R 0, r /∈ V,
holds for any continuous function f (r). By direct differentiation we have

2 1
∇ = 0 for r = 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

1
1
2 2
f (r )∇ dV = lim f (r )∇ dV .
V R ε→0 V ε R
Since f is continuous at r = r, we have by the mean value theorem

1
1

2 2
f (r )∇ dV = f (r) lim ∇ dV .
V R ε→0 V ε R
2
The integral over V ε can be computed using ∇ (1/R) =∇ ·∇ (1/R) and the divergence

theorem:

2 1 1
∇ dV = ˆ n ·∇ dS ,
R R
V ε S ε
ˆ
where S ε bounds V ε . Noting that ˆ n =−R, using (57), and writing the integral in

spherical coordinates (ε, θ, φ)centered at the point r,wehave
ˆ

1
2π π R
2 ˆ 2
f (r )∇ dV = f (r) lim −R · 2 ε sin θ dθ dφ =−4π f (r).
V R ε→0 0 0 ε
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
1

G(r|r ) = . (3.60)

4π|r − r |
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