Page 383 - Electromagnetics

so

˜ i
¯ ˜ i
˜ i

∇∇ · J (r ,ω)G(r|r ; ω) dV = P.V. ∇ ∇G(r|r ; ω) · J (r ,ω) dV − L · J (r,ω).
V V
¯
Here L is usually called the depolarizing dyadic [113]. Its value depends on the shape of
V δ , as considered below.
We may nowwrite
1

˜
¯
¯ ˜ i
˜

E(r,ω) =− jω ˜µ(ω) P.V. G e (r|r ; ω) · J(r ,ω) dV − L · J (r,ω). (5.86)

c
V jω˜ (ω)
We may also incorporate both terms into a single dyadic Green’s function using the
notation
1
¯ ¯ ¯
G(r|r ; ω) = P.V. G e (r|r ; ω) − Lδ(r − r ).
k 2
Hence when we compute

˜
¯
˜ i

E(r,ω) =− jω ˜µ(ω) G(r|r ; ω) · J (r ,ω) dV
V
1

¯
˜ i
¯

=− jω ˜µ(ω) P.V. G e (r|r ; ω) − 2 Lδ(r − r ) · J (r ,ω) dV
V k
we reproduce (5.86). That is, the symbol P.V. on G e indicates that a principal value
integral must be performed.
¯
Our final task is to compute L from (5.85). When we remove the excluded region
from the principal value computation we leave behind a hole in the source region. The
contribution to the field at r by the sources in the excluded region is found from the
i
scalar potential produced by the surface distribution ˆ n · J . The value of this correction
term depends on the shape of the excluding volume. However, the correction term always
adds to the principal value integral to give the true field at r, regardless of the shape of
the volume. So we must always match the shape of the excluded region used to compute
the principal value integral with that used to compute the correction term so that the
true field is obtained. Note that as V δ → 0 the phase factor in the Green’s function
becomes insignificant, and the values of the current on the surface approach the value at
i
r (assuming J is continuous at r). Thus we may write
˜ i
J (r,ω) · ˆ n
¯ ˜ i

lim ∇ dS = L · J (r,ω).

V δ →0 4π|r − r |
S δ
This has the form of a static field integral. For a spherical excluded region we may com-
pute the above quantity quite simply by assuming the current to be uniform throughout
V δ and by aligning the current with the z-axis and placing the center of the sphere at the
origin. We then compute the integral at a point r within the sphere, take the gradient,
and allow r → 0. We thus have for a sphere
J cos θ
˜ i
¯
˜ i

lim ∇ dS = L · [ˆ zJ (r,ω)].

V δ →0 S 4π|r − r |
This integral has been computed in § 3.2.7 with the result given by (3.103). Using this
we find

˜ i
1 J
˜ i
¯
˜ i
lim ∇ J z = ˆ z = L · [ˆ zJ (r,ω)]
V δ →0 3 3
r=0
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