Page 193 - Electromagnetics
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the shell thickness by = b − a, we find that K ≈ 6µ /a when /a 1.Thus
r
3 1
κ =
2 µ r
a
describes the coefficient of shielding for a highly permeable spherical enclosure, valid
when µ r 1 and /a 1. A shell for which µ r = 10, 000 and a/b = 0.99 can reduce
the enclosure field to 0.15% of the applied field.
3.4 Static field theorems
3.4.1 Mean value theorem of electrostatics
The average value of the electrostatic potential over a sphere is equal to the potential
at the center of the sphere, provided that the sphere encloses no electric charge. To see
this, write
ˆ
1 ρ(r ) 1 R ∇ (r )
(r) = dV + − (r ) + · dS ;
4π V R 4π S R 2 R
put ρ ≡ 0 in V , and use the obvious facts that if S is a sphere centered at point r then
ˆ
(1) R is constant on S and (2) ˆ n =−R:
1 1
(r) = 2 (r ) dS − E(r ) · dS .
4π R S 4π R S
The last term vanishes by Gauss’s law, giving the desired result.
3.4.2 Earnshaw’s theorem
It is impossible for a charge to rest in stable equilibrium under the influence of elec-
trostatic forces alone. This is an easy consequence of the mean value theorem of electro-
statics, which precludes the existence of a point where can assume a maximum or a
minimum.
3.4.3 Thomson’s theorem
Static charge on a system of perfect conductors distributes itself so that the electric
stored energy is a minimum. Figure 3.23 shows a system of n conducting bodies held at
potentials 1 ,..., n . Suppose the potential field associated with the actual distribution
of charge on these bodies is , giving
W e = E · E dV = ∇ ·∇ dV
2 V 2 V
for the actual stored energy. Now assume a slightly different charge distribution, resulting
in a new potential = +δ that satisfies the same boundary conditions (i.e., assume
δ = 0 on each conducting body). The stored energy associated with this hypothetical
situation is
W = W e + δW e = ∇( + δ ) ·∇( + δ ) dV
e
2 V
© 2001 by CRC Press LLC