Page 191 - Electromagnetics

and for r ≥ a

∞

−(n+1)
m2 (r,θ) = B n r P n (cos θ). (3.202)
n=0
The boundary condition (3.154)at r = a requires that

∞ ∞
n −(n+1)

A n a P n (cos θ) = B n a P n (cos θ);
n=0 n=0
upon application of the orthogonality of the Legendre functions, this becomes
n
A n a = B n a −(n+1) . (3.203)
We can write (3.155)as

∂ m1 ∂ m2
− + =−ρ Ms
∂r ∂r
so that at r = a
∞ ∞

n−1
−(n+2)
− A n na P n (cos θ) − B n (n + 1)a P n (cos θ) =−M 0 cos θ.
n=0 n=0
After application of orthogonality this becomes

A 1 + 2B 1 a −3 = M 0 , (3.204)
na n−1 A n =−(n + 1)B n a −(n+2) , n = 1. (3.205)

Solving (3.203)and (3.204)simultaneously for n = 1 we find that
M 0 M 0 3
A 1 = , B 1 = a .
3 3
We also see that (3.203)and (3.205)are inconsistent unless A n = B n = 0, n = 1.
Substituting these results into (3.201)and (3.202), we have

M 0 r cos θ, r ≤ a,
3
m =
M 0 a 3 2 cos θ, r ≥ a,
3 r
which is (3.200).
3.3.8 Bodies immersed in an impressed magnetic field: magnetostatic
shielding

A highly permeable enclosure can provide partial shielding from external magnetostatic
fields. Consider a spherical shell of highly permeable material (Figure 3.22); assume it
is immersed in a uniform impressed field H 0 = H 0 ˆ z. We wish to determine the internal
field and the factor by which it is reduced from the external applied field. Because there
are no sources (the applied field is assumed to be created by sources far removed), we
may use magnetic scalar potentials to represent the fields everywhere. We may represent
the scalar potentials using a separation of variables solution to Laplace’s equation, with
a contribution only from the n = 1 term in the series. In region 1 we have both scattered

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