Page 161 - Electromagnetics

where r < = min{r, a} and r > = max{r, a}. Using orthogonality of the exponentials we
find that only the m = 0 terms contribute:

∞ Y n0 (θ, φ) r n π

∗

F(r) = 2πa 2 < f (θ )Y (θ ,φ ) sin θ dθ .

n+1 n0
n=0 2n + 1 r > 0
Finally, since

2n + 1
Y n0 = P n (cos θ)
4π
we have
∞
1 2
r < n π

F(r) = a P n (cos θ) f (θ )P n (cos θ ) sin θ dθ . (3.101)
2 r > n+1 0
n=0
As an example, suppose f (θ) = cos θ = P 1 (cos θ). Then
∞
1 2
r < n π

F(r) = a P n (cos θ) n+1 P 1 (cos θ )P n (cos θ ) sin θ dθ .
2 0
n=0 r >
The orthogonality of the Legendre polynomials can be used to show that
π 2

P 1 (cos θ )P n (cos θ ) sin θ dθ = δ 1n ,
0 3
hence
a 2 r <
F(r) = cos θ . (3.102)
2
3 r >
3.2.7 Field produced by a permanently polarized body
Certain materials, called electrets, exhibit polarization in the absence of an external
electric field. A permanently polarized material produces an electric field both internal
and external to the material, hence there must be a charge distribution to support the
fields. We can interpret this charge as being caused by the permanent separation of
atomic charge within the material, but if we are only interested in the macroscopic field
then we need not worry about the microscopic implications of such materials. Instead, we
can use the Maxwell–Boffi equations and find the potential produced by the material by
using (3.99). Thus, the field of an electret with known polarization P occupying volume
region V in free space is dipolar in nature and is given by

1 −∇ · P(r ) 1 ˆ n · P(r )

(r) = dV + dS

4π 0 V |r − r | 4π 0 S |r − r |
where ˆ n points out of the volume region V .
As an example, consider a material sphere of radius a, permanently polarized along
its axis with uniform polarization P(r) = ˆ zP 0 . We have the equivalent source densities
ρ p =−∇ · P = 0, ρ Ps = ˆ n · P = ˆ r · ˆ zP 0 = P 0 cos θ.
Then

1 ρ Ps (r ) 1 P 0 cos θ

(r) = dS = dS .

4π 0 S |r − r | 4π 0 S |r − r |
© 2001 by CRC Press LLC