Page 167 - Electromagnetics

at r = a. This gives
∞ ∞

−n−2
n−1
− 0 E 0 cos θ + 0 [−(n + 1)B n ]a P n (cos θ) = [nA n ]a P n (cos θ).
n=0 n=0
By orthogonality of the Legendre functions we have

− 0 E 0 − 2 0 B 1 a −3 = A 1 , (3.113)
− 0 (n + 1)B n a −n−2 = nA n a n−1 , n = 1. (3.114)
Equations (3.112)and (3.114)cannot hold simultaneously unless A n = B n = 0 for n = 1.
Solving (3.111)and (3.113)we have

3 0 3 − 0
A 1 =−E 0 , B 1 = E 0 a .
+ 2 0 + 2 0
Hence
3 0 3 0
1 (r) =−E 0 r cos θ =−E 0 z , (3.115)
+ 2 0 + 2 0
3
a − 0
2 (r) =−E 0 r cos θ + E 0 2 cos θ. (3.116)
r + 2 0
Interestingly, the electric field
3 0
E 1 (r) =−∇ 1 (r) = ˆ zE 0
+ 2 0
inside the sphere is constant with position and is aligned with the applied external field.
However, it is weaker than the applied field since > 0 . To explain this, we compute
the polarization charge within and on the sphere. Using D = E = 0 E + P we have

3 0
P 1 = ˆ z( − 0 )E 0 . (3.117)
+ 2 0
The volume polarization charge density −∇ · P is zero, while the polarization surface
charge density is

3 0
ρ Ps = ˆ r · P = ( − 0 )E 0 cos θ.
+ 2 0
Hence the secondary electric field can be attributed to an induced surface polarization
charge, and is in a direction opposing the applied field. According to the Maxwell–Boffi
viewpoint we should be able to replace the sphere by the surface polarization charge
immersed in free space, and use the formula (3.61)to reproduce (3.115)and (3.116).
This is left as an exercise for the reader.

3.3 Magnetostatics

The large-scale forms of the magnetostatic field equations are

H · dl = J · dS, (3.118)
S

B · dS = 0, (3.119)
S

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