Tensors for Physics

172 10 Multipole Potentials

The hexadecapole moment, corresponding to=4, and the higher even moments

are also non-zero. However, due to extra factors(a/r)^2 ,(a/r)^4 ..., they can be dis-

regarded in the electrostatic potential, whena/r 1. On the other hand, there exist

charge distributions where the hexadecapole is the lowest non-vanishing multipole

moment. Examples are charges on the corners of a cube or of a regular octahedron,

compensated by an appropriate opposite charge in the center.

10.3.5 Connection with Legendre Polynomials

The reciprocal distance|r−r′|−^1 occurring in (10.24), is equal to

|r−r′|−^1 =r−^1

[

1 − 2 (r′/r)cosθ+(r′/r)^2

]− 1 / 2

, (10.35)

with cosθ=ˆr·rˆ′. Clearly, the angle betweenrandr′is denoted byθ. Assuming
r>r′, the expression[...]−^1 /^2 in (10.35) is the generating function of the Legendre
polynomialsP=P(cosθ). The resulting expansion of|r−r′|−^1 with respect to
Legendre polynomials reads

|r−r′|−^1 =r−^1

∑∞

= 0

(

r′

r

)

P(cosθ), r>r′. (10.36)

Due toXμ 1 μ 2 ···μ=( 2 − 1 )!!r−(^2 +^1 )rμ 1 rμ 2 ···rμ, see also (10.10) and (10.11),

comparison with (10.24) yields

P(cosθ)=

1

!

Yμ 1 μ 2 ···μ̂rμ′ 1 ̂r′μ 2 ···̂r′μ=

( 2 − 1 )!!

!

̂rμ 1 ̂rμ 2 ···̂rμr̂μ′ 1 r̂′μ 2 ···r̂μ′,

(10.37)

which is equivalent to (9.10) with (9.11).

10.4 Further Applications in Electrodynamics

10.4.1 Induced Dipole Moment of a Metal Sphere

Consider a piece of metal placed into an electric field. The conduction electrons

inside the metal feel the force caused by the electric field. When the piece of metal

is electrically isolated, the ‘free’ charges are displaced just within the metal, such

that an electric dipole is induced. This dipole modifies the surrounding electric field.

For the special case of a metallic sphere with radiusR, the induced electric dipole

momentpindis computed as follows.