Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES

+ + + + + + + +

+

+

+

+

−

−

−

−

−

−

−

−

−

−

−

−

θ

a

Figure 21.6 Induced charge and field lines associated with a conducting

sphere placed in an initially uniform electrostatic field.

A hollow split conducting sphere of radiusais placed at the origin. If one half of its

surface is charged to a potentialv 0 and the other half is kept at zero potential, find the

potentialvinside and outside the sphere.

Let us choose the top hemisphere to be charged tov 0 and the bottom hemisphere to be
at zero potential, with the plane in which the two hemispheres meet perpendicular to the
polar axis; this is shown in figure 21.7. The boundary condition then becomes

v(a, θ, φ)=

{

v 0 for 0<θ<π/2(0<cosθ<1),

0forπ/ 2 <θ<π (− 1 <cosθ<0). (21.50)

The problem is clearly axially symmetric and so we may setm= 0. Also, we require the
solution to be finite on the polar axis and so it cannot containQ(cosθ). Therefore the
general form of the solution to (21.38) is

v(r, θ, φ)=

∑∞

=0

(Ar+Br−(+1))P(cosθ). (21.51)

Inside the sphere (forr<a) we require the solution to be finite at the origin and so
B=0forallin (21.51). Imposing the boundary condition atr=awe must then have

v(a, θ, φ)=

∑∞

=0

AaP(cosθ),

wherev(a, θ, φ) is also given by (21.50). Exploiting the mutual orthogonality of the Legendre
polynomials, the coefficients in the Legendre polynomial expansion are given by (18.14)
as (writingμ=cosθ)

Aa=

2 +1

2

∫ 1

− 1

v(a, θ, φ)P(μ)dμ

=

2 +1

2

v 0

∫ 1

0

P(μ)dμ,