Answer to Question #298582 in Electricity and Magnetism for Ralph

Question

The potential at the surface of a sphere (radius R) is given by

V_{0}=k \cos 6 \theta

V0

​=kcos6θ ,

where k is a constant. Find the potential inside and outside the sphere, as well as the surface charge density σ (θ) on the sphere. (Assume there’s no charge inside or outside the sphere.)

Answer 1

We know that

Poisson equation

$\nabla^2V=-\frac{\rho}{\epsilon_0}$

Charge density is zero r>R,r<R

$\rho=0$

$\nabla^2V=0$

$V(r,\theta)=\Sigma A_l r^lP_l(cos\theta)$ r<R

$V(r,\theta)=\Sigma B_l r^{-l+1}P_l(cos\theta)$ r>R

$V_0=kcos6\theta$

Comparison the gives potential with legendre's polynomial equation

$V(r,\theta)=kP_l(cos6\theta)$

$A_1=\frac{k}{R}\\B_1={kR^2}$

Now

$V_1(r_1,\theta)=\frac{kR^2}{r^2}cos6\theta$ r>R

$V(r,\theta)=\frac{Kr}{R}cos6\theta$ r<R