Electric potential inside a hollow sphere with non-uniform charge

[RodolfoM]

Homework Statement

The electric potential on the surface of a hollow spherical shell of radius 𝑅 is 𝑉0 𝑐𝑜𝑠𝜃, where 𝑉0 is a constant. In this problem we use spherical coordinates with origin at the center of the shell. What is the potential inside the shell?

Answer: 𝑉(𝑟,𝜃) = 𝑟/𝑅 𝑉0 𝑐𝑜𝑠𝜃

Relevant Equations

Gauss's Law, Point charge potential.

I tried to find the charge distribution using the given potential but couldn't produce the correct result. Also, Gauss's Law doesn't help, as the electric flux is 0 but we don't have any symmetry. Can someone please shine a light on this? Thanks in advance..

 

[Dr Transport]

expand the potential in terms of Legendre polynomials.