Potential due to a spherical surface charge

Question

The potential at the surface of an insulating sphere (radius R) is given by
$$V(R,\theta) = k \cos(3\theta)$$ where $k$ is a constant. Use separation of variables to find the potential inside the sphere (r $\leq$ R) and outside the sphere (r $\geq$ R), and then use your answer to determine the surface charge density $\sigma(\theta)$ on the sphere. Assume there is no charge inside or outside the sphere.

My approach: I will set my reference point at infinity, therefore the potential at infinity is 0, to set that as a boundary condition. Essentially, the approach in finding the surface charge density would be to find the potential function outside the sphere, then integrate that with the surface area. I am following Griffiths for E&M, but he doesn't include a specific example I can refer to for the separation of variables, and how to apply it. I can follow his examples for following potentials pretty easily, but cannot understand the first steps to set up this problem with SOV.

Answer 1

Started with Griffith's potential form, for inside and outside the sphere.

$$\Phi (r,\theta) = E_{l=0}^\infty A_lr^lP_lcos(\theta), r \leq R$$ $$\Phi (r,\theta) = E_{l=0}^\infty B_lr^{-(l+1)}P_lcos(\theta), r \geq R$$

The potential is continuous for the tangential component, therefore:

$$B_l = A_l R^{2l+1}$$

But the normal component is discontinuous (Griffiths explains this to converge to $\sigma\over\epsilon_0$) , therefore:

$$\sum((2l+1)A_lR^{-1-2}-A_llR^{l-1})P_lcos(\theta)=-\sigma(\theta)/\epsilon_0$$

and find $A_l$ as: $$A_l = 1/(2\epsilon_0R^{l-1})*\int\sigma(\theta)P_lcos(\theta)sin(\theta)d\theta $$

I used this identity along with the Legendre Polynomial I solved for: $$ V(\theta) = kcos(3\theta) = (k/5)(8P_3cos(\theta)-3P_1cos(\theta))$$

and this Legendre can be analyzed in the two different conditions.

The charge density: $$\sigma(\theta) = \epsilon_o[d\Phi/dr(R_+)-d\Phi/dr(R_-)$$

$$\epsilon_0(k/5R)(56P_3cos(\theta)-9P_1cos(\theta))$$