Electric potential inside and outside spherical capacitator using laplacian

[MellyC]

Homework Statement

Find the electric potential inside and outside a spherical capacitor, consisting of two hemispheres
of radius 1 m. joined along the equator by a thin insulating strip, if the upper hemisphere is kept
at 220 V and the lower hemisphere is grounded

Homework Equations

u (r,θ) = ρ(r)y(θ)
Laplacian in spherical parts: -Δ^2 = $\frac{1}{sin(θ)}$ $\frac{∂}{∂θ}$(sinθ $\frac{∂}{∂θ}$) + $\frac{1}{sin^2(θ)}$$\frac{∂^2}{d\phi^2}$
where we can assume that this does not depend on $\phi$ because rotation is symmetric

The Attempt at a Solution

Two equations:

$\frac{1}{r^2}$$\frac{∂}{∂r}$ (r^2 ρ'(r)) + λy=0

$\frac{1}{sin(θ)}$$\frac{∂}{∂θ}$ (sin(θ)$\frac{∂y}{∂θ}$) - λy=0

I believe that I am supposed to somehow convert this into Legendre's equation, but I'm not sure how to do this.

 

[ehild]

What is the charge density in the free space inside and outside of this capacitor?

ehild