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 = [itex]\frac{1}{sin(θ)}[/itex] [itex]\frac{∂}{∂θ}[/itex](sinθ [itex]\frac{∂}{∂θ}[/itex]) + [itex]\frac{1}{sin^2(θ)}[/itex][itex]\frac{∂^2}{d\phi^2}[/itex]
where we can assume that this does not depend on [itex]\phi[/itex] because rotation is symmetric

The Attempt at a Solution

Two equations:

[itex]\frac{1}{r^2}[/itex][itex]\frac{∂}{∂r}[/itex] (r^2 ρ'(r)) + λy=0

[itex]\frac{1}{sin(θ)}[/itex][itex]\frac{∂}{∂θ}[/itex] (sin(θ)[itex]\frac{∂y}{∂θ}[/itex]) - 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