EM Laplace Equation Homework: Solving for Potential in Gap

[rabbit44]

Homework Statement

Hi, I've attached the question as I don't know how to write equations on here without them looking awful.

Homework Equations

Laplace -> del^2 V=0

The Attempt at a Solution

I've done the first bit (expression for Q0).

For the next bit I tried to solve laplace's equation to find the potential in the gap. I called the general radius of a point u, as r is already taken. I was then going to use the boundary condition that
-dV/du (at u=a) = charge density/permittivity of free space

Which is from the boundary condition for perpendicular field components, and the field must be zero inside the solid sphere.

I took delta to be along the z-axis, so the problem has azimuthal symmetry and we can use the usual general solution of

V = sum[Pl(cost) (Al u^l + Bl u^-(l+1))]

Where t is theta.

Looking at the answer given for charge density, I thought this V would only include an l=0 and l=1 term. So I used them and then used the boundary conditions that

V=0 at u=r=b + delta*cost
V=V0 at u=a

However this did not give me the result.

Any help? Thanks

 

[turin]

Did you try the method of images?