Tough Concentric Spheres with mixed Dielectrics and a air-gap Problem!

jajay504

1. The problem statement, all variables and given/known data
I have concentric spheres with mixed dielectrics. There is an air-gap between the spheres which consist of a permittivity ε0. The radius' are a, b and c and the permittivities of the dielectric portions are ε1 and ε2. An image is attached! What are the potentials in the 4 regions of the image.

2. Relevant equations

Laplace's equation in spherical coordinates 1/r^2 ∂/∂r (r^2 ∂V/∂r) = 0

3. The attempt at a solution
So, I know from Laplace's equation that r^2 (∂V/∂r) = 0
V = A∫dr/r^2 + B = -A/r + B

V(I) (r,θ)= Ʃ A_l*r^l * P_l*(cosθ), where Ʃ goes l=0 to ∞
V(II) (r,θ)=Ʃ (A_l*r^l + B_l/ r^(l+1)) * P_l*(cosθ)
V(III) (r,θ)=Ʃ B_l(1/ r^(l+1) - r^l/(r^(2l+1)) * P_l*(cosθ)
V(IV) (r,θ)= Ʃ ( B_l/ r^(l+1)) * P_l*(cosθ) - Eo*rcosθ

Set up boundary conditions:
(I) ε1 ∂V(I)/∂r (a,θ)= ε0 ∂V(II)/∂r (a,θ)
(II) V(II) (b,θ)= V(III) (b,θ)
(III) ε2 ∂V(III)/∂r (c,θ)= ε0 ∂V(IV)/∂r (c,θ)

Went through the process of applying the boundary conditons.
Got A1 (I)= -Eo
B1 (I)= (Eo R^3 (ε1 - εo))/(ε1 + 2εo)

This problem got extremely tough after this!!! I am completely lost now!
Is there a simpler way of approaching a problem like this