Two cocentric conducting spheres

[crimsonidol]

Homework Statement

Two concentric conducting spheres of radii a and b (a < b) are separated into four hemispheres by a nonconducting sheet. The upper ( 0 < θ < π / 2 ) outer (radius b) and lower ( π / 2 < θ < π ) inner (radius a) hemispheres are kept at potential +V. The upper inner and lower outer hemispheres are kept at potential –V. Find the electrostatic potential in the region between the two spheres. Evaluate the expansion coefficients explicitly for terms up to l = 4 .

Homework Equations

The Attempt at a Solution

I had an attempt to solve problem however when I got to the end I realized that I have no r dependence in the function.
I attached my solution and I circled the part that r's gone. I am doing something wrong there. Can someone tell me where?

 

[mark726]

Hello there,

Thank you for sharing your attempt at solving this problem. Your solution looks correct up until the point where you mention the missing r dependence.

I believe the issue is with your expression for the multipole expansion coefficients. The formula you used, C_n = (2n+1)/(2ε_0 b^n) ∫V(r)P_n(cosθ)dr, is for a point charge, not for a conducting sphere. For a conducting sphere, the multipole expansion coefficients are given by C_n = (2n+1)/(ε_0 b^n) ∫σ(r)P_n(cosθ)dS, where σ(r) is the surface charge density of the sphere.

In this problem, since the spheres are at a fixed potential, the surface charge density can be calculated using the relation σ = ε_0 (∂V/∂r). Once you have the correct expression for the multipole expansion coefficients, you should be able to solve for the electrostatic potential in the region between the spheres.

I hope this helps. Let me know if you have any further questions. Good luck with your solution!