Potential difference inside a spherical shell

[warfreak131]

Homework Statement

A top half of a spherical shell has radius R and uniform charge density sigma. Find the potential difference V(b)-V(a) between point b at the north pole, and point a at the center of the sphere.

Homework Equations

The Attempt at a Solution

\oint E ds = \frac{\int \sigma ds}{\epsilon_{0}}

I integrated sigma ds from 0 to 2pi and 0 to pi/2, and got the total surface area as 2pi r^2. Now I want to solve for E, and then use -\int E dl to give me the potential difference, but what does the left hand side of my original equation equal?

E is uniform along the surface, so I can pull it out of the integral and be left with the integral of ds. But wouldn't that be 2pir^2 as well? I'd be left with E=sigma/epsilon0

 

[Spinnor]

Would it be easier to integrate the potential directly?

V(r) = C * ∫ σ(r')*da'/[r - r']