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Potential difference inside a spherical shell

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. The attempt at a solution

[itex]\oint E ds = \frac{\int \sigma ds}{\epsilon_{0}}[/itex]

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 [itex]-\int E dl[/itex] 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 wouldnt that be 2pir^2 as well? I'd be left with E=sigma/epsilon0


Gold Member
Would it be easier to integrate the potential directly?

V(r) = C * ∫ σ(r')*da'/[r - r']

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