Electric Potential of a Spherical Shell

1. The problem statement, all variables and given/known data
A conducting spherical shell has inner radius a, outer radius b, and has a +Q point charge at the center. A charge of -Q is put on the conductor.

a) What is the charge on the inside and outside of the shell?

b) What is the electric field both inside and outside of the shell?

c) What is the electric potential both inside and outside of the shell?

2. Relevant equations
Gauss’s Law
V = -∫E[itex]\bullet[/itex]dl

3. The attempt at a solution
a) From conservation of charge and the fact that it’s a conductor, the inside of the shell is -Q and the outside is 0. In addition, there is no charge inside the shell.

b) For 0 ≤ r ≤ a, Gauss’s Law gives E = kQ/(r^2)

For a ≤ r ≤ b, E = 0

For r ≥ b, E = 0

c) This is where I’m stuck. E = 0 for r ≥ b, so the potential difference is 0. But the question asks for the actual potential function. Normally, I would use V = kQ/r. But Q = 0 in this case, so V = 0. So that would mean V = 0 for a ≤ r ≤ b. Lastly, for 0 ≤ r ≤ a,
V(r) = V(a) – kQ((1/a)-(1/r)) = 0 – kQ((1/a)-(1/r)) = kQ((1/r)-(1/a)).

I believe this is what the answer should be. Please, can anyone tell me if they agree?

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