Two charged nonconducting sphere shells

[JSGandora]

Homework Statement

One of two nonconducting spherical shells of radius a carries a charge Q uniformly distributed over its surface, the other a charge -Q, also uniformly distributed. The spheres are brought together until they touch. What does the electric field look like, both outside and inside the shells? How much work is needed to move them far apart?

Homework Equations

$U = \epsilon_0 \int E_1 \cdot E_2 dv$

The Attempt at a Solution

I tried using $U = \epsilon_0 \int E_1 \cdot E_2 dv$ to bash out the work it takes to move the two spheres apart to infinity but it is way too messy.

 

[TSny]

Consider the electric field (inside and outside) of a single uniformly charged shell. For two shells, think superposition. Can you relate the problem to 2 point charges?

 

[JSGandora]

So the electric field by one of the spheres, say the shell of charge Q, is the same field as caused by a point charge Q at the center of that shell. So can we say that the potential outside the shell satisfies Laplace's Equation? Then the average value of the potential over the shell of charge -Q is the same as the potential at the center of the shell, which is $\frac{1}{4\pi\epsilon_o}\frac{Q}{2a}$, so the energy needed to move these two shells infinitely apart would be $\frac{Q^2}{8\pi\epsilon_o a}$. Is that correct?

 

[TSny]

Yes, I believe that's right.

Another way to approach the problem is to consider the force between the two shells. That should be the same as the force between two point charges with charges Q and -Q. So, the work that you want to calculate is just the change in potential energy of the two point charges as they are separated from a distance of 2a to infinity.