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## Homework Statement

Ok, so I've read many of the threads on here and they all say the same thing. I think I understand the Gauss Law and the theory behind the spherical shell.

The question is this:

Find the WORK done bringing a test charge q from infinity to the center of a spherical shell of thickness T, radius R, and surface charge [itex]\sigma[/itex]. Assume the charge passes through an infinitesimal hole in the shell.

## Homework Equations

[itex]W = q[\phi(inf)-\phi(r)] [/itex]

## The Attempt at a Solution

[itex]W_{\infty, R+T} = q[\phi(\infty)-\phi(R+T)] = -\phi(R+T) [/itex]

and

[itex]W_{R+T,0} = q[\phi(R+T)-\phi(0)] = \phi(R+T) [/itex]

My second equation takes the 'fact' that [itex]E(r) = 0[/itex] inside the shell, resulting in a zero potential.

The final result is that the total WORK = 0

This is for a graduate class, and this result seems somewhat trivial. My other assumption is that the test charge induces an electric field inside the shell, but I do not think work can be done by moving the test charge through its own electric field? I could be totally wrong, and that's why I'm asking this question.

Any help clarifying my result would be greatly appreciated.