Work done moving a test charge into conducting shell

[KleZMeR]

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 $\sigma$. Assume the charge passes through an infinitesimal hole in the shell.

Homework Equations

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

The Attempt at a Solution

$W_{\infty, R+T} = q[\phi(\infty)-\phi(R+T)] = -\phi(R+T)$
and
$W_{R+T,0} = q[\phi(R+T)-\phi(0)] = \phi(R+T)$

My second equation takes the 'fact' that $E(r) = 0$ 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.

 

[DelcrossA]

The electric field being zero inside the shell does not mean the potential is zero. Recall that electric field is the negative gradient of potential.