Work required to move a charge A particle of charge q is moved from the infinity to the center of a shell sphere with radius r and thickness t, through a small hole on the shell. What is the work required?
I think what i did is wrong, but i am not sure why.
$U_{o} = \frac{Q^2}{8 \pi \epsilon r}\\U_{f} = \frac{(Q+q)^2}{8 \pi \epsilon r} + \frac{q^2}{8 \pi \epsilon (r-t)}\\W = U_{o}-U_{f}$
Where Uo and Uf are the initial and final energy of the field.
Now the work would simply be the difference of this, but i could not take off Q (The sphere's charge, the problem does not give us). So probably my approach is wrong, but i would appreciate to know why (Please does not give me the answer to the question, just point where is wrong).
 A: Your basic approach is correct: thinking about $W = U_f - U_i$ (sign depending on whether you are thinking about the work done by the field or the work done by the agent moving the particle). But you should think about this in two steps, bringing the particle from infinity to the hole in the shell ($W_1$), and then bringing the particle from the hole to the center of the shell ($W_2$). The work you are looking for is $W = W_1 + W_2$.
Edit: actually, you should think about this in three steps. I didn't see the thing about the thickness of the shell. This means there is a middle step of moving the particle from the outside of the shell to the inside of the shell.
In the first step, think about what the initial energy is if the particle starts infinitely far away. You should know the EXACT value of the initial energy when $r = \infty$. This should give you ($W_1$).
In the second step, think about the electric field inside the shell. If it is a spherically symmetric shell, you should know the value for $\vec{E}$ inside the shell. Since $\vec{F} = q\vec{E}$ and $W = \int \vec{F} \cdot d\vec{x}$, this should help you figure out the ($W_2$).
