# Charge and conducting thick slab

I have thought over this problem but I haven't found the solution: There is an electric charge $q$ at a distance $d$ from a conducting slab with thickness $t$, the problem is to find the potential everywhere, assumming that the slab is grounded. So, on the side of the slab where the charge is I think one may use the image method solution to find the potential in this region of space. Inside the slab $\phi =0$, but the question remains how to find the potential on the other side? The method of images does not seem to work on that side(?), and other solutions seem difficult to implement. Charge conservation seems to imply that the total charge induced on each surface of the slab is $\pm q$. Is there any easy or semi-easy way to do it? I guess my question would be: what is the most straightforward way to do it? Any help is greatly appreciated.

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Let the "other side" exist in the lower-half space $z<0$. The conductor imposes the following boundary condition on the potential $\Phi = \Phi(x,y,z)$: \begin{align} \Phi(x,y,0) = 0. \end{align} The charge density vanishes for $z<0$, so we simply need to solve the Laplace equation $\nabla^2\Phi = 0$ for the region $z<0$ with this boundary condition at $z=0$. Note that $\Phi = 0$ is a solution to this equation satisfying the desired boundary condition, so this is the solution by uniqueness; \begin{align} \Phi(x,y,z) = 0, \qquad \text{for all $z<0$}. \end{align}
Considering a spherical conducting shell of radius $R$ and thickness $t$, with a charge $q$ at a distance $d$ outside. So, what is the potential inside the shell? Extending $R\to\infty$ now and you get it.