I want to solve the Poisson equation for a thin slab of charge held above a grounded plane at $z=0$. The problem is somewhat reminiscent of the classical image problem of a point charge above an infinite grounded plane but differs because we are using a slab of charge instead which extends above the whole grounded plane. I will assume that the sheet and plane extend over a large region in the x- and $y$-plane so that we can assume translational invariance in these directions. Poisson's equation then reads

$$ \begin{cases} \dfrac{d^2\phi}{dz^2} = \dfrac{\rho_{0}}{\epsilon_{0}} &a \leq z \leq b \\ \\ \dfrac{d^2\phi}{dz^2} = 0 &\text{Elsewhere} \end{cases} $$

My question is now: What should be the boundary conditions for this equation? In the classical Image problem you assume that $\phi(x,y,z)=0$ and that $\phi \rightarrow 0$ for $\sqrt{x^2+y^2+z^2}\rightarrow \infty$. But if you impose this condition for $\phi(z)$ in my problem you get a non-physical result, which predicts that there is a field infinitely far away from the plane. This is understandable since we have assumed translational invariance, which is effectively the same as saying that we have an infinite plane of charge from which the electric field is constant when you move away. In reality, of course, the plane is not infinite and one should solve the full 3D problem with the boundary condition above. This seems however like a lot of effort. Is there a way to translate the boundary condition for the full 3D problem into a boundary condition for the 1D problem with translational invariance?



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