## An Electric Charge in front of a Dielectric Interface

The method of image charges can also be applied to problems including dielectrics. In this problem, a charge in front of an interface between two dielectrics is considered. Find out how to solve it and what happens if the charge is put in between the dielectrics.

## Problem Statement

Let us assume two dielectric media with permittivities $$\varepsilon_1$$ for $$z<0$$ and $$\varepsilon_2$$ for $$z>0$$. In $$\mathbf{r}_q = \left(x_q,y_q,d\right)$$ may be a charge $$q$$. Calculate the electrostatic potential $$\phi\left(\mathbf{r}\right)$$ for the whole space. Calculate $$\phi$$ in the limit $$d\rightarrow 0$$ and discuss your result. On the right is a schematic of the situation: