A Dipole in front of a flat Metallic surface
The Green's function can be used to find the electric field or electrostatic potential for arbitrary charge distributions. You will learn that the polarization of a dipole in front of a flat metallic surface enormously affects the overall field.
A Metallic Sphere subject to a Point Charge
In this problem we will encounter the main physical features of dielectric spheres - their induced field and polarizability. The result is of great interest: understanding the interaction of light with small particles is one of the main concerns of nanophotonics. It leads to astonishing applications like cloaking, improvement of solar cells and lithography with extreme resolution.