# Conducting cylinder by dielectric interface

To help me with a project I'm working on, I attempted to solve what I thought was an easy problem. There is an infinite, conducting cylinder of radius $$R$$ at some potential $$V$$, located a distance $$b$$ from a dielectric interface. Rudimentary image included.

I want to find the potential in the half space including the cylinder. My first thought was to solve using a charged wire, using the image method, but the solution that gives does not have circular equipotential surfaces. I though perhaps to write the potential as a Fourier sum, but the different geometries (of the interface and cylinder) are making it very difficult for me to write boundary conditions.

The way I would proceed is first to exchange the spherical electrode by a point charge at the centre : $$Q = \frac{VR}{K}$$ With : $$K = \frac{1}{4 \pi \epsilon_0}$$