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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.

Problem Layout

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.

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Ok, it was obviously not apparent to people reading this question - there is a perfectly conducting, infinite cylinder (with a circular cross-section) at some distance from the plane of separation. After some research, I am convinced there is not analytic solution to the problem, and instead used numeric methods to receive an approximate solution.

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

Then that charge would have an image across the dielectric surface, the method has been described quite well in introduction to electrodynamics or here

The image charge will itself have a image charge inside the wire, see here and here on how to calculate it.

The you just repeat the process until the solutions converges, the potential will be the one induced by all the image charges. (See my answer here for more details on the process)

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