# How to calculate the potential of a charge in a plate capacitor?

This seems like a simple enough problem, but I am still struggling with it: Suppose you have two infinite, uncharged, metallic plates. I am now interested in the electrostatic potential (more specifically the energy) of the system if a point charge is introduced with some distance $$a$$ to the plane between both plates.

Now this sounds like it should be solvable by introducing image charges, like one learns in basic electrostatics education. The problem is though, that the image charge has a mirror image again and that image has another one and so on. One can still construct a series describing the potential, but I did not succeed in calculating the limit of said series. I switched to searching for literature on the problem and found the following paper describing how to apply the Sommerfeld-Watson transformation to the series to obtain an integral expression. The solution that the authors show in their paper, however, diverges when approaching the axis going perpendicular through the charge and the plates ($$\rho = 0$$). I am specifically interested in calculating the energy of the system and thus have to have a solution that is defined on the axis.

Has anybody got an idea on how I can calculate this? It really seems so easy and when working on the problem I expected to find a lot of textbook solutions, but the literature is really sparse. Does anybody now where I can find more information on how to solve the problem? Any other ideas on how to calculate the energy?

Edit: I found another paper that has a analytical solution in case anybody is interested in the same question Paper can be found here.

• Is the charge midway between the two plates? Dec 4, 2019 at 14:04
• No, it can be anywhere between both plates. The way I framed the problem I introduced the variable $a$ to indicate the distance to the midway-plane. Then the force would be 0 due to symmetry and the problem is not that hard. Dec 4, 2019 at 14:15
• The fact that the charge isn't in the middle is one of the many reasons this problem is hard. This means that the image charge distribution isn't symmetric about the midplane and is hard to even compute properly. Dec 4, 2019 at 14:18
• @Iconstrife How do you determine the energy of a system? Dec 4, 2019 at 16:35
• Well it should be the electrostatic potential without the 1/r-Coulomb term corresponding to the "original charge". Dec 5, 2019 at 9:43