On the motion of a point charge in a plate capacitor considering influence effects

A point charge between the plates of a capacitor generates an influence charge distribution on the plates that modify the electric field acting upon the point charge. This effect is described by the well-known Dirichlet Green's function for the two parallel conducting plate problem for which we derive an infinite mirror charge series representation. At the line perpendicular to the plates and passing through the point charge this Green's function and hence the total force can be explicitly evaluated in terms of the psi function. For the motion of the point charge we develop an analytical approximation and compare it with the numerical integration of the exact equations of motion. The correction due to influence effects is shown to be of order $O(-\lambda \log \lambda)$ where $\lambda$ denotes the relative strength of the Green's function compared with the pure capacitor potential.