It is known that the following integral equation describes the electrostatic field produced by a capacitor consisting of two parallel circular plates, derived in this paper (download for free)
in which $\kappa$ is the distance between the plates and when dimensionless variables are taken so that the plates have a unit radius. This is the relevant equation when the potentials of the plates are equal in magnitude but opposite in sign. Its numerical solution can be used to determine the field line and thus the edge effect could be determined.
Now, I need an analog for the electrostatic field produced by a capacitor consisting of two parallel 1D plates of different lengths, as shown in the following figure (sorry for the crude drawing), in which the lower plate is grounded and the upper shorter one is charged at a high voltage.
Actually, I try to use such an equation to plot the potential lines and then estimate the length of a significant influence of the electric field on the lower plate, that is $l+2\delta$ in the figure. This problem is also related to the previous one. Please see the comments there. So, is there any paper or lecture note about such a configuration? If you know, please share it. Thank you for any suggestions!