Electic potential due to finite rectangular plate

Question

I am trying to find the potential at any point (x,y,z) due to a rectangular plate with a constant surface charge density. Let's assume the plate is centered on the X-Y plane and extends from -n to n in the x direction and from -m to m in the y direction. Here is what I can work out so far:

$$V(R)=\frac{1}{4 \pi \epsilon_0}\int_{s'}\frac{\rho_s(R')ds'}{|R-R'|}$$

With constant charge density, this simplifies to the following

$$V(R)=\frac{\rho_s}{4 \pi \epsilon_0}\int_{s'}\frac{1}{|R-R'|}\,ds'$$ $$V(x,y,z)=\frac{\rho_s}{4 \pi \epsilon_0}\int_{s'}\frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}\,ds'$$ $$V(x,y,z)=\frac{\rho_s}{4 \pi \epsilon_0}\int_{-n}^{n}\int_{-m}^{m}\frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}\,dy\, dx$$ $$V(x,y,z)=\frac{\rho_s}{4 \pi \epsilon_0}\int_{-n}^{n}\int_{-m}^{m}\frac{1}{\sqrt{(x-x')^2+(y-y')^2+z^2}}\,dy\, dx$$

Can the integral be evaluated from here? I have tried with mathematica, but the evaluation never seems to complete. Is there any better way to approach the problem?

Answer 1

The general calculation to this potential is far from trivial and its not known in the present literature, since no symmetry can be invoked. However, in the particular case of point lying above the plate along the symmetry axis passing through the center of the plate, at distance z, the exact solution follows:

$$V(z)= \frac{\sigma}{\pi\epsilon_0}\left[a \sinh^{-1}\left( \frac{b}{\sqrt{z^2+a^2}}\right)+b \sinh^{-1}\left( \frac{a}{\sqrt{z^2+b^2}}\right) -z\tan^{-1}\left( \frac{ab}{z\sqrt{z^2+a^2+b^2}}\right)\right]$$

More details about the (long) calculation can be found in this paper: https://iopscience.iop.org/article/10.1088/1361-6404/ac362f

Cheers,

D.A. Fagundes