Analytic formula for finite parallel plate capacitor with arbitrary plate distance Is there an analytic formula for the electric field in a  (rectangular as well as circular) parallel plate capacitor with finite plate size for arbitrary plate distances?
 A: The 1982 paper "An analytic solution for the potential due to a circular parallel plate capacitor" derives two exact formulas for the potential. From them you can derive the field by taking the negative gradient. One formula is an integral, and the other is an infinite series.
Let $a$ be the radius of the circular plates and $2L$ their separation. Introduce cylindrical coordinates $(\rho, \phi, z)$ such that the plates are in the planes $z=\pm L$ and centered on the $z$-axis. Let the upper plate be at potential $V$ and the lower plate at potential $-V$. The azimuthal symmetry of the system means that the potential depends on $\rho$ and $z$ but not on $\phi$.
The integral formula for the potential is
$$\Phi(\rho,z)=\frac{2V}{\pi}\int_0^\infty\frac{1}{1-e^{-2kL}}\frac{\sin{ka}}{k}(e^{-k|z-L|}-e^{-k|z+L|})J_0(k\rho)dk$$
where $J_0(z)$ is a Bessel function of the first kind.
The series formula is
$$\Phi(\rho,z)=\frac{2V}{\pi}\sum_{n=0}^\infty\left(\sin^{-1}\frac{a}{x_{n_-}}-\sin^{-1}\frac{a}{x_{n_+}}\right)$$
where
$$x_{n_\pm}\equiv\frac12\{[b_{n_\pm}^2+(\rho+a)^2]^{1/2}+[b_{n_\pm}^2+(\rho-a)^2]^{1/2}\},$$
and
$$b_{n_\pm}\equiv 2nL+|z\pm L|.$$
I found the integral formula faster to evaluate in Mathematica. Here is a plot of the field when $a=L$. To compute the field, I took the gradient of the potential by analytically differentiating inside the integral and then numerically integrating. The arrows show the field direction, and their color shows the field strength. The fringing field is obvious.

The lack of azimuthal symmetry for square plates probably makes that problem much harder. I have not found a nice formula for that potential. It might be necessary to numerically solve Poisson’s equation.
ADDENDUM:
After writing this, I noticed that appended to Atkinson's 1982 paper (cf., pages 6 and 7 of Atkinson 1982) is a 1984 Comment by another author, Hughes, who shows that these formulas are wrong! Its abstract says “A recently proposed analytic solution to a celebrated problem in classical potential theory is shown to be incorrect” and the Comment by Hughes concludes that (as of 1984), “The reduction of the circular parallel plate capacitor problem to quadratures remains an open problem.”
Further investigation reveals that this 1949 paper by E. R. Love seems to be authoritative and cited in recent studies. Its formula for the potential involves solving a Fredholm integral equation, and there are various papers (such as this 2020 one) regarding how to do this.
An amusing coincidence mentioned in the 2020 paper is that Love’s integral equation also arises in the quantum Lieb-Liniger model.
