The capacitance references suggested by John, two days ago, do not belong
to what I can read with pleasure. Why? Because they remind me how little
mathematics I know beyond what is commonly expected in an introductory
calculus course. I believe the authors and hope that one day I may be
able to verify what they say in going from one line to another. This is
my personal problem; I suspect that I am not alone in this situation.
In any case the conclusions of Carlson and Illman (AJP, Dec. 1994, 1099)
are interesting. They show how C of a circular disk parallel plate capacitor
(axially symmetric) depends on the distance between the plates. The data
are tabulated in terms of % differences between what is numerically
calculated and what one would get by using the usual ignore-the-fringe-
effect approximation. As one would expect, the % goes up when the ratio
of L/a (distance over the radius) increases. Here are some illustrations:
Keep in mind that this refers to a traditional case with |Q1|=|Q2|=Q=const.
Also note that by knowing how C changes with distance we can calculate dV=Q/C.
But all this does not apply directly to the example we are discussing. We have
a non-zero net charge and the flux of the escaping field lines exceeds the
flux of lines which go from plate to plate. And they say nothing about how
charge distributions become non-uniform when L/a increases.