According to my analytic calculations the capacitance (to infinity)
of a circular disk of radius R and thickness 0 is a solvable problem, and
the *exact* answer is:
C = 8*[epsilon]_0*R
or equivalently, the capacitance of a disk of 1 meter diameter is about
35.41675127... pF.
This makes the capacitance of the disk 27.32395...% greater than half
that of a sphere of the same radius.
BTW, in case anyone is interested, I solved the problem by making a
transformation to a coordinate system whose surfaces of constant
coordinate were a nested family of confocal oblate spheroids that were
locally orthogonal to another confocal nested set of hyperboloids of
revolution (of one sheet). The third coordinate was the usual azimuthal
angle about the symmetry axis of these surfaces. The problem was
solvable because the innermost spheroid was degenerate with an
eccentricity of 1-- yet having a finite semimajor axis. This innermost
degenerate surface was the disk of interest (which had to be an
equipotential surface). The overall potential in space depended only
on the coordinate that labelled the confocal spheroids, and each one was
a surface of constant potential. The also automatically means, BTW, that
the problem of the capacitance of any oblate spheroid of arbitrary
eccentricity is also exactly solvable. If anyone is interested I can
post the potential function (after it is reconverted back to spherical
coordinates), but it is quite a mess to try to reduce to ASCII notation,
so I hesitate to post it unless there is sufficient curiosity about it.