The formula used by Mma for the fringing field of a capacitor was first given in Valluri, Jeffrey, Corless: Canadian J. Physics, vol 78, pp823-831 (2000) rwg@sdf.lonestar.org wrote:
The (mostly impressive) Mma 6.0 doc gives, under applications of EllipticF, "Parametrization of a mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):", followed by three definitions and a call to ParametricPlot3D that makes an "M&M". Inelastic mylar would crinkle at the equator, so the true solution would be more like two shallow paper cupcake molds.
More convincing, the doc of LambertW (which they insist on calling ProductLog, probably because a bully named Lambert beat up Wolfram in grade school or something), claims that the equipotentials of the fringing field of a plate capacitor are given by
\[Phi][{x_, y_}] := With[{z = x + I y}, Im[z - 1 - ProductLog[Ceiling[(y - Pi)/(2 Pi)], Exp[z - 1]]]]
ContourPlot[\[Phi][{x, y}], {x, -2, 2}, {y, -4, 4}, Epilog -> {Red, Thickness[0.02], Line[{{-2, Pi}, {0, Pi}}], Line[{{-2, -Pi}, {0, -Pi}}]}, ContourShading -> False, Contours -> 20]
This would make a nice closure. AT MIT, they always told us to neglect fringing because it was too difficult. --rwg