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Re: capacitance of a disk

At 08:27 PM 2/8/01 -0600, brian whatcott wrote:
As R2 grows very large, the influence of asperities
on the inner shape declines,

Well, that's narrowly true, but misleading. There's a difference between
"declines" and "declines to zero".

It's true that asperities on object 1 are more significant when object 2 is
nearby, but they remain nontrivial even when object 2 is removed to infinity.

R2 is a red herring. The question asked about an isolated object, and it
is simplest and best to think of it in those terms. All the integrals
converge just fine in the absence of other objects, so there's nothing to
gain by introducing them.

so that its projected area dominates the expression.

Projected area is not the whole story. Different shapes can have
significantly different capacitance per unit area. For instance, an
annulus will have much higher capacitance than a disk with the same area.

And to focus on the task Carl set for us, John M. proved that the
capacitance-per-unit-area of a disk is slightly more than the
capacitance-per-unit-area of a sphere. (This is a rigorous bound, and a
much tighter bound than the one I gave).