Re: capacitance of a disk

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Bob Sciamanda's comments:

> Dave, your method is a version of Smythe's "oblate spheroidal harmonics"
> method where he, too, makes a transformation to boundary matching
> coordinates (his section 5.271)

Oh yeah? I would have thought that the phrase "oblate spheroidal
harmonics" had referred to making an expansion in an infinite
countable orthogonal basis set where each basis function was a product
of 3 1-dimensional basis functions for each of the 3 "conicoidal"
coordinates.  I would have expected that the expansion coefficients in
the expansion would have been found by requiring that the expansion
satisfy the relevant boundary conditions.  I suppose you could say
that my matching conditions are a special case of this since the
independence of the potential over the v and A coordinates means that
only their (presumably constant) zero-order harmonics are needed to
match the functional behavior of the potential w.r.t. these coordinates.

> His handling of the disc (pg 111-114, section 5.00 & ff) is quite
> different and would interest you. Because you'll probably have difficulty
> finding Smythe, let me try to outline at least the beginning of his
> method:
>
> The equation of a surface is F(x,y,z) = C;  for each value of C this will
> be used as an equipotential surface:  V = f(C).
>
> He then applies Laplace's equation DEL^2 V = 0 and gets
>
> DEL^2 V = f''(C)*{Grad C}^2 + f'(C)*DEL^2 C=0  =>
>
> DEL^2 C/{Grad C}^2 = - f''(C)/f'(C) = G(C)   That this is a function only
> of C is then the requirement that F(x,y,z)=C can be an equipotential .

This is a nice trick for generic equipotential surfaces.  I wish I had
thought of it.

> He then integrates this to get the potential (much skipped):
>
> V = f(C) = A INT exp{-INT G(C) dC} dC + B.   A & B are determined by
> specifying the potential on two of the surfaces.
>
> He then applies this to the "nonintersecting confocal conicoids"
>
> x^2/(a^2+C) + y^2/(b^2+C) + z^2/c^2+C) = 1  and explicitly solves the disc
> as a special case.

So he does the generic triaxial conicoids?  I'm impressed.

> In his section 5.271 he develops the "oblate spheroidal coordinates" by
> taking
> b=c , y = r*cos(phi) and z = r* sin(phi) . . . .
>
> I hope you can get Smythe, cuz I'm sure the above is over-simplified and
> bungled.  (I could mail you a photocopy, if needed.)

Can you email a scanned version?  It might be easier (& quicker).

David Bowman
David_Bowman@georgetowncollege.edu