Re: capacitance of a disk

[Bob Sciamanda <trebor@VELOCITY.NET>]

Bravisssimo!
I can confirm that this result agrees with an exact calculation by a
completely different method:
W. Smythe  "Static and Dynamic Electricity" , 2nd ed, McGraw Hill, 1939,
pg 114.

Smythe's work covers the general conducting ellipsoid of which disks
(circular and elliptical) are special cases.  He also gives the surface
charge distribution.  Gotta run - more details later

Bob


Bob Sciamanda
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor
----- Original Message -----
From: "David Bowman" <David_Bowman@GEORGETOWNCOLLEGE.EDU>
To: <PHYS-L@lists.nau.edu>
Sent: Saturday, February 10, 2001 02:32 PM
Subject: Re: capacitance of a disk


> 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.
>
> David Bowman
> David_Bowman@georgetowncollege.edu