Charge distrbtn on a disk

["Prof. John P. Ertel (wizard)" <jpe@nadn.navy.mil>]

On Thu, 3 Apr 1997, LUDWIK KOWALSKI wrote:
>                                                  And there is a clear 
> link between this philosophy and the elementary calculation I am trying to 
> finish. It shows that the final distribution of Q over a metallic disk can 
> not possibly remain uniform. Why? Because I found a distribution for which 
> potential energy (the sum of all the k*dQ*dq/r components) is smaller than 
> for the uniform one. I will be more specific during the weekend.

LUDWIK,

You might not want to spend too much time on the "distribution of charge
on a conducting disk" problem --- it's already been done by Sir W. Thomson
(AKA Lord Kelvin) in
   "Papers On Electrostatics And Magnetism" QC7.5, K29, R4 
   and again in David Jackson's E&M text (Sect 3.12, 1962ed)

As noted in David Jackson's E&M text, the equilibrium charge density is

                    q           1
     sigma(R) =  ------ * ---------------      ,   for all R<a
                 2 pi a   Sqrt(a^2 - R^2)

where a is the radius of the disk, R is where you are on the disk, and the
thickness of the disk is assumed to be negligible as compared to a.

In a section in the _Method_of_Inversion_, Jackson has also noted:

"A very interesting use of inversion was made by Lord Kelvin in 1847.  He
calculated the charge densities on the inner and outer surfaces of a thin,
charged conducting bowl made from a sphere with a cap cut out of it.  The
potential distribution which he inverted was that of a thin, flat,
charged, circular disk.  As the shape of the bowl is varied from a shallow
watch glass-like shape to an almost closed sphere, the charge densities go
from those of the disk to those of a closed sphere, in one limit being
almost the same inside and out, but concentrated on the edges of the bowl,
and in the other limit being almost zero on the inner surface and uniform
on the outer surface."

And we sometimes think we're good at math today --- WOW!


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