Re: capacitance of a disk

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Bob Sciamanda's comment:

> This surface charge density result is just twice Smythe's result: pg 114;
> eq (3).
> Perhaps there is a discrepancy in what "Q" stands for; in Smythe it is the
> total charge, added up over all disck surfaces.

I explained this already.   The surface charge density *is* 1/2 of what I
quoted below *if* you merely take the e --> 1 limit from the generic
conducting spheroid case.  *After* this limit is taken, however, and the
spheroid collapses to a zero-thickness circular disk, the surface charge
on the bottom merges with the surface charge on the top and this doubles
the surface charge.  Before the limit is taken the total charge is found
by integrating the surface charge over the entire surface (top *and*
bottom) of the spheroid.  *After* the e --> 1 limit is taken and the
surface charge density is doubled, then the total charge is found by just
integrating over *one* side of the disk since it has zero thickness and
both the top charge and the bottom charge can be counted together.  If
you insist on separately integrating over *both* the top and the bottom
of the disk as two separate surfaces--even when they are on top of each
other, then you are not supposed to double the surface charge density,
and just keep the limiting value you get from the e --> 1 limit of the
conducting spheroid case.  Apparently, this is what Smythe does.

> David, I have been able to spend only sporadic moments on these things -
> but I will "shortly" e-mail you some scanned pages of Smythe in TIFF
> format.
>
> Bob

Thanks. I appreciate it.

> ----- Original Message -----
> From: "David Bowman" <David_Bowman@GEORGETOWNCOLLEGE.EDU>
> To: <PHYS-L@lists.nau.edu>
> Sent: Sunday, February 11, 2001 01:44 PM
> Subject: Re: capacitance of a disk
>
> > Regarding Ludwik's question:
> > . . .
> > Yes. In my most recent post I showed how to calculate the surface
> > charge density for this limiting flat disk case as well as for the
> > generic spheroid conductor case.  To recap the result for the disk
> > the surface charge density is:
> >
> > [sigma](r) = Q/(2*[pi]*R^2*sqrt(1 - (r/R)^2))
> > . . .

David Bowman
David_Bowman@georgetowncollege.edu