Re: [Phys-L] Charge division on Spheres.

[John Denker <jsd@av8n.com>]

On 05/05/2017 04:44 AM, Carl Mungan wrote:
> I’m afraid that’s a very tough geometry. The rims of the two ends of
> the can are quite close to the outer pot and will dominate the
> capacitance in a tricky fashion.

Yes.

> If we try my simple model:

Why are we pretending this is a physics problem?  It's not.

The original question was grossly ill-posed, and obviously
so.  It was intended as an exercise in equation hunting,
with no connection to real-world physics.

In particular, a question that depends on "the" capacitance
is obviously ill-posed in a situation where there are N
self-capacitances and N(N-1)/2 mutual capacitances.

There is a "rule" that says anything not mentioned in the
statement of the problem must be negligible.  This "rule"
works in the lowest levels of the ivory tower and nowhere
else, but must be applied here;  otherwise the question is
unanswerable.

The question is not worth taking seriously, but hypothetically
if it were worth anything at all, the only appropriate
simplification would be to neglect *all* of the mutual
capacitances and neglect any effects coming from extraneous
pots and chassis grounds ... retaining only the two self-
capacitances, i.e. the diagonal elements of the capacitance
matrix.

As for the physics, it should be obvious from dimensional
analysis alone that the self-capacitance of a sphere scales
like the radius.  The permittivity of the vacuum is 8.85
picofarads per meter ... not meter squared or anything
like that.
  https://en.wikipedia.org/wiki/Vacuum_permittivity

Continuing down that road:  As always, dimensional analysis
is just a poor man's version of a scaling argument.  It
should be obvious from Gauss's law (conservation of flux
lines) plus the symmetry of the situation that the E field
of an isolated charged sphere falls off like 1/r^2.  You
can integrate that dr (at constant charge) to find that
the voltage goes like 1/r.  The smaller sphere has more
voltage per unit charge, i.e. less capacitance.

  I would hope that all students would be able to carry
  out this scaling analysis in the heads, in less time
  than it takes to tell about it.  This sort of thing is
  more interesting and (not coincidentally) vastly more
  useful than any amount of equation-hunting.

The original question should be stuck some place where
the sun doesn't shine.  The only thing sillier than a
spherical cow is /two/ spherical cows.