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Re: [Phys-l] Electric potential of charged spheres



On 11/28/2011 09:24 AM, Paul Nord wrote:
Since they have the same capacitance, sphere 1 will have 9 units of
charge and sphere 2 will have 1 unit. They'll share that equally
once they touch. After you move them far apart again they will each
have a potential of 5 V.

That's a good way of looking at it, and a good first step.

The capacitance of two spheres in contact is a much more complex
question. They would have a larger capacitance than the sum of the
two spheres individually. The voltage would therefore be lower than
5 V.

I wouldn't have said that.

One could express the energy as

E = 0.5 C V^2 [1]

... but that is not directly helpful, because we know the
charge but not the voltage. It is better to write

E = 0.5 (1/C) Q^2 [2]
= 0.5 P Q^2

where P is called the elastance.

At small separation, the energy is clearly greater than at
large separation ... which means the capacitance is *smaller*
in accordance with equation [2].

=================================================

Hint: Here is a scheme for calculating the mutual capacitance
of two side-by-side spheres:

http://www.iue.tuwien.ac.at/phd/wasshuber/node77.html

Once you know the mutual capacitance and the self capacitance,
you can grind out the complete solution.

=================================================

Note that the original problem is slightly ill-posed.

It is tempting to assume that the two spheres that were mentioned
are the only objects in the universe, but this assumption is not
tenable, because it would violate charge neutrality. All those
field lines coming off the two spheres have to terminate somewhere.

Therefore we have to assume there is some sort of charge-sink aka
counter-electrode somewhere far, far away.

This is not just a technicality, because if you want to analyze
the problem in any serious way, you need to treat it as a
three-terminal capacitor: sphere + sphere + counter-electrode.

Along the same lines, note that the original problem is stated
in a way that is not gauge-invariant. It is reasonably obvious
what gauge is intended, but as soon as you try to analyze the
problem in any serious way, you need to pin this down.

=======================

Note that the definition of elastance at
http://en.wikipedia.org/wiki/Capacitance#Capacitance_matrix

is incomplete and incomprehensible (or utter nonsense, depending
on how you interpret it), for reasons discussed at
http://www.av8n.com/physics/capacitance.htm
especially
http://www.av8n.com/physics/capacitance.htm#eq-elastance-deriv

I find this particularly annoying, because wikipedia cites me
as the "authority" for this, even though it strongly diverges
from what I actually said.