Re: [Phys-L] The status of Kirchhoff''s laws

[John Denker <jsd@av8n.com>]

On 03/02/2013 08:25 PM, Bruce Sherwood wrote:
> An
> underappreciated aspect of capacitors in circuits is the role of the fringe
> field.

Indeed, the fringe field is important and often under-appreciated.

[lots of good stuff snipped]

> Now for the punch line. Suppose in a circuit that more conventional current
> is flowing onto the positive plate of a capacitor than is flowing off the
> negative plate. In that case the field in the neighboring wires contributed
> by the capacitor, just outside the capacitor, is no longer the tiny fringe
> field of a device with zero net charge. Now field in the wires can be
> large, because the capacitor has a nonzero net charge, in this case a
> positive net charge. What will this large field do? It will slow down the
> conventional current approaching the positive plate, and it will increase
> the conventional current leaving the negative plate. There is a built-in
> feedback mechanism that will bring the two currents back to being equal.

That's a good description of the ideal situation.  That describes
the way in which Kirchhoff's laws are enforced ... /assuming/ the
laws actually work.

That's fine as far as it goes, but in keeping with the spirit of 
this thread, we should also consider the cases where Kirchhoff's
laws do not work.

In particular, suppose I grab a generic .1 μF capacitor from the
stockroom, and hook /both/ leads to a Van de Graaff generator.
Then /both/ capacitor plates will pick up a charge of the same
sign, and there will be no "feedback mechanism" to bring them 
back to having equal-and-opposite charges.

Things get even weirder if I hook only /one/ of the leads to the 
VdG generator, and leave the other lead floating, not grounded.

At this point conventional terminology fails us.  Talking about
the «charge» on the capacitor is annoyingly ambiguous.  To solve
this problem, I recommend introducing the term /gorge/.  When a
capacitor has +q on one plate and -q on the other plate, we say
it has a /gorge/ equal to q.  Specifically, we define the gorge
to be

       G := (q1 - q2) / q
while -- as always -- the honest-to-goodness charge is
       Q := (q1 + q2)

To describe a capacitor (or a battery) in situations where Kirchhoff's
law has failed, you need to specify both the gorge and the charge.
Note that according to the laws of physics, charge is strictly
conserved ... whereas gorge is not conserved.  A battery can
disgorge itself via internal leakage processes.

For more discussion, including diagrams, see
  http://www.av8n.com/physics/electrophorus.htm#sec-gorge