Re: [Phys-l] defn of capacitance

[John Denker <jsd@av8n.com>]

This is a really important thread.  There is some deep and
beautiful physics involved.



On 02/06/2008 11:10 PM, John Mallinckrodt wrote:

> There was a discussion on Phys-L back in March of 1997 that began  
> with a question from Ludwik titled "How many volts?" and led into  
> this same territory.
>
> See
>
> <https://carnot.physics.buffalo.edu/archives/1997/03_1997/msg00476.html>
>
> and the subsequent discussion.

That's true ... but IMHO that is a long, confused, and confusing
discussion, so rather than ask each phys-l subscriber to wade
through it I will try to summarize the high points.

1) There was a long discussion of how to solve a specific problem.
Only at the end did the discussion segue to how to understand the
basic physics.  To make a long story short, the solution for this
whole class of problems is to consider the _capacitance matrix_
so that
   Qi = Cij Vj                [1]

as in http://www.av8n.com/physics/laplace.htm#eq-q-of-v

Note that equation [1] is a linear equation.


2) Symmetry is important!  Symmetry is at the core of physics.
John M. gets the award for pouncing on the symmetry of the
problem,   Way back on Tue, 25 Mar 1997 13:36:10 -0800 (PST)
he wrote:

> I think Martha is on to something critical here.  I believe that I can
> show that the potential difference of any two isolated conductors carrying
> charges Q1 and Q2 will be given by
>  
>              Q1 - Q2   Q1 + Q2
>   delta V =  ------- + -------           [2]
>               2 Cb      2 Cc
>
> where we might call Cb the "balanced mode capacitance" and Cc the "common
> mode capacitance." 

Micro-quibble: I do *not* want to argue about terminology;  you
can use whatever terminology you like.  I merely remark that 
electrical engineers uniformly use the terms "common mode" (as
above) and "differential mode" (in contrast to "balanced mode").

> Cb corresponds to the usual definition of capacitance; 

Sorry, I can't agree with that.  Cb is the part of the capacitance
than upholds Kirchhoff's "laws" while Cc is the part that does not.
But capacitance is capacitance.  The capacitance _matrix_ is "the"
capacitance.  You shouldn't pick out one matrix element and call 
it "the" capacitance.

Note that the transformation from (Q1, Q2) to (Q1-Q2, Q1+A2) is
a linear transformation.  This transformation is tantamount to
a change-of-basis for the capacitance matrix.  So equation [2]
does not in any way conflict with equation [1];  in fact it is
a consequence of equation [1].  It's just one row of equation [1],
in a particular basis.


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


3) The following point deserves more attention than it has
as yet received.

The problems being discussed here are fundamentally 
*three-terminal* capacitor problems.

The statement of the exercise calls for placing Q1 on the first 
terminal and Q2 on the second terminal, where Q1+Q2 is nonzero.  
Alas, that's unphysical.  How are you going to do that?  Where 
are you going to get the "excess" charge i.e. Q1+Q2?  By 
conservation of charge, you have to get it from 	somewhere.

The smart way to get out of this trap is to introduce a third
terminal with charge Q3 such that Q1+Q2+Q3=0.  You can, if you
want, make the third terminal large and far away, so that the
details of its shape don't matter much ... but you should not
imagine for a moment that terminal 3 doesn't exist at all.  It
exists and is important.  The *full* capacitance matrix for 
this exercise is a 3x3 matrix.  The full capacitance matrix is
manifestly gauge-invariant and manifestly charge-conserving.

There are a lot of useful things you can do with a reduced
capacitance matrix, which is 2x2 in this case ... but you
must not confuse that with the full 3x3 capacitance matrix.

For more on this, see
  http://www.av8n.com/physics/laplace.htm