Re: [Phys-l] defn of capacitance

[John Denker <jsd@av8n.com>]

On 02/11/2008 03:24 PM, Bob Sciamanda wrote:
> I don't understand.  I see two different questions:

Agreed.  There are two different questions ... and the
following is a good way to untangle things:

> 1) Given the matrix Cij, can one find its inverse (call it Xij) , such that 
> the matrix product C*X yields the identity matrix?  This is the problem of 
> inverting a MATRIX.
>
> 2) Given the matrix Cij such that Qi = Cij Vj ,  can one detemine the 
> matrix Pij such that Vi  = Pij Qj ?  This is the problem of inverting a 
> MATRIX EQUATION.
>
> Are you saying connecting/equating these two problems ?

I hope I'm not equating them.  



Here's how I would say it:

1) In one direction, we have not just a linear equation, but a
*system* of linear equations:
       Vi = Pij Qj + G                     [1a]
       ∑ Qj = 0                            [1b]

where G expresses gauge invariance and where equation [1b]
expresses charge neutrality.  Note that G means equation [1a]
is a linear equation but not simply a proportionality.
Having valid values for Pij does not suffice to tell the 
whole story.

   Beware there are conflicting definitions of "linear"
   http://www.av8n.com/physics/weird-terminology.htm#main-linear


2) In the other direction, things are simpler:
     Qi = Cij Vj                           [2]

If you have valid values for Cij, the internal structure 
of Cij implements gauge invariance and charge neutrality,
so no additional equations are necessary.


3) Summarizing item (1) and item (2):  the role of Pij is
not the mirror image of the role of Cij.


4) Note that equation [1b] effectively reduces the dimensionality
of the problem by one.  The mapping in equation [1a] is
singular (non-invertible) when considered a mapping in N
dimensions, but is nonsingular when considered a mapping
in N-1 dimensions (if we ignore G).

This probably isn't the end of the story.  I need to think
about this some more.  I've never had to worry about Pij
before;  Cij has always been sufficient for my needs.