Re: [Phys-l] defn of capacitance

["Bob Sciamanda" <trebor@winbeam.com>]

I don't understand.  I see two different questions:

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 ?

----- Original Message ----- 
From: "John Denker" <jsd@av8n.com>
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Sent: Monday, February 11, 2008 4:42 PM
Subject: Re: [Phys-l] defn of capacitance


> On 02/11/2008 02:12 PM, Bob Sciamanda wrote:
> > Wangness' text "Electromagnetic Fields" begins with  Vi = Pij Qj , shows
> > that the Pij's are functions only of geometry, and finishes his treatment 
> > by
> > asking the reader (in an end of chapter problem) to show that this can be
> > inverted to yield Qi=Cij Vj .  Your paragraph 1.7 seems to say that this 
> > is
> > impossible. . .  What am I misunderstanding?
>
> I said Cij is not invertible.  No doubt about that.
>
> Inverting Pij is a lot easier than inverting Cij.
>
> Cij has to enforce gauge invariance and charge
> neutrality;  Pij doesn't.
>
> Starting from Pij seems like a cop-out.  It solves
> the easy, unimportant half of the problem, and conceals
> the difficulty and importance of the other half of the
> problem.  In real world applications, such as Laplace
> equation solvers, you start with fixed potentials and
> solve for the charge distribution, not vice versa so
> you naturally get a Cij matrix ... and if you need to
> find an inverse (one of the all-too-many inverses),
> you can't just plug Cij into your spreadsheet's matrix-
> inversion routine.  You have to be smart about it.  I
> explain how to do this at
>  http://www.av8n.com/physics/capacitance.htm#sec-inverse-cap
>
>
> _______________________________________________
> Forum for Physics Educators
> Phys-l@carnot.physics.buffalo.edu
> https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l