Are Coefficients of Potential Non-Negative?

[N2O2]

Homework Statement

I want to prove that coefficients of potential are non-negative. i.e. P_ij ≥ 0

Homework Equations

By coefficients of potential method we know that in a system of conductors the potential on i_th conductor is related to charges on conductors by:

\Phi_i = \sum P_ij Q_j

The Attempt at a Solution

It is written in Reitz and Milford that it is hard to prove in a rigorous way. I think that we should use P_ii ≥ P_ij.

 

[ardie]

how is the system arranged?

 

[N2O2]

This is a general statement. Of course we should set a specific arrangement to obtain a result. for example put Q_i on i_th conductor and assume j_th conductor to be grounded and Q on other conductors to be zero.
But I haven't found the correct conditions.

 

[ardie]

in your scenario, should the conductors be sufficiently a long way away, their potentials will not effect each other (to a certain approximation) hence defeating the equation connecting the potential of one to the charge of another.