Coefficients of Potential

In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric:


\begin{matrix}
\phi_1 = p_{11}Q_1 + \cdots + p_{1n}Q_n \\
\phi_2 = p_{21}Q_1 + \cdots + p_{2n}Q_n \\
\vdots \\
\phi_n = p_{n1}Q_1 + \cdots + p_{nn}Q_n
\end{matrix}.

where Qi is the surface charge on conductor i. The coefficients of potential are the coefficients pij. φi should be correctly read as the potential due to charge 1, and hence "" is the p due to charge 2 on charge 1.

or more formally

Note that:

  1. pij = pji, by symmetry, and
  2. pij is not dependent on the charge,

The physical content of the symmetry is as follows:

if a charge Q on conductor j brings conductor i to a potential φ, then the same charge placed on i would bring j to the same potential φ.

In general, the coefficients is used when describing system of conductors, such as in the capacitor.

Read more about Coefficients Of Potential:  Theory, Example, Related Coefficients

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