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 $Q_{i}$ is the surface charge on conductor $i$. The coefficients of potential are the coefficients $p_{ij}$. $&phi;_{i}$ should be correctly read as the potential on the $i$-th conductor, and hence "$$p_{21}$$" is the $p$ due to charge 1 on conductor 2.
 * $$p_{ij} = {\partial \phi_i \over \partial Q_j} = \left({\partial \phi_i \over \partial Q_j} \right)_{Q_1,...,Q_{j-1}, Q_{j+1},...,Q_n}.$$

Note that:
 * 1) $p_{ij} = p_{ji}$, by symmetry, and
 * 2) $p_{ij}$ 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 $&phi;$, then the same charge placed on $i$ would bring $j$ to the same potential $&phi;$.

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

Theory
System of conductors. The electrostatic potential at point $P$ is $$\phi_P = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{j}}$$.

Given the electrical potential on a conductor surface $S_{i}$ (the equipotential surface or the point $P$ chosen on surface $i$) contained in a system of conductors $j = 1, 2, ..., n$:
 * $$\phi_i = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{ji}} \mbox{ (i=1, 2..., n)},$$

where $R_{ji} = |r_{i} - r_{j}|$, i.e. the distance from the area-element $da_{j}$ to a particular point $r_{i}$ on conductor $i$. $&sigma;_{j}$ is not, in general, uniformly distributed across the surface. Let us introduce the factor $f_{j}$ that describes how the actual charge density differs from the average and itself on a position on the surface of the $j$-th conductor:
 * $$\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j,$$

or
 * $$\sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j.$$

Then,
 * $$\phi_i = \sum_{j = 1}^n\frac{Q_j}{4\pi\epsilon_0S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}.$$

It can be shown that $$\int_{S_j}\frac{f_j da_j}{R_{ji}}$$ is independent of the distribution $$\sigma_j$$. Hence, with
 * $$p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}},$$

we have
 * $$\phi_i=\sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1, 2, ..., n)}. $$

Example
In this example, we employ the method of coefficients of potential to determine the capacitance on a two-conductor system.

For a two-conductor system, the system of linear equations is

\begin{matrix} \phi_1 = p_{11}Q_1 + p_{12}Q_2 \\ \phi_2 = p_{21}Q_1 + p_{22}Q_2 \end{matrix}.$$

On a capacitor, the charge on the two conductors is equal and opposite: $Q = Q_{1} = -Q_{2}$. Therefore,

\begin{matrix} \phi_1 = (p_{11} - p_{12})Q \\ \phi_2 = (p_{21} - p_{22})Q \end{matrix},$$ and
 * $$\Delta\phi = \phi_1 - \phi_2 = (p_{11} + p_{22} - p_{12} - p_{21})Q.$$

Hence,
 * $$ C = \frac{1}{p_{11} + p_{22} - 2p_{12}}.$$

Related coefficients
Note that the array of linear equations
 * $$\phi_i = \sum_{j = 1}^n p_{ij}Q_j \mbox{   (i = 1,2,...n)}$$

can be inverted to
 * $$Q_i = \sum_{j = 1}^n c_{ij}\phi_j \mbox{   (i = 1,2,...n)}$$

where the $c_{ij}$ with $i = j$ are called the coefficients of capacity and the $c_{ij}$ with $i &ne; j$ are called the coefficients of electrostatic induction.

For a system of two spherical conductors held at the same potential,
 * $$Q_a=(c_{11}+c_{12})V,  \qquad  Q_b=(c_{12}+c_{22})V$$

$$Q =Q_a+Q_b =(c_{11}+2c_{12}+c_{bb})V$$

If the two conductors carry equal and opposite charges,
 * $$\phi_1=\frac{Q(c_{12}+c_{22})}  , \qquad  \quad   \phi_2=\frac{-Q(c_{12}+c_{11})} $$

$$ \quad C =\frac{Q}{\phi_1-\phi_2}= \frac{c_{11}c_{22} - c_{12}^2}{c_{11} + c_{22} + 2c_{12}}$$

The system of conductors can be shown to have similar symmetry $c_{ij} = c_{ji}$.