# 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$_{ij}$. φ$_i$ 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:
* p$_{ij}$ = p$_{ji}$, by symmetry, and
* 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 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.

## 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. σ$_j$ 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,
    $\phi _{i}=\sum _{j=1}^{n}{\frac {Q_{j}}{4\pi \epsilon _{0}S_{j}}}\int _{S_{j}}{\frac {f_{j}da_{j}}{R_{ji}}}.$
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 ≠ 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,
    $\phi _{1}={\frac {Q(c_{12}+c_{22})}{(c_{11}c_{22}-c_{12}^{2})}},\qquad \quad \phi _{2}={\frac {-Q(c_{12}+c_{11})}{(c_{11}c_{22}-c_{12}^{2})}}$
$\quad C={\frac {Q}{\phi _{1}-\phi _{2}}}={\frac {c_{11}c_{22}-c_{12}^{2}}{c_{11}+c_{22}+2c_{12}}}$
The system of conductors can be shown to have similar symmetry c$_{ij}$ = c$_{ji}$.