# Coefficients of potential

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

${\displaystyle {\begin{matrix}\phi _{1}=p_{11}Q_{1}+\cdots +p_{1n}Q_{n}\\\phi _{2}=p_{21}Q_{1}+\cdots +p_{2n}Q_{n}\\\vdots \\\phi _{n}=p_{n1}Q_{1}+\cdots +p_{nn}Q_{n}\end{matrix}}.}$

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 on the i-th conductor, and hence "${\displaystyle p_{21}}$" is the p due to charge 1 on conductor 2.

${\displaystyle p_{ij}={\partial \phi _{i} \over \partial Q_{j}}=\left({\partial \phi _{i} \over \partial Q_{j}}\right)_{Q_{1},...,Q_{j-1},Q_{j+1},...,Q_{n}},}$

or more formally

${\displaystyle p_{ij}={\frac {1}{4\pi \epsilon _{0}S_{j}}}\int _{S_{j}}{\frac {f_{j}da_{j}}{R_{ji}}}.}$

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.

## Theory

System of conductors. The electrostatic potential at point P is ${\displaystyle \phi _{P}=\sum _{j=1}^{n}{\frac {1}{4\pi \epsilon _{0}}}\int _{S_{j}}{\frac {\sigma _{j}da_{j}}{R_{j}}}}$.

Given the electrical potential on a conductor surface Si (the equipotential surface or the point P chosen on surface i) contained in a system of conductors j = 1, 2, ..., n:

${\displaystyle \phi _{i}=\sum _{j=1}^{n}{\frac {1}{4\pi \epsilon _{0}}}\int _{S_{j}}{\frac {\sigma _{j}da_{j}}{R_{ji}}}{\mbox{ (i=1, 2..., n)}},}$

where Rji = |ri - rj|, i.e. the distance from the area-element daj to a particular point ri on conductor i. σj is not, in general, uniformly distributed across the surface. Let us introduce the factor fj that describes how the actual charge density differs from the average and itself on a position on the surface of the j-th conductor:

${\displaystyle {\frac {\sigma _{j}}{\langle \sigma _{j}\rangle }}=f_{j},}$

or

${\displaystyle \sigma _{j}=\langle \sigma _{j}\rangle f_{j}={\frac {Q_{j}}{S_{j}}}f_{j}.}$

Then,

${\displaystyle \phi _{i}=\sum _{j=1}^{n}{\frac {Q_{j}}{4\pi \epsilon _{0}S_{j}}}\int _{S_{j}}{\frac {f_{j}da_{j}}{R_{ji}}}}$

can be written in the form

${\displaystyle \phi _{i}=\sum _{j=1}^{n}p_{ij}Q_{j}{\mbox{ (i = 1, 2, ..., n)}},}$

i.e.

${\displaystyle p_{ij}={\frac {1}{4\pi \epsilon _{0}S_{j}}}\int _{S_{j}}{\frac {f_{j}da_{j}}{R_{ji}}}.}$

## 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

${\displaystyle {\begin{matrix}\phi _{1}=p_{11}Q_{1}+p_{12}Q_{2}\\\phi _{2}=p_{21}Q_{1}+p_{22}Q_{2}\end{matrix}}.}$

On a capacitor, the charge on the two conductors is equal and opposite: Q = Q1 = -Q2. Therefore,

${\displaystyle {\begin{matrix}\phi _{1}=(p_{11}-p_{12})Q\\\phi _{2}=(p_{21}-p_{22})Q\end{matrix}},}$

and

${\displaystyle \Delta \phi =\phi _{1}-\phi _{2}=(p_{11}+p_{22}-p_{12}-p_{21})Q.}$

Hence,

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

## Related coefficients

Note that the array of linear equations

${\displaystyle \phi _{i}=\sum _{j=1}^{n}p_{ij}Q_{j}{\mbox{ (i = 1,2,...n)}}}$

can be inverted to

${\displaystyle Q_{i}=\sum _{j=1}^{n}c_{ij}\phi _{j}{\mbox{ (i = 1,2,...n)}}}$

where the cij with i = j are called the coefficients of capacity and the cij with i ≠ j are called the coefficients of electrostatic induction.[1]

The capacitance of the above two-conductor example system can be expressed in terms of the cij as

${\displaystyle C={\frac {c_{11}c_{22}-c_{12}^{2}}{c_{11}+c_{22}+2c_{12}}}}$

(the system of conductors can be shown to have similar symmetry cij = cji.)

## References

1. ^ L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Course of Theoretical Physics, Vol. 8), 2nd ed. (Butterworth-Heinemann, Oxford, 1984) p. 4.