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 due to charge 1, and hence "p₂₁" is the p due to charge 2 on charge 1.

$p_{ij} = {\part \phi_i \over \part Q_j} = \left({\part \phi_i \over \part Q_j} \right)_{Q_1,...,Q_{j-1}, Q_{j+1},...,Q_n},$

or more formally

$p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}.$

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 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:

$\frac{\sigma_j}{<\sigma_j>} = f_j,$

or

$\sigma_j = <\sigma_j>f_j = \frac{Q_j}{S_j}f_j.$

Then,

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

can be written in the form

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

i.e.

$p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}.$

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

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

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

$\begin{matrix} \phi_1 = (p_{11} - p_{12})Q \\ \phi_2 = (p_{21} - p_{22})Q \end{matrix},$

and

Δϕ = ϕ₁ − ϕ₂ = (p₁₁ + p₂₂ − p₁₂ − p₂₁)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 c_ii is called the coefficients of capacitance and the c_ij with i ≠ j is called the coefficients of induction.

The capacitance of this system can be expressed as

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

(the system of conductors can be shown to have similar symmetry c_ij = c_ji.)