Coefficients of potential

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

$\begin{array}{c}{\varphi }_1=p_{11}{Q}_1+\cdots +p_{1n}{Q}_n\\ {\varphi }_2=p_{21}{Q}_1+\cdots +p_{2n}{Q}_n\\ ⋮\\ {\varphi }_n=p_{n1}{Q}_1+\cdots +p_{nn}{Q}_n\end{array}.$

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.

$p_{ij}=\frac{\partial {\varphi }_i}{\partial Q_j}={\left(\frac{\partial {\varphi }_i}{\partial Q_j}\right)}_{Q_1,...,Q_{j-1},Q_{j+1},...,Q_n}.$

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 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

File:System of conductors.png
System of conductors. The electrostatic potential at point P is ${\varphi }_P=\sum _{j=1}^n\frac{1}{4\pi {ϵ}_0}\underset{S_j}{\int }\frac{{\sigma }_{j}d{a}_j}{R_j}$.

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:

${\varphi }_i=\sum _{j=1}^n\frac{1}{4\pi {ϵ}_0}\underset{S_j}{\int }\frac{{\sigma }_{j}d{a}_j}{R_{ji}}\text{ (i=1, 2..., n)},$

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,

${\varphi }_i=\sum _{j=1}^n\frac{Q_j}{4\pi {ϵ}_0{S}_j}\underset{S_j}{\int }\frac{f_{j}d{a}_j}{R_{ji}}.$

It can be shown that $\underset{S_j}{\int }\frac{f_{j}d{a}_j}{R_{ji}}$ is independent of the distribution ${\sigma }_j$. Hence, with

$p_{ij}=\frac{1}{4\pi {ϵ}_0{S}_j}\underset{S_j}{\int }\frac{f_{j}d{a}_j}{R_{ji}},$

we have

${\varphi }_i=\sum _{j=1}^n{p}_{ij}{Q}_j\text{ (i = 1, 2, ..., n)}.$

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

$\begin{array}{c}{\varphi }_1=p_{11}{Q}_1+p_{12}{Q}_2\\ {\varphi }_2=p_{21}{Q}_1+p_{22}{Q}_2\end{array}.$

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

$\begin{array}{c}{\varphi }_1=(p_{11}-p_{12})Q\\ {\varphi }_2=(p_{21}-p_{22})Q\end{array},$

and

$\mathrm{\Delta }\varphi ={\varphi }_1-{\varphi }_2=(p_{11}+p_{22}-p_{12}-p_{21})Q.$

Hence,

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

Related coefficients

Note that the array of linear equations

${\varphi }_i=\sum _{j=1}^n{p}_{ij}{Q}_j\text{ (i = 1,2,...n)}$

can be inverted to

$Q_i=\sum _{j=1}^n{c}_{ij}{\varphi }_j\text{ (i = 1,2,...n)}$

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,

$Q_a=(c_{11}+c_{12})V,Q_b=(c_{12}+c_{22})V$

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

If the two conductors carry equal and opposite charges,

${\varphi }_1=\frac{Q(c_{12}+c_{22})}{(c_{11}{c}_{22}-c_{12}^2)},{\varphi }_2=\frac{-Q(c_{12}+c_{11})}{(c_{11}{c}_{22}-c_{12}^2)}$

$C=\frac{Q}{{\varphi }_1-{\varphi }_2}=\frac{c_{11}{c}_{22}-c_{12}^2}{c_{11}+c_{22}+2c_{12}}$

The system of conductors can be shown to have similar symmetry c_ij = c_ji.

References

• James Clerk Maxwell (1873) A Treatise on Electricity and Magnetism, § 86, page 89.