Coefficients of potential

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

[math]\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}. }[/math]

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 "[math]\displaystyle{ p_{21} }[/math]" is the p due to charge 1 on conductor 2.

[math]\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}. }[/math]

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

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:

[math]\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)}, }[/math]

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:

[math]\displaystyle{ \frac{\sigma_j}{\langle\sigma_j\rangle} = f_j, }[/math]

or

[math]\displaystyle{ \sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j. }[/math]

Then,

[math]\displaystyle{ \phi_i = \sum_{j = 1}^n\frac{Q_j}{4\pi\epsilon_0S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}. }[/math]

It can be shown that [math]\displaystyle{ \int_{S_j}\frac{f_j da_j}{R_{ji}} }[/math] is independent of the distribution [math]\displaystyle{ \sigma_j }[/math]. Hence, with

[math]\displaystyle{ p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}, }[/math]

we have

[math]\displaystyle{ \phi_i=\sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1, 2, ..., n)}. }[/math]

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

[math]\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}. }[/math]

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

[math]\displaystyle{ \begin{matrix} \phi_1 = (p_{11} - p_{12})Q \\ \phi_2 = (p_{21} - p_{22})Q \end{matrix}, }[/math]

and

[math]\displaystyle{ \Delta\phi = \phi_1 - \phi_2 = (p_{11} + p_{22} - p_{12} - p_{21})Q. }[/math]

Hence,

[math]\displaystyle{ C = \frac{1}{p_{11} + p_{22} - 2p_{12}}. }[/math]

Related coefficients

Note that the array of linear equations

[math]\displaystyle{ \phi_i = \sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1,2,...n)} }[/math]

can be inverted to

[math]\displaystyle{ Q_i = \sum_{j = 1}^n c_{ij}\phi_j \mbox{ (i = 1,2,...n)} }[/math]

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.^[1]

For a system of two spherical conductors held at the same potential,^[2]

[math]\displaystyle{ Q_a=(c_{11}+c_{12})V , \qquad Q_b=(c_{12}+c_{22})V }[/math]

[math]\displaystyle{ Q =Q_a+Q_b =(c_{11}+2c_{12}+c_{bb})V }[/math]

If the two conductors carry equal and opposite charges,

[math]\displaystyle{ \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)}} }[/math]

[math]\displaystyle{ \quad C =\frac{Q}{\phi_1-\phi_2}= \frac{c_{11}c_{22} - c_{12}^2}{c_{11} + c_{22} + 2c_{12}} }[/math]

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.

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