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>

\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 Template:Math is the surface charge on conductor Template:Math. The coefficients of potential are the coefficients Template:Math. Template:Math should be correctly read as the potential on the Template:Math-th conductor, and hence "<math>p_{21}</math>" is the Template:Math due to charge 1 on conductor 2.

<math>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. Template:Math, by symmetry, and
2. Template:Math is not dependent on the charge.

The physical content of the symmetry is as follows:

if a charge Template:Math on conductor Template:Math brings conductor Template:Math to a potential Template:Math, then the same charge placed on Template:Math would bring Template:Math to the same potential Template:Math.

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 Template:Math is <math>\phi_P = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{j}}</math>.

Given the electrical potential on a conductor surface Template:Math (the equipotential surface or the point Template:Math chosen on surface Template:Math) contained in a system of conductors Template:Math:

<math>\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 Template:Math, i.e. the distance from the area-element Template:Math to a particular point Template:Math on conductor Template:Math. Template:Math is not, in general, uniformly distributed across the surface. Let us introduce the factor Template:Math that describes how the actual charge density differs from the average and itself on a position on the surface of the Template:Math-th conductor:

<math>\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j,</math>

or

<math>\sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j.</math>

Then,

<math>\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>\int_{S_j}\frac{f_j da_j}{R_{ji}}</math> is independent of the distribution <math>\sigma_j</math>. Hence, with

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

we have

<math>\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>

\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: Template:Math. Therefore,

<math>

\begin{matrix} \phi_1 = (p_{11} - p_{12})Q \\ \phi_2 = (p_{21} - p_{22})Q \end{matrix},</math> and

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

Hence,

<math> C = \frac{1}{p_{11} + p_{22} - 2p_{12}}.</math>

Related coefficients

Note that the array of linear equations

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

can be inverted to

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

where the Template:Math with Template:Math are called the coefficients of capacity and the Template:Math with Template:Math are called the coefficients of electrostatic induction.<ref>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.</ref>

For a system of two spherical conductors held at the same potential,<ref>Template:Cite journal</ref>

<math>Q_a=(c_{11}+c_{12})V , \qquad Q_b=(c_{12}+c_{22})V</math>

<math>Q =Q_a+Q_b =(c_{11}+2c_{12}+c_{bb})V</math>

If the two conductors carry equal and opposite charges,

<math>\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> \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 Template:Math.

References

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