Coefficients of potential

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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 Qi is the surface charge on conductor i. The coefficients of potential are the coefficients pij. φi 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. 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.

[edit] Theory

Image:System_of_conductors.png
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 Si (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 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:

\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}}.

[edit] 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}}.

[edit] 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 cii is called the coefficients of capacitance and the cij 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 cij = cij.)