Coefficients of potential

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

where φ_i is the electrical potential on a conductor surface S_i and Q_i is the surface charge on conductor i. The coefficients of potential is the coefficients p_ij.

or more formally

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 follow:

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.

Table of contents

1 Theory 2 Example 3 Related coefficients

Theory

System of conductors. The electrostatic potential at point P is .

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:

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 jth conductor:

or

Then,

can be written in the form

i.e.

Example

For a two-conductor system, the system of linear equations is

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

and

Hence,

Related coefficients

Note that the array of linear equations

can be inverted to

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

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