Coefficients of potential


In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential, which is purely geometric:
where is the surface charge on conductor. The coefficients of potential are the coefficients. should be correctly read as the potential on the -th conductor, and hence "" is the due to charge 1 on conductor 2.
Note that:
  1. , by symmetry, and
  2. is not dependent on the charge.
The physical content of the symmetry is as follows:
In general, the coefficients is used when describing system of conductors, such as in the capacitor.

Theory


Image:System of conductors.png
System of conductors. The electrostatic potential at point is.

Given the electrical potential on a conductor surface contained in a system of conductors :
where, i.e. the distance from the area-element to a particular point on conductor. is not, in general, uniformly distributed across the surface. Let us introduce the factor that describes how the actual charge density differs from the average and itself on a position on the surface of the -th conductor:
or
Then,
It can be shown that is independent of the distribution. Hence, with
we have

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
On a capacitor, the charge on the two conductors is equal and opposite:. Therefore,
and
Hence,

Related coefficients

Note that the array of linear equations
can be inverted to
where the with are called the coefficients of capacity and the with are called the coefficients of electrostatic induction.
For a system of two spherical conductors held at the same potential,
If the two conductors carry equal and opposite charges,
The system of conductors can be shown to have similar symmetry.