# Coefficients of potential

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

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 "" is the *p* due to charge 1 on conductor 2.

Note that:

*p*_{ij}=*p*_{ji}, by symmetry, and*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

[edit]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 j-th conductor:

or

Then,

It can be shown that is independent of the distribution . Hence, with

we have

## Example

[edit]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: *Q* = *Q*_{1} = -*Q*_{2}. Therefore,

and

Hence,

## Related coefficients

[edit]Note that the array of linear equations

can be inverted to

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]}

If the two conductors carry equal and opposite charges,

The system of conductors can be shown to have similar symmetry *c*_{ij} = *c*_{ji}.

## References

[edit]**^**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.**^**Lekner, John (2011-02-01). "Capacitance coefficients of two spheres".*Journal of Electrostatics*.**69**(1): 11–14. doi:10.1016/j.elstat.2010.10.002.

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