Coefficients of potential

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

ϕ1=p11Q1+⋯+p1nQnϕ2=p21Q1+⋯+p2nQn⋮ϕn=pn1Q1+⋯+pnnQn.

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 on the i-th conductor, and hence "p21" is the p due to charge 1 on conductor 2.

pij=∂ϕi∂Qj=(∂ϕi∂Qj)Q1,...,Qj−1,Qj+1,...,Qn.

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.

## Contents

* 1 Theory
* 2 Example
* 3 Related coefficients
* 4 References

## Theory

Image:System of conductors.png
System of conductors. The electrostatic potential at point P is ϕP=∑j=1n14πϵ0∫SjσjdajRj.

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:

ϕi=∑j=1n14πϵ0∫SjσjdajRji (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:

σj⟨σj⟩=fj,

σj=⟨σj⟩fj=QjSjfj.

Then,

ϕi=∑j=1nQj4πϵ0Sj∫SjfjdajRji.

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

pij=14πϵ0Sj∫SjfjdajRji,

we have

ϕi=∑j=1npijQj (i = 1, 2, ..., n).

## 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

ϕ1=p11Q1+p12Q2ϕ2=p21Q1+p22Q2.

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

ϕ1=(p11−p12)Qϕ2=(p21−p22)Q,

and

Δϕ=ϕ1−ϕ2=(p11+p22−p12−p21)Q.

Hence,

C=1p11+p22−2p12.

## Related coefficients

Note that the array of linear equations

ϕi=∑j=1npijQj (i = 1,2,...n)

can be inverted to

Qi=∑j=1ncijϕj (i = 1,2,...n)

where the cij with i = j are called the coefficients of capacity and the cij with i ≠ j are called the coefficients of electrostatic induction.[1]

For a system of two spherical conductors held at the same potential,[2]

Qa=(c11+c12)V,Qb=(c12+c22)V

Q=Qa+Qb=(c11+2c12+cbb)V

If the two conductors carry equal and opposite charges,

ϕ1=Q(c12+c22)(c11c22−c122),ϕ2=−Q(c12+c11)(c11c22−c122)

C=Qϕ1−ϕ2=c11c22−c122c11+c22+2c12

The system of conductors can be shown to have similar symmetry cij = cji.

## References

1. ↑ 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.
2. ↑ 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.

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