Coefficients of potential

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

${\displaystyle {\begin{matrix}\phi _{1}=p_{11}Q_{1}+\cdots +p_{1n}Q_{n}\\\phi _{2}=p_{21}Q_{1}+\cdots +p_{2n}Q_{n}\\\vdots \\\phi _{n}=p_{n1}Q_{1}+\cdots +p_{nn}Q_{n}\end{matrix}}.}$

where 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 "${\displaystyle p_{21}}$" is the p due to charge 1 on conductor 2.

${\displaystyle p_{ij}={\partial \phi _{i} \over \partial Q_{j}}=\left({\partial \phi _{i} \over \partial Q_{j}}\right)_{Q_{1},...,Q_{j-1},Q_{j+1},...,Q_{n}}.}$

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 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

System of conductors. The electrostatic potential at point P is ${\displaystyle \phi _{P}=\sum _{j=1}^{n}{\frac {1}{4\pi \epsilon _{0}}}\int _{S_{j}}{\frac {\sigma _{j}da_{j}}{R_{j}}}}$.

Given the electrical potential on a conductor surface S_i (the equipotential surface or the point P chosen on surface i) contained in a system of conductors j = 1, 2, ..., n:

${\displaystyle \phi _{i}=\sum _{j=1}^{n}{\frac {1}{4\pi \epsilon _{0}}}\int _{S_{j}}{\frac {\sigma _{j}da_{j}}{R_{ji}}}{\mbox{ (i=1, 2..., n)}},}$

where 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:

${\displaystyle {\frac {\sigma _{j}}{\langle \sigma _{j}\rangle }}=f_{j},}$

or

${\displaystyle \sigma _{j}=\langle \sigma _{j}\rangle f_{j}={\frac {Q_{j}}{S_{j}}}f_{j}.}$

Then,

${\displaystyle \phi _{i}=\sum _{j=1}^{n}{\frac {Q_{j}}{4\pi \epsilon _{0}S_{j}}}\int _{S_{j}}{\frac {f_{j}da_{j}}{R_{ji}}}.}$

It can be shown that ${\displaystyle \int _{S_{j}}{\frac {f_{j}da_{j}}{R_{ji}}}}$ is independent of the distribution ${\displaystyle \sigma _{j}}$. Hence, with

${\displaystyle p_{ij}={\frac {1}{4\pi \epsilon _{0}S_{j}}}\int _{S_{j}}{\frac {f_{j}da_{j}}{R_{ji}}},}$

we have

${\displaystyle \phi _{i}=\sum _{j=1}^{n}p_{ij}Q_{j}{\mbox{ (i = 1, 2, ..., n)}}.}$

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

${\displaystyle {\begin{matrix}\phi _{1}=p_{11}Q_{1}+p_{12}Q_{2}\\\phi _{2}=p_{21}Q_{1}+p_{22}Q_{2}\end{matrix}}.}$

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

${\displaystyle {\begin{matrix}\phi _{1}=(p_{11}-p_{12})Q\\\phi _{2}=(p_{21}-p_{22})Q\end{matrix}},}$

and

${\displaystyle \Delta \phi =\phi _{1}-\phi _{2}=(p_{11}+p_{22}-p_{12}-p_{21})Q.}$

Hence,

${\displaystyle C={\frac {1}{p_{11}+p_{22}-2p_{12}}}.}$

Related coefficients

Note that the array of linear equations

${\displaystyle \phi _{i}=\sum _{j=1}^{n}p_{ij}Q_{j}{\mbox{ (i = 1,2,...n)}}}$

can be inverted to

${\displaystyle Q_{i}=\sum _{j=1}^{n}c_{ij}\phi _{j}{\mbox{ (i = 1,2,...n)}}}$

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]

${\displaystyle Q_{a}=(c_{11}+c_{12})V,\qquad Q_{b}=(c_{12}+c_{22})V}$

${\displaystyle Q=Q_{a}+Q_{b}=(c_{11}+2c_{12}+c_{bb})V}$

If the two conductors carry equal and opposite charges,

${\displaystyle \phi _{1}={\frac {Q(c_{12}+c_{22})}{(c_{11}c_{22}-c_{12}^{2})}},\qquad \quad \phi _{2}={\frac {-Q(c_{12}+c_{11})}{(c_{11}c_{22}-c_{12}^{2})}}}$

${\displaystyle \quad C={\frac {Q}{\phi _{1}-\phi _{2}}}={\frac {c_{11}c_{22}-c_{12}^{2}}{c_{11}+c_{22}+2c_{12}}}}$

The system of conductors can be shown to have similar symmetry c_ij = c_ji.

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.