coefficients of potential

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

: $\begin{matrix}
\phi_1 = p_{11}Q_1 + \cdots + p_{1n}Q_n \\
\phi_2 = p_{21}Q_1 + \cdots + p_{2n}Q_n \\
\vdots \\
\phi_n = p_{n1}Q_1 + \cdots + p_{nn}Q_n
\end{matrix}.$

where {{math|Q_i}} is the surface charge on conductor {{math|i}}. The coefficients of potential are the coefficients {{math|p_ij}}. {{math|φ_i}} should be correctly read as the potential on the {{math|i}}-th conductor, and hence " $p_{21}$ " is the {{math|p}} due to charge 1 on conductor 2.

: $p_{ij} = {\partial \phi_i \over \partial Q_j} = \left({\partial \phi_i \over \partial Q_j} \right)_{Q_1,...,Q_{j-1}, Q_{j+1},...,Q_n}.$

Note that:

{{math|1=p_ij = p_ji}}, by symmetry, and
{{math|1=p_ij}} is not dependent on the charge.

The physical content of the symmetry is as follows:

: if a charge {{math|Q}} on conductor {{math|j}} brings conductor {{math|i}} to a potential {{math|φ}}, then the same charge placed on {{math|i}} would bring {{math|j}} to the same potential {{math|φ}}.

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 {{math|P}} is $\phi_P = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{j}}$ .

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

: $\phi_i = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{ji}} \mbox{ (i=1, 2..., n)},$

where {{math|1=R_ji = {{!}}r_i - r_j{{!}}}}, i.e. the distance from the area-element {{math|da_j}} to a particular point {{math|r_i}} on conductor {{math|i}}. {{math|σ_j}} is not, in general, uniformly distributed across the surface. Let us introduce the factor {{math|f_j}} that describes how the actual charge density differs from the average and itself on a position on the surface of the {{math|j}}-th conductor:

: $\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j,$

: $\sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j.$

Then,

: $\phi_i = \sum_{j = 1}^n\frac{Q_j}{4\pi\epsilon_0S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}.$

It can be shown that $\int_{S_j}\frac{f_j da_j}{R_{ji}}$ is independent of the distribution $\sigma_j$ . Hence, with

: $p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}},$

we have

: $\phi_i=\sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1, 2, ..., n)}.$

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

: $\begin{matrix}
\phi_1 = p_{11}Q_1 + p_{12}Q_2 \\
\phi_2 = p_{21}Q_1 + p_{22}Q_2
\end{matrix}.$

On a capacitor, the charge on the two conductors is equal and opposite: {{math|1=Q = Q₁ = -Q₂}}. Therefore,

: $\begin{matrix}
\phi_1 = (p_{11} - p_{12})Q \\
\phi_2 = (p_{21} - p_{22})Q
\end{matrix},$

and

: $\Delta\phi = \phi_1 - \phi_2 = (p_{11} + p_{22} - p_{12} - p_{21})Q.$

Hence,

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

Related coefficients

Note that the array of linear equations

: $\phi_i = \sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1,2,...n)}$

can be inverted to

: $Q_i = \sum_{j = 1}^n c_{ij}\phi_j \mbox{ (i = 1,2,...n)}$

where the {{math|c_ij}} with {{math|1=i = j}} are called the coefficients of capacity and the {{math|c_ij}} with {{math|i ≠ j}} are called the coefficients of electrostatic induction.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.

For a system of two spherical conductors held at the same potential,{{Cite journal|last=Lekner|first=John|date=2011-02-01|title=Capacitance coefficients of two spheres|journal=Journal of Electrostatics|volume=69|issue=1|pages=11–14|doi=10.1016/j.elstat.2010.10.002}}

: $Q_a=(c_{11}+c_{12})V , \qquad Q_b=(c_{12}+c_{22})V$

$Q =Q_a+Q_b =(c_{11}+2c_{12}+c_{bb})V$

If the two conductors carry equal and opposite charges,

: $\phi_1=\frac{Q(c_{12}+c_{22})}{{(c_{11}c_{22}-c_{12}^2)}} , \qquad \quad \phi_2=\frac{-Q(c_{12}+c_{11})}{{(c_{11}c_{22}-c_{12}^2)}}$

$\quad C =\frac{Q}{\phi_1-\phi_2}= \frac{c_{11}c_{22} - c_{12}^2}{c_{11} + c_{22} + 2c_{12}}$

The system of conductors can be shown to have similar symmetry {{math|1=c_ij = c_ji}}.

References

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

Category:Electrostatics