In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric:
:<math>
\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}.</math>
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 "<math>p_{21}</math>" is the due to charge 1 on conductor 2.
:<math>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}.</math>
Note that:
- , by symmetry, and
- is not dependent on the charge.
The physical content of the symmetry is as follows:
: if a charge on conductor brings conductor to a potential , then the same charge placed on would bring to the same potential .
In general, the coefficients is used when describing system of conductors, such as in the capacitor.
Theory
<div style="float:right; text-align:center;">
Image:System of conductors.png
<br>System of conductors. The electrostatic potential at point is <math>\phi_P = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{j</math>.
</div>
Given the electrical potential on a conductor surface (the equipotential surface or the point chosen on surface ) contained in a system of conductors :
:<math>\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)},</math>
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:
:<math>\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j,</math>
or
: <math>\sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j.</math>
Then,
:<math>\phi_i = \sum_{j = 1}^n\frac{Q_j}{4\pi\epsilon_0S_j}\int_{S_j}\frac{f_j da_j}{R_{ji.</math>
It can be shown that <math>\int_{S_j}\frac{f_j da_j}{R_{ji</math> is independent of the distribution <math>\sigma_j</math>. Hence, with
:<math>p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji,</math>
we have
:<math>\phi_i=\sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1, 2, ..., n)}. </math>
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
:<math>
\begin{matrix}
\phi_1 = p_{11}Q_1 + p_{12}Q_2 \\
\phi_2 = p_{21}Q_1 + p_{22}Q_2
\end{matrix}.</math>
On a capacitor, the charge on the two conductors is equal and opposite: . Therefore,
:<math>
\begin{matrix}
\phi_1 = (p_{11} - p_{12})Q \\
\phi_2 = (p_{21} - p_{22})Q
\end{matrix},</math>
and
:<math>\Delta\phi = \phi_1 - \phi_2 = (p_{11} + p_{22} - p_{12} - p_{21})Q.</math>
Hence,
: <math> C = \frac{1}{p_{11} + p_{22} - 2p_{12.</math>
Related coefficients
Note that the array of linear equations
:<math>\phi_i = \sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1,2,...n)}</math>
can be inverted to
:<math>Q_i = \sum_{j = 1}^n c_{ij}\phi_j \mbox{ (i = 1,2,...n)}</math>
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,
:<math>Q_a=(c_{11}+c_{12})V , \qquad Q_b=(c_{12}+c_{22})V</math>
<math>Q =Q_a+Q_b =(c_{11}+2c_{12}+c_{bb})V</math>
If the two conductors carry equal and opposite charges,
:<math>\phi_1=\frac{Q(c_{12}+c_{22})} , \qquad \quad \phi_2=\frac{-Q(c_{12}+c_{11})} </math>
<math> \quad C =\frac{Q}{\phi_1-\phi_2}= \frac{c_{11}c_{22} - c_{12}^2}{c_{11} + c_{22} + 2c_{12</math>
The system of conductors can be shown to have similar symmetry .
References
- James Clerk Maxwell (1873) A Treatise on Electricity and Magnetism, § 86, page 89.