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2014-01-29T17:06:22Z

216.26.134.251: /* Applications */

In [[electrostatics]], the '''[[coefficients]] of potential''' determine the relationship between the [[electric charge|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 ''Q''i is the surface charge on conductor ''i''. The coefficients of potential are the coefficients ''p''ij. φi should be correctly read as the potential due to charge 1, and hence "<math>p_{21}</math>" is the p due to charge 2 on charge 1.
:<math>p_{ij} = {\part \phi_i \over \part Q_j} = \left({\part \phi_i \over \part Q_j} \right)_{Q_1,...,Q_{j-1}, Q_{j+1},...,Q_n},</math>
or more formally
:<math>p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}.</math>

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==
<div style="float:right; text-align:center;">
[[Image:System of conductors.png]]
 ''System of conductors. The electrostatic potential at point P 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 ''S''i (the [[equipotential surface]] or the point ''P'' chosen on surface i) contained in a system of conductors j = 1, 2, ..., ''n'':
:<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 ''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. σj 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:
:<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>
can be written in the form
:<math>\phi_i=\sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1, 2, ..., n)}, </math>
i.e.
:<math>p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}.</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: ''Q'' = ''Q''1 = -''Q''2. 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 ''c''ii is called the ''[[coefficients of capacitance]]'' and the ''c''ij with i ≠ j is called the ''[[coefficients of induction]]''.

The [[capacitance]] of this system can be expressed as
:<math>C = \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 ''c''ij = ''c''ji.)

[[Category:Electrostatics]]

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