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In [[electrostatics]], the '''[[coefficients]] of potential''' determine the relationship between the [[electric charge|charge]] and [[electrostatic potential]] ([[electrical potential]]), which is purely geometric:
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:<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''<sub>i</sub> is the surface charge on conductor ''i''. The coefficients of potential are the coefficients ''p''<sub>ij</sub>. &phi;<sub>i</sub> 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''<sub>ij</sub> = ''p''<sub>ji</sub>, by symmetry, and
# ''p''<sub>ij</sub> 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 &phi;, then the same charge placed on i would bring j to the same potential &phi;.
:
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 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''<sub>i</sub> (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''<sub>ji</sub> = |'''r'''<sub>i</sub> - '''r'''<sub>j</sub>|, i.e. the distance from the area-element ''da''<sub>j</sub> to a particular point '''r'''<sub>i</sub> on conductor i. &sigma;<sub>j</sub> is not, in general, uniformly distributed across the surface. Let us introduce the factor ''f''<sub>j</sub> 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''<sub>1</sub> = -''Q''<sub>2</sub>. 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''<sub>ii</sub> is called the ''[[coefficients of capacitance]]'' and the ''c''<sub>ij</sub> with i &ne; 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''<sub>ij</sub> = ''c''<sub>ji</sub>.)
 
[[Category:Electrostatics]]

Revision as of 03:26, 21 February 2014

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