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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]]

Latest revision as of 22:35, 15 December 2014

Have we ever heard which someone mentioned "My computer is getting slower, Help me?" Are we looking techniques on how do I accelerate my computer? Are you tired of wasting too much time because the loading procedure is absolutely slow? If you wish to understand how to improve plus speed up computer performance, then this article may assist show you several concepts plus strategies "What is the cause?" plus How to prevent your computer getting slower?

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How to fix this problem is to initially reinstall the program(s) causing the mistakes. There are a great deal of different programs which use this file, nevertheless one can have placed their own faulty variation of the file onto a system. By reinstalling any programs that are causing the error, you'll not just enable your PC to run the program properly, but a unique file is placed onto a system - exiting a computer running as smoothly because possible again. If you try this, plus discover it refuses to function, then you need to look to update a program & any software we have on the PC. This will likely update the Msvcr71.dll file, permitting the computer to read it properly again.

Many tuneup utilities 2014 s allow you to download their product for free, thus you can scan the computer oneself. That means you can see how many mistakes it finds, where it finds them, plus how it may fix them. A awesome registry cleaner might remove a registry issues, plus optimize plus accelerate a PC, with small effort on the piece.

S/w related error handling - If the blue screen bodily memory dump occurs following the installation of s/w application or perhaps a driver it might be that there is system incompatibility. By booting into secure mode and removing the software we can rapidly fix this error. You may also try out a "program restore" to revert to an earlier state.

Your registry is the destination all your significant configurations for hardware, software and user profile configurations plus needs are stored. Every time one of these items is changed, the database then starts to expand. Over time, the registry could become bloated with unnecessary files. This causes a general slow down but inside extreme instances could result important tasks plus programs to stop working all together.

There are numerous businesses that offer the service of troubleshooting the PC every time we call them, all you need to do is signal up with them plus for a little fee, we could have the machine constantly working effectively and serve you better.

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