Editing Capacitance (section)

==Mutual capacitance==
A common form is a parallel-plate [[capacitor]], which consists of two conductive plates insulated from each other, usually sandwiching a [[dielectric]] material. In a parallel plate capacitor, capacitance is very nearly proportional to the surface area of the conductor plates and inversely proportional to the separation distance between the plates.

If the charges on the plates are <math display="inline">+q</math> and <math display="inline">-q</math>, and <math display="inline">V</math> gives the [[voltage]] between the plates, then the capacitance <math display="inline">C</math> is given by <math display="block">C = \frac{q}{V},</math>
which gives the voltage/[[electric current|current]] relationship
<math display="block">i(t) = C \frac{dv(t)}{dt} + V\frac{dC}{dt},</math>
where <big><math display="inline">\frac{dv(t)}{dt}</math></big> is the instantaneous rate of change of voltage, and <big><math display="inline">\frac{dC}{dt}</math></big> is the instantaneous rate of change of the capacitance. For most applications, the change in capacitance over time is negligible, so the formula reduces to:
<math display="block">i(t) = C \frac{dv(t)}{dt},</math>

The energy stored in a capacitor is found by [[integral|integrating]] the work <math display="inline">W</math>:
<math display="block"> W_\text{charging} = \frac{1}{2}CV^2.</math>

===Capacitance matrix===
The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition <math>C = Q/V</math> does not apply when there are more than two charged plates, or when the net charge on the two plates is non-zero. To handle this case, [[James Clerk Maxwell]] introduced his ''[[coefficients of potential]]''. If three (nearly ideal) conductors are given charges <math>Q_1, Q_2, Q_3</math>, then the voltage at conductor 1 is given by
<math display="block">V_1 = P_{11}Q_1 + P_{12} Q_2 + P_{13}Q_3, </math>
and similarly for the other voltages. [[Hermann von Helmholtz]] and [[Sir William Thomson]] showed that the coefficients of potential are symmetric, so that <math>P_{12} = P_{21}</math>, etc. Thus the system can be described by a collection of coefficients known as the ''elastance matrix'' or ''reciprocal capacitance matrix'', which is defined as:
<math display="block">P_{ij} = \frac{\partial V_{i}}{\partial Q_{j}}.</math>

From this, the mutual capacitance <math>C_{m}</math> between two objects can be defined<ref name=Jackson1999>{{cite book |last=Jackson |first=John David |title=Classical Electrodynamic |publisher=John Wiley & Sons |year=1999 |edition=3rd |page=43 |isbn=978-0-471-30932-1}}</ref> by solving for the total charge <math display="inline">Q</math> and using <math>C_{m}=Q/V</math>.

<math display="block">C_m = \frac{1}{(P_{11} + P_{22})-(P_{12} + P_{21})}.</math>

Since no actual device holds perfectly equal and opposite charges on each of the two "plates", it is the mutual capacitance that is reported on capacitors.

The collection of coefficients <math>C_{ij} = \frac{\partial Q_{i}}{\partial V_{j}}</math> is known as the ''capacitance matrix'',<ref name=maxwell>{{cite book| last =Maxwell | first =James | author-link =James Clerk Maxwell | title = A treatise on electricity and magnetism |volume=1 | publisher = Clarendon Press | year = 1873 | chapter =3 | at =p. 88ff | chapter-url = https://archive.org/details/electricandmagne01maxwrich}}</ref><ref>{{Cite web |title=Capacitance: Charge as a Function of Voltage |url=http://www.av8n.com/physics/capacitance.htm |website=Av8n.com |access-date=20 September 2010}}</ref><ref>{{cite journal |last1= Smolić |first1= Ivica |last2= Klajn |first2= Bruno |date= 2021 |title= Capacitance matrix revisited |url= https://www.jpier.org/PIERB/pier.php?paper=21011501 |journal= Progress in Electromagnetics Research B |volume= 92 |pages= 1–18 |doi= 10.2528/PIERB21011501|arxiv=2007.10251 |access-date= 4 May 2021|doi-access= free }}</ref> and is the [[matrix inverse|inverse]] of the elastance matrix.