Editing Capacitance (section)

==Capacitance of conductors with simple shapes ==
Calculating the capacitance of a system amounts to solving the [[Laplace equation]] <math display="inline">\nabla^2\varphi=0</math> with a constant potential <math display="inline">\varphi</math> on the 2-dimensional surface of the conductors embedded in 3-space. This is simplified by symmetries. There is no solution in terms of elementary functions in more complicated cases.

For plane situations, analytic functions may be used to map different geometries to each other. See also [[Schwarz–Christoffel mapping]].

{| class="wikitable"
|+ Capacitance of simple systems
! Type !! Capacitance !! Diagram and definitions
|-
! Parallel-plate capacitor
| <math>\ \mathcal{C} = \frac{\ \varepsilon A\ }{d}\ </math>
| [[Image:Plate CapacitorII.svg|125px]]
*<math display="inline">\varepsilon</math>: [[Permittivity]]
|-
! Concentric cylinders
| <math>\ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\ \ln \left( R_{2}/R_{1}\right)\ }\ </math>
| [[Image:Cylindrical CapacitorII.svg|130px]]
*<math display="inline">\varepsilon</math>: [[Permittivity]]
|-
! Eccentric cylinders<ref>{{cite journal |last=Dawes |year=1973 |first=Chester L. |title=Capacitance and potential gradients of eccentric cylindrical condensers |doi=10.1063/1.1745162 |journal=Physics |volume=4 |issue=2 |pages=81–85 |url=https://aip.scitation.org/doi/abs/10.1063/1.1745162}}</ref>
| <big><math>\ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\ \operatorname{arcosh}\left(\frac{R_{1}^2 + R_{2}^2 - d^2}{2 R_{1} R_{2}}\right)\ }\ </math></big>
| [[Image:Eccentric capacitor.svg|130px]]
*<math display="inline">\varepsilon</math>: [[Permittivity]]
*<math display="inline">R_1</math>: Outer radius
*<math display="inline">R_2</math>: Inner radius 
*<math display="inline">d</math>: Distance between center
*<math display="inline">\ell</math>: Wire length
|-
! Pair of parallel wires<ref name="Jackson 1975 80">{{cite book |last=Jackson |first=J. D. |year=1975 |title=Classical Electrodynamics |publisher=Wiley |page=80}}</ref>
| <big><math>\ \mathcal{C} = \frac{\pi \varepsilon \ell}{\ \operatorname{arcosh}\left( \frac{d}{2a}\right)\ } = \frac{\pi \varepsilon \ell}{\ \ln \left( \frac{d}{\ 2a\ } + \sqrt{\frac{d^2}{\ 4a^2\ } -1\ }\right)\ }\ </math></big>
|[[Image:Parallel Wire Capacitance.svg|130px]]
|-
! Wire parallel to wall<ref name="Jackson 1975 80"/>
| <big><math>\ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\ \operatorname{arcosh}\left( \frac{d}{a}\right)\ } = \frac{2\pi \varepsilon \ell}{\ \ln \left( \frac{\ d\ }{a}+\sqrt{\frac{\ d^2\ }{a^2} - 1\ }\right)\ }\ </math></big>
|
*<math display="inline">a</math>: Wire radius
*<math display="inline">d</math>: Distance, <math display="inline">d > a</math>
*<math display="inline">\ell</math>: Wire length
|-
! Two parallel<br/>coplanar strips<ref>{{cite book | last1 = Binns | last2 = Lawrenson | year = 1973 | title = Analysis and computation of electric and magnetic field problems | publisher = Pergamon Press | isbn = 978-0-08-016638-4}}<!--| access-date = 4 June 2010 --></ref>
| <math>\ \mathcal{C} = \varepsilon \ell\ \frac{\ K\left( \sqrt{1-k^2\ } \right)\ }{ 2 K\left( k \right) }\ </math>
|
*<math display="inline">d</math>: Distance
*<math display="inline">\ell</math>: Length
*<math display="inline">w_1, w_2</math>: Strip width
*<math display="inline">\ k_1 = \left( \tfrac{\ 2 w_1\ }{d} + 1 \right)^{-1}\ </math><br/><math>\ k_2 = \left( \tfrac{\ 2 w_2\ }{d} + 1 \right)^{-1}\ </math><math>\ k = \sqrt{ k_1\ k_2\ }\ </math>
*<math display="inline">K</math>: [[Elliptic integral#Complete elliptic integral of the first kind|Complete elliptic integral of the first kind]]
|-
! Concentric spheres
| <math>\ \mathcal{C} = \frac{4\pi \varepsilon}{\ \frac{1}{R_1} - \frac{1}{R_2}\ }\ </math>
| [[Image:Spherical Capacitor.svg|97px]]
*<math display="inline">\varepsilon</math>: [[Permittivity]]
|-
! Two spheres,<br/>equal radius<ref name="Maxwell 1873 266 ff">{{Cite book |last=Maxwell |first=J.;C. |year=1873 |title=A Treatise on Electricity and Magnetism |publisher=Dover |page=266&nbsp;ff |isbn=978-0-486-60637-8}}</ref><ref>{{Cite journal |last=Rawlins |first=A.D. |year=1985 |title=Note on the capacitance of two closely separated spheres |journal=IMA Journal of Applied Mathematics |volume=34 |issue=1 |pages=119–120 |doi=10.1093/imamat/34.1.119}}</ref>
| <math>\begin{align}
\ \mathcal{C}\ = &\ {} 2 \pi \varepsilon a\ \sum_{n=1}^{\infty }\frac{\sinh \left( \ln \left( D+\sqrt{D^2-1}\right) \right) }{\sinh \left( n\ln \left( D+\sqrt{ D^2-1}\right) \right) }  \\
={}&{}2\pi \varepsilon a\left[ 1+\frac{1}{2D}+\frac{1}{4D^2}+\frac{1}{8D^3}+\frac{1}{8D^4}+\frac{3}{32D^5}+ \mathcal{O}\left( \frac{1}{D^6} \right) \right] \\
={}&{} 2\pi \varepsilon a\left[ \ln 2+\gamma -\frac{1}{2}\ln \left( 2D-2\right) + \mathcal{O}\left( 2D-2\right) \right] \\
={}&{} 2\pi \varepsilon a \,\frac{\sqrt{D^2 - 1}}{\log(q)}\left[\psi_q\left(1+\frac{i\pi}{\log(q)}\right) - i\pi - \psi_q(1)\right]
\end{align}\ </math>
|
*<math display="inline">a</math>: Radius
*<math display="inline">d</math>: Distance, <math display="inline">d > 2a</math>
*<math display="inline">D = d/2a, D > 1</math>
*<math display="inline">\gamma</math>: [[Euler–Mascheroni constant|Euler's constant]]
*<math>q = D + \sqrt{D^2 - 1}</math>
*<math>\psi_q(z)=\frac{\partial_z\Gamma_q(z)}{\Gamma_q(z)}</math>: the q-digamma function
*<math>\Gamma_q(z)</math>: the [[q-gamma function]]<ref>{{Cite book| last1 = Gasper | last2 = Rahman | title = Basic Hypergeometric Series | year = 2004 | publisher = Cambridge University Press |at = p.20-22 | isbn = 978-0-521-83357-8}}</ref>
See also [[Basic hypergeometric series]].
|-
! Sphere in front of wall<ref name="Maxwell 1873 266 ff"/>
| <math>\ \mathcal{C} = 4\pi \varepsilon a\sum_{n=1}^{\infty }\frac{\sinh \left( \ln \left( D+\sqrt{D^{2}-1}\right) \right) }{\sinh \left( n\ln \left( D+\sqrt{ D^{2}-1}\right) \right) }\ </math>
|
*<math>\ a\ </math>: Radius
*<math>\ d\ </math>: Distance, <math>d > a</math>
*<math>D=d/a</math>
|-
! Sphere
| <math>\ \mathcal{C} = 4 \pi \varepsilon a\ </math>
|
*<math>a</math>: Radius
|-
! Circular disc<ref name="Jackson 1975 128">{{cite book |last=Jackson |first=J.D. |year=1975 |title=Classical Electrodynamics |publisher=Wiley |page=128, problem 3.3 }}</ref>
| <math>\ \mathcal{C} = 8 \varepsilon a\ </math>
|
* <math>a</math>: Radius
|-
! Thin straight wire,<br/> finite length<ref>{{cite journal |last=Maxwell |first=J. C. |year=1878 |title=On the electrical capacity of a long narrow cylinder and of a disk of sensible thickness |journal=Proceedings of the London Mathematical Society |volume=IX |pages=94–101 |doi=10.1112/plms/s1-9.1.94 |url=https://zenodo.org/record/1447764 }}</ref><ref>{{Cite journal |last=Vainshtein |first=L. A. |year=1962 |title=Static boundary problems for a hollow cylinder of finite length. III Approximate formulas |journal=[[Zhurnal Tekhnicheskoi Fiziki]] |volume=32 |pages=1165–1173}}</ref><ref>{{cite journal |last=Jackson |first=J. D. |year=2000 |title=Charge density on thin straight wire, revisited |journal=American Journal of Physics |volume=68 |issue=9 |pages=789–799 |doi=10.1119/1.1302908 |bibcode = 2000AmJPh..68..789J }}</ref>
| <math>\ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\Lambda }\left[ 1+\frac{1}{\Lambda }\left( 1-\ln 2\right) +\frac{1}{\Lambda ^{2}}\left( 1+\left( 1-\ln 2\right)^2 - \frac{\pi ^{2}}{12}\right) + \mathcal{O}\left(\frac{1}{\Lambda ^{3}}\right) \right]\ </math>
|
*<math>a</math>: Wire radius
*<math>\ell</math>: Length
*<math>\ \Lambda = \ln \left( \ell/a \right)\ </math>
|}