SherbyNet - /page/wikibook/entry.mu

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<title>Capacitance</title>

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Capacitance

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</style><div role="note" class="hatnote navigation-not-searchable">For capacitance of blood vessels, see <a href="Compliance_(physiology)" title="Compliance (physiology)">Compliance (physiology)</a>.</div>

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</style><table class="infobox"><tbody><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data">C</td></tr><tr><th scope="row" class="infobox-label"><a href="SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="Farad" title="Farad">farad</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data">μF, nF, pF</td></tr><tr><th scope="row" class="infobox-label">In <a href="SI_base_unit" title="SI base unit">SI base units</a></th><td class="infobox-data">F = A2 s4 kg−1 m−2</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from other quantities</div></th><td class="infobox-data">C = <a href="Electric_charge" title="Electric charge">charge</a> / <a href="Voltage" title="Voltage">voltage</a></td></tr><tr><th scope="row" class="infobox-label"><a href="Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}^{-2}{\mathsf {M}}^{-1}{\mathsf {T}}^{4}{\mathsf {I}}^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}^{-2}{\mathsf {M}}^{-1}{\mathsf {T}}^{4}{\mathsf {I}}^{2}}</annotation>

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</math><img src="./686139104bb585c59a841f6f8c2ceb824475eb8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.297ex; height:2.676ex;" alt="{\displaystyle {\mathsf {L}}^{-2}{\mathsf {M}}^{-1}{\mathsf {T}}^{4}{\mathsf {I}}^{2}}" loading="lazy"></td></tr></tbody></table>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks em-sidebar"><tbody><tr><th class="sidebar-title"><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Electricity" title="Electricity">Electricity</a></li>

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<li><a href="Displacement_current" title="Displacement current">Displacement current</a></li>

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<li><a href="Poynting_vector" title="Poynting vector">Poynting vector</a></li>

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<ul><li><a href="Alternating_current" title="Alternating current">Alternating current</a></li>

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<ul><li><a href="AC_motor" title="AC motor">AC motor</a></li>

<li><a href="DC_motor" title="DC motor">DC motor</a></li>

<li><a href="Electric_machine" title="Electric machine">Electric machine</a></li>

<li><a href="Electric_motor" title="Electric motor">Electric motor</a></li>

<li><a href="Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">Gyrator–capacitor</a></li>

<li><a href="Induction_motor" title="Induction motor">Induction motor</a></li>

<li><a href="Linear_motor" title="Linear motor">Linear motor</a></li>

<li><a href="Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a></li>

<li><a href="Permeance" title="Permeance">Permeance</a></li>

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<li><a href="Rotor_(electric)" title="Rotor (electric)">Rotor</a></li>

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<ul><li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li>

<li><a href="Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Electromagnetism and special relativity</a></li>

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<li><a href="Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li>

<li><a href="Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li>

<li><a href="Maxwell's_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell equations in curved spacetime</a></li>

<li><a href="Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li>

<li><a href="Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li>

<li><a href="Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>

<li><a href="Michael_Faraday" title="Michael Faraday">Faraday</a></li>

<li><a href="Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>

<li><a href="Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li>

<li><a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li>

<li><a href="Joseph_Henry" title="Joseph Henry">Henry</a></li>

<li><a href="Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li>

<li><a href="John_Hopkinson" title="John Hopkinson">Hopkinson</a></li>

<li><a href="Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li>

<li><a href="James_Prescott_Joule" title="James Prescott Joule">Joule</a></li>

<li><a href="Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li>

<li><a href="Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li>

<li><a href="Joseph_Larmor" title="Joseph Larmor">Larmor</a></li>

<li><a href="Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li>

<li><a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li>

<li><a href="James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li>

<li><a href="Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li>

<li><a href="Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li>

<li><a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li>

<li><a href="John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li>

<li><a href="William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li>

<li><a href="F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li>

<li><a href="George_Singer" title="George Singer">Singer</a></li>

<li><a href="Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li>

<li><a href="Nikola_Tesla" title="Nikola Tesla">Tesla</a></li>

<li><a href="J._J._Thomson" title="J. J. Thomson">Thomson</a></li>

<li><a href="Alessandro_Volta" title="Alessandro Volta">Volta</a></li>

<li><a href="Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li>

<li><a href="Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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Capacitance is the ability of an object to store <a href="Electric_charge" title="Electric charge">electric charge</a>. It is measured by the change in charge in response to a difference in <a href="Electric_potential" title="Electric potential">electric potential</a>, expressed as the ratio of those quantities. Commonly recognized are two closely related notions of capacitance: self capacitance and mutual capacitance.<a href="#cite_note-Harrington_2003-1">[1]</a>: 237–238 An object that can be electrically charged exhibits self capacitance, for which the electric potential is measured between the object and ground. Mutual capacitance is measured between two components, and is particularly important in the operation of the <a href="Capacitor" title="Capacitor">capacitor</a>, an elementary <a href="Linear_circuit" title="Linear circuit">linear</a> <a href="Electronic_component" title="Electronic component">electronic component</a> designed to add capacitance to an <a href="Electric_circuit" class="mw-redirect" title="Electric circuit">electric circuit</a>.

The capacitance between two <a href="Electrical_resistance_and_conductance" title="Electrical resistance and conductance">conductors</a> depends only on the geometry; the opposing surface area of the conductors and the distance between them; and the <a href="Permittivity" title="Permittivity">permittivity</a> of any <a href="Dielectric" title="Dielectric">dielectric</a> material between them. For many dielectric materials, the permittivity, and thus the capacitance, is independent of the potential difference between the conductors and the total charge on them.

The <a href="SI" class="mw-redirect" title="SI">SI</a> unit of capacitance is the <a href="Farad" title="Farad">farad</a> (symbol: F), named after the English physicist <a href="Michael_Faraday" title="Michael Faraday">Michael Faraday</a>.<a href="#cite_note-NSW-2">[2]</a> A 1 farad capacitor, when charged with 1 <a href="Coulomb" title="Coulomb">coulomb</a> of electrical charge, has a potential difference of 1 <a href="Volt" title="Volt">volt</a> between its plates.<a href="#cite_note-3">[3]</a> The reciprocal of capacitance is called <a href="Elastance" title="Elastance">elastance</a>.

<div class="mw-heading mw-heading2"><h2 id="Self_capacitance">Self capacitance</h2></div>

In discussing electrical circuits, the term capacitance is usually a shorthand for the mutual capacitance between two adjacent conductors, such as the two plates of a capacitor. However, every isolated conductor also exhibits capacitance, here called self capacitance. It is measured by the amount of electric charge that must be added to an isolated conductor to raise its <a href="Electric_potential" title="Electric potential">electric potential</a> by one unit of measurement, e.g., one <a href="Volt" title="Volt">volt</a>.<a href="#cite_note-4">[4]</a> The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, with the conductor centered inside this sphere.

Self capacitance of a conductor is defined by the ratio of charge and electric potential:

<mfrac>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle C={\frac {q}{V}},}</annotation>

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where

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<annotation encoding="application/x-tex">{\textstyle q}</annotation>

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</math><img src="./a32efd4de244331181b8fa97862015251f65da5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\textstyle q}" loading="lazy"> is the charge held,</li>

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<annotation encoding="application/x-tex">{\textstyle V={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\sigma }{r}}\,dS}</annotation>

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</math><img src="./cc842c312f8260c957eac5dd4557f5fe4a13f643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.154ex; height:4.009ex;" alt="{\textstyle V={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\sigma }{r}}\,dS}" loading="lazy"></big> is the electric potential,</li>

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<annotation encoding="application/x-tex">{\textstyle \sigma }</annotation>

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</math><img src="./8ca336f17488923d11206e131f7eb2a569e8dde2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle \sigma }" loading="lazy"> is the surface charge density,</li>

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<annotation encoding="application/x-tex">{\textstyle dS}</annotation>

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</math><img src="./a4d5150354609d35245fc96ee526f91f45d66838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.715ex; height:2.176ex;" alt="{\textstyle dS}" loading="lazy"> is an infinitesimal element of area on the surface of the conductor, over which the surface charge density is integrated,</li>

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<annotation encoding="application/x-tex">{\textstyle r}</annotation>

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</math><img src="./f2dfb06630b52c9e18fcc0a4688da10774206729.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\textstyle r}" loading="lazy"> is the length from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle dS}">

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<annotation encoding="application/x-tex">{\textstyle dS}</annotation>

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</math><img src="./a4d5150354609d35245fc96ee526f91f45d66838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.715ex; height:2.176ex;" alt="{\textstyle dS}" loading="lazy"> to a fixed point M on the conductor,</li>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>

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</math><img src="./acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"> is the <a href="Vacuum_permittivity" title="Vacuum permittivity">vacuum permittivity</a>.</li></ul>

Using this method, the self capacitance of a conducting sphere of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R}">

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<annotation encoding="application/x-tex">{\textstyle R}</annotation>

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</math><img src="./197e66194eb64577670e2a100026bff6fb15d236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\textstyle R}" loading="lazy"> in free space (i.e. far away from any other charge distributions) is:<a href="#cite_note-NSW-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle C=4\pi \varepsilon _{0}R.}</annotation>

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</math>

Example values of self capacitance are:

<ul><li>for the top "plate" of a <a href="Van_de_Graaff_generator" title="Van de Graaff generator">van de Graaff generator</a>, typically a sphere 20 cm in radius: 22.24 pF,</li>

<li>the planet <a href="Earth" title="Earth">Earth</a>: about 710 μF.<a href="#cite_note-5">[5]</a></li></ul>

The inter-winding capacitance of a <a href="Electromagnetic_coil" title="Electromagnetic coil">coil</a> is sometimes called self capacitance,<a href="#cite_note-6">[6]</a> but this is a different phenomenon. It is actually mutual capacitance between the individual turns of the coil and is a form of stray or <a href="Parasitic_capacitance" title="Parasitic capacitance">parasitic capacitance</a>. This self capacitance is an important consideration at high frequencies: it changes the <a href="Electrical_impedance" title="Electrical impedance">impedance</a> of the coil and gives rise to parallel <a href="Electrical_resonance" title="Electrical resonance">resonance</a>. In many applications this is an undesirable effect and sets an upper frequency limit for the correct operation of the circuit.

<div class="mw-heading mw-heading2"><h2 id="Mutual_capacitance">Mutual capacitance</h2></div>

A common form is a parallel-plate <a href="Capacitor" title="Capacitor">capacitor</a>, which consists of two conductive plates insulated from each other, usually sandwiching a <a href="Dielectric" title="Dielectric">dielectric</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle +q}">

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<annotation encoding="application/x-tex">{\textstyle +q}</annotation>

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</math><img src="./e01b6be93d1af857b742f24acc35a9d1e975e7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.878ex; height:2.343ex;" alt="{\textstyle +q}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle -q}">

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<annotation encoding="application/x-tex">{\textstyle -q}</annotation>

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</math><img src="./fdd664c9a379cb62fe3d5e3927e79f7651c8851c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.878ex; height:2.343ex;" alt="{\textstyle -q}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

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<annotation encoding="application/x-tex">{\textstyle V}</annotation>

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</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> gives the <a href="Voltage" title="Voltage">voltage</a> between the plates, then the capacitance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C}</annotation>

</semantics>

</math><img src="./6dca76d9ff4b48256b6a4a99bcb234b64b2fa72b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\textstyle C}" loading="lazy"> is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C={\frac {q}{V}},}">

<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C={\frac {q}{V}},}</annotation>

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which gives the voltage/<a href="Electric_current" title="Electric current">current</a> relationship

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle i(t)=C{\frac {dv(t)}{dt}}+V{\frac {dC}{dt}},}</annotation>

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where <big><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {dv(t)}{dt}}}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {dv(t)}{dt}}}</annotation>

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</math><img src="./1296c62ad33ac50df8569d468b43c941416cf458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.366ex; height:4.343ex;" alt="{\textstyle {\frac {dv(t)}{dt}}}" loading="lazy"></big> is the instantaneous rate of change of voltage, and <big><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {dC}{dt}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\textstyle {\frac {dC}{dt}}}</annotation>

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</math><img src="./4cea8d9be913286386806ebdc1f9c4934dc7ed7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.945ex; height:3.843ex;" alt="{\textstyle {\frac {dC}{dt}}}" loading="lazy"></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:

<mfrac>

<mrow>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle i(t)=C{\frac {dv(t)}{dt}},}</annotation>

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</math>

The energy stored in a capacitor is found by <a href="Integral" title="Integral">integrating</a> the work <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle W}</annotation>

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</math><img src="./e95737ee2530885a10b104e9cd5331077e1c88d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\textstyle W}" loading="lazy">:

<msub>

<mtext>charging</mtext>

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<annotation encoding="application/x-tex">{\displaystyle W_{\text{charging}}={\frac {1}{2}}CV^{2}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Capacitance_matrix">Capacitance matrix</h3></div>

The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=Q/V}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle C=Q/V}</annotation>

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</math><img src="./865af670ac05c87761922dc6685480d11f616622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.653ex; height:2.843ex;" alt="{\displaystyle C=Q/V}" loading="lazy"> 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, <a href="James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> introduced his <a href="Coefficients_of_potential" title="Coefficients of potential">coefficients of potential</a>. If three (nearly ideal) conductors are given charges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1},Q_{2},Q_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{1},Q_{2},Q_{3}}</annotation>

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</math><img src="./9ddf0d0c3a5913a13ec2b1f92a68ac2f0e035a7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.746ex; height:2.509ex;" alt="{\displaystyle Q_{1},Q_{2},Q_{3}}" loading="lazy">, then the voltage at conductor 1 is given by

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<annotation encoding="application/x-tex">{\displaystyle V_{1}=P_{11}Q_{1}+P_{12}Q_{2}+P_{13}Q_{3},}</annotation>

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</math>

and similarly for the other voltages. <a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a> and <a href="Sir_William_Thomson" class="mw-redirect" title="Sir William Thomson">Sir William Thomson</a> showed that the coefficients of potential are symmetric, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{12}=P_{21}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{12}=P_{21}}</annotation>

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</math><img src="./9fa70d2158b9f7cc1e64b1f4a56a15d84d257b4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.835ex; height:2.509ex;" alt="{\displaystyle P_{12}=P_{21}}" loading="lazy">, 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:

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<annotation encoding="application/x-tex">{\displaystyle P_{ij}={\frac {\partial V_{i}}{\partial Q_{j}}}.}</annotation>

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From this, the mutual capacitance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{m}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle C_{m}}</annotation>

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</math><img src="./957e2166e45b11d83b27710666eb7aa38f5e4b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.337ex; height:2.509ex;" alt="{\displaystyle C_{m}}" loading="lazy"> between two objects can be defined<a href="#cite_note-Jackson1999-7">[7]</a> by solving for the total charge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle Q}</annotation>

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</math><img src="./131bb2da649dd6c113517c5ae8c26370654ee8fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\textstyle Q}" loading="lazy"> and using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{m}=Q/V}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle C_{m}=Q/V}</annotation>

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</math><img src="./d24afcadedbbc56d62143a03b465e321d6a43f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.223ex; height:2.843ex;" alt="{\displaystyle C_{m}=Q/V}" loading="lazy">.

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<annotation encoding="application/x-tex">{\displaystyle C_{m}={\frac {1}{(P_{11}+P_{22})-(P_{12}+P_{21})}}.}</annotation>

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</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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ij}={\frac {\partial Q_{i}}{\partial V_{j}}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle C_{ij}={\frac {\partial Q_{i}}{\partial V_{j}}}}</annotation>

</semantics>

</math><img src="./5531170cf5666234fc34baadfa12b176321997e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.03ex; height:6.176ex;" alt="{\displaystyle C_{ij}={\frac {\partial Q_{i}}{\partial V_{j}}}}" loading="lazy"> is known as the capacitance matrix,<a href="#cite_note-maxwell-8">[8]</a><a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> and is the <a href="Matrix_inverse" class="mw-redirect" title="Matrix inverse">inverse</a> of the elastance matrix.

<div class="mw-heading mw-heading2"><h2 id="Capacitors">Capacitors</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Capacitor" title="Capacitor">Capacitor</a></div>

The capacitance of the majority of capacitors used in electronic circuits is generally several orders of magnitude smaller than the <a href="Farad" title="Farad">farad</a>. The most common units of capacitance are the <a href="Micro-" title="Micro-">microfarad</a> (μF), <a href="Nano-" title="Nano-">nanofarad</a> (nF), <a href="Pico-" class="mw-redirect" title="Pico-">picofarad</a> (pF), and, in microcircuits, <a href="Femto-" class="mw-redirect" title="Femto-">femtofarad</a> (fF). Some applications also use <a href="Supercapacitors" class="mw-redirect" title="Supercapacitors">supercapacitors</a> that can be much larger, as much as hundreds of farads, and parasitic capacitive elements can be less than a femtofarad. Historical texts use other, obsolete submultiples of the farad, such as "mf" and "mfd" for microfarad (μF); "mmf", "mmfd", "pfd", "μμF" for picofarad (pF).<a href="#cite_note-11">[11]</a><a href="#cite_note-12">[12]</a>

The capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. Capacitance is proportional to the area of overlap and inversely proportional to the separation between conducting sheets. The closer the sheets are to each other, the greater the capacitance.

An example is the capacitance of a capacitor constructed of two parallel plates both of area <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A}">

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<annotation encoding="application/x-tex">{\textstyle A}</annotation>

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</math><img src="./a118c6ad00742b3f5dccd2f0e74b5e369df6fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}" loading="lazy"> separated by a distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d}">

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<annotation encoding="application/x-tex">{\textstyle d}</annotation>

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</math><img src="./252135f29da0e9f9e130ff2d53be5df2f7044d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\textstyle d}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d}">

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<annotation encoding="application/x-tex">{\textstyle d}</annotation>

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</math><img src="./252135f29da0e9f9e130ff2d53be5df2f7044d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\textstyle d}" loading="lazy"> is sufficiently small with respect to the smallest chord of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A}">

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<annotation encoding="application/x-tex">{\textstyle A}</annotation>

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</math><img src="./a118c6ad00742b3f5dccd2f0e74b5e369df6fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}" loading="lazy">, there holds, to a high level of accuracy:

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<annotation encoding="application/x-tex">{\displaystyle \ C=\varepsilon {\frac {A}{d}};}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon =\varepsilon _{0}\varepsilon _{r},}</annotation>

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where

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<annotation encoding="application/x-tex">{\textstyle C}</annotation>

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</math><img src="./6dca76d9ff4b48256b6a4a99bcb234b64b2fa72b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\textstyle C}" loading="lazy"> is the capacitance, in farads;</li>

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<annotation encoding="application/x-tex">{\textstyle A}</annotation>

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</math><img src="./a118c6ad00742b3f5dccd2f0e74b5e369df6fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}" loading="lazy"> is the area of overlap of the two plates, in square meters;</li>

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<annotation encoding="application/x-tex">{\textstyle \varepsilon _{0}}</annotation>

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</math><img src="./1b8b05479dc2dccbc42ee2f8d1b45d99a49b0a61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\textstyle \varepsilon _{0}}" loading="lazy"> is the <a href="Vacuum_permittivity" title="Vacuum permittivity">electric constant</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varepsilon _{0}\approx 8.854\times 10^{-12}~\mathrm {F{\cdot }m^{-1}} }">

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<annotation encoding="application/x-tex">{\textstyle \varepsilon _{0}\approx 8.854\times 10^{-12}~\mathrm {F{\cdot }m^{-1}} }</annotation>

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</math><img src="./1476af647eaf4a8f4365e1240cfe0332840b1762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.867ex; height:3.009ex;" alt="{\textstyle \varepsilon _{0}\approx 8.854\times 10^{-12}~\mathrm {F{\cdot }m^{-1}} }" loading="lazy">);</li>

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<annotation encoding="application/x-tex">{\textstyle \varepsilon _{r}}</annotation>

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</math><img src="./6aeb9464ac233fab92afdb47a3cd78da84aef5e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.057ex; height:2.009ex;" alt="{\textstyle \varepsilon _{r}}" loading="lazy"> is the <a href="Relative_permittivity" title="Relative permittivity">relative permittivity</a> (also dielectric constant) of the material in between the plates (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varepsilon _{r}\approx 1}">

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<annotation encoding="application/x-tex">{\textstyle \varepsilon _{r}\approx 1}</annotation>

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</math><img src="./1ee1c1044ffe74ee870df838a5f2a5ed2e77e622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.318ex; height:2.509ex;" alt="{\textstyle \varepsilon _{r}\approx 1}" loading="lazy"> for air); and</li>

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<annotation encoding="application/x-tex">{\textstyle d}</annotation>

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</math><img src="./252135f29da0e9f9e130ff2d53be5df2f7044d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\textstyle d}" loading="lazy"> is the separation between the plates, in meters.</li></ul>

The equation is a good approximation if d is small compared to the other dimensions of the plates so that the <a href="Electric_field" title="Electric field">electric field</a> in the capacitor area is uniform, and the so-called fringing field around the periphery provides only a small contribution to the capacitance.

Combining the equation for capacitance with the above equation for the energy stored in a capacitor, for a flat-plate capacitor the energy stored is:

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<annotation encoding="application/x-tex">{\displaystyle W_{\text{stored}}={\frac {1}{2}}CV^{2}={\frac {1}{2}}\varepsilon {\frac {A}{d}}V^{2}.}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle W}">

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<annotation encoding="application/x-tex">{\textstyle W}</annotation>

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</math><img src="./e95737ee2530885a10b104e9cd5331077e1c88d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\textstyle W}" loading="lazy"> is the energy, in joules; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C}">

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</math><img src="./6dca76d9ff4b48256b6a4a99bcb234b64b2fa72b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\textstyle C}" loading="lazy"> is the capacitance, in farads; and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

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<annotation encoding="application/x-tex">{\textstyle V}</annotation>

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</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> is the voltage, in volts.

<div class="mw-heading mw-heading2"><h2 id="Stray_capacitance">Stray capacitance</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Parasitic_capacitance" title="Parasitic capacitance">Parasitic capacitance</a></div>

Any two adjacent conductors can function as a capacitor, though the capacitance is small unless the conductors are close together for long distances or over a large area. This (often unwanted) capacitance is called parasitic or stray capacitance. Stray capacitance can allow signals to leak between otherwise isolated circuits (an effect called <a href="Crosstalk_(electronics)" class="mw-redirect" title="Crosstalk (electronics)">crosstalk</a>), and it can be a limiting factor for proper functioning of circuits at <a href="High_frequency" title="High frequency">high frequency</a>.

Stray capacitance between the input and output in amplifier circuits can be troublesome because it can form a path for <a href="Feedback#Electronic_engineering" title="Feedback">feedback</a>, which can cause instability and <a href="Parasitic_oscillation" title="Parasitic oscillation">parasitic oscillation</a> in the amplifier. It is often convenient for analytical purposes to replace this capacitance with a combination of one input-to-ground capacitance and one output-to-ground capacitance; the original configuration – including the input-to-output capacitance – is often referred to as a pi-configuration. Miller's theorem can be used to effect this replacement: it states that, if the gain ratio of two nodes is <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠1/K⁠, then an <a href="Electrical_impedance" title="Electrical impedance">impedance</a> of Z connecting the two nodes can be replaced with a ⁠Z/1 − K⁠ impedance between the first node and ground and a ⁠KZ/K − 1⁠ impedance between the second node and ground. Since impedance varies inversely with capacitance, the internode capacitance, C, is replaced by a capacitance of KC from input to ground and a capacitance of ⁠(K − 1)C/K⁠ from output to ground. When the input-to-output gain is very large, the equivalent input-to-ground impedance is very small while the output-to-ground impedance is essentially equal to the original (input-to-output) impedance.

<div class="mw-heading mw-heading2"><h2 id="Capacitance_of_conductors_with_simple_shapes">Capacitance of conductors with simple shapes</h2></div>

Calculating the capacitance of a system amounts to solving the <a href="Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \nabla ^{2}\varphi =0}">

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<annotation encoding="application/x-tex">{\textstyle \varphi }</annotation>

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</math><img src="./99015519246670af1cb5592e439ad64a27fb4830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\textstyle \varphi }" loading="lazy"> 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 <a href="Schwarz%E2%80%93Christoffel_mapping" title="Schwarz–Christoffel mapping">Schwarz–Christoffel mapping</a>.

<caption>Capacitance of simple systems

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<th>Capacitance</th>

<th>Diagram and definitions

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<th>Parallel-plate capacitor

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<annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {C}}={\frac {\ \varepsilon A\ }{d}}\ }</annotation>

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</math><img src="./7735e66558544f1da2bcc9f59f04ce6b053e5058.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.323ex; height:5.509ex;" alt="{\displaystyle \ {\mathcal {C}}={\frac {\ \varepsilon A\ }{d}}\ }" loading="lazy">

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<annotation encoding="application/x-tex">{\textstyle \varepsilon }</annotation>

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</math><img src="./2193f5fc8a6dd05f24e01d0789b78ec2b57515ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\textstyle \varepsilon }" loading="lazy">: <a href="Permittivity" title="Permittivity">Permittivity</a></li></ul>

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<th>Concentric cylinders

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<annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {C}}={\frac {2\pi \varepsilon \ell }{\ \ln \left(R_{2}/R_{1}\right)\ }}\ }</annotation>

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</math><img src="./b06b7319a2b03715d547260b5f0983f32f0b6f11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.431ex; height:6.176ex;" alt="{\displaystyle \ {\mathcal {C}}={\frac {2\pi \varepsilon \ell }{\ \ln \left(R_{2}/R_{1}\right)\ }}\ }" loading="lazy">

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<annotation encoding="application/x-tex">{\textstyle \varepsilon }</annotation>

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</math><img src="./2193f5fc8a6dd05f24e01d0789b78ec2b57515ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\textstyle \varepsilon }" loading="lazy">: <a href="Permittivity" title="Permittivity">Permittivity</a></li></ul>

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<th>Eccentric cylinders<a href="#cite_note-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {C}}={\frac {2\pi \varepsilon \ell }{\ \operatorname {arcosh} \left({\frac {R_{1}^{2}+R_{2}^{2}-d^{2}}{2R_{1}R_{2}}}\right)\ }}\ }</annotation>

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</math><img src="./d569a9ab5e37276c71038dacfd91e9486ab3e051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:27.026ex; height:9.343ex;" alt="{\displaystyle \ {\mathcal {C}}={\frac {2\pi \varepsilon \ell }{\ \operatorname {arcosh} \left({\frac {R_{1}^{2}+R_{2}^{2}-d^{2}}{2R_{1}R_{2}}}\right)\ }}\ }" loading="lazy"></big>

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<annotation encoding="application/x-tex">{\textstyle \varepsilon }</annotation>

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</math><img src="./2193f5fc8a6dd05f24e01d0789b78ec2b57515ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\textstyle \varepsilon }" loading="lazy">: <a href="Permittivity" title="Permittivity">Permittivity</a></li>

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</math><img src="./554d626bee80ffdb3ab924a78a2d05c5a9e642db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\textstyle \ell }" loading="lazy">: Wire length</li></ul>

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<th>Pair of parallel wires<a href="#cite_note-Jackson_1975_80-14">[14]</a>

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<annotation encoding="application/x-tex">{\displaystyle \ {\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)\ }}\ }</annotation>

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</math><img src="./d2d59c44af29370e08de9cd6d0c90bdbb50973fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:48.465ex; height:9.676ex;" alt="{\displaystyle \ {\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)\ }}\ }" loading="lazy"></big>

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<th>Wire parallel to wall<a href="#cite_note-Jackson_1975_80-14">[14]</a>

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<annotation encoding="application/x-tex">{\displaystyle \ {\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)\ }}\ }</annotation>

</semantics>

</math><img src="./6dfa22478b0ba56a15df895367f2cdab038f3aa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:45.981ex; height:9.676ex;" alt="{\displaystyle \ {\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)\ }}\ }" loading="lazy"></big>

</td>

<td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle a}</annotation>

</semantics>

</math><img src="./7a503f107a7c104e40e484cee9e1f5993d28ffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\textstyle a}" loading="lazy">: Wire radius</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle d}</annotation>

</semantics>

</math><img src="./252135f29da0e9f9e130ff2d53be5df2f7044d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\textstyle d}" loading="lazy">: Distance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d>a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle d>a}</annotation>

</semantics>

</math><img src="./2f0cadff95880044e8c805b6e1a9410d5057ea53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.544ex; height:2.176ex;" alt="{\textstyle d>a}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \ell }</annotation>

</semantics>

</math><img src="./554d626bee80ffdb3ab924a78a2d05c5a9e642db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\textstyle \ell }" loading="lazy">: Wire length</li></ul>

</td></tr>

<tr>

<th>Two parallel coplanar strips<a href="#cite_note-15">[15]</a>

</th>

</mrow>

<mfrac>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {C}}=\varepsilon \ell \ {\frac {\ K\left({\sqrt {1-k^{2}\ }}\right)\ }{2K\left(k\right)}}\ }</annotation>

</semantics>

</math><img src="./afdfbdf222551282a32f3596c49ca6c4311f8222.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.918ex; height:8.343ex;" alt="{\displaystyle \ {\mathcal {C}}=\varepsilon \ell \ {\frac {\ K\left({\sqrt {1-k^{2}\ }}\right)\ }{2K\left(k\right)}}\ }" loading="lazy">

</td>

<td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle d}</annotation>

</semantics>

</math><img src="./252135f29da0e9f9e130ff2d53be5df2f7044d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\textstyle d}" loading="lazy">: Distance</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \ell }</annotation>

</semantics>

</math><img src="./554d626bee80ffdb3ab924a78a2d05c5a9e642db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\textstyle \ell }" loading="lazy">: Length</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle w_{1},w_{2}}</annotation>

</semantics>

</math><img src="./04e1e5ec5dd45d8357be9a5ad9a8d336df9e6bdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.471ex; height:2.009ex;" alt="{\textstyle w_{1},w_{2}}" loading="lazy">: Strip width</li>

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \ k_{1}=\left({\tfrac {\ 2w_{1}\ }{d}}+1\right)^{-1}\ }</annotation>

</semantics>

</math><img src="./2f17a636489192f61f85471d60083f28e08388b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.464ex; height:5.176ex;" alt="{\textstyle \ k_{1}=\left({\tfrac {\ 2w_{1}\ }{d}}+1\right)^{-1}\ }" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k_{2}=\left({\tfrac {\ 2w_{2}\ }{d}}+1\right)^{-1}\ }">

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ k_{2}=\left({\tfrac {\ 2w_{2}\ }{d}}+1\right)^{-1}\ }</annotation>

</semantics>

</math><img src="./20d72d42a4f60b759ff3c1f444643c63ed898226.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.464ex; height:5.176ex;" alt="{\displaystyle \ k_{2}=\left({\tfrac {\ 2w_{2}\ }{d}}+1\right)^{-1}\ }" loading="lazy"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k={\sqrt {k_{1}\ k_{2}\ }}\ }">

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ k={\sqrt {k_{1}\ k_{2}\ }}\ }</annotation>

</semantics>

</math><img src="./8bc82f8cc7cc9e6062c8967f136bd49d7cf649f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.487ex; height:3.343ex;" alt="{\displaystyle \ k={\sqrt {k_{1}\ k_{2}\ }}\ }" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle K}</annotation>

</semantics>

</math><img src="./985dcc2532a2d4d91b9a9610139216c63cf832d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\textstyle K}" loading="lazy">: <a href="Elliptic_integral#Complete_elliptic_integral_of_the_first_kind" title="Elliptic integral">Complete elliptic integral of the first kind</a></li></ul>

</td></tr>

<tr>

<th>Concentric spheres

</th>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {C}}={\frac {4\pi \varepsilon }{\ {\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\ }}\ }</annotation>

</semantics>

</math><img src="./68b476cbf348bba38db1406ab1b40278996a8bfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:16.167ex; height:7.009ex;" alt="{\displaystyle \ {\mathcal {C}}={\frac {4\pi \varepsilon }{\ {\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\ }}\ }" loading="lazy">

</td>

<td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varepsilon }</annotation>

</semantics>

</math><img src="./2193f5fc8a6dd05f24e01d0789b78ec2b57515ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\textstyle \varepsilon }" loading="lazy">: <a href="Permittivity" title="Permittivity">Permittivity</a></li></ul>

</td></tr>

<tr>

<th>Two spheres, equal radius<a href="#cite_note-Maxwell_1873_266_ff-16">[16]</a><a href="#cite_note-17">[17]</a>

</th>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ {\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{aligned}}\ }</annotation>

</semantics>

</math><img src="./b633a8a25dbe55de0e323c5ca869ab2e6c78b1e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:66.507ex; height:29.843ex;" alt="{\displaystyle {\begin{aligned}\ {\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{aligned}}\ }" loading="lazy">

</td>

<td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle a}</annotation>

</semantics>

</math><img src="./7a503f107a7c104e40e484cee9e1f5993d28ffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\textstyle a}" loading="lazy">: Radius</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle d}</annotation>

</semantics>

</math><img src="./252135f29da0e9f9e130ff2d53be5df2f7044d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\textstyle d}" loading="lazy">: Distance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d>2a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle d>2a}</annotation>

</semantics>

</math><img src="./814d16cd28ad50bd8b83a9cf68d15667342e63f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.707ex; height:2.176ex;" alt="{\textstyle d>2a}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle D=d/2a,D>1}</annotation>

</semantics>

</math><img src="./90498f0ee5fc403de434b4e329c2d2255635bf6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.012ex; height:2.843ex;" alt="{\textstyle D=d/2a,D>1}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \gamma }</annotation>

</semantics>

</math><img src="./079fa214f0d7f7dcf92af4c520c95aa7ca787d1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\textstyle \gamma }" loading="lazy">: <a href="Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler's constant</a></li>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=D+{\sqrt {D^{2}-1}}}</annotation>

</semantics>

</math><img src="./3b27f407e95b1c34a20dc36666c8722c19a3fa0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.238ex; height:3.509ex;" alt="{\displaystyle q=D+{\sqrt {D^{2}-1}}}" loading="lazy"></li>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{q}(z)={\frac {\partial _{z}\Gamma _{q}(z)}{\Gamma _{q}(z)}}}</annotation>

</semantics>

</math><img src="./6a2d70840f1445b3601ac1112cc7f2e83f2a6292.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.909ex; height:6.509ex;" alt="{\displaystyle \psi _{q}(z)={\frac {\partial _{z}\Gamma _{q}(z)}{\Gamma _{q}(z)}}}" loading="lazy">: the q-digamma function</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma _{q}(z)}</annotation>

</semantics>

</math><img src="./1013a7187dcbf7518dd1514b1c2de420c9f555cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.339ex; height:3.009ex;" alt="{\displaystyle \Gamma _{q}(z)}" loading="lazy">: the <a href="Q-gamma_function" title="Q-gamma function">q-gamma function</a><a href="#cite_note-18">[18]</a></li></ul>

See also <a href="Basic_hypergeometric_series" title="Basic hypergeometric series">Basic hypergeometric series</a>.

</td></tr>

<tr>

<th>Sphere in front of wall<a href="#cite_note-Maxwell_1873_266_ff-16">[16]</a>

</th>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\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)}}\ }</annotation>

</semantics>

</math><img src="./a4c499a0a2bd13db98ec1cc5ba4f27ccb0ef98a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:42.762ex; height:10.176ex;" alt="{\displaystyle \ {\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)}}\ }" loading="lazy">

</td>

<td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ a\ }</annotation>

</semantics>

</math><img src="./8124de742ae987fe73be9ca9d3d4ba8586e28b11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.391ex; height:1.676ex;" alt="{\displaystyle \ a\ }" loading="lazy">: Radius</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ d\ }</annotation>

</semantics>

</math><img src="./3035fc0c2be984029a7ef81eb60f13c26b0c6ed5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.377ex; height:2.176ex;" alt="{\displaystyle \ d\ }" loading="lazy">: Distance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d>a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d>a}</annotation>

</semantics>

</math><img src="./5a63ce3b01161f8f4a757d6d8e0408dcdbe5b478.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.544ex; height:2.176ex;" alt="{\displaystyle d>a}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=d/a}</annotation>

</semantics>

</math><img src="./557cc492f07e952b74c8d82406d38cbaeb9a4dab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.631ex; height:2.843ex;" alt="{\displaystyle D=d/a}" loading="lazy"></li></ul>

</td></tr>

<tr>

<th>Sphere

</th>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {C}}=4\pi \varepsilon a\ }</annotation>

</semantics>

</math><img src="./efd5eb04590b5ea11104a39d181e781086360d6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.307ex; height:2.176ex;" alt="{\displaystyle \ {\mathcal {C}}=4\pi \varepsilon a\ }" loading="lazy">

</td>

<td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">: Radius</li></ul>

</td></tr>

<tr>

<th>Circular disc<a href="#cite_note-Jackson_1975_128-19">[19]</a>

</th>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {C}}=8\varepsilon a\ }</annotation>

</semantics>

</math><img src="./f55ed23a0dd0dbcd71b5dd780c08fe6725ceb68e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.975ex; height:2.176ex;" alt="{\displaystyle \ {\mathcal {C}}=8\varepsilon a\ }" loading="lazy">

</td>

<td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">: Radius</li></ul>

</td></tr>

<tr>

<th>Thin straight wire, finite length<a href="#cite_note-20">[20]</a><a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a>

</th>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\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]\ }</annotation>

</semantics>

</math><img src="./3bcd1ed9c0ea246c62f8705fb162f3e2b50ccd85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:72.829ex; height:6.343ex;" alt="{\displaystyle \ {\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]\ }" loading="lazy">

</td>

<td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">: Wire radius</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>

</semantics>

</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy">: Length</li>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ \Lambda =\ln \left(\ell /a\right)\ }</annotation>

</semantics>

</math><img src="./60e1931b394c02c33a3fd4153d2997590c865dfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.983ex; height:2.843ex;" alt="{\displaystyle \ \Lambda =\ln \left(\ell /a\right)\ }" loading="lazy"></li></ul>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Energy_storage">Energy storage</h2></div>

The <a href="Energy" title="Energy">energy</a> (measured in <a href="Joule" title="Joule">joules</a>) stored in a capacitor is equal to the work required to push the charges into the capacitor, i.e. to charge it. Consider a capacitor of capacitance C, holding a charge +q on one plate and −q on the other. Moving a small element of charge dq from one plate to the other against the potential difference V = q/C requires the work dW:

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} W={\frac {q}{C}}\,\mathrm {d} q,}</annotation>

</semantics>

</math>

where W is the work measured in joules, q is the charge measured in coulombs and C is the capacitance, measured in farads.

The energy stored in a capacitor is found by <a href="Integral" title="Integral">integrating</a> this equation. Starting with an uncharged capacitance (q = 0) and moving charge from one plate to the other until the plates have charge +Q and −Q requires the work W:

<msub>

<mtext>charging</mtext>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msub>

<mtext>stored</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{\text{charging}}=\int _{0}^{Q}{\frac {q}{C}}\,\mathrm {d} q={\frac {1}{2}}{\frac {Q^{2}}{C}}={\frac {1}{2}}QV={\frac {1}{2}}CV^{2}=W_{\text{stored}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Nanoscale_systems">Nanoscale systems</h2></div>

The capacitance of nanoscale dielectric capacitors such as <a href="Quantum_dots" class="mw-redirect" title="Quantum dots">quantum dots</a> may differ from conventional formulations of larger capacitors. In particular, the electrostatic potential difference experienced by electrons in conventional capacitors is spatially well-defined and fixed by the shape and size of metallic electrodes in addition to the statistically large number of electrons present in conventional capacitors. In nanoscale capacitors, however, the electrostatic potentials experienced by electrons are determined by the number and locations of all electrons that contribute to the electronic properties of the device. In such devices, the number of electrons may be very small, so the resulting spatial distribution of equipotential surfaces within the device is exceedingly complex.

<div class="mw-heading mw-heading3"><h3 id="Single-electron_devices">Single-electron devices</h3></div>

The capacitance of a connected, or "closed", single-electron device is twice the capacitance of an unconnected, or "open", single-electron device.<a href="#cite_note-Tsu-23">[23]</a> This fact may be traced more fundamentally to the energy stored in the single-electron device whose "direct polarization" interaction energy may be equally divided into the interaction of the electron with the polarized charge on the device itself due to the presence of the electron and the amount of potential energy required to form the polarized charge on the device (the interaction of charges in the device's dielectric material with the potential due to the electron).<a href="#cite_note-LaFave-DCD-24">[24]</a>

<div class="mw-heading mw-heading3"><h3 id="Few-electron_devices">Few-electron devices</h3></div>

The derivation of a "quantum capacitance" of a few-electron device involves the thermodynamic <a href="Chemical_potential" title="Chemical potential">chemical potential</a> of an N-particle system given by

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (N)=U(N)-U(N-1),}</annotation>

</semantics>

</math>

whose energy terms may be obtained as solutions of the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. The definition of capacitance,

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {1 \over C}\equiv {\Delta V \over \Delta Q},}</annotation>

</semantics>

</math>

with the potential difference

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta V={\Delta \mu \, \over e}={\mu (N+\Delta N)-\mu (N) \over e}}</annotation>

</semantics>

</math>

may be applied to the device with the addition or removal of individual electrons,

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta N=1}</annotation>

</semantics>

</math> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta Q=e.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta Q=e.}</annotation>

</semantics>

</math>

The "quantum capacitance" of the device is then<a href="#cite_note-25">[25]</a>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{Q}(N)={\frac {e^{2}}{\mu (N+1)-\mu (N)}}={\frac {e^{2}}{E(N)}}.}</annotation>

</semantics>

</math>

This expression of "quantum capacitance" may be written as

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{Q}(N)={e^{2} \over U(N)},}</annotation>

</semantics>

</math>

which differs from the conventional expression described in the introduction where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\text{stored}}=U}">

<msub>

<mtext>stored</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{\text{stored}}=U}</annotation>

</semantics>

</math><img src="./4e9781e1e1b0249d5008608bad4f9282ba9d4c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.705ex; height:2.509ex;" alt="{\displaystyle W_{\text{stored}}=U}" loading="lazy">, the stored electrostatic potential energy,

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C={Q^{2} \over 2U},}</annotation>

</semantics>

</math>

by a factor of ⁠1/2⁠ with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=Ne}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q=Ne}</annotation>

</semantics>

</math><img src="./b233ed65ba48267c49f94df7c9a0e43c7bda72dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.084ex; height:2.509ex;" alt="{\displaystyle Q=Ne}" loading="lazy">.

However, within the framework of purely classical electrostatic interactions, the appearance of the factor of ⁠1/2⁠ is the result of integration in the conventional formulation involving the work done when charging a capacitor,

<msub>

<mtext>charging</mtext>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{\text{charging}}=U=\int _{0}^{Q}{\frac {q}{C}}\,\mathrm {d} q,}</annotation>

</semantics>

</math>

which is appropriate since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} q=0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} q=0}</annotation>

</semantics>

</math><img src="./663c1866d01f78fe9a3c27af4741fcd7854936fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.623ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} q=0}" loading="lazy"> for systems involving either many electrons or metallic electrodes, but in few-electron systems, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} q\to \Delta \,Q=e}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} q\to \Delta \,Q=e}</annotation>

</semantics>

</math><img src="./7292acec444e6c365e2d528aa35cb5cac79fcf4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.319ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} q\to \Delta \,Q=e}" loading="lazy">. The integral generally becomes a summation. One may trivially combine the expressions of capacitance

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q=CV}</annotation>

</semantics>

</math>

and electrostatic interaction energy,

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U=QV,}</annotation>

</semantics>

</math>

to obtain

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C=Q{1 \over V}=Q{Q \over U}={Q^{2} \over U},}</annotation>

</semantics>

</math>

which is similar to the quantum capacitance. A more rigorous derivation is reported in the literature.<a href="#cite_note-26">[26]</a> In particular, to circumvent the mathematical challenges of spatially complex equipotential surfaces within the device, an average electrostatic potential experienced by each electron is utilized in the derivation.

Apparent mathematical differences may be understood more fundamentally. The potential energy, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(N)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U(N)}</annotation>

</semantics>

</math><img src="./75bc0531863b76e4cb6239ccfc14804a5a817d1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.656ex; height:2.843ex;" alt="{\displaystyle U(N)}" loading="lazy">, of an isolated device (self-capacitance) is twice that stored in a "connected" device in the lower limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N=1}</annotation>

</semantics>

</math><img src="./85982022b9eb1f295b44de55023687a490db0a39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=1}" loading="lazy">. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> grows large, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(N)\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U(N)\to U}</annotation>

</semantics>

</math><img src="./c52cf0c8c3c8dd0bccdc7a8ed618eb8e5aae4790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.052ex; height:2.843ex;" alt="{\displaystyle U(N)\to U}" loading="lazy">.<a href="#cite_note-LaFave-DCD-24">[24]</a> Thus, the general expression of capacitance is

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(N)={(Ne)^{2} \over U(N)}.}</annotation>

</semantics>

</math>

In nanoscale devices such as quantum dots, the "capacitor" is often an isolated or partially isolated component within the device. The primary differences between nanoscale capacitors and macroscopic (conventional) capacitors are the number of excess electrons (charge carriers, or electrons, that contribute to the device's electronic behavior) and the shape and size of metallic electrodes. In nanoscale devices, <a href="Nanowires" class="mw-redirect" title="Nanowires">nanowires</a> consisting of metal atoms typically do not exhibit the same conductive properties as their macroscopic, or bulk material, counterparts.

<div class="mw-heading mw-heading2"><h2 id="Capacitance_in_electronic_and_semiconductor_devices">Capacitance in electronic and semiconductor devices</h2></div>

In electronic and semiconductor devices, transient or frequency-dependent current between terminals contains both conduction and displacement components. Conduction current is related to moving charge carriers (electrons, holes, ions, etc.), while displacement current is caused by a time-varying electric field. Carrier transport is affected by electric fields and by a number of physical phenomena - such as carrier drift and diffusion, trapping, injection, contact-related effects, impact ionization, etc. As a result, device <a href="Admittance" title="Admittance">admittance</a> is frequency-dependent, and a simple electrostatic formula for capacitance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=q/V,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C=q/V,}</annotation>

</semantics>

</math><img src="./febc2618b31fcb6b4cb43cea4ad7f7439c945535.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.531ex; height:2.843ex;" alt="{\displaystyle C=q/V,}" loading="lazy"> is not applicable. A more general definition of capacitance, encompassing electrostatic formula, is:<a href="#cite_note-LauxCapacitance-27">[27]</a>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C={\frac {\operatorname {Im} (Y(\omega ))}{\omega }},}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(\omega )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y(\omega )}</annotation>

</semantics>

</math><img src="./f58497f00333ec38be62fdc7c4220d005bf06ad1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.028ex; height:2.843ex;" alt="{\displaystyle Y(\omega )}" loading="lazy"> is the device admittance, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> is the angular frequency.

In general, capacitance is a function of frequency. At high frequencies, capacitance approaches a constant value, equal to "geometric" capacitance, determined by the terminals' geometry and dielectric content in the device.

A paper by Steven Laux<a href="#cite_note-LauxCapacitance-27">[27]</a> presents a review of numerical techniques for capacitance calculation. In particular, capacitance can be calculated by a Fourier transform of a transient current in response to a step-like voltage excitation:

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(\omega )={\frac {1}{\Delta V}}\int _{0}^{\infty }[i(t)-i(\infty )]\cos(\omega t)dt.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Negative_capacitance_in_semiconductor_devices">Negative capacitance in semiconductor devices</h2></div>

Usually, capacitance in semiconductor devices is positive. However, in some devices and under certain conditions (temperature, applied voltages, frequency, etc.), capacitance can become negative. Non-monotonic behavior of the transient current in response to a step-like excitation has been proposed as the mechanism of negative capacitance.<a href="#cite_note-JonscherNegCap-28">[28]</a> Negative capacitance has been demonstrated and explored in many different types of semiconductor devices.<a href="#cite_note-29">[29]</a>

<div class="mw-heading mw-heading2"><h2 id="Measuring_capacitance">Measuring capacitance</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Capacitance_meter" title="Capacitance meter">Capacitance meter</a></div>

A <a href="Capacitance_meter" title="Capacitance meter">capacitance meter</a> is a piece of <a href="Electronic_test_equipment" title="Electronic test equipment">electronic test equipment</a> used to measure capacitance, mainly of discrete <a href="Capacitor" title="Capacitor">capacitors</a>. For most purposes and in most cases the capacitor must be disconnected from <a href="Electronic_circuit" title="Electronic circuit">circuit</a>.

Many DVMs (<a href="Voltmeter" title="Voltmeter">digital volt meters</a>) have a capacitance-measuring function. These usually operate by charging and discharging the <a href="Device_under_test" title="Device under test">capacitor under test</a> with a known <a href="Electric_current" title="Electric current">current</a> and measuring the rate of rise of the resulting <a href="Voltage" title="Voltage">voltage</a>; the slower the rate of rise, the larger the capacitance. DVMs can usually measure capacitance from <a href="Farad" title="Farad">nanofarads</a> to a few hundred microfarads, but wider ranges are not unusual. It is also possible to measure capacitance by passing a known <a href="High-frequency" class="mw-redirect" title="High-frequency">high-frequency</a> <a href="Alternating_current" title="Alternating current">alternating current</a> through the device under test and measuring the resulting <a href="Volt" title="Volt">voltage</a> across it (does not work for polarised capacitors).

More sophisticated instruments use other techniques such as inserting the capacitor-under-test into a <a href="Bridge_circuit" title="Bridge circuit">bridge circuit</a>. By varying the values of the other legs in the bridge (so as to bring the bridge into balance), the value of the unknown capacitor is determined. This method of indirect use of measuring capacitance ensures greater precision. Through the use of <a href="Four-terminal_sensing" title="Four-terminal sensing">Kelvin connections</a> and other careful design techniques, these instruments can usually measure capacitors over a range from picofarads to farads.

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<ul><li><a href="Capacitive_displacement_sensor" title="Capacitive displacement sensor">Capacitive displacement sensor</a></li>

<li><a href="Capacity_of_a_set" title="Capacity of a set">Capacity of a set</a></li>

<li><a href="Displacement_current" title="Displacement current">Displacement current</a></li>

<li><a href="Gauss_law" class="mw-redirect" title="Gauss law">Gauss law</a></li>

<li><a href="LCR_meter" title="LCR meter">LCR meter</a></li>

<li><a href="Magnetocapacitance" title="Magnetocapacitance">Magnetocapacitance</a></li>

<li><a href="Quantum_capacitance" title="Quantum capacitance">Quantum capacitance</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFHarrington2003" class="citation book cs1"><a href="Roger_F._Harrington" title="Roger F. Harrington">Harrington, Roger F.</a> (2003). Introduction to Electromagnetic Engineering (1st ed.). Dover Publications. p. 43. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-43241-6</bdi>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFMassariniKazimierczuk1997" class="citation journal cs1">Massarini, A.; Kazimierczuk, M. K. (1997). "Self capacitance of inductors". IEEE Transactions on Power Electronics. 12 (4): 671–676. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997ITPE...12..671M">1997ITPE...12..671M</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.205.7356">10.1.1.205.7356</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F63.602562">10.1109/63.602562</a>: example of the use of the term 'self capacitance'.</cite><code class="cs1-code">{{cite journal}}</code>: CS1 maint: postscript (link)

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<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFErshovLiuLiBuchanan1998" class="citation journal cs1">Ershov, M.; Liu, H.C.; Li, L.; Buchanan, M.; Wasilewski, Z.R.; Jonscher, A.K. (October 1998). "Negative capacitance effect in semiconductor devices". IEEE Trans. Electron Devices. 45 (10): 2196–2206. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9806145">cond-mat/9806145</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998ITED...45.2196E">1998ITED...45.2196E</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F16.725254">10.1109/16.725254</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:204925581">204925581</a>.</cite>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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</style><div class="refbegin" style="">

<ul><li>Tipler, Paul (1998). Physics for Scientists and Engineers: Vol. 2: Electricity and Magnetism, Light (4th ed.). W. H. Freeman. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-57259-492-6</bdi></li>

<li>Serway, Raymond; Jewett, John (2003). Physics for Scientists and Engineers (6th ed.). Brooks Cole. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-40842-7</bdi></li>

<li>Saslow, Wayne M.(2002). Electricity, Magnetism, and Light. Thomson Learning. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-619455-6</bdi>. See Chapter 8, and especially pp. 255–259 for coefficients of potential.</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Capacitance" class="extiw external" title="commons:Category:Capacitance">Capacitance</a> at Wikimedia Commons</li></ul>

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