Array ( [0] => {{Short description|Ability of a body to store an electrical charge}} [1] => {{For|capacitance of blood vessels|Compliance (physiology)}} [2] => {{Use dmy dates|date=June 2020}} [3] => {{Infobox physical quantity [4] => | name = [5] => | width = [6] => | background = [7] => | image = [8] => | caption = [9] => | unit = [[farad]] [10] => | otherunits =μF, nF, pF [11] => | symbols = {{math|''C''}} [12] => | baseunits = F = A2 s4 kg−1 m−2 [13] => | dimension = wikidata [14] => | derivations = ''C'' = '''[[electric charge|charge]]''' / '''[[voltage]]''' [15] => }} [16] => {{Electromagnetism |Network}} [17] => '''Capacitance''' is the capability of a material object or device to store [[electric charge]]. It is measured by the charge in response to a difference in [[electric potential]], expressed as the ratio of those quantities. Commonly recognized are two closely related notions of capacitance: ''self capacitance'' and ''mutual capacitance''.{{cite book |last=Harrington |first=Roger F. |author-link=Roger F. Harrington |title=Introduction to Electromagnetic Engineering |publisher=Dover Publications |year=2003 |edition=1st |page=43 |isbn=0-486-43241-6}}{{rp|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 [[capacitor]], an elementary [[Linear circuit|linear]] [[electronic component]] designed to add capacitance to an [[electric circuit]]. [18] => [19] => The capacitance between two [[Electrical resistance and conductance|conductors]] is a function only of the geometry; the opposing surface area of the conductors and the distance between them, and the [[permittivity]] of any [[dielectric]] 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. [20] => [21] => The [[SI]] unit of capacitance is the [[farad]] (symbol: F), named after the English physicist [[Michael Faraday]]. A 1 farad capacitor, when charged with 1 [[coulomb]] of electrical charge, has a potential difference of 1 [[volt]] between its plates.{{cite web |url=http://www.collinsdictionary.com/dictionary/english/farad |title=Definition of 'farad' |publisher=Collins}} The reciprocal of capacitance is called [[elastance]]. [22] => [23] => ==Self capacitance== [24] => 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 [[electric potential]] by one unit of measurement, e.g., one [[volt]].{{cite book|author=William D. Greason| title=Electrostatic discharge in electronics|url=https://books.google.com/books?id=404fAQAAIAAJ|year=1992|publisher=Research Studies Press|isbn=978-0-86380-136-5 |page=48}} The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, with the conductor centered inside this sphere. [25] => [26] => Self capacitance of a conductor is defined by the ratio of charge and electric potential: [27] => C = \frac{q}{V}, [28] => where [29] => *q is the charge held, [30] => *V = \frac{1}{4\pi\varepsilon_0}\int \frac{\sigma}{r}\,dS is the electric potential, [31] => *\sigma is the surface charge density, [32] => *dS is an infinitesimal element of area on the surface of the conductor, [33] => *r is the length from dS to a fixed point ''M'' on the conductor, [34] => *\varepsilon_0 is the [[vacuum permittivity]]. [35] => [36] => Using this method, the self capacitance of a conducting sphere of radius R in free space (i.e. far away from any other charge distributions) is:{{cite web|archive-url= https://web.archive.org/web/20090226225105/http://www.phys.unsw.edu.au/COURSES/FIRST_YEAR/pdf%20files/5Capacitanceanddielectr.pdf|archive-date=2009-02-26|url=http://www.phys.unsw.edu.au/COURSES/FIRST_YEAR/pdf%20files/5Capacitanceanddielectr.pdf|title=Lecture notes: Capacitance and Dieletrics|publisher=University of New South Wales}} [37] => C = 4 \pi \varepsilon_0 R. [38] => [39] => Example values of self capacitance are: [40] => *for the top "plate" of a [[van de Graaff generator]], typically a sphere 20 cm in radius: 22.24 pF, [41] => *the planet [[Earth]]: about 710 μF.{{cite book | last1 = Tipler | first1 = Paul | last2 = Mosca | first2 = Gene | title = Physics for Scientists and Engineers | publisher = Macmillan | year = 2004 | edition = 5th | page = 752 | isbn = 978-0-7167-0810-0 }} [42] => [43] => The inter-winding capacitance of a [[electromagnetic coil|coil]] is sometimes called self capacitance,{{cite journal| title=Self capacitance of inductors|doi=10.1109/63.602562 |last1=Massarini |first1=A. |last2=Kazimierczuk |first2=M. K. |year=1997 |volume=12 |issue=4 |pages=671–676 |journal=IEEE Transactions on Power Electronics |postscript=: example of the use of the term 'self capacitance'.|bibcode=1997ITPE...12..671M |citeseerx=10.1.1.205.7356 }} 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 [[parasitic capacitance]]. This self capacitance is an important consideration at high frequencies: it changes the [[Electrical impedance|impedance]] of the coil and gives rise to parallel [[Electrical resonance|resonance]]. In many applications this is an undesirable effect and sets an upper frequency limit for the correct operation of the circuit.{{citation needed|date=May 2017}} [44] => [45] => ==Mutual capacitance== [46] => A common form is a parallel-plate [[capacitor]], which consists of two conductive plates insulated from each other, usually sandwiching a [[dielectric]] material. In a parallel plate capacitor, capacitance is very nearly proportional to the surface area of the conductor plates and inversely proportional to the separation distance between the plates. [47] => [48] => If the charges on the plates are +q and -q, and V gives the [[voltage]] between the plates, then the capacitance C is given by C = \frac{q}{V}, [49] => which gives the voltage/[[electric current|current]] relationship [50] => i(t) = C \frac{dv(t)}{dt} + V\frac{dC}{dt}, [51] => where \frac{dv(t)}{dt} is the instantaneous rate of change of voltage, and \frac{dC}{dt} is the instantaneous rate of change of the capacitance. For most applications, the change in capacitance over time is negligible, so you can reduce to: [52] => i(t) = C \frac{dv(t)}{dt}, [53] => [54] => The energy stored in a capacitor is found by [[integral|integrating]] the work W: [55] => W_\text{charging} = \frac{1}{2}CV^2. [56] => [57] => ===Capacitance matrix=== [58] => The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition C = Q/V does not apply when there are more than two charged plates, or when the net charge on the two plates is non-zero. To handle this case, [[James Clerk Maxwell]] introduced his ''[[coefficients of potential]]''. If three (nearly ideal) conductors are given charges Q_1, Q_2, Q_3, then the voltage at conductor 1 is given by [59] => V_1 = P_{11}Q_1 + P_{12} Q_2 + P_{13}Q_3, [60] => and similarly for the other voltages. [[Hermann von Helmholtz]] and [[Sir William Thomson]] showed that the coefficients of potential are symmetric, so that P_{12} = P_{21}, 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: [61] => P_{ij} = \frac{\partial V_{i}}{\partial Q_{j}}. [62] => [63] => From this, the mutual capacitance C_{m} between two objects can be defined{{cite book |last=Jackson |first=John David |title=Classical Electrodynamic |publisher=John Wiley & Sons |year=1999 |edition=3rd |page=43 |isbn=978-0-471-30932-1}} by solving for the total charge Q and using C_{m}=Q/V. [64] => [65] => C_m = \frac{1}{(P_{11} + P_{22})-(P_{12} + P_{21})}. [66] => [67] => 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. [68] => [69] => The collection of coefficients C_{ij} = \frac{\partial Q_{i}}{\partial V_{j}} is known as the ''capacitance matrix'',{{cite book| last =Maxwell | first =James | author-link =James Clerk Maxwell | title = A treatise on electricity and magnetism |volume=1 | publisher = Clarendon Press | year = 1873 | chapter =3 | at =p. 88ff | chapter-url = https://archive.org/details/electricandmagne01maxwrich}}{{Cite web |title=Capacitance: Charge as a Function of Voltage |url=http://www.av8n.com/physics/capacitance.htm |website=Av8n.com |access-date=20 September 2010}}{{cite journal |last1= Smolić |first1= Ivica |last2= Klajn |first2= Bruno |date= 2021 |title= Capacitance matrix revisited |url= https://www.jpier.org/PIERB/pier.php?paper=21011501 |journal= Progress in Electromagnetics Research B |volume= 92 |pages= 1–18 |doi= 10.2528/PIERB21011501|arxiv=2007.10251 |access-date= 4 May 2021|doi-access= free }} and is the [[matrix inverse|inverse]] of the elastance matrix. [70] => [71] => ==Capacitors== [72] => {{Main|Capacitor}} [73] => The capacitance of the majority of capacitors used in electronic circuits is generally several orders of magnitude smaller than the [[farad]]. The most common units of capacitance are the [[micro-|micro]]farad (μF), [[nano-|nano]]farad (nF), [[pico-|pico]]farad (pF), and, in microcircuits, [[femto-|femto]]farad (fF). Some applications also use [[supercapacitors]] 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).{{cite web |url=http://www.justradios.com/MFMMFD.html |title=Capacitor MF-MMFD Conversion Chart |website=Just Radios}}{{cite book |url=https://archive.org/details/FundamentalsOfElectronics93400A1b |title=Fundamentals of Electronics |volume=1b – Basic Electricity – Alternating Current |publisher=Bureau of Naval Personnel |year=1965 |page=[https://archive.org/details/FundamentalsOfElectronics93400A1b/page/n58 197]}} [74] => [75] => 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. [76] => [77] => An example is the capacitance of a capacitor constructed of two parallel plates both of area A separated by a distance d. If d is sufficiently small with respect to the smallest chord of A, there holds, to a high level of accuracy: [78] => \ C=\varepsilon\frac{A}{d}; [79] => [80] => \varepsilon=\varepsilon_0 \varepsilon_r, [81] => [82] => where [83] => *C is the capacitance, in farads; [84] => *A is the area of overlap of the two plates, in square meters; [85] => *\varepsilon_0 is the [[vacuum permittivity|electric constant]] {{nowrap|(\varepsilon_0 \approx 8.854\times 10^{-12} ~ \mathrm{F{\cdot}m^{-1}});}} [86] => *\varepsilon_r is the [[relative permittivity]] (also dielectric constant) of the material in between the plates {{nowrap|(\varepsilon_r \approx 1}} for air); and [87] => *d is the separation between the plates, in meters. [88] => [89] => The equation is a good approximation if ''d'' is small compared to the other dimensions of the plates so that the electric field in the capacitor area is uniform, and the so-called ''fringing field'' around the periphery provides only a small contribution to the capacitance. [90] => [91] => 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: [92] => W_\text{stored} = \frac{1}{2} C V^2 = \frac{1}{2} \varepsilon \frac{A}{d} V^2. [93] => where W is the energy, in joules; C is the capacitance, in farads; and V is the voltage, in volts. [94] => [95] => ==Stray capacitance== [96] => {{Main|Parasitic capacitance}} [97] => 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 [[Crosstalk (electronics)|crosstalk]]), and it can be a limiting factor for proper functioning of circuits at [[high frequency]]. [98] => [99] => Stray capacitance between the input and output in amplifier circuits can be troublesome because it can form a path for [[Feedback#Electronic engineering|feedback]], which can cause instability and [[parasitic oscillation]] 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 {{sfrac|1|''K''}}, then an [[electrical impedance|impedance]] of ''Z'' connecting the two nodes can be replaced with a {{sfrac|''Z''|1 − ''K''}} impedance between the first node and ground and a {{sfrac|''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 {{sfrac|(''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. [100] => [101] => ==Capacitance of conductors with simple shapes == [102] => Calculating the capacitance of a system amounts to solving the [[Laplace equation]] \nabla^2\varphi=0 with a constant potential \varphi 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. [103] => [104] => For plane situations, analytic functions may be used to map different geometries to each other. See also [[Schwarz–Christoffel mapping]]. [105] => [106] => {| class="wikitable" [107] => |+ Capacitance of simple systems [108] => ! Type !! Capacitance !! Comment [109] => |- [110] => ! Parallel-plate capacitor [111] => | \ \mathcal{C} = \frac{\ \varepsilon A\ }{d}\ [112] => | [[Image:Plate CapacitorII.svg|125px]] [113] => *\varepsilon: [[Permittivity]] [114] => |- [115] => ! Concentric cylinders [116] => | \ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\ \ln \left( R_{2}/R_{1}\right)\ }\ [117] => | [[Image:Cylindrical CapacitorII.svg|130px]] [118] => *\varepsilon: [[Permittivity]] [119] => |- [120] => ! Eccentric cylinders{{cite journal |last=Dawes |year=1973 |first=Chester L. |title=Capacitance and potential gradients of eccentric cylindrical condensers |doi=10.1063/1.1745162 |journal=Physics |volume=4 |issue=2 |pages=81–85 |url=https://aip.scitation.org/doi/abs/10.1063/1.1745162}} [121] => | \ \mathcal{C} = \frac{2\pi \varepsilon \ell}{\ \operatorname{arcosh}\left(\frac{R_{1}^2 + R_{2}^2 - d^2}{2 R_{1} R_{2}}\right)\ }\ [122] => | [[Image:Eccentric capacitor.svg|130px]] [123] => *\varepsilon: [[Permittivity]] [124] => *R_1: Outer radius [125] => *R_2: Inner radius [126] => *d: Distance between center [127] => *\ell: Wire length [128] => |- [129] => ! Pair of parallel wires{{cite book |last=Jackson |first=J. D. |year=1975 |title=Classical Electrodynamics |publisher=Wiley |page=80}} [130] => | \ \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)\ }\ [131] => |[[Image:Parallel Wire Capacitance.svg|130px]] [132] => |- [133] => ! Wire parallel to wall [134] => | \ \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)\ }\ [135] => | [136] => *a: Wire radius [137] => *d: Distance, d > a [138] => *\ell: Wire length [139] => |- [140] => ! Two parallel
coplanar strips{{cite book | last1 = Binns | last2 = Lawrenson | year = 1973 | title = Analysis and computation of electric and magnetic field problems | publisher = Pergamon Press | isbn = 978-0-08-016638-4}} [141] => | \ \mathcal{C} = \varepsilon \ell\ \frac{\ K\left( \sqrt{1-k^2\ } \right)\ }{ K\left( k \right) }\ [142] => | [143] => *d: Distance [144] => *\ell: Length [145] => *w_1, w_2: Strip width [146] => *\ k_1 = \left( \tfrac{\ 2 w_1\ }{d} + 1 \right)^{-1}\
\ k_2 = \left( \tfrac{\ 2 w_2\ }{d} + 1 \right)^{-1}\ \ k = \sqrt{ k_1\ k_2\ }\ [147] => *K: [[Elliptic integral#Complete elliptic integral of the first kind|Complete elliptic integral of the first kind]] [148] => |- [149] => ! Concentric spheres [150] => | \ \mathcal{C} = \frac{4\pi \varepsilon}{\ \frac{1}{R_1} - \frac{1}{R_2}\ }\ [151] => | [[Image:Spherical Capacitor.svg|97px]] [152] => *\varepsilon: [[Permittivity]] [153] => |- [154] => ! Two spheres,
equal radius{{Cite book |last=Maxwell |first=J.;C. |year=1873 |title=A Treatise on Electricity and Magnetism |publisher=Dover |page=266 ff |isbn=978-0-486-60637-8}}{{Cite journal |last=Rawlins |first=A.D. |year=1985 |title=Note on the capacitance of two closely separated spheres |journal=IMA Journal of Applied Mathematics |volume=34 |issue=1 |pages=119–120 |doi=10.1093/imamat/34.1.119}} [155] => | \begin{align} [156] => \ \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) } \\ [157] => ={}&{}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] \\ [158] => ={}&{} 2\pi \varepsilon a\left[ \ln 2+\gamma -\frac{1}{2}\ln \left( 2D-2\right) + \mathcal{O}\left( 2D-2\right) \right] \\ [159] => ={}&{} 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] [160] => \end{align}\ [161] => | [162] => *a: Radius [163] => *d: Distance, d > 2a [164] => *D = d/2a, D > 1 [165] => *\gamma: [[Euler–Mascheroni constant|Euler's constant]] [166] => *q = D + \sqrt{D^2 - 1} [167] => *\psi_q(z)=\frac{\partial_z\Gamma_q(z)}{\Gamma_q(z)}: the q-digamma function [168] => *\Gamma_q(z): the q-Gamma function{{Cite book| last1 = Gasper | last2 = Rahman | title = Basic Hypergeometric Series | year = 2004 | publisher = Cambridge University Press |at = p.20-22 | isbn = 978-0-521-83357-8}} [169] => See also [[Basic hypergeometric series]]. [170] => |- [171] => ! Sphere in front of wall [172] => | \ \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) }\ [173] => | [174] => *\ a\ : Radius [175] => *\ d\ : Distance, d > a [176] => *D=d/a [177] => |- [178] => ! Sphere [179] => | \ \mathcal{C} = 4 \pi \varepsilon a\ [180] => | [181] => *a: Radius [182] => |- [183] => ! Circular disc{{cite book |last=Jackson |first=J.D. |year=1975 |title=Classical Electrodynamics |publisher=Wiley |page=128, problem 3.3 }} [184] => | \ \mathcal{C} = 8 \varepsilon a\ [185] => | [186] => * a: Radius [187] => |- [188] => ! Thin straight wire,
finite length{{cite journal |last=Maxwell |first=J. C. |year=1878 |title=On the electrical capacity of a long narrow cylinder and of a disk of sensible thickness |journal=Proceedings of the London Mathematical Society |volume=IX |pages=94–101 |doi=10.1112/plms/s1-9.1.94 |url=https://zenodo.org/record/1447764 }}{{Cite journal |last=Vainshtein |first=L. A. |year=1962 |title=Static boundary problems for a hollow cylinder of finite length. III Approximate formulas |journal=[[Zhurnal Tekhnicheskoi Fiziki]] |volume=32 |pages=1165–1173}}{{cite journal |last=Jackson |first=J. D. |year=2000 |title=Charge density on thin straight wire, revisited |journal=American Journal of Physics |volume=68 |issue=9 |pages=789–799 |doi=10.1119/1.1302908 |bibcode = 2000AmJPh..68..789J }} [189] => | \ \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]\ [190] => | [191] => *a: Wire radius [192] => *\ell: Length [193] => *\ \Lambda = \ln \left( \ell/a \right)\ [194] => |} [195] => [196] => ==Energy storage== [197] => The [[energy]] (measured in [[joule]]s) 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 d''q'' from one plate to the other against the potential difference {{nowrap|1=''V'' = ''q''/''C''}} requires the work d''W'': [198] => \mathrm{d}W = \frac{q}{C}\,\mathrm{d}q, [199] => where ''W'' is the work measured in joules, ''q'' is the charge measured in coulombs and ''C'' is the capacitance, measured in farads. [200] => [201] => The energy stored in a capacitor is found by [[integral|integrating]] this equation. Starting with an uncharged capacitance ({{nowrap|1=''q'' = 0}}) and moving charge from one plate to the other until the plates have charge +''Q'' and −''Q'' requires the work ''W'': [202] => 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}. [203] => [204] => ==Nanoscale systems== [205] => The capacitance of nanoscale dielectric capacitors such as [[quantum dots]] 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. [206] => [207] => ===Single-electron devices=== [208] => The capacitance of a connected, or "closed", single-electron device is twice the capacitance of an unconnected, or "open", single-electron device.{{Cite book | pages=312–315 | title=Superlattice to Nanoelectronics | isbn = 978-0-08-096813-1 | author=Raphael Tsu | publisher=Elsevier | year=2011 }} 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).{{Cite journal | author=T. LaFave Jr. | title=Discrete charge dielectric model of electrostatic energy | arxiv=1203.3798|journal=J. Electrostatics | year=2011 | volume=69 | issue=6 | pages=414–418 | doi=10.1016/j.elstat.2011.06.006 | s2cid=94822190 }} [209] => [210] => ===Few-electron devices=== [211] => The derivation of a "quantum capacitance" of a few-electron device involves the thermodynamic chemical potential of an ''N''-particle system given by [212] => \mu(N) = U(N) - U(N-1), [213] => [214] => whose energy terms may be obtained as solutions of the Schrödinger equation. The definition of capacitance, [215] => {1\over C} \equiv {\Delta V\over\Delta Q}, [216] => with the potential difference [217] => \Delta V = {\Delta \mu \,\over e} = {\mu(N + \Delta N) -\mu(N) \over e} [218] => [219] => may be applied to the device with the addition or removal of individual electrons, [220] => \Delta N = 1 and \Delta Q = e. [221] => [222] => The "quantum capacitance" of the device is then{{cite journal [223] => |author1=G. J. Iafrate |author2=K. Hess |author3=J. B. Krieger |author4=M. Macucci |year=1995 [224] => |title=Capacitive nature of atomic-sized structures [225] => |journal=Phys. Rev. B [226] => |volume=52 [227] => |issue=15 [228] => |pages=10737–10739 |doi=10.1103/physrevb.52.10737 [229] => |pmid=9980157 |bibcode = 1995PhRvB..5210737I }} [230] => C_Q(N) = \frac{e^2}{\mu(N+1)-\mu(N)} = \frac{e^2}{E(N)}. [231] => [232] => This expression of "quantum capacitance" may be written as [233] => C_Q(N) = {e^2\over U(N)}, [234] => which differs from the conventional expression described in the introduction where W_\text{stored} = U, the stored electrostatic potential energy, [235] => C = {Q^2\over 2U}, [236] => by a factor of {{sfrac|2}} with Q = Ne. [237] => [238] => However, within the framework of purely classical electrostatic interactions, the appearance of the factor of {{sfrac|2}} is the result of integration in the conventional formulation involving the work done when charging a capacitor, [239] => W_\text{charging} = U = \int_0^Q \frac{q}{C} \, \mathrm{d}q, [240] => [241] => which is appropriate since \mathrm{d}q = 0 for systems involving either many electrons or metallic electrodes, but in few-electron systems, \mathrm{d}q \to \Delta \,Q= e. The integral generally becomes a summation. One may trivially combine the expressions of capacitance [242] => Q=CV [243] => and electrostatic interaction energy, [244] => U = Q V , [245] => to obtain [246] => C = Q{1\over V} = Q {Q \over U} = {Q^2 \over U}, [247] => [248] => which is similar to the quantum capacitance. A more rigorous derivation is reported in the literature.{{cite journal [249] => |author1 = T. LaFave Jr [250] => |author2 = R. Tsu [251] => |date = March–April 2008 [252] => |title = Capacitance: A property of nanoscale materials based on spatial symmetry of discrete electrons [253] => |url = http://www.pagesofmind.com/FullTextPubs/La08-LaFave-2008-capacitance-a-property-of-nanoscale-materials.pdf [254] => |access-date = 12 February 2014 [255] => |journal = Microelectronics Journal [256] => |volume = 39 [257] => |issue = 3–4 [258] => |pages = 617–623 [259] => |doi = 10.1016/j.mejo.2007.07.105 [260] => |url-status = dead [261] => |archive-url = https://web.archive.org/web/20140222131652/http://www.pagesofmind.com/FullTextPubs/La08-LaFave-2008-capacitance-a-property-of-nanoscale-materials.pdf | archive-date = 22 February 2014}} 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. [262] => [263] => Apparent mathematical differences may be understood more fundamentally. The potential energy, U(N), of an isolated device (self-capacitance) is twice that stored in a "connected" device in the lower limit N = 1. As N grows large, U(N)\to U. Thus, the general expression of capacitance is [264] => C(N) = {(Ne)^2 \over U(N)}. [265] => [266] => 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, [[nanowires]] consisting of metal atoms typically do not exhibit the same conductive properties as their macroscopic, or bulk material, counterparts. [267] => [268] => ==Capacitance in electronic and semiconductor devices== [269] => [270] => 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 [[admittance]] is frequency-dependent, and a simple electrostatic formula for capacitance C = q/V, is not applicable. A more general definition of capacitance, encompassing electrostatic formula, is:{{cite journal |first=S.E. |last=Laux |title=Techniques for small-signal analysis of semiconductor devices |journal=IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems |volume=4 |issue=4 |pages=472–481 |doi=10.1109/TCAD.1985.1270145 |date=Oct 1985|s2cid=13058472 }} [271] => C = \frac{\operatorname{Im}(Y(\omega))}{\omega} , [272] => where Y(\omega) is the device admittance, and \omega is the angular frequency. [273] => [274] => 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. [275] => A paper by Steven Laux 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: [276] => C(\omega) = \frac{1}{\Delta V} \int_0^\infty [i(t)-i(\infty)] \cos (\omega t) dt. [277] => [278] => ==Negative capacitance in semiconductor devices== [279] => [280] => 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.{{cite journal |first=A.K. |last=Jonscher |title=The physical origin of negative capacitance |journal=J. Chem. Soc. Faraday Trans. II |volume=82 |pages=75–81 |doi=10.1039/F29868200075 |date=1986}} Negative capacitance has been demonstrated and explored in many different types of semiconductor devices.{{cite journal |first1=M. |last1=Ershov |first2=H.C. |last2=Liu |first3=L. |last3=Li |first4=M. |last4=Buchanan |first5=Z.R. |last5=Wasilewski |first6=A.K. |last6=Jonscher |title=Negative capacitance effect in semiconductor devices |journal=IEEE Trans. Electron Devices |volume=45 |issue=10 |pages=2196–2206 |date=Oct 1998 |doi=10.1109/16.725254|arxiv=cond-mat/9806145 |bibcode=1998ITED...45.2196E |s2cid=204925581 }} [281] => [282] => == Measuring capacitance == [283] => {{Main|Capacitance meter}} [284] => A [[capacitance meter]] is a piece of [[electronic test equipment]] used to measure capacitance, mainly of discrete [[capacitor]]s. For most purposes and in most cases the capacitor must be disconnected from [[electronic circuit|circuit]]. [285] => [286] => Many DVMs ([[Voltmeter|digital volt meter]]s) have a capacitance-measuring function. These usually operate by charging and discharging the [[Device under test|capacitor under test]] with a known [[Electric current|current]] and measuring the rate of rise of the resulting [[voltage]]; the slower the rate of rise, the larger the capacitance. DVMs can usually measure capacitance from [[Farad|nanofarads]] to a few hundred microfarads, but wider ranges are not unusual. It is also possible to measure capacitance by passing a known [[high-frequency]] [[alternating current]] through the device under test and measuring the resulting [[volt]]age across it (does not work for polarised capacitors). [287] => [288] => [[Image:AH2700 cap br.jpg|thumb|right|An [http://www.andeen-hagerling.com Andeen-Hagerling] 2700A capacitance bridge]] [289] => [290] => More sophisticated instruments use other techniques such as inserting the capacitor-under-test into a [[bridge circuit]]. 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 [[four-terminal sensing|Kelvin connection]]s and other careful design techniques, these instruments can usually measure capacitors over a range from picofarads to farads. [291] => [292] => ==See also== [293] => {{div col begin|colwidth=14em}} [294] => * [[Capacitive displacement sensor]] [295] => * [[Capacity of a set]] [296] => * [[Displacement current]] [297] => * [[Gauss law]] [298] => * [[LCR meter]] [299] => * [[Magnetocapacitance]] [300] => * [[Quantum capacitance]] [301] => {{div col end}} [302] => [303] => ==References== [304] => {{reflist|25em}} [305] => [306] => ==Further reading== [307] => {{Refbegin}} [308] => *Tipler, Paul (1998). ''Physics for Scientists and Engineers: Vol. 2: Electricity and Magnetism, Light'' (4th ed.). W. H. Freeman. {{ISBN|1-57259-492-6}} [309] => *Serway, Raymond; Jewett, John (2003). ''Physics for Scientists and Engineers'' (6th ed.). Brooks Cole. {{ISBN|0-534-40842-7}} [310] => *Saslow, Wayne M.(2002). ''Electricity, Magnetism, and Light''. Thomson Learning. {{ISBN|0-12-619455-6}}. See Chapter 8, and especially pp. 255–259 for coefficients of potential. [311] => {{Refend}} [312] => [313] => ==External links== [314] => *{{Commonscatinline|Capacitance}} [315] => [316] => {{Authority control}} [317] => [318] => [[Category:Capacitance| ]] [319] => [[Category:Scalar physical quantities]] [320] => [[Category:Electricity]] [321] => [[Category:Electromagnetic quantities]] [] => )
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Capacitance

Capacitance is the ability of a body or system to store an electric charge. It is measured in units called farads and is represented by the symbol C.

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It is measured in units called farads and is represented by the symbol C. The concept of capacitance was first introduced by the 18th-century German scientist Ewald Georg von Kleist and was further developed by other scientists such as Benjamin Franklin and Alessandro Volta. Capacitors, which are devices specifically designed to store electric charge, are essential components in electronic circuits. They consist of two conductive plates separated by an insulating material called a dielectric. When a voltage difference is applied across the plates, an electric field forms, causing the accumulation of opposite charges on the capacitor. The amount of charge that a capacitor can store for a given voltage is directly proportional to its capacitance. The capacitance of a capacitor depends on various factors such as the area of the plates, their separation distance, and the properties of the dielectric material. Different types of capacitors exist, including electrolytic capacitors, ceramic capacitors, and tantalum capacitors, each with their own characteristics and applications. Capacitance plays a crucial role in various fields, from power distribution and filtering in electronic devices to energy storage in devices like batteries and supercapacitors. It also affects the behavior of electrical circuits, influencing parameters like frequency response, time constant, and impedance. Understanding capacitance is fundamental in electrical engineering and physics because it allows for the analysis and design of electrical circuits and systems. The Wikipedia page on capacitance provides a comprehensive overview of this concept, covering its theoretical background, historical development, practical applications, and related mathematical equations and formulas.

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