Capacitance - Wikipedia, the free encyclopediaPage 1 of 10 Capacitance From Wikipedia, the free encyclopedia In electromagnetism and electronics, capacitance is the ability of a body to hold an electrical charge. Capacitance is also a measure of the amount of electrical energy stored (or separated) for a given electric potential. A common form of energy storage device is a parallel-plate capacitor. In a parallel plate capacitor, capacitance is directly 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 +Q and −Q, and V gives the voltage between the plates, then the capacitance is given by Electromagnetism Electricity · Magnetism Electrostatics Electric charge · Coulomb's law · Electric field · Electric flux · Gauss's law · Electric potential · Electrostatic induction · Electric dipole moment · Polarization density Magnetostatics Ampère's law · Electric current · Magnetic field · Magnetization · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss's law for magnetism Electrodynamics Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy current Electrical Network Electrical conduction · Electrical resistance · Capacitance · Inductance · Impedance · Resonant cavities · Waveguides Covariant formulation Electromagnetic tensor · EM Stress-energy tensor · The SI unit of capacitance is the farad; 1 farad is 1 coulomb per volt. The energy (measured in joules) stored in a capacitor is equal to the work done to charge it. Consider a 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: 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 capacitance is found by integrating 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: Contents ■ 1 Capacitors ■ 1.1 Voltage dependent capacitors ■ 1.2 Frequency dependent capacitors ■ ■ ■ ■ ■ ■ ■ 2 Capacitance matrix 3 Self-capacitance 4 Elastance 5 Stray capacitance 6 Capacitance of simple systems 7 See also 8 References http://en.wikipedia.org/wiki/Capacitance 2/22/2011 In CGS units the equation has the form: [1] where C in this case has the units of length.Wikipedia. ε0 is the electric constant (ε0 ≈ 8. in farads.Capacitance . the greater the capacitance. where W is the energy.org/wiki/Capacitance 2/22/2011 .wikipedia. The most common subunits of capacitance in use today are the millifarad (mF). nanofarad (nF) and picofarad (pF). in volts. εr is the relative static permittivity (sometimes called the dielectric constant) of the material between the plates (for a vacuum. the free encyclopedia Page 2 of 10 ■ 9 Further reading Four-current · Electromagnetic four-potential Scientists Ampère · Coulomb · Faraday · Gauss · Heaviside · Henry · Hertz · Lorentz · Maxwell · Tesla · Volta · Weber · Ørsted Capacitors Main article: Capacitor The capacitance of the majority of capacitors used in electronic circuits is several orders of magnitude smaller than the farad. and V is the voltage. For example. A is the area of overlap of the two plates. in joules. C is the capacitance. The capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. for a flatplate capacitor the energy stored is: . The equation is a good approximation if d is small compared to the other dimensions of the plates so the field in the capacitor over most of its area is uniform. εr = 1). http://en. and Capacitance is proportional to the area of overlap and inversely proportional to the separation between conducting sheets. and the so-called fringing field around the periphery provides a small contribution. −12 F m–1).854 × 10 d is the separation between the plates. the capacitance of a parallel-plate capacitor constructed of two parallel plates both of area A separated by a distance d is approximately equal to the following: where C is the capacitance. The closer the sheets are to each other. microfarad (µF). Combining the SI equation for capacitance with the above equation for the energy stored in a capacitance. then the polarization of the dielectric cannot follow the signal. the dipole response is limited and the dielectric constant diminishes. so the capacitance for these devices is more complex. This field polarizes the dielectric. A changing dielectric constant with frequency is referred to as dielectric dispersion.[2][3] Corresponding to the voltage-dependent capacitance. C(V). in charging such a capacitor the differential increase in voltage with charge is governed by: where the voltage dependence of capacitance.[4] Another example of voltage dependent capacitance occurs in semiconductor devices such as semiconductor diodes.org/wiki/Capacitance 2/22/2011 .wikipedia. for example ferroelectric materials. stems from the field. By the same token.Capacitance . where the voltage dependence stems not from a change in dielectric constant but in a voltage dependence of the spacing between the charges on the two sides of the capacitor. As an example of the origin of this mechanism.[5] Frequency dependent capacitors If a capacitor is driven with a time-varying voltage that changes rapidly enough. the energy stored in the capacitor now is given by Integrating: where interchange of the order of integration is used. The nonlinear capacitance of a microscope probe scanned along a ferroelectric surface is used to study the domain structure of ferroelectric materials. which in a large area parallel plate device is given by ε = V/d. Under transient conditions. the internal microscopic dipoles contributing to the dielectric constant cannot move instantly. the free encyclopedia Page 3 of 10 Voltage dependent capacitors The dielectric constant for a number of very useful dielectrics changes as a function of the applied electrical field.Wikipedia. the displacement field can be expressed as (see electric susceptibility): http://en. translates into a capacitance that is a nonlinear function of the voltage causing the field. and so as frequency of an applied alternating voltage increases. is a nonlinear Sshaped function of field. and is governed by dielectric relaxation processes. which. such as Debye relaxation. which polarization. For example. in the case of a ferroelectric. in the case of a large area parallel plate device. to charge the capacitor to voltage V an integral relation is found: which agrees with Q = CV only when C is voltage independent. [12] Another example of frequency dependent capacitance occurs with MOS capacitors. V(ω) is the voltage component at angular frequency ω. Z(ω) is the complex impedance. A Fourier transform in time then results in: where εr(ω) is now a complex function. such as dichroism. the dielectric constant may exhibit a resonance at certain frequencies corresponding to characteristic response frequencies (excitation energies) of contributors to the dielectric constant. called the conductance. linear response function.[13][14] http://en. as in deep-level transient spectroscopy. Fourier transforming Gauss's law with this form for displacement field: where j is the imaginary unit.org/wiki/Capacitance 2/22/2011 . of the dipole behavior in the dielectric. For practical purposes. and C0 is the capacitance without the dielectric. also exhibits this frequency behavior. when measurement errors are taken into account. or simply a calculation of C0. See permittivity. for example.[6][7] The integral extends over the entire past history up to the present time. Z(ω) is the complex impedance with the dielectric present. with an imaginary part related to absorption of energy from the field by the medium. for example.[10]) Using this measurement method. C(ω) is the so-called complex capacitance with the dielectric present. while at low frequencies both types of carrier respond.[11] Alternatively. the free encyclopedia Page 4 of 10 indicating the lag in response by the time dependence of εr. G(ω) is the real part of the current. These resonances are the basis for a number of experimental techniques for detecting defects.Wikipedia. See. the time response of the capacitance can be used directly.[8][9] (Measurement "without the dielectric" in principle means measurement in free space. often a measurement in terrestrial vacuum.Capacitance . an unattainable goal inasmuch as even the quantum vacuum is predicted to exhibit nonideal behavior. the measurement of dielectric properties of the medium is based upon the relation: where a single prime denotes the real part and a double prime the imaginary part. is sufficiently accurate. The conductance method measures absorption as a function of frequency. When a parallel-plate capacitor is filled with a dielectric. being proportional to the dielectric constant. and C(ω) determines the imaginary part of the current and is the capacitance. calculated in principle from an underlying microscopic analysis. where the slow generation of minority carriers means that at high frequencies the capacitance measures only the majority carrier response. The capacitance.wikipedia. [15] Capacitance matrix The discussion above is limited to the case of two conducting plates. Hermann von Helmholtz and Sir William Thomson showed that the coefficients of potential are symmetric. The collection of coefficients elastance matrix. non-zero. The definition C=Q/V still holds for a single plate given a charge. the term capacitance is usually a shorthand for the mutual capacitance between two adjacent conductors.Capacitance . centered on the conductor. the self-capacitance of a conducting sphere of radius R is given by:[19] http://en. which is the amount of electrical charge that must be added to an isolated conductor to raise its electrical potential by one volt. so that P12 = P21. To handle this case.Q3.Q2. although of arbitrary size and shape. or when the net charge on the two plates is V1 = P11Q1 + P12Q2 + P13Q3 . Thus the system can be described by a collection of coefficients known as the elastance matrix or reciprocal capacitance matrix. Sophisticated modulation spectroscopy measurement methods based upon modulating the crystal structure by pressure or by other stresses and observing the related changes in absorption or reflection of light have advanced our knowledge of these materials. and similarly for the other voltages.[17][18] and is the inverse of the Self-capacitance In electrical circuits. the free encyclopedia Page 5 of 10 At optical frequencies. There also exists a property called self-capacitance. in which case the field lines produced by that charge terminate as if the plate were at the center of an oppositely charged sphere at infinity. Maxwell introduced his coefficients of potential.Wikipedia. The reference point for this potential is a theoretical hollow conducting sphere.org/wiki/Capacitance 2/22/2011 . of infinite radius. the mutual capacitance Cm between two objects can be defined[16] by solving for the total charge Q and using Cm = Q / V. etc. is known as the capacitance matrix. then the voltage of plate 1 is given by C = Q / V does not apply when there are more than two charged plates. Since no actual device holds perfectly equal and opposite charges on each of the two "plates". in semiconductors the dielectric constant exhibits structure related to the band structure of the solid. such as the two plates of a capacitor.wikipedia. If three plates are given charges Q1. it is the mutual capacitance that is reported on capacitors. which is defined as: From this. Using this method. typically a sphere 20 cm in radius: 20 pF ■ the planet Earth: about 710 μF[20] Elastance The inverse of capacitance is called elastance.Wikipedia. Miller's theorem states that. if the gain ratio of two nodes is 1/K. the internode capacitance. 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.) Miller's theorem can be used to effect this replacement. Capacitance of simple systems Calculating the capacitance of a system amounts to solving the Laplace equation ∇2φ=0 with a constant potential φ on the surface of the conductors. 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. will be seen to have been replaced by a capacitance of KC from input to ground and a capacitance of (K-1)C/K from output to ground. Stray capacitance can allow signals to leak between otherwise isolated circuits (an effect called crosstalk). (Since impedance varies inversely with capacitance. Capacitance of simple systems http://en.wikipedia. This is trivial in cases with high symmetry. The unit of elastance is the daraf. then an impedance 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. and it can be a limiting factor for proper functioning of circuits at high frequency.) When the input-to-output gain is very large. See also Schwarz-Christoffel mapping.org/wiki/Capacitance 2/22/2011 . There is no solution in terms of elementary functions in more complicated cases. (The original configuration — including the input-to-output capacitance — is often referred to as a piconfiguration. the free encyclopedia Page 6 of 10 Example values of self-capacitance are: ■ for the top "plate" of a van de Graaff generator. This (often unwanted) effect is termed "stray capacitance". Stray capacitance is often encountered in amplifier circuits in the form of "feedthrough" capacitance that interconnects the input and output nodes (both defined relative to a common ground). although the capacitance will be small unless the conductors are close together for long. Stray capacitance Any two adjacent conductors can be considered a capacitor.Capacitance . C. For quasi two-dimensional situations analytic functions may be used to map different geometries to each other. the free encyclopedia Page 7 of 10 Type Parallel-plate capacitor Capacitance Comment A: Area d: Distance ε: Permittivity a1: Inner radius a2: Outer radius l: Length a: Wire radius d: Distance. equal radius [23][24] http://en. w2: Strip width ki: d/(2wi+d) k2: k1k2 K: Elliptic integral l: Length a1: Inner radius a2: Outer radius a: Radius d: Distance. d >a l: Wire length d: Distance w1.Capacitance . d > 2a l: Length of pair a: Wire radius d: Distance.Wikipedia.org/wiki/Capacitance 2/22/2011 . d > 2a D = d/2a γ: Euler's constant Coaxial cable Pair of parallel wires [21] Wire parallel to wall[21] Two parallel coplanar strips[22] Concentric spheres Two spheres.wikipedia. George W Taylor (Editors) (2001). Science and Technology of Integrated Ferroelectrics: Selected Papers from Eleven Years of the Proceedings of the International Symposium on Integrated Ferroelectrics (http://books. 504. U Böttger & S Tiedke. ^ Solomon Musikant (1991).google.google.com/books?id=KVM2onr8_QYC&pg=PA648) 2. http://books.M1) . finite length [26][27][28] See also ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Ampère's law Capacitor Capacitive Displacement Sensors Conductance Conductor Displacement current Electromagnetism Electricity Electronics Hydraulic analogy Inductor Inductance Quantum capacitance References 1. Ramamoorthy Ramesh. Figure 25. Wiley. 1986. ISBN 0824784987. http://en.com/books? id=Jc8xRdgdH38C&pg=PA44&dq=nonlinear+capacitance+ferroelectric&lr=&as_brr=0#PPA43. ^ Carlos Paz de Araujo.M1) (in Polar Oxides. http://books. ISBN 3527405321.google. http://books. CRC Press. p.). Kwok K.wikipedia. Google books link (http://books.Capacitance .M1. 5. ^ The Physics Problem Solver.com/books? id=wQ09DhMBJroC&pg=PA304&dq=nonlinear+capacitance+ferroelectric&lr=&as_brr=0#PPA303. ISBN 9056997041. Scanning Nonlinear Dielectric Microscope (http://books.com/books? id=QMlOkeJ4qN4C&pg=PA504&dq=nonlinear+capacitance+ferroelectric&lr=&as_brr=0) . p.google. Wiley-VCH. Sze.google.google.org/wiki/Capacitance 2/22/2011 . d >a D = d/a a: Radius a: Radius a: Wire radius l: Length Λ: ln(l/a) Sphere Circular disc [25] Thin straight wire.Wikipedia. R Waser. ^ Yasuo Cho (2005).M1. Ng (2006). p. CRC Press.google. What Every Engineer Should Know about Ceramics (http://books.com/books? id=o4unkmHBHb8C&pg=PA217&dq=high+low+frequency+C-V&lr=&as_brr=0#PPA121.9.M1. the free encyclopedia Page 8 of 10 Sphere in front of wall [23] a: Radius d: Distance. 4. http://books.com/books? id=QMlOkeJ4qN4C&pg=PA504&dq=nonlinear+capacitance+ferroelectric&lr=&as_brr=0.google.M1) (3rd Edition ed. Figure 3.). editors ed.google.com/books? id=o4unkmHBHb8C&pg=PA217&dq=high+low+frequency+C-V&lr=&as_brr=0#PPA121.com/books? id=Jc8xRdgdH38C&pg=PA44&dq=nonlinear+capacitance+ferroelectric&lr=&as_brr=0#PPA43. ^ Simon M. 121.com/books? id=wQ09DhMBJroC&pg=PA304&dq=nonlinear+capacitance+ferroelectric&lr=&as_brr=0#PPA303. Chapter 16. 3. Figure 2. ISBN 0470068302. 43. Physics of Semiconductor Devices (http://books. com/books? id=kFkIKRfgUpsC&pg=PA538&dq=%22linear+response+theory% 22+capacitance+OR+conductance&lr=&as_brr=0#PPA111. ^ J.google. ISBN 0470068302. A treatise on electricity and magnetism. A Treatise on Electricity and Magnetism. ^ Binns. Volume 1. J. http://books. "3" (http://www. Semiconductor Material and Device Characterization (http://books. http://www.av8n.google. 17.doi. Macmillan. 425. Kalmykov (2006). ISBN 3540207856.com/books?id=kFkIKRfgUpsC&pg=PA538&dq=%22linear+response+theory% 22+capacitance+OR+conductance&lr=&as_brr=0#PPA111. Yu. ^ Simon M. ^ "Capacitance" (http://www. 7. Nozaki.google.M1) . 12. Springer.M1.com/books? id=mgtQslaXBc4C&pg=PA18&dq=%22dielectric+relaxation+function%22&lr=&as_brr=0#PPA17. 22. pp. 2005. Fundamentals of Semiconductors (http://books. Wiley. p. 111. ISBN 0521874998. P. 8. Springer Handbook of Electronic and Photonic Materials (http://books. http://books. p. p.google. Tetsuya Saito. ISBN 0471739065.org/details/electricandmagne01maxwrich) .google. p. J.google. Mosca. 15. Figure 20.M1 13. Gene (2004).com/books? id=o4unkmHBHb8C&pg=PA217&dq=high+low+frequency+C-V&lr=&as_brr=0) (3rd Edition ed.M1.). USA: John Wiley & Sons. 16-19 May 2005. ISBN 978-0-47130932-1 17. §6. ed. 158. Giovanni Vignale (2005).google. ^ a b Maxwell.phys.org/10. 10. Lawrenson (1973).org/xpl/freeabs_all.archive.).google.. Leslie Smith (2006). Ng (2006). 217. Peter Capper (2006).1604368 (http://dx. ISBN 0-486-60637-6. 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Kasap.google. http://books. 266 ff.M1 (3rd Edition ed. pp.com/books? id=rVVW22pnzhoC&pg=PA425&dq=high+low+frequency+C-V&lr=&as_brr=0. p. Springer. Wiley.edu. http://books.M1. Dover. Quantum Theory of the Electron Liquid (http://books. ^ Maxwell. p.ieee. (1975). Ottawa ISBN 0-7803-8879-8 doi:10.1350-1353.M1.1109%2FIMTC.). pp. Clarendon Press. Fractals. Proceedings of the IEEE: Instrumentation and Measurement Technology Conference.com/books?id=A7TbrAm5Wq0C&pg=PR6&dq=%22linear+response+theory% 22+capacitance+OR+conductance&lr=&as_brr=0#PPA158. Classical Electrodynamics. Pergamon Press.google.htm) . C.22.com/books?id=8lANaR-Pqi4C&pg=RA1PA475&dq=%22the+dielectric+permittivity+is+defined%22&lr=&as_brr=0.org/details/electricandmagne01maxwrich 18.Part A (http://books. p.google. Wiley.pdf) 20. http://books.jsp?arnumber=1604368) .org/wiki/Capacitance 2/22/2011 .google. p. John David (1999).google.google. ^ Dieter K Schroder (2006).com/books?id=mgtQslaXBc4C&pg=PA18&dq=%22dielectric+relaxation+function% 22&lr=&as_brr=0#PPA17. Physics for scientists and engineers (5th ed.M1) . 14.). the free encyclopedia Page 9 of 10 6. ISBN 9780716708100 21. Obrzut. ISBN 0470046074. ^ Tipler.. pp. p.com/books? id=W9pdJZoAeyEC&pg=PA244&dq=isbn=3540254706#PPA315.M1.google. Paul.archive. 9.com/books?id=o4unkmHBHb8C&pg=PA217&dq=high+low+frequency+CV&lr=&as_brr=0.com/physics/capacitance.2005. A.Capacitance .). pp. Cambridge University Press. ^ Lecture notes. 80. Inc. Classical Electrodynamic (3rd. diffusion and relaxation in disordered complex systems. ^ Gabriele Giuliani.com/books? id=OX2cHKJWCKgC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA347. ISBN 978-0 -08-016638-4. ^ Horst Czichos. D. ^ Jørgen Rammer (2007). p. http://books.av8n.M1) (3rd Edition ed. http://books.2005. 270 ff. 43. 347 ff.M1) (3rd Edition ed. Quantum Field Theory of Non-equilibrium States (http://books.au/COURSES/FIRST_YEAR/pdf% 20files/5Capacitanceanddielectr.google. Springer.com/books? id=OX2cHKJWCKgC&pg=PA305&dq=capacitance+conductance+measurement+spectroscopy&lr=&as_brr=0#PPA271. ^ a b Jackson.com/books? id=OX2cHKJWCKgC&printsec=frontcover&source=gbs_summary_r&cad=0#PPA347. 16.com/books? id=W9pdJZoAeyEC&pg=PA244&dq=isbn=3540254706#PPA315. 475. Kwok K. Anopchenko and R.com/physics/capacitance. James (1873)..com/books?id=8lANaR-Pqi4C&pg=RA1-PA475&dq=%22the+dielectric+permittivity+is+defined% 22&lr=&as_brr=0) .htm.M1) . 88ff. 752. ISBN 3540254706.). http://en. Analysis and computation of electric and magnetic field problems.. 23. http://www.Wikipedia. http://books. ^ Rawlins.org/10. Soc. IMA Journal of Applied Mathematics 34 (1): 119–120.wikipedia. Raymond. (1878). See Chapter 8.119 (http://dx. IX: 94–101.wikipedia. a non-profit organization. See Terms of Use for details. "Charge density on thin straight wire.1093/imamat/34.1302908) . Wayne M.). "On the electrical capacity of a long narrow cylinder and of a disk of sensible thickness". Zh. Wikipedia® is a registered trademark of the Wikimedia Foundation. doi:10. problem 3. J.1093%2Fimamat%2F34. Physics for Scientists and Engineers: Vol. 2: Electricity and Magnetism.doi. revisited". ISBN 0-12-619455-6. ^ Jackson.119) .1112%2Fplms%2Fs1-9. Tekh. Proc. A. D.1119%2F1. H. London Math. Fiz. 26. the free encyclopedia Page 10 of 10 24. A. and Light.. Magnetism.doi.org/wiki/Capacitance" Categories: Fundamental physics concepts | Physical quantities | Electricity | Electronics terms ■ This page was last modified on 14 February 2011 at 14:13. 28. Brooks Cole. 32: 1165–1173. Further reading ■ Tipler. D. doi:10. John (2003). Jewett. p. Thomson Learning. Phys 68 (9): 789–799.org/10. 25. Retrieved from "http://en. and especially pp. C. Freeman. ■ Text is available under the Creative Commons Attribution-ShareAlike License.Capacitance .Wikipedia. L.3. J. http://en.94 (http://dx. (1985). Paul (1998). ^ Maxwell. 128. J. (1975). ^ Vainshtein. 27.org/10. J.1. "Static boundary problems for a hollow cylinder of finite length. "Note on the Capacitance of Two Closely Separated Spheres". ISBN 1-57259-492-6 ■ Serway. Inc.org/wiki/Capacitance 2/22/2011 . Wiley.doi. D.1119/1. W. Classical Electrodynamics.1. doi:10.1. 255–259 for coefficients of potential.1112/plms/s1-9. (2000). Am. Light (4th ed.).94) . (1962). Electricity. Physics for Scientists and Engineers (6 ed. ^ Jackson. additional terms may apply.1. III Approximate formulas".1302908 (http://dx.(2002). ISBN 0-534-40842-7 ■ Saslow.