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Common symbols | C |
---|---|

SI unit | farad |

Other units | μF, nF, pF |

In SI base units | F = A^{2} s^{4} kg^{−1} m^{−2} |

Derivations from other quantities | C = charge / voltage |

Dimension | M^{−1}L^{−2}T^{4}I^{2} |

**Capacitance** is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential. There are two closely related notions of capacitance: *self capacitance* and *mutual capacitance*. Any object that can be electrically charged exhibits *self capacitance*. A material with a large self capacitance holds more electric charge at a given voltage than one with low capacitance. The notion of *mutual capacitance* is particularly important for understanding the operations of the capacitor, one of the three elementary linear electronic components (along with resistors and inductors).

**Electric charge** is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: *positive* and *negative*. Like charges repel and unlike attract. An object with an absence of net charge is referred to as *neutral*. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.

An **electric potential** is the amount of work needed to move a unit of charge from a reference point to a specific point inside the field without producing an acceleration. Typically, the reference point is the Earth or a point at infinity, although any point can be used.

**Voltage**, **electric potential difference**, **electric pressure **or **electric tension** is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named *volt*. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for *volt* uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by ∆*V*, but more often simply as *V*, for instance in the context of Ohm's or Kirchhoff's circuit laws.

- Self capacitance
- Mutual capacitance
- Capacitance matrix
- Capacitors
- Stray capacitance
- Capacitance of conductors with simple shapes
- Energy storage
- Nanoscale systems
- Single-electron devices
- Few-electron devices
- Capacitance in electronic and semiconductor devices
- Negative capacitance in semiconductor devices
- Measuring capacitance
- See also
- References
- Further reading

The capacitance is a function only of the geometry of the design (e.g. area of the plates and the distance between them) and the permittivity of the dielectric material between the plates of the capacitor. 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.

In electromagnetism, **absolute permittivity**, often simply called **permittivity**, usually denoted by the Greek letter *ε* (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a given medium. A charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

A **dielectric** is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing **dielectric polarization**. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

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.^{ [1] } The reciprocal of capacitance is called elastance.

The **farad** is the SI derived unit of electrical capacitance, the ability of a body to store an electrical charge. It is named after the English physicist Michael Faraday.

**Michael Faraday** FRS was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis.

The **coulomb** is the International System of Units (SI) unit of electric charge. It is the charge transported by a constant current of one ampere in one second:

In 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, for an isolated conductor, there also exists a property called *self capacitance*, which is the amount of electric charge that must be added to an isolated conductor to raise its electric potential by one unit (i.e. one volt, in most measurement systems).^{ [2] } The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, with the conductor centered inside this sphere.

**Mutual capacitance** is intentional or unintentional capacitance that occurs between two charge-holding objects or conductors, in which the current passing through one passes over into the other. In transmission lines, when conductors are closely spaced together, the air or material separating the lines acts as a dielectric, and the conductors act as a capacitors plates.

Mathematically, the *self capacitance* of a conductor is defined by

where

*q*is the charge held by the conductor,- is the electric potential,
- σ is the surface charge density.
*dS*is an infinitesimal element of area,*r*is the length from dS to a fixed point*M*within the plate- is the vacuum permittivity

Using this method, the self capacitance of a conducting sphere of radius *R* is:^{ [3] }

Example values of self capacitance are:

- for the top "plate" of a van de Graaff generator, typically a sphere 20 cm in radius: 22.24 pF,
- the planet Earth: about 710 µF.
^{ [4] }

A **Van de Graaff generator** is an electrostatic generator which uses a moving belt to accumulate electric charge on a hollow metal globe on the top of an insulated column, creating very high electric potentials. It produces very high voltage direct current (DC) electricity at low current levels. It was invented by American physicist Robert J. Van de Graaff in 1929. The potential difference achieved by modern Van de Graaff generators can be as much as 5 megavolts. A tabletop version can produce on the order of 100,000 volts and can store enough energy to produce a visible spark. Small Van de Graaff machines are produced for entertainment, and for physics education to teach electrostatics; larger ones are displayed in some science museums.

**Earth** is the third planet from the Sun and the only astronomical object known to harbor life. According to radiometric dating and other sources of evidence, Earth formed over 4.5 billion years ago. Earth's gravity interacts with other objects in space, especially the Sun and the Moon, Earth's only natural satellite. Earth orbits around the Sun in 365.26 days, a period known as an Earth year. During this time, Earth rotates about its axis about 366.26 times.

The inter-winding capacitance of a coil is sometimes called self capacitance,^{ [5] } 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 impedance of the coil and gives rise to parallel resonance. In many applications this is an undesirable effect and sets an upper frequency limit for the correct operation of the circuit.^{[ citation needed ]}

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

If the charges on the plates are +*q* and −*q*, and *V* gives the voltage between the plates, then the capacitance *C* is given by

which gives the voltage/current relationship

The energy stored in a capacitor is found by integrating the work *W*:

The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition 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, Maxwell introduced his * coefficients of potential *. If three (nearly ideal) conductors are given charges , then the voltage at conductor 1 is given by

and similarly for the other voltages. Hermann von Helmholtz and Sir William Thomson showed that the coefficients of potential are symmetric, so that , 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:

From this, the mutual capacitance between two objects can be defined^{ [6] } by solving for the total charge *Q* and using .

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 is known as the *capacitance matrix*,^{ [7] }^{ [8] } and is the inverse of the elastance matrix.

The capacitance of the majority of capacitors used in electronic circuits is generally several orders of magnitude smaller than the farad. The most common subunits of capacitance in use today are the microfarad (µF), nanofarad (nF), picofarad (pF), and, in microcircuits, femtofarad (fF). However, specially made supercapacitors can be much larger (as much as hundreds of farads), and parasitic capacitive elements can be less than a femtofarad. In the past, alternate subunits were used in historical electronic books; "mfd" and "mf" for microfarad (µF); "mmfd", "mmf", "µµF" for picofarad (pF); but are rarely used any more.^{ [9] }^{ [10] }

Capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. A qualitative explanation for this can be given as follows.

Once a positive charge is put unto a conductor, this charge creates an electrical field, repelling any other positive charge to be moved onto the conductor; i.e., increasing the necessary voltage. But if nearby there is another conductor with a negative charge on it, the electrical field of the positive conductor repelling the second positive charge is weakened (the second positive charge also feels the attracting force of the negative charge). So due to the second conductor with a negative charge, it becomes easier to put a positive charge on the already positive charged first conductor, and vice versa; i.e., the necessary voltage is lowered.

As a quantitative example consider 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:

where

*C*is the capacitance, in farads;*A*is the area of overlap of the two plates, in square meters;*ε*_{0}is the electric constant (*ε*_{0}≈ 8.854×10^{−12}F⋅m^{−1}); and*d*is the separation between the plates, in meters;

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. 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.

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

where *W* is the energy, in joules; *C* is the capacitance, in farads; and *V* is the voltage, in volts.

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), and it can be a limiting factor for proper functioning of circuits at high frequency.

Stray capacitance between the input and output in amplifier circuits can be troublesome because it can form a path for 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 1/*K*, 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. 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.

Calculating the capacitance of a system amounts to solving the Laplace equation *∇ ^{2}φ = 0* with a constant potential

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

Type | Capacitance | Comment |
---|---|---|

Parallel-plate capacitor |
| |

Concentric cylinders |
| |

Pair of parallel wires^{ [11] } | ||

Wire parallel to wall^{ [11] } | a: Wire radius d: Distance, d > aℓ: Wire length | |

Two parallel coplanar strips ^{ [12] } | d: Distancew: Strip width_{1}, w_{2}k: _{m}d/(2w_{m}+d)
k_{1}k_{2}K: Elliptic integral ℓ: Length | |

Concentric spheres |
| |

Two spheres, equal radius ^{ [13] }^{ [14] } | a: Radiusd: Distance, d > 2aD = d/2a, D > 1γ: Euler's constant | |

Sphere in front of wall^{ [13] } | : Radius : Distance, | |

Sphere | : Radius | |

Circular disc^{ [15] } | : Radius | |

Thin straight wire, finite length ^{ [16] }^{ [17] }^{ [18] } | : Wire radius : Length |

The energy (measured in joules) 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 *V* = *q/C* requires the work d*W*:

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 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*:

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, however, the resulting spatial distribution of equipotential surfaces within the device are exceedingly complex.

The capacitance of a connected, or "closed", single-electron device is twice the capacitance of an unconnected, or "open", single-electron device.^{ [19] } 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).^{ [20] }

The derivation of a "quantum capacitance" of a few-electron device involves the thermodynamic chemical potential of an *N*-particle system given by

whose energy terms may be obtained as solutions of the Schrödinger equation. The definition of capacitance,

- ,

with the potential difference

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

- and .

Then

is the "quantum capacitance" of the device.^{ [21] }

This expression of "quantum capacitance" may be written as

which differs from the conventional expression described in the introduction where , the stored electrostatic potential energy,

by a factor of 1/2 with .

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,

which is appropriate since for systems involving either many electrons or metallic electrodes, but in few-electron systems, . The integral generally becomes a summation. One may trivially combine the expressions of capacitance and electrostatic interaction energy,

- and ,

respectively, to obtain,

which is similar to the quantum capacitance. A more rigorous derivation is reported in the literature.^{ [22] } In particular, to circumvent the mathematical challenges of the spatially complex equipotential surfaces within the device, an *average* electrostatic potential experiences by *each* electron is utilized in the derivation.

The reason for apparent mathematical differences is understood more fundamentally as the potential energy, , of an isolated device (self-capacitance) is twice that stored in a "connected" device in the lower limit *N*=1. As *N* grows large, .^{ [20] } Thus, the general expression of capacitance is

- .

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.

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 time-varying electric field. Carrier transport is affected by electric field 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 is not applicable. A more general definition of capacitance, encompassing electrostatic formula, is:^{ [23] }

where is the device admittance, and is the angular frequency.

In general case, capacitance is a function of frequency. At high frequencies, capacitance approached a constant value, equal to "geometric" capacitance, determined by the terminals' geometry and dielectric content in the device. A paper by Steven Laux^{ [23] } 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:

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.^{ [24] } Negative capacitance has been demonstrated and explored in many different types of semiconductor devices.^{ [25] }

A capacitance meter is a piece of electronic test equipment used to measure capacitance, mainly of discrete capacitors. For most purposes and in most cases the capacitor must be disconnected from circuit.

Many DVMs (digital volt meters) have a capacitance-measuring function. These usually operate by charging and discharging the capacitor under test with a known 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 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 voltage across it (does not work for polarised capacitors).

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 Kelvin connections and other careful design techniques, these instruments can usually measure capacitors over a range from picofarads to farads.

**Electrical impedance** is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term *complex impedance* may be used interchangeably.

In physics, the **dissipation factor** (DF) is a measure of loss-rate of energy of a mode of oscillation in a dissipative system. It is the reciprocal of quality factor, which represents the "quality" or durability of oscillation.

In physics, the **electric displacement field**, denoted by **D**, is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "**D**" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter square (C⋅m^{−2}).

In semiconductor physics, the **depletion region**, also called **depletion layer**, **depletion zone**, **junction region**, **space charge region** or **space charge layer**, is an insulating region within a conductive, doped semiconductor material where the mobile charge carriers have been diffused away, or have been forced away by an electric field. The only elements left in the depletion region are ionized donor or acceptor impurities.

**Electric potential energy**, or **electrostatic potential energy**, is a potential energy that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An *object* may have electric potential energy by virtue of two key elements: its own electric charge and its relative position to other electrically charged *objects*.

**Comb-drives** are actuators, often used as linear actuators electrostatic forces that act between two electrically conductive combs. Comb drive actuators typically operate at the micro- or nanometer scale and are generally manufactured by bulk micromachining or surface micromachining a silicon wafer substrate.

In mesoscopic physics, a **Coulomb blockade** (**CB**), named after Charles-Augustin de Coulomb's electrical force, is the decrease in electrical conductance at small bias voltages of a small electronic device comprising at least one low-capacitance tunnel junction. Because of the CB, the conductance of a device may not be constant at low bias voltages, but disappear for biases under a certain threshold, i.e. no current flows.

Capacitors are manufactured in many forms, styles, lengths, girths, and from many materials. They all contain at least two electrical conductors separated by an insulating layer. Capacitors are widely used as parts of electrical circuits in many common electrical devices.

**Parasitic capacitance**, or **stray capacitance** is an unavoidable and usually unwanted capacitance that exists between the parts of an electronic component or circuit simply because of their proximity to each other. When two electrical conductors at different voltages are close together, the electric field between them causes electric charge to be stored on them; this effect is parasitic capacitance. All actual circuit elements such as inductors, diodes, and transistors have internal capacitance, which can cause their behavior to depart from that of 'ideal' circuit elements. Additionally, there is always non-zero capacitance between any two conductors; this can be significant at higher frequencies with closely spaced conductors, such as wires or printed circuit board traces. Parasitic capacitance is a significant problem in high frequency circuits and is often the factor limiting the operating frequency and bandwidth of electronic components and circuits.

A **capacitor** is a device that stores electrical energy in an electric field. It is a passive electronic component with two terminals.

A **ceramic capacitor** is a fixed-value capacitor where the ceramic material acts as the dielectric. It is constructed of two or more alternating layers of ceramic and a metal layer acting as the electrodes. The composition of the ceramic material defines the electrical behavior and therefore applications. Ceramic capacitors are divided into two application classes:

**Differential capacitance** in physics, electronics, and electrochemistry is a measure of the voltage-dependent capacitance of a nonlinear capacitor, such as an electrical double layer or a semiconductor diode. It is defined as the derivative of charge with respect to potential.

**Quantum capacitance**, also called **chemical capacitance** and **electrochemical capacitance** , is a quantity first introduced by Serge Luryi (1988), and is defined as the variation of electrical charge respect to the variation of electrochemical potential , i.e., .

An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An **LC circuit** is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:

A **micro-channel plate** (**MCP**) is a planar component used for detection of single particles and low intensity impinging radiation. It is closely related to an electron multiplier, as both intensify single particles or photons by the multiplication of electrons via secondary emission. However, because a microchannel plate detector has many separate channels, it can additionally provide spatial resolution.

This article provides a more detailed explanation of p–n diode behavior than that found in the articles p–n junction or diode.

**Double-layer capacitance** is the important characteristic of the electrical double layer which appears, for example, at the interface between a conductive electrode and an adjacent liquid electrolyte. At this boundary two layers of charge with opposing polarity form, one at the surface of the electrode, and one in the electrolyte. These two layers, electrons on the electrode and ions in the electrolyte, are typically separated by a single layer of solvent molecules that adhere to the surface of the electrode and act like a dielectric in a conventional capacitor. The amount of electric charge stored in double-layer capacitor depends on the applied voltage. The unit of capacitance is the farad.

In semiconductor electrochemistry, a **Mott–Schottky plot** describes the reciprocal of the square of capacitance versus the potential difference between bulk semiconductor and bulk electrolyte. In many theories, and in many experimental measurements, the plot is linear. The use of Mott–Schottky plots to determine system properties is termed Mott–Schottky analysis.

Performance modelling is the abstraction of a real system into a simplified representation to enable the prediction of performance. The creation of a model can provide insight into how a proposed or actual system will or does work. This can, however, point towards different things to people belonging to different fields of work.

- ↑ "Definition of 'farad'". Collins.
- ↑ William D. Greason (1992).
*Electrostatic discharge in electronics*. Research Studies Press. p. 48. ISBN 978-0-86380-136-5 . Retrieved 4 December 2011. - ↑ Lecture notes; University of New South Wales
- ↑ Tipler, Paul; Mosca, Gene (2004).
*Physics for Scientists and Engineers*(5th ed.). Macmillan. p. 752. ISBN 978-0-7167-0810-0. - ↑ Massarini, A.; Kazimierczuk, M.K. (1997). "Self capacitance of inductors".
*IEEE Transactions on Power Electronics*.**12**(4): 671–676. Bibcode:1997ITPE...12..671M. CiteSeerX 10.1.1.205.7356 . doi:10.1109/63.602562: example of the use of the term 'self capacitance'. - ↑ Jackson, John David (1999).
*Classical Electrodynamic*(3rd ed.). John Wiley & Sons. p. 43. ISBN 978-0-471-30932-1. - ↑ Maxwell, James (1873). "3".
*A treatise on electricity and magnetism*.**1**. Clarendon Press. p. 88ff. - ↑ "Capacitance : Charge as a Function of Voltage".
*Av8n.com*. Retrieved 20 September 2010. - ↑ "Capacitor MF-MMFD Conversion Chart".
*Just Radios*. - ↑
*Fundamentals of Electronics*. Volume 1b — Basic Electricity — Alternating Current. Bureau of Naval Personnel. 1965. p. 197. - 1 2 Jackson, J. D. (1975).
*Classical Electrodynamics*. Wiley. p. 80. - ↑ Binns; Lawrenson (1973).
*Analysis and computation of electric and magnetic field problems*. Pergamon Press. ISBN 978-0-08-016638-4. - 1 2 Maxwell, J. C. (1873).
*A Treatise on Electricity and Magnetism*. Dover. p. 266ff. ISBN 978-0-486-60637-8. - ↑ Rawlins, A. D. (1985). "Note on the Capacitance of Two Closely Separated Spheres".
*IMA Journal of Applied Mathematics*.**34**(1): 119–120. doi:10.1093/imamat/34.1.119. - ↑ Jackson, J. D. (1975).
*Classical Electrodynamics*. Wiley. p. 128, problem 3.3. - ↑ Maxwell, J. C. (1878). "On the electrical capacity of a long narrow cylinder and of a disk of sensible thickness".
*Proc. London Math. Soc*.**IX**: 94–101. doi:10.1112/plms/s1-9.1.94. - ↑ Vainshtein, L. A. (1962). "Static boundary problems for a hollow cylinder of finite length. III Approximate formulas".
*Zh. Tekh. Fiz*.**32**: 1165–1173. - ↑ Jackson, J. D. (2000). "Charge density on thin straight wire, revisited".
*Am. J. Phys*.**68**(9): 789–799. Bibcode:2000AmJPh..68..789J. doi:10.1119/1.1302908. - ↑ Raphael Tsu (2011).
*Superlattice to Nanoelectronics*. Elsevier. pp. 312–315. ISBN 978-0-08-096813-1. - 1 2 T. LaFave Jr. (2011). "Discrete charge dielectric model of electrostatic energy".
*J. Electrostatics*.**69**(6): 414–418. arXiv: 1203.3798 . doi:10.1016/j.elstat.2011.06.006. - ↑ G. J. Iafrate; K. Hess; J. B. Krieger; M. Macucci (1995). "Capacitive nature of atomic-sized structures".
*Phys. Rev. B*.**52**(15): 10737–10739. Bibcode:1995PhRvB..5210737I. doi:10.1103/physrevb.52.10737. - ↑ T. LaFave Jr; R. Tsu (March–April 2008). "Capacitance: A property of nanoscale materials based on spatial symmetry of discrete electrons" (PDF).
*Microelectronics Journal*.**39**(3–4): 617–623. doi:10.1016/j.mejo.2007.07.105. Archived from the original (PDF) on 22 February 2014. Retrieved 12 February 2014. - 1 2 Laux, S.E. (October 1985). "Techniques for small-signal analysis of semiconductor devices".
*IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems*.**4**(4): 472–481. doi:10.1109/TCAD.1985.1270145. - ↑ Jonscher, A.K. (1986). "The physical origin of negative capacitance".
*J. Chem. Soc. Faraday Trans. II*.**82**: 75–81. doi:10.1039/F29868200075. - ↑ 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. arXiv: cond-mat/9806145 . Bibcode:1998ITED...45.2196E. doi:10.1109/16.725254.

- Tipler, Paul (1998).
*Physics for Scientists and Engineers: Vol. 2: Electricity and Magnetism, Light*(4th ed.). W. H. Freeman. ISBN 1-57259-492-6 - Serway, Raymond; Jewett, John (2003).
*Physics for Scientists and Engineers*(6th ed.). Brooks Cole. ISBN 0-534-40842-7 - 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.

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