The phenomenon of Capacitance is the ability of a body to hold an electrical charge.
Capacitance is also a measure of the amount of electric charge stored (or separated) for a given electric potential. A common form of charge storage device is a two-plate capacitor. If the charges on the plates are +Q and −Q, and V gives the voltage between the plates, then the capacitance is given by
The SI unit of capacitance is the farad; 1 farad = 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
We can find the energy stored in a capacitance 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:
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The physicist James Clerk Maxwell invented the concept of displacement current in his 1861 paper On Physical Lines of Force in connection with the displacement of electrical particles.
| “ | Electromotive force acting on a dielectric produces a state of polarization of its parts...capable of being described as a state in which every particle has its poles in opposite conditions. | ” |
| “ | ...we may conceive that the electricity in each molecule is so displaced that one side is rendered positively, and the other negatively electrical, but that the electricity remains entirety connected with the molecule, and does not pass from one molecule to another. | ” |
| “ | This displacement does not amount to a current, because when it attains a certain value it remains constant, but it is the commencement of a current, and its variations constitute currents in the positive of negative direction, according as the displacement is increasing or diminishing. | ” |
| “ | ...when the electromotive force varies, the electric displacement also varies. But a variation of displacement is equivalent to a current, and this current must be taken into account... | ” |
He then added displacement current to Ampère's law. Maxwell's correction to Ampère's law remains valid.
Capacitance is related to displacement current I through the equation
where C is capacitance, Q is charge, and V is voltage. In the simplest case, the capacitance is time-independent and the first term is zero.
Maxwell never used the term electric charge, but he did refer to the "distribution of electricity in a body" and to the "quantity of electricity". Capacity C was defined in his equation (138) for two surfaces bearing equal and opposite quantities of electricity e and electric tensions or potentials ψ1 and ψ2 as the ratio C = e / (ψ1 - ψ2).[1] The modern equation, Q = CV, has an emphasis on the capacity for storage of charge, whereas Maxwell had an emphasis more on elasticity.[clarify]
The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition C=Q/V still holds if only one plate is given a charge, provided that we recognize that the field lines produced by that charge terminate as if the plate were at the center of an oppositely charged sphere at infinity.
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, Maxwell introduced his "coefficients of potential". If three plates are given charges Q1,Q2,Q3, then the voltage of plate 1 is given by
and similarly for the other voltages. Maxwell showed that the coefficients of potential are symmetric, so that p12 = p21, etc.
In mathematical terms, the ideal capacitance can be considered as an inverse of the ideal inductance, because the voltage-current equations of the two phenomena can be transformed into one another by exchanging the voltage and current terms.
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. There also exists a property called self-capacitance, which is the amount of electrical charge that must be added to an isolated conductor to raise its electrical potential by one volt. The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, centred on the conductor. Using this method, the self-capacitance of a conducting sphere of radius R is given by:
Typical values of self-capacitance are:
The inverse of capacitance is called elastance, and its unit is the reciprocal farad.
Any two adjacent conductors can be considered as a capacitor, although the capacitance will be small unless the conductors are close together or long. This (unwanted) effect is termed "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 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). 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. Miller's theorem 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, 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.) 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.
The capacitance of the majority of capacitors used in electronic circuits is several orders of magnitude smaller than the farad. The most common subunits of capacitance in use today are the millifarad (mF), microfarad (µF), the nanofarad (nF) and the picofarad (pF) (also known as a "puff")
The capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. For example, 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:
(in SI units)where
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, and the so-called fringing field region provides a small contribution. In CGS units the equation has the form:
where C in this case has the units of length.
Combining the SI equation for capacitance with the above equation for the energy stored in a capacitance, for a flat-plate capacitor the energy stored is:
.where
The dielectric constant for a number of very useful dielectrics changes as a function of the applied electrical field, for example ferroelectric materials, so the capacitance for these devices is more complex. For example, in charging such a capacitor the differential increase in voltage with charge is governed by:
where the voltage dependence of capacitance stems from the field, which in a large area parallel plate device is given by ε = V/d. Thus, to charge the capacitor to voltage V an integral relation is found:
which agrees with Q = CV only when C is voltage independent.
By the same token, the energy stored in the capacitor now is given by
![dW =Q dV =\left[ \int_0^V\ dV' \ C(V') \right] \ dV \ .](http://upload.wikimedia.org/math/2/2/c/22c8e7f9c2fada07ec3e7b5db9595335.png)
Integrating:
where interchange of the order of integration is used.
Another example of voltage dependent capacitance occurs in semiconductor devices such as semiconductor diodes, 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.
If a capacitor is driven with a sinusoidal voltage, the dielectric constant, more accurately referred to as the relative static permittivity, is a function of frequency. As an example of the origin of this mechanism, the internal microscopic dipoles contributing to the dielectric constant cannot move instantly, and so as frequency increases, the dipole response is limited and the dielectric constant diminishes. A changing dielectric constant with frequency is referred to as dielectric dispersion, and is governed by dielectric relaxation processes, such as Debye relaxation.
As an example using this phenomena, the dielectric constant may exhibit a resonance at certain frequencies corresponding to characteristic response frequencies of defects that contribute to the dielectric constant. These resonances are the basis for a number of experimental techniques for detecting defects, for example, deep-level transient spectroscopy.
