Capacitance · aiwiki.page

Capacitance is a physical quantity describing the relationship between stored electric charge and electric potential. For an ideal linear, two-conductor system carrying equal and opposite charges, it is the magnitude of the charge on either conductor divided by the potential difference between them. Capacitance depends on the conductors’ geometry and the electrical properties of their surroundings. A capacitor is a component designed to provide capacitance, but capacitance also exists between conductors that are not deliberately assembled into a capacitor. (openstax.org)

Definition and units

For a linear capacitor,

C=\frac{Q}{V},\qquad Q=CV,

where $C$ is capacitance, $Q$ is the magnitude of the charge on either conductor, and $V$ is the potential difference between them. The two conductors carry $+Q$ and $-Q$ : “stored charge” in this context describes charge separation, not a net charge of $Q$ on the complete capacitor. Within the ideal linear model, increasing the voltage increases the charge proportionally without changing the capacitance. (openstax.org)

The unit of capacitance in the International System of Units is the farad, symbol F:

1\ \mathrm{F}=1\ \mathrm{C}/\mathrm{V}.

Thus, a capacitance of one farad corresponds to one coulomb of charge per volt of potential difference. Its expression in SI base units is

\mathrm{F}=\mathrm{kg}^{-1}\mathrm{m}^{-2}\mathrm{s}^{4}\mathrm{A}^{2}.

Common subdivisions include the microfarad ( $\mu\mathrm{F}=10^{-6}\mathrm{F}$ ), nanofarad ( $\mathrm{nF}=10^{-9}\mathrm{F}$ ), and picofarad ( $\mathrm{pF}=10^{-12}\mathrm{F}$ ). (nist.gov)

Capacitance is not, by itself, a maximum permissible charge or a voltage rating. Those limits also depend on insulation strength and other physical constraints. The defining ratio $Q/V$ applies within the operating conditions for which the linear model is valid. (openstax.org)

Physical basis and geometrical dependence

In electrostatics, charge on an electrical conductor redistributes until the conductor is at a uniform potential. Separated charges establish an electric field, whose distribution determines the potential difference. Calculating capacitance therefore involves finding the field and potential for a specified conductor geometry and then evaluating $Q/V$ . (openstax.org)

Several geometries have particularly useful analytical expressions:

Configuration	Capacitance	Conditions
Parallel plates	$C\approx \varepsilon A/d$	Plate area $A$ , separation $d$ ; edge effects neglected
Concentric spherical conductors	$C=4\pi\varepsilon ab/(b-a)$	Inner radius $a$ , outer radius $b$
Long coaxial cylinders	$C\approx 2\pi\varepsilon L/\ln(b/a)$	Length $L$ , inner radius $a$ , outer radius $b$ ; end effects neglected

Here $\varepsilon$ is the permittivity of a homogeneous, linear insulating medium. These expressions show that capacitance can be increased by enlarging the effective conducting surfaces, reducing their separation, or increasing the medium’s permittivity. The parallel-plate approximation is most accurate when the separation is small compared with the plates’ lateral dimensions. (openstax.org)

Dielectric effects

A dielectric placed between conductors undergoes electric polarization: its bound charges respond to the applied field. In the usual static, linear case, this reduces the potential difference for a given free charge, thereby increasing capacitance. For a capacitor completely filled by a uniform dielectric,

C=\varepsilon_r C_0,

where $\varepsilon_r$ is relative permittivity and $C_0$ is the capacitance of the same geometry in vacuum. (openstax.org)

The consequences depend on what is held constant. For an isolated capacitor, inserting a dielectric leaves the free charge unchanged while reducing voltage. If a voltage source remains connected, the voltage stays fixed and additional charge enters the capacitor. At sufficiently high fields, an insulating material can undergo electrical breakdown, limiting usable voltage even when its capacitance is large. (openstax.org)

Self-capacitance and systems of conductors

Self-capacitance describes an individual conductor’s charge relative to its potential, conventionally taking the potential at infinity as zero. For an isolated conducting sphere of radius $R$ in a uniform medium,

C_{\mathrm{self}}=4\pi\varepsilon R.

This is the limiting form of the spherical-capacitor expression as the outer conductor recedes to infinity. Nearby conductors alter the field and therefore alter the charge–potential relationship; self-capacitance is not independent of the environment. (openstax.org)

For several conductors, a single number generally cannot describe all their electrical interactions. In a linear medium, their charges and potentials obey

Q_i=\sum_j C_{ij}V_j.

The coefficients form a capacitance matrix. They describe both each conductor’s response to its own potential and its coupling to the other conductors. With a specified reference potential, this formulation generalizes the familiar two-terminal capacitor relation. Terminology requires care: an off-diagonal matrix coefficient is not necessarily the same quantity as the positive capacitance assigned to a capacitor connected between two circuit nodes. (ocw.mit.edu)

Energy storage

Charging a capacitor requires work to separate charges against the developing electric field. For a capacitor charged reversibly from an uncharged state, the stored energy is

U=\int_0^Q V(q)\,dq.

For constant capacitance, this becomes

U=\frac{Q^2}{2C} =\frac12 QV =\frac12 CV^2.

The energy is associated with the electric field rather than with the mere presence of charge on a conductor. (openstax.org)

In vacuum, the field energy per unit volume is

u=\frac12\varepsilon_0 E^2.

For a linear, nondispersive dielectric, the corresponding expression is $\tfrac12\mathbf E\cdot\mathbf D$ , where $\mathbf D$ is electric displacement. These field descriptions connect capacitance to energy distributed throughout the space around the conductors, including regions outside the nominal gap when fringing fields are significant. (openstax.org)

The distinction between fixed charge and fixed voltage is important: increasing capacitance decreases $Q^2/(2C)$ at fixed charge but increases $CV^2/2$ at fixed voltage. In the latter case, an attached source exchanges energy with the capacitor. (openstax.org)

Capacitance in electric circuits

Series and parallel combinations

Ideal capacitors connected in parallel share the same voltage, and their charges add:

C_{\mathrm{eq}}=\sum_i C_i.

For a series chain with initially uncharged, isolated intermediate nodes, the capacitors carry equal charge magnitudes and their voltages add:

\frac{1}{C_{\mathrm{eq}}}=\sum_i\frac{1}{C_i}.

Consequently, a parallel combination has greater capacitance than any individual member, whereas a series combination has less capacitance than the smallest member. These rules assume that unintended coupling between the components can be neglected. (openstax.org)

Current and transient response

Using the passive sign convention, electric current entering a capacitor terminal is the rate at which charge accumulates on that terminal’s conductor. For constant capacitance,

i(t)=C\frac{dv(t)}{dt}.

A capacitor therefore draws current while its voltage changes, but an ideal capacitor carries no steady conduction current under a constant voltage. (openstax.org)

When a capacitor charges through an electrical resistance $R$ , the characteristic time is

\tau=RC.

For an initially uncharged capacitor connected through $R$ to a constant source $V_s$ ,

v(t)=V_s\left(1-e^{-t/(RC)}\right).

During discharge through the resistance,

v(t)=V_0e^{-t/(RC)}.

These exponential responses underlie timing circuits and many filtering functions. (openstax.org)

Alternating-current response

For sinusoidal alternating current, an ideal capacitor has electrical impedance

Z_C=\frac{1}{j\omega C},

where $j^2=-1$ and $\omega=2\pi f$ . Its impedance magnitude, conventionally called capacitive reactance, is

X_C=\frac{1}{\omega C}.

Current leads voltage by $90^\circ$ . The decreasing impedance with increasing frequency enables capacitors to separate steady and varying signal components and to participate in frequency-selective circuits. (openstax.org)

Nonlinear capacitance and practical limitations

For a nonlinear charge–voltage relation, the ratio $Q/V$ and the local slope are different quantities. Differential capacitance is

C_{\mathrm{diff}}=\frac{dQ}{dV}.

It describes the response to a sufficiently small voltage change around a particular operating point. Such charge derivatives are important in models of semiconductor devices, including depletion capacitances in transistors. The constant-capacitance energy formula cannot simply be applied to every nonlinear system. (edadownload.software.keysight.com)

Real capacitance can depend on temperature, frequency, applied direct-current bias, and measurement amplitude. High-permittivity ceramic capacitors, for example, can lose a substantial fraction of their nominal capacitance under direct-current bias, whereas temperature-compensating ceramic types have much smaller bias dependence. A marked capacitance value therefore describes specified test conditions rather than an invariant property under all circumstances. (murata.com)

Practical capacitors also exhibit leakage, dielectric losses, equivalent series resistance, and parasitic inductance. Near their self-resonant frequency, capacitive and inductive effects balance; above that frequency, the component may behave predominantly inductively. Thus, capacitance alone does not fully describe a component’s high-frequency behavior. (murata.com)

Measurement and applications

Capacitance is measured by observing a charge–voltage relationship, a charging transient, or an alternating-current impedance. Precision capacitance meters commonly apply a known alternating signal and measure the resulting current and phase. Capacitance–voltage measurements vary the bias to investigate material properties and device structure, making them important in semiconductor characterization. Cable effects, fixture capacitance, and the chosen equivalent-circuit model can affect the result. (openstax.org)

In metrology, calculable capacitors provide a connection between precisely known dimensions and capacitance. The US National Institute of Standards and Technology uses a calculable-capacitor facility as part of its realization of the SI farad and its system for accurate impedance measurements. (nist.gov)

Applications include electrical energy storage, pulse delivery, timing, signal coupling, and filtering. Unintended or parasitic capacitance also influences circuits: conductors separated by insulation retain capacitive coupling even when no discrete capacitor is installed. At high frequencies, both these unintended capacitances and the nonideal properties of deliberately installed capacitors become important to circuit behavior. (openstax.org)

Historical development

An early practical demonstration of electrical charge storage was the Leyden jar, independently developed by Ewald Georg von Kleist in 1745 and Pieter van Musschenbroek in Leiden in 1746. The device used a glass vessel to separate conducting regions and became an important instrument for experiments with static electricity. (collection.sciencemuseumgroup.org.uk)

Modern implementations include ceramic, film, aluminum-electrolytic, tantalum, and silicon capacitors. Their different materials and structures provide different combinations of capacitance, size, losses, voltage capability, and stability; the underlying charge–potential relationship remains the organizing concept. (murata.com)

References

1 Capacitors and Capacitance — University Physics Volume 2openstax.org
5 Capacitors and Dielectrics — Physicsopenstax.org
NIST Guide to the SI, Chapter 4: The Two Classes of SI Units and the SI Prefixesnist.gov
4 Capacitor with a Dielectric — University Physics Volume 2openstax.org
Electromagnetism II, Lecture Notes 4ocw.mit.edu
3 Energy Stored in a Capacitor — University Physics Volume 2openstax.org
Ch. 8 Summary — University Physics Volume 2openstax.org
5 RC Circuits — University Physics Volume 2openstax.org
11 Reactance, Inductive and Capacitive — College Physics 2eopenstax.org
Nonlinear Devices — Keysight Product Documentationedadownload.software.keysight.com
Exact Calculation of the Capacitance and the Electrostatic Potential Energy for a Nonlinear Parallel-Plate Capacitor in a Two-Parameter Modification of Born-Infeld Electrodynamicsarxiv.org
Does the capacitance change when a DC voltage is applied to ceramic capacitors?murata.com