Capacitance ($C$)

The capacitance of a linear two-terminal capacitor is the constant relating its separated electric charge to its terminal electric potential difference:Here $Q$ is the electric charge on one electrical conductor, the other carries $-Q$, and $V$ is their electric potential difference with the same ordering. Reversing the terminal ordering reverses both $Q$ and $V$ and leaves capacitance unchanged. An ideal passive capacitor has $C\ge 0$; an ordinary pair of separated electrical conductors with positive permittivity has $C>0$. At $V=0$, use $Q=CV$ rather than the undefined ratio $0/0$.

For example, electric charge magnitude $Q=0.8854\mathrm{nC}$ at terminal electric potential difference $V=10\mathrm{V}$ gives capacitance $C=88.54\mathrm{pF}$. At $20\mathrm{V}$, the same linear capacitor has $Q=1.7708\mathrm{nC}$ and the same capacitance. The plots in capacitor energy show that capacitance is the slope of the electric charge versus terminal electric potential difference graph.

For fixed geometry and linear material response, capacitance depends on dimensions and permittivity, rather than the source's terminal electric potential difference. This is why capacitance is a useful property of a component: once it is known, $Q=CV$ predicts the separated electric charge for each operating terminal electric potential difference. The SI unit is the farad, $1\mathrm{F}=1C/V$. ^[ref].

For combinations of capacitors, the same terminal definition gives an equivalent capacitance. Two identical capacitors of capacitance $C$ connected in parallel share the same terminal electric potential difference, and their electric charges add, giving $C_{\mathrm{eq}}=2C$. Connected in series with an initially neutral isolated junction, they carry equal plate electric charge magnitudes and their terminal electric potential differences add, giving $C_{\mathrm{eq}}=C/2$. ^[ref].

For an ideal two-plate capacitor with opposite charges, a stated inside-minus-outside charge difference is twice the inner plate charge. The voltage is inner plate charge divided by capacitance. Confusing a plate charge with the difference of plate charges creates a factor-two error in inferred membrane voltage.

Two isolated conducting drops of radius $a$ and equal charge merge at fixed total charge and volume. The new radius is $2^{1/3}a$, so its capacitance grows by only $2^{1/3}$ while total charge doubles. Relative to the sum of the initially separated self energies, the final electric field energy grows by $2^{2/3}$. Decreased surface energy and/or applied mechanical work can supply that increase. Sufficiently large like charges need not coalesce spontaneously; energy conservation is not a guarantee of merging. A maintained external voltage would define a different problem.

The capacitive coupling between a control gate and an electronic channel or island. In a single-island model the gate lever arm is ${\alpha }_g=C_g/C_{\Sigma }$. A charging-only gate period is $e/C_g$, whereas an orbital gap enlarges that period.

Capacitance normalized by a stated mass or volume. Mass-specific differential capacitance is $C_{\mathrm{sp}}=m^{-1}dQ/dV$; electrode and full-device mass conventions must be distinguished.

For a parallel-plate capacitor of area $A$ and separation $d$, filled by a spatially varying positive scalar permittivity, neglect edge fields. At fixed terminal voltage, the physical electric potential minimizes $\int \epsilon |\nabla V{|}^{2}d\mathcal{V}/2$. A linear trial potential gives the arithmetic upper bound. At fixed plate charge, minimizing $\int |\mathbf{D}{|}^2/(2\epsilon )d\mathcal{V}$ over divergence-free electric displacement fields gives the harmonic lower bound by using a uniform trial displacement. The averages are volume averages. Layers parallel to the field attain the arithmetic bound; layers perpendicular to the field attain the harmonic bound. Equal volumes of vacuum and relative permittivity $ϵ$ therefore give $2ϵC_0/(1+ϵ)\le C\le (1+ϵ)C_0/2$, where $C_0={\epsilon }_{0}A/d$.

For a collection of fixed electrical conductors in linear electrostatics, the capacitance matrix relates conductor charges to potentials relative to a specified reference: $Q_i={\sum }_j{C}_{ij}{V}_j$. It includes mutual electrostatic effects omitted when each conductor is treated as an isolated capacitor. Its electrostatic energy is $U=\frac{1}{2}{\mathbf{V}}^{\mathsf{T}}\mathbf{CV}=\frac{1}{2}{\sum }_i{Q}_i{V}_i$. Geometry and separation are needed to determine its entries.

The electronic contribution to differential capacitance arising from a change of chemical potential as particles are added. For noninteracting electrons at zero temperature, it is $e^2$ times the density of states of the whole sample at the Fermi energy. The quantum capacitance of a graphene cone gives a worked example.

The farad, symbol $\mathrm{F}$, is the SI unit of capacitance:A capacitor of capacitance $1\mathrm{F}$ carries separated electric charge magnitude $1\mathrm{C}$ at terminal electric potential difference $1\mathrm{V}$. A picofarad is $10^{-12}\mathrm{F}$, so the worked parallel-plate capacitor's capacitance $88.54\mathrm{pF}$ is $8.854\times 10^{-11}\mathrm{F}$. ^[ref].

An isolated conducting sphere of radius $R$ in a medium of permittivity $ϵ$ has surface electrostatic potential $V=Q/(4\pi ϵR)$ relative to infinity. Thus its capacitance is $C=Q/V=4\pi ϵR$.

For an isolated smooth conducting sphere of radius $a$, Gauss's law gives surface electric field $E_s=V/a$. Combining its capacitance $4\pi {ϵ}_{0}a$ with capacitor energy $C{V}^2/2$ gives the displayed bound when the surrounding medium has dielectric strength $E_{\mathrm{bd}}$ and approximately vacuum permittivity. Nearby conductors, surface roughness and discharge geometry can lower the practical limit.

For a conducting sphere of radius $R$ at center height $h>R$ above a grounded infinite plane,The distance $h$ is measured to the center, so the surface gap is $h-R$. Relative to the capacitance of an isolated conducting sphere, the enhancement is $c(R/h)=C(R,h)/(4\pi ϵR)$.

The method of images gives a constructive proof. At fixed sphere potential $V_s$, start with $q_0=4\pi ϵR{V}_s$ at height $z_0=h$ and place its opposite image below the plane. Cancel that image's nonconstant sphere potential by the successive chargeswith opposite mirror charges below the plane at every step. The resulting sphere potential is $V_s$ and the plane potential is zero. Since $q_n=4\pi ϵR{V}_s\sinh \alpha /\sinh ((n+1)\alpha )$, summing the sphere charges gives the displayed capacitance. In particular $c(y)=1+y/2+O(y^2)$ for small $y$.

The exact sphere-plane boundary-value problem is also treated in Behunin and colleagues' analysis of sphere-plane electrostatics.

Capacitance per unit length is the charge per unit length divided by the potential difference. A vacuum coaxial cable with radii $a<b$ has

Replace two well-separated cylindrical conductors by opposite axial line charges. Their surface-to-surface potential difference is $\lambda [\ln (d/a_1)+\ln (d/a_2)]/(2\pi ϵ)$, giving the displayed capacitance per unit length. The approximation needs $d\gg a_1,a_2$ and a homogeneous surrounding permittivity.