= Parallel-plate capacitor
{title2=$C=\epsilon A/d$}
A <parallel-plate capacitor> consists of two parallel <electrical conductors> with overlapping <area> $A$ and separation $d$. If the gap is small compared with both plate widths and is filled by a uniform material of positive <permittivity> $\epsilon$, its <capacitance> is approximately
$$
C\simeq\frac{\epsilon A}{d}.
$$
This neglects the <fringing field of a capacitor> near the plate edges. The interior <electric field> points from the plate of positive <electric charge> to the plate of negative <electric charge>, and is approximately uniform. https://openstax.org/books/university-physics-volume-2/pages/8-1-capacitors-and-capacitance[OpenStax University Physics 2, §8.1: the parallel-plate approximation]{ref}.
For a worked vacuum example, stipulate $A=0.01\,\mathrm{m^2}$, $d=0.001\,\mathrm m$, and <vacuum permittivity> $\epsilon_0\approx8.854\times10^{-12}\,\mathrm{F/m}$. This rounds the value in <Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others>. Within the approximation,
$$
C\simeq\frac{(8.854\times10^{-12})(0.01)}{0.001}\,\mathrm F
=8.854\times10^{-11}\,\mathrm F=88.54\,\mathrm{pF}.
$$
At terminal <electric potential difference> $V=10\,\mathrm V$, the <electric field> magnitude is $E\simeq V/d=10^4\,\mathrm{V/m}$, the plate <electric charge> magnitude is $Q=CV=0.8854\,\mathrm{nC}$, and the <capacitor energy> is $U=CV^2/2=4.427\,\mathrm{nJ}$. These numbers use the stipulated rounded <vacuum permittivity>; finite plates also have corrections from the <fringing field of a capacitor>.
\Image[/parallel-plate-capacitor-field.png]
{title=Uniform <electric field> in a <parallel-plate capacitor>}
The formula makes design changes easy to predict: doubling the overlapping <area> doubles the <capacitance>, and halving the gap doubles the <capacitance>, with <permittivity> fixed and the <fringing field of a capacitor> neglected. Doubling <permittivity> also doubles <capacitance> if the material uniformly fills the gap and remains linear. This lets a compact device store more separated <electric charge> at the same terminal <electric potential difference>.
The short derivation uses <surface charge density> $\sigma=Q/A$. In the interior, <Gauss's law> gives <electric field> magnitude $E\simeq\sigma/\epsilon=Q/(\epsilon A)$. The <electric potential difference> is $V\simeq Ed=Qd/(\epsilon A)$, so dividing $Q$ by $V$ gives $C\simeq\epsilon A/d$. A narrow curved <capacitor> gap has the same leading local approximation when its width is much less than the curvature radius.
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