Formulas/physics/Electrostatic Potential/Capacitance (Definition)

Capacitance (Definition)

Capacitance is the charge stored per unit potential difference. SI unit: farad (F = C V⁻¹). A property of the conductor geometry, not of Q or V individually.

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Derivation

Observation

For a conductor, experiment shows that the potential $V$ is always proportional to the charge $Q$ placed on it:

Q \propto V \implies Q = CV

The constant of proportionality $C$ is the capacitance:

\boxed{C = \frac{Q}{V}}

Why C is a geometric property

$V$ due to charge $Q$ on a conductor depends on the geometry (shape, size, nearby conductors) but is always linearly proportional to $Q$ . Doubling $Q$ doubles the field everywhere, which doubles $V$ . The ratio $Q/V$ cancels the charge dependence — it depends only on geometry.

Units

[C] = \frac{\text{C}}{\text{V}} = \text{F (farad)}

$1\,\text{F}$ is a very large capacitance. Practical values: $\mu\text{F}$ ( $10^{-6}$ F), $\text{nF}$ ( $10^{-9}$ F), $\text{pF}$ ( $10^{-12}$ F).

Isolated sphere

For a sphere of radius $R$ : $V = Q/4\pi\varepsilon_0 R$ , so:

C = \frac{Q}{V} = 4\pi\varepsilon_0 R

For Earth ( $R \approx 6400$ km): $C_{Earth} \approx 711\,\mu\text{F}$ .

Note

Capacitance increases with size (larger conductors can hold more charge at the same potential) and decreases with isolation from other conductors (nearby grounded conductors increase capacitance by induction).

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