What is Capacitance? — Physics

Definition 27.12 University Physics — Year 1 · Chapter 27 — Electric Potential; Conductors and Capacitors

An isolated conductor at potential $V$ carries a charge $Q$ proportional to $V$ : $Q = CV$ , with $C$ its capacitance (farad, $\mathrm{F}$ ); for a sphere $C = 4\pi\varepsilon_0R$ . A capacitor is a pair of conductors (plates) carrying opposite charges $\pm Q$ and facing each other so that the field is confined between them; its capacitance is $C = Q/U$ with $U = V_1 - V_2$ the voltage between the plates. The symbol and the circuit law $i = C\,\dd u/\dd t$ were used in Chapter 7.

Plane, spherical and dielectric-filled capacitors. The field (orange) is confined between the plates except for the fringing at the edges, neglected when d is small; the energy 1/2 _0E2 per unit volume lives in that field. — Plane, spherical and dielectric-filled capacitors. The field (orange) is confined between the plates except for the fringing at the edges, neglected when $d$ is small; the energy $\tfrac12\varepsilon_0E^2$ per unit volume lives in that field.

Examples

Example 27.15 (Three capacitors and their energies)

A square metre of foil pair $1\,\mathrm{mm}$ apart: $C = 8.85 \times 10^{-12}/ 10^{-3} = 8.9\,\mathrm{nF}$ — the farad is a huge unit. A defibrillator’s $150\,\text{µ}\mathrm{F}$ at $2\,\mathrm{kV}$ : $300\,\mathrm{J}$ , delivered in milliseconds. A supercapacitor of $3000\,\mathrm{F}$ at $2.7\,\mathrm{V}$ : $11\,\mathrm{kJ}$ in a can the size of a soda bottle — its capacitance comes from a molecular gap $d$ and an enormous porous $S$ . The energy density of a field at the breakdown of air, $\tfrac12\varepsilon_0(3 \times 10^6)^2 = 40\,\mathrm{J}/\mathrm{m}^{3}$ , is why capacitors store so little compared with batteries.

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Examples

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