Parallel-plate capacitor

Two parallel metal plates of area $A$ separated by a distance $D$ form a parallel-plate capacitor, the elementary device for storing charge and, with it, electrostatic energy. Each plate carries charge $\pm Q$, distributed with surface density $\sigma = Q/A$.

The geometry is chosen cleverly: the fields of the two plates add up in the interior region and cancel outside. All the field is thus confined between the plates, uniform and predictable:

$E_{\text{interno}} = \frac{\sigma}{\varepsilon_0} \qquad E_{\text{esterno}} = 0$

This is the key to the capacitor’s efficiency: no field dispersion towards the outside, all the energy collected in a well-defined volume $A\,D$.

Principle — Capacitance

Capacitance measures how much charge the capacitor accumulates for each volt of applied potential difference: $\ev{C = \frac{Q}{V} = \frac{\varepsilon_0\,A}{D}}$ It is measured in farads ($\mathrm{F} = \mathrm{C/V}$). It depends only on geometry: large, close plates give large capacitance.

Key formula

For the parallel-plate capacitor the following chain of relations holds: $\begin{aligned} E &= \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A} \\ \Delta V &= E\,D = \frac{Q\,D}{\varepsilon_0 A} \\ C &= \frac{\varepsilon_0 A}{D} \\ U &= \tfrac{1}{2}C(\Delta V)^2 = \frac{Q^2}{2C} \end{aligned}$

The capacitor stores energy in the electric field, not on the charges. Rewriting in terms of the known quantities gives both the total energy and its density per unit volume:

$U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} \qquad u = \frac{U}{A D} = \ev{\tfrac{1}{2}\varepsilon_0 E^2} \quad [\text{J/m}^3]$

Note

The energy density $u = \tfrac{1}{2}\varepsilon_0 E^2$ holds for any electric field, not just the parallel-plate capacitor. It is a property of the field itself: wherever there is a field, energy resides there. This is the idea that opens the way to the energy density of the electric field in general.

Links

Topics: Electric field and potential Concepts: Capacitance and capacitor Objects: Parallel-plate capacitor

Related exercises: Energy and power of a defibrillator · Field energy in the capacitor · Capacitance and charge of a capacitor