Result

Energy density

(E4.4.2) E4.4 · Energy Stored in a Charged Capacitor →

u_{E} = \frac{1}{2} ε_{0} E^{2}

Statement

Energy per cubic metre stored in the field itself.

Symbols and units

$u_{E}$	energy per unit volume of field	J/m³
$E$	field strength	V/m
$ε_{0}$	permittivity of free space	8.85×10⁻¹² C²/(N·m²)

When it applies

Any electric field. With a dielectric present, multiply by κ.

Don't use when

The bare ½ε₀E² form is for vacuum or air — with a dielectric present, the density is ½κε₀E² instead, using the material's dielectric constant.

Derivation

Take a parallel-plate capacitor and write its energy using the field:

U = \frac{1}{2} C (Δ V)^{2} = \frac{1}{2} \frac{ε _{0} A}{d} (E d)^{2}

Simplify:

U = \frac{1}{2} ε_{0} E^{2} (A d)

Ad is exactly the volume between the plates, so divide it out:

u_{E} = \frac{U}{A d} = \frac{1}{2} ε_{0} E^{2}

Worked examples

Energy density at breakdown in air

u_{E} = \frac{1}{2} (8.85 \times 1 0^{- 12}) (3 \times 1 0^{6})^{2} \approx 40 J/m^{3}

Petrol stores about 3 × 10¹⁰ J/m³. Capacitors deliver power, not capacity.

Watch out for

Derived from a capacitor, but it turns out to hold for any electric field anywhere — including the field of a light wave. With a dielectric present, use ½κε₀E².

Connects to

Stored energy Infinite sheet of charge