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Formulas/physics/Electrostatic Potential/Energy Density of Electric Field

Energy Density of Electric Field

Energy stored per unit volume in an electric field. Derived from the parallel plate capacitor energy. Applies to any electric field in vacuum — not just capacitors.
Class 12
Derivation

Derivation from parallel plate capacitor

For a parallel plate capacitor: C=ε0A/dC = \varepsilon_0 A/d, field E=V/dE = V/d, so V=EdV = Ed.

Energy stored:

U=12CV2=12⋅ε0Ad⋅E2d2=12ε0E2⋅AdU = \frac{1}{2}CV^2 = \frac{1}{2}\cdot\frac{\varepsilon_0 A}{d}\cdot E^2 d^2 = \frac{1}{2}\varepsilon_0 E^2 \cdot Ad

Volume between the plates: Vol=Ad\text{Vol} = Ad.

Energy per unit volume:

u=UVol=12ε0E2⋅AdAdu = \frac{U}{\text{Vol}} = \frac{\frac{1}{2}\varepsilon_0 E^2 \cdot Ad}{Ad} u=12ε0E2\boxed{u = \frac{1}{2}\varepsilon_0 E^2}

Universality

Although derived for a uniform field in a capacitor, this result holds for any electric field configuration in vacuum. The total energy stored in any electric field distribution is:

U=∫all space12ε0E2 dVU = \int_{\text{all space}} \frac{1}{2}\varepsilon_0 E^2\, dV

With dielectric

In a dielectric medium with permittivity ε=Kε0\varepsilon = K\varepsilon_0:

u=12εE2=12Kε0E2u = \frac{1}{2}\varepsilon E^2 = \frac{1}{2}K\varepsilon_0 E^2
Note
The energy is distributed throughout the field — not localised on the charges. This field-theoretic view of energy is essential in electromagnetism: EM waves carry energy in their fields even in the absence of charges.