Energy Density Formula: u = ½ε₀E², Unit J/m³, Examples

The energy density formula for an electric field is $u = \frac{1}{2}\varepsilon_0E^2$ : the energy stored per unit volume equals half the permittivity of free space times the field squared. It answers a simple question, how many joules are packed into each cubic metre of space where a field exists.

The magnetic partner is $u_B = \frac{B^2}{2\mu_0}$ . Together they describe where a charged capacitor or a current carrying coil keeps its energy: not on the plates or wires, but in the field itself. Class 12 students use the electric form in the capacitors chapter, and JEE problems often mix the two.

Quick facts

Electric form
$u_E = \frac{1}{2}\varepsilon_0E^2$
Magnetic form
$u_B = \frac{B^2}{2\mu_0}$
In a dielectric
$u = \frac{1}{2}K\varepsilon_0E^2$
SI unit
joule per cubic metre (J/m³), same as the pascal
Dimensional formula
[M¹ L⁻¹ T⁻²]
Type
Scalar

What the symbols in the energy density formula mean

Terms in u = ½ε₀E²
Symbol	Quantity	Value or unit
$u$	Energy per unit volume	J/m³
$\varepsilon_0$	Permittivity of free space	8.854 × 10⁻¹² C²/(N m²)
$E$	Electric field magnitude	V/m or N/C
$K$	Dielectric constant of a medium, if present	no unit

The field appears squared, so doubling E stores four times as much energy in the same volume. A negative field direction makes no difference: energy density is always positive.

Derivation from a parallel plate capacitor

A parallel plate capacitor of plate area A and gap d has $C = \frac{\varepsilon_0A}{d}$ . Its stored energy is $U = \frac{1}{2}CV^2$ , and the field between the plates is $E = \frac{V}{d}$ , so $V = Ed$ . Substitute:

$U = \frac{1}{2}\cdot\frac{\varepsilon_0A}{d}\cdot E^2d^2 = \frac{1}{2}\varepsilon_0E^2(Ad)$

The product $Ad$ is the volume between the plates. Dividing gives $u = \frac{U}{Ad} = \frac{1}{2}\varepsilon_0E^2$ . The result holds for any electric field, not only between plates. See the capacitance formula page for C itself.

Worked examples

Example 1: field near air breakdown

Air starts to conduct at about 3.0 × 10⁶ V/m. Find the energy density at that field.

$u = \frac{1}{2}(8.854\times10^{-12})(3.0\times10^{6})^2 = \frac{1}{2}(8.854\times10^{-12})(9.0\times10^{12}) \approx 40\ \text{J/m}^3$

Example 2: total energy in a capacitor gap

A capacitor has plates of area 0.010 m², 1.0 mm apart, and the field between them is 2.0 × 10⁵ V/m. Find the energy stored.

$u = \frac{1}{2}(8.854\times10^{-12})(2.0\times10^{5})^2 = 0.177\ \text{J/m}^3$

Volume = 0.010 × 1.0 × 10⁻³ = 1.0 × 10⁻⁵ m³, so U = 0.177 × 1.0 × 10⁻⁵ ≈ 1.8 × 10⁻⁶ J.

Check: C = 8.85 × 10⁻¹¹ F and V = Ed = 200 V, so ½CV² ≈ 1.8 × 10⁻⁶ J as well.

Example 3: magnetic energy density

Find the energy density in a magnetic field of 0.50 T.

$u_B = \frac{(0.50)^2}{2(4\pi\times10^{-7})} = \frac{0.25}{2.513\times10^{-6}} \approx 9.9\times10^{4}\ \text{J/m}^3$

A modest lab magnet stores far more energy per cubic metre than the strongest electric field air can hold.

Units and dimensions of energy density

Joules per cubic metre reduce to $\frac{\text{N m}}{\text{m}^3} = \frac{\text{N}}{\text{m}^2}$ , which is the pascal. So energy density has the same dimensions as pressure and stress:

$[u] = \frac{[M L^2 T^{-2}]}{[L^3]} = [M L^{-1} T^{-2}]$

This is no coincidence. Field energy density equals the pressure the field exerts on a surface, which is why radiation pressure has the same form. The dimensional formula of energy density page compares it with pressure in more detail.

Other meanings of energy density

In chemistry and engineering, energy density can also mean energy per unit volume of a fuel or a battery, still in J/m³, while energy per kilogram is called specific energy, in J/kg. In an electromagnetic wave, the electric and magnetic parts carry equal average energy densities, and the total is $\varepsilon_0E_{\text{rms}}^2$ .

If a question gives a dielectric, multiply by K: u = ½Kε₀E². For the same charge on the plates, E drops by a factor of K.

Mistakes to watch for

Dropping the factor ½.
Using μ₀ on top in the magnetic form. It is B² divided by 2μ₀.
Forgetting to square E after converting kV/m to V/m.
Giving total energy when the question asks for energy per unit volume, or the reverse.

Related field formulas sit on the electricity formulas page, including the electric field and magnetic field pages.

Frequently asked questions

What is the formula of energy density in an electric field?

u = ½ε₀E². In a dielectric of constant K, use u = ½Kε₀E².

What is the formula of energy density in a magnetic field?

u = B² / (2μ₀), with μ₀ = 4π × 10⁻⁷ T m/A.

Which SI unit does energy density use?

Joule per cubic metre (J/m³). It is dimensionally the same as the pascal, [M¹ L⁻¹ T⁻²].

Does energy density have a direction?

No. Energy density is a single number at each point, with no direction attached, so it is a scalar.

Why does energy density depend on E squared?

Both the charge on a capacitor and the voltage across it grow with the field, and stored energy depends on their product, so the energy scales as E².