Energy stored per cubic metre has the energy density dimensional formula [M¹ L⁻¹ T⁻²]; shortened, [ML⁻¹T⁻²]. That is exactly the bracket of pressure, and the match is no coincidence.
Energy density u is energy divided by the volume it occupies, u = E/V. It describes the energy packed into an electric field between capacitor plates, a magnetic field inside a coil, a compressed gas or a beam of light. This page derives the bracket, shows why J/m³ equals the pascal, checks the electric and magnetic field forms and flags the per kilogram trap. For the full set of energy density equations, go to the energy density formula.
Quick facts
Dimensional formula
[M¹ L⁻¹ T⁻²], short form [ML⁻¹T⁻²]Defining formula
Field forms
,SI unit
J/m³, equal to PaScalar or vector
ScalarSame dimensions as
Pressure, stress, elastic moduli
Deriving the energy density dimensional formula
- Definition: u = E/V.
- Any energy reduces to [ML²T⁻²].
- Volume is a length cubed, [L³].
- Divide: L² over L³ leaves L⁻¹, so u is [ML⁻¹T⁻²], with mass and time unchanged.
Nothing about the type of energy enters the working. Electric, magnetic, elastic or thermal, the energy is still [ML²T⁻²], and per unit volume it is still [ML⁻¹T⁻²]. The energy page shows the top line in detail.
J/m³ and why it equals the pascal
As base units, the bracket spells kg m⁻¹ s⁻². Written the natural way, it is the joule per cubic metre, J/m³.
Now regroup: a joule is a newton metre, so J/m³ = N m/m³ = N/m², which is the pascal. Energy per volume and force per area are the same unit. That is why a gas at pressure P stores energy of the order of P in every cubic metre, and why light that carries energy also pushes on surfaces it hits.
| Quantity | Usual unit |
|---|---|
| Energy density | J/m³ |
| Pressure | Pa = N/m² |
| Stress | Pa |
| Young's modulus, bulk modulus | Pa |
Compare with the dimensional formula of pressure, which reaches the same bracket from F/A.
Field formulas checked with dimensions
Check: electric field energy density u = ½ε₀E²
Electric field E = F/q is [MLT⁻²] ÷ [IT] = [MLT⁻³I⁻¹], so E² is [M²L²T⁻⁶I⁻²].
Bring in ε₀ at [M⁻¹L⁻³T⁴I²]; the exponents combine into M¹, L⁻¹, T⁻², I⁰.
Result [ML⁻¹T⁻²], the energy density bracket. Numbers: in a field of 1.0 × 10⁶ V/m, u = 0.5 × 8.854 × 10⁻¹² × 10¹² = 4.4 J/m³.
Check: magnetic field energy density u = B²/(2μ₀)
From F = qvB, B is [MLT⁻²] ÷ ([IT][LT⁻¹]) = [MT⁻²I⁻¹]. So B² is [M²T⁻⁴I⁻²].
Divide by μ₀, [MLT⁻²I⁻²]: M²⁻¹ = M¹, L⁻¹, T⁻⁴⁺² = T⁻², I⁰. Again [ML⁻¹T⁻²].
Numbers: for B = 1.0 T, u = 1 ÷ (2 × 4π × 10⁻⁷) = 3.98 × 10⁵ J/m³, far more than the electric example.
Derivations of the E and B brackets: electric field, magnetic field.
Per volume or per kilogram
The phrase energy density is sometimes used loosely for energy per kilogram, especially for fuels and batteries. That is a different quantity, often called specific energy.
| Quantity | Definition | Dimensions | Unit |
|---|---|---|---|
| Energy density | E ÷ V | [ML⁻¹T⁻²] | J/m³ |
| Specific energy | E ÷ m | [L²T⁻²] | J/kg |
Specific energy has the bracket of a speed squared. If a question gives a unit, let the unit decide which one is meant. In Class 12 electrostatics and magnetism, energy density always means per unit volume.
Slips on energy density
- Writing [ML²T⁻²]. That is energy itself. Dividing by volume removes three powers of length.
- Writing [ML⁻²T⁻²]. That divides by area, not volume.
- Forgetting that E² squares the current power too. I⁻¹ becomes I⁻², which ε₀'s I² then cancels.
- Mixing up per volume and per mass. J/m³ and J/kg have different brackets.
Field and fluid brackets are listed together in the dimensional formulas collection.
Frequently asked questions
Why do energy density and pressure share a bracket?
Energy per volume is (force × length) ÷ length³, which simplifies to force ÷ area. Both are [ML⁻¹T⁻²] and both can be written in pascals.
Which SI unit does energy density take?
J/m³, joule per cubic metre. It is identical to the pascal, N/m².
Is energy density a scalar?
Yes. It is energy, a scalar, divided by volume, another scalar.
What is the dimensional formula for energy density in an electric field?
The same as any energy density, [M¹ L⁻¹ T⁻²]. The form ½ε₀E² reduces to it once ε₀ and E are written in base quantities.
Is energy per kilogram also called energy density?
Sometimes, for fuels, but strictly it is specific energy, [L²T⁻²] in J/kg. Energy density in physics problems means per unit volume.