How this instrument works
Energy density here means joules stored per cubic metre of the space an electric field occupies, not joules stored in a charge or a wire. The formula u = ½E²ε₀ falls straight out of capacitor bookkeeping: a parallel-plate capacitor with plate area A and gap d holds W = ½CV² of energy, and substituting C = ε₀A/d with V = Ed turns that into W = (½ε₀E²)(Ad) — energy per unit volume times the volume the field fills. Strip away the geometry and ½ε₀E² is what remains.
Treating the field as the thing that holds the energy, rather than the charges that made it, was the genuinely new idea in this equation. Coulomb's law lets you tally energy by pairs of charges and their separation; Maxwell's field picture lets you tally the same total by integrating this density over every point in space, charges present or not. Both totals agree, but only the field version survives once you ask where the energy is while a wave is travelling through vacuum with no charges anywhere nearby.
The formula assumes a linear, non-dispersive medium — vacuum or an ordinary dielectric obeying D = εE. Push the field toward a material's breakdown strength and that linearity fails: dry air ionises near 3 megavolts per metre, at which point charge starts moving through what was supposed to be an insulator and the neat ½ε₀E² bookkeeping stops describing what happens next.
- Enter the field strength into "Electric field strength", in volts per metre; switch the unit menu to kV/mm or MV/m for stronger fields.
- No second input is needed — the instrument squares that value and multiplies by half the permittivity of free space automatically.
- Read the result in "Energy density", reported in joules per cubic metre (J/m³).
- Switch the result's unit to µJ/m³ for everyday field strengths, or compare it against a target such as air's breakdown density to judge how close a design sits to that limit.
Worked example — a 1,000 V/m capacitor field
A field of 1,000 V/m is a realistic figure inside an air-gap capacitor charged to a modest voltage — a 1 mm gap at 1,000 V, for instance. Squaring it gives 1,000,000, and 8.854187817 × 10⁻¹² × 1,000,000 × 0.5 works out to 4.427 × 10⁻⁶ J/m³: roughly four and a half millionths of a joule stashed in every cubic metre that field fills.
That sounds negligible, and for a 1 mm-gap capacitor of modest area it is — the field only occupies a few cubic millimetres, so the total stored energy is a tiny fraction of a joule. Push the same air gap toward its breakdown strength of about 3 × 10⁶ V/m instead, and the same formula returns roughly 39.8 J/m³, because doubling the field quadruples the density: energy climbs with the square of E, not with E itself.
Questions
Why does an electric field carry energy at all?
Because moving or removing the charges that made the field takes work, and that work has to go somewhere — Maxwell's field theory locates it in the field itself, spread through space at ½ε₀E² per cubic metre, rather than pinned to the charges. It is exactly this stored energy that lets a charged capacitor light an LED after the battery is disconnected.
Where does the one-half in the formula come from?
From integrating the work of assembling the field gradually. Building a capacitor's field by moving charge in small increments, from zero up to the final value, averages to half the final force times the final displacement — the same ½ that appears in ½CV² and ½kx². It is not a rounding choice; it falls out of the calculus of assembling the field step by step.
Does this instrument include the magnetic field's contribution?
No — this calculator computes only the electric-field term, ½ε₀E². A magnetic field stores energy density B²/(2μ₀) by the same logic. For a travelling electromagnetic wave in vacuum the two densities are equal at every instant, so the wave's total energy density is twice the value this instrument reports for E alone.
What changes inside a dielectric instead of vacuum?
Replace ε₀ with the material's permittivity ε = εᵣε₀. A dielectric with relative permittivity εᵣ = 5 stores five times the energy per cubic metre at the same field strength, which is precisely why high-εᵣ ceramics let engineers pack large capacitance into small components.
Is there an upper limit on the field this formula stays valid for?
It holds as long as the medium responds linearly, which for dry air at sea level ends around 3 × 10⁶ V/m — the field where air ionises and starts conducting instead of insulating. Past that breakdown point the material's behaviour turns nonlinear and ½ε₀E² no longer tracks the true stored energy.
How does this relate to a capacitor's stored energy, W = ½CV²?
They describe the same physics from two angles. Multiply this instrument's density by the volume the field occupies — plate area times gap, for a parallel-plate capacitor — and the geometry cancels algebraically back down to ½CV². The field-density view is the one that still works once the geometry is not a tidy pair of plates.