Energy Density of Electric and Magnetic Fields Calculator
Enter an electric field strength and a magnetic field strength to get the energy density stored in each field and the combined electromagnetic total. You can also include the volume of a region to find the total energy stored, or switch to a material mode to account for a dielectric or ferromagnetic medium. The reverse-solve mode lets you work backwards from a known energy density to find the field that produced it.
Formula
Worked example
A parallel-plate capacitor has an internal electric field of E = 1000 V/m in vacuum (B = 0): uE = 0.5 × 8.854 × 10⁻¹² × (1000)² = 4.427 × 10⁻⁶ J/m³. A toroidal inductor core with B = 0.001 T in vacuum adds uB = (0.001)² / (2 × 4π × 10⁻⁷) = 3.979 × 10⁻¹ J/m³. If both fields coexist in a 0.001 m³ region the total stored energy is U = (4.427 × 10⁻⁶ + 0.3979) × 0.001 = 3.979 × 10⁻⁴ J.
What is energy density of a field?
When an electric or magnetic field exists in a region of space it stores energy. The energy density (symbol u, unit J/m³) describes how much energy is packed into each cubic metre of that region. This is important in capacitor and inductor design, antenna near-field analysis, microwave engineering, and plasma physics, because it tells you not just how much total energy a device stores but how concentrated that energy is, which determines breakdown risk, heating, and radiation efficiency.
The formulas explained
For a pure electric field in a medium with permittivity ε: uE = (1/2) ε E², where E is the field strength in V/m and ε = ε₀ εr. The factor of 1/2 appears because building up a field from zero against the restoring force of the medium requires integrating over all intermediate states. For a magnetic field in a medium with permeability μ: uB = B² / (2μ), where B is the flux density in tesla and μ = μ₀ μr. When both fields are present at the same point the densities add: u = uE + uB. Multiplying by the volume of the region gives the total stored energy U = u × V. In vacuum, ε = ε₀ = 8.8542 × 10⁻¹² F/m and μ = μ₀ = 4π × 10⁻⁷ H/m.
Electromagnetic waves: equal energy in both fields
In a self-sustaining plane wave traveling through vacuum the electric and magnetic energy densities are always exactly equal: uE = uB. This follows directly from the wave equations and leads to the simple relations u = ε₀E² = B²/μ₀, and intensity I = uc where c is the speed of light. The Poynting vector S = E × B / μ₀ (W/m²) describes the directional power flux and equals I for a plane wave. Sunlight arriving at Earths surface carries about 1000 W/m², corresponding to an electromagnetic energy density of roughly 3.3 × 10⁻⁶ J/m³.
How medium properties change stored energy
A dielectric material with relative permittivity εr > 1 stores εr times more electric energy per unit volume than vacuum at the same field strength. This is why high-permittivity ceramics (BaTiO₃, εr up to 10 000) are used in capacitors: the same physical volume stores vastly more energy. Similarly, a magnetic core with relative permeability μr > 1 stores more magnetic energy per unit volume, though most practical inductors operate below saturation to keep the core linear. The reverse-solve mode in this calculator lets you find what field is needed in a given medium to reach a target energy density, useful for optimising capacitor plate spacing or inductor gap design.
Typical energy density values in real devices
| System | Dominant field | Typical energy density | Notes |
|---|---|---|---|
| Earth's surface electric field (~100 V/m) | Electric | ~4.4 × 10⁻⁸ J/m³ | Fair-weather atmosphere |
| Earth's magnetic field (~50 µT) | Magnetic | ~1.0 × 10⁻³ J/m³ | Near the surface |
| Ceramic capacitor (100 kV/m field) | Electric | ~22 J/m³ | εr ≈ 5 |
| Power-line magnetic field (~0.1 mT) | Magnetic | ~4 × 10⁻³ J/m³ | 50/60 Hz, near conductors |
| Iron-core transformer (B = 1.5 T) | Magnetic | ~9 × 10⁵ J/m³ | Saturation regime |
| Sunlight (plane wave, 1000 W/m²) | EM wave | ~3.3 × 10⁻⁶ J/m³ | uE = uB for a plane wave |
| Pulse-power capacitor (MV/m field) | Electric | ~4.4 × 10⁶ J/m³ | High-energy physics |
| Strong MRI (3 T, air core) | Magnetic | ~3.6 × 10⁶ J/m³ | Superconducting coil |
Order-of-magnitude figures for common electromagnetic systems. Actual values depend on geometry, material grade, and operating conditions.
Frequently asked questions
What is the unit of electromagnetic energy density?
Energy density is measured in joules per cubic metre (J/m³). Because 1 J/m³ = 1 Pa (pascal), energy density is numerically equal to a pressure, which is useful in radiation-pressure and magnetic-confinement calculations.
How does energy density relate to field strength?
Electric energy density scales with the square of the electric field: doubling E quadruples uE. The same quadratic relationship holds for the magnetic field: doubling B quadruples uB. This strong dependence is why small increases in field strength can lead to large increases in stored energy or breakdown risk.
Are the electric and magnetic energy densities always equal?
No. They are equal only in a plane electromagnetic wave propagating through a uniform medium. In a capacitor the field is almost entirely electric; in an inductor it is almost entirely magnetic. In a resonant cavity or coupled LC circuit the split between uE and uB oscillates with time at the resonant frequency, but the time-averaged values are equal.
Why does using a dielectric material increase the stored energy for the same voltage?
The energy density is uE = (1/2) ε₀ εr E². At a fixed plate voltage the field E stays the same but εr multiplies the whole expression, so a material with εr = 10 stores 10x more energy per unit volume than vacuum at the same field. In practice the field is limited by the dielectric breakdown strength, so the goal is to maximise the product εr × Ebreakdown².
What is the energy density of a magnetic field of 1 T in vacuum?
uB = B² / (2μ₀) = (1)² / (2 × 4π × 10⁻⁷) ≈ 397 887 J/m³, roughly 0.4 MJ/m³. For comparison, a strong MRI machine at 3 T stores about 3.6 MJ/m³ in its bore, which is why quenching (sudden loss of superconductivity) releases enough energy to boil hundreds of litres of liquid helium.
How do I find the total energy stored, not just the density?
Multiply the energy density by the volume of the region: U = u × V. For a parallel-plate capacitor with plate area A and gap d, the volume is A × d, giving U = (1/2) ε E² × A × d = (1/2) ε E² × volume. This recovers the familiar U = (1/2) CV² when you substitute C = ε A/d and V = E d.