Energy Density of Electric & Magnetic Fields Calculator
How much energy is stored per cubic metre in an electric or magnetic field? Solve every direction — u from E/B, E from u, B from u — combine both fields, and see the plane-wave proof that u_E = u_B.
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For a plane electromagnetic wave in vacuum the amplitudes satisfy E = c·B (c = 299,792,458 m/s), which forces u_E = u_B. Enter either amplitude and watch both densities come out equal.
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Worked examples (independently verified)
| Scenario | Fields | Energy density |
|---|---|---|
| Strong lab field | E = 1 MV/m | u_E = 4.4271 J/m³ |
| 1 T magnet | B = 1 T | u_B = 397,887.36 J/m³ |
| Capacitor + coil | E = 1 kV/m, B = 1 mT | u = 0.397892 J/m³ |
| Fair-weather atmosphere | E = 100 V/m | u_E = 4.4271×10⁻⁸ J/m³ |
| Earth's magnetic field | B = 50 µT | u_B = 9.9472×10⁻⁴ J/m³ |
| MRI scanner | B = 3 T | u_B = 3.581×10⁶ J/m³ |
| Sunlight at Earth | S = 1,361 W/m² | u = S/c = 4.5398×10⁻⁶ J/m³ |
| 1 litre at 0.397892 J/m³ | V = 0.001 m³ | U = 3.9789×10⁻⁴ J |
All values recomputed from the definitions with 50-digit arithmetic; the calculator uses full float precision and these rounded figures are display-only.
Methodology & constants
Formulas (vacuum / free space): u_E = ½·ε₀·E² and u_B = B²/(2·μ₀), with total u = u_E + u_B and stored energy U = u·V. Inverted: E = √(2u/ε₀), B = √(2·μ₀·u). For a plane wave in vacuum, E = c·B gives u_E = u_B exactly and u = ε₀E² = B²/μ₀.
Constants: ε₀ = 8.8541878128×10⁻¹² F/m (CODATA 2018). μ₀ = 4π×10⁻⁷ H/m — the exact pre-2019 definition value; since the 2019 SI redefinition it is a measured quantity differing by ~2×10⁻¹⁰, which changes no displayed digit here. c = 299,792,458 m/s (exact, SI defining constant). This tool computes in vacuum; inside linear materials the densities use the material ε and μ instead.
Frequently asked questions
What is the formula for the energy density of an electric field?
The electric energy density in vacuum is u_E = ½·ε₀·E², where E is the electric field strength in V/m and ε₀ = 8.8541878128×10⁻¹² F/m is the vacuum permittivity (CODATA 2018). Example: a field of E = 1 MV/m (10⁶ V/m) stores u_E = ½ × 8.8541878128×10⁻¹² × (10⁶)² = 4.4271 J/m³.
What is the formula for the energy density of a magnetic field?
The magnetic energy density in vacuum is u_B = B²/(2·μ₀), where B is the magnetic flux density in tesla and μ₀ = 4π×10⁻⁷ H/m is the vacuum permeability (the pre-2019 exact definition value — see the FAQ on μ₀ below). Example: B = 1 T gives u_B = 1/(2 × 4π×10⁻⁷) = 397,887.36 J/m³.
Are the electric and magnetic energy densities equal in an electromagnetic wave?
Yes — and here is the proof. In a plane wave in vacuum the field amplitudes satisfy E = c·B. Substituting into the electric density: u_E = ½·ε₀·E² = ½·ε₀·(cB)² = ½·ε₀·c²·B². Since c² = 1/(ε₀·μ₀), this becomes ½·(1/μ₀)·B² = B²/(2·μ₀) = u_B. The two contributions are exactly equal, and the total density is u = u_E + u_B = ε₀E² = B²/μ₀. (With the post-2019 SI definitions of the constants, the equality holds to about 5×10⁻¹⁰ — exact for any practical purpose. Try the “Prove the equality” tab above with B = 1 µT: it returns E = 299.792458 V/m and u_E = u_B.)
What units is field energy density measured in?
Joules per cubic metre (J/m³) in SI. Useful conversions: 1 J/m³ = 0.001 mJ/cm³ (because 1 mJ/cm³ = 1000 J/m³) and 1 J/m³ = 10 erg/cm³. This calculator lets you enter E in V/m or kV/m, B in T, mT, µT or gauss (1 G = 10⁻⁴ T), and read u in J/m³, mJ/cm³ or erg/cm³.
Is μ₀ exactly 4π×10⁻⁷ H/m?
Honestly: it was exact before 2019, when the SI was redefined and the ampere was decoupled from μ₀. Since then μ₀ is a measured quantity; the current experimental value differs from 4π×10⁻⁷ by only about 2×10⁻¹⁰, which is negligible for every engineering and teaching calculation. This calculator uses 4π×10⁻⁷ H/m and labels it as the pre-2019 definition value — a choice that changes no displayed digit in any realistic scenario.
How do I find the electric field from a given energy density?
Invert the formula: E = √(2u/ε₀). Example: u = 4.4270939064 J/m³ gives E = √(2 × 4.4270939064 / 8.8541878128×10⁻¹²) = 1.0×10⁶ V/m exactly. Use the “E from u” tab above.
How do I find the magnetic field from a given energy density?
Invert the formula: B = √(2·μ₀·u). Example: u = 397,887.3577 J/m³ gives B = √(2 × 4π×10⁻⁷ × 397,887.3577) = 1.0000 T. Use the “B from u” tab above.
What are some real-world field energy densities?
Fair-weather atmospheric electric field (~100 V/m): u_E = 4.4271×10⁻⁸ J/m³. Earth’s magnetic field (~50 µT): u_B = 9.9472×10⁻⁴ J/m³. A 3 T MRI scanner magnet: u_B = 3.581×10⁶ J/m³ — about 3.6 billion times the energy density of Earth’s field. Sunlight at Earth (1,361 W/m²): u = S/c = 4.5398×10⁻⁶ J/m³, split equally between its electric and magnetic parts.
How much total energy is stored in a given volume?
Multiply the density by the volume: U = u·V. Example: a combined density of u = 0.397892 J/m³ (E = 1 kV/m plus B = 1 mT) filling one litre (0.001 m³) stores U = 0.397892 × 0.001 = 3.9789×10⁻⁴ J. Enter a volume in the “Both fields” tab to get U alongside u.
Do these formulas apply to static fields, or only to waves?
They apply to static (and quasi-static) fields too: the energy density at any point in a static electric or magnetic field in vacuum is still ½·ε₀·E² and B²/(2·μ₀). Only the equality u_E = u_B is special to plane waves in vacuum, where E = c·B ties the two amplitudes together.
How does this relate to the capacitor energy formula ½CV²?
It is the same physics. Integrating u_E = ½·ε₀·E² over the volume between the plates of a parallel-plate capacitor gives exactly U = ½CV². Energy density is the local version of that global formula — useful when the field is not uniform or when no capacitance is defined.
Does this calculator work for fields inside materials like dielectrics or iron cores?
No — this tool uses the vacuum (free-space) formulas, which is the correct boundary for textbook problems, waves in air/vacuum, and most “prove the equality” exercises. Inside a linear dielectric or magnetic material the densities become u_E = ½·ε·E² and u_B = B²/(2·μ) with the material’s permittivity ε and permeability μ. That is a different tool.
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