Hong Life CalculatorsUnderstanding the field energy density calculator — concept and principle
This calculator finds how much energy per cubic metre is stored in an electric field and a magnetic field, adds them, and gives the total energy in a chosen volume. It is used in electromagnetism courses and for quick estimates in capacitors, magnets, MRI scanners and pulsed-power devices.
How it is calculated
u_E = ½ ε₀ ε_r E²
u_B = B² ÷ (2 μ₀ μ_r)
u = u_E + u_B; energy in the volume = u × V
- E is entered in kV/m (× 1,000 to V/m), B in tesla, the volume in litres (÷ 1,000 to m³).
- ε₀ = 8.8541878188 × 10⁻¹² F/m and μ₀ = 1.25663706127 × 10⁻⁶ N/A² (CODATA 2022).
The principle
Building up a field takes work: charging a capacitor means pushing charge against the field already there, and building a current in a coil means working against the induced voltage. That work is stored in the field itself, spread through space in proportion to the square of the field strength. For a parallel-plate capacitor, ½CV² divided by the volume between the plates gives exactly ½ε₀E². Since 1 J/m³ equals 1 Pa, energy density is also a pressure: a 1 T field exerts a magnetic pressure of about 4 bar.
Worked example
Default inputs: E = 3,000 kV/m, B = 0.5 T, ε_r = μ_r = 1, volume 1 L.
- u_E = ½ × 8.854 × 10⁻¹² × (3 × 10⁶)² = 39.84 J/m³.
- u_B = 0.5² ÷ (2 × 1.2566 × 10⁻⁶) = 99,470 J/m³.
- Total u = 99,510 J/m³; the magnetic share is 99.96%.
- In 1 L = 0.001 m³: 99,510 × 0.001 = 99.51 J.
| Magnetic flux density B (T) | Total energy density (J/m³) |
|---|---|
| 0.125 | 6,257 |
| 0.25 | 24,910 |
| 0.375 | 55,990 |
| 0.5 | 99,510 |
| 0.625 | 155,500 |
| 0.75 | 223,900 |
| 1 | 397,900 |
How to read the result
The example shows how differently the two fields store energy. An electric field of 3 MV/m, close to the level at which dry air breaks down, stores only about 40 J/m³, while a moderate 0.5 T magnet stores about 2,500 times more. This is why large amounts of field energy are usually stored magnetically, and why capacitors use thin, high-permittivity dielectrics.
Common mistakes
- Entering E in V/m when the field asks for kV/m.
- Forgetting that both terms scale with the square of the field.
- Using the volume total for fields that are not uniform.
- Leaving ε_r or μ_r at 1 inside a dielectric or magnetic material.
Frequently asked questions
Why is J/m³ the same as a pascal?
1 J = 1 N·m, so 1 J/m³ = 1 N/m², which is 1 Pa.
How much energy does a 1 T field store?
About 400,000 J/m³, or 0.4 J per cubic centimetre.
Does doubling the field double the energy?
No, it quadruples it, because energy density depends on the square of the field.
Sources: CODATA 2022 recommended values (ε₀, μ₀), NIST · OpenStax, University Physics Volume 2 (2016), sections on capacitor and inductor energy