Formula and Method for Energy Density of Electric and Magnetic Fields
Any region of space that contains an electric field or a magnetic field stores energy, even in a vacuum with no charges or currents present. The electric field energy density is uE = ½ɛ0ɛrE², and the magnetic field energy density is uB = B²/(2μ0μr), where E is the electric field strength (V/m), B is the magnetic flux density (T), ɛ0 = 8.8541878128×10⁻¹² F/m is the permittivity of free space, and μ0 = 4π×10⁻⁷ T·m/A is the permeability of free space. The relative permittivity ɛr and relative permeability μr scale these formulas for a surrounding medium other than vacuum (both equal 1 for air or free space). Adding the two contributions gives the total electromagnetic energy density, u = uE + uB, in joules per cubic meter (J/m³).
How the calculation works
Enter the electric field strength and its unit, then the magnetic flux density and its unit; the calculator converts both to SI base units (volts per meter and tesla) before computing. It squares the electric field and multiplies by ½ɛ0ɛr to get uE, squares the magnetic flux density and divides by 2μ0μr to get uB, then sums them for the total. This derives directly from the work needed to assemble a field: for a linear dielectric, the electric displacement is D = ɛ0ɛrE and the stored energy density is ½E·D; for a linear magnetic material, the field intensity is H = B/(μ0μr) and the stored energy density is ½B·H — both reduce to the formulas above.
Common mistakes
- Forgetting the medium: in a dielectric or magnetic material, ɛr and μr are rarely exactly 1 — using vacuum values inside a capacitor's dielectric or an inductor's core will understate the stored energy.
- Mixing field units: convert kV/m or MV/m to V/m, and mT, μT, or gauss to tesla, before comparing results — a factor-of-1000 unit slip changes the energy density by a factor of a million (since both formulas involve a squared term).
- Using peak instead of average field for AC sources: for a sinusoidally varying field, the time-averaged energy density uses the RMS value of E or B, not the peak amplitude (Erms = Epeak/√2).
Real-world applications
- Capacitor design uses uE to estimate how much energy a dielectric volume can store before approaching its breakdown field.
- Inductor and electromagnet design uses uB to estimate the energy stored in a magnetic core or air gap.
- Electromagnetic wave propagation (radio, light, radar) splits its energy equally between uE and uB in vacuum, and the total energy density relates directly to the wave's intensity via the Poynting vector, S = uc.
- MRI and particle-accelerator magnet design accounts for the very large uB stored in strong (multi-tesla) magnetic fields, since that stored energy must be safely contained if the magnet quenches.
Frequently Asked Questions
What is the formula for electric field energy density?
Electric field energy density is uE = ½ɛ0ɛrE², where E is the electric field strength in volts per meter, ɛ0 = 8.854×10⁻¹² F/m is the permittivity of free space, and ɛr is the relative permittivity of the surrounding medium (ɛr = 1 for vacuum or air). The result is in joules per cubic meter (J/m³).
What is the formula for magnetic field energy density?
Magnetic field energy density is uB = B² / (2μ0μr), where B is the magnetic flux density in tesla, μ0 = 4π×10⁻⁷ T·m/A is the permeability of free space, and μr is the relative permeability of the medium (μr = 1 for vacuum, air, or non-magnetic materials). The result is in joules per cubic meter (J/m³).
How do you find the total electromagnetic energy density?
Add the electric and magnetic contributions: u = uE + uB. This total represents the energy stored per unit volume in the combined electric and magnetic fields at a point in space, which is the basis for calculating the energy stored in capacitors, inductors, and propagating electromagnetic waves.
Why does a 1 tesla magnetic field store far more energy than a strong electric field?
Because μ0 (about 1.2566×10⁻⁶) is many orders of magnitude smaller than 1/ɛ0. A 1 T field, easily produced by a strong permanent magnet or MRI coil, stores about 398,000 J/m³, while air's dielectric breakdown field of roughly 3 MV/m stores only about 40 J/m³. Achievable magnetic fields simply pack far more energy per unit volume than achievable electric fields.
Common Tip Percentages
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| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
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| Needs | 50% | Rent, groceries, utilities, insurance |
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |