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Vacuum Electric and Magnetic Field Energy per Volume

Relate nonnegative electric-field magnitude, magnetic-field magnitude and vacuum energy density, with inverse controls and physical energy-per-volume units.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

E = 1000 N/C and B = 0.001 T give approximately 0.397892 J/m³ of vacuum field energy density.

u = ε₀E²/2 + B²/(2μ₀), with ε₀ = 8.8541878188 ×10⁻¹² F/m and μ₀ = 1.25663706127 ×10⁻⁶ N/A² (2022 CODATA central values).

Worked example

Enter these known values and leave the other values blank.

Electric field magnitude
1000 N/C
Magnetic field magnitude
1 mT
Vacuum field energy density
0.3979 J / m³

Assumptions and limitations

  • This computes instantaneous energy per volume for field magnitudes in vacuum. The result is energy divided by the selected volume unit (J/m³ by default); it is not total energy in joules.
  • Electric and magnetic contributions add. The tool does not impose the plane-wave relationship E = cB or compute a sinusoidal time average, wave intensity or propagation direction.
  • Field magnitudes and energy density are nonnegative. Either field may be zero. Recovering one field requires total density and the other field; a square-root inverse returns a magnitude rather than a field sign.
  • Total energy density cannot be smaller than the contribution of an already known field. Impossible inverse combinations produce an input error instead of a complex field value. A relative 10⁻⁶⁰ arithmetic margin handles conversion round-off at a zero-field boundary.
  • The constants are measured 2022 CODATA central values. Standard uncertainties are 1.4 ×10⁻²¹ F/m for ε₀ and 2.0 ×10⁻¹⁶ N/A² for μ₀; neither constant is an exact SI defining value.
  • The formula does not model material permittivity or permeability, losses, nonlinear media or a total storage volume.

Common questions

Does this result give the total stored energy?

It gives energy per selected volume unit, with J/m³ as the default. Total energy requires integrating the density over the relevant volume; multiplying by volume works only for a uniform density.

Can I determine both fields from one energy-density reading?

In general, many pairs give the same total. Enter one field magnitude as well to recover the other within this vacuum model.

References

Accept zero field magnitudes and zero energy density, and require total density to cover each known field contribution within a relative 10^-60 arithmetic margin. Preserve the captured energy numerator and reversed volume denominator selectors. Sign-based comparisons enforce the declared bound without the shared arithmetic library's relative equality tolerance. Use the NIST 2022 CODATA central constants throughout the forward and principal-magnitude inverse graph; guard both square-root domains and clamp only the accepted round-off residual to zero with sign-based arithmetic, preserving positive residuals above the declared margin.

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