Energy Density of Fields Calculator
See how much energy per cubic metre is stored in an electric field, a magnetic field, or both, and find the field that stores a chosen amount.
🔋 What is the Energy Density of Fields?
The energy density of a field is the amount of energy stored per unit volume of space where an electric or magnetic field is present. Fields are not just a bookkeeping device. Charging a capacitor or switching on a current in a coil takes work, and that work is stored in the field itself, spread through the space around the charges and currents. For an electric field the energy density is uE = ½ ε0 E², and for a magnetic field it is uB = B² / (2 μ0). Both are measured in joules per cubic metre, the same unit as pressure in pascals.
The idea is used in many practical settings. Capacitor and inductor designers use it to relate stored energy to volume and to compare technologies. MRI and fusion magnet engineers use it to estimate the enormous stored energy and the magnetic pressure on a coil. Insulation and gas engineers use it to see how much energy a dielectric breakdown can release. Astrophysicists compare the magnetic and electric energy density in a cloud or a pulsar wind, and radio engineers connect the energy density of a wave to its intensity, which is the energy density times the speed of the wave.
Some points deserve care. The energy density depends on the square of the field, so doubling a field quadruples the energy stored per volume. The electric and magnetic terms are separate, and in vacuum they are equal when E = cB, as in a light wave. A typical strong electric field is far weaker in energy terms than a typical strong magnetic field, because c is large. Finally, the formulas assume a linear medium with no hysteresis, and for materials you scale by the relative permittivity and permeability.
This calculator works in two directions. In the first mode you enter E and B, and it finds both energy densities, the electric share, the energy in a chosen volume, and the fields of the other kind with the same energy. In the second mode you enter an energy density and pick a field type, and it returns the field strength needed. Each mode draws a log-log chart so that the very wide range of values is easy to read.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — An MRI Magnet
A uniform 3 T magnetic field in vacuum over 1 m³, with no electric field
Example 2 — Electric Field at Air Breakdown
An electric field of 3,000,000 V/m in air (εr = 1) over 1 m³, with no magnetic field
Example 3 — Matched Electric and Magnetic Parts
E = 300,000,000 V/m and B = 1 T together, filling 2 m³
Example 4 — Field Inside Water
E = 1,000,000 V/m in a medium with εr = 80, over 1 m³
Example 5 — Magnetic Field for a Chosen Energy Density
What magnetic field holds 1,000,000 J/m³?
Example 6 — Sunlight at the Top of the Atmosphere
Energy density 4.53981×10−6 J/m³, shared equally between E and B in a light wave
Example 7 — Electric Field for a Chosen Energy Density
What electric field holds 100 J/m³?
❓ Frequently Asked Questions
🔗 Related Calculators
What is the energy density of an electric field?
The energy per unit volume in an electric field is u_E = ½ ε0 E², with E in V/m and ε0 = 8.8541878128e-12 F/m. In air at the breakdown field of 3 MV/m the energy density is 39.8438 J/m³.
What is the energy density of a magnetic field?
The magnetic energy density is u_B = B²/(2μ0), with B in tesla and μ0 = 1.25663706212e-6 N/A². A 3 T MRI field stores 3.5810e6 J/m³, which is about 3.58 MJ per cubic metre.
How do I find the total energy density of an electromagnetic field?
Add the two contributions, u = ½ ε0 E² + B²/(2μ0). For E = 3e8 V/m and B = 1 T the two parts are 3.9844e5 and 3.9789e5 J/m³, giving 7.9633e5 J/m³ with the electric share at 50.03 percent.
Why is there a factor of one half?
The field is built up from zero, and the energy stored while charging or magnetising grows with the square of the field. Integrating the work gives half of the product, as in a capacitor energy ½CV², where the same factor appears.
How much energy is stored in a given volume of field?
Multiply the energy density by the volume, U = u V, for a uniform field. A 3 T field filling 1 m³ holds 3.5810e6 J. For non-uniform fields, you integrate the energy density over the volume.
How do I find the field strength from an energy density?
Invert the formula. For an electric field E = sqrt(2u/ε0), and for a magnetic field B = sqrt(2 μ0 u). An energy density of 1e6 J/m³ needs B = 1.5853 T or E = 4.7527e8 V/m.
What is the energy density of sunlight?
At the top of the atmosphere the solar intensity is about 1361 W/m², so the energy density is the intensity divided by c, about 4.54e-6 J/m³. In the wave mode this corresponds to an rms electric field of 716.05 V/m and an rms magnetic field of 2.3885e-6 T.
Is energy density the same as pressure?
They share units, since a joule per cubic metre equals a pascal. A magnetic field of 3 T exerts a magnetic pressure of 3.5810e6 Pa, which is about 35.3 atm. An electric field exerts a similar pressure ½ε0E² on conductor surfaces.
How do dielectrics and magnetic materials change the energy density?
In a linear medium, u_E = ½ ε0 εr E² and u_B = B²/(2 μ0 μr). For water with εr = 80 and E = 1e6 V/m, the electric energy density is 354.1675 J/m³, which is 80 times the vacuum value.
Where do electric and magnetic energy densities come from in Maxwell's theory?
They come from Poynting's theorem, which states that the rate of change of field energy plus the energy flow out through the surface equals minus the work done on charges. The energy flow is the Poynting vector S = E × B / μ0.