Magnetic Field Energy Density Calculator

How magnetic fields store energy

A magnetic field stores energy in space (and in materials) in a way that is directly analogous to the energy stored in an electric field. For many engineering calculations—coils, magnets, transformers, MRI systems, and plasma confinement—the most useful quantity is the magnetic energy density u, i.e., how many joules of energy are stored per cubic meter of field region (J/m³). Once you know u and the relevant volume V, you can estimate the total stored magnetic energy U via a simple multiplication.

This calculator is designed for the common “uniform-field” estimate: you provide any two of the variables and compute the others (with the special case that supplying B alone is enough to compute u, because u depends only on B for the free-space formula). The relationships are nonlinear because energy density scales with B²: doubling the magnetic flux density quadruples the energy density.

Formulas (SI units)

In free space (vacuum/air to a good approximation), the magnetic energy density is:

Energy density: u = B² / (2μ₀)

In MathML form:

Total energy stored in a region of approximately uniform field is:

Total energy: U = u · V

Where:

• B = magnetic flux density (tesla, T)
• u = energy density (joules per cubic meter, J/m³)
• V = field volume (cubic meters, m³)
• U = total stored energy (joules, J)
• μ₀ = permeability of free space (≈ 4π × 10⁻⁷ H/m)

Solving for the missing quantity

Depending on which values you know, you can rearrange the same relationships:

• If you know B: compute u directly from u = B²/(2μ₀).
• If you know u: compute B from B = √(2μ₀u).
• If you know u and V: compute U from U = uV.
• If you know U and V: compute u from u = U/V.
• If you know U and u: compute V from V = U/u.
• If you know B and V: compute U by first finding u, then multiplying by V.

Materials (relative permeability)

If the magnetic field energy is primarily stored in a material with permeability μ (instead of free space), a commonly used variant is:

Energy density in a linear medium: u = B² / (2μ) where μ = μ₀μ_r

However, real magnetic cores often have μ_r that varies with B (nonlinear), and energy storage may be dominated by an air gap rather than the core itself. In those cases, the uniform, single-μ estimate can be misleading; see limitations and assumptions below.

Interpreting the results

The calculator returns up to four related quantities:

• Magnetic field B (T): how strong the flux density is.
• Energy density u (J/m³): energy stored per unit volume of the field region.
• Volume V (m³): the region over which you assume the field is roughly uniform.
• Total energy U (J): total stored energy estimate, U = uV.

A helpful physical interpretation is that energy density in J/m³ has the same units as pressure in pascals (Pa), because 1 Pa = 1 N/m² = 1 J/m³. For magnetostatics in free space, magnetic pressure is often written as p = u. This means a high-field magnet can exert mechanical stresses comparable to substantial fluid pressures, which is why coil reinforcement and structural design are so important for superconducting systems.

Worked example

Problem: Estimate the energy density and total stored energy for a strong magnet region with B = 3 T and an approximately uniform-field volume of V = 0.50 m³.

1. Compute energy density

   Use u = B²/(2μ₀).

   μ₀ ≈ 4π × 10⁻⁷ H/m, so:

   u ≈ 3² / (2 · 4π × 10⁻⁷) = 9 / (8π × 10⁻⁷) ≈ 3.58 × 10⁶ J/m³

2. Compute total energy

   U = uV ≈ (3.58 × 10⁶ J/m³)(0.50 m³) ≈ 1.79 × 10⁶ J

Interpretation: 3.58 MJ/m³ corresponds to about 3.58 MPa of equivalent pressure. The total energy of ~1.8 MJ is significant (comparable to the kinetic energy of a large vehicle at highway speed). In real magnets, not all the space has uniform field; this is an order-of-magnitude estimate for design intuition and quick checks.

Comparison table: how fast energy density grows with B

The quadratic dependence is easier to see in a table. The values below use the free-space formula u = B²/(2μ₀). “Equivalent pressure” is numerically the same as u in SI units.

Using the calculator effectively

• Enter values in SI units: tesla (T), joules (J), cubic meters (m³), and J/m³.
• For total energy, the most direct pair is u and V (or U and V to back out u).
• If you only know B, you can still compute u (energy density depends only on B in the free-space model).
• If you know U and V, you can compute u, then compute an equivalent B that would produce that u in free space.

Limitations and assumptions

This calculator is intentionally simple and is best used for quick estimates, sanity checks, and educational intuition. Keep the following assumptions in mind:

• Uniform-field assumption: The step U = uV assumes u is approximately constant over the volume. Real magnetic fields vary with position; a more accurate computation uses U = ∫ u(r) dV.
• Edge/fringing fields: In coils, gaps, and near magnet ends, fringing fields can add significant energy outside the “obvious” volume. Choosing V too small underestimates U.
• Material nonlinearity and saturation: The replacement μ = μ₀μ_r assumes a linear medium (constant permeability). Ferromagnetic cores have B–H curves with saturation and hysteresis, so the true stored energy is not captured by a single constant μ.
• Where energy is actually stored: In many inductors/transformers, most magnetic energy is stored in the air gap (low permeability) rather than in the high-μ core. If you use a core’s high μ_r in the formula everywhere, you may substantially mis-estimate energy.
• Time-varying fields and losses: This page computes stored field energy, not dissipated losses (eddy currents, hysteresis, resistive heating, radiation). For AC devices, thermal design often depends more on loss models than on stored energy alone.
• Geometry-specific designs: For tight tolerances (e.g., MRI quench energy accounting, accelerator magnets, high-power pulsed magnets), use manufacturer data, inductance-based methods (U = ½LI²), or finite-element analysis (FEA) that resolves the full geometry and materials.