Magnetic Field Energy Density Calculator

How magnetic fields store energy density

A magnetic field stores energy in the space around it, and the most compact way to describe that storage is the magnetic energy density u, measured in joules per cubic meter (J/m³). In coil, magnet, transformer, MRI, and air-gap calculations, that density is the key quantity because it lets you estimate total stored energy U once you know the volume V that the field occupies.

This calculator is set up for the uniform-field approximation. Enter whichever two quantities you know and it solves the others; if you start with B alone, the free-space formula gives u directly because magnetic energy density grows with the square of the flux density. That square-law means small changes in B can cause large changes in stored energy.

Magnetic energy-density formulas (SI units)

For free space, or for air when the approximation is acceptable, 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 B, u, V, or U

Because the calculator is built around one magnetic energy-density relation, you can rearrange it in several directions depending on whether you begin with B, u, V, or U.

• 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 and relative permeability

If the field energy sits mainly in a material instead of free space, the calculator uses the linear-medium form with permeability μ. That is often fine for a first look at a simple magnetic circuit or air gap, but it becomes shaky once the material response is no longer linear.

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 magnetic energy-density results

The calculator returns up to four linked quantities, so the cleanest way to read the output is to think of B as the cause, u as the density, V as the region, and U as the total stored magnetic energy.

• 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: 3 T field across 0.50 m³

Problem: Estimate the magnetic energy density and total stored energy for a 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 about 1.8 MJ is large enough to matter for supports, quench protection, and fault analysis, even though the field is only approximated as uniform here.

Comparison table: how magnetic 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 this magnetic-field 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 because the free-space model uses B².
• 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 for magnetic energy density

This calculator is intentionally simple and is best used for quick magnetic-energy checks, teaching, and first-pass design intuition. Keep the following assumptions in mind before using the result for hardware decisions:

• Uniform-field assumption: The step U = uV assumes u is approximately constant over the chosen 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.