Magnetic energy density

Every magnetic field carries energy, and that energy is distributed through space at a density u_B = B²/(2μ₀) (in vacuum or non-magnetic material), or more generally u_B = ½ B·H inside linear magnetic media. This parallels the electric case — an electric field stores ε₀E²/2 per unit volume — and once you take both together, u_total = ε₀E²/2 + B²/(2μ₀), you have the energy density of the electromagnetic field in every point of space, independent of whether any circuit is nearby.

The derivation comes from tracking the energy delivered to a coil as its current is built up from zero. For an inductor, the work done by the external source against the back-EMF is dW = V I dt = (L dI/dt) I dt = L I dI, integrated from 0 to I gives U = ½ L I². Substitute L in terms of the coil geometry and I in terms of B inside the coil, and U expressed as a volume integral becomes exactly ∫ (B²/2μ₀) dV — the field-localised picture. The energy, you discover, isn't really "in the coil"; it's in the space around the coil, stored in the magnetic field itself.

Practical implications: a solenoid running 10 T (an MRI-scale superconducting magnet) stores B²/(2μ₀) ≈ 4 × 10⁷ J/m³ — forty megajoules per cubic metre, enough energy in a one-litre bore to vaporise a kilogram of water. This is why superconducting magnet quenches are dangerous: the stored magnetic energy has to go somewhere on timescales set by the quench physics. Inductive energy storage systems, including proposed superconducting magnetic energy storage (SMES) for grid applications, exploit this high energy density — the trade-off is that keeping B high requires keeping the superconductor cold, which costs its own energy. In pulsed-power applications (capacitor banks feeding pulsed magnets, Z-pinch experiments), the inductance's ½LI² stored during current ramp-up becomes the dominant energy reservoir, and managing its release on discharge is the main engineering problem.