# Magnetic energy

Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet of magnetic moment $\mathbf {m}$ in a magnetic field $\mathbf {B}$ is defined as the mechanical work of the magnetic force (actually magnetic torque) on the re-alignment of the vector of the Magnetic dipole moment and is equal to:

$E_{\rm {p,m}}=-\mathbf {m} \cdot \mathbf {B}$ while the energy stored in an inductor (of inductance $L$ ) when a current $I$ flows through it is given by:

$E_{\rm {p,m}}={\frac {1}{2}}LI^{2}$ .

This second expression forms the basis for superconducting magnetic energy storage.

Energy is also stored in a magnetic field. The energy per unit volume in a region of space of permeability $\mu _{0}$ containing magnetic field $\mathbf {B}$ is:

$u={\frac {1}{2}}{\frac {B^{2}}{\mu _{0}}}$ More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates $\mathbf {B}$ and $\mathbf {H}$ , then it can be shown that the magnetic field stores an energy of

$E={\frac {1}{2}}\int \mathbf {H} \cdot \mathbf {B} \ \mathrm {d} V$ where the integral is evaluated over the entire region where the magnetic field exists.