# Magnetic energy

Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet of magnetic moment ${\displaystyle \mathbf {m} }$ in a magnetic field ${\displaystyle \mathbf {B} }$ 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:

${\displaystyle E_{\rm {p,m}}=-\mathbf {m} \cdot \mathbf {B} }$

while the energy stored in an inductor (of inductance ${\displaystyle L}$) when a current ${\displaystyle I}$ flows through it is given by:

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

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 ${\displaystyle \mu _{0}}$ containing magnetic field ${\displaystyle \mathbf {B} }$ is:

${\displaystyle u={\frac {1}{2}}{\frac {B^{2}}{\mu _{0}}}}$

More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates ${\displaystyle \mathbf {B} }$ and ${\displaystyle \mathbf {H} }$, then it can be shown that the magnetic field stores an energy of

${\displaystyle E={\frac {1}{2}}\int \mathbf {H} \cdot \mathbf {B} \ \mathrm {d} V}$

where the integral is evaluated over the entire region where the magnetic field exists.[1]

## References

1. Jackson, John David (1998). Classical Electrodynamics. New York: Wiley.