# Magnetic energy

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

${\displaystyle E_{\rm }=-\mathbf \cdot \mathbf }$

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

${\displaystyle E_{\rm }={\frac }LI^}$.

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

${\displaystyle u={\frac }{\frac }{\mu _}}}$

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

${\displaystyle E={\frac }\int \mathbf \cdot \mathbf \ \mathrm V}$

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.