Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet or 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:

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

${\displaystyle L}$

${\displaystyle I}$

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

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 the magnetization ${\displaystyle \mathbf {H} }$, then it can be shown that the magnetic field stores an energy of

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

For a magnetostatic system of currents in free space, the stored energy can be found by imagining the process of linearly turning on the currents and their generated magnetic field, arriving at a total energy of:^[1]

$${\displaystyle E={\frac {1}{2))\int \mathbf {J} \cdot \mathbf {A} \,\mathrm {d} V}$$

${\displaystyle \mathbf {J} }$

${\displaystyle \mathbf {A} }$

${\textstyle {\frac {1}{2))\int \rho \phi \,\mathrm {d} V}$