Magnetic energy

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Magnetic energy and electric 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 magnetic force (actually of magnetic torque) on re-alignment of the vector of the Magnetic dipole moment, and is equal:

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

while the energy stored in an inductor (of inductance, ${\displaystyle L}$) when current, ${\displaystyle I}$, is passing via it is:

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

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

${\displaystyle E={\frac {1}{2}}\int \mathbf {H} \cdot \mathbf {B} \ dV}$

where the integral is evaluated over the whole region where the magnetic field exists