Energy from the work of a magnetic force
The potential magnetic energy of a magnet or magnetic moment
in a magnetic field
is defined as the mechanical work of the magnetic force 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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4354636e2e549eacf30a58ff82dc686f9c6ada)
while the energy stored in an inductor (of inductance
) when a current
flows through it is given by:
![{\displaystyle E_{\text{p,m))={\frac {1}{2))LI^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d32e5b4623f56e8e05147bb7854b9561891012)
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
containing magnetic field
is:
![{\displaystyle u={\frac {1}{2)){\frac {B^{2)){\mu _{0))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/e67fb7bb3123fd60804ed1f0a2f1c8a51e480db9)
More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates
and the magnetization
, 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/635ff1ed2c3b9a5c490bb38f6f427cbcf038bce9)
where the integral is evaluated over the entire region where the magnetic field exists.[1]
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f3283bbb7c86b86b99b572e228e4c73aafcb911)
where
is the current density field and
is the magnetic vector potential. This is analogous to the electrostatic energy expression
; note that neither of these static expressions apply in the case of time-varying charge or current distributions.[2]