Magnetic energy
The potential magnetic energy of a magnet or magnetic moment in a magnetic field is defined as the work of the magnetic force on the re-alignment of the vector of the magnetic [dipole moment] and is equal to: The work is done by a torque :
which will act to "realign" the magnetic dipole with the magnetic field.
In an electronic circuit the energy stored in an inductor when a current flows through it is given by:
This expression forms the basis for superconducting magnetic energy storage. It can be derived from a time average of the product of current and voltage across an inductor.
Energy is also stored in a magnetic field itself. The energy per unit volume in a region of free space with vacuum permeability containing magnetic field is:
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
where the integral is evaluated over the entire region where the magnetic field exists.
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:
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.