Magnetic energy

Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet or magnetic moment formula_1 in a magnetic field formula_2 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: 
while the energy stored in an inductor (of inductance formula_4) when a current formula_5 flows through it is given by:
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 formula_7 containing magnetic field formula_2 is:
More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates formula_2 and formula_11, 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 formula_14 is the current density field and formula_15 is the magnetic vector potential. This is analogous to the electrostatic energy expression formula_16; note that neither of these static expressions do apply in the case of time-varying charge or current distributions.