Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet or magnetic moment in a magnetic field 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:
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:
More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates and , 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.