Magnetic energy

The potential magnetic energy of a magnet or magnetic moment $$\mathbf{m}$$ in a magnetic field $$\mathbf{B}$$ 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: $$E_\text{p,m} = -\mathbf{m} \cdot \mathbf{B}$$ while the energy stored in an inductor (of inductance $$L$$) when a current $$I$$ flows through it is given by: $$E_\text{p,m} = \frac{1}{2} LI^2.$$ 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 $$\mu _0$$ containing magnetic field $$\mathbf{B}$$ is: $$u = \frac{1}{2} \frac{B^2}{\mu_0}$$

More generally, if we assume that the medium is paramagnetic or diamagnetic so that a linear constitutive equation exists that relates $$\mathbf{B}$$ and the magnetization $$\mathbf{H}$$, then it can be shown that the magnetic field stores an energy of $$E = \frac{1}{2} \int \mathbf{H} \cdot \mathbf{B} \, \mathrm{d}V$$ 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: $$E = \frac{1}{2} \int \mathbf{J} \cdot \mathbf{A}\, \mathrm{d}V$$ where $$\mathbf{J}$$ is the current density field and $$\mathbf{A}$$ is the magnetic vector potential. This is analogous to the electrostatic energy expression $\frac{1}{2}\int \rho \phi \, \mathrm{d}V$ ; note that neither of these static expressions apply in the case of time-varying charge or current distributions.