### Magnetic energy and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet of magnetic moment $${\mathbf {m}}$$ in a magnetic field $$\mathbf {B}$$ 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:

$$E_{\rm {p,m}}=-\mathbf {m} \cdot \mathbf {B} }$$

while the energy stored in an inductor (of inductance L) when a current I flows through it is given by:

E $$E_{\rm {p,m}}={\frac {1}{2}}LI^{2}.}$$

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 $$\mathbf{H}$$ , 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.
References

Jackson, John David (1998). Classical Electrodynamics. New York: Wiley.

Magnetic Energy, Richard Fitzpatrick Professor of Physics The University of Texas at Austin.

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