Zeeman energy

Zeeman energy, or the external field energy, is the potential energy of a magnetised body in an external magnetic field. It is named after the Dutch physicist Pieter Zeeman, primarily known by the Zeeman effect. In SI units, it is given by

E_{Zeeman} =-\mu_{0} \int_V\,\textbf M\cdot \textbf H_{Ext} \, \mathrm dV

where H_Ext is the external field, M the local magnetisation, and the integral is done over the volume of the body. This is the statistical average (over a unit volume macroscopic sample) of a corresponding microscopic Hamiltonial (energy) for each individual magnetic moment m, which is however experiencing a local induction B:

H =-\textbf m \cdot \textbf B

References

d'Aquino, M, Phd thesis, http://wpage.unina.it/mdaquino/PhD_thesis/main/node16.html
Hubert, A. and Schäfer, R. Magnetic domains: the analysis of magnetic microstructures, Springer-Verlag, 1998