Curie law

In a paramagnetic material the magnetization of the material is (approximately) directly proportional to an applied magnetic field. However, if the material is heated, this proportionality is reduced: for a fixed value of the field, the magnetization is (approximately) inversely proportional to temperature. This fact is encapsulated by Curie's law:

where

This relation was discovered experimentally (by fitting the results to a correctly guessed model) by Pierre Curie. It only holds for high temperatures, or weak magnetic fields. As the derivations below show, the magnetization saturates in the opposite limit of low temperatures, or strong fields.

A simple model of a paramagnet concentrates on the particles which compose it which do not interact with each other. Each particle has a magnetic moment given by ${\displaystyle {\vec {\mu }}}$. The energy of a magnetic moment in a magnetic field is given by

To simplify the calculation, we are going to work with a 2-state particle: it may either align its magnetic moment with the magnetic field, or against it. So the only possible values of magnetic moment are then ${\displaystyle \mu }$ and ${\displaystyle -\mu }$. If so, then such a particle has only two possible energies

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