The magnetic field is a vector field that describes magnetic effects, and is defined as the field that correctly predicts the Lorentz force on a particle with electric charge :
If an object is placed in a magnetic field , and wants to rotate to align itself with the field, then its magnetic dipole moment is defined from the aligning torque :
Where has units of . From this, the magnetization is defined as follows, and roughly represents the moments per unit volume:
If has the same magnitude and orientation throughout the body, then , where is the volume. Therefore, has units of .
A nonzero complicates things, since it contributes to the field and hence modifies . We thus define the “free” auxiliary field from the “bound” field and the “net” field :
Where the magnetic permeability of free space is a known constant. It is important to point out some inconsistencies here: contains a factor of , and thus measures flux density, while and do not contain , and therefore measure field intensity. Note that this convention is the opposite of the analogous electric fields , and . Also note that has the opposite sign convention of .
Some objects, called ferromagnets or permanent magnets, have an inherently nonzero . Others objects, when placed in a -field, may instead gain an induced .
When is induced, its magnitude is usually proportional to the applied field strength :
Where is the volume magnetic susceptibility, and and are the relative permeability and absolute permeability of the medium, respectively. Materials with intrinsic magnetization, i.e. ferromagnets, do not have a well-defined .
If , the medium is paramagnetic, meaning it strengthens the net field . Otherwise, if , the medium is diamagnetic, meaning it counteracts the applied field .
For , as is often the case, the magnetization can be approximated by: