Two field quantities are used to describe the shape of a magnetic field, the
*B-field*
and the
*H-field*.
Traditionally, they are named
*magnetic field*, $\vec{H}$,
and
*magnetic flux density*
or
*magnetic induction*, $\vec{B}$
but this suggests a causal relationship of the two rather than one of interaction, which is more appropriate. The
H-field represents the field produced by a current running in a conductor; it is therefore measured in A/m. The B-field is
measured in T(esla). In vacuum, the two are linked by a universal constant of proportionality, the
*permeability*, $\mu_0$ = 1.26×10^{-6} H/m (=Tm/A),
which takes care of the conversion of units:
$$\vec{B}=\mu_0\vec{H}\qquad.$$

When a material is placed in a magnetic field, the field interacts with the electrons in the material. Like all moving
charges, the electrons are subject to the
*Lorentz force*,
which accelerates (*i.e.* deflects) them at right angles
with both their trajectory and the field. We can envisage a slab of material as consisting of many small cells. Then there will
be a small eddy current in each of those cells because the external field forces the electrons to circulate around it. Since
these
*bound currents*
cancel each other out where neighbouring eddies are in contact, the only macroscopic current is
a surface current around the outer surface of the object.

The H-field is induced only by macroscopic (net) currents, while bound currents are also sources of the B-field. This means that,
in a material, the simple relationship between $\vec{H}$ and $\vec{B}$ breaks down. While the component of the H-field
tangential to a long magnetic rod does not change at its surface (bottom half of figure), the tangential component of the B-field
has a step there, indicated in the figure by a different density of B-field lines inside and outside the material. The graph
shows cross sections of both fields across the rod. The difference between the two fields is the
*magnetisation*, $\vec{M}$:
$$\vec{B}=\mu_0(\vec{H}+\vec{M})$$
Like the H-field, magnetisation is a vector and is measured in A/m. It is the vector sum of the magnetic moments of each atom
in the sample, normalised to the sample volume:
$$\vec{M}=\frac{\sum_i\vec{p}_{m,i}}{V}$$

Since the magnitude of the magnetisation depends on the material, it is logical to restore the link between B-field and H-field
by introducing a material constant which describes to what extent B-field lines are drawn into or out of a material. This is
the (dimensionless)
*relative permeability*, $\mu_{rel}$,
of the material:
$$\vec{B}=\mu_0\mu_{rel}\vec{H}$$
Since relative permeabilities are very close to 1 for many materials, it is often easier to consider the material's
*susceptibility*, $\chi:=\mu_{rel}-1$
instead:
$$\vec{B}=\mu_0(1+\chi)\vec{H}$$

Depending on the value of the susceptibility, materials are grouped into three classes:

*diamagnetics*- negative suseptibility, $\chi\sim -10^{-5}$*paramagnetics*- small positive susceptibility, $\chi\sim +10^{-5}$*ferromagnetics*- very large positive susceptibility, $\chi\sim +10^{+2}$

It is important to note that these different material classes are not identical with the three underlying mechanisms, which will be discussed in the following. All materials are diamagnetic, but in para- and ferromagnetics the para- or ferromagnetic effect overcompensates the diamagnetism. In order to be able to show ferromagnetism, a material must be paramagnetic.

Over the next few sections, we will look at dia- and paramagnetism in more detail, see how conduction electrons in metals affect paramagnetism, and investigate why magnetism in ferromagnets is so much stronger.