Magnetization

All matter is built up out of atoms, and each atom consists of electrons in motion. The currents associated with this motion are termed atomic currents. Each atomic current is a tiny closed circuit of atomic dimensions, and may therefore be appropriately described as a magnetic dipole. If the atomic currents of a given atom all flow in the same plane then the atomic dipole moment is directed normal to the plane (in the sense given by the right-hand rule), and its magnitude is the product of the total circulating current and the area of the current loop. More generally, if ${\bf j}({\bf r})$ is the atomic current density at the point ${\bf r}$ then the magnetic moment of the atom is

$${\bf m} = \frac{1}{2}\int {\bf r}\times{\bf j} d^3{\bf r},$$ (856)

where the integral is over the volume of the atom. If there are $N$ such atoms or molecules per unit volume then the magnetization ${\bf M}$ (i.e., the magnetic dipole moment per unit volume) is given by ${\bf M} = N {\bf m}$. More generally,

$${\bf M} ({\bf r}) = \sum_i N_i \langle {\bf m}_i\rangle,$$ (857)

where $\langle {\bf m}_i\rangle$ is the average magnetic dipole moment of the $i$th type of molecule in the vicinity of point ${\bf r}$, and $N_i$ is the average number of such molecules per unit volume at ${\bf r}$.

Consider a general medium which is made up of molecules which are polarizable and possess a net magnetic moment. It is easily demonstrated that any circulation in the magnetization field ${\bf M}({\bf r})$ gives rise to an effective current density ${\bf j}_m$ in the medium. In fact,

$${\bf j}_m = \nabla\times{\bf M}.$$ (858)

This current density is called the magnetization current density, and is usually distinguished from the true current density, ${\bf j}_t$, which represents the convection of free charges in the medium. In fact, there is a third type of current called a polarization current, which is due to the apparent convection of bound charges. It is easily demonstrated that the polarization current density, ${\bf j}_p$, is given by

$${\bf j}_p = \frac{\partial{\bf P}}{\partial t}.$$ (859)

Thus, the total current density, ${\bf j}$, in the medium takes the form

$${\bf j} = {\bf j}_t + \nabla\times{\bf M} + \frac{\partial{\bf P}}{\partial t}.$$ (860)

It must be emphasized that all terms on the right-hand side of the above equation represent real physical currents, although only the first term is due to the motion of real charges (over more than atomic dimensions).

The differential form of Ampère's law is

$$\nabla\times{\bf B} = \mu_0 {\bf j} + \mu_0\epsilon_0 \frac{\partial {\bf E}} {\partial t},$$ (861)

which can also be written

$$\nabla\times{\bf B} = \mu_0 {\bf j}_t +\mu_0\nabla\times{\bf M} + \mu_0\frac{\partial{\bf D}}{\partial t},$$ (862)

where use has been made of the definition ${\bf D} = \epsilon_0 {\bf E} + {\bf P}$. The above expression can be rearranged to give

$$\nabla\times{\bf H} = {\bf j}_t + \frac{\partial{\bf D}}{\partial t},$$ (863)

where

$${\bf H} = \frac{\bf B}{\mu_0} - {\bf M}$$ (864)

is termed the magnetic intensity, and has the same dimensions as ${\bf M}$ (i.e., magnetic dipole moment per unit volume). In a steady-state situation, Stokes' theorem tell us that

$$\oint_C {\bf H}\!\cdot\!d{\bf l} = \int_S {\bf j}_t\!\cdot\!d{\bf S}.$$ (865)

In other words, the line integral of ${\bf H}$ around some closed curve is equal to the flux of true current through any surface attached to that curve. Unlike the magnetic field ${\bf B}$ (which specifies the force $q {\bf v}\times {\bf B}$ acting on a charge $q$ moving with velocity ${\bf v}$), or the magnetization ${\bf M}$ (the magnetic dipole moment per unit volume), the magnetic intensity ${\bf H}$ has no clear physical meaning. The only reason for introducing it is that it enables us to calculate fields in the presence of magnetic materials without first having to know the distribution of magnetization currents. However, this is only possible if we possess a constitutive relation connecting ${\bf B}$ and ${\bf H}$.