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There is, however, a third class of magnetic materials called ferromagnetic materials. Such materials are characterized by a possible permanent magnetization, and generally have a profound effect on magnetic fields (i.e., $\mu/\mu_0\gg 1$). Unfortunately, ferromagnetic materials do not exhibit a linear dependence between ${\bfm M}$ and ${\bfm H}$ or ${\bfm B}$ and ${\bfm H}$, so that we cannot employ Eqs. (3.124) and (3.125) with constant values of $\chi_m$ and $\mu$. It is still expedient to use Eq. (3.125) as the definition of $\mu$, with $\mu = \mu({\bfm H})$, however this practice can lead to difficulties under certain circumstances. The permeability of a ferromagnetic material, as defined by Eq. (3.125), can vary through the entire range of possible values from zero to infinity, and may be either positive or negative. The most sensible approach is to consider each problem involving ferromagnetic materials separately, try to determine which region of the ${\bfm B}$-${\bfm H}$ diagram is important for the particular case in hand, and then make approximations appropriate to this region.

Figure 2: Magnetization curve and relative permeability of commercial iron (annealed)
\begin{figure}\epsfysize =3in

First, let us consider an unmagnetized sample of ferromagnetic material. If the magnetic intensity, which is initially zero, is increased monotonically, then the ${\bfm B}$-${\bfm H}$ relationship traces out a curve such as that shown in Fig. 2. This is called a magnetization curve. It is evident that the permeabilities $\mu$ derived from the curve (according to the rule $\mu = B/H$) are always positive, and show a wide range of values. The maximum permeability occurs at the ``knee'' of the curve. In some materials this maximum permeability is as large as $10^5\,\mu_0$. The reason for the knee in the curve is that the magnetization ${\bfm M}$ reaches a maximum value in the material, so that

{\bfm B} = \mu_0({\bfm H} + {\bfm M})
\end{displaymath} (548)

continues to increase at large ${\bfm H}$ only because of the $\mu_0{\bfm H}$ term. The maximum value of ${\bfm M}$ is called the saturation magnetization of the material.

Next, consider a ferromagnetic sample magnetized by the above procedure. If the magnetic intensity ${\bfm H}$ is decreased, the ${\bfm B}$-${\bfm H}$ relation does not follow back down the curve of Fig. 2, but instead moves along a new curve, shown in Fig. 3, to the point $R$. The magnetization, once established, does not disappear with the removal of ${\bfm H}$. In fact, it takes a reversed magnetic intensity to reduce the magnetization to zero. If ${\bfm H}$ continues to build up in the reversed direction, then ${\bfm M}$ (and hence ${\bfm B}$) becomes increasingly negative. Finally, when ${\bfm H}$ increases again the operating point follows the lower curve of Fig. 3. Thus, the ${\bfm B}$-${\bfm H}$ curve for increasing ${\bfm H}$ is quite different to that for decreasing ${\bfm H}$. This phenomenon is known as hysteresis.

Figure 3: Typical hysteresis loop of a ferromagnetic material
\begin{figure}\epsfysize =2.5in

The curve of Fig. 3 is called the hysteresis loop of the material in question. The value of ${\bfm B}$ at the point $R$ is called the retentivity or remanence. The magnitude of ${\bfm H}$ at the point $C$ is called the coercivity. It is evident that $\mu$ is negative in the second and fourth quadrants of the diagram and positive in the first and third quadrants. The shape of the hysteresis loop depends not only on the nature of the ferromagnetic material but also on the maximum value of ${\bfm H}$ to which the material is subjected. However, once this maximum value, ${\bfm H}_{\rm max}$, becomes sufficient to produce saturation in the material the hysteresis loop does not change shape with any further increase in ${\bfm H}_{\rm max}$.

Ferromagnetic materials are used either to channel magnetic flux (e.g., around transformer circuits) or as sources of magnetic field (permanent magnets). For use as a permanent magnet, the material is first magnetized by placing it in a strong magnetic field. However, once the magnet is removed from the external field it is subject to a demagnetizing ${\bfm H}$. Thus, it is vitally important that a permanent magnet should possess both a large remanence and a large coercivity. As will become clear later on, it is generally a good idea for the ferromagnetic materials used to channel magnetic flux around transformer circuits to possess small remanences and small coercivities.

next up previous
Next: Boundary conditions for and Up: The effect of dielectric Previous: Magnetic susceptibility and permeability
Richard Fitzpatrick 2002-05-18