Magnetization currents
In this chapter we will discuss some materials in which the net effect of the magnetic moments in the material is much greater than in the case of paramagnetism or diamagnetism. The phenomenon is called ferromagnetism. In paramagnetic and diamagnetic materials the induced magnetic moments are usually so weak that we don’t have to worry about the additional fields produced by the magnetic moments. For ferromagnetic materials, however, the magnetic moments induced by applied magnetic fields are quite enormous and have a great effect on the fields themselves. In fact, the induced moments are so strong that they are often the dominant effect in producing the observed fields. So one of the things we will have to worry about is the mathematical theory of large induced magnetic moments. That is, of course, just a technical question. The real problem is, why are the magnetic moments so strong—how does it all work? We will come to that question in a little while. Finding the magnetic fields of ferromagnetic materials is something like the problem of finding the electrostatic field in the presence of dielectrics. You will remember that we first described the internal properties of a dielectric in terms of a vector field $\FLPP$, the dipole moment per unit volume. Then we figured out that the effects of this polarization are equivalent to a charge density $\rho_{\text{pol}}$ given by the divergence of $\FLPP$: \begin{equation} \label{Eq:II:36:1} \rho_{\text{pol}}=-\FLPdiv{\FLPP}. \end{equation}  The total charge in any situation can be written as the sum of this polarization charge plus all other charges, whose density we write1 $\rho_{\text{other}}$. Then the Maxwell equation which relates the divergence of $\FLPE$ to the charge density becomes \begin{equation*} \FLPdiv{\FLPE}=\frac{\rho}{\epsO}= \frac{\rho_{\text{pol}}+\rho_{\text{other}}}{\epsO}, \end{equation*}  or \begin{equation*} \FLPdiv{\FLPE}=-\frac{\FLPdiv{\FLPP}}{\epsO}+ \frac{\rho_{\text{other}}}{\epsO}. \end{equation*}  We can then pull out the polarization part of the charge and put it on the other side of the equation, to get the new law \begin{equation} \label{Eq:II:36:2} \FLPdiv{(\epsO\FLPE+\FLPP)}=\rho_{\text{other}}. \end{equation}  The new law says the divergence of the quantity $(\epsO\FLPE+\FLPP)$ is equal to the density of the other charges. Pulling $\FLPE$ and $\FLPP$ together as in Eq. (36.2), of course, is useful only if we know some relation between them. We have seen that the theory which relates the induced electric dipole moment to the field was a relatively complicated business and can really only be applied to certain simple situations, and even then as an approximation. We would like to remind you of one of the approximate ideas we used. To find the induced dipole moment of an atom inside a dielectric, it is necessary to know the electric field that acts on an individual atom. We made the approximation—which is not too bad in many cases—that the field on the atom is the same as it would be at the center of the small hole which would be left if we took out the atom (keeping the dipole moments of all the neighboring atoms the same). You will also remember that the electric field in a hole in a polarized dielectric depends on the shape of the hole. We summarize our earlier results in Fig. 36–1. For a thin, disc-shaped hole perpendicular to the polarization, the electric field in the hole is given by \begin{equation*} \FLPE_{\text{hole}}=\FLPE_{\text{dielectric}}+\frac{\FLPP}{\epsO}, \end{equation*}  which we showed by using Gauss’ law. On the other hand, in a needle-shaped slot parallel to the polarization, we showed—by using the fact that the curl of $\FLPE$ is zero—that the electric fields inside and outside of the slot are the same. Finally, we found that for a spherical hole the electric field was one-third of the way between the field of the slot and the field of the disc: \begin{equation} \label{Eq:II:36:3} \FLPE_{\text{hole}}=\FLPE_{\text{dielectric}}+\frac{1}{3}\, \frac{\FLPP}{\epsO}\:(\text{spherical hole}). \end{equation}  This was the field we used in thinking about what happens to an atom inside a polarized dielectric. Now we have to discuss the analog of all this for the case of magnetism. One simple, short-cut way of doing this is to say the $\FLPM$, the magnetic moment per unit volume, is just like $\FLPP$, the electric dipole moment per unit volume, and that, therefore, the negative of the divergence of $\FLPM$ is equivalent to a “magnetic charge density” $\rho_m$—whatever that may mean. The trouble is, of course, that there isn’t any such thing as a “magnetic charge” in the physical world. As we know, the divergence of $\FLPB$ is always zero. But that does not stop us from making an artificial analog and writing \begin{equation} \label{Eq:II:36:4} \FLPdiv{\FLPM}=-\rho_m, \end{equation}  where it is to be understood that $\rho_m$ is purely mathematical. Then we could make a complete analogy with the electrostatic case and use all our old equations from electrostatics. People have often done something like that. In fact, historically, people even believed that the analogy was right. They believed that the quantity $\rho_m$ represented the density of “magnetic poles.” These days, however, we know that the magnetization of materials comes from circulating currents within the atoms—either from the spinning electrons or from the motion of the electrons in the atom. It is therefore nicer from a physical point of view to describe things realistically in terms of the atomic currents, rather than in terms of a density of some mythical “magnetic poles.” Incidentally, these currents are sometimes called “Ampèrian” currents, because Ampère first suggested that the magnetism of matter came from circulating atomic currents.  The actual microscopic current density in magnetized matter is, of course, very complicated. Its value depends on where you look in the atom—it’s large in some places and small in others; it goes one way in one part of the atom and the opposite way in another part (just as the microscopic electric field varies enormously inside a dielectric). In many practical problems, however, we are interested only in the fields outside of the matter or in the average magnetic field inside of the matter—where we mean an average taken over many, many atoms. It is only for such macroscopic problems that it is convenient to describe the magnetic state of the matter in terms of $\FLPM$, the average dipole moment per unit volume. What we want to show now is that the atomic currents of magnetized matter can give rise to certain large-scale currents which are related to $\FLPM$.  What we are going to do, then, is to separate the current density $\FLPj$—which is the real source of the magnetic fields—into various parts: one part to describe the circulating currents of the atomic magnets, and the other parts to describe what other currents there may be. It is usually most convenient to separate the currents into three parts. In Chapter 32 we made a distinction between the currents which flow freely on conductors and the ones which are due to the back and forth motions of the bound charges in dielectrics. In Section 32–2 we wrote \begin{equation*} \FLPj=\FLPj_{\text{pol}}+\FLPj_{\text{other}}, \end{equation*}  where $\FLPj_{\text{pol}}$ represented the currents from the motion of the bound charges in dielectrics and $\FLPj_{\text{other}}$ took care of all other currents. Now we want to go further. We want to separate $\FLPj_{\text{other}}$ into one part, $\FLPj_{\text{mag}}$, which describes the average currents inside of magnetized materials, and an additional term which we can call $\FLPj_{\text{cond}}$ for whatever is left over. The last term will generally refer to currents in conductors, but it may also include other currents—for example the currents from charges moving freely through empty space. So we will write for the total current density: \begin{equation} \label{Eq:II:36:5} \FLPj=\FLPj_{\text{pol}}+\FLPj_{\text{mag}}+\FLPj_{\text{cond}}. \end{equation}  Of course it is this total current which belongs in the Maxwell equation for the curl of $\FLPB$: \begin{equation} \label{Eq:II:36:6} c^2\FLPcurl{\FLPB}=\frac{\FLPj}{\epsO}+\ddp{\FLPE}{t}. \end{equation}   Now we have to relate the current $\FLPj_{\text{mag}}$ to the magnetization vector $\FLPM$. So that you can see where we are going, we will tell you that the result is going to be that \begin{equation} \label{Eq:II:36:7} \FLPj_{\text{mag}}=\FLPcurl{\FLPM}. \end{equation}  If we are given the magnetization vector $\FLPM$ everywhere in a magnetic material, the circulation current density is given by the curl of $\FLPM$. Let’s see if we can understand why this is so. First, let’s take the case of a cylindrical rod which has a uniform magnetization parallel to its axis. Physically, we know that such a uniform magnetization really means a uniform density of atomic circulating currents everywhere inside the material. Suppose we try to imagine what the actual currents would look like in a cross section of the material. We would expect to see currents something like those shown in Fig. 36–2. Each atomic current goes around and around in a little circle, with all the circulating currents going around in the same direction. Now what is the effective current of such a thing? Well, in most of the bar there is no effect at all, because right next to each current there is another current going in the opposite direction. If we imagine a small surface—but one still quite a bit larger than a single atom—such as is indicated in Fig. 36–2 by the line $\overline{AB}$, the net current through such a surface is zero. There is no net current anywhere inside the material. Note, however, that at the surface of the material there are atomic currents which are not cancelled by neighboring currents going the other way. At the surface there is a net current always going in the same direction around the rod. Now you see why we said earlier that a uniformly magnetized rod is equivalent to a long solenoid carrying an electric current.  How does this view fit with Eq. (36.7)? First, inside the material the magnetization $\FLPM$ is constant, so all its derivatives are zero. This agrees with our geometric picture. At the surface, however, $\FLPM$ is not really constant—it is constant up to the edge and then suddenly collapses to zero. So, right at the surface there are terrific gradients which, according to (36.7), will give a high current density. Suppose we look at what happens near the point $C$ in Fig. 36–2. Taking the $x$- and $y$-directions as in the figure, the magnetization $\FLPM$ is in the $z$-direction. Writing out the components of Eq. (36.7), we have \begin{equation} \begin{aligned} \ddp{M_z}{y}&=(j_{\text{mag}})_x,\\[1ex] -\ddp{M_z}{x}&=(j_{\text{mag}})_y. \end{aligned} \label{Eq:II:36:8} \end{equation}  At the point $C$, the derivative $\ddpl{M_z}{y}$ is zero, but $\ddpl{M_z}{x}$ is large and positive. Equation (36.7) says that there is a large current density in the minus $y$-direction. This agrees with our picture of a surface current going around the bar. Now we want to find the current density for a more complicated case in which the magnetization varies from point to point in a material. It is easy to see qualitatively that if the magnetization is different in two neighboring regions, there will not be a perfect cancellation of the circulating currents so that there will be a net current in the volume of the material. It is this effect that we want to work out quantitatively.  First, we need to recall the results of Section 14–5 that a circulating current $I$ has a magnetic moment $\mu$ given by \begin{equation} \label{Eq:II:36:9} \mu=IA, \end{equation}  where $A$ is the area of the current loop (see Fig. 36–3). Now let’s consider a small rectangular block inside of a magnetized material, as sketched in Fig. 36–4. We take the block so small that we can consider that the magnetization is uniform inside it. If this block has a magnetization $M_z$ in the $z$-direction, the net effect will be the same as a surface current going around on the vertical faces, as shown. We can find the magnitude of these currents from Eq. (36.9). The total magnetic moment of the block is equal to the magnetization times the volume: \begin{equation*} \mu=M_z(abc), \end{equation*}  from which we get (remembering that the area of the loop is $ac$) \begin{equation*} I=M_zb. \end{equation*}  In other words, the current per unit length (vertically) on each of the vertical surfaces is equal to $M_z$. Now suppose that we imagine two such little blocks next to each other, as shown in Fig. 36–5. Because block $2$ is slightly displaced from block $1$, it will have a slightly different vertical component of magnetization, which we call $M_z+\Delta M_z$. Now on the surface between the two blocks there will be two contributions to the total current. Block $1$ will produce a current $I_1$ flowing in the positive $y$-direction, and block $2$ will produce a surface current $I_2$ flowing in the negative $y$-direction. The total surface current in the positive $y$-direction is the sum: \begin{align*} I&=I_1-I_2=M_zb-(M_z+\Delta M_z)b\\[1ex] &=-\Delta M_zb. \end{align*}  We can write $\Delta M_z$, as the derivative of $M_z$ in the $x$-direction times the displacement from block $1$ to block $2$, which is just $a$: \begin{equation*} \Delta M_z=\ddp{M_z}{x}\,a. \end{equation*}  The current flowing between the two blocks is then \begin{equation*} I=-\ddp{M_z}{x}\,ab. \end{equation*}  To relate the current $I$ to an average volume current density $\FLPj$, we must realize that this current $I$ is really spread over a certain cross-sectional area. If we imagine the whole volume of the material to be filled with such little blocks, one such side face (perpendicular to the $x$-axis) can be associated with each block.2 Then we see that the area to be associated with the current $I$ is just the area $ab$ of one of the front faces. We get the result \begin{equation*} j_y=\frac{I}{ab}=-\ddp{M_z}{x}. \end{equation*}  We have at least the beginning of the curl of $\FLPM$. There should be another term in $j_y$ from the variation of the $x$-component of the magnetization with $z$. This contribution to $\FLPj$ will come from the surface between two little blocks stacked one on top of the other, as shown in Fig. 36–6. Using the same arguments we have just made, you can show that this surface will contribute to $j_y$ the amount $\ddpl{M_x}{z}$. These are the only surfaces which can contribute to the $y$-component of the current so we have that the total current density in the $y$-direction is \begin{equation*} j_y=\ddp{M_x}{z}-\ddp{M_z}{x}. \end{equation*}  Working out the currents on the remaining faces of a cube—or using the fact that our $z$-direction is completely arbitrary—we can conclude that the vector current density is indeed given by the equation \begin{equation*} \FLPj=\FLPcurl{\FLPM}. \end{equation*}  So if we choose to describe the magnetic situation in matter in terms of the average magnetic moment per unit volume $\FLPM$, we find that the circulating atomic currents are equivalent to an average current density in matter given by Eq. (36.7). If the material is also a dielectric, there may be, in addition, a polarization current $\FLPj_{\text{pol}}=\ddpl{\FLPP}{t}$. And if the material is also a conductor, we may have a conduction current $\FLPj_{\text{cond}}$ as well. We can write the total current as \begin{equation} \label{Eq:II:36:10} \FLPj=\FLPj_{\text{cond}}+\FLPcurl{\FLPM}+\ddp{\FLPP}{t}. \end{equation} 