The magnetization curve
Now we will look at some simple situations in which the magnetic field is constant, or in which the fields change slowly enough that we can neglect $\ddpl{\FLPD}{t}$ in comparison with $\FLPj_{\text{cond}}$. Then the fields obey the equations \begin{gather} \label{Eq:II:36:16} \FLPdiv{\FLPB}=0,\\[1.5ex] \label{Eq:II:36:17} \FLPcurl{\FLPH}=\FLPj_{\text{cond}}/\epsO c^2,\\[1.5ex] \label{Eq:II:36:18} \FLPH=\FLPB-\FLPM/\epsO c^2. \end{gather}  Suppose we have a torus (a donut) of iron wrapped with a coil of copper wire, as shown in Fig. 36–7(a). A current $I$ flows in the wire. What is the magnetic field? The magnetic field will be mainly inside the iron; there, the lines of $\FLPB$ will be circles, as drawn in Fig. 36–7(b). Since the flux of $\FLPB$ is continuous, its divergence is zero, and Eq. (36.16) is satisfied. Next, we write Eq. (36.17) in another form by integrating around the closed loop $\Gamma$ drawn in Fig. 36–7(b). From Stokes’ theorem, we have that \begin{equation} \label{Eq:II:36:19} \oint_\Gamma\FLPH\cdot d\FLPs=\frac{1}{\epsO c^2} \int_S\FLPj_{\text{cond}}\cdot\FLPn\,da, \end{equation}  where the integral of $\FLPj$ is to be carried out over any surface $S$ bounded by $\Gamma$. This surface is cut once by each turn of the winding. Each turn contributes the current $I$ to the integral, and, if there are $N$ turns in all, the integral is $NI$. From the symmetry of our problem, $\FLPB$ is the same all around the curve $\Gamma$; if we assume that the magnetization, and therefore, the field $H$ is also constant along $\Gamma$, Eq. (36.19) becomes \begin{equation*} Hl=\frac{NI}{\epsO c^2}, \end{equation*}  where $l$ is the length of the curve $\Gamma$. So, \begin{equation} \label{Eq:II:36:20} H=\frac{1}{\epsO c^2}\,\frac{NI}{l}. \end{equation}  It is because $\FLPH$ is directly proportional to the magnetizing current in cases like this one that $\FLPH$ is sometimes called the magnetizing field. Now all we need is an equation which relates $\FLPH$ to $\FLPB$. But there isn’t any such equation! There is, of course, Eq. (36.18), but it is no help because there is no direct relation between $\FLPM$ and $\FLPB$ for a ferromagnetic material like iron. The magnetization $\FLPM$ depends on the whole past history of the iron, and not only on what $\FLPB$ is at the moment. All is not lost, though. We can get solutions in certain simple cases. If we start out with unmagnetized iron—let’s say with iron that has been annealed at high temperatures—then in the simple geometry of the torus, all the iron will have the same magnetic history. Then we can say something about $\FLPM$—and therefore about the relation between $\FLPB$ and $\FLPH$—from experimental measurements. The field $\FLPH$ in the torus is, from Eq. (36.20), given as a constant times the current $I$ in the winding. The field $\FLPB$ can be measured by integrating over time the emf in the coil (or in an extra coil wound over the magnetizing coil shown in the figure). This emf is equal to the rate of change of the flux of $\FLPB$, so the integral of the emf with time is equal to $\FLPB$ times the cross-sectional area of the torus. Figure 36–8 shows the relation between $\FLPB$ and $\FLPH$, observed with a torus of soft iron. When the current is first turned on, $\FLPB$ increases with increasing $\FLPH$ along the curve $a$. Note the different scales on $\FLPB$ and $\FLPH$; initially, it takes only a relatively small $\FLPH$ to make a large $\FLPB$. Why is $\FLPB$ so much larger with the iron than it would be with air? Because there is a large magnetization $\FLPM$ which is equivalent to a large surface current on the iron—the field $\FLPB$ comes from the sum of this current and the conduction current in the winding. Why $\FLPM$ should be so large, we will discuss later.  At higher values of $\FLPH$, the magnetization curve levels off. We say that the iron saturates. With the scales of our figure, the curve appears to become horizontal. Actually, it continues to rise slightly—for large fields, $\FLPB$ becomes proportional to $\FLPH$, and with a unit slope. There is no further increase of $\FLPM$. Incidentally, we should point out that if the torus were made of some nonmagnetic material, $\FLPM$ would be zero and $\FLPB$ would equal $\FLPH$ for all fields. The first thing we notice is that curve $a$ in Fig. 36–8—which is the so-called magnetization curve—is highly nonlinear. But it’s worse than that. If, after reaching saturation, we decrease the current in the coil to bring $\FLPH$ back to zero, the magnetic field $\FLPB$ falls along curve $b$. When $\FLPH$ reaches zero, there is still some $\FLPB$ left. Even with no magnetizing current there is a magnetic field in the iron—it has become permanently magnetized. If we now turn on a negative current in the coil, the $B$-$H$ curve continues along $b$ until the iron is saturated in the negative direction.  If we then bring the current back to zero again, $\FLPB$ goes along curve $c$. If we alternate the current between large positive and negative values, the $B$-$H$ curve goes back and forth along very nearly the curves $b$ and $c$. If we vary $\FLPH$ in some arbitrary way, however, we can get more complicated curves which will, in general, lie somewhere between the curves $b$ and $c$. The loop made by repeated oscillation of the fields is called a hysteresis loop of the iron.  We see then that we cannot write a functional relationship like $B=f(H)$, because the value of $\FLPB$ at any instant depends not only on what $\FLPH$ is at that time, but on its whole past history. Naturally, the magnetization and hysteresis curves are different for different substances. The shape of the curves depends critically on the chemical composition of the material, and also on the details of its preparation and subsequent physical treatment. We will discuss some of the physical explanations for these complications in the next chapter.