Magnetization and Magnetic Intensity

Section 11.3.1 showed that atoms with unpaired electrons carry a net magnetic dipole moment. A bulk material is made of an enormous number of such atoms, each contributing its own moment; but ordinarily these atomic moments point in random directions, so the material's net moment is zero. In some materials (Fe$_3$O$_4$ is one example given in the text), however, the vector sum of all these atomic moments is genuinely nonzero -- such materials are said to have a net magnetic moment, and the ratio of this net moment to the material's volume is called its magnetization: $M=\dfrac{m_{net}}{V}$. Magnetization is a vector quantity, with SI unit A m$^{-1}$ (the same unit as the intensity H introduced below).

To connect magnetization to a measurable field, consider a rod of such a material with some net magnetization, placed inside a solenoid of n turns per unit length carrying current I. In the absence of the rod, the solenoid alone produces a field $B_0=\mu_0nI$. With the material rod present, its own magnetization contributes an additional field inside the solenoid, $B_m=\mu_0M$ (observed to be directly proportional to M, with $\mu_0$, the permeability of free space, as the constant of proportionality), so the total field becomes $B=B_0+B_m=\mu_0(nI+M)$.

To separate the field's dependence on the SOLENOID (via its current and turns) from its dependence on the MATERIAL placed inside, a new quantity, the magnetic field intensity, is defined as $H=nI$ -- notice H depends only on the solenoid's current and geometry, not at all on what material (if any) is placed inside it. In terms of H, the total field becomes $B=\mu_0(H+M)$, so H and M share the same unit and dimensions ([L$^{-1}$A], i.e. A m$^{-1}$), and the field actually produced inside the material (B) depends on both.

When H is not too strong, the magnetization M induced in a material is found to be directly proportional to H: $M=\chi H$, where the dimensionless constant $\chi$ is called the magnetic susceptibility -- a measure of how strongly, and in which sense, a given material responds magnetically to an external field (it is the ratio of two quantities sharing the same unit, A m$^{-1}$, and so is itself unitless). Substituting $M=\chi H$ into $B=\mu_0(H+M)$ gives $B=\mu_0(1+\chi)H$. Defining the relative permeability $\mu_r=1+\chi$ and the (absolute) permeability $\mu=\mu_0\mu_r$, this can be written compactly as $B=\mu H$. …