Magnetic susceptibility and permeability

In a large class of materials, there exists an approximately linear relationship between ${\bf M}$ and ${\bf H}$. If the material is isotropic then

$${\bf M} = \chi_m {\bf H},$$ (866)

where $\chi_m$ is called the magnetic susceptibility. If $\chi_m$ is positive then the material is called paramagnetic, and the magnetic field is strengthened by the presence of the material. On the other hand, if $\chi_m$ is negative then the material is diamagnetic, and the magnetic field is weakened in the presence of the material. The magnetic susceptibilities of paramagnetic and diamagnetic materials are generally extremely small. A few sample values are given in Table. 2.¹

Table 2:

Material	$\chi_m$
Aluminium	$2.3\times 10^{-5}$
Copper	$-0.98\times 10^{-5}$
Diamond	$-2.2\times 10^{-5}$
Tungsten	$6.8\times 10^{-5}$
Hydrogen (1 atm)	$-0.21\times 10^{-8}$
Oxygen (1 atm)	$209.0\times 10^{-8}$
Nitrogen (1 atm)	$-0.50\times 10^{-8}$

A linear relationship between ${\bf M}$ and ${\bf H}$ also implies a linear relationship between ${\bf B}$ and ${\bf H}$. In fact, we can write

$${\bf B} = \mu {\bf H},$$ (867)

where

$$\mu = \mu_0 (1+ \chi_m)$$ (868)

is termed the magnetic permeability of the material in question. (Likewise, $\mu_0$ is termed the permeability of free space.) Note that $\mu$ has the same units as $\mu_0$. It is clear from Table 2 that the permeabilities of common diamagnetic and paramagnetic materials do not differ substantially from the permeability of free space. In fact, to all intents and purposes, the magnetic properties of such materials can be safely neglected (i.e., $\mu =\mu_0$).