Magnetic Permeability and Susceptibility

When a material is placed inside a magnetising field, described by the field intensity vector $\vec{H}$ (SI unit ampere/metre, A/m -- distinct from $\vec{B}$, which includes the material's own response), the material develops some degree of magnetisation and modifies the total field inside it. Two related quantities measure how strongly a given material responds.

Magnetic permeability $\mu$. Defined as the ratio of the resulting magnetic flux density $B$ inside the material to the magnetising field intensity $H$ that produced it:

with SI unit tesla-metre/ampere (T$\cdot$m/A), equivalent to henry/metre (H/m). For vacuum (equivalently, air, to a very good approximation), this ratio has the fixed value $\mu_0 = 4\pi\times10^{-7}\ \text{T}\cdot\text{m/A}$, the permeability of free space -- the magnetic counterpart of $\epsilon_0$ in electrostatics. The relative permeability of a material,

is a dimensionless number comparing the material's permeability to that of vacuum: $\mu_r=1$ for vacuum exactly, close to 1 (slightly below or above) for diamagnetic and paramagnetic materials, and very much greater than 1 for ferromagnetic materials (Section 1.14).

Magnetic susceptibility $\chi$. Defined as the ratio of the intensity of magnetisation $M$ (Section 1.10) that the material acquires to the magnetising field intensity $H$ that produced it:

a dimensionless number (since both $M$ and $H$ carry the same unit, A/m), positive for materials whose magnetisation reinforces the applied field and negative for materials whose magnetisation opposes it.

The relation $\mu_r=1+\chi$. The total flux density inside the material has two contributions: the field that would exist in vacuum, $\mu_0H$, plus the extra field due to the material's own magnetisation, $\mu_0M$:

$B = \mu_0(H+M)$ …