Electromagnetism – Difference Between Magnetic Susceptibility and Permeability

Question

From what I have read it seems magnetic susceptibility and magnetic permeability both represent the level of a material's magnetic response to an external field.

Fundamentally, what is the difference between them?

Answer 1

The important equation here is the difference between $\mathbf H$ and $\mathbf B$ fields (using boldface for vectors):

$$\mathbf B = \mu_0(\mathbf H +\mathbf M),$$

where $\mathbf M$ is the magnetization of a sample under the applied magnetic field $\mathbf H$, $\mu_0$ is the permeability of vacuum and $\mathbf B$ you can see it as the real magnetic field in the room (both due to $\mathbf H$ and $\mathbf M$). The $\mathbf B$ field is the one that appears in the Maxwell law $\nabla \cdot \mathbf B =0$ (Gauss' law or no monopoles law).

For simplicity let us suppose that it is a 1D problem (or that every vector points in the same direction) so $$B=\mu_0 (H + M), \tag{1}$$ (where italicized $B,H,M$ are the magnitudes of $\mathbf B$, $\mathbf H$ and $\mathbf M$).

The magnetic susceptibility $\chi$ tells you how the magnetization changes when you change $H$: $$\chi=\frac{\mathrm{d} M}{\mathrm d H}.$$

IF and only IF $M$ is proportional to $H$ ($\chi$ is a constant of $H$, that is $M=\chi H$), then you can write equation ($1$) as

$$B=\mu_0(H+\chi H)=\mu H$$

where $\mu=\mu_0(1+\chi)$ is the permeability. This is often the case for paramagnetic/diamagnetic materials under small $H$.

Again when dealing with systems where everything is proportional and $\chi$ is a constant then $\chi$ and $\mu$ measure the same thing (with a difference of 1). In magnetic materials like iron or other ferromagnets this is not necessarily true.

Take away, $\chi$ is the relation between $H$ and $M$, and $\mu$ is the relation between $H$ and $B$ (when everything is proportional).