What's the difference between magnetic susceptibility and permeability? 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?
 A: 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).
A: There's a different definition for each one,
Magnetic permeability has to do with the property of a magnetic material to have its parts in a aligned magnetization
$$B=\mu H.$$
Magnetic susceptibility have to do with other properties of the material, as refractive index or absorption
$$ M= \chi H.$$
A: I hope my answer is still relevant.
I'm just studying this myself and they are very similar.
$\mu _r = \chi + 1$
Where $\mu _r$ is relative permeability (The ratio of the permeability of a material to the permeability of free space $\mu _0$),
and $\chi$ is the magnetic susceptibility
That is their difference mathematically speaking but physically speaking, magnetic susceptibility measures the induced magnetic field as a factor of the applied magnetic field.
You might know of diamagnetism, paramagnetism and ferromagnetism
Diamagnetic materials repel, very weakly, applied magnetic fields and so their magnetic susceptibility is negative but  very tiny. For example, $\chi _{copper} = -10^{-5}$. So this figure tells us that the induced magnetic field inside copper is $-10^{-5}$ times the applied magnetic field.
Now speaking of permeability, this measures the NET magnetic field as a factor of the applied magnetic field. That is the difference, magnetic susceptibility is to do with induced magnetic field but permeability is to do with net magnetic field (the vector sum of the applied and induced magnetic field)!!!
So to finish it off, $\mu _{copper} = -10^{-5} + 1$ which is slightly less than 1 (from the equation above)
Note: above is the relative permeability of copper
Hope this helps Connor.
