Magnetic field in materials with non-constant magnetic susceptibility

Question

I'm quite lost what $B$ and $H$ is. It seams to me that most of the texts I read do quite poor job in explaining them properly. They are explained only in cases when magnetic susceptibility is constant in the space.

First law says that $M = \chi H$, $M$ is magnetization and to my understanding $H$ is something like external magnetic field (for example Earth magnetic field when I do experiment in some lab).

Next law says that magnetic field is $B = \mu_0(M+H) = \mu H$. This has to be wrong when $\chi$ is not constant because than we are not guarantied that divergence of $B$ is zero. $$ \nabla \cdot B = \nabla \cdot (\mu H ) \neq \mu \nabla \cdot H = 0 $$

Can you please direct me to some book when this is explained properly?

My aim: I want to calculate magnetic field around paramagnetic material. I cannot use formula $B = \mu H$ because outside paramagnet is $\mu = \mu_0$ so magnetic field would be unchanged.

Answer 1

Hopefully this should dispel some of your confusion. In general, the fields $\textbf{B}$ and $\textbf{H}$ are related by

$$ \textbf{H} \equiv \frac{1}{\mu_0}(\textbf{B} - \textbf{M}) $$

This is always true, regardless of the materials involved. We define linear media as materials whose fields satisfy

$$ \textbf{B} = \frac{1}{\chi}\textbf{M}. $$

In this case, for relative permeability $\mu = \mu_0(1+\chi)$, the relation $\textbf{B} = \mu\textbf{H}$ also holds. For nonlinear media, as you describe, it does not. Also, taking the divergence of both sides of the first equation yields

$$ \nabla \cdot \textbf{H} = -\frac{1}{\mu_0}\nabla \cdot \textbf{M}. $$

Thus, the divergence of $\textbf{H}$ is not always 0. To help you with your "aim," it might be useful to note that most paramagnetic materials, for low magnetism, can be approximated as linear media ($\mu$ is constant).

Answer 2

The concept of magnetic field H and magnetic flux density B could be confusing. These are the names commonly known in textbooks. Obviously the naming is not unique either. At least these are the case I see in most books. Although H is called magnetic field, it is B that directly determines Lorentz force. So in many experiments B is measured. It then makes more sense to call B magnetic field. Nonexistence of magnetic charge leads to the divergence free property of B. And H is determined by both B and the magnetic properties of materials. In this sense B corresponds to E and H corresponds to D: D = \epsilon E, H = (1 / \mu) B.

Hope it helps somewhat instead of causing more confusion...