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I'm confused about the most fundamental and universally correct relationship between the magnetic fields $B$ (magnetic flux density) and $H$ (magnetic field strength) inside a hard magnetic material. I've encountered multiple expressions in textbooks and literature, and they are contradictory to each other.

The common forms are:

  1. This formula is universally valid in all media: $B = μH$, where $μ = μ_{0}μ_{r}$ ($μ$ is the absolute permeability), therefore $B = μ_{0}μ_{r}H$. (This form is right.)
  2. This formula is universally valid in all media: $B = μ_{0}H + μ_{0}M$, where $M$ is the magnetization. (This form is right.)

But in some literatures, there is : $B = μ_{0}μ_{r}H + μ_{0}M$.

For the hard magnetic materials such as the permanent magnet material, what is the correct? We can begin with the postulate $B = μ_{0}(H + M)$ (or equivalently $H = B/μ_{0} - M$) and then $B = μ_{0}μ_{r}H$ and $B = μ_{0}H + μ_{0}M$ are all valid for any materials, $μ_{r}-1=χ_{m}=M/H$ is also valid for all materials (The assumptions of isotropic and anisotropic materials are not discussed here).

But for the hard magnetic materials, how $B = μ_{0}μ_{r}H + μ_{0}M$ is valid?

I'm looking for a clear explanation.

Update

For the relative permeability of permanent magnet materials, you can try searching and find that the relative permeability of permanent magnet materials is around 1, such as 1.03, but that is not the main point. The main focus of this topic is: for all materials (including hard magnetic materials like permanent magnets), is it $B = μ_{0} H + μ_{0}M$ or $B = μ_{0}μ_{r} H + μ_{0}M$ that holds true for all materials?

Additionally, some have mentioned anisotropy and nonlinearity. I have clarified that this topic does not discuss anisotropy and assumes isotropy by default. As for the relative permeability $μ_{r}$ not being a constant, it can be treated as a variable, which does not affect the validity of $B = μH $.

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  • $\begingroup$ You should give some examples of the "some literature" that is giving the bad form. Anyway, they are wrong. When you have a strongly magnetic material, the $\mu_r$ concept stops being useful, and any literature that invokes that concept is suspect. Of course, there are still some good literature that uses that concept in the strongly magnetic case, but they are very careful in handling them and making sure that the concept is used correctly. $\endgroup$ Commented Jan 6 at 8:34
  • $\begingroup$ I apologize for not listing the specific literature references. It should be noted that this discussion focuses specifically on hard magnetic materials. While the relative permeability $u_{r}$ values may differ between hard magnetic materials (such as permanent magnets) and soft magnetic materials, both can be characterized using relative permeability $u_{r}$. Therefore, could you clarify in what context you consider the $u_{r}$ concept stops being useful？ $\endgroup$ Commented Jan 6 at 8:53
  • $\begingroup$ Can I assume that the constitutive relation $B = μH$, i.e., $B = μ_{0}μ_{r}H$, holds universally for all materials including hard magnetic materials? $\endgroup$ Commented Jan 6 at 8:57
  • $\begingroup$ When there is no externally imposed magnetic field / current causing the magnetism, then $H=0$ but $B\neq0$, and thus there is no way to discuss a permanent magnet using $\mu_r$ $\endgroup$ Commented Jan 6 at 9:55
  • $\begingroup$ > "This formula is universally valid in all media: B=μH" Not true. There is no such generally valid relation. $B$ may point in different direction than $H$, and it need not be linear function of $H$. $\endgroup$ Commented Jan 6 at 11:52

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