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 somne 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 $.

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 $.

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 somne 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.

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 somne 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 $.