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Now before saying anything, I have seen the similar questions in this page regarding $\vec H$ but no one has fully convinced me yet, since I will try to give another perspective to this question.

We know that the total contribution to a magnetic field is $$\vec B_{total}=\vec B_{material}+\vec B_0$$ with $\vec B_{material}$ and $\vec B_0$ being the field magnetized in the material (like a cylinder or whatever figure with no field generated from itself) and the field generated in by a common source (like a magnet or a spiral) respectively. The topic of magnetic fields in the matter corresponds to $\vec B_{material}$ and the magnetic fields on the void corresponds to $\vec B_0$. But from the topic of magnetic fields in the matter, we know $$\vec B_{material}=\mu_0 (\vec H+\vec M)$$ which generates a huge doubt for me since I've always associated $\vec H$ with the "cause" of the $\vec B_{material}$ and $\vec M$ with the "consecuence" of the $\vec B_{material}$, but now we remember that for a $\vec B_{material}$ to exist, it has to be an original magnetic field that magnetizes the material, because by itself it is unable to generate such field, and I thought we associated that field to $\vec B_0$, but now I think about it and it could also be $\vec H$, isn't it? And the "response" of the material I thought it would be $\vec B_{material}$ but then again it could also be $\vec M$, since it is associated to the consecuence of a magnetic field, what's going on?

I think my head is too confused so maybe I contradicted myself a couple times but it just doesn't make sense to me intuitively what letter corresponds to what contribution.

And the whole thing can also be compared to the electric field, with $\vec E=\vec E_0+\vec E_{material}$, where we have $\vec D=\epsilon \vec E+\vec P$, what is the contribution to what or what is the response of the material to an external field? It's all very confusing.


EDITED PART FOR THE BOUNTY UPDATE: The comments here suggest that the $\vec B_0$ and $\vec B_m$ refer to the $\vec H$ and $\vec M$ and no more. But today, my teacher explained to us with the example of a problem that was like so and actually didn't agree with the comments here:

A sphere of radius R of a linear magnetic material of permeability $\mu$ is located in a region of empty space where a uniform magnetic field $B_0$ exists. a) Knowing that the magnetization that appears on the sphere is uniform, calculate $M$, the dipole moment induced in the sphere and the $B$ field at all points in space.

And my teacher did the following:

After finding the magnetic potential vector $\vec A$, we can find the $\vec B_m=\nabla\times \vec A=\frac 23 \mu_0 \vec M$, and since $\vec B_m=\mu_0(\vec H_m+\vec M)$ we get $$\vec H_m=\frac{\vec B_m}{\mu_0}-\vec M=-\frac{\vec M}{3}$$ we can find the total magnetic field by using the expression I was told in the comments to be untrue: $$\vec B_{total}=\vec B_0+\vec B_m=\mu \vec H_{total}$$ and since $\vec H_{total}=\vec H_m+\vec H_0$ and $\vec B_0=\mu_0 \vec H_0$ and finally we got earlier $\vec B_m=\frac 23 \mu_0 \vec M$ we can substitute and find $\vec M$: $$\vec M=\frac{3(\frac{\mu}{\mu_0}-1)\vec B_0}{2\mu_0+\mu}$$ Which again the result is irrelevant for my purposes, since I'm looking, again, to understand how there are so many different contributions that I've lost sense of what letter means what, I thought $\vec M$ was the magnetic field caused in the material as a response of an external field $\vec H$, but now it turns out we have $B_0$ and $B_m$, and I'm quite lost.

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    $\begingroup$ You've written things like field magnetized in the material, I've associated $\vec{H}$ with the "cause" of $B_{material}$, and $\vec B_{total}=\mu_0(\vec{H} + \vec{M})$, which reveal you have serious misconceptions about these concepts and their relations in material media. It is actually the total field which obeys $\vec{B}_{total} = \mu_0\vec{H} + \mu_0 \vec{M}$ both in vacuum and material medium. $\endgroup$ Nov 24, 2023 at 0:57
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    $\begingroup$ You basically make a broad request "I'm confused, explain this stuff to me" - this is not a good question. I recommend forgetting what you wrote above and learning this properly from scratch from EM theory textbooks, like Griffiths: Introduction to Electrodynamics, Landau & Lifshitz: Electrodynamics of Continuous Media and others. $\endgroup$ Nov 24, 2023 at 0:57
  • $\begingroup$ It might be best to begin your study of this topic with electric field $\vec{E}$, polarization $\vec{P}$ and displacement field $\vec{D}$ in dielectrics. Magnetic quantities in various magnetic materials are somewhat harder to understand, and it gets easier when you already know how things work in dielectrics. For detailed dedicated explanation of what $\vec{E},\vec{P},\vec{D}$ mean in material medium, see e.g. Böttcher: Theory of Electric Polarization. $\endgroup$ Nov 24, 2023 at 1:11
  • $\begingroup$ Your problem is, "I've always associated H⃗ with the 'cause'" is wrong, as JL started. $\bf H$ causes nothing. It is just a somewhat confusing shorthand for writing ${\bf B}-4\pi{\bf M}$ (in Gaussian units) or ${\bf B}/\mu_0-{\bf M}$ in SI. I recommend sections 8.1 and 8.2 of my EM text book. $\endgroup$ Nov 24, 2023 at 11:59

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Here the problem is that you are using too many notations and you are changing them constantly.

  1. The master equation in electrostatics : $$ \vec{D}= \epsilon_0 \vec{E} + \vec{P}.$$

(i) $ \vec{D} $ is known as the electric displacement vector. This represents the electric field (But it's really not a true electric field. Check its dimension.) in a system due to the free charges. For example, the a charged conductor or ions embedded in a dielectric material are the free charges of this system. We can control these free charges and consequently we have full control over $ \vec{D}$.

(ii) $ \vec{P} $ is the polarisation vector. It is defined as the electric dipole moment per unit volume of a system. Polarised atoms or atoms having permanent dipole moment in the system create tiny dipoles. These tiny dipole moments constitute this polarisation vector. Also note that these tiny dipoles create the bound charges in the system. The value of these bound charges can be obtained from the polarisation vector itself.

(iii) $ \vec{E}$ is the total electric field of a system. Means it’s the field due to both free and bound charges present in the system.

(iv) $\epsilon_0$ is obviously the permittivity of free space. Now for linear dielectrics (Dielectrics in which polarisation varies linearly with electric field, $ \vec{E}$.), the defining equation is, $$ \vec{P} = \epsilon_0 \chi_e \vec{E}. $$

Note that in RHS we are putting total field $ \vec{E}$ not $ \vec{D}$. This is the definition. By using this convention everything works out well. This also leads to the equation $ \vec{D} = \epsilon \vec{E} $. Where $ \epsilon $ is the permittivity of the dielectric material.

Exactly similar quantities appear in magnetostatics too.

  1. The master equation in magnetostatics: $$ \vec{H} = (\vec{B}/\mu_0) - \vec{M}. $$

(i) Here $ \vec{H} $ is known as the Auxiliary field. This represents the magnetic field (not true magnetic field) in a system due to the free currents. For example a constant current carrying wire embedded in a paramagnetic material provides the free current to this system. We can control these free currents and consequently we have full control over $ \vec{H}$.

(ii) $ \vec{M} $ is the magnetisation vector. It is defined as the magnetic dipole moment per unit volume of a system. The atoms in the system act as tiny magnetic dipoles. These tiny dipole moments constitute this magnetisation vector. Also note that these tiny dipoles create the bound currents in the system. The value of these bound currents can be obtained from the magnetisation vector itself.

(iii) $ \vec{B}$ is the total magnetic field of the system. Means it’s the field due to both free and bound currents present in the system.

(iv) $\mu_0$ is obviously the permeability of free space. Now for linear magnetic materials (materials in which magnetisation varies linearly with auxiliary field, $ \vec{H}$.), the defining equation is, $$ \vec{M} = \chi_m \vec{H}. $$

This also leads to the equation $ \vec{H} = \vec{B}/\mu $. Where $ \mu $ is the permeability of the material. Now let’s come to your problem.

A sphere of radius R of a linear magnetic material of permeability μ is located in a region of empty space where a uniform magnetic field B0 exists. a) Knowing that the magnetization that appears on the sphere is uniform, calculate M, the dipole moment induced in the sphere and the B field at all points in space.

Here, the free currents or the current you can control is producing a magnetic field $\vec{B_0}$, outside the given system. So, the auxiliary field will be $ \vec{H} = B_0/\mu_0$ outside the material. Using the equations mentioned above you need to find magnetisation vector $ \vec{M} $ and total magnetic field $ \vec{B}$ inside the material. As you have mentioned the total magnetic field and auxiliary field inside the material due to the constant magnetisation vector only are $ 2\mu_0 \vec{M}/3$ and $ - \vec{M}/3$, respectively. Therefore in this case the total auxiliary field inside the material is $ \vec{H} = (\vec{B_0}/\mu_0) - (\vec{M}/3)$.

So, by putting these in the magnetostatics master equation and using $ \vec{H} = \vec{B}/\mu $ we get, $$ \frac{\vec{B_0}}{\mu_0} - \frac{\vec{M}}{3} = \frac{\mu}{\mu_0}(\frac{\vec{B_0}}{\mu_0} - \frac{\vec{M}}{3}) - \vec{M}. $$

Simplify the above equation to get your desired result.

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    $\begingroup$ To know more about this read magnetostatics chapter from Griffiths electrodynamics. $\endgroup$ Dec 4, 2023 at 13:32

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