Does $\mathbf{B}$ field decrease based on materials in its path to magnetic field source, or only the material at a point?

Question

My understanding of magnetic fields was that the strength of the field was "path dependent". I.E. the $\mathbf{B}$ field decreased more when it went through materials with relative permeability close to 1 as compared to ferrous materials with much higher relative permeability. (I.E. magnet -> wood -> some point in space would have a lower B field than magnet -> iron -> some point in space). I guess I got this idea because outside of electromagnetism permeability means "the state or quality of a material or membrane that causes it to allow liquids or gases to pass through it." So I assumed it was the way the magnetic field passed through a material, and how the magnetic field turned out on the other side of that material.

But then I started reading https://www.maxwells-equations.com, and now my whole world is turned upside down. Specifically because of this line: "In Equation 1, permeability is the permeability of the medium (material) where we are measuring the fields." (emphasis added). So it doesn't matter what is between the magnet and the point? It only depends on the distance and the material itself?

This Question and answer seem to indicate the magnetic field is indeed "path dependent". But I'm not sure if they're talking about the H field or the B field. In other words, are the "insulators" mentioned in the question messing with the H field, and then to get the B field you must multiply by mu of the last material, or are they "insulating" by reducing the B field (in which case, are the B and H fields really distinct, or just a mathematical construct?).

Answer 1

Here's my understanding (although I am not too qualified): the magnetic flux density $\text{d}\bf{B}$ due to a current element $I\text{d}\bf{s}$ at a position $\bf{r}$ relative to this current element is given by the Biot-Savart Law

$\text{d}\textbf{B} = \frac{\mu_0 I}{4\pi r^3} (\text{d}\textbf{s} \times \textbf{r})$

which is indeed independent of any material which lies between the current element and the point of interest $\textbf{r}$. It only depends on the current elements present and their distance from your point of interest. (However, if there is a magnetisable material nearby, this B field will produce induced magnetic dipoles within that material, which will then in turn affect the overall B field.)

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As for your second question: the H field is defined by

$\textbf{H} = \frac{\textbf{B}}{{\mu_0}} - \textbf{M}$

Where $\textbf{M}$ is the magnetisation per unit volume of the material. It is material dependent and I'm sure there's lots of unusual materials that have weird magnetisations. However for the most part we assume that all 3 of these fields are parallel, and that $\textbf{M}$ is linearly proportional to $\text{H}$. That allows you to write $\textbf{B} = \mu \textbf{H}$ for some constant $\mu$ which we call the permeability of the medium.

So to answer your question: if one had full knowledge of how the material you are within behaved, i.e. you fully understood its magnetisation, then the H field would be 'distinct' from the B field (in the sense that they are not just constants away from each other).

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P.S. I'm not too sure how mu-metals work. There is an example in Bleaney & Bleaney of a sphere within a H field & I assume you could generalise this to a sphere within a shell within a field, and then set the permeability of the shell to be very high, in order to get the H field (and then also the B field - although this derivation does make some assumptions about the magnetisation which may not be valid.)

P.P.S. Reading Bleaney & Bleaney is always a good idea if you wish to understand EM more.

Source for Biot-Savart formula: Electricity & Magnetism by Bleaney & Bleaney 3rd edition Section 4.1 e.g. page 100 here

Answer 2

I emailed a professor and got this reply:

"Joseph,

"You are right. To find B you just multiply H by mu, and this is a local calculation. In the absence of ferromagnetic materials, like with an electromagnet, H is calculated from true currents only (not magnetization currents that arise from magnetic materials.) If you have a material with zero permeability (I'm not sure what this would be) the B inside it would be zero, but H would not be zero. The H would be the same as if the material weren't there. The magnetization of the material would cancel out the H when creating B, because B=mu0(H+M).

"To have a region with zero H, you would have to divert the field lines around it. This would require a real current to flow in the material, and then that current would modify the H. That current would then be part of the sources that need to be included to calculate H at other locations as well. The only situation I can think of like this is the Meissner effect where magnetic fields are excluded from inside a superconductor. They do that through superconducting surface currents.

"The linear relationship between B and H is only true for non-ferromagnetic materials. The relationship between B and H is complicated in ferromagnetic materials and depends on history, since ferromagnetic materials have a permanent magnetization. Depending on where you are in the hysteresis curve, mu can be either positive or negative.

"I hope this makes sense to you. Magnetic fields have some subtle things associated with them."

Thank you professor hart for the wonderful answer!