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In many electrodynamics/magnetism textbooks it is stated that the magnetic induction $\vec{B}$ is related to the magnetic field $\vec{H}$ via $\vec{B}=\mu_0(\vec{H}+\vec{M})$. It is then argued that $\vec{H}$ is an important quantity because it is given by the free currents only, so that $\nabla \times \vec{H} = \vec{j}_\text{free}$.
Free currents are something we control in experiments and so tuning $\vec{H}$ via $\vec{j}_\text{free}$ is straightforward. In addition, it is often argued that $\vec{H}$ is independent of the medium, so it really makes sense to think about everything in terms of $\vec{H}$. All of this gave me the impression that $\vec{H}$ can be thought of as some sort of an external field we impose on our sample, which then responds through some magnetization $\vec{M}$ and then the net field inside the sample would be $\vec{H}+\vec{M}$. I never really understood why we needed to introduce a new quantity $\vec{H}$, instead of just writing the induction as $\vec{B}_\text{tot} = \vec{B}_\text{ext} + \mu_0\vec{M}$ where $\vec{B}_\text{tot}$ would be the total induction in the sample and $\vec{B}_\text{ext}$ would be the induction imposed externally, that we would measure if the sample was not there.

But then I learned about shape anisotropy and demagnetizing fields which show that the non-uniformities (specifically, divergence) of $\vec{M}$ can also result in an extra $\vec{H}$ field. If I'm correct, $\vec{j}_\text{free}$ controls the curl and $\nabla \cdot\vec{M}$ controls the divergence of $\vec{H}$. However, if that's the case, $\vec{H}$ is really not material independent. What would be a good, intuitive way to think about $\vec{H}$ then? And what is all the fuss about $\vec{H}$ if it's i) material dependent and ii) cannot even be measured or controlled in the material, independently of $\vec{M}$?

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    $\begingroup$ "However, if that's the case, $\vec{H}$ is really not material independent"—exactly right. $\vec{B}$ is the fundamental magnetic field. In a lot of situation, $\vec{H}$ is more convenient, but its ultimate meaning is simply that it is a convenient linear combination of $\vec{B}$ and $\vec{M}$. $\endgroup$
    – Buzz
    Commented Apr 1, 2022 at 20:55

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I think there's no fundamental meaning of $H$, although some people would disagree. I've never understood their point of view, so I won't attempt to paraphrase it.

But if that's the case, where did $H$ come from? Maxwell introduced it in A Dynamical Theory of the Electromagnetic Field with the following meaning:

... the force acting on a unit magnetic pole placed at the given point ...

(Maxwell didn't call it $H$, by the way. He called it $(\alpha, \beta, \gamma)$. It was Heaviside who rewrote Maxwell's equations in vector form.)

Maxwell regarded the $H$ field as more fundamental than $B$. He used $(\mu \alpha, \mu\beta, \mu\gamma)$ to denote the $B$ field.

Although Maxwell's model was incorrect (in the sense that, in nature, all magnetic forces are actually due to the $B$ field acting upon currents, and even point magnetic dipoles should be viewed as infinitesimal current loops), his equations are still correct, and that includes the ones that involve $H$. So $H$ survives in the theory but without a fundamental meaning.

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It is not true in general that $B_{tot}=B_{ext}+\mu_0 M$ unless $M$ represents a permanent magnet whose dipole moment density is independent of external influences. In general, $\mu_0 H \ne B_{ext}$, becasue there is also the effect of the surface poles induced by the external field and these poles demagnetize it, so the effective internal $H$ is less than $H_{ext}=B_{ext}/\mu_0$.

In fact, $B$, $H$ and $M$ may all be variable and depend on the external bias field that is generated by fixing the fields of currents and permanent magnets in which we place the polarizable matter.

Added to this mutual dependence there is the complication that the polarization and induced fields all depend also on the geometry of the polarizable matter. This is because the effective strength (density and distribution) of the induced surface poles all naturally depend on the geometric shape of the material. There are some special geometric configurations that lead to simple homogeneous or almost homogeneous internal fields inside the polarized material. When such arrangement is used there is a directly measurable relationship between these fields $B,H,M$ and the external bias field $B_{ext}=\mu_o H_{ext}$. Such examples of special geometry are

  1. toroidal
  2. spherical or prolate spheroidal
  3. thin & long needle

They all have one thing common: it is reasonable easy to induce a homogeneous or almost homogeneous field in them. If you take either (1) or (3) and wrap a tightly wound coil around them, or (2) place a spheroidal magnet in an externally homogeneous bias field then the internal fields inside the magnetic materials will be essentially homogeneous and directly dependent on the external bias strength. Moreover, for (1) and (3) the internal field is directly calculable from the solenoidal current wrapped around. For the spheroidal you should know that the internal field is aligned and parallel internally but not necessarily aligned with $H_{ext}$ unless the latter is also parallel with the prolate axis.

Especially cases (1) and (3) are easier to deal with in practice than with (2), and the flux can be measured via Faraday's law in having the magnet fashioned as transformer between two coupled coils wrapped around it. Obviously, both the geometrical arrangement of the coils wound around a spheroidal magnet and the magnet's precise manufacturing are more difficult than it is for a needle or toroid.

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