Why are electric and magnetic susceptibilities defined in such an unintuitive way?

Question

When I studied electromagnetism the $\mathbf{B}$ and $\mathbf{E}$ fields were introduced as fundamental quantities to me, and the $\mathbf{H}$ and $\mathbf{D}$ fields were introduced as something of an afterthought in order to more easily work with free currents and charge densities in materials (I understand that this may not be the way maxwell originally thought about these, but it makes perfect sense nowadays). It was always clear to me that the analogous quantity to $\mathbf{B}$ was $\mathbf{E}$ and the analogous quantity to $\mathbf{H}$ was $\mathbf{D}$, since the equations $\nabla \cdot \mathbf{E} =4\pi \rho$ and $c\nabla \times \mathbf{B} = 4\pi\mathbf{J}$ (for non time-varying electric fileds) turn into $\nabla \cdot \mathbf{D} = 4\pi\rho_{f}$ and $c\nabla \times \mathbf{H} = 4\pi\mathbf{J}_{f}$ in materials (I am using gaussian units), however, the definitions of the electric and magnetic susceptibilities don't make this clear at all, it actually seems like they are defined as if the analog to $\mathbf{E}$ was actually $\mathbf{H}$. Why is this the case? From my point of view I can't really see any reason for identifying $\mathbf{E}$ with the "electric analogue" of $\mathbf{H}$. The definitions I am talking about are as follows (for linear, isotropic materials):

$$ \mathbf{M} = \chi_v \mathbf{H}$$ $$ \mathbf{P} = \chi_e \mathbf{E}$$

I have also heard $\mathbf{H}$ being called the "magnetic field" and $\mathbf{B}$ the "magnetic induction" which is also pretty confusing. Is this just a historical anachronism (in which case it is quite a confusing and impractical one) or is there a deeper reason why this is the case?

To clarify: what puzzles me is the fact that $\mathbf{H}$ is used instead of $\mathbf{B}$ to define the magnetic susceptibility, which makes as a consequence the actual meaning of the susceptibilities different in a magnetic and an electric context, since in one case a fundamental quantity, $\mathbf{E}$, is being used, whereas in the other case a less fundamental quantity, $\mathbf{H}$, is being used in order to define constants that are called with the exact same name and could easily be defined symmetrically.

Answer 1

This is an interesting question that calls for a proper historical analysis, but here is one pragmatic reason.

The vectors $\mathbf E$ and $\mathbf H$ are used because they are more directly related to what is an independent variable (chosen by experimenter) when the measurement of behaviour of the medium is done by common methods.

Polarization in a medium due to electric field can be measured with a parallel plate capacitor that has a slab of that medium inside and where voltage is controlled and measured by the experimenter. This voltage is directly related to electric field inside the medium, due to relation between electric work and voltage. Polarization is not measured directly, but other related things are, such as electric current during the process of charging the capacitor. Electric displacement $\mathbf D$ can then be determined by calculation from the measured current and voltage and then dielectric constant or susceptibility can be determined as well. This makes $\mathbf D$ dependent variable; it is determinable only by a calculation from other quantities and thus is not intuitive to use it in the role of an independent variable in the relation between response and field set up by the experimenter.

The same is true for $\mathbf H$ and magnetization. Magnetization in a medium due to magnetic field can be measured with help of a toroidal coil which has torus of that medium inside and where current in the coil is controlled and measured. This current is directly related to magnetic strength $\mathbf H$ inside the body, due to Ampere law. Magnetization is not measured directly, but other things are, such as voltage on the coil or (more often) voltage on a second coil wrapped around the torus, which is related to magnetic induction $\mathbf B$ in the torus. Or, if static field is to be measured, the torus may have an air gap in which magnetic induction can be measured by a Hall probe. From these, magnetic permeability or susceptibility can be determined. However $\mathbf B$ is determined, it is more difficult to do than to determine $\mathbf H$, so it is $\mathbf H$ that is taken as the variable that experimenter can reliably control and measure.

Answer 2

Your puzzlement is appropriate. It all goes back to periodic units conferences where these issues were settled democratically, with engineers and telegraphers outnumbering physicists. H was chosen because a technician looked at a dial where the current was given as H, using~NI/L. Since H_tangential is continuous, H inside a ferromagnetic rod was equal to either H or B outside, when put in a solenoid. Physicists, but not engineers, knew that H and B were equal outside. Thus the engineers put everything onto H and won the day. This issue was further sealed by the strange fact that H and B are considered quite different, even in vacuum, by SI units. Their use has completely messed up electromagnetism to this day, and the foreseeable future.

Answer 3

In free space H and B are the same. Fundamentally H only arises in a medium (non-vacuum). If you read a lot of theoretical physics you can find whole books in which H is never mentioned (e.g., Cohen-Tannoudji's book on QED). At the micro-level you can always talk about B and you end up with essentially random wave propagation. M and D are the response of a medium to an applied field (B and E, respectively). Further, in CGS units H and B have the same units, D and E have the same units and the response functions are dimensionless. So, if you're doing pure theory, then B and E are all you need. But in material, you need both micro fields and macro fields. H is only defined macroscopically as the non-local response to an applied B field. Also, keep in mind that a lot of texts play fast and loose by mixing the time domain Maxwell equations with the Fourier domain response law, since convolutions are annoying. Linear response laws must be causal and this is tricky in the frequency domain (the real and imaginary parts of the response functions are constrained by Kramers-Kronig). If you just arbitrarily set the imaginary parts to zero you will violate causality. For me, B and E are the fundamental fields. And they are what I measure in the lab usually. If you launch an incoming E field at a material and measure the amplitude and phase of the outgoing E field you can infer the permittivity without ever dealing with D, for example.