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

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.

• Duplicate of physics.stackexchange.com/questions/309951/…; see also physics.stackexchange.com/questions/343241/… – tparker May 4 '18 at 23:02
• @tparker The question is not an exact duplicate, since the poster in that question asks about the exclusion of $\mu_0$ in the definition of the magnetic suceptibility, whereas I am asking about the use of $H$ instead of $B$ if this is unclear I can edit the question accordingly. – Ignacio May 4 '18 at 23:08
• You're right. But I think in all cases the answer is the same - a combination of historical accident (with the "basic-looking" quantities being easier to measure or discovered earlier for some other reason) and the general suckiness of SI units. – tparker May 4 '18 at 23:13
• @tparker thanks, I actually hope the answer is historical accident, but if that is indeed the case to be convinced I would like to know with some more detail what the historical reasons are, it seems arbitrary and very inconvenient to me to define these things like this. SI units I don't really care for, that is why I set $\epsilon_0$ and $\mu_0$ to 1, to avoid noise =) – Ignacio May 4 '18 at 23:24

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.

• I really like this answer and, upon thinking about it for a while I feel like it is possibly correct, as tparker said in a comment. It doesn't directly address susceptibility, but I can infer from what you say that it is/was intuitive for experimentalists to think of the $\mathbf{E}$ field as being a cause and the $\mathbf{P}$ field an effect in the case of electricity, and to think of the $\mathbf{H}$ field as a cause and the $\mathbf{M}$ field as an effect in the other case. Upon thinking about it a bit more from that point of view I believe the meaning of both objects is not that different. – Ignacio May 6 '18 at 22:45
• I have a small tangential question though, you imply in your answer that you can measure the $\mathbf{B}$ field by using a Hall probe in an air gap, but wouldn't the $\mathbf{H}$ and $\mathbf{B}$ fields be equal in an air gap inside the torus? (assuming air is basically vacuum). I'm sorry if the question is kind of dumb but I am almost completely unfamiliar with experimental physics. (I now also see that you actually did address susceptibility directly in the anwser, my bad) – Ignacio May 6 '18 at 22:52
• @Ignacio, if the gap is thin, magnetic induction in the gap is the same as inside the torus (due to Gauss law for $\mathbf B$). You are right that in the gap the two fields are "the same" (in SI, $\mathbf B_{vac}=\mu_0\mathbf H_{vac}$). However, this $\mathbf H_{vac}$ is not the same as $\mathbf H$ near the gap in the torus, so the Hall probe or coil would really measure induction $\mathbf B$, not $\mathbf H$ that is in the torus. – Ján Lalinský May 7 '18 at 0:45
• It is important that the gap be thin in the direction of magnetic lines and wide across (the gap has a shape of thin disk). If the Hall probe was in a cavity that had a shape of cigar oriented along the magnetic lines, then it would measure $\mathbf H$ inside the medium, but that is not needed since $\mathbf H$ is easily determined from the Ampere law. – Ján Lalinský May 7 '18 at 0:49

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.

• Maybe in engineering, but Gaussian units are still widely used in physics. It's nice to never have to worry about B and H having different units. Much easier to teach too. – JohnS May 5 '18 at 22:46
• @JohnScales Gaussian units seem to have been falling out of favor even in physics in recent years, though. – Chris May 5 '18 at 22:50
• Well Jackson swore he would never use SI in the second edition. Then he comes out with a mishmash in the 3rd. Sigh. – JohnS May 5 '18 at 23:00
• Unfortunately, it's too late to ask Jackson, but I think his publisher forced him to use SI units in a third edition. I think he hints at that in the third paragraph of his preface. Engineers buy many more physics books and do physicists. I think in the standard contract, the publisher can choose a co-author to write a new edition if the original author refuses. – Jerrold Franklin May 7 '18 at 0:40

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.

• Aren't the units of $B$ and $H$ in cgs gauss and oersted, respectively? – Chris May 5 '18 at 23:05
• It also occurs to me that this doesn't really answer the question, but just agrees with the OP that the way susceptibilities are defined is weird. – Chris May 5 '18 at 23:11
• 1 Gauss produces 1 Oersted in vacuum. That they call these things differently is a historical anachronism. Also, my main point was that the definitions make perfect sense if you think in terms of the response of a medium to an applied field. So I was giving a contrarian point of view (relative to other answers) that they are only obscure because of they way they are often taught at the undergraduate level. In a graduate level course you start with the micro-B/E fields, do some averaging to get the macro fields, then look at the response functions to get H and D. What's so weird? – JohnS May 5 '18 at 23:16
• I disagree that it's a historical anachronism, since 1 gauss doesn't always produce 1 oersted. How can you have a system of units where 1 gauss can produce 2 gauss? (Well, clearly you can, since English customary units totally do that, but that's usually seen as a negative feature, isn't it?) – Chris May 5 '18 at 23:21
• Apart from a compulsion to honor dead physicists, I think the Oersted-Gauss anachronism resulted from an early misunderstanding that has been continue in SI units. Because H appears in Ampere's law and B in the force law, they were thought of as two different things, even in the vacuum. Actually, I think all four, E, B, D, H should have the same unit Gauss. This was originally proposed by Mel Schwartz. – Jerrold Franklin May 7 '18 at 0:56