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I've been delving a bit beyond just using what's typically given as the "electric field" and "magnetic field" in various problems, and finding out about the various more fundamental quantities and how they relate to one another; but as I've been reading that relationships between the magnetic quantities on the one hand and the electric quantities on the other should be completely symmetric, I struggle to reconcile that with what I read about how these quantities are in fact related.

First there's the magnetic field intensity denoted by $\mathbf{H}$ and the magnetization field $\mathbf{M}$, which together appear to relate to the magnetic flux density $\mathbf{B}$ by the formula $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$, where $\mu_0$ of course is the vacuum permeability.

Then there's the electric field intensity denoted by $\mathbf{E}$ and the polarization field $\mathbf{P}$, which together appear to relate to the electric flux density $\mathbf{D}$ (which seems to typically be called the "displacement field") by the formula $\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}$, where $\varepsilon_0$ of course is the vacuum permittivity.

The difference here might be minor, but apparently for the magnetic quantities the vacuum permeability is multiplied by both $\mathbf{H}$ and $\mathbf{M}$ added together, while for the electric quantities the vacuum permittivity is only multiplied by $\mathbf{E}$ and not to the polarization field $\mathbf{P}$. Am I just missing something fundamental here (I would think so), or is there actually an underlying asymmetry?

As a small separate point, it's also somewhat confusing that $\mathbf{B}$ seems to be used as the magnetic equivalent to $\mathbf{E}$ in most cases, when the two aren't really (i.e. that $\mathbf{E}$ seems to be used instead of $\mathbf{D}$, which I suppose isn't unexpected if it's simply due to $\mathbf{E}$ corresponding better to the term "electric field" and being used for that reason).

Hope someone can clear up my confusion.

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    $\begingroup$ Nothing stops you from writing it as $H = \frac{B}{\mu_0} - M$ in analogy to the equation for $D$. You're kind of comparing apples and oranges by showing a definition for D compared to a definition for B. $\endgroup$
    – Triatticus
    Commented Mar 20 at 23:34
  • $\begingroup$ @Triatticus: Well, that would be fine, and that looks perfectly antisymmetric, which is a neat form of symmetry too, but that would mean that B should be taken as the fundamental quantity that H is computed from, but everywhere I read, including here on SE, answers state that H is the more fundamental quantity, analogous to E, and that H represents the magnetic field intensity that is what it is independent of materials and medium and such, while B is the actual magnetic flux density, and thus rather analogous to D. That's why I'm confused. $\endgroup$
    – Outis Nemo
    Commented Mar 21 at 10:53
  • $\begingroup$ I agree with all that Vincent has stated, I usually work in vacuum situations and take E and B to be the fundamental quantities ala Griffiths and Jackson. $\endgroup$
    – Triatticus
    Commented Mar 21 at 15:54

1 Answer 1

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Functionally, $\mathbf{E}$ and $\mathbf{B}$ are analagous because they both represent the total field, taking into account both free and bound sources.

However, mathematically, we have $\mathbf{E}\sim\mathbf{H}$ and $\mathbf{D}\sim\mathbf{B}$ because when there is no free charge or current, $\nabla\times\mathbf{E} = \nabla\times\mathbf{H} = \mathbf{0}$ while $\nabla\cdot\mathbf{D} = \nabla\cdot\mathbf{B} = 0$. Therefore, both $\mathbf{E}$ and $\mathbf{H}$ admit scalar potentials. This is rooted in the fact that electrostatic fields have zero curl while magnetostatic fields have zero divergence. Sources contribute to the divergence of $\mathbf{E}$ but the curl of $\mathbf{B}$. In addition, from an experimental point of view, $\mathbf{E}$ and $\mathbf{H}$ are the more useful quantities. From Griffiths' Introduction to Electrodynamics, section 6.3:

$\mathbf{H}$ plays a role in magnetostatics analogous to $\mathbf{D}$ in electrostatics: Just as $\mathbf{D}$ allowed us to write Gauss's law in terms of the free charge alone, $\mathbf{H}$ permits us to express Ampere's law in terms of the free current alone-and free current is what we control directly. Bound current, like bound charge, comes along for the ride - the material gets magnetized, and this results in bound currents; we cannot turn them on or off independently, as we can free currents.

As it turns out, $\mathbf{H}$ is a more useful quantity than $\mathbf{D}$. In the laboratory, you will frequently hear people talking about $\mathbf{H}$ (more often even than $\mathbf{B}$), but you will never hear anyone speak of $\mathbf{D}$ (only $\mathbf{E}$). The reason is this: To build an electromagnet you run a certain (free) current through a coil. The current is the thing you read on the dial, and this determines $\mathbf{H}$ (or at any rate, the line integral of $\mathbf{H}$); $\mathbf{B}$ depends on the specific materials you used and even, if iron is present, on the history of your magnet. On the other hand, if you want to set up an electric field, you do not plaster a known free charge on the plates of a parallel plate capacitor; rather, you connect them to a battery of known voltage. It's the potential difference you read on your dial, and that determines $\mathbf{E}$ (or rather, the line integral of $\mathbf{E}$); $\mathbf{D}$ depends on the details of the dielectric you're using. If it were easy to measure charge, and hard to measure potential, then you'd find experimentalists talking about $\mathbf{D}$ instead of $\mathbf{E}$. So the relative familiarity of $\mathbf{H}$, as contrasted with $\mathbf{D}$, derives from purely practical considerations; theoretically, they're on an equal footing. Many authors call $\mathbf{H}$, not $\mathbf{B}$, the "magnetic field". Then they have to invent a new word for $\mathbf{B}$: the "flux density," or magnetic "induction" (an absurd choice, since that term already has at least two other meanings in electrodynamics). Anyway, $\mathbf{B}$ is indisputably the fundamental quantity, so I shall continue to call it the "magnetic field," as everyone does in the spoken language. $\mathbf{H}$ has no sensible name: just call it "H".

The other difference between the two expressions is due to the convention of the Coulomb and Biot-Savart laws: $$\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\int\ \rho(\mathbf{r}') \frac{\mathbf{r}-\mathbf{r}'}{\left|\mathbf{r}-\mathbf{r}'\right|^3}\mathrm{d}^3\mathbf{r}' \\ \mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}')\times\left(\mathbf{r}-\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|^3}\mathrm{d}^3\mathbf{r}'$$ Notice how the constants are on opposite sides of the fraction. The magnetic field can be rewritten as $$\mathbf{H} = \frac{1}{\mu_0}\mathbf{B} - \mathbf{M}$$ which will be completely analogous to the electric case except for the sign. The convention was chosen such that the greater the permittivity, the smaller the electric field (because more polarization is "permitted" which opposes the electric field, reducing it) while the greater the permeability, the greater the magnetic field. So the permeability was moved to the other side to match. As for the difference in sign, I suspect that it is because a polarization $\mathbf{P}$ creates an $\mathbf{E}$ that points opposite to it while, on the other hand, a magnetization $\mathbf{M}$ creates a $\mathbf{B}$ that points in the same direction. This is again a matter of convention.

More discussion can be found here.

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