How did Maxwell combine Faraday's equation with the Lorentz force law?

Question

The table at this page lists the equations that Maxwell wrote down in his treatise “A Dynamical Theory of the Electromagnetic Field” in his original notation. They are 20 component-wise equations, which, when simplified using the divergence, curl, and gradient operators (which Maxwell was not aware of), reduce to eight equations. When translated into modern notation, seven of these equations map directly to standard equations from classical electromagnetism as currently understood, including three of the four modern "Maxwell's equations" (or close equivalents).

But there are two missing "standard" equations: the Lorentz force law and Faraday's law. Instead, there is a strange equation which, in modern notation, looks like this:

$${\bf E} = \mu \left( \frac{d{\bf x}}{dt} \times {\bf H} \right) - \frac{\partial {\bf A}}{\partial t} - {\bf \nabla} \phi.$$

This seems to be combining the Lorentz force law and Faraday's law in a way that I can't disentangle. The ${\bf E} - {\bf v} \times \mu {\bf H} = {\bf E} - {\bf v} \times {\bf B}$ seems to be getting at the Lorentz force law, while the equation ${\bf E} = - \frac{\partial {\bf A}}{\partial t} - {\bf \nabla} \phi$, which is similar to the one that Maxwell wrote, is essentially equivalent to Faraday's law (as can be seen by taking the curl of the former equation and using the equation ${\bf B} = {\bf \nabla} \times {\bf A}$, which is another one of the equations that Maxwell wrote down). (Note that Maxwell used some different sign conventions than is standard today, so I'm not worrying at all about signs in this question.)

However, the equation above does not seem to actually be correct as far as I can tell, although it does combine together different "pieces" of correct equations.

Is the equation above actually correct (when interpreted correctly)? If so, how should it be interpreted? Or did Maxwell simply get Faraday's law and the Lorentz force law slightly wrong?

The equation looks plainly incorrect to me (although "close" to correct) - but on the other hand, I suspect I would have heard about it at some point if Maxwell had gotten any part of his final theory of electromagnetism flat-out wrong.

I believe that this questions belongs on Physics SE rather than on History of Science and Mathematics SE, because I'm not asking about when Maxwell discovered what, or who influenced him, etc. Instead, I'm asking about the substantive content of the physics itself. Is his equation reproduced above actually correct? If so, how?

Answer 1

but on the other hand, I suspect I would have heard about it at some point if Maxwell had gotten any part of his final theory of electromagnetism flat-out wrong.

It is flat-out wrong. And basically nobody understood Maxwell, including the definitive translator Heaviside. Basically everybody read Heaviside, and it is his versions of Maxwell's equations that people referred to. There are many quirks and oddities and rewrites of Maxwell's equations in his original treatises that are problematic. In fact, the earlier versions tended to be more correct than the later rewrites.

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Maxwell introduced $\vec D$ and $\vec H$ fields so that he could write fewer instances of $\mu$ and $\varepsilon$, and so you are correct to note that his $\mu(\vec v\times\vec H)$ is really $\vec v\times\vec B$. The extra term is most obviously wrong when you consider Gauß's law: If $\vec E=\vec v\times\vec B-\partial_t \vec A-\vec\nabla\phi$ then $$ \begin{align} \tag1\rho=\vec\nabla\cdot\vec D=\vec\nabla\cdot(\varepsilon\vec E) &=\varepsilon\vec\nabla\cdot(\vec v\times\vec B)+\rho\\ \tag2\text{but}\qquad\vec\nabla\cdot(\vec v\times\vec B) &=\vec B\cdot(\vec\nabla\times\vec v)-\vec v\cdot(\vec\nabla\times\vec B)\\ \tag3&=\vec B\cdot(\vec\nabla\times\vec v)-\mu\vec v\cdot(\vec\jmath+\partial_t\vec D) \end {align} $$ And then it is easy to see that this last expression can be easily chosen to be non-zero, leading to a contradiction. So, this expression must be a mistake and cannot be accepted as-is.

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Another way to see that it has to be wrong, is to consider Faraday's law from it. This might be even more straightforward: $$ \begin{align} \tag4\vec\nabla\times\vec E=\vec\nabla\times(\vec v\times\vec B)-\partial_t\vec B\neq-\partial_t\vec B \end {align} $$ And you can immediately see that this new addition, which came later than Maxwell already having acknowledged Faraday's law earlier, thus rewrote Faraday's law into a mistaken form.

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Oliver Heaviside only removed the $\vec v\times\vec B$ for convenience, and so technically he had to put it back in. That state of affairs persisted until Lorentz discovered that it belonged on the force law, that now bears his name. But he also didn't give it a lot of thought, and so it fell onto Einstein, and Minkowski, to make sense of the theory under SR. Between Lorentz, Einstein and Minkowski, and Abraham and Dirac, the topic is settled, but it gets difficult to know who contributed what.

As you can see, it was a heroic effort just to recast Maxwell's equations into a form that we can use, and to remove all the mistakes, and figure out all the inconsistent units.

The history in this answer is taken from KathyLovesPhysics. I am eagerly awaiting her video lecture (from which I took them) to exit Patreon status and go onto her youtube page.

Answer 2

Regarding Maxwell's error this is what Whittaker writes in his monumental "A History of the Theories of Aether and Electricity," Vol1, page 259.

The memoir of 1864 contained an extension of the equations to the case of bodies in motion ; the consideration of which naturally revives the question as to whether the aether is in any degree carried along with a body which moves through it. Maxwell did not formulate any express doctrine on this subject ; but his custom was to treat matter as if it were merely a modification of the aether, distinguished only by altered values of such constants as the magnetic permeability and the specific inductive capacity; so that his theory may be said to involve the assumption that matter and aether move together. In deriving the equations which are applicable to moving bodies, he made use of Faraday's principle that the electromotive force induced in a body depends only on the relative motion of the body and the lines of magnetic force, whether one or the other is in motion absolutely. From this principle it may be inferred that the equation which determines the electric force$^1$ in terms of the potentials, in the case of a body which is moving with velocity $\mathbf w,$ is $$\mathbf E = \frac{1}{c} \mathbf w \times \mathbf {\mu H}-\frac{1}{c}\frac{\partial \mathbf A}{\partial t}+\mathbf {grad} \psi$$ Maxwell thought that the scalar quantity $\psi$ in this equation represented the electrostatic potential; but the researches of other investigators$^2$ have indicated that it represents the sum of the electrostatic potential and the quantity $\tfrac{1}{c}\mathbf A\cdot \mathbf w$.

and the footnotes:

$^1$ It may be here remarked that later writers distinguished between the electric force in a moving body and the electric force in the aether through which the body is moving, and that $\mathbf E$ in the present equation corresponds to the former of these vectors.
$^2$ Helmholtz, Jour. for Math. lxxviii (1874), p. 309; H. W. Watson, Phil. Mag. (5), XXV (1888), P· 271

Answer 3

The reference is wrong and Maxwell never came up with any Lorentz force law. That's a widespread myth that's been propagated on the net. It was Lorentz who came up with it (as I explain in more detail in the link below). Instead, the various forms of the force law that Maxwell took a stab at in his treatise were much more involved and (in fact) he never settled on any specific formulation for the force law at all.

First, let's rewrite the equation you put down, $$𝐄 = μ \left(\frac{d𝐱}{dt} × 𝐇\right) - \frac{∂𝐀}{∂t} - ∇φ,$$ as $$𝐄_M = 𝐆 × 𝐁 + 𝐄,\quad 𝐄 = \frac{∂𝐀}{∂t} - ∇φ,\quad 𝐁 = μ𝐇,$$ and then clarify the terms in it, and put everything back into context. Maxwell's version of the electric field $𝐄_M$ included a part that we now term the electric field $𝐄$ and a contribution arising from a velocity $𝐆$ that (in the context of his development) referred to the frame in which the constitutive laws are isotropic. In the literature of the time, equations set in that frame were referred to as the "stationary" form of Maxwell's equations, while those set in other frames were the "moving" form of Maxwell's equations. (In some parts of his treatise, Maxwell actually did use $𝐆$ for this velocity.)

This is why, for instance, the title of Einstein's landmark paper on Special Relativity was "On the electrodynamics of moving bodies". It was a wake-up call to the people of the time that he was going to reconcile the seeming discrepancy of having a reference to a fixed frame in the equations. It's why his theory was called "Relativity" (even against Einstein's own objections), despite the presence of an axiom within it that made reference to an absolute speed. In the opening discussions of Einstein's 1905 Special Relativity paper, he makes reference to this velocity, but does not call it out by name. So, from your vantage point - here in the 21st century - with all the extra context of that time removed, it passes by unnoticed.

Lorentz, Heaviside and Helmholtz also made similar distinctions between the "stationary" and "moving" forms of Maxwell's equations. A similar dichotomy in Lorentz' treatment exists between the $𝐆$ velocity, that's relative to the frame where the constitutive laws are isotropic, versus the velocity $𝐯$ relative to your reference. (He actually used $-𝐆$ instead of $𝐆$ and different letters for everything.)

Maxwell included the $𝐆$-dependent term in his definition $𝐄_M$ of the $𝐄$ field in order to make the constitutive law $𝐃 = ε 𝐄_M$ hold in all frames. In terms of what we now understand to be the $𝐄$ field, we would - instead - write it as: $$𝐃 = ε (𝐄 + 𝐆×𝐁).$$

In his earliest treatments in his pre-treatise days, Maxwell did not make any distinction between $𝐁$ and $𝐇$, though the context made it clear that there was a tacit distinction being made. In particular, where today we would write $𝐁$ he had a tendency to write it as a diglyph $μ𝐇$. This is all up to a change in notation; remembering that before the treatise, he didn't use vector notation but wrote everything component-wise. So, the components of our $𝐁$ were written as diglyphs formed of the components of $μ𝐇$. In addition, there was also an extra factor of $4π$ in $𝐇$ ... as well as in his version of $μ$. That's why (until a couple years ago) the SI value of $μ_0$ had an extra $4π$ in it as $μ_0 = 4π×10^{-7} \text{Henri/meter}$.

Maxwell's confusion of $𝐁$ and $μ𝐇$ was an error and led to the wrong form of the constitutive law. Whereas he wrote $𝐁 = μ𝐇$ (and tacitly assumed it in his pre-treatise days), the correct form should have been $$𝐁 = μ(𝐇 - 𝐆×𝐃).$$ This correction was made by Heaviside and Thomson and was present in Lorentz' treatment.

Together, the Maxwell equations $$ 𝐁 = ∇×𝐀,\quad 𝐄 = -∇φ - \frac{∂𝐀}{∂t},\quad⇒\quad ∇·𝐁 = 0,\quad ∇×𝐄 + \frac{∂𝐁}{∂t} = 𝟎,\\ ∇·𝐃 = ρ,\quad ∇×𝐇 - \frac{∂𝐃}{∂t} = 𝐉,\quad⇒\quad ∇·𝐉 + \frac{∂ρ}{∂t} = 0, $$ along with the constitutive laws for isotropic media $$ 𝐃 = ε(𝐄 + 𝐆×𝐁),\quad 𝐁 = μ(𝐇 - 𝐆×𝐃), $$ that may be termed the Maxwell-Heaviside-Thomson-Lorentz relations are Galilean-covariant.

In contrast, Einstein and Laub in 1907-1908 and Minkowski in his landmark paper on Minkowski geometry came up with what we would today write $$ 𝐃 + \frac{1}{c^2}𝐆×𝐇 = ε(𝐄 + 𝐆×𝐁),\quad 𝐁 - \frac{1}{c^2}𝐆×𝐄 = μ(𝐇 - 𝐆×𝐃), $$ which are known as the Maxwell-Minkowski relations. Taken together with the Maxwell equations, they are Lorentz-covariant.

The key property of the Maxwell-Minkowski relations are that in any medium where $εμ = 1/c^2$ and $|𝐆| < c$, the "moving" and "stationary" forms of the relations are equivalent, thereby making $𝐆$ "superfluous", to use Einstein's characterization of the velocity.

Maxwell never wrote down any Lorentz force law. Instead, it was Lorentz who did - which is why it's named after him. The actual form he wrote it down as was not $$𝕱 = ρ𝐄 + 𝐉×𝐁$$ but with his convention $𝐉 = ρ𝐯$ as a force $𝐅_1$ per unit charge: $$𝐅_1 = 𝐄 + 𝐯×𝐁,$$ and his $𝐯$ was not $𝐆$! Instead, he used $𝐩$ for $-𝐆$. This is the force law for a unit charge. More generally, for a charge $e$ this leads to the Lorentz force $𝐅 = e𝐅_1$, which is given by $$𝐅 = e(𝐄 + 𝐯×𝐁).$$

I laid out the Rosetta Stone, here: Lorentz' Force Law showing, in detail, the equivalence of the key equations in Lorentz' treatment with the Maxwell-Heaviside-Thomson-Lorentz relations and explaining how the Lorentz force law came about.

Answer 4

According to the paper Electromagnetic induction: How the “flux rule” has superseded Maxwell's general law, Maxwell confusingly used the same letter to denote both the electric field (which he called "electric intensity") and the force per unit charge (i.e. the thing which in modern language, we integrate to compute an electromotive force). So in modern notation, your equation is actually $$\frac{\mathbf{F}}{q} = \mathbf{v} \times \mathbf{B} + \mathbf{E}$$ which is perfectly correct. Of course, this is just the Lorentz force law, and Faraday's law plays no role here. The linked paper explains that Maxwell originally derived this equation in a more complicated form which was intimately tied to considering induced emfs.