In a recent exchange it was pointed out that in certain situations Maxwell's equations need Lorentz force as a patch (the latter not being part of those equations neither is it derivable from them).
It appears that the equations proposed by H. Hertz lack that deficiency while retaining Lorentz invariance. What might be the reason for using Maxwell's equations instead?.
The content of the Maxwell equations are named after Maxwell because he derived their wave character, and theoretically predicted the induction of magnetic fields by changing electric fields (the Maxwell term). This is the reason they are named for Maxwell--- he took the known field relations from empirical laws and turned them into a comprehensive theory.
The motion of point charges was not so much of a concern at that time. I would assume that the discovery of the electron added an impetus to figuring out how points move. Bulk material properties are more interesting to physicists in Maxwell's time, and I wouldn't be surprised if bulk forces appear in Maxwell which reproduce the Lorentz force when you take the limit of a small charged sphere. I don't know, but it doesn't matter at all.
Any other system of equations is presumably equivalent in physical content to the Maxwell equations, predicting the same set of electric and magnetic fields in response to the same currents. If Hertz's theory predicts the same as Maxwell's, it is equivalent, and if it doesn't then it is a new theory, and probably a wrong one.
The question of which particular shape of symbols to express a physical law is a very inane one. It doesn't matter so long as the predictions for experimental quantities are the same. This prediction can be written as Maxwell's equations, as Hertz's equations, as differential forms, or as a C-code to give the fields from the sources. It really doesn't make any difference, it's still Maxwell's theory.
I have no personal opinion and I will report what I found interesting in the net about the subject.
The Hertz equations (I was not aware of them) had originally an error, here corrected by Pechenkov
A little error was eliminated from Hertz’s equations ... Therefore Hertz’s electrodynamics is the alternative to Einstein’s electrodynamics.
A discussion of Hertz versus Maxwell eqs can be seen here, by Petrovic Branko and the more interesting is his conclusions about 'Detectors' - 'Conformity to the first relativity postulate' - 'Denying Newton's Third Law'.
I will have to study more on this subject.
What is described in the query is filled with false myths and misunderstandings that need to be called out and addressed. The short answer is that Hertz didn't make any substantial change to the equations Maxwell wrote, other than deletions (i.e. no potentials), changes of notation and the inclusion of Thomson's correction.
So, let's take this one step at a time.
First, Maxwell never added any patch related to with the "Lorentz force" in his expressions. That's a widespread folklore myth which has propagated primarily on and through the net and has no basis in anything other than superficial resemblances of equations or expressions people lifted from books or papers without properly contextualizing what they were or said.
Second, what Maxwell did do is the same as what Lorentz and - yes Hertz, too, as you're about to see - did. He added $×$ as a contribution to $$ to account for the motion relative to the background medium, where $$ is the magnetic induction, $$ the electric field and $$ a velocity having to do with the speed of the background. A distinction is drawn between "moving" and "stationary" based on this.
Third, Maxwell never posed any Lorentz force law; and his attempts at finding any cohesive force law (partly obscured by what he did with $$) were badly muddled, even though the answer was staring him in the face by way of the very kind of dimensional analysis and study of the behavior of the fields under Galilean transform that he had already done later in his treatise. The Lorentz force was probably Lorentz' doing and was there in Lorentz papers, as noted below.
Fourth, it is not a "deficiency" at all, but a necessity - even in Relativity, with only one exception. The frame of reference, in question, given by $$ is the unique frame of reference in which the constitutive laws relating these fields to the displacement field $$ and magnetic field $$ assume a particularly simple form. It appears in Lorentz' equations (he uses the opposite sign for the velocity) ... and also in Hertz' equations. Moreover, it is also present in the Relativistic versions, when dealing with the constitutive law inside a medium other than a vacuum, and arises in the equations written by Minkowski in 1908 and by Einstein, himself, with Laub in the same year.
The Relativistic version of the constitutive laws that uses this "adjustment" is the Maxwell-Minkowski relations.
Maxwell puts the adjustment directly in with the $$ field. There is also an adjustment of $-×$ required for the $$ field, which he failed to add, even though he considered the possibility even as far back as his early 1860's papers. It was added by others later as a correction. Hertz does, in fact, do the same, and has both corrections directly added into the fields. In contrast, neither Lorentz, nor Minkowski or Einstein/Laub, do so and kept them separate from the fields.
The versions by Maxwell, Hertz are both non-relativistic, as is the version by Lorentz. The versions by Minkowski, Einstein and Laub are relativistic, by virtue of also including corrections of the form $×/c^2$ and $-×/c^2$, respectively to $$ and $$ that are entirely relativistic in nature.
Because of the additional terms, the "one exception" alluded to above may arise, where there is no longer a unique frame and the $$ terms simply drop out. That's we now refer to the vacuum form of the constitutive laws: $$ = μ_0 , \hspace 1em = ε_0 ,$$ with "vacuum" values $μ_0$ and $ε_0$, respectively, for permeability $μ$ and permittivity $ε$. In other media, where $0 < εμ = 1/V^2$ yields a wave speed $V$ that is not the absolute speed $c$ of Relativity, and in a non-relativistic setting, the $$-dependency remains, and the constitutive laws assume the form: $$ = μ ( - ×), \hspace 1em = ε ( + ×),$$ for in the non-relativistic case, and $$ - \frac{×}{c^2} = μ ( - ×), \hspace 1em + \frac{×}{c^2} = ε ( + ×),$$ in the relativistic case.
Lorentz' equations - along with their conversion to contemporary form - are laid out in my reply here.
Lorentz force law in Newtonian relativity
and yield the non-relativistic case.
There, you can also see the distinction between the $×$ correction versus what actually is the Lorentz force, where the addition made $×$, is with a velocity $$ that refers to the motion of the charge relative to you, not relative to any background medium.
In both Lorentz' and Hertz' papers, different treatments - and even different section names and chapters - are given based on the "stationary" versus "moving" form of the equations, based on whether you're moving with respect to the medium or not.
Fifth, and most importantly, Hertz did make the distinction and did write down the equivalent of the non-relativistic version of the constitutive laws. He didn't "remove" any "deficiency" at all.
As a common point of reference, these are Maxwell's equations such as we would write them out today: $$∇·, \hspace 1em ∇× + \frac{∂}{∂t} = , \hspace 1em ∇· = ρ, \hspace 1em ∇× - \frac{∂}{∂t} = .$$ The source terms include the charge density $ρ$ and current density $$.
These equations apply in both non-relativistically and relativistically ... and even in the curved coordinates and space-times of General Relativity. It is independent of paradigm and geometry.
Hertz used and described the following fields:
They are laid out in the section
"XIII. On The Fundamental Equations Of Electromagnetics For Bodies At Rest"
in his book whose English version is "Electric Waves" in 1893, forwarded by the same Lord Kelvin who provided the $-×$ correction to $$ that Maxwell was missing.
When dealing in the "moving" version of the equations, in the section
"XIV. On The Fundamental Equations Of Electromagnetics For Bodies In Motion"
he also introduces the fields
The vector notation is mine, for the sake of brevity, not Hertz'. He uses only scalars and component forms. The "line densities" for $$ and $$ correspond to what we would now write as differential forms $·d$ and $·d$, matching what we would today write as $·d$ and $·d$, where $d = (dx, dy, dz)$. The "surface densities" would correspond to the differential forms $·d$ and $·d$, which we would today write as $·d$ and $·d$, where $d = (dz∧dy, dx∧dz, dy∧dx)$ in Hertz' coordinate frame (which is backwards from ours), and $d = (dy∧dz, dz∧dx, dx∧dy)$, in our coordinate frame.
In addition, he also uses the following, throughout:
I mark them as $\hat{ε}$ and $\hat{μ}$ to distinguish them from our versions $ε$ and $μ$. He never explicitly writes $\hat{ε}_0$ and $\hat{μ}_0$, nor $ε_0$ and $μ_0$, but just simply drops out the factor, when dealing with the case of the vacuum.
Hertz doesn't put any expressions or equations in profile for the charge density, though he describes it in the narrative. That's the same gap and oversight seen in Einstein's 1905 "On The Electrodynamics Of Moving Bodies" (emphasis on "Moving" mine). As an additional footnote, Einstein's term "Moving" refers to the above sense of "Moving", and the paper drops the hammer that in a vacuum: the "Moving" form = the "Stationary" form ... provided one takes $1/\sqrt{ε_0μ_0}$ (what we now call $c$) as an absolute speed.
I'm using vector form for his equations partly, also, to encapsulate his convention, which differs from ours - as he, himself, acknowledged. His $(x,y,z)$ coordinates are oriented as North-East-Up, which is backwards from our $(x,y,z)$ coordinates which are oriented as North-West-Up. So, the vector-cross product in his orientation would read: $$× = \left(a_z b_y - a_y b_z, a_x b_z - a_z b_x, a_y b_x - a_x b_y\right),$$ for vectors $ = \left(a_x, a_y, a_z\right)$ and $ = \left(b_x, b_y, b_z\right)$, while we would write it as $$× = \left(a_y b_z - a_z b_y, a_z b_x - a_x b_z, a_x b_y - a_y b_x\right).$$
That includes its use with the "curl" operator $∇×(\_)$, where $$∇ = \left(\frac{∂}{∂x}, \frac{∂}{∂y}, \frac{∂}{∂z}\right).$$ So, if you just lift his equations out of his book, without paying proper attention to the context, you'll get everything backwards.
Also, he use the same notation for total and partial derivatives. In particular, his $d/dt$ is our $∂/∂t$.
With these things in mind, this is the vector form of what he wrote: $$\begin{align} A\left(\frac{∂}{∂t} - ∇×(×) + ∇·\right) &= -∇×, &\hspace 1em (1a) \\ A\left(\frac{∂}{∂t} - ∇×(×) + ∇·\right) &= +∇× - 4πA. &\hspace 1em (1b) \end{align}$$ In other contexts, he called out the following in narrative form $∇· = 4πe$ and cited the following as the energy density for the field $$\frac{· + ·}{8π},$$ which corresponds to our $½ (· + ·)$. He also calls out the conductivity law (by inference in his equation 6b in section XIII, comparing it to equation 1b in section XIV) $ = λ$, which we now write as $ = σ$. For the equation $∇· = 0$, he hedged his bets a little and never put it in profile.
His $A$ is our $1/c$ - and is attributed to as a property of the vacuum, itself. He states that $$ can be variable, particularly for material media, but that it seems to be arbitrary for the vacuum, even though it shouldn't have been. It literally took an Einstein to resolve that one, in 1905, and, with Laub, to show in 1908 where it will continue to make a difference.
The correspondences are: $$ \left(A, \frac{d}{dt}, \left(\frac{d}{dx}, \frac{d}{dy}, \frac{d}{dz}\right), \hat{ε}, \hat{μ}\right) ⇔ \left(\frac{1}{c}, \frac{∂}{∂t}, ∇ = \left(\frac{∂}{∂x}, \frac{∂}{∂y}, \frac{∂}{∂z}\right), \frac{ε}{ε_0}, \frac{μ}{μ_0}\right), \\ \left(, , , \right) ⇔ \left(\sqrt{\frac{4π}{μ_0}}, \sqrt{\frac{4π}{ε_0}}, \sqrt{4πμ_0}( - ×), \sqrt{4πε_0}( + ×)\right), \\ \left(, , e, λ\right) ⇔ \left(, \frac{ - ρ}{\sqrt{4πε_0}}, \frac{ρ}{\sqrt{4πε_0}}, \frac{σ}{4πε_0}\right). \\ $$ Under these correspondences, his equations become Maxwell's equations in the reference form I cited, plus the non-relativistic version of the constitutive laws, plus the conductivity law $ = σ$.
