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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?.

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Whether or not Hertz thinks this or that is absolutely immaterial. What really matters is what follows from the equations. Does it really follow that Hertz' equations accout for all known experimental facts in EM while Maxwell's don't? – ganzewoort Oct 12 '11 at 8:04
Unnecessarily combative comments deleted. – dmckee Oct 12 '11 at 13:30
It seems I am wrong from "Finally, in 1892, Hendrik Lorentz derived the modern day form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields." I guess "Lorentz" is the clue ;) So it certainly wasn't Maxwell, but neither was it Hertz, unless ganzewoort can provide the page number of his reference. – Larry Harson Oct 12 '11 at 22:52
@ganzewoort: You might want to put the actual equations in the question itself. – Dan Oct 13 '11 at 1:11
You think much too complicated! Hertz did his experiments, and wrote a paper using some equations he thought to be appropriate. Some years later he learned about Maxwells work and realized that this was the right thing. You must not think that information was spread as fast as today, moreover there were only very few physicists like Maxwell, Boltzmann or Clausius or Hertz who had enough math knowledge to understand Maxwells equations at that time. It is very unlikely that Hertz learnd about Maywell in study. – Georg Oct 13 '11 at 9:40

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.

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It is Maxwell, who really provided the twist in the real story of electromagnetism by introducing the concept of "displacement current" to make the set of equations, discovered by previous giants, self consistent. Symbols does not matter of course, content matters and the credit is got to be given to that great original thinker named Maxwell. – user1355 Oct 13 '11 at 4:36
@sb1: I mentioned the displacement current in the first paragraph, although not by name. Perhaps I should say the name of the effect. But I never liked the term "displacement current", because it is derived from Maxwell's ether-thinking, which I believe he rejected later in life. – Ron Maimon Oct 13 '11 at 6:21
Yes, indeed you mentioned. It is my fault to somehow missed it. I am sorry for that. You certainly deserve a +1 for this comprehensive answer. – user1355 Oct 13 '11 at 7:02
The question still remains. Hertz' theory accounts for the voltage observed in the unipolar generator, Maxwell's theory doesn't. Is there anything which Maxwell's theory accounts for while Hertz' doesn't? If these theories are equivalent in this respect then Hertz' is superior becuse it accounts for all known facts from physical experiments. – ganzewoort Oct 13 '11 at 9:07

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.

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