By: Adrian (a.delete@this.acm.org), December 16, 2021 3:51 pm

Room: Moderated Discussions

While I mostly agree with what you said, I have to nit pick some parts, where what is stated might be confusing.

> - Weber presented an alternative view of electromagnetism to (and before) Maxwell, in which relative

> acceleration is part of the force law. You can explain some things with this in a way that's different

> from Maxwell, but pretty much every physicist would say that you don't get as far, or as conveniently.

While Weber indeed presented the electromagnetism laws in an alternative way than Maxwell, the alternative was mostly in the presentation form not in the content.

The fact that Weber added the acceleration-dependent term in the force is absolutely essential, because without it the conservation of energy is not observed and there is no electric induction or wave radiation.

In the approximation when the relative velocities are small in comparison with the speed of light, the formulas from Weber are absolutely correct and any relativistic expression of the electromagnetic forces that does not transform into the Weber form at small velocities is just wrong and there are many relativistic presentations full of wrong formulas.

Some 20 years after Weber first introduced his expressions for the force and for the electromagnetic potential energy, which contained the present relative distances, velocities and accelerations of all other particles, regardless how far they might be (which nonetheless still produced correct results for most phenomena, like the propagation of the electromagnetic waves with the speed of light), Bernhard Riemann, Carl Neumann and Ludvig Lorenz have introduced the idea of retarded potentials and Neumann has shown that the Weber potential energy can be derived from simpler retarded potentials and this formulation removed the possibility that the electromagnetic interaction might happen instantaneously at a distance. So already by then, 40 years before Lorentz, Poincare and Einstein, there already existed a formulation of electromagnetism that was essentially unchanged by the later theory of "relativity".

While the Weber formulation of electromagnetism was in fact equivalent with that of Maxwell, because the Maxwell equations can be deduced from the electric charge conservation plus the Weber potential energy or force law, in practice the Maxwell form was more convenient.

In the Weber form, intermediate variables like electromagnetic potentials or electromagnetic fields were not used. That meant that you had to compute integrals of the distributions of charges and currents. Those were not normally known, so you actually obtained integral equations that had to be solved.

Today, with computers, there are actually cases when solving integral equations is preferable, especially for radiation problems, e.g. by using methods with boundary elements. In the 19th century solving integral equations was much harder.

In the Maxwell formulation, solving a problem was decomposed in 2 steps, computing the electromagnetic potentials or the electromagnetic fields and then computing the quantities of interest.

Not only the resulting equations were simpler, but for the common cases with linear media it was easy to rewrite the equations for different media just by changing some constants in the equations (the permittivities/permeabilities) while in the Weber formulation it was far less obvious how to introduce the influence of dielectric or ferromagnetic materials.

While Maxwell's formulation happened to be superior for solving practical problems, that was not why Maxwell created his theory. Following Faraday, Maxwell did not believe that it is possible to describe the world by interactions from a distance, but only by contact interactions, so there must be an intermediate medium which is influenced by bodies and eventually the influence reaches the other bodies that interact only with the intermediate medium. On the continent, the same philosophical ideas lead to the development of the method of the retarded potentials, which is mathematically equivalent with Maxwell's approach, but less convenient in most cases.

So the Maxwell formulation won, but unfortunately, not all of it was preserved unscarred, because people like Heaviside thought that Maxwell's theory is too complex and it must be simplified for students.

Regarding the force due to the relative acceleration, that is of course retained by Maxwell, it just is written with a different notation. With Maxwell, the force is computed from the electric and magnetic potentials. The same 3 terms from Weber are present, but the electric force is proportional with the gradient of the electric potential, the magnetic force is proportional with the curl of the magnetic potential and the inductive force is proportional with the time derivative of the magnetic potential, which is determined by the relative acceleration of the charges.

The Maxwell equations as written by Maxwell were written in an integral form, which is generally valid. Unfortunately that form is almost never shown in standard manuals.

Most manuals show a form of Maxwell's equations for dummies, in a differential form due to Heaviside.

The familiar differential form is valid only in simple cases, when the distributions of sources and materials are very smooth and when nothing moves.

When there are solid mobile parts, or worse, when there are fluid parts, writing the correct derivatives of the original Maxwell equations becomes a complex problem of differential geometry and many good and important physicists, including Hertz and Heaviside, have failed to compute correctly the derivatives, which resulted in formulas that did not match the experiments.

It was a pity that Maxwell died much too young, because he would have had the opportunity to expand and improve his work, which proved to be a difficult task for his successors.

I am always triggered to comment about such things like the theory of electromagnetism, because I am always annoyed to remember how much time I have wasted when I was young, both at the University and at various libraries reading countless manuals of physics, only to understand after many years that a much too large part of them was just garbage, because of contradictions, incompleteness and errors.

Since then, I have learned that I must always go to the original sources and almost never trust modern authors when they say that e.g. Maxwell or Weber or Einstein said something, because very often they have actually said something very different and too often their successors, in attempts to simplify their theories, have actually lost important parts, which were no longer transmitted correctly.

Reading the original publications from the 19th century or early 20th century or even from the 18th century, can be more instructive than many of the modern sources. However, it is true that this approach also has it difficulties, as you have also mentioned, because when reading ancient authors you must be already experienced to distinguish the parts where they were wrong, and later authors found the right ways, from the parts that are still valid today.

The good part is that thanks to archive.org, Google and a few large libraries or universities, many old books have been scanned, so many ancient authors have become much more accessible than when I was young.