# Is Gauss's electrodynamics law for inertial motion correct at all?

My question refers to a fragment on electrodynamics written by Gauss at around 1835. In this short note (see Gauss's Werke, volume V, p. 616-617) Gauss wrote down a fundamental equation which describes the general electromagnetic interaction between moving charges with respect to each other, i.e his formula is a generalization of Coulomb's law for moving charges. I find this formula to be the most insightful part of his physical work - Gauss wrote it in the context of his meditations on the electromagnetic force, after an idea came to his mind that that "two elements of electricity attract or repel each other in a different way than if they were in a relative state of rest" and that one needs to replace the "action at a distance" theory of charge-charge interaction with a finite speed of propagation of forces. Gauss's equation is:

$$F = \frac {{q_1q_2}}{{r^2}}(1 + \frac {{1}}{{c^2}}(u^2 - \frac {{3}}{{2}}\dot r^2))$$

where $$c$$ is the speed of light, $$u$$ the relative velocity of the two charges and $$\dot r$$ the radial component of the relative movement of the charges. Actually the Gauss's Nachlass contains an additional such basic law (see Schaefer's treatise on Gauss physical works), but "Gauss's first electrodynamics law" isn't the subject of this question.

Now, I found very confusing statements about Gauss's equation in the literature - all the sources agree that this equation doesn't settle completely with Maxwell's description of electromagnetism, but some state that it's consistent with Maxwell in many aspects (though not completely).

According to those books and articles, Gauss's formula violates the conservation of energy for general motions (accelerations) and therefore fails to account for the phenomenas of induction - a failure Maxwell noticed in his comments on Gauss's equation. Later, in 1845, Weber generalized Gauss's equation to include accelerations, resulting in what is now called "Weber's electrodynamics".

Despite these limitations, Gauss's equation makes him the originator of retarded potentials theory, which was a mainstream approach in physics until the publication of Maxwell's "Dynamical theory of electromagnetism".

Now, my questions are:

• Is Gauss's equation correct at all? I checked the general relativistic formula for the complete charge to charge force, and it's not the same as Gauss's, even for inertial motions (so why I still find sources which claim correctness of it for inertial motions?)
• Are there any surviving documents that give a clue of the reasoning he used to deduce this equation? In particular, was his derivation just a formal manipulation on Ampere's force law for current-current interaction? Ampere's force equation is the only formula I found that resembles Gauss's (in the factor 3/2).
• How this equation preempt the later development of retarded potentials theory? How does it relate to more modern conceptions (like Lorenz's ideas, for example) ?
• Have you looked at the non-quantum laws of electrodynamics for particles moving at ${u}$ a significant fraction of $c$ ? Also, just the appearance of $\frac{u^2}{c^2}$ doesn't mean it has anything to do with Lorenz. – Carl Witthoft Jul 2 '18 at 11:54
• Do you know if this formula is correct? – user2554 Jul 2 '18 at 11:57
• i think Gauss's formula is inconsistent with Gauss's law for electric flux - i performed integration of Gauss's formula for electric field over the whole sphere and the result wasn't $q/(4\pi \epsilon_0)$. – user2554 Jul 2 '18 at 13:15
• it's funny since Gauss himself formulated "Gauss's law" for electric flux. – user2554 Jul 2 '18 at 13:16
• On the second question take a look at links in What 19th century developments contributed to the General theory of Relativity? and Hecht's Suppressed Electrodynamics of Ampère-Gauss-Weber. It is cranky but gives some rarely mentioned historical details and original references. – Conifold Jul 5 '18 at 18:42