# Are Gauss' electrodynamics laws for charge-charge interaction correct at all?

My question refers to a fragment on electrodynamics written by Gauss at around 1835. In this short note (see Gauss' Werke, volume V, p.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' 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. The Gauss' Nachlass contains an additional such basic law (see Schaefer's treatise on Gauss physical works):

$$F = -\frac{u}{r^3} - \frac{\alpha}{u r}(\frac{du}{dt})^2 - \frac{\beta}{r}\frac{d^2 u}{dt^2}$$

where $$u$$ is a kind of "distance vector" (Gauss terms it "tricomplex number"), $$r$$ is the scalar distance, $$t$$ time, and $$\alpha, \beta$$ are two constants which are related by: $$\beta = 2\alpha$$. Gauss defines $$\alpha, \beta$$ in terms of a third constant $$\epsilon$$ which has units of speed: $$\alpha = \frac{1}{8\epsilon^2}, \beta = \frac{1}{4\epsilon^2}$$. Gauss' first electrodynamics law is therefore very similar to Weber's general law, by that the first term is the Coulomb's Law, the second depends on the squared relative speed, and the third is an acceleration term.

In the article "The Suppressed Electrodynamics of Ampère-Gauss-Weber", note 8, the author says:

More than a decade before the publication of Weber’s 1846 paper, one can find an 1835 entry in Gauss’s Notebooks, showing a hypothesized form of the electrodynamic force law, dependent on relative velocity and acceleration, that is essentially equivalent to that which Weber used in the 1846 publication. Interestingly, the Gauss formulation appears on the same page as an alternative formulation, which was the one James Clerk Maxwell chose to use in his text Treatise on Electricity and Magnetism to falsely imply a difference in electrodynamic views among the three collaborators, Gauss, Weber, and Riemann.

I guess that the author refers to Gauss' first electrodynamics law, which appears on p. 616 of the same volume.

Now, I found very confusing statements about Gauss' 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).

I'm not to trying to be a promoter of lost and obscure physical theories, (obviously if Maxwell's theory eventually superseded Gauss-Weber approach, then there must be a rational historical reason for that), but I'm impressed by the insightfulness of the two electrodynamics laws written down by Gauss.

So I think that despite these limitations, Gauss' equation makes him the originator of retarded potentials theory, which is a very significant conceptual innovation.

Now, my questions are:

• Are Gauss' equations correct at all, at least for non-relativistic speeds? I checked the general relativistic formula for the complete charge to charge force, and it's not the same as Gauss'.
• Are there any articles 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' (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. Jul 2 '18 at 11:54
• Do you know if this formula is correct? 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 $q/(4\pi \epsilon_0)$. Jul 2 '18 at 13:15
• it's funny since Gauss himself formulated "Gauss's law" for electric flux. 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. Jul 5 '18 at 18:42