# Are Gauss' electrodynamics laws for charge-charge interaction correct in the limit of low-speeds?

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 checked the modern relativistic formula for the electric field produced by a uniformly moving charge:

$$E(r,\theta) = \frac{kq}{r^2}\frac{1-\beta^2}{(1-\beta^2\mathbb{sin}^2\theta)^{3/2}}\hat{r}$$

which in general is obviously not of the same mathematical form of Gauss's equation for non-accelerating charges. Here $$\beta=u/c$$ and $$\theta$$ is the angle between the line of sight to the charge and the direction of relative motion. But I still tried to check if the two expressions are equivalent for low speeds ($$\beta<<1$$) by using first-order approximation of the relativistic formula:

$$cos\theta = \frac{\dot{r}}{u} \implies \frac{1-\beta^2}{(1-\beta^2\mathbb{sin}^2\theta)^{3/2}}= \frac{1-(u/c)^2}{(1-(u/c)^2\cdot (\frac{u^2-\dot{r}^2}{u^2}))^{3/2}}\approx (1-(u/c)^2)\cdot(1+\frac{3}{2}(\frac{u^2-\dot{r}^2}{c^2}))\approx 1+\frac{1}{c^2}(\frac{1}{2}u^2-\frac{3}{2}\dot{r}^2)$$

so there is no agreement of the two expressions even in the limit of low speeds. This is interesting, because apparently (according to articles I found) Gauss's electrodynamics law settles completely with Ampere's force law for current-current interaction, with the last law being an established scientific fact (Ampere derived it from detailed experiments).

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 really think i miss a piece of the puzzle in my historical understanding of pre-maxwellian electrodynamics.

• 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, 2018 at 11:54
• Do you know if this formula is correct? Jul 2, 2018 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, 2018 at 13:15
• it's funny since Gauss himself formulated "Gauss's law" for electric flux. Jul 2, 2018 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, 2018 at 18:42