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

Question

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.

Answer 1

Gauss' formula is very probably correct in the non-relativistic regime, as long as the accelerations are negligible. However, the formula is in fact not fully compatible with Maxwell's equations.

From a historical point of view, one needs to start with Ampere. Around 1820, he measured the magnetic forces between parallel and non-parallel conductors. He found a force law that does not correspond to the formula that is known today as Ampère's force law. In 1835, Gauss realized that this original force law could be explained by imagining a current element to be composed of two point charges. For Gauss and later for Weber, magnetism was only an effect and not a fundamental force (the Lorentz force can be deduced and there is no need to postulate it).

In 1870, there was unfortunately an incompatible break with the interpretation of magnetism, because Maxwell assumed in his derivations that the B fields are generated by currents in closed conductor loops and not by single moving point charges (see page 162: Maxwell, "Treatise on Electricity and Magnetism", Vol. 2 (2 ed.), Clarendon Press, Oxford, 1881). By means of this postulate, he was able to carry out some formula transformations and arrived at the Biot-Savart law. However, Maxwell now had to include the Lorentz force as an additional equation, since it could not be deduced any longer from the field of a point charge. Maxwell's electrodynamics was very successful, as it allowed the modeling of electromagnetic free-space waves for the first time. The electrodynamics of Ampere, Gauss and Weber was almost completely forgotten.

Interestingly, it is relatively easy to merge the two electrodynamics. The approach is as follows: First one solves Maxwell's equations for two point charges in the rest frame of the recipient. The Lorentz force can therefore be simplified to $\vec{F} = q\,\vec{E}$ and all velocity-dependent terms go into the current density. Once the solution for the E-field has been found, a Galilean transformation is carried out, which is allowed if one is only interested in non-relativistic velocities. As it turns out, the solution is compatible with Gauss' formula if one uses the equation $\vec{F} = q\, \gamma(v)\,\vec{E}$ instead of $\vec{F} = q\,\vec{E}$. An interesting aspect of this method is that B-field and Lorentz force become obsolete, as all magnetic effects can now be interpreted in the same way as Gauss did. In addition, one does not lose the advantages of Maxwell's equations.

The final formula for the electromagnetic force $\vec{F}$ that a point charge $q_s$ with the trajectory $\vec{r}_s(t)$ exerts on another point charge $q_d$ with the trajectory $\vec{r}_d(t)$ is \begin{equation} \vec{F} = \frac{q_d\,q_s\,\gamma(v)}{4\,\pi\,\varepsilon_0}\,\left(\frac{\left(\vec{r}\,c + r\,\vec{v}\right)\,\left(c^2-v^2 - \vec{r}\cdot\vec{a}\right)}{\left(r\,c + \vec{r}\cdot\vec{v}\right)^3} + \frac{\vec{a}\,r}{\left(r\,c + \vec{r}\cdot\vec{v}\right)^2} \right), \end{equation} with the retarded distance vector \begin{equation} \vec{r} = \vec{r}_d(\tau) - \vec{r}_s(\tau), \end{equation} the retarded difference velocity \begin{equation} \vec{v} = \dot{\vec{r}}_d(\tau) - \dot{\vec{r}}_s(\tau), \end{equation} and the retarded difference acceleration \begin{equation} \vec{a} = \ddot{\vec{r}}_d(\tau) - \ddot{\vec{r}}_s(\tau). \end{equation} The retarded time can be obtained by solving the equation \begin{equation} \tau = t - \frac{r}{c}. \end{equation} (for example with Newton's method, note that $r = \Vert\vec{r}\Vert$ etc.)

The force formula is very useful for computer simulations of electromagnetic fields and waves (https://github.com/StKuehn/OpenWME), since there is no longer any need to solve differential equations. Incidentally, it also fulfills Newton's third law, i.e. one can easily show that the total momentum of an isolated system of point charges is a conserved quantity. In addition, the force formula is invariant in all inertial frames and complies with Einstein's postulates without the need to perform a Lorentz transformation (of course, this only applies to the non-relativistic regime). Because of these properties, the formula is particularly useful in electrical engineering.

The derivation from Maxwell's equations and the proof that the force formula agrees with Gauss' formula for very small accelerations and velocities can be found in "The Importance of Weber–Maxwell Electrodynamics in Electrical Engineering", IEEE Transactions on Antennas and Propagation, vol. 71, no. 8, 2023 and "Weber-Maxwell electrodynamics: classical electromagnetism in its most compact and pure form", Techrxiv, 2024.