Weber Beats Lorentz!
An earth shattering breakthrough has appeared in the July issue of Nature. Measurements of trajectories of electrons passing through a flattened solenoid strongly favor Weber’s force law and Weber Electrodynamics as opposed to Lorentz force law, which is a corner stone of classical electromagnetism:
Overall, the field model is found to have an average accuracy of 67.4%, followed by CPO with 73.2% and the Weber model with 91.7 % accuracy for the modelled deflections relative to the experimental values.
Baumgärtel, C., Smith, R.T. & Maher, S. Accurately predicting electron beam deflections in fringing fields of a solenoid. Sci Rep 10, 10903 (2020). https://doi.org/10.1038/s41598-020-67596-0
Does this mean that there resides a deeply-rooted problem in classical (and quantum) electrodynamics, which was just now brought to light? The proper answer is no.
Theories Are Models
We must not forget that all theories – regardless of how successful and how well established – are just mathematical abstractions with limited domain of validity. A useful theory gives a successful approximation of the infinite complexity of nature in a certain special case or domain. A useful theory must be clear about it’s domain of applicability and provide us with computational tools that (when used correctly) would accurately predict the outcomes of measurements.
In this sense every theory has a deeply rooted problem or problems – physics is far from done, the four forces of nature are not yet unified and we do not have a satisfactory ‘theory of everything’. We find theories useful not solely on the basis of predictive accuracy, but also on the basis of ‘ease of use’, which closely parallels the ‘ease of understanding’ and the ‘ease of learning’.
Classical Electrodynamics
This is why classical electrodynamics (which comes with the Lorentz force law) is an immensely useful theory, which is simple enough to be taught at middle school in some places. However, when we use classical electrodynamics we must remember that the precision of numerical predictions of classical electrodynamics will always be limited since this theory pays no attention to relativistic effects. Also, the classical electrodynamics has ‘usability’ issues related to the apparent violation of Newton’s third law (Lorentz force is not symmetric when it comes to action and reaction) and it fails to offer a simple explanation for unipolar induction. Thus even in simple scenarios we are often forced to resort to the concepts of fields, electromagnetic radiation and even relativity in order to avoid running into paradoxes and arriving at outright incorrect conclusions (e.g. qualitatively incorrect conclusions, rather than quantitatively inaccurate predictions). It appears that a ‘wholesome’ and ‘safe’ theory of classical electrodynamics must be relativistic and must introduce the concept of electric and magnetic fields (as governed by the Maxwell’s equations) and electromagnetic radiation from the onset. This places the classical relativistic electrodynamics firmly into the undergraduate curriculum and therefore drastically limits its usefulness. But even in this all-encompassing formulation Faraday paradoxes arise and confusion regarding complimentarity of electric and magnetic fields perpetuates (e.g. we are still incorrectly taught that time-varying electric fields causes magnetic fields and vice versa; in fact there is no causal relationship between the electric and magnetic fields as these are inseparable and therefore caused simultaneously by moving electric charges and varying currents as eloquently described by Jefimenko equations).
Weber Electrodynamics
This is where Weber electrodynamics comes in handy. Because, unlike the Lorentz’s force the Weber force has terms that depend on relative velocity between charges, and because there is a term due to relative acceleration between the charges the resulting force is strictly compatible with Newton’s third law: every electromagnetic action comes with equal and opposite reaction. Consequently in Weber electrodynamics charges interact via a force that is best described as ‘action at a distance’. This force conserves total linear and total angular momentum without the need to resort to fields or electromagnetic radiation. In a way the dependence of Weber force (and Weber potential) on relative velocity makes Weber electrodynamics semi-relativistic as predictions of Weber electrodynamics fall somewhere in between the classical non-relativistic and classical relativistic electrodynamics in terms of accuracy. This is a very big deal, especially in view of the fact that Weber electrodynamics is easier to learn since the concept of fields is not involved.
Limitations of Weber Electrodynamics
Ironically, these very benefits of Weber electrodynamics are also it’s shortcomings: because in it’s original formulation the theory is only ‘partially’ relativistic, Weber electrodynamics must be extended in order to make correct predictions about charges moving with velocities approaching that of light (the original formulation of Weber electrodynamics predicts infinite velocities for electrons in particle accelerators, which are clearly not observed). Obviously, a fully relativistic extension of Weber electrodynamics will result in the concept of fields and radiation being introduced into the theory and thus likely take away from it’s original beauty and simplicity.
Conclusions
- Weber electrodynamics promises to be a much more useful theory than classical (non-relativistic) electrodynamics because it’s force law is compatible with the principle of action and reaction and therefore conserves total linear and angular momentum without the need to resort to the concept of fields.
- Therefore Weber electrodynamics is easier to learn and understand; Weber electrodynamics easily explains common phenomena of electromagnetic induction without causing paradoxes.
- Numerical predictions of Weber electrodynamics appear to be much more accurate than the predictions of classical (non-relativistic) electrodynamics,
- However, in its original formulation Weber electrodynamics apparently fails to establish a speed and therefore predicts infinite velocities for electrons accelerated in electrostatic fields, which are of course not observed. Still this seems like a minor and well-defined limitation that hardly detracts from the usability of Weber electrodynamics.
- Electromagnetic radiation is also outside of the scope of Weber electrodynamics.
Thus, if we start teaching Weber electrodynamics to a broader base we will enter a new and exciting era where more engineers are capable of wielding electromagnetic calculations with fewer mistakes. New inventions will follow. And this is what makes it exciting!