Weber electrodynamics is a theory of electromagnetism that preceded Maxwell electrodynamics and was replaced by it by the end of the 19th century. Weber electrodynamics is mainly based on the contributions of André-Marie Ampère, Carl Friedrich Gauss and Wilhelm Eduard Weber. In this theory, Coulomb's law becomes velocity and acceleration dependent. Weber electrodynamics is only applicable for electrostatics, magnetostatics and for the quasistatic approximation. Weber electrodynamics is not suitable for describing electromagnetic waves and for calculating the forces between electrically charged particles that move very rapidly or that are accelerated more than insignificantly. The outstanding feature of Weber electrodynamics is that it makes it possible to describe magnetic forces between direct currents, low-frequency alternating currents, and permanent magnets without a magnetic field.
History Around 1820, André-Marie Ampère carried out numerous systematic experiments with direct currents. Eventually in 1823 he developed the force law
which can be used to calculate the force d 2 F {\displaystyle d^{2}\mathbf {F} } that a current element I 2 d s 2 {\displaystyle I_{2}\,d\mathbf {s} _{2}} exerts on another current element I 1 d s 1 {\displaystyle I_{1}\,d\mathbf {s} _{1}} . Here, r {\displaystyle \mathbf {r} } is the vector that points from the current element I 2 d s 2 {\displaystyle I_{2}\,d\mathbf {s} _{2}} to the current element I 1 d s 1 {\displaystyle I_{1}\,d\mathbf {s} _{1}} . A current element I d s {\displaystyle I\,d\mathbf {s} } should be interpreted as a very short segment of the length s {\displaystyle s} of a conductor with a direct current I {\displaystyle I} flowing in the direction of s {\displaystyle \mathbf {s} } . In 1835, Carl Friedrich Gauss realized that Ampère's force law can be interpreted by a minor generalization of Coulomb's law. He postulated that the electric force exerted by a point charge q 2 {\displaystyle q_{2}} on another point charge q 1 {\displaystyle q_{1}} depends not only on the distance r = r 1 − r 2 {\displaystyle \mathbf {r} =\mathbf {r} _{1}-\mathbf {r} _{2}} , but also on the relative velocity v = r ˙ 1 − r ˙ 2 {\displaystyle \mathbf {v} ={\dot {\mathbf {r} }}_{1}-{\dot {\mathbf {r} }}_{2}} :
Importantly, Gauss's force law is a significant generalization of Ampere's force law, since moving point charges do not represent direct currents. In fact, today, Ampere's force law is no longer presented in its original form, as there are equivalent representations for direct currents such as the Biot–Savart law in combination with the Lorentz force. This is the point at which Weber electrodynamics and Maxwell electrodynamics take different paths, because James Clerk Maxwell decided to base his theory on the Biot–Savart law, which was originally also only valid for closed conductor loops. Wilhelm Eduard Weber's contribution to Weber electrodynamics was that he extended Gauss's force formula in such a way that it was possible to provide a formula for the potential energy. He presented his formula in 1848 which reads
with r ˙ {\displaystyle {\dot {r}}} being the radial velocity. Weber also carried out numerous experiments and documented the state of knowledge at this time in his substantial work. Weber electrodynamics and Gauss's hypothesis fell gradually into oblivion after the introduction of the displacement current around 1870, since the full set of Maxwell equations made it possible to describe electromagnetic waves for the first time. From around 1880, experiments such as the Michelson–Morley experiment showed that electromagnetic waves propagate at the speed of light regardless of the state of motion of the transmitter or receiver in a vacuum, which is not consistent with the predictions of Maxwell's equations, since these describe wave propagation in a medium. To overcome this problem, the Lorentz transformation was developed. As a result, Gauss's hypothesis that the electric force depends on the relative velocity was added back in a modified form.
Mathematical description
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