structural:summarization, linguistic:paraphrasing
Interpretation of magnetism by means of relative velocities

In the years after 1820, André-Marie Ampère carried out numerous experiments in which he measured the forces between direct currents. In particular, he also studied the magnetic forces between non-parallel wires. The final result of his work was a force law that is now named after him. In 1835, Carl Friedrich Gauss realized that Ampere's force law in its original form can be explained by a generalization of Coulomb's law.

Gauss's force law states that the electromagnetic force F 1 {\textstyle \mathbf {F} _{1}} experienced by a point charge, q 1 {\displaystyle q_{1}} with trajectory r 1 (t) {\displaystyle \mathbf {r} _{1}(t)}, in the vicinity of another point charge, q 2 {\displaystyle q_{2}} with trajectory r 2 (t) {\displaystyle \mathbf {r} _{2}(t)}, in a vacuum is equal to the central force

where r = r 1 (t) − r 2 (t) {\textstyle \mathbf {r} =\mathbf {r} _{1}(t)-\mathbf {r} _{2}(t)} is the distance between the charges and v = r ˙ 1 (t) − r ˙ 2 (t) {\textstyle \mathbf {v} ={\dot {\mathbf {r} }}_{1}(t)-{\dot {\mathbf {r} }}_{2}(t)} is the relative velocity. Wilhelm Eduard Weber confirmed Gauss's hypothesis in numerous experiments. By means of Weber electrodynamics it is possible to explain the static and quasi-static effects in the non-relativistic regime of classical electrodynamics without magnetic field and Lorentz force.

Since 1870, Maxwell electrodynamics has been developed, which postulates that electric and magnetic fields exist. In Maxwell's electrodynamics, the actual electromagnetic force can be calculated using the Lorentz force, which, like the Weber force, is speed-dependent. However, Maxwell's electrodynamics is not fully compatible with the work of Ampère, Gauss and Weber in the quasi-static regime. In particular, Ampère's original force law and the Biot-Savart law are only equivalent if the field-generating conductor loop is closed. Maxwell's electrodynamics therefore represents a break with the interpretation of magnetism by Gauss and Weber, since in Maxwell's electrodynamics it is no longer possible to deduce the magnetic force from a central force.