Onoochin’s Paradox

Vladimir Onoochin has drawn our attention to a paradox related to the apparatus shown in the figure below. Two electric charges −q are forced to execute uniform circular motion with speed v such that their azimuthal angles are always 90 ◦ apart. The apparatus, including the motors which drive the motion, is mounted on a frictionless platform such that there are no external forces on the system that have components in the horizontal (x-y) plane. Therefore, we expect that the center of mass of the system would remain at rest. However, the magnetic Lorentz forces of the moving charges on each other are not equal and opposite, so the sum of the internal forces is nonzero. In particular, the total (magnetic) Lorentz force has a component in the −x direction at all times during the motion. Hence, we are led to conclude that the center of mass of this system is accelerated monotonically in the −x direction by its own internal forces, in contradiction to the principles of Newtonian mechanics. The sum of the (magnetic) Lorentz forces on the two charges is (in Gaussian units) of order q2v2/c2r2, where c is the speed of light and r is the average distance between the two charges. For any reasonable tabletop apparatus the apparently offending Lorentz force is extremely tiny. Thus, the issue is primarily one of principle: Are the laws of electromagnetism are compatible with the laws of mechanics?