My understanding is that the apparent violation of Newton's Third Law by the Lorentz Force necessitates a description of the system that describes the "missing" momentum as being absorbed/carried by the magnetic field itself. What is not clear to me, however, is why this apparent violation cannot yield an apparent unidirectional force to the system.
For example: In a system with 2 magnets aligned across from one another and a current carrying rod placed parallel to the stage between them (perpendicular to the $B$ field lines), application of current to the rod can result in the Lorentz force vector pointing upwards acting on the rod. If the rod were to be affixed to the magnets themselves by a non-conducting support, preventing the rod from moving upward out of the $B$ field between the magnets, would the upward force acting on the rod not be transmitted to the magnets themselves? This seems to imply that the entire system (rod, electrons within the rod, and the magnets supplying the $B$ field) would experience an upward force.
This can't possibly be the case because that'd basically produce an anti-gravity device, but the math seems to suggest it, meaning I'm missing something fundamental somewhere. If the back-reaction from the Lorentz Force is not being transmitted to the magnets (pushing them "down" to counter the the upward vector force applied to the rod) but instead is being absorbed by the $B$ field, what prevents the system in this setup from experiencing a net upward force?
Best Answer
The moment you rigidly connect the magnet (or two magnets in your example) to the rod, they become one body.
Therefore a force between them should be treated as an internal force, which means that it cannot move the body as a whole, due to conservation of momentum.