Moving Magnet and Conductor Problem - Modification of Dynamics For Consistency With Maxwell's Equations

Modification of Dynamics For Consistency With Maxwell's Equations

The Lorentz force has the same form in both frames, though the fields differ, namely:

See Figure 1. To simplify, let the magnetic field point in the z-direction and vary with location x, and let the conductor translate in the positive x-direction with velocity v. Consequently, in the magnet frame where the conductor is moving, the Lorentz force points in the negative y-direction, perpendicular to both the velocity, and the B-field. The force on a charge, here due only to the B-field, is

while in the conductor frame where the magnet is moving, the force is also in the negative y-direction, and now due only to the E-field with a value:

The two forces differ by the Lorentz factor γ. This difference is expected in a relativistic theory, however, due to the change in space-time between frames, as discussed next.

Relativity takes the Lorentz transformation of space-time suggested by invariance of Maxwell's equations and imposes it upon dynamics as well (a revision of Newton's laws of motion). In this example, the Lorentz transformation affects the x-direction only (the relative motion of the two frames is along the x-direction). The relations connecting time and space are ( primes denote the moving conductor frame ) :

These transformations lead to a change in the y-component of a force:

That is, within Lorentz invariance, force is not the same in all frames of reference, unlike Galilean invariance. But, from the earlier analysis based upon the Lorentz force law:

which agrees completely. So the force on the charge is not the same in both frames, but it transforms as expected according to relativity.