Moving magnet and conductor problem
The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity.
[1][better source needed] However, according to Maxwell's equations, the charges in the conductor experience a magnetic force in the frame of the magnet and an electric force in the frame of the conductor.
[2] Einstein's 1905 paper that introduced the world to relativity opens with a description of the magnet/conductor problem:[3] It is known that Maxwell's electrodynamics – as usually understood at the present time – when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena.
For if the magnet is in motion and the conductor at rest, there arises in the neighborhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated.
In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise – assuming equality of relative motion in the two cases discussed – to electric currents of the same path and intensity as those produced by the electric forces in the former case.
Consistency is an issue because Newtonian mechanics predicts one transformation (so-called Galilean invariance) for the forces that drive the charges and cause the current, while electrodynamics as expressed by Maxwell's equations predicts that the fields that give rise to these forces transform differently (according to Lorentz invariance).
Observations of the aberration of light, culminating in the Michelson–Morley experiment, established the validity of Lorentz invariance, and the development of special relativity resolved the resulting disagreement with Newtonian mechanics.
Special relativity revised the transformation of forces in moving reference frames to be consistent with Lorentz invariance.
A description that uses scalar and vector potentials φ and A instead of B and E avoids the semantical trap.
A Lorentz-invariant four vector Aα = (φ / c, A) replaces E and B[5] and provides a frame-independent description (albeit less visceral than the E– B–description).
[6] An alternative unification of descriptions is to think of the physical entity as the electromagnetic field tensor, as described later on.
The existence of classical electromagnetic fields can be inferred from the motion of charged particles, whose trajectories are observable.
Electromagnetic fields do explain the observed motions of classical charged particles.
This requirement places constraints on the nature of electromagnetic fields and on their transformation from one reference frame to another.
It also places constraints on the manner in which fields affect the acceleration and, hence, the trajectories of charged particles.
Perhaps the simplest example, and one that Einstein referenced in his 1905 paper introducing special relativity, is the problem of a conductor moving in the field of a magnet.
Newton's law of motion, however, had to be modified to provide consistent particle trajectories.
these relations hold, and therefore the Lorentz force equation is also valid if the magnetic field in the conductor frame is not varying in time.
At relativistic velocities a correction factor is needed, see below and Classical electromagnetism and special relativity and Lorentz transformation.
A similar sort of argument can be made if the magnet's frame also contains electric fields.
By plugging these transformation rules into the full Maxwell's equations, it can be seen that if Maxwell's equations are true in one frame, then they are almost true in the other, but contain incorrect terms proportional to the quantity v/c raised to the second or higher power.
Accordingly, these are not the exact transformation rules, but are a close approximation at low velocities.
This result is a consequence of requiring that observers in all inertial frames arrive at the same form for Maxwell's equations.
In particular, all observers must see the same speed of light c. That requirement leads to the Lorentz transformation for space and time.
Special relativity modifies space and time in a manner such that the forces and fields transform consistently.
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.
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:
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).
The relations connecting time and space are ( primes denote the moving conductor frame ) :[11]
