The magnet and conductor problem.

[spaghetti3451]

I am trying to understand the magnet and conductor problem that Einstein mentions at the beginning of one of his 1905 papers.

In the frame of reference of the conductor, the magnet moves. Why does this mean that Faraday's law is applicable and not Lorentz force law?

In the frame of reference of the magnet, the conductor moves. Why mean that Lorentz force law is applicable and not Faraday's law?

The experiment generates the same current if either one of the items moves. This is taken as evidence that we can't tell if the magnet is at rest or the conductor, if any. I don't understand how?

 

[vanhees71]

If the conductor is at rest and the magnet is moving, the current is produced by the electric field, given by Faraday's Law (one of Maxwell's equations, which are the first relativistic field theory, but that was unknown to its discoverers, Faraday and Maxwell).

If the magnet is at rest and the magnet is moving, the current is induced due to the Lorentz force on the moving electrons. Since the (special) principle of relativity is valid, although not in its Galilean form, observers from both reference frames will measure the same current. The Galileo transformation, however, doesn't give the correct result (or better said only approximately for small relative velocities between magnet and conductor).

An understanding of the issue of how to transform correctly from one frame to the other has been reached after a lot of work by Lorentz, FitzGerald, Poincare, H. Hertz through Einstein's seminal paper from 1905 and finally through the mathematical analysis by Minkowski in 1907.

 

[Philip Wood]

Hope this will complement the post of VanHees71 ...

The Lorentz force on charge q moving with velocity v is F = q(E + vxB). So only if the charge is moving can it experience a magnetic force, i.e. a force due to B. So in the magnet's frame, in the case you raised, the conductor, and hence the charges in it, are moving and will experience magnetic forces.

But in the conductor's frame, v = 0, so there's no magnetic force on the charges in the conductor, (i.e. no forces due to B). But there is a force on the charges due to an electric field, E. [This is, strictly, still called a Lorentz force.]

But, you may ask, where does this E come from? It turns out that the B in the magnet's frame transforms to a B and an E in the conductor's frame. This can be seen by applying the Lorentz space-time transforms to Maxwell's equations. The equations stay the same but the value of the E and B components differ in two frames in relative motion.

It's worth noting that if we consider the conductor to be part of a closed loop, the induced emf is proportional to the rate of change of magnetic flux linking the loop, and this applies whether the conductor or the magnet is moving.

 

[vanhees71]

I highly recommend the following nice article on the subject:

P. J. Scanlon, R. N. Henriksen, J. R. Allen, Approaches to Electromagnetic Induction, Am. Jour. Phys. 37, 698 (1969)

 

[Philip Wood]

I wish I had access to this article! I take your recommendations seriously.