How do Relativity explains electromagnetic induction in the case of changing $\bf{B}$ with both loop and magnet at rest?

Question

Here is how Griffiths lists the experimental evidences of electromagnetic induction.

Case 1 can be explained with Lorentz Force. I understood how Special Relativity relates case 2 to case 1, predicting the presence of an electric field in the frame of reference of the circuit, in place of what Lorentz Force would have created.

Nevertheless I do not see how Special Relativity relates case 3 to the other two: roughly speaking there is "nothing moving" in case 3, so how do I use Lorentz transformations of fields?

It can be said that, from the point of view of the loop, case 3 is just equal to case 2 since in both cases the magnetic field is changing and that's it. But I would like to know if there is a different way by which the transformation of fields relates case 3 to case 1.

In other words, do trasformations of fields predicts that, in case 3 an electric field will appear because of the changing magnetic field? If so, how do it predict that?

Answer 1

You do not need relativity to analyze the third case: all the causal actors in this scene are in the same inertial frame. Here it is simply a case of applying Faraday's law in its wonted for $\nabla\times\vec{E}=-\partial_t\,\vec{B}$.

However, relativity is tied in with this problem insofar that Maxwell's equations are of course covariant with respect to the Lorentz transformation. Therefore, you could realize an equivalent effect by shifting a graded magnetic field relative to the loop (as happens in an AC induction motor, for example), making it equivalent to case (b) and you'd get the same answer for the EMF around the loop.

Answer 2

If you know how a particle that is moving due to relativistic effects, it will create what we call magnetic field around itself that will rotate. With this, you can relate relativity with the induction and actually explain this experiment.

Since relativistic effects gave rise to this rotating magnetic field, it will have an angular momentum.

if we see the wire... free electrons in it, even if there is no electricity, are always moving from one atom to another randomly. So they all have a magnetic field that is circulating around them.

Now, when an external magnetic field is applied on the wire, each particle will try to conserve its own magnetic angular momentum since the external magnetic field will make them experience a torque on their own field. The particles will start to ''wobble around themselves'' or precess. The rate of precession can be calculated by the Larmor equation.

So now we have the particle precessing around themselves plus the fact that they where initially moving. This precession plus initial movement will make them go in a circular path, just like what the Lorentz force predicts and this is the physical why behind it.

A good analogy will be a spinning toy. The toy, if it doesn't spin, it falls down instantly and doesn't move. Just like the particle, if it's not initially moving, then the external magnetic field will not apply Lorentz force on it. But now when you spin the toy it will not fall. Instead, over time, as it loses its energy, it will start to wobble around its axis and with the way you throw the toy initially (assuming you threw it in a straight line), the initial tendency to go in straight line plus the wobbling of the toy (which is called precession if we want to get more technical) will result in the toy going in a circular path just like the electron did. We say that the precession of the toy opposes gravity and since the same thing happens to the electron, then the electron precess to oppose the external magnetic field that is making the torque on them to conserve their angular momentum.

So with this electrons will experience a Lorentz force when the magnet is placed out of the wire. But the question is if the electrons where initially moving randomly, shouldn't the addition of the precession to them make them go in a new random path and oppose the external magnetic field in a randomized way, or in other words, each electron experiences random Lorentz force? Well what the external magnetic field did is changing the electrons from being each a system of its own to unify them under one bigger system. With this unification, randomness tends to wash out. This is called scale invariance where the little randomized effects of every electron tend to wash out and they will all have a net effect that will make them oppose the external magnetic field in one specific manner. Since now the external magnetic field made the electrons a part of one bigger system, their precession and motion tendency will give rise to an averaged out Lorentz force so they'll move to oppose the external magnetic field to conserve their own angular momentum, and since now we have electrons placed in one place more than the other in the inductor then there will be voltage difference thus the inductor now can play the role as a generator (so we can say that the external magnetic field makes the voltage by unifying the electrons under one bigger system so when they try to precess and preserve their angular momentum in a determined manner, they will do it in a way where they average out their behavior and become displaced at the macroscopic level creating the voltage) so what relativity did here was to bring this rotational effect that is the magnetic field and by this relativity bring with it the concept of conservation of angular momentum that then explains everything as I said earlier.