Confusion about one of Maxwell's equations

[Ax_xiom]

TL;DR

Why does a rotating wire in a magnetic field only cause a current within that wire

So Faraday's Law of induction states this: $$ \nabla \times E = - \frac {\partial B} {\partial t} $$
Or if we write it in it's integral form: $$ \int E \cdot dl = - \frac {d \Phi_B} {dt} $$ which (to my understanding) means that the magnitude of the EMF around a coil of wire will be equal to the magnitude of the change flux through that wire. I get confused in some cases as when we have a generator like this:

Nowhere in the region between the 2 magnets is the field actually changing, so (with my limited understanding) no electric field should be generated.

If we make the coil stationary and rotate the magnet instead, everywhere in between the magnets is experiencing a change in flux, so everywhere should have an electric field, but one is only generated within the wire

So what is wrong with my understanding?

 

[anuttarasammyak]

TL;DR: Why does a rotating wire in a magnetic field only cause a current within that wire

If we make the coil stationary and rotate the magnet instead, everywhere in between the magnets is experiencing a change in flux, so everywhere should have an electric field

Yes, loop integral of electric field is generated with no regard to the wire.

but one is only generated within the wire

Free electrons in the conductor wire on the loop are required for generation of the force which are charge times electric field.

 

[Dale]

TL;DR: Why does a rotating wire in a magnetic field only cause a current within that wire

Nowhere in the region between the 2 magnets is the field actually changing, so (with my limited understanding) no electric field should be generated.

That is correct. The EMF in the wire is not due to an E field. The EMF in the wire is due to the Lorentz force on the charges in the wire.

 

[Ax_xiom]

That is correct. The EMF in the wire is not due to an E field. The EMF in the wire is due to the Lorentz force on the charges in the wire.

So in most cases, is the induction from a coil in a magnetic field not due to Faraday's Law of Induction?

Free electrons in the conductor wire on the loop are required for generation of the force which are charge times electric field.

So the electric field is generated everywhere, but a current is only generated within the wire due to it having free electrons?

 

[anuttarasammyak]

So the electric field is generated everywhere, but a current is only generated within the wire due to it having free electrons?

Yes. No charge, no current.

 

[Dale]

So in most cases, is the induction from a coil in a magnetic field not due to Faraday's Law of Induction?

I wouldn’t say “most cases”. In very many cases you do have changing magnetic fields.

 

[Ax_xiom]

I wouldn’t say “most cases”. In very many cases you do have changing magnetic fields.

But not in the case of a coil rotating in a magnetic field (if so, are the underlying physics causing a current to be generated in a rotating wire in a magnetic field different to the physics of the same thing happening in a wire surrounded by a rotating magnet? If so, why?)

 

[Dale]

But not in the case of a coil rotating in a magnetic field

Correct.

are the underlying physics causing a current to be generated in a rotating wire in a magnetic field different to the physics of the same thing happening in a wire surrounded by a rotating magnet? If so, why?

It can be slightly different because in one case an accelerometer will detect that one is accelerating and in the other case the other will be accelerating. Rotation is non inertial.

To understand how an experiment or device works you need to consider Maxwell’s equations in a reference frame that is inertial, meaning it is not itself accelerating or spinning.