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We all know that moving a magnet through a loop of wire induces a current, like in this youtube experiment here. Similarly, we know that moving one solenoid (with a battery hooked up) through a larger solenoid (with no battery) will induce a current in the larger solenoid, just like in Faraday's original experiment setup below. Call the small solenoid S1 and the large one S2.

enter image description here

But for the charges in the S2's wire to move, they must experience a force from V cross B (where V is velocity of S1, and B is magnetic field of S1). Since V points along S1, there must be some B component that points radially. See image below; is it safe to say that S2's induction is due to the field lines outside S1, not inside?

enter image description here

If yes, then what would happen if you used truly infinite solenoids? Then you would have no field lines outside S1, so would there be no induced current at all?

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Faradays law is that the induced voltage in a coil is

$\epsilon = -N\frac{\Delta \phi}{\Delta t}$

Where $N$ is the number of turns in the coil, $\Delta \phi$ is the change of magnetic flux through it and $\Delta t$ is the change in time.

For infinite coils the $\Delta \phi$ part would be zero, so you are right that there would be no induced voltage or current at all. It's the change in the magnetic flux that matters...

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