According to Faraday's law of induction, volts = -Number of coils in a solenoid * change in strength of magnet / change in time. This doesn't take into account distance or speed, only time. If amps = volts / ohms, and ohms is 0, it seems like amps should be infinity. If there are infinite amps, then wouldn't the alternator just generate a constant voltage without the magnet needing to move at all?

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You just asked a question about Ohm's law in superconductor. The answer you accepted states that Ohm's law does not apply in superconductors and therefore no infinite current. What are you asking then? – Siyuan Ren Jul 9 '12 at 8:10
the superconductor blocks magnetic and electric fields from entering the interior too, not just magnetic fields. – Ron Maimon Jul 9 '12 at 9:28

You seem a bit confused on your standing conditions, so we'll go at this step by step.

From this, we can easily deduce that $V=0$

Now, $V=IR$, and we have $R=0$ as well. Thus, $0=I\times 0$, and then $I$ can be any value. (In case of a non-perfect superconductor, we have $R \approx 0$, and we can derive $I=0$, since $0=IR$ and $R\neq 0$)

OK, so you got a constant current. What of it? The current will only be able to course forever in the superconductor ring. You cannot try to draw energy off it, because all ways of drawing energy off current increase the resistance. If you add a load coil/etc, there will obviously be resistance. If you try to harness its magnetic field, the fluctuations in the field will induce a reverse current on the coil, acting as a resistance.

If amps = volts / ohms, and ohms is 0, it seems like amps should be infinity.

You completely forgot the other alternative--volts can be zero instead. Then you can easily see that the current can be any value.

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Now what if the magnet does move? V is no longer 0. If there were no resistance, would the voltage persistently increase as the magnet continued to move? – Waffle Jul 9 '12 at 5:46
No superconductor is perfect?????? – Siyuan Ren Jul 9 '12 at 5:58
Even if the superconductor isn't perfect, that isn't the point. If a superconductor can last to the end of the universe before significant voltage is converted into heat, then that is good enough for me. – Waffle Jul 9 '12 at 6:04
Also, why does V = 0 if the magnet is not moving? Faraday's law of induction does not say that. – Waffle Jul 9 '12 at 6:13
@Waffle: (1) If the magnet moves, we get $V>0$. If $R=0$, then obviously current is infinite. Nothing wrong with that. (2) No, in this case there is a big difference between $R=0$ and $R\approx 0$. In the former, we get $I=\text{any value}$. In the latter, we get $I=0$. (Though it can be zero in the former case as well). (3) I'm assuming you're using a permanent magnet. To have a change of flux, you must have relative motion between the magnet and coil. If it is not moving, there is no change of flux, and thus $V=0$ – Manishearth Jul 9 '12 at 6:56