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Does the magnitude of current induced in a shorted superconducting coil depend on the external flux's rate of change $\frac{d\Phi_{EXT}}{dt}$ ?

Assume that initially the flux through the coil is zero and the initial current circulating in that coil is zero, too, but at some later time, an external flux source attempts to change the flux threading that shorted ideal coil to $\Phi_{EXT}$.

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Since the total flux through the superconducting loop remains constant at $0$ (the reason for this is that any change in flux requires a nonzero emf around the loop, which requires in infinite current, so magnetic flux through the loop cannot change), the flux from the self-inductance $L$ of the loop must be equal and opposite to the external flux. We conclude that $I=\frac{\Phi}{L}$.

To answer the question, no: current only depends on the flux, not the rate of change.

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  • $\begingroup$ Note this phenomenon, described by @DanDan0101 is independent on how fast the flux changes; it can be days, it can be nanoseconds. If the changes were on picosecond scale, the corresponding energy ($\hbar/\tau$) could get similar to the pseudogap and some induced energy could be lost in the semiconductor. $\endgroup$
    – dominecf
    Oct 14, 2020 at 19:32
  • $\begingroup$ What physical law says that the flux generated by the self-inductance of the coil must be equal and opposite to the external flux? When I read about the Lenz Law on Wikipedia, it only talks about the opposite DIRECTION of the induced current. It says nothing about its magnitude being such, that it opposes the entire external flux. It also says that the Lenz law is not a quantitative law. $\endgroup$ Oct 14, 2020 at 20:30
  • $\begingroup$ @GeorgeRobinson superconducting loop has frozen magnetic flux, because it is superconductive. Any external induced electric field is counteracted by induced electric field of the loop and the result is total electric field inside is zero. $\endgroup$ Oct 14, 2020 at 21:04
  • $\begingroup$ @Jan. Why are you bringing up electric field when coils are concerned primarily with the magnetic field ? Capacitors work with electric field, don't they? $\endgroup$ Oct 14, 2020 at 21:33
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    $\begingroup$ Because total electric field equals zero, hence magnetic flux cannot change. This is Faraday's law of EM induction. $\endgroup$ Oct 15, 2020 at 0:37

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