Suppose, we have a bulky ring of aluminum. Above the superconducting transition temperature, we place a magnet inside the ring so that there´s a finite magnetic flux through its hole. Now, we cool below the critical temperature and pull the ring from the hole. We will induce a circular supercurrent which, in turn, induces a magnetic field such that the magnetic flux through the hole persists.

So far so good - what about the dynamics?

Apparently, the supercurrent can adopt 'simultanesouly' (that is, leaving relativity aside for a moment) to changes of an external magnetic field that tries to change the flux through the hole. Or, equivalently, a magnet flux-pinned to a type-II superconductor will immediately generate a restoring force upon relocation by virtue of inducing a suited supercurrent and magnetic field.

Now if we consider basic electrical wisdom, we will find that a current we want to pass through a wire loop by applying a voltage $V_0$ to its ends will reach its final value only asymptotically


with a time constant $\tau=L/R$ set by the inductance $L$ and ohmic resistance $R$ of the loop. Obviously, there´s some trouble lurking here if we want to apply this reasoning to our supercoducting ring.

But first the wire loop in real life: if we make the major part of it superconducting, $R$ will of course drop significantly, but not to zero as there will always be some ohmic leads, contacts, etc connecting to the voltage source. We can calculate a well defined $\tau$, use above equation and all´s good

But how does this reasoning apply to our entirely superconduting ring? Here, $R=0$ and $\tau$ diverges. It would take an infinitely long time to induce the supercurrent - which is clearly at odds with our experiments. Even if we consider the finite ac conductivity for time-dependent currents (Mattis and Bardeen, 1958) and thus the vanishingly small but finite resistance $R$, we would end up with $\tau$ being much bigger than any experimental time scale. But the reaction and hence the suppercurrent is there with essentially no delay!

After letting thoughts run freely (e.g. can we, after all, define a meaningful $L$ for a superconducting ring without any voltage drop? Must we set $L=0$ as the circular supercurrent runs perfectly symmetric and thus nothing changes with time in a continuum limit? ...) I have not settled to a conclusive solution to this conundrum and I´m happy for any suggestions.


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