A superconducting ring can carry a current for an indefi nite length of time. Isn’t...

Perpetual motion machine is a kind of machine which can do work without any energy input.
A Superconductor is an hypothetical material which offers zero resistance. The entropy change in a superconducting material is zero as it is a highly ordered material.
From Second law of thermodynamics:
$$\begin{gathered} \mathrm{△}S=\frac{\mathrm{△}Q}{T} \\ 0=\frac{\mathrm{△}Q}{T} \\ \mathrm{△}Q=0 \end{gathered}$$
From first law of thermodynamics:
$$\begin{gathered} \mathrm{△}Q=\mathrm{△}U+\mathrm{△}W \\ 0=\mathrm{△}U+\mathrm{△}W \\ \mathrm{△}W=-\mathrm{△}U \end{gathered}$$
Since internal energy of a superconductor is constant.
$\mathrm{△}U=0$
Then,
Work done, $\mathrm{△}W=0$
But, in this case the superconductor is carrying the current for indefinite amount of time so work done can't be zero. So it is violating the First Law of Thermodynamics.