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The behavior of magnetic flux in the ring-shaped finite-gap superconductors is explored from the view-point of the flux-conservation theorem which states that under the variation of external magnetic field "the magnetic flux through the ring remains constant" (see, e.g., [L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuos Media, vol. 8 (New York, Pergamon Press, 1960), Section 42]). Our results, based on the time-dependent Ginzburg-Landau equations and COMSOL modeling, made it clear that in the general case, this theorem is incorrect. While for rings of macroscopic sizes the corrections are small, for micro and nanorings they become rather substantial. The physical reasons behind the effect are discussed. The dependence of flux deviation on ring sizes, bias temperature, and the speed of external flux evolution are explored. The detailed structure of flux distribution inside of the ring opening, as well as the electric field distribution inside the ring's wire cross section are revealed. Our results and the developed finite element modeling approach can assist in elucidating various fundamental topics in superconducting nanophysics and in the advancement of nanosize superconducting circuits prior to time-consuming and costly experiments.


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