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When a force is applied to a superconductor to move it in a direction that it resists because it is flux pinned, where does the additional energy used to overcome the flux pinning resistance go? E.g. is it dissipated as radiation, current, lateral/rotational motion or something else?

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The energy you supply by the force is transformed into changing the flux tube configuration. Microscopically this means that certain parts of the superconductor go from superconducting to "normal" in a small areas (flux tubes) to allow the magnetic flux through and the energy supplied is used to rearrange these regions. In superconductors that stay fixed above a magnet the flux tubes are generally stuck to defects in the lattice. In this case it costs energy to move the superconductor away since energy is needed to create flux tubes not near a defect.

As a side note, I love STM movies of flux tubes: https://www.youtube.com/watch?v=vwFm7d_0GsA

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