One very simple method for measuring the critical temperature of a superconductor is by using the Meissner Effect. Remember when the temperature of a superconductor is lowered to below the critical temperature, (Tc), the superconductor will "push" the field out of itself. It does this by creating surface currents in itself which produces a magnetic field exactly countering the external field, producing a "magnetic mirror". The superconductor becomes perfectly diamagnetic, canceling all magnetic flux in its interior. This perfect diamagnetic property of superconductors is perhaps the most fundamental macroscopic property of a superconductor. When your material is superconducting the magnet will begin to float above the superconductor. If you monitor the temperature as the Meissner effect occurs you can obtain a good approximation of the critical temperature.
Remember the disk will not be warming equally throughout. The top of the superconductor will warm up before the bottom of the disk which is still in liquid nitrogen. Because YBCO disks are bulk superconductors, the Meissner effect will diminish as the top of the disk warms until the entire superconductor is above the critical temperature.
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