Critical Current Related to Magnetic Field

The critical current density in a superconductor can be found from V x H = J, the differential form of Ampere s law (or the Maxwell equation in the absence of E-fields). Integrating this quantity over a surface and applying Stokes theorem. 

Assume a wire with radius r. The field at the surface of the wire is 

Since the field only penetrates to depth A, the current density will only exist to depth A. Therefore, if A r, JJJ ds = iTirXJc- Therefore from Equation 26.33, the critical field is 

We see that the limiting current is not affected by the wire size provided r A. If the wire with radius r A is carrying current /, the critical field at the surface will be reached when Jc = 2Hclr. In this case, the critical current density increases inversely with r, but since the current I = nr, the critical current will be directly proportional to r. 

The temperature dependence of the critical field. He, was given by Equation 26.38 as He (T)/Hc (0) 1 — (T/Tc). Since both A and Jc are directly proportional to He, they follow the same temperature dependence. 

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