If a superconductive wire can hold a current indefinitely, does that mean you can keep adding to the current and if so, how do you determine the capacity of a given length of superconductive cable? : askscience
the exact mechanism is unknown, but when the cooper pairs is given enough energy it will break apart into two electrons resulting in the end of superconductivity. If the current is increased in a superconductor you don’t end up with more carriers, but rather each carrier moves faster (remember I=dQ/dt). This mechanism means that at a high enough current we expect any cooper pairs to break apart. An interesting addendum is that the critical current is often of the order of magnitude to the current necessary to generate the critical magnetic field for a superconductor.
If you are familiar with basic E&M, you will recall that currents produce magnetic fields. The critical current density corresponds to the critical magnetic field at which the superconducting state is broken.
The next logical question is why the magnetic field breaks the superconducting state. A key property of (Type I) superconductors is the Meissner effect, which is their ability to expel magnetic flux completely (up to a small penetration depth at the surface) from the bulk to the superconducting material. This effect, and the superconducting state, are broken when the magnetic field reaches a certain critical value.