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WHAT IS SUPERCONDUCTIVITY ?
Superconductivity is a phenomenon that is occurring at low temperature based on quantum mechanical pairing up of the electrons in metals. The precise explanation of the phenomenon is quite complicated and will not be attempted here.
Some basic facts
A superconductor becomes superconducting when it is cooled down under its critical temperature Tc. The highest critical temperature measured so far is of order -110C in an oxide superconductor.
Most metals are superconducting. Exceptions are noble metals, magnetic metals and most good conductors like copper. However, aluminium is a superconductor.
The critical temperature itself is dependent on the magnetic field to which the superconductor is exposed to. For technical superconductors the critical temperature becomes zero at the upper critical field Hc2.
The presence of a transport current also affects superconductivity, however not necessarily in a straight forward fashion. Typically operating current densities are limited to the critical current density, which is dependent on temperature and magnetic field.
Some of the properties
Electrical currents are transported loss-less in ideal superconductors. The electrical resistance becomes zero. This is the most interesting property for the use in power cables.
The current densities at which the loss-less transport of current occurs is 100 to 1000 times higher than what is possible in normal metals like copper. This enables very high power densities in power cables, and generation of magnetic fields 5 to 10 times higher than those found in permanent magnets based on ferromagnetism.
The superconductor is perfectly or partially diamagnetic. This property is the basis for other applications of superconductivity, for example magnetic levitation.
Use of superconductors
The most common place to find a superconductor today is in hospitals. Most magnetic resonance body scanners contain a magnet in the superconducting state in which current circulates forever due to the loss-less transportation of current!

Superconductivity
is electric current without losses