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A superconducting wire has no electrical resistance and as such it does not heat up when current passes through it. Non-superconducting wires can be damaged by too much current, because they get too hot and melt. But since current is the flow of electrons, surely unlimited electrons can't flow?

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There is a critical current density for every superconductor where the superconductor acts as an ordinary conductor and a voltage difference can be measured between its ends.

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As a side not current is not necessarily the flow of electrons, take AC for instance, which is better pictured as the flow of electromagnetic waves. –  Cem Nov 18 '10 at 14:01
    
And this implies that there is critical current for any piece of superconductor? Determined by the critical current density and its geometry? –  endolith Nov 18 '10 at 15:18
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It depends on many factors but yes, there is no such matter with infinite conductivity for all current values. –  Cem Nov 18 '10 at 15:23
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@endolith: I just read your question again and yes there is also such a thing called critical current which is exactly what you have defined =). –  Cem Nov 18 '10 at 15:39
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In addition to the fundamental limits noted by others there is the practical engineering issue of dealing with the increasing heating at the point where the superconductor joins the normally conducting parts of your circuit.

Get it wrong, and "Hello, Quench City!"

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The problem with trying to pass huge amounts of current through a superconductor is that any flowing current creates a magnetic field circling around it (Ampere's law). A superconductor also expels all magnetic fields from inside itself (Meissner effect), so what you get is a lot of magnetic field lines all bunched up just outside the surface of the superconductor, which makes a strong magnetic field there.

Magnetic fields kill superconductivity. Every superconductor has a "critical field" at which it stops superconducting (which actually depends strongly on temperature). So at some point the magnetic field produced by the current will be too much for the superconductor to handle, and it will "quench", which means it abruptly stops superconducting, and then the energy in the flowing current is very rapidly turned into heat.

A superconducting magnet quench can produce a sizeable explosion!

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Note that you can talk about a "critical current density", but the real physical origin of this is as I've described. The critical current density is certianly NOT due to a shortage of electrons, or a limitation on their speed. –  Keenan Pepper Nov 18 '10 at 16:02
    
Agreed. I was merely stating the result, not the procedure how. –  Cem Nov 18 '10 at 17:23
    
+1 I wanted to post the same. –  mbq Nov 18 '10 at 17:24
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"A superconducting magnet quench can produce a sizeable explosion!" True but this has nothing to do with superconductivity itself. –  Cedric H. Nov 18 '10 at 19:29
    
"A superconductor also expels all magnetic fields from inside itself (Meissner effect)," This is true only for type I SC or for type II below the first critical field. Usually magnets are type II operating between the critical fields, in that case the field penetrates the material partially. –  Cedric H. Nov 18 '10 at 19:31
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