Doubts about basics of superconductivity.

[ShreyasR]

I am just introduced to basics of superconductivity in Engineering Physics.
My lecturer asked a question: Is the resistance of a superconductor zero or close to zero?
And my answer was close to zero. I have read that the resistance of a superconductor is exactly zero, but then when I put some thought into it, I still feel that the resistance of a superconductor is so low that it cannot be measured, may be in the order of 10^-30Ω or so (or even lesser).
Reason being:
Electrons/charge carriers/cooper pairs will have a certain mass, and when they are moving, they will have some amount of kinetic energy. If we consider that there will be no collisions, and if we consider that the charge carriers move in a closed circular superconductor loop. There is a continuous change in direction (tangentially along the circular loop) There is no actual centripetal force causing the change in direction as there is NOTHING at the center of the circular loop. Hence my conclusion is work is being done continuously. Hence superconductors will be having some resistance.
One more interpretation is from Ohm's law:

I=V/R

If R is 0, V should be 0 if there is going to be some current flowing.
But Mathematically 0/0 is not defined. where as
Lim (V/R) = a finite value of I
V,R-->0

So R must be so small that it cannot be measured, and P=I²R also cannot be measured. This would have given us the result that Resistance of a superconductor is 0.

If I am wrong please correct me.

 

[mfb]

You don't have to do work to keep electrons moving in a circle, not even in a classical description. They do not change their speed. You just need some potential border on the outside. The borders of the cable provide that (a very classical description!).

But Mathematically 0/0 is not defined.

That just means that you used a wrong formula. Use RI=V. R=0, V=0, everything is fine. And you can NOT divide this by R for a superconductor, as R=0.

 

[daveyrocket]

Ohm's law definitely isn't something you should look to for any sort of proof about unusual phenomena. It breaks down for all sorts of common non-superconducting devices (diodes, transistors, incandescent light bulbs, etc.)

 

[ShreyasR]

You don't have to do work to keep electrons moving in a circle, not even in a classical description. They do not change their speed. You just need some potential border on the outside. The borders of the cable provide that (a very classical description!)..

Can you please elaborate? If there is a potential border, doesn't it mean that V≠0?

 

[mfb]

It has nothing to do with the electric potential along the line. This border is the border of the cable (perpendicular to the current flow) - and you do not have current outside the cable.
Anyway, don't try to extend that model too far. As I said, it is a very classical view, and this simplified model will break down if you look at it too closely.

 

[Dali]

...If I am wrong please correct me.

Superconductivity is a many-body quantum mechanical effect, and can not be understood in terms of semi-classical pictures. The reason a superconductor has zero resistance is because the groundstate of the many-body system of interacting electrons has a finite energy gap to its first possible excited state. Provided the gap is larger than the available thermal energy, the system simply has no way to get rid of any extra kinetic energy. So regardless of what you do to the system, it cannot dissipate any energy. That is why the resistance is zero regardless of impurities or detailed shape of potentials confining it etc.

 

[ShreyasR]

Ok! Thanks everyone :)