Superconductivity & the Meissner Effect

[Jimmy87]

Hi,
I am new to superconductivity and have been doing a lot of reading to try and become familiar with it. I have come across a few questions that I would be grateful is someone could answer. I am confused about the mechanism that describes the expulsion of magnetic fields from inside a superconductor. The source of my confusion came from this YouTube clip from an MIT professor where he talks about superconductivity at 30mins 30seconds in.



He argues that a superconductor expels magnetic fields because if one did penetrate it then an emf would be generated across it and if the resistance is zero, the current would go to infinity according to Ohm's Law. If this is true then what about if you take a superconducting material above the critical temperature and expose it to a magnetic field. This would cause a magnetic field to go through the superconducting material. If you then cool it until it becomes superconducting then all the sources I have read say that the magnetic field is expelled. Would this not be considered a change in magnetic field and therefore generate an infinite current using this logic from the lecturer? What is the reason magnetic fields are expelled from superconductors? Is it to satisfy the infinite current caused by a changing flux?

Thanks.

 

[DrDu]

The point is that you have to accelerate the cooper pairs and by Newtons law this acceleration is proportional to E/2m, where E is the electric field due to the changing magnetic field and m is the electron mass. As E is zero before the condensation of Cooper pairs and after expulsion of the magnetic field, E only acts for a finite time and will only lead to a finite velocity of the cooper pairs, i.e. a finite current, which is precisely large enough to expell the exterior magnetic field.

 

[Jimmy87]

The point is that you have to accelerate the cooper pairs and by Newtons law this acceleration is proportional to E/2m, where E is the electric field due to the changing magnetic field and m is the electron mass. As E is zero before the condensation of Cooper pairs and after expulsion of the magnetic field, E only acts for a finite time and will only lead to a finite velocity of the cooper pairs, i.e. a finite current, which is precisely large enough to expell the exterior magnetic field.

Thanks. What is the reason for the expulsion of the magnetic field inside a superconductor? Is it argued like the professor in terms of conservation of energy where the magnetic field is expelled so that an electric field is not generated inside the superconductor which would lead to an infinite current?

 

[DrDu]

There are various ways to understand the expulsion of the magnetic field in a superconductor and this energetic argument is certainly valid.
The most interesting interpretation of the expulsion of the magnetic field is the Anderson-Higgs mechanism which explains the expulsion with the Cooper pairs acting as a Higgs field which gives mass to the effective electromagnetic field inside the superconductor, so that the field has to decay exponentially at the surface. This mechanism was first developed for superconductors before it was applied in high energy physics. However, the mathematical details are rather advanced.

 

[Jimmy87]

There are various ways to understand the expulsion of the magnetic field in a superconductor and this energetic argument is certainly valid.
The most interesting interpretation of the expulsion of the magnetic field is the Anderson-Higgs mechanism which explains the expulsion with the Cooper pairs acting as a Higgs field which gives mass to the effective electromagnetic field inside the superconductor, so that the field has to decay exponentially at the surface. This mechanism was first developed for superconductors before it was applied in high energy physics. However, the mathematical details are rather advanced.

Thanks again. If you say the energy argument is valid could you explain some confusion I have with it:

I totally get that if you approach a superconductor below its critical temperature with a magnet then it will expel the magnetic fields so that the rate of change of flux is zero to give no emf and therefore no electric field. The small problem I have with this is that surely an emf (and electric field) must be induced to drive the eddy currents which expel the magnetic field. Does this not violate energy conservation if R=0 but V is non zero?

The other problem is when you let a magnetic field penetrate the material above its critical temperature. You then cool it and the magnetic field is expelled. Since it had a flux which has now gone to zero isn't there a change in flux which again would give a non zero V with R=0?

 

[DrDu]

Thanks again. If you say the energy argument is valid could you explain some confusion I have with it:

I totally get that if you approach a superconductor below its critical temperature with a magnet then it will expel the magnetic fields so that the rate of change of flux is zero to give no emf and therefore no electric field. The small problem I have with this is that surely an emf (and electric field) must be induced to drive the eddy currents which expel the magnetic field. Does this not violate energy conservation if R=0 but V is non zero?

The other problem is when you let a magnetic field penetrate the material above its critical temperature. You then cool it and the magnetic field is expelled. Since it had a flux which has now gone to zero isn't there a change in flux which again would give a non zero V with R=0?

I thought we had already cleared this point. What is V, btw? Voltage? A magnetic field cannot be described in terms of voltage.
Your problem may be that you don't completely understand the implications of vanishing resistance.
Consider a massive sphere in a viscous medium. If you pull with constant force, then, after a long time, the velocity will be proportional to the force and inversely proportional to viscosity. This is Ohm's law, basically. But when you start pulling, the sphere will accelerate. This acceleration phase will be the longer the lesser the viscosity of the medium. I.e., the time after which Ohm's law is applicable will be the longer the lesser the resistance. In a medium with zero resistance, Ohm's law will never hold because the charge carriers will experience constant acceleration. In the case of a superconductor, this acceleration will continue until the induced magnetic field, which is proportional to the velocity will compensate the external magnetic field inside the superconductor. Hence you end up with a finite current density in the material.

 

[Jimmy87]

I thought we had already cleared this point. What is V, btw? Voltage? A magnetic field cannot be described in terms of voltage.
Your problem may be that you don't completely understand the implications of vanishing resistance.
Consider a massive sphere in a viscous medium. If you pull with constant force, then, after a long time, the velocity will be proportional to the force and inversely proportional to viscosity. This is Ohm's law, basically. But when you start pulling, the sphere will accelerate. This acceleration phase will be the longer the lesser the viscosity of the medium. I.e., the time after which Ohm's law is applicable will be the longer the lesser the resistance. In a medium with zero resistance, Ohm's law will never hold because the charge carriers will experience constant acceleration. In the case of a superconductor, this acceleration will continue until the induced magnetic field, which is proportional to the velocity will compensate the external magnetic field inside the superconductor. Hence you end up with a finite current density in the material.

Ok I think I get you. So there is an induced electric field inside the superconductor but it only lasts long enough to accelerate the electrons to a velocity that opposes the external field?

 

[DrDu]

Yes, exactly