# Lorentz force in superconductors

## Main Question or Discussion Point

Hi, everyone.

In a course on superconducting materials, my lecturer has suggested that in a Type I(one) superconductor, any normalconducting region containing trapped magnetic flux will feel a Lorentz force per unit volume $$F_L = J \times B$$, where $$J$$ is the transport current density (vector!) that the material is carrying, and $$B$$ is a vaguely-defined magnetic flux density.

He goes on to define the "critical current density" $$J_c$$ by the equality $$J_c \times B = -F_p$$ where $$F_p$$ is a force per unit volume due to pinning of the flux lines on some kind of material defect.

My problems with this are:
• Concept - how can a non-charged body feel a Lorentz force? (This is probably solved by thinking about a supercurrent that surrounds the flux-containing region...)
• What is $$B$$, given that the Meissner effect excludes magnetic fields in superconducting regions? Could is be the externally-applied field, measured at a distance? Or the (higher) flux density inside the flux-containin region of the material?
• How can I reconcile my lecturer's definition of $$J_c$$ with the more generally-available definition: "the maximum current a superconductor can carry before making a transition back to normal conduction"?

Any help would be much appreciated, as I'm very stuck on this concept and I can't find any online resources which mention this particular phenomenon - thanks!

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seycyrus
Hi, everyone.

In a course on superconducting materials, my lecturer has suggested that in a Type I(one) superconductor, any normalconducting region containing trapped magnetic flux will feel a Lorentz force per unit volume $$F_L = J \times B$$, where $$J$$ is the transport current density (vector!) that the material is carrying, and $$B$$ is a vaguely-defined magnetic flux density.
This is the way it is defined in type II superconductors. I've never seen it discussed this way in terms of type I, but I'll take your word for it. I'll be talking about things from the perspective of type IIs in the abrikosov state.

B here means either a single quantum of magnetic flux or in the case of a flux bundle, some multiple of it.

My problems with this are:
• Concept - how can a non-charged body feel a Lorentz force? (This is probably solved by thinking about a supercurrent that surrounds the flux-containing region...)

• That's one way of thinking about it. It's probably the more rigorously correct way. The other is to think about it in terms of equal and opposite forces... a moving charge feels a force due to it's interaction with a nearby field, therefore...

[*] What is $$B$$, given that the Meissner effect excludes magnetic fields in superconducting regions? Could is be the externally-applied field, measured at a distance? Or the (higher) flux density inside the flux-containin region of the material?
This is where I start to wonder about the effect in type Is. Are you sure he was talking about type I SCs? There is a mixed state, comprised of laminar regions in type I SCs, but i've never seen a treatment on flux pinning in this state before.

[*] How can I reconcile my lecturer's definition of $$J_c$$ with the more generally-available definition: "the maximum current a superconductor can carry before making a transition back to normal conduction"?
In type IIs, they are roughly equivalent. The pinning forces dictate the flux gradient which in turn controls the flux motion. Once the fluxoids start to move, they start to dissipate energy, which removes the "lossless' current flow concept.

This is the way it is defined in type II superconductors. I've never seen it discussed this way in terms of type I, but I'll take your word for it. I'll be talking about things from the perspective of type IIs in the abrikosov state.

...
This is where I start to wonder about the effect in type Is. Are you sure he was talking about type I SCs? There is a mixed state, comprised of laminar regions in type I SCs, but i've never seen a treatment on flux pinning in this state before.
This treatment really is about Type I, in the laminar "intermediate" state. Of course, the same effects can be observed in Type II superconductors, where the normal regions are usually single quanta of flux. It seems my lecturer has made an unusual choice, discussing these effects before even introducing Type II.

B here means either a single quantum of magnetic flux or in the case of a flux bundle, some multiple of it.
B is a flux density (= flux per unit area), so it is a continuously varying vector field. I guess my question is: If these "Lorentz force effects" are due to forces on the net supercurrents (transport + fluxon circulation) in the area, then what B-field do the charge carriers actually see locally?

This is the problem of the "nearby field", I suppose - classically, moving charges don't respond to nearby fields, only ones that are right on top of them. However, my handy "anatomy of a fluxon" diagram tells me that (at least when there is no transport current), the fluxon's "paramagnetic supercurrents" are in regions of non-zero B. So I can start to see how this all works!

In type IIs, the (definitions of J_c) are roughly equivalent. The pinning forces dictate the flux gradient which in turn controls the flux motion. Once the fluxoids start to move, they start to dissipate energy, which removes the "lossless' current flow concept.
Again, I'm beginning to see how this works - certainly to the detail level I'm required to. Many thanks =)

seycyrus
Tiresome,

I agree that your instructor's approach is a bit unusual. Is he using a text, if so which one?

Has he shown you any magnetization data of a type I (either theoretical or experimental), that includes the effects of this mixed state?

Most texts that I have seen, simply have a single critical field for type Is.