Magnetic field exclusion and retention in superconductors

I'm studying the behaviour of superconductors in different cases, and I can't understand this.

We have a superconducting cylinder with a coaxial hole inside (vacuum). When the cylinder is cooled below it's critical temperature, and later a magnetic field is applied (case a), the superconductor will exclude the field even from the hole (B=0 inside the hole). However, if we first apply the magnetic field and then we cool down the cylinder (case b), the magnetic field will be excluded from the superconductor but not from the empty hole (as seen in the image below). Thus, depending on the procedure, we have two possible final states.

My question is, why is there this difference? Why doesn't the superconductor allow a magnetic field inside the hole in case a? It would still be excluding it from all the superconducting volume.

(Sorry, I don't know the source of the image, I took it from my professor's notes)

• Hello, may I ask which is the source of the image? Some book, or lecture notes? It could be useful also to read the original source. Thanks! Commented Nov 15, 2021 at 15:01
• @Quillo oof, I have no idea, I asked this question like 8 years ago. Sorry. Commented Nov 16, 2021 at 21:22

Since the magnetic field lines have to close themselves, when it transverses the superconductor it has to do it in a continuous fashion. This means, since the superconductor expels the field when in the SC state, the field gets trapped because it has no way of transverse the cylinder ring without opening the magnetic field lines (some geometric imagination is useful here ;).

Hope it's clear enough.

• Yes, I see why it gets trapped, but my question is why doesn't the exact same thing happen when the magnetic field is applied after the phase transition from normal to superconductor qnd then turned off? Commented Jan 12, 2014 at 18:01
• Because as I said, it has to go to the hollow part through the SC part first, and it has to be done continuously without opening the magnetic field lines which cannot happen since the SC part expels the field not letting it go through. You would see it happening in a very thin cylindrical shell (thinner than the penetration depth). Commented Jan 12, 2014 at 18:05
• You are absolutely right! I just got what you mean. Thank you very much! ^^ Commented Jan 12, 2014 at 18:09

The hole in the superconductor ought to be very small I believe. How the superconductor is cooled matters here as well. Essentially if you are a magnetic field line, you want to avoid diamagnets, avoid magnets in the wrong orientation and go towards magnets in the correct orientation.

If some large bulk superconductor is suddenly dropped in a field, even with a tiny hole in its center, its easier to go around it than just through it (to go through the hole requires the creation of super currents which takes some amount of energy).

If the superconductor is warm and in an existing field, and its cooled from the outside in then as it cools, the outer layers become diamagnetic expelling all magnetic fields and the inner hole still contains a field (which is easy to see since the region around here is not superconducting).

By the time the region around the center hole reaches the transition temperature, in order for these center magnetic field lines to leave the superconductor they need to basically drag across a massive diamagnetic bulk (very much not energetically favorable) so instead they don't move at all (just squeeze together to avoid the superconductor from all sides) and the supercondutor forms eddy currents to repel the material (I don't know for sure but I think the currents loop around the hole see here)

Now a different picture occurs if we cool the superconductor from left to right I believe. As the superconductor cools from its left side, all magnetic fields are expelled and as we go from left to right cooling everything in a line by the time we reach the center, the magnetic field lines can very easily just travel to the right (which is not superconducting yet) and so I suspect cooling this way will result in NO field lines in the center even though they were there originally.

It seems in your professors diagram, given we don't specify HOW the cooling occurs, the assumption is the entire superconductor suddenly transitions.