Magnetic field (B-field) tangential to surface of Superconductor

• Magnetic field (B-field) is always tangential to the surface of a Superconductor

We are going to prove that magnetic fields (B-fields) are always tangential to the surface of a superconductor.

First, let`s briefly discuss what superconductivity is. Superconductivity is the sudden disappearance of all electrical resistance when the material is cooled below a certain temperature, called the critical temperature. Apart from the disappearance of all electrical resistance, what makes superconductors to be of great interest is that superconductors have extraordinary magnetic properties.

There are 2 types of superconductivity:

• Superconductivity of type I (discovered by Omnes),
• Superconductivity of type II (discovered by Bednorz and Muller)

Let`s now look at the properties of superconductors of type I.

Properties of superconductive material of type I

• There is an absence of the magnetic field (B-field) inside the superconductor. In fact, the B-field is expelled from inside of the material (B = 0 inside - Meissner-Ochsenfeld effect). This is independent of how the transition to the superconductor state occurs. (There is a loss of history) Note that in superconducting alloys, the Meissner-Ochsenfeld effect is not sharp and the field can penetrate the superconductor.
• There is a persistent current inside the material even in the absence of an electromotive force (battery).
• There is no magnetization of the material.
• Current flows along a skin (thickness ~ 10^-7m) around the boundary of the material.

Let`s now explain how the B-field is always tangential to the surface of a superconductor.

From Maxwell`s equations, we know that the divergence of B is zero (div(B) = 0). It follows that the magnetic field lines are continuous and closed. Therefore, the components of B normal to the surface, Bn must be equal on both sides of the surface, that is, both inside and outside the material.

However, from the first property mentioned above, the B-field is zero inside (so, it is also zero outside the material). It follows that the normal component at the external field at the surface outside the superconductor is also zero. This means that the magnetic field lines are tangential to the surface of the superconductor. The magnetic field lines surround the superconductor.



Additionally, as B = 0 inside, this means that the mean density of current = 0. Therefore, there is no magnetization field and no H-field.