# Meissner effect

**CLAIMED BY AUSTIN CALE (austincale96) [FALL 2016]**

**Edited by David Wiens (Fall 2017)**

## What is the Meissner Effect

The Meissner effect is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state. Below the transition temperature, superconductors cancel nearly all interior magnetic fields actively. The effect that occurs, leading to a zero net magnetic effect in the material, can be attributed to diamagnetism.

The Meissner effect is most often demonstrated by a magnate suspended above a superconductor. Because of the Meissner effect, a magnetic field is created in the superconductor that exactly mirrors the magnetic field of the magnet. Thus, the magnetic force on the magnet is upwards, in the opposite direction of gravity. When the magnet reaches a distance far enough above the superconductor, the magnetic force from the superconductor is equal to and opposite of the force of gravity. Thus, the net force is zero so the magnet remains suspended above the superconductor at a constant position, appearing to levitate.

### What is diamagnetism?

Some materials tend to expel a magnetic field, materials that do this are called diamagnetic, but the effects of this diamagnetism are weak. For example, water and the human body are diamagnetic materials. Diamagnetism is a weak repulsion from a magnetic field. It is a form of magnetism that is only exhibited by a substance in the presence of an externally applied magnetic field. It results from changes in the orbital motion of electrons. Applying a magnetic field creates a magnetic force on a moving electron in the form of F = Qv × B. This force changes the centripetal force on the electron, causing it to either speed up or slow down in its orbital motion. This changed electron speed modifies the magnetic moment of the orbital in a direction opposing the external field.

In superconducting material the Meissner effect creates currents which completely oppose the magnetic field applied by a magnet. In other words, they will repel a magnet, causing it to levitate. This consequently makes a superconductor in the Meissner state a perfect diamagnet.

## How does it Work?

A superconductor with little or no magnetic field within it is said to be in the Meissner state and breaks down when the magnetic field is larger than the critical magnetic field.

A superconductor is fundamentally different from a conductor, because Faraday’s law of induction alone does not explain magnetic repulsion by a superconductor. At a temperature below its Critical Temperature, Tc, a superconductor will not allow any magnetic field to freely enter it. This is because microscopic magnetic dipoles are induced in the superconductor that oppose the applied field. This induced field then repels the source of the applied field, and will consequently repel the magnet associated with that field.

This implies that if a magnet was placed on top of the superconductor when the superconductor was above its Critical Temperature, and then it was cooled down to below Tc, the superconductor would then exclude the magnetic field of the magnet. This means that a magnet already levitating above a superconductor does not demonstrate the Meissner effect, while a magnet that is initially stationary and then repelled by a superconductor as it is cooled through its critical temperature does.

### Type I and Type II Superconductors

There are two categories of superconductors that relate to the Meissner effect: Type I and Type II. Type I superconductors consist of pure metals that act as superconductors, while Type II superconductors are alloys that act as superconductors.

**Type I superconductors** exclude magnetic fields completely due to the Meissner effect when they are cooled to a certain temperature and when the magnetic field is not extremely high. Type I superconductors are usually metals and only become superconductors at very low degrees Kelvin. The temperature at which a metal becomes a superconductor is referred to as its Critical Temperature or Tc. One example of a Type I superconductor is Mercury, which has a critical point of 4.2 K. The other important characteristic of a Type I superconductor is the magnitude of the magnetic field at which the metal no longer can exclude the magnetic field completely. This magnetic field magnitude is referred to as the critical magnetic field or Hc. Mercury has a critical magnetic field of .04 T.

**Type II superconductors**, on the other hand, are typically a mixture of normal and superconducting regions. Type II superconductors do not obey the Meissner effect perfectly. Instead, they only partially exclude the magnetic and slowly lose their superconductivity. The magnitude of the magnetic field at which a type II superconductor begins to partially lose its superconductivity is referred to as the lower critical magnetic field or Hc1. The magnitude of the magnetic field at which a type II superconductor completely loses its superconductivity is referred to as the upper critical magnetic field or Hc2.

### A Mathematical Model

It was mentioned that when a superconductor is exposed to a magnetic field greater than the critical magnetic field, it breaks down. Through experimentation, the critical temperature can be found, as well as the critical magnetic field at 0 kelvin. After these two values are known, the relationship between critical magnetic field and temperature can be found in the following equation.

- [math]\displaystyle{ B_c \approx B_{c0} \left [ 1 - \left ( \frac{T}{T_c} \right ) ^2 \right ] }[/math]

As shown by the formula, at absolute zero, superconductors have a certain initial critical magnetic field that decreases with increasing temperature.

### A Computational Model

File:Users/DavidWiens/Desktop/MeissnerEffect.jpg

## Examples

### Simple

Mercury is a Type I superconductor with an initial critical magnetic field of .041 tesla and a critical temperature of 4.15 kelvin. What would be the largest applied magnetic field you could put on mercury at a temperature of 3K that it could still completely exclude?

**Solution**

We know that the applied magnetic field cannot be greater than the critical magnetic field at a given temperature. Therefore, to find the maximum applied magnetic field, we can simply solve for the critical magnetic field. In order to do so, we can use the equation

- [math]\displaystyle{ B_c \approx B_{c0} \left [ 1 - \left ( \frac{T}{T_c} \right ) ^2 \right ] }[/math].

We are already given [math]\displaystyle{ B_{c0} }[/math], [math]\displaystyle{ T }[/math], and [math]\displaystyle{ T_c }[/math], so we can solve the equation for [math]\displaystyle{ B_c }[/math].

- [math]\displaystyle{ B_c \approx (.041T) \left [ 1 - \left ( \frac{3K}{4.15K} \right ) ^2 \right ] \approx .01957T }[/math]

So we can see that the largest applied magnetic field you can place on mercury at 3K is approximately .01957 tesla, in order for the magnetic field to be completely excluded due to the Meissner effect.

### Medium

### Difficult

## History

The most important discovery leading up to the discovery of the Meissner effect was the discovery of superconductivity in 1911 by Heike Onnes. He discovered superconductivity by observing the resistance of mercury as he cooled it. When cooled mercury to 4.2 K, he measured the resistance to be 0. Thus, mercury was conducting electricity perfectly and was a "super" conductor.

The knowledge of superconductors was expanded upon in 1933 when two German physicists, Robert Ochsenfeld and Walther Meissner discovered the Meissner effect. They were able to do discover the electric field created in the superconductor in the presence of another electric field, not by measuring the electric field directly, but by measuring the associated flux through the superconductor.

The theory of the Meissner effect led to the phenomenological theory of superconductivity by Frits London and Heinz London in 1935. This theory explained resistance less transport and the Meissner effect, and allowed the first theoretical predictions for superconductivity to be made as seen below.

By using the London equations and Maxwell equations, one can predict how the magnetic field and surface current vary with distance from the surface of a superconductor.

## Connectedness

The Meissner effect causes superconductors to exclude magnetic fields. Because of this exclusion, superconductors will repel magnets. Assuming the magnet is small enough that the repulsion can overcome the force of gravity, the magnet can be capable of levitating. This magnetic levitation has numerous applications, one of the most well-known being maglev trains. Maglev trains are convenient for many reasons; however, one of the most important advantages of maglev systems is the lack of friction between the train and the rails. This lack of friction creates benefits such as faster and quieter transportation, and also less maintenance required, because less friction means parts will last longer.

### Further reading

Albert Einstein (1922). "Theoretical remark on the superconductivity of metals". arXiv:physics/0510251v2. Bibcode:2005physics..10251E. Fritz Wolfgang London (1950). "Macroscopic Theory of Superconductivity". Superfluids. Structure of matter series 1. OCLC 257588418.. Revised 2nd edition, Dover (1960) ISBN 978-0-486-60044-4. By the man who explained the Meissner effect. pp. 34–37 gives a technical discussion of the Meissner effect for a superconducting sphere.

### External links

Videos to aid in the understanding of the concept.

https://www.youtube.com/watch?v=44mVZdnR6Yc

https://www.youtube.com/watch?v=bnyB-PInFA4

## References

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/meis.html http://www.supraconductivite.fr/en/index.php?p=supra-levitation-meissner-more http://lrrpublic.cli.det.nsw.edu.au/lrrSecure/Sites/Web/physics_explorer/physics/lo/superc_12/superc_12_02.htm http://www.chm.bris.ac.uk/webprojects2006/Truscott/paged_r.html http://www.imagesco.com/articles/superconductors/superconductor-meissner-efsfect.html http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/maglev.html https://www.britannica.com/technology/maglev-train http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/scond.html http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/scbc.html#c3

See link to the right to read more on superconductors. [[1]]