The London theory, also known as the London equations, is a fundamental concept in the field of superconductivity. Superconductivity is a remarkable state of matter where certain materials, when cooled to very low temperatures, can conduct electric current with zero resistance. The London theory provides an explanation for some of the key behaviours observed in superconductors.

Here's a simplified explanation of the London theory and its relevance to superconductivity:

Zero Resistance: In normal materials, when you send an electric current through them, they encounter resistance, which generates heat and causes energy loss. However, in superconductors, there's no resistance at all. This means that if you were to start a current flowing in a superconducting loop, it would keep flowing indefinitely without any energy loss.
Perfect Diamagnetism: When a magnetic field is applied to a superconductor, it actively tries to expel the magnetic field from its interior. This behaviour, known as perfect diamagnetism, is a hallmark of superconductors. Even if you move a magnet close to a superconducting material, it will seemingly repel the magnet due to this effect.

The London theory, proposed by brothers Fritz London and Heinz London in 1935, explains these properties using the idea of "Cooper pairs." Cooper pairs are pairs of electrons that team up and move together through the lattice of atoms in a superconductor. The London equations provide a mathematical framework to describe the behaviour of these Cooper pairs and their response to external electric and magnetic fields.

London Theory and the Meissner Effect: The London brothers, Fritz and Heinz London, proposed a theory in 1935 to explain the Meissner effect and other properties of superconductors. This theory provides a deeper understanding of why superconductors exhibit the Meissner effect and how they can maintain zero electrical resistance.

The London theory introduces the concept of "Cooper pairs," which are pairs of electrons that form in the superconducting state. These pairs of electrons are bound together by attractive interactions arising from lattice vibrations (phonons) in the material. In the absence of lattice vibrations and at low temperatures, these Cooper pairs can move through the lattice without scattering, leading to zero electrical resistance.

When an external magnetic field is applied to a superconductor, the Cooper pairs are influenced by this field. The London theory explains that Cooper pairs respond to the magnetic field by exerting a force to counteract the field's penetration into the superconductor. This action results in the expulsion of the magnetic field from the interior of the superconductor, causing the Meissner effect.

According to Meissner effect

For super conducting state i = 0

But the current density

states that current density is directly proportional to the vector potential.

which is London's penetration

Also From Maxwell's equation

Taking curl on both sides

This is also London's equation in other form.

In a nutshell, the London theory helps us understand why superconductors have zero resistance and exhibit perfect diamagnetism. It's a crucial piece of the puzzle in the field of superconductivity and has paved the way for the development of technologies like superconducting magnets used in medical imaging (MRI) and in particle accelerators.

However, it's important to note that while the London theory provides a useful explanation for certain aspects of superconductivity, it's not a complete theory that can explain all the details of high-temperature superconductors and some other exotic superconducting behaviours. Modern theories, like the BCS theory and more advanced models, build upon the concepts from the London theory to provide a more comprehensive understanding of superconductivity.

This note is a part of the Physics Repository.

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