Derive London equations.

Name: Problem 8, Chapter 27: Engineering Physics
Uploaded: 2021-08-06T05:04:06-08:00
Duration: 7 min 13 s
Channel: Mirza Aslam Beig
Description: Derive London equations.

Key Concepts

Superconductivity

Superconductivity is a physical state of matter characterized by exactly zero electrical resistance and the expulsion of magnetic fields (the Meissner effect). It typically emerges below a critical temperature and involves the collective behavior of electrons, leading to phenomena that cannot be explained by classical Ohm's law. Understanding superconductivity is essential for appreciating why the London equations were developed to describe the macroscopic electromagnetic behavior of superconductors.

London Equations

The London equations are phenomenological relations introduced to explain the electrodynamics of superconductors. They relate the supercurrent density to the electromagnetic fields and incorporate the behavior of superconducting electrons without recourse to a detailed microscopic theory. These equations help account for the Meissner effect and are foundational in describing how magnetic fields are expelled from superconducting materials.

Maxwell's Equations

Maxwell's equations govern classical electromagnetism and provide the framework within which the London equations are applied. In the context of superconductivity, they are used in conjunction with the London equations to derive properties such as the penetration of magnetic fields into a superconductor and the dynamics of supercurrents, ensuring consistency of the theoretical description with the established laws of electromagnetism.

London Penetration Depth

The London penetration depth is a key parameter that arises from the London equations, representing the characteristic distance over which an external magnetic field can penetrate a superconductor. It is determined by the density of superconducting carriers and the effective mass of these carriers, thus linking microscopic properties of the superconducting state to macroscopic electromagnetic behavior.

Gauge Invariance and Vector Potential

Gauge invariance is a fundamental principle in field theory, ensuring the physical observables remain unchanged under specific transformations of the electromagnetic potentials. In the derivation of the London equations, the vector potential is often used to relate the phase of the superconducting order parameter to the supercurrent. This connection is crucial for formulating the electromagnetic response in superconductors and underpins the theoretical consistency of the London model.

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