London equations
Introduction
The London equations, named after brothers Fritz and Heinz London, are a pair of phenomenological equations that were first introduced in 1935 to describe the electromagnetic response of superconductors. These equations, along with the Ginzburg-Landau theory and the BCS theory, form the theoretical foundation of superconductivity.
London Equations
The London equations are expressed as follows:
1. ∂j/∂t = nse^2/m * (E + (1/c) * v × B) 2. ∇ × j = - nse^2/m * B
where j is the superconducting current density, E is the electric field, B is the magnetic field, v is the velocity of the superconducting electrons, ns is the density of superconducting electrons, e is the elementary charge, m is the electron mass, and c is the speed of light.
These equations essentially describe two fundamental properties of superconductors: the absence of electrical resistance and the expulsion of magnetic fields (the Meissner effect).
Derivation of London Equations
The London equations can be derived from a phenomenological consideration of the behaviour of superconducting electrons under the influence of electromagnetic fields. The derivation is based on two assumptions:
1. The superconducting electrons move as a whole without scattering, i.e., they form a coherent quantum state. 2. The superconducting electrons respond to electromagnetic fields as a single entity, i.e., they exhibit collective behavior.
Implications of London Equations
The London equations have profound implications for the understanding of superconductivity. They provide a mathematical description of the Meissner effect, one of the defining characteristics of superconductors. The Meissner effect refers to the expulsion of a magnetic field from a superconductor below its critical temperature. This effect cannot be explained by classical physics and is a direct consequence of quantum mechanics.
The London equations also imply that the superconducting current is a constant of motion. This means that once a superconducting current is set up in a closed loop of superconducting wire, it will continue indefinitely without any applied voltage, a phenomenon known as persistent currents.
Limitations of London Equations
While the London equations have been instrumental in the understanding of superconductivity, they have their limitations. They are phenomenological in nature and do not provide a microscopic explanation of superconductivity. They also fail to explain why some materials become superconducting at low temperatures while others do not.
Furthermore, the London equations do not account for the effects of impurities and defects in the superconducting material, which can significantly affect the superconducting properties. These limitations were later addressed by the BCS theory and the Ginzburg-Landau theory.