# Introduction to London Equations

London equations are a set of differential equations that describe the motion of superconducting electrons in a superconductor. They were introduced by Fritz London and his brother Heinz London in 1935. These equations are used to elucidate the macroscopic properties of superconductors. They provide insight into the fundamental processes that take place in a superconductor, including the expulsion of magnetic fields and the perfect conductivity of the material.

The London equations are essential for understanding the properties of superconductors, especially the Meissner effect. The Meissner effect is the expulsion of magnetic fields from a superconductor as it transitions to a superconducting state. In other words, a superconductor is a perfect diamagnet. The London equations explain this effect by describing how the superconducting electrons screen the magnetic field from the interior of the material. These equations remain a cornerstone of the theory of superconductivity even today.

# Derivation of London Equations

The London equations are derived from the Ginzburg-Landau theory of superconductivity. This theory describes the behavior of the superconducting wave function, which is a complex function that characterizes the state of the superconductor. The Ginzburg-Landau theory predicts that the superconducting wave function is a constant inside the superconductor and decays exponentially outside the material. This means that there is no electric field inside the superconductor and a perfect diamagnetism is expected.

The London equations are a direct result of the Ginzburg-Landau theory. They describe the motion of the superconducting electrons in terms of the vector potential of the magnetic field. They show that the superconducting electrons respond to the magnetic field by generating a current that flows parallel to the direction of the magnetic field. This current, called the London current, screens the magnetic field from the interior of the superconductor.

# Applications of London Equations

The London equations have numerous applications in superconductivity research. They are used to model the behavior of superconductors under various conditions, including the presence of magnetic fields, temperature changes, and current flow. They are also used to design superconducting devices, such as superconducting magnets, superconducting microwave filters, and superconducting quantum interference devices.

Moreover, the London equations provide a foundation for the understanding of the Josephson effect, which is a phenomenon observed in superconducting junctions. The Josephson effect is the flow of an electrical current across a non-conducting barrier between two superconductors without any voltage applied. The London equations describe the behavior of the superconducting electrons in the junction and provide a theoretical framework for the Josephson effect.

# Example of London Equations in Superconductivity

One example of the application of the London equations is in the design of superconducting magnets. Superconducting magnets are used in many applications, including particle accelerators, magnetic resonance imaging (MRI) machines, and nuclear magnetic resonance (NMR) spectrometers. The London equations are used to calculate the critical current density of the superconductor, which is the maximum amount of electrical current that can flow through the material without losing its superconducting properties.

The critical current density is an important parameter in the design of superconducting magnets, as it determines the maximum magnetic field that can be generated by the magnet. The London equations show that the critical current density is proportional to the strength of the magnetic field and inversely proportional to the temperature of the superconductor. This means that higher magnetic fields and lower temperatures lead to higher critical current densities, which allows for the design of superconducting magnets with stronger magnetic fields.