London Equation Overview

The London equations are a set of fundamental equations in the theory of superconductivity, developed by Fritz and Heinz London in 1935. These equations describe the electromagnetic response of superconductors and are crucial for understanding phenomena like the Meissner effect, where superconductors expel magnetic fields.

There are two primary London equations:

• First London Equation: Relates the supercurrent density ($\vec{J}_s$) to the electric field ($\vec{E}$). It can be written as: $$ \frac{\partial \vec{J}_s}{\partial t} = \frac{n_s q^2}{m} \vec{E} $$ where $n_s$ is the number density of superconducting charge carriers, $q$ is the charge of the carriers, and $m$ is their mass.
• Second London Equation: Relates the supercurrent density to the magnetic field ($\vec{B}$). It is given by: $$ \nabla \times \vec{J}_s = -\frac{n_s q^2}{m} \vec{B} $$ This equation, when combined with Maxwell's equations, leads to a differential equation for the magnetic field inside the superconductor.

Magnetic Field London Equation

The second London equation, combined with Ampere's law and the absence of ordinary currents (only supercurrents), leads to a differential equation that describes the spatial variation of the magnetic field ($\vec{B}$) within a superconductor. This equation is typically expressed in terms of the Laplacian of the magnetic field.

The form of the London equation provided in the question, specifically for the magnetic field $B$, is:

$$ \nabla^2 B = B \frac{mc^2}{4\pi nq^2} $$

Let's break down the terms in this important equation:

• $ \nabla^2 $ (nabla squared): This is the Laplacian operator, which describes the spatial variation of the magnetic field.
• $ B $: Represents the magnetic field strength inside the superconductor.
• $ m $: Denotes the effective mass of the charge carriers in the superconductor (often considered to be the mass of an electron or a Cooper pair).
• $ c $: Represents the speed of light in vacuum.
• $ 4\pi $: A numerical constant, often appearing in formulas when using CGS (centimeter-gram-second) units.
• $ n $: Represents the number density of superconducting charge carriers (superconducting electrons or Cooper pairs).
• $ q $: Denotes the charge of the superconducting charge carriers (e.g., the elementary charge $e$ for electrons, or $2e$ for Cooper pairs).

London Penetration Depth and Significance

This differential equation implies that the magnetic field does not abruptly drop to zero at the surface of a superconductor but instead decays exponentially over a characteristic distance. This distance is known as the London penetration depth ($ \lambda_L $).

The general form of the magnetic field London equation is often written as:

$$ \nabla^2 B = \frac{1}{\lambda_L^2} B $$

By comparing this general form with the given equation, it implies that the term $ \frac{mc^2}{4\pi nq^2} $ corresponds to $ \frac{1}{\lambda_L^2} $. Therefore, the square of the London penetration depth $ \lambda_L^2 $ would be $ \frac{4\pi nq^2}{mc^2} $ based on this specific representation of the equation.

This equation is fundamental because it quantifies the Meissner effect, explaining how magnetic fields are expelled from superconductors, except for a thin layer near the surface, due to the supercurrents that are generated to screen the magnetic field.