Explore the London equations: their origin, significance in superconductivity, limitations, and a practical calculation example.
Introduction to London Equations
London equations, proposed by brothers Fritz and Heinz London in 1935, form the backbone of our understanding of superconductivity. These two phenomenological equations accurately describe the electromagnetic properties of superconductors, making them invaluable in fields like physics and engineering.
Origins and Significance
With the aim to explain the Meissner effect – the expulsion of a magnetic field from a material as it transitions into a superconducting state – the London brothers formulated these equations. As an enduring testament to their impact, the London equations remain a central feature in the theoretical framework of superconductivity.
London’s First Equation
The first London equation establishes a linear relationship between the current density J and the vector potential A. This directly relates the supercurrent to changes in the magnetic vector potential, represented as ∂A/∂t, and indicates that electric fields inside a superconductor are proportional to the rate of change of magnetic fields.
London’s Second Equation
The second London equation takes this one step further by linking the magnetic field to the current density. It’s critical in elucidating how a superconductor responds to an applied magnetic field, leading to the phenomena of perfect diamagnetism and zero electrical resistance.
The Implications of London Equations
The London equations represent one of the early successful attempts to explain superconductivity on a microscopic level. Although they predate the BCS theory, a more comprehensive quantum mechanical explanation, their intuitive appeal and effectiveness continue to make them useful in understanding basic superconducting behavior.
Limitations
While the London equations have provided many insights, they do have limitations. For example, they can’t describe phenomena related to superconducting phase transitions, or handle cases where the supercurrent isn’t a single-valued function of the vector potential. To tackle these complex scenarios, we often turn to the more comprehensive Ginzburg-Landau theory.
Conclusion
In conclusion, the London equations play a pivotal role in our understanding of superconductivity. Despite their limitations, they provide a remarkable framework for interpreting the electromagnetic properties of superconductors and have left an indelible mark on the field of condensed matter physics.
An Illustrative Calculation Using London’s Second Equation
Let’s go through a brief example calculation using the second London equation, which states that the curl of the supercurrent density J is proportional to the magnetic field B in a superconductor. Mathematically, this is represented as ∇ × J = –2λ2 (B – B0), where λ is the London penetration depth and B0 is the initial magnetic field.
Assume a cylindrical superconductor with a radius larger than the London penetration depth λ, subjected to a uniform magnetic field B0. As it transitions into the superconducting state, it will expel the applied magnetic field. Let’s calculate the magnetic field inside the superconductor, B(r).
From symmetry, the supercurrent will circulate azimuthally and will only depend on the radial position r. Therefore, the curl of J is along the z-axis, and its magnitude can be written as (∂Jφ/∂r – Jφ/r). The magnetic field B will also be along the z-axis.
Setting the curl of J equal to –2λ2(B – B0) and solving this differential equation, we find that the magnetic field inside the superconductor decays exponentially from its surface value B(0) = B0, given by B(r) = B0 exp(-r/λ).
This means that the magnetic field is essentially zero for r > λ, demonstrating the Meissner effect. This example highlights how the London equations help us quantitatively understand the behavior of superconductors.
