London equations

Explore the London equations, their significance in understanding superconductivity, and a calculation example in this concise article.

Understanding the London Equations

The London equations, formulated by brothers Fritz and Heinz London in 1935, are a pair of mathematical expressions that describe the behavior of superconductors. These equations provided the first theoretical foundation for understanding superconductivity and remain a vital tool for physicists and engineers working in this field.

A Brief Overview of Superconductivity

Superconductivity is a phenomenon that occurs in certain materials, wherein they exhibit zero electrical resistance and expel magnetic fields when cooled below a specific critical temperature. This unique property allows for the creation of highly efficient electrical devices, such as transformers, MRI machines, and magnetic levitation trains, among others.

The London Equations: A Closer Look

The London equations consist of two key expressions, which can be written as:

1. ∇ x J_s = – (1/λ²) B
2. ∇ x B = μ₀ J_s

Here, J_s represents the supercurrent density, B is the magnetic field, λ is the London penetration depth, and μ₀ is the vacuum permeability.

The first equation, also known as the London penetration depth equation, describes how a superconductor expels magnetic fields, a phenomenon known as the Meissner effect. The penetration depth (λ) represents the distance over which the magnetic field decays exponentially within the superconductor, and it depends on the material’s properties and temperature.

The second equation, known as the London constitutive equation, relates the supercurrent density (J_s) to the magnetic field (B) and demonstrates how the supercurrents generate magnetic fields. This equation highlights the interdependence between magnetic fields and supercurrents, which is crucial for understanding the behavior of superconductors.

Significance of the London Equations

The London equations were groundbreaking at the time of their introduction, as they offered a theoretical foundation for understanding the microscopic origins of superconductivity. They provided a means for calculating key properties of superconducting materials, such as the penetration depth and critical magnetic field, which proved essential for developing practical applications.

Although more advanced theories, such as the BCS theory and the Ginzburg-Landau theory, have since been developed, the London equations remain a valuable tool in the study of superconductivity. They offer a relatively simple and intuitive way to describe the behavior of superconductors, making them an excellent starting point for students and researchers alike.

Conclusion

In summary, the London equations have played a pivotal role in shaping our understanding of superconductivity since their introduction in 1935. By describing the fundamental relationship between supercurrents and magnetic fields, these equations have provided invaluable insights into the behavior of superconducting materials and laid the groundwork for many practical applications that continue to transform our world today.

An Example of London Equations Calculation

Let’s consider a simple example to illustrate the use of London equations in determining the London penetration depth (λ) and magnetic field (B) for a superconducting material.

Suppose we have a type-I superconductor with the following known properties:

• Superconducting critical temperature (T_c): 5 K
• Operating temperature (T): 4 K
• Supercurrent density (J_s): 10⁶ A/m²
• London penetration depth at 0 K (λ₀): 50 nm
• Material’s coherence length (ξ): 100 nm

First, we calculate the London penetration depth (λ) at the given temperature using the following relation:

λ(T) = λ₀ / √(1 – (T/T_c)²)

Substituting the given values:

λ(4 K) = 50 nm / √(1 – (4 K/5 K)²)

λ(4 K) ≈ 80 nm

Now, we can use the first London equation to find the magnetic field (B) within the superconductor:

∇ x J_s = – (1/λ²) B

As the curl (∇ x) of J_s is equal to zero for a uniform supercurrent, we can rewrite the equation as:

B = – λ² ∇ x J_s

In this example, let’s assume the supercurrent density (J_s) is uniform and parallel to the surface of the superconductor, and the magnetic field (B) is perpendicular to the surface. We can then simplify the expression:

B = – λ² (∂J_s / ∂z)

Since J_s is uniform and does not change with the position, the gradient (∂J_s / ∂z) is zero. Therefore, the magnetic field (B) within the superconductor is also zero:

B = 0 T

This result demonstrates the Meissner effect, where a superconductor expels magnetic fields and maintains a zero magnetic field within itself. In this example, we have used the London equations to calculate the London penetration depth and the magnetic field for a given type-I superconductor at a specific temperature.