London EquationsEdit
The London equations are a pair of macroscopic, phenomenological relations formulated to describe how superconductors respond to electromagnetic fields. Introduced in the 1930s by brothers Fritz London and Heinz London to account for the Meissner effect—the expulsion of magnetic fields from the interior of a superconducting sample—the equations provide a simple, robust framework for understanding how superconducting currents organize themselves in the presence of electric and magnetic fields. They establish a direct link between the current density and the electromagnetic potentials, and they predict a characteristic distance over which magnetic fields penetrate into a superconductor, known as the London penetration depth. The London equations remain a cornerstone of superconductivity theory, especially as a bridge between experimental observations and more detailed microscopic theories.
Beyond their historical role, the London equations remain a useful tool in describing a wide range of superconducting phenomena at a macroscopic level. They are closely tied to the concept of a coherent superconducting state in which electrons form a single quantum-mechanical phase-coherent condensate. When combined with Maxwell’s equations, the London relations imply that static magnetic fields die out exponentially inside a superconductor, a result that is experimentally observed as the Meissner effect. The equations also imply a form of infinite conductivity for DC currents in the idealized limit, reinforcing the notion of perfect conductivity that distinguishes superconductors from ordinary metals. The core parameter in the London theory is the density of superconducting carriers, n_s, occasionally referred to as the superfluid density, which governs how strongly the material expels magnetic fields and how deeply fields can intrude.
London equations
Origins and formulation
The London relations can be written in a form that highlights their macroscopic nature. The first London equation states that the time rate of change of the current density J is proportional to the applied electric field E: ∂J/∂t = (n_s e^2 / m) E The second London equation asserts a direct relation between the current density and the magnetic field B: ∇ × J = - (n_s e^2 / m) B Here e is the electron charge, m is an effective mass for the conduction carriers, and n_s is the density of superconducting carriers. In many treatments these equations are discussed in conjunction with the vector potential A, through J ≈ - (n_s e^2 / m) A in an appropriate gauge. When these relations are inserted into Maxwell’s equations (in the low-frequency, quasi-static regime where displacement currents can be neglected), one obtains a differential equation for B with a characteristic decay length: ∇^2 B = B / λ_L^2, with λ_L^2 = m / (μ0 n_s e^2) in SI units. This yields the Meissner screening of magnetic fields, B(z) ∼ B0 e^{-z/λ_L}, inside the superconductor. The London penetration depth λ_L thus encapsulates how deeply magnetic fields can penetrate a superconducting body.
Mathematical structure and consequences
The London equations capture two key aspects of superconductivity: (1) a rigid, phase-coherent current that responds to electromagnetic fields without dissipation in the stationary limit, and (2) a screening mechanism that prevents steady magnetic fields from penetrating the bulk. The equations imply that the current is intimately tied to the electromagnetic potentials and, in particular, to the magnetic field, giving rise to the characteristic exponential attenuation of B within the material. The temperature dependence of n_s means that λ_L grows as the system nears the superconducting transition temperature Tc, signaling the gradual loss of superconducting rigidity.
Physical interpretation and limitations
Physically, n_s represents the density of electrons participating in the superconducting condensate. As temperature increases toward Tc, fewer electrons contribute to the coherent current, and the London penetration depth increases. While the London picture gives a clean macroscopic description of the Meissner effect and static screening, it does not supply a microscopic mechanism for pairing. For that, one turns to microscopic theories such as BCS theory and related frameworks. The London equations are most reliable for conventional, low-frequency, and relatively clean superconductors where nonlocal effects are small. In more complex materials or under strong fields, nonlocality and other phenomena become important, and extensions are needed.
Microscopic connections and extensions
The London equations are fundamentally phenomenological. They are compatible with the microscopic understanding of superconductivity provided by the formation of bound Cooper pairs in many conventional superconductors. In particular, microscopic theories show how a condensate of paired electrons can give rise to a well-defined superfluid density that governs the macroscopic electrodynamics captured by the London relations. The two-fluid model, developed earlier, provides a bridge between normal excitations and the superconducting component and can be reconciled with the London picture in appropriate limits. For a more complete description near and above Tc, or in materials with unusual pairing, researchers turn to the Ginzburg–Landau theory, which treats the superconducting order parameter as a complex field, and to nonlocal generalizations such as Pippard’s theory, which relaxes the local relation between J and A. See, for example, discussions of Ginzburg–Landau theory and Pippard nonlocal electrodynamics for broader context.
Applications and limitations
The London framework informs many practical aspects of superconducting technology. It underpins the understanding of magnetic shielding, stability of superconducting magnets, and the behavior of superconducting materials in modest field regimes. The concept of a finite penetration depth is essential in designing superconducting cavities, shields, and thin-film devices. However, the London equations alone cannot describe vortex dynamics in type II superconductors, where magnetic flux penetrates in quantized vortices, nor can they capture flux creep, finite resistivity under real-world conditions, or complex behavior in high-temperature superconductors. In those regimes, approaches such as Ginzburg–Landau theory and the broader microscopic theories provide the necessary tools.