London Equations Summary

London Equations Overview

• Maxwell’s Limitations:

  • Cannot explain superconductors' electrodynamics.

• Types of Electrons:

  • Two types in superconductors: normal electrons and superconducting electrons.

  • At 0K: only superconducting electrons exist.

  • As temperature rises, normal electrons increase until Tc, where all electrons are normal.

• Electron Densities:

  • Total electron density, n = ns (superconducting) + nn (normal).

• Current Densities:

  • Supercurrent Js and normal current Jn are parallel.

  • Total current density, J = Js + Jn.

  • Js flows without resistance; Jn flows with resistance.

• Response to Electric Field:

  • Force on super electrons: F = dP/dt.

  • Equation: -eE = (m * dVs/dt), where Vs = velocity of superconducting electrons.

  • Leads to the first London equation: dJs/dt = (ns * e^2 * E)/m.

• Steady Current:

  • In absence of electric field, normal current Jn=0; only Js can exist.

• Modification of Maxwell’s Equation:

  • Original: ∇ × E = - ∂B/∂t.

  • Revised for superconductors: dJs/dt = (ns * e^2 * E)/m leading to the second London equation: ∇ × Js = - (ns * e^2 / m) B.

• Magnetic Flux Penetration:

  • Magnetic field does not drop to zero but decreases exponentially.

  • London penetration depth denoted by λL, calculated as:

  • λL^2 = m/(μ0 * ns * e^2).

  • In 1D, diffusion equation gives:

  • B(x) = B0 * exp(-x/λL).

• Temperature Dependence of Penetration Depth:

  • Relation: λL(T) = λ0 * (1 - (T/Tc)^4)^1/2 with λ0 as penetration depth at 0K.

  • At Tc, ns = 0 leads to infinite λL.

• Electron Behavior at Different Temperatures:

  • At 0K: ns/n0 = 1 (all are superconducting electrons).

  • As temp increases, ns decreases, reaching Tc where all are normal electrons.

Maxwell's equations do not explain superconductors' electrodynamics. Superconductors have two types of electrons: normal and superconducting. At 0K, all electrons are superconducting; as temperature increases to TC, normal electrons appear. The total electron density is represented as n = ns + nn, where ns is superconducting and nn is normal.

Supercurrent (Js) flows without resistance, while normal current (Jn) flows with resistance. The total current density is J = Js + Jn. In response to an electric field, the force on super electrons leads to the first London equation: dJs/dt = (ns * e^2 * E)/m. Without an electric field, only Js exists (Jn = 0).

Maxwell's original equation ∇ × E = -∂B/∂t is modified for superconductors as dJs/dt = (ns * e^2 * E)/m, resulting in the second London equation: ∇ × Js = - (ns * e^2 / m) B. Magnetic fields penetrate superconductors but decrease exponentially, defined by the London penetration depth λL, calculated as λL^2 = m/(μ0 * ns * e^2), with the magnetic field's diffusion represented as B(x) = B0 * exp(-x/λL).

The temperature dependence of λL is given by λL(T) = λ0 * (1 - (T/Tc)^4)^(1/2). At Tc, the penetration depth becomes infinite as ns reaches 0. As temperature rises from 0K