In superconductivity, the London equation can be written as: ∇ × j = -1/(μ₀λ_L²) B, where λ_L is the London penetration depth, and Ampere's law can be written as: ∇ × B = μ₀j. Using these equations, Gauss's law and the fact that ∇ × ∇ × B = ∇(∇ ⋅ B) - ∇²B, derive an equation that describes the Meissner effect in terms of only the parameters B and λ_L.
Show that B(x) = C exp(-x/λ_L) is a solution of the equation you have derived and state what x and C refer to, and so justify why the equation describes the Meissner effect.