Derivation of London equation $\vec{\nabla}\times\vec{j}=-\frac{n_se^2}{m}\vec{B}$

Question

London's first equation $$\frac{d}{dt}\vec{j}=\frac{n_se^2}{m}\vec{E}$$ where $j=-en_s\vec{v}_s$, $n_s$ is the number density of electrons that contribute to the supercurrent and $\vec{v}_s$ is their mean velocity, coupled with Faraday's law of induction, $$\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}$$ we promptly obtain $$\frac{\partial}{\partial t}\big(\vec{\nabla}\times\vec{j}+\frac{n_se^2}{m}\vec{B}\Big)=0.$$ This equation, only tells that $$\vec{\nabla}\times\vec{j}+\frac{n_se^2}{m}\vec{B}={\rm any~ function~ of~ position~ only,~in~ general.}$$ But the solution is taken to be $$\vec{\nabla}\times\vec{j}+\frac{n_se^2}{m}\vec{B}=0.$$ Please explain, why. I have followed the derivation of Aschroft and Mermin's solid state physics book. They just mention but did not show what fixes the right hand side to be zero. The answer to this question notes that it does not necessarily follow from Maxwell's equations; but I would like to know how it can be justified.

Answer 1

In a superconductor the velocity of the condensate of paired electrons is given by $$ {\bf v}= \frac 1 {m^*}(\nabla \phi-e^*{\bf A}) $$ where $\nabla \equiv {\rm grad}$. Here $m^* \approx 2m_e$ and $e^*=2e$ are the effective mass and charge of the Cooper pairs and $\phi(x)$ is the local phase of the order parameter $\langle \psi \psi \rangle = |\psi|^2 e^{i\phi}$. On taking the curl of the equation for ${\bf v}$ we get $$ m^* {\boldsymbol{\omega}} +e^*{\bf B}=0, $$ where ${\boldsymbol{\omega}}= \nabla\times {\bf v}$ is the vorticity. We have used $\nabla \times (\nabla \phi)=0$ and $\nabla\times {\bf A}={\bf B}$. A similar but not identical equation holds for any inviscid charged fluid because changing ${\bf B}$ spins up the fluid via $$ \nabla\times {\bf E}= -\frac{\partial {\bf B}}{\partial t}, $$ but the inviscid charged fluid equation is only $$ m^* {\boldsymbol{\omega}} +e^*{\bf B}=\hbox{time independent}, $$ It is the zero on the RHS of the superconductor equation that is crucial for excluding (rather than merely trapping) the magnetic field. The zero comes from the fact that $\nabla \times (\nabla \phi)=0$, i.e the existence of a well-defined local phase for the order parameter.