# London equation problems

London's acceleration equation.

$$E=\frac{m}{ne^2} \frac{DJ}{dt}$$

is derived from the definition of current density and $$F = ma$$

However, why is the magnetic contribution of force ignored for the derivation?

As it is simply taken $$F =eE$$

## 1 Answer

It's because of the Meissner relation $${\boldsymbol \omega}m+ e{\bf B}=0$$ which follows from taking the curl of Fritz and Heinz London's formula for the momentum $$m {\bf v}= \hbar \nabla\left (\phi-\frac{e}{\hbar} {\bf A}\right).$$ Here $${\boldsymbol \omega}= \nabla\times {\bf v}$$ is the fluid vorticity and $$\phi$$ is the superfluid order parameter phase. This means that when you use a vector identity to write the fluid dynamics Euler equation $$\left(\frac{\partial {\bf v}}{\partial t}+ ({\bf v}\cdot \nabla) {\bf v}\right) = \frac{e}{m}({\bf E}+ {\bf v}\times {\bf B})- \nabla P$$ in the Bernouli form $$\left(\frac{\partial {\bf v}}{\partial t}+ {\boldsymbol \omega}\times {\bf v}\right)= \frac{e}{m}({\bf E}+ {\bf v}\times {\bf B})- \nabla\left(P +\frac 12 |{\bf v}|^2\right)$$ the $${\bf v}\times {\bf B}$$ force cancels against the vorticity term on the LHS.

It is the Messner relation that causes a magnetic field to be expelled from a superconductor. In the fluid dymamics of an inviscid charged fluid you can show that the quantity $${\boldsymbol \omega}m+ e{\bf B}={\rm constant}$$ because changing the magnetic field causes an $${\bf E}$$ field that generates vortcicity. What is special about the charged superfluid is that the constant is zero. This means that a $${\bf B}$$ field comes with non-zero $${\boldsymbol \omega}$$ and hence costs kinetic energy. If the constant where not zero, the magnetic field would just be trapped in the fluid, as it is in highly condutive plasmas.