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 $F =eE$


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.


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