Quote by Enthalpy
In the second equation, E must be complemented by dA/dt, possibly with some sign if you like.
This is allimportant in induction. For instance in a generator, copper wires shall have low loss, meaning E~0, but you get a V at the terminals thanks to the induction dA/dt summed over the conductor path (or d phi / dt if you prefer). Or even, E=0 in a superconductor, which is considered practically for generators and motors, at orientable pods for boats for instance. Though there, it would supposedly be a type II superconductor, which has a resistance.

Thanks for the answers. Where exactly should A be included ? In the first equation, ∇×E=j ω B, I would put [itex]E=\nabla Vj \omega A[/itex] to give
[tex]
\oint\left(\nabla V+j\omega A\right)\cdot dl=j\omega \Phi
[/tex]
So does this mean that my voltage is
[tex]
V=j\omega\Phij\omega\oint A\cdot dl
[/tex]
?