Induced voltage in a coil
In the second equation, E must be complemented by dA/dt, possibly with some sign if you like.
This is all-important 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.