You do not need neither a coil, nor core. The vacuum is a pretty good conductor of the magnetic field. Cores and coils only complicate the things. Consider a single loop of current. Surprise: It also has a magnetic field! Now, which microscopic level
Individual electrons are you talking about?susceptible to the electric field, that is induced by the changing magnetic field in the usual way: $F = q * E$. $\vec B$ circles around the wire, $E$ is in the direction of wire.
Might be you need to look at quantum electrodynamics?