Hello. Let's say a current i(t) flows through an inductor of solenoid type. Time-varying i(t) generates a magnetic field B(t) inside the magnetic core of the inductor. As B(t) is time-varying, EMF, which is the line-integral of the induced electric field E(t) along a coil (which carries i(t)) around the core, is made (In fact, I'm not convinced yet that E(t) is really to be called "induced" one, as there is no clue that E(t) is "caused" by B(t) in the Maxwell equation. E(t) and B(t) here are may be better called "dual". But for convience, I'll keep call it "induced"). My question is why EMF has to be voltage applied on the inductor? If EMF-field E(t) is the only electric field present on the inductor coil, then EMF is deserved to be called a voltage. But...Is EMF-field only electric field existing on the coil? The current density is expressed as J = σE' where σ is the conductivity of the conductor. We know that voltage on the inductor and current flowing on it are not in-phase (there is actuall 90 degree phase differece between the volage and current on the inductor), so E' is not necessarilly equal to EMF-fiedl E(t), I think. (If E' = E, then voltage and current are in-phase, but it is not true in the inductor). I thought EMF-field E is so much larger than E' in J = σE in the inductor so that E' can be ignored in the voltage calculation. But if this is true, the current should flows the same direction of E' but this is not always true. Thanks for reading this post I hope there would be some comments on this.