Is this explanation of electromagnetic induction correct? More specifically interpreting Lenz Law

When the magnet, starting with the north pole, goes towards a coil of wire (maybe a solenoid???) the magnetic field within the wire changes polarity in order to repel the north pole. This is so that work has to be done against the magnetic field in order for that work done to be converted into electrical energy. When the magnet pulls away from the coil (so the south pole is the closest face towards the coil), the polarity of the magnetic field within the coil of wire changes so that the south pole is attracted. Thus work has to be done again to pull the magnet out of the field and thus the work done is converted into electrical energy so an emf is induced.

What I don't understand about my own explanation is that how can a coil have a magnetic field unless it is a solenoid? And if the coil is a solenoid, then it must already have an emf going through it because a current-carrying coil is used to produce a magnetic field and I don't know....

I know Faraday's law explains the magnitude of the emf induced but I'm trying to make sense of the negative (contributed by Lenz) in: induced emf is directly proportional to (-) rate of change of magnetic flux linkage.

I don't even know if I'm explaining my confusion correctly...

• "What I don't understand about my own explanation is that how can a coil have a magnetic field unless it is a solenoid?" What are you thinking is the difference between a solenoid and a coil of wire? Sep 13, 2020 at 11:52
• @BioPhysicist I believe a solenoid is a coil of wire with a current going through it? To be clearer: is the current in the solenoid produced by the work done of pushing the magnet through the coil, or is the current already there in the first place in order for the solenoid to have a magnetic field? Sep 13, 2020 at 13:09
• It could be either; it's your example. What do you want it to be? Do you want current to be going through the cool e before you bring the magnet in or not? Sep 13, 2020 at 13:37