When we push current through a coil with voltage, a magnetic field is produced. This is interesting in of itself because if we resist the magnetic field it has no effect on the voltage or current. i.e. it does not matter if the magnetic field is placing a force upon something else such as attracting or repelling, it does not alter voltage or current. Having said that, if a magnetic field is moving within our magnetic field an effect is seen, and this seems to be the source of BEMF in motors.
When our voltage is removed is when another special event happens
Provided there is a conductor for the current to flow forming a path from one end of the coil to the other, a current continues to flow. Just like when we push a car, when we stop pushing the car continues to move. Of course the car slows and stops after a short time and this is also true of the current in the coil. This slowing is due to resistance.If the path for the current has high resistance, the voltage across the now un-powered coil now rises above the voltage that we originally applied. If there is no path the voltage may rise many 10s or even 100s of times higher than the original voltage
I compare this to the brakes on a car, depending how quickly we want the car to stop the braking force can be 10s or 100s of times greater than the force used to push the car forward in the first place.This phenomenon is very important in many of the devices that have been discussed on this forum and in many is the source of the whole output power.
In truth it is not well understood. Some have compared it to momentum, such as that in a car, others have said that it is something entering into the coil from the environment.
I am not really sure, it could be both. I would go as far as to say that many people on this forum know more about it than the so called experts in our universities
and there is more to it than can be explained by this simple analogy.