Move a magnet near a coil and the needle of the meter wired to it swings. No battery is connected anywhere, yet a current is flowing. Faraday found this in 1831, and it is called electromagnetic induction.
Bringing the magnet closer and pulling it away swing the needle opposite ways. The magnet in the figure runs back and forth, and the current reverses every time the motion does.
Move it faster and the needle swings further. What counts is not how close the magnet was placed but how quickly it was moved.
Hold the magnet right beside the coil and stop your hand, and the needle drops to zero. The magnet is still there and the field through the coil is still strong, but no current flows.
So it is not the magnetic field that drives the current. It is the field changing. A strong field standing still does nothing, and a weak field that is changing does drive a current.
Written out, this is . The right side is the rate at which the magnetic field changes in time. With no change it is zero, and no electric field appears on the left.
Take the coil away and a changing magnetic field still creates an electric field around itself. The field that appears curls, running in loops around the magnetic field as the figure shows.
All the coil ever did was supply electrons free to move under that electric field. The field is there whether or not a wire is present. Put a wire in it and the field shows up as a current, and that is all.
The loops run one way while the magnetic field grows and the other way while it shrinks. The marks in the figure show whether the magnetic field points into the page or out of it, and the loops turn over whenever those marks do.
Move the same magnet the same distance, faster, and you get a stronger electric field. The figure changes nothing between the two rows but the speed of the motion, and the lower needle swings further.
What sets the strength of the induced field is not what the magnetic field is but how much it changes per unit time. The notation says exactly that.
This is why electromagnetic waves keep working up to very high frequencies. The faster the oscillation, the larger the rate of change, and the stronger the field it induces.