What you described IS the process. As the magnet moves in and out of the coil, a current is induced in the coil, one peak for each movement on the magnet.
Suppose you had lots of magnets and lots of coils and could move them relative to each other very quickly for a long time. That is what they do to make most of the electricity that comes out of our power sockets. Instead of magnets going in and out of a coil of wire, the coils are arranged around a set of revolving magnets in a device called a generator.
When the magnetic lines of force in the ⊥ field are cut by a coil of wire (charge), a current is produced in the coil due to induction.
According to Lenz's Law the induced current in the coil produces a magnetic field to oppose the change in magnetic field which produced it.
Faraday's law: When an emf is generated by a change in magnetic flux according to Faraday's Law, the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant
variation of magnetic flux with time. this happens in turbine. This is the principle in all electricity generators.
Electromagnetic induction - When a coil of wire and a bar magnet are moved in relation to each other, an electric current is produced. This current is produced because the strength of the field at the location of the coil changes.
Induction, i.e. relative motion between a wire and a magnetic field produced by the magnet, able to generate across the wire an induced e.m.f. E ; if such wire is a closed path having impedance Z, then a current will take place in it and worth E/Z
To the other answerers: having a name (“induction”) for something doesn’t explain a thing... I hope the asker wants to know the microscopic mechanism by which a changing magnetic field causes a current, not the name of the effect (“inducing a current”).
A microscopic explanation, is any of you taking up that challenge?
Electricity is produced in a conductor wire if a force line of magnetism cuts across it at a right angle----If you have a COIL of wire and move a magnet through it, obviously more lines of magnetic force will cut more lengths of the wire at right angles, producing a detectable flow of electricity. That's what electricity is---A flow of electrons. And that's what magnetism IS--- the force generated by electrons as they MOVE AROUND an atomic nucleus. (some metals are aligned to concentrate this force and make magnets) So magnetism and electricity are part of the same thing ----- A moving Magnetic force will cause an electron to move AND a moving electron will make a magnetic force (at right angles to itself)