How is there self-induction at the moment of closing the circuit? I have just learned about self-induction and I'm confused about something.
There was a problem where there was a battery connected in parallel with a bulb both connected in parallel with a coil, the question was what happens at the moment of closing the switch the answer was "At the moment of closing the switch no current passes in the coil due to back induced emf in the coil and all current passes in the lamp so it gives high brightness" my question here is how was self induction produced in the coil although current didn't pass through it.
 A: The inductance of the coil resists sudden changes in the current flow. At the instant you close the switch, the current-versus-time curve has a very sharp kink in it right at its origin (at time = zero) indicating a very very sudden, discontinuous change in the current, and the inductor fights back against that change.
This is entirely analogous to the situation where you apply a sudden force to a mass and then measure its velocity over time. At the instant when you first apply the force, the velocity is zero but the acceleration is not, and because of that, the mass presses back on you as you apply the force.
A: The statement

At the moment of closing the switch no current passes in the coil due to back induced emf in the coil...

is a good approximation of what happens but not enough for the description what happens in detail. What happens in detail?

*

*Electrons begin to move along all the wires.

*The higher resistance in the thin filament of the bulb leads to an increase in moving electrons per area, the number of collisions increases and the filament begins to glow.

*The electrons in the coil start to generate a magnetic field. But the induction of a field needs energy and the only energy available is the kinetic energy of the moving electrons. So the electrons slow down, which means an increase in the resistance of the coiled wire.

< my question here is how was self induction produced in the coil although current didn't pass through it.
So your intuition was right. Without current, induction does not take place. It is the increasing resistance in the coil that leads to lower current in the coil and a higher current in the bulb (very well imaginable by the flow of water instead of electrons).
