Question

derive the self inductance of the solenoid ?

Answer 1

Solenoid with N total number of turns and l length with r radius is shown below:-

Now,

magnetic induction inside the solenoid is given by

Magnetic flux through any cross section of solenoid is  

Then the total magnetic flux linked with all the turns of the solenoid is given by

thus

we know that the magnetic flux though any coil is proportional to the current

or

By comparison we can say that

Where L is the coefficient of self inductance for an ideal solenoid.