In the following problem, we will walk through the derivation of what is known as the self$-$inductance of a solenoid.

a$.)$ Consider a solenoid of radius b$,$ length L$,$ and total number of turns N$.$ When a current of magnitude I is run through the solenoid, what is the total magnetic flux through the solenoid?

b$.)$ Now let$$s say the current is changing at a rate $\backslash$Delta I$/\backslash$Delta t$.$ What is the corresponding rate of change of the flux through the solenoid?

c$.)$ What is the emf through the solenoid? In which direction will the induced current flow relative to the change in the current being pushed through?

d$.)$ The self$-$inductance of a solenoid is defined as the absolute value of the emf divided by $\backslash$Delta I$/\backslash$Delta t$.$ What is this in terms of L$,$ N$,$ b$,$ and any relevant constants?