Question

Determine the expression for the inductance of the solenoid. (Your answer is approximate because B is actually smaller at the ends than at the center of the solenoid. For this reason, your answer is actually an upper limit on the inductance.)

Answer 1

For the case of an infinitely long solenoid the magnetic field

B = µ0ni

where n is the number of turns per unit length.

A finite solenoid has a finite length l and finite number of turns N.

B = µ0ni =  µ0Ni / l

For self inductance the flux trough the area enclosed by the turns of wire in the solenoid is

φB = B(NA) = µ0AN^2i / l

the flux φB is always proportional to the current i. The proportionality constant is the self inductance L

L = µ0AN^2 / l