A very long solenoid is made out of a wire with n turns per units length. The radius of the cylinder is a and is negligible compared to its length l. The interior of the cylinder is filled with materials such that the linear magnetic permeability varies with the distance r from axis according to `mu(r)={{:(mu_(1)="constant , for "0ltrlt b),(mu_(2)="constant , for "b lt r lt a):}}`
The self - inductance of the solenoid is