A very long solenoid is made out of a wire with $n$ turns per unit 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
$\mu \left(r\right)=\left\{\begin{array}{c}{\mu }_1=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t};\mathrm{f}\mathrm{o}\mathrm{r} 0<r<b\\ {\mu }_2=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t};\mathrm{f}\mathrm{o}\mathrm{r} b<r<a\end{array}\right\}$
The self-inductance of the solenoid is