A wire of fixed lengths is wound on a solenoid of length $$\ell $$ and radius $$r.$$ Its self inductance is found to be $$L.$$ Now if same wire is wound on a solenoid of length $$\frac{\ell }{2}$$ and radius $$\frac{r}{2},$$ then the self inductance will be -

Solution :
$$L = \frac{{{\mu _0}{N^2}\pi {r^2}}}{\ell }$$
Length of wire $$ = N\,2\pi r = {\text{constant}}\left( { = C,{\text{suppose}}} \right)$$
$$\eqalign{ & \therefore L = {\mu _0}{\left( {\frac{C}{{2\pi r}}} \right)^2}\frac{{\pi {r^2}}}{\ell } \cr & \therefore L \propto \frac{1}{\ell } \cr} $$
$$\therefore $$ Self inductance will become $$2L.$$