A long solenoid of length L and radius r_1 having N_1 turns is surrounded symmetrically by a coil of radius r_2(> r_1) having N_2 turns ( N_2 ≪ N_1)…

A long solenoid of length L and radius $\displaystyle \mathrm{r}_{1}$ having $\displaystyle \mathrm{N}_{1}$ turns is surrounded symmetrically by a coil of radius $\displaystyle \mathrm{r}_{2}\left(>\mathrm{r}_{1}\right)$ having $\displaystyle \mathrm{N}_{2}$ turns $\displaystyle \left(\mathrm{N}_{2} \ll \mathrm{~N}_{1}\right)$ around its mid-point. Derive an expression for the mutual inductance of solenoid and coil. Is $\displaystyle \mathrm{M}_{12}=\mathrm{M}_{21}$ valid in this case?