Consider two long co-axial solenoids S_1 and S_2, each of length l(≫ r_2) and of radius r_1 and r_2( r_2> r_1). The number of turns per unit length…

Let $\displaystyle N_{1}$ and $\displaystyle N_{2}$ be the total number of turns of coils $\displaystyle S_{1}$ and $\displaystyle S_{2}$ respectively. When current $\displaystyle I_{2}$ is set up in $\displaystyle S_{2}$, flux linkage with solenoid $\displaystyle S_{1}$ is: $\displaystyle N_{1} \phi_{1}=M_{12} I_{2}$ --(i) $\displaystyle N_{1} \phi_{1}=\left(n_{1} l\right)\left(\pi r_{1}^{2}\right)\left(\mu_{0} n_{2} I_{2}\right)$ $\displaystyle N_{1} \phi_{1}=\mu_{0} n_{1} n_{2} \pi r_{1}^{2} l I_{2}$ --(ii) From (i) and (ii): $\displaystyle M_{12}=\mu_{0} n_{1} n_{2} \pi r_{1}^{2} l$ Considering the reverse case, when $\displaystyle I_{1}$ current is set up in $\displaystyle S_{1}$, flux linkage with $\displaystyle S_{2}$ is: $\displaystyle N_{2} \phi_{2}=M_{21} I_{1}$ --(iii) $\displaystyle N_{2} \phi_{2}=\left(n_{2} l\right)\left(\pi r_{1}^{2}\right)\left(\mu_{0} n_{1} I_{1}\right)$ --(iv) From (iii) and (iv): $\displaystyle M_{21}=\mu_{0} n_{1} n_{2} \pi r_{1}^{2} l$ $\displaystyle \therefore M_{12}=M_{21}$