A straight solenoid has 50 turns per cm in primary and total 200 turns in the secondary. The area of cross-section of the solenoids is $4c{m}^2$. The mutual inductance between them is $k\pi \times {10}^{-5}H.$ Find the value of k?

The magnetic filed at any point inside the straight solenoid of primary with $n_1$ turns per-unit length carrying a current $i_1$ is given by relation.

$B={\mu }_0{n}_1{i}_1$

$\therefore M=\frac{N_2{\varphi }_2}{i_1}={\mu }_0{n}_1{N}_{2}S$

Substituting the values, we get, $M=(4\pi \times {10}^{-7})\left(\frac{50}{{10}^{-2}}\right)\left(200\right)(4\times {10}^{-4})=5.0\times {10}^{-4}H$