Two co axial solenoids having number of turns per unit length 50 and 200, respectively have same length 50 cm. The area of cross-section of the two solenoids are $4\, cm^2$ and $6\, cm^2$, respectively. The mutual inductance of the two solenoids is

$8π × 10^{-3} H$

The correct answer is Option (3) → $8π × 10^{-7} H$

Given:

Number of turns per unit length: $n_1 = 50 \, \text{turns/cm} = 5000 \, \text{turns/m}$

$n_2 = 200 \, \text{turns/cm} = 20000 \, \text{turns/m}$

Length: $l = 50 \, cm = 0.5 \, m$

Areas: $A_1 = 4 \, cm^2 = 4 \times 10^{-4} \, m^2$, $A_2 = 6 \, cm^2 = 6 \times 10^{-4} \, m^2$

Mutual inductance formula for co-axial solenoids: $M = \mu_0 n_1 n_2 A l$

Use the smaller cross-sectional area as common area: $A = 4 \times 10^{-4} \, m^2$

$M = (4 \pi \times 10^{-7}) (5000) (20000) (4 \times 10^{-4}) (0.5)$

$M = 4 \pi \times 10^{-7} \times 5000 \times 20000 \times 2 \times 10^{-4}$

$M = 4 \pi \times 10^{-7} \times 20 \times 10^3$

$M = 8 \pi \times 10^{-3} \, H$

Answer: The mutual inductance of the solenoids is $8 \pi \times 10^{-3} \, H$.