Two solenoids each of length 7 m are placed co-axially, the outer solenoid has 1000 turns and the inner solenoid has 1200 turns. The diameter of the inner solenoid is 8 cm. The mutual inductance between the two solenoids is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 1.08\times10^{-3}\,\mathrm{H} \)
Let the outer solenoid (\( N_2 = 1000 \)) carry a current \( I \). It produces a field \( B = \mu_0 N_2 I/l \) inside, and this field passes through every turn of the inner solenoid. The flux linked with the inner solenoid (\( N_1 = 1200 \) turns, area \( A \) of the inner one) is \( N_1BA = MI \), which gives the mutual inductance
\( \displaystyle M = \frac{\mu_0 N_1 N_2 A}{l} \)
The area is that of the inner solenoid, radius \( r = 4\,\mathrm{cm} = 0.04\,\mathrm{m} \):
\( A = \pi r^{2} = \pi(0.04)^{2} = 5.03\times10^{-3}\,\mathrm{m^{2}} \)
\( \displaystyle M = \frac{4\pi\times10^{-7}\times1200\times1000\times5.03\times10^{-3}}{7} \)
\( M \approx 1.08\times10^{-3}\,\mathrm{H} \)