Case: Read the passage and answer the question.
Consider two long co-axial solenoids, each of length l. We denote the radius of the inner solenoid S1 by \( r_1 \) and its number of turns per unit length by \( n_1 \). The corresponding quantities for the outer solenoid S2 are \( r_2 \) and \( n_2 \), respectively. Let \( N_1 \) and \( N_2 \) be the total numbers of turns of coils S1 and S2, respectively.
Consider the following statements:
(A) For the same \( n_1 \), \( n_2 \) and \( r_1 \), the mutual inductance is larger for longer solenoids.
(B) If both \( n_1 \) and \( n_2 \) are doubled, the mutual inductance becomes four times.
(C) For this pair, \( M_{12} \) is equal to \( M_{21} \).
(D) Mutual inductance is a vector quantity.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (C) only
For two long co-axial solenoids in air, \( M=\mu_0n_1n_2\pi r_1^{2}l \). The value depends only on the geometry of the pair and the medium inside.
(A) For the same \( n_1 \), \( n_2 \) and \( r_1 \), the mutual inductance is larger for longer solenoids. — True. \( M \) is proportional to the length \( l \).
(B) If both \( n_1 \) and \( n_2 \) are doubled, the mutual inductance becomes four times. — True. \( M \) is proportional to \( n_1n_2 \). Doubling both makes it \( 2\times2=4 \) times.
(C) For this pair, \( M_{12} \) is equal to \( M_{21} \). — True. The flux linkage of S1 per unit current in S2 is equal to the flux linkage of S2 per unit current in S1.
(D) Mutual inductance is a vector quantity. — False. Inductance is the ratio of flux linkage to current. Both are scalars, so inductance has no direction.
NCERT (2026-27):
Mutual inductance – “For these simple co-axial solenoids it is possible to calculate M12. …”
\( M_{12}=\mu_0n_1n_2\pi r_1^{2}l \)\( \qquad (6.9) \)
Mutual inductance – “… we get”
\( M_{12}=M_{21}=M\ \text{(say)} \)\( \qquad (6.12) \)
Inductance – “Inductance is a scalar quantity. …”