The mutual inductance M_12 of coil 1 with respect to coil 2:

The correct answer is Option (2) → increases when the coils are brought nearer

Mutual inductance is the flux linked with one coil per unit current in the other, \( N_1\Phi_1 = M_{12}I_2 \). It depends only on the geometry of the two coils (number of turns, size, separation, relative orientation) and on the medium, and \( M_{12} = M_{21} \).

Option (1) depends on the current passing through the coil — Incorrect. The flux linked is proportional to the current, so their ratio \( M \) does not depend on the current.

Option (2) increases when the coils are brought nearer — Correct. Bringing the coils closer makes more of the flux of one coil pass through the other, so the flux linked per unit current, and hence \( M \), increases.

Option (3) \( M_{21} \) of coil 2 with respect to the coil 1 is not same — Incorrect. By the reciprocity of mutual inductance, \( M_{12} = M_{21} \) for any pair of coils, so the two are always the same.

Option (4) increases when any one coil is rotated about its axis — Incorrect. Turning a coil about its own axis does not change the arrangement of the two coils, so the flux linkage and \( M \) stay the same.

NCERT (2026-27):
Mutual Inductance – “… It is also important to know that the mutual inductance of a pair of coils, solenoids, etc., depends on their separation as well as their relative orientation.”