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

The correct answer is Option (4) → is same as \( M_{21} \) of coil 2 with respect to coil 1.

The mutual inductance of a pair of coils depends only on their geometry (number of turns, area, separation, relative orientation) and the medium between them. It is defined by the flux linked with one coil per unit current in the other.

Option (1) — Incorrect. When the coils are brought nearer, more of the flux of one links with the other, so the mutual inductance increases.

Option (2) — Incorrect. The mutual inductance is the ratio of flux linkage to current, so the current cancels; it does not depend on the current.

Option (3) — Incorrect. The mutual inductance depends on the relative orientation of the coils, but rotating a coil does not always increase it: it can increase or decrease, depending on the starting orientation, and a rotation about the coil's own axis changes nothing.

Option (4) — Correct. For any pair of coils \( M_{12} = M_{21} \).

NCERT (2026-27):
Mutual Inductance – “… M₁₂ is called the mutual inductance of solenoid S₁ with respect to solenoid S₂.”

Mutual Inductance – “… Using Eq. (6.9) and Eq. (6.10), we get

    \( \displaystyle M_{12} = M_{21} = M \text{ (say)} \qquad (6.12) \)

We have demonstrated this equality for long co-axial solenoids. However, the relation is far more general.”