Self-inductance of a solenoid is

The correct answer is Option (3) → directly proportional to its area of cross-section

The self-inductance of a long solenoid of length \( l \), cross-sectional area \( A \) and \( n \) turns per unit length is

    \( \displaystyle L = \mu_0\,n^{2}\,A\,l \)

It depends only on the geometry of the coil and the core, not on the current.

Option (1) directly proportional to the current flowing through the coil — Incorrect. \( L \) is a property of the coil and does not depend on the current flowing through it.

Option (2) directly proportional to the number of turns per unit length — Incorrect. \( L \propto n^{2} \), so it is proportional to the square of the number of turns per unit length, not directly proportional to \( n \).

Option (3) directly proportional to its area of cross-section — Correct. \( L \propto A \), so it is directly proportional to the area of cross-section.

Option (4) inversely proportional to its area of cross-section — Incorrect. \( L \) increases with \( A \); it is not inversely proportional to \( A \).

NCERT (2026-27):
Self-Inductance – “… Let us calculate the self-inductance of a long solenoid of cross-sectional area A and length l, having n turns per unit length. … Thus, the self-inductance is,

    \( \displaystyle L = \frac{N\Phi_B}{I} = \mu_0 n^2 A l \qquad (6.15) \)”

Self-Inductance – “… The self-inductance of the coil depends on its geometry and on the permeability of the medium.”