Self-Inductance & Energy question from MHT CET 2020 (16 Oct, Shift 2)

Answer: (a)

1. With wire length $w$: $L = \dfrac{\mu_0 w^2}{4\pi I}$ (from $L = \dfrac{\mu_0 N^2\pi r^2}{I}$ and $w = 2\pi r N$).
2. Wire cross-section $a$: $R = \dfrac{\rho w}{a}$ and $m = D w a$, so $Rm = \rho D w^2$.
3. $w^2 = \dfrac{Rm}{\rho D} \Rightarrow L = \dfrac{\mu_0}{4\pi I}\left(\dfrac{Rm}{\rho D}\right)$
4. So the answer is (a).