Two circular loops P and Q of radii 'r' and 'nr' are made respectively from a uniform wire. Moment of inertia of loop Q about its axis is 4 times that of loop P about its axis. The value of n is

Linear mass density $\lambda$ is constant for both loops. The mass of each loop equals $\lambda$ times its circumference.

For loop P with radius $r$, mass $M_P = \lambda \cdot 2\pi r$ and moment of inertia about its axis $I_P = M_P r^2 = 2\pi\lambda r^3$.

For loop Q with radius $nr$, mass $M_Q = \lambda \cdot 2\pi nr$ and moment of inertia $I_Q = M_Q (nr)^2 = 2\pi\lambda n^3 r^3$.

Given $I_Q = 4I_P$, substituting yields $2\pi\lambda n^3 r^3 = 4(2\pi\lambda r^3)$.

Canceling $2\pi\lambda r^3$ from both sides gives $n^3 = 4$, so $n = 4^{1/3} = (2^2)^{1/3} = 2^{2/3}$.

The value $n = 2^{2/3}$ corresponds to option B.