A needle of length l and density $\rho$ will float on a liquid of surface tension $\sigma$ if its radius r is less than or equal to

Force per unit area exerted on the needle due to liquid is

                               $P_1=\frac{\sigma }{r}$

Pressure exerted by needle on the liquid is

                    $P_2=\frac{mg}{l\left(2r\right)}=\frac{{\mathrm{\pi r}}^2\mathrm{l\rho g}}{2lr}=\frac{\mathrm{\pi r\rho g}}{2}$

For equilibrium, P₁ = P₂, i.e. 

                   $\frac{\sigma }{r}=\frac{\mathrm{\pi r\rho g}}{2}$

or                $r=\sqrt{\frac{2\sigma }{\pi \rho g}}$

Hence the correct choice is (d).