Two separate wires A and B are stretched by 2 mm and 4 mm respectively, when they are subjected to a force of 2 N. Assume that both the wires are made up of same material and the radius of wire B is 4 times that of the radius of wire A. The length of the wires A and B are in the ratio of a : b. Then a/b can be expressed as 1/x where x is _________.

${\rho _A} = {\rho _B}$<br><br>${r _B} = 4{r _A}$<br><br>$\Delta {l_a} = 2$ mm<br><br>$\Delta {l_B} = 4$ mm<br><br>$y = {{stress} \over {strain}} = {{F/A} \over {\Delta l/l}}$<br><br>$\Rightarrow$ y${{\Delta l} \over l} = {F \over {A}}$<br><br>$\Rightarrow$ $l = {{Ay\Delta l} \over F}$<br><br>$\Rightarrow$ $${{{I_a}} \over {{I_b}}} = {{\pi r_a^2 \times y \times \Delta {I_a} \times F} \over {\pi r_b^2 \times y \times \Delta {I_b} \times F}}$$<br><br>$\Rightarrow$ $${{{I_a}} \over {{I_b}}} = {{r_a^2 \times \Delta {I_a}} \over {r_b^2 \times \Delta {I_b}}} = {{r_a^2 \times 2} \over {{{(4{r_a})}^2} \times 4}} = {{r_a^2} \over {16r_a^2 \times 2}}$$<br><br>$\Rightarrow$ ${{{I_a}} \over {{I_b}}} = {1 \over {32}}$<br><br>$\therefore$ $x = 32$