Two separate wires $A$ and $B$ are stretched by $2 \mathrm{mm}$ and $4 \mathrm{mm}$ respectively, when they are subjected to a force of $2 \mathrm{N}.$ 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 $\frac{a}{b}$ can be expressed as $\frac{1}{x}$, where $x$ is _____.