A wire of cross-sectional area $A$, modulus of elasticity $2 \times 10^{11} \, \text{Nm}^{-2}$, and length $2 \, \text{m}$ is stretched between two vertical rigid supports.
When a mass of $2 \, \text{kg}$ is suspended at the middle, it sags lower from its original position making angle $\theta = \frac{1}{100}$ radian on the points of support.
The value of $A$ is ____ $\times 10^{-4} \, \text{m}^2$ (consider $x \ll l$).
(Given: $g = 10 \, \text{m/s}^2$)
