In an experiment to determine the Young's modulus, steel wires of five different lengths (1, 2, 3, 4 and 5 m) but of same cross-section area $$(2 \ mm^2)$$ were taken and the curves between extension and load were obtained. The slope (extension/load) of the curves were plotted with the wire length and the following graph was obtained. If the Young's modulus of the given steel wire is $$x \times 10^{11} \ N m^{-2}$$, then the value of $$x$$ is _____

$$Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L}$$

$$\frac{\Delta L}{F} = \left( \frac{1}{AY} \right) \cdot L$$

This is a linear equation ($$y = m \cdot x_{axis}$$) where the slope is $$\frac{1}{AY}$$.

At $$L = 4\text{ m}$$, the value on the $$y$$-axis is $$1.0 \times 10^{-5} \text{ N}^{-1} \text{ m}$$.

$$Slope = \frac{\text{Extension/Load}}{\text{Length}} = \frac{1.0 \times 10^{-5}}{4} = 0.25 \times 10^{-5} \text{ N}^{-1}$$

$$Y = \frac{1}{A \cdot Slope}$$

$$Y = \frac{1}{(2 \times 10^{-6}) \times (0.25 \times 10^{-5})}$$

$$Y = \frac{1}{0.5 \times 10^{-11}} = 2 \times 10^{11} \text{ N m}^{-2}$$

$$x = 2$$