A mild steel wire of length 2L and cross-sectional area A is...

Increase in length&

$\Delta L = BO + OC - \left(BC\right)</p><p>$= 2BO - 2BD$</p><p>$⇒ \Delta L = 2BO - 2L$</p><p>$= 2 \left[L^{2} + x^{2}\ \right]\dfrac{1}{2} -2L$</p><p>or \Delta$ L = 2L \left[1 + \dfrac{x2}{L2}\right]$\dfrac{1}{2} -$2L$[$∵$using Bionomial theorem]</p><p>$⇒ $\Delta$ L \approx2L \left[1 + \dfrac{1}{2}\dfrac{x2}{L2}-1\right]

=\dfrac{x2}{L} & \left[∵ x < < L\right]

$∴$ Strain =\dfrac{\Delta L}{2L}$</p><p>$= \dfrac{1}{2L} \times ∆ L = \dfrac{1}{2L} \times \dfrac{x2}{L} =\dfrac{x2}{2L^{2}}.