If a wire is extended to a new length l , the work done is

$Y=\frac{F / A}{\frac{\Delta l}{l}} \therefore F=\frac{Y A}{l} \cdot \Delta l=k \cdot \Delta l$ Force $\propto$ extension If the extension is $x$, work done in extending by $d x \therefore F=\frac{Y A}{l} \cdot \Delta l=k \cdot \Delta l$ If $x$ is \left(l^{\prime}-l\right), W=\frac{1}{2} k\left(l^{\prime}-l\right)^2 i.e. work done is \frac{1}{2} \frac{Y A}{l}\left(l-l^{\prime}\right)^2.