$w=\frac{\lambda q}{2 \pi \epsilon_{0}} \int_{r_{1}}^{r_{2}} \frac{d r}{r}$

$=\frac{\lambda q}{2 \pi \epsilon_{0}} \ln \left(\frac{r_{2}}{r_{1}}\right)$

Now, lets us it as formula $W N=W x+W y+W z$

$\left[W_{i}: \text { Work done by wire in } i \text { direction }\right]$

$=\frac{2 \lambda_{1}(1)}{2 \pi \epsilon_{0}} \ln \left(\frac{\sqrt{2}}{\sqrt{2}}\right)+\frac{3 \lambda(1)}{2 \pi \epsilon_{0}} \ln \left(\frac{1}{\sqrt{2}}\right)+\frac{\lambda}{2 \pi \epsilon_{0}} \ln \left(\frac{1}{\sqrt{2}}\right)$

$=0+\frac{\lambda}{2 \pi \epsilon_{0}}\left[\frac{-3}{2} \ln 2-\frac{1}{2} \ln 2\right]$

$=-\frac{2 \lambda \ln 2}{2 \pi \epsilon_{0}}=\frac{-\lambda \ln 2}{\pi \epsilon_{0}}$

$W_{e x t}=-W N=\frac{\lambda \ln 2}{\pi \epsilon_{0}}$