Two semicircular rings lying in the same plane of uniform linear charge density $\lambda$ have radii r and 2r. They are joined using two straight uniformly charged wires of linear charge density $\lambda$ and length r as shown in the figure. The magnitude of electric field at common center of semi-circular rings is 

The electric field due to both straight wires shall cancel at common center O. The electric field due to larger and smaller semi-circular rings at O be E and E' respectively.

$$\begin{gathered} At center of Arc : E=\frac{2k\lambda }{r}\sin \left(\raisebox{1ex}{$\varphi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right) \\ For semicircle, take \varphi =\pi \\ \Rightarrow E=\frac{2k\lambda }{r} \end{gathered}$$

$$\begin{gathered} Due to Larger Semicircular ring : E=\frac{1}{4\pi {\epsilon }_o}\frac{2\lambda }{2r} \\ Due to Larger Semicircular ring : E\text{'}=\frac{1}{4\pi {\epsilon }_o}\frac{2\lambda }{r} \end{gathered}$$

$\therefore$Magnitude of electric field at O is

$\Rightarrow E_{net}=\frac{1}{4\pi {\epsilon }_o}\left(\frac{2\lambda }{r}-\frac{\lambda }{r}\right)=\frac{1}{4\pi {\epsilon }_o}\frac{\lambda }{r}$