A straight long uniformly charged wire of charged density $\lambda$  has a circular arc portion AB (of angular thickness $74°$  and radius R) having a point charge q placed at arc centre. The wire is supported at A and B by identical strings. Tension in either string, if the wire is in equilibrium, is  $\frac{xk\lambda q}{R}$.Find $x$ . (Where  $K=\frac{1}{4\pi {\epsilon }_0}$)
             

A. Field at O due to segment (1),
 $E_1=\frac{k\lambda }{r}(\cos 37°-\cos 90°)\widehat{i}+\frac{k\lambda }{r}(\sin 90°-\sin 37°)\widehat{j}$
B. Field at O due to segment (2),
 $E_2=\frac{k\lambda }{r}(\cos 90°-\cos 37°)\widehat{i}+\frac{k\lambda }{r}(\sin 90°-\sin 37°)\widehat{j}$
C. Field at O due to segment (3),
 $E_3=\frac{2k\lambda }{R}\sin 37°\widehat{j}$
 $E_{net}=(\frac{4k\lambda }{5r}+\frac{6k\lambda }{5R})\widehat{j}=\frac{2.2k\lambda }{R}\widehat{j}$
Force on  $q=\frac{2.2k\lambda q}{R}\widehat{j}$
By Newton’s 3rd law, wire experiences equal but opposite force which is balanced by tension.
 $$\begin{gathered} \Rightarrow 2T=\frac{2.2k\lambda q}{R} \\ \Rightarrow T=\frac{1.1k\lambda q}{R} \end{gathered}$$