Three quarter circles (each of radius $R$ ) made of insulating material are uniformly charged with linear charge densities $- λ,$ $λ a n d - λ$ and arranged in space as shown. Then net electric field at Centre is

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$\frac{λ}{2 {πε}_{0} R} \hat{k}$

$\frac{λ}{2 {πε}_{0} R} \hat{i}$

$\frac{λ}{2 {πε}_{0} R} \hat{j}$

$\frac{λ}{2 {πε}_{0} R} \hat{j} + \hat{k}$

answer is A.

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Detailed Solution

$E_{q u a r t e r c i r c l e} = \frac{λ}{2 {πε}_{0} R} \sin 45 ° a l o n g t h e l i n e j o i n i n g t h e m i d p o i n t o n t h e w i r e a n d c e n t r e .$

${\vec{E}}_{n e t} = \frac{1}{4 {πε}_{0} R} \{λ (\hat{i} + \hat{j}) + λ (- \hat{j} - \hat{k}) + λ (\hat{i} + \hat{k})\} = \frac{λ}{2 {πε}_{0} R} \hat{i}$