A uniformly charged wire of linear charge density λ consists of a quarter-circular arc of radius R joining two mutually perpendicular semi-infinite straight portions tangentially. The centre of the quarter circle is O.With the horizontal semi-infinite part extending to the right and the vertical part extending downward, the electric field at O has magnitude:
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Question

A uniformly charged wire of linear charge density λ consists of a quarter-circular arc of radius R joining two mutually perpendicular semi-infinite straight portions tangentially. The centre of the quarter circle is O.With the horizontal semi-infinite part extending to the right and the vertical part extending downward, the electric field at O has magnitude:

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G1 Physics · Accepted Answer

The quarter-circle arc contributes E⃗_arc = (kλ/R)(î - ĵ). The horizontal and vertical semi-infinite straight portions give equal and opposite resultant fields and cancel each other. Hence, E⃗_net = (kλ/R)(î - ĵ), so |E⃗_net| = √2 kλ/R.