A) \[\frac{\lambda }{2\pi {{\varepsilon }_{0}}R}\]
B) \[\frac{\lambda }{\pi {{\varepsilon }_{0}}R}\]
C) \[\frac{\sqrt{2}\lambda }{2\pi {{\varepsilon }_{0}}R}\]
D) none of these
Correct Answer: A
Solution :
The electric fields due to a quadrant (of radius R having charge per unit length \[\lambda \]) is \[E=\int_{-\pi /4}^{+\pi /4}{\frac{i}{4\pi {{\varepsilon }_{0}}}\,\frac{dq}{{{R}^{2}}}\cos \theta }\] \[=\frac{\lambda }{4\pi {{\varepsilon }_{0}}R}\,\int_{-\pi /4}^{+\pi /4}{\cos \theta d\theta \,=\frac{\sqrt{2}\lambda }{4\pi {{\varepsilon }_{0}}R}}\] The electric fields due to the four quadrants given in the question are shown with their direction in figure. The resultant of there fields is towards left equal to \[{{E}_{net}}\,=\frac{\lambda }{2\pi {{\varepsilon }_{0}}R}\]You need to login to perform this action.
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