question_answer

The charge per unit length of the four quadrants of the ring of radius R are \[2\lambda ,-2\lambda ,\lambda \] and \[-\lambda \], respectively as shown in figure. The magnitude of electric field at the centre is:

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 \[{{E}_{net}}\,=\frac{\lambda }{2\pi {{\varepsilon }_{0}}R}\]