question_answer

A uniformly charged and infinitely long line having a linear charge density \[\lambda \] is placed at a normal distance y from a point O. Consider an imaginary sphere of radius R with O as center and R>y. Electric flux through the surface of the sphere is

A) zero      

B) \[\frac{2\lambda R}{{{\varepsilon }_{0}}}\]

C) \[\frac{2\lambda \sqrt{{{R}^{2}}-{{y}^{2}}}}{{{\varepsilon }_{0}}}\]

D) \[\frac{\lambda \sqrt{{{R}^{2}}+{{y}^{2}}}}{{{\varepsilon }_{0}}}\]

Correct Answer: C

Solution :

[c] Charge on length \[AB=2\sqrt{{{R}^{2}}-{{y}^{2}}}\times \lambda \] Electric flux \[\oint{\vec{E}.\overrightarrow{ds}}=\frac{2\lambda \sqrt{{{R}^{2}}-{{y}^{2}}}}{{{\varepsilon }_{0}}}\]