Question

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

• A. zero
• B. $\frac{2\lambda R}{{\epsilon }_0}$
• C. $\frac{2\lambda \sqrt{R^2-y^2}}{{\epsilon }_0}$
• D. $\frac{\lambda \sqrt{R^2-y^2}}{{\epsilon }_0}$

Answer 1

The correct option is C $\frac{2\lambda \sqrt{R^2-y^2}}{{\epsilon }_0}$

From triangle OAO', $O^{\prime }A=\sqrt{R^2-y^2}$
The charge on the length AB is $q_{ab}=2O^{\prime }A\times \lambda =2\sqrt{R^2-y^2}\times \lambda$
By Gauss's law, the electric flux $,\varphi =\oint \overrightarrow{E}.\overrightarrow{dS}=\frac{q_{enclosed}}{{ϵ}_0}=\frac{q_{ab}}{{ϵ}_0}=\frac{2\lambda \sqrt{R^2-y^2}}{{ϵ}_0}$