Question

A uniformly charged and infinitely long line having a linear charge density $‘{\lambda }^{\prime }$ is placed at a normal distance $y$ from a point $O$. Consider a sphere of radius $R$ with $O$ as centre 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 $\frac{2\lambda \sqrt{R^2-y^2}}{{\epsilon }_0}$

Electric flux ${\oint }_S\overrightarrow{E}.\overrightarrow{ds}=\frac{q_{in}}{{ϵ}_0}...\left(1\right)$
$q_{in}$ is the charge enclosed by the Gaussian-surface which, in the present case, is the surface of given sphere. As shown, length AB of the line lies inside the sphere.


In
$\Delta O{O}^{\prime }A,R^2=y^2+(O^{\prime }A{)}^2$
$\therefore O^{\prime }A=\sqrt{R^2-y^2}$
and $AB=2\sqrt{R^2-y^2}$
Charge on length $AB=2\sqrt{R^2-y^2}\times \lambda$
From equation (1),
$\therefore$ electric flux $={\oint }_S\overrightarrow{E}.\overrightarrow{ds}=\frac{2\lambda \sqrt{R^2-y^2}}{{\epsilon }_0}$