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 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

Answer: (C)

Electric flux, ${\oint }_S\vec{E}.d\vec{s}=\frac{q_{\text{in }}}{{\epsilon }_0}$
$q_{in}$ is the charge enclosed by the Gaussian surface which, in the present case, is the surface of given sphere. As shown, length $AB$ of line lies inside the sphere.

In $△O{O}^{\prime }A$
$R^2=y^2+{\left(O^{\prime }A\right)}^2$
$\therefore O^{\prime }A=\sqrt{R^2-y^2}$
and $AB=2\sqrt{R^2-y^2}$
Charge on length $AB=2\sqrt{R^2-y^2}\times \lambda$
$\therefore$ Electric flux $={\oint }_S\vec{E}\cdot d\vec{s}=\frac{2\lambda \sqrt{R^2-y^2}}{{\epsilon }_0}$