Question

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 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 ${\oint }_S\vec{E}\cdot \overrightarrow{dS}=\frac{q_{\text{in }}}{{\epsilon }_0},q_{\text{in }}$ 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 $\Delta O{O}^{\prime }A{R}^2=y^2+{\left(O^{\prime }A\right)}^2$
$\therefore O^{\prime }A=\sqrt{R^2-y^2}$
and and $AB=2\sqrt{R^2-y^2}$
Charge on length $AB=2\sqrt{R^2-y^2}\times \lambda$
$\therefore$ electric flux $={\oint }_S\vec{E}\cdot \overrightarrow{dS}=\frac{2\lambda \sqrt{R^2-y^2}}{{\epsilon }_0}$