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. $\frac{2 \lambda R}{\varepsilon_{0}}$
• C. $\frac{2 \lambda \sqrt{R^{2}-y^{2}}}{\varepsilon_{0}}$
• D. $\frac{\lambda \sqrt{R^{2}+y^{2}}}{\varepsilon_{0}}$

Answer 1

Answer: (C)

Electric flux, $\oint_{S} \vec{E} . d \vec{s}=\frac{q_{\text {in }}}{\varepsilon_{0}}$
$q_{ in }$ is the charge enclosed by the Gaussian surface which, in the present case, is the surface of given sphere. As shown, length $A B$ of line lies inside the sphere.

In $\triangle O O^{\prime} A$
$R^{2}=y^{2}+\left(O^{\prime} A\right)^{2}$
$\therefore O^{\prime} A=\sqrt{R^{2}-y^{2}}$
and $A B=2 \sqrt{R^{2}-y^{2}}$
Charge on length $A B=2 \sqrt{R^{2}-y^{2}} \times \lambda$
$\therefore $ Electric flux $=\oint_{S} \vec{E} \cdot d \vec{s}=\frac{2 \lambda \sqrt{R^{2}-y^{2}}}{\varepsilon_{0}}$