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. $\frac{2\lambda R}{\varepsilon_0}$
• 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 $∮\limits_S \vec{E} \cdot \vec{d} s = \frac{q_{in}}{\varepsilon_0}$

$q_{in}$ is the charge enclosed by the Gaussian surface, which, in the present case, is the surface of the given sphere. As shown, length $AB$ of line lies inside the sphere. In $\Delta OO' A$,
$R^2 = Y^2 + (O'A)^2$
$\therefore O'A = \sqrt{R^2 - y^2}$
and $AB = 2\sqrt{R^2 - y^2}$
Charge on length $AB$ is $2\sqrt{R^2 - y^2} \times \lambda$
Therefore, electric flux is
$∮\limits_S \vec{E} \cdot \vec{d} s = \frac{2\lambda\sqrt{R^2 - y^2}}{\varepsilon_0}$