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

$\text{ Electric flux }{\text{∮}}_S\overrightarrow{E}\cdot \overrightarrow{d}s=\frac{q_{\text{in }}}{{\epsilon }_0}$

q_in, is the charge enclosed by the Gaussian surface, which, in the present case, is the surface of the given sphere. As shown, length AB of line lies inside the sphere. In $\mathrm{\Delta }O{O}^{\mathrm{\prime }}A$,

$\begin{array}{l}{R}^2=Y^2+{\left(O^{\mathrm{\prime }}A\right)}^2\\ \therefore O^{\mathrm{\prime }}A=\sqrt{R^2-y^2}\\ \text{ and }AB=2\sqrt{R^2-y^2}\end{array}$

$\text{ Charge on length }AB\text{ is }2\sqrt{R^2-y^2}\times \lambda$

Therefore, electric flux is

${\text{∮}}_S\overrightarrow{E}\cdot \overrightarrow{d}s=\frac{2\lambda \sqrt{R^2-y^2}}{{\epsilon }_0}$