A uniformly charged and infinitely long line having a linear charge density $\mathrm{\lambda }$ 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

Electric flux $\underset{\mathrm{S}}{\oint }\overrightarrow{\mathrm{E}}\cdot \overrightarrow{\mathrm{dS}}=\frac{{\mathrm{q}}_{\mathrm{in}}}{{\mathrm{\epsilon }}_0}$

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 the line lies inside the sphere. ln ${\mathrm{\Delta OO}}^{\mathrm{\prime }}\mathrm{A}$

${\mathrm{R}}^2={\mathrm{y}}^2+{\left({\mathrm{O}}^{\mathrm{\prime }}\mathrm{A}\right)}^2$

or ${\mathrm{O}}^{\mathrm{\prime }}\mathrm{A}=\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}\text{ and }\mathrm{AB}=2\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}$

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

Electric flux $\underset{\mathrm{S}}{\oint }\overrightarrow{\mathrm{E}}\cdot \overrightarrow{\mathrm{dS}}=\frac{2\mathrm{\lambda }\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}{{\mathrm{\epsilon }}_0}$