Question

A point charge $+q$ is placed on the axis of a closed cylinder of radius $R$ and height $25R/12$ as shown. If electric flux coming out from the curved surface of the cylinder is $xq/10{ϵ}_o$, then calculate $x$.

Answer 1

Flux through the curved surface of the cylinder is given by,

$\varphi =\frac{q}{{ϵ}_o}-[\frac{q(1-\cos {\theta }_1)}{2{ϵ}_0}+\frac{q(1-\cos {\theta }_2)}{2{ϵ}_0}]$

Where ${\theta }_1$ and ${\theta }_2$ are the angles made by the charge with the edges of the cylinder, as shown in the figure.

$\therefore \varphi =\frac{q}{{ϵ}_o}-[\frac{q(1-\cos {37}^{\circ })}{2{ϵ}_0}+\frac{q(1-\cos {53}^{\circ })}{2{ϵ}_0}]$

$\Rightarrow \varphi =\frac{q}{{ϵ}_o}-[\frac{q(1-4/5)}{2{ϵ}_0}+\frac{q(1-3/5)}{2{ϵ}_0}]$

$\Rightarrow \varphi =\frac{q}{{ϵ}_o}-\frac{3q}{10{ϵ}_o}=\frac{7q}{10{ϵ}_o}$

Comparing with the data given in the question, we get, $x=7$