Question

A uniform electric field $E$ is parallel to the axis of a hollow hemisphere of radius $r$. The electric flux through the hemispherical surface is

• A. $4\pi r^{2}E$
• B. $\pi r^{2}E$
• C. $\frac{3}{8}\pi r^{2}E$
• D. $2\pi r^{2}E$

Answer 1

The correct option is B $\pi r^{2}E$

Since number of field lines entering from circular side is equal to number of field lines leaving the hemispherical surface, net flux is $0.$
${\varphi }_{\text{net}}=0$

The total flux through the hemishphere is the sum of the flux through curved surface and circular face.i.e.,
${\varphi }_{\text{circular face}}+{\varphi }_{\text{curved surface}}=0....\left(1\right)$

Flux through the circular face,${\varphi }_{\text{circular face}}=|\overrightarrow{E}||\overrightarrow{A}|\cos \theta ....\left(2\right)$

Here, $A=\pi r^2$ and $\theta ={180}^{\circ }$

Substituting the values in equation $\left(2\right)$, we get
${\varphi }_{\text{cirular face}}=\pi r^{2}E\cos {180}^{\circ }....\left(3\right)$

From equation $\left(1\right)$ and $\left(3\right)$,$\Rightarrow {\varphi }_{\text{curved surface}}=\pi r^{2}E$

Hence, option (c) is correct answer.