Question

A hypothetical butterfly net is in a uniform electric field of magnitude $E=3.0\hat{i}{\text{mNC}}^{-1}$. The circular rim of radius $R=11\text{cm}$, is aligned perpendicular to the field as shown in the diagram. If the butterfly net contains no net charge then find the magnitude of electric flux $(|\varphi |)$ passing through the circular rim of net.

• A. $1.114\times {10}^{-4}{\text{Nm}}^2{\text{C}}^{-1}$
• B. $3.342\times {10}^{-4}{\text{Nm}}^2{\text{C}}^{-1}$
• C. $0{\text{Nm}}^2{\text{C}}^{-1}$
• D. $2.228\times {10}^{-4}{\text{Nm}}^2{\text{C}}^{-1}$

Answer 1

The correct option is A $1.114\times {10}^{-4}{\text{Nm}}^2{\text{C}}^{-1}$

Given,
$\overrightarrow{E}=3\hat{i}{\text{mNC}}^{-1}=0.003\hat{i}{\text{NC}}^{-1}$

Radius of the rim, $R=11\text{cm}\text{or}0.11\text{m}$

Flux $\left(\varphi \right)$ through the net,
$\varphi =\oint \overrightarrow{E}.\overrightarrow{dS}$

$\Rightarrow \varphi =\oint EdS\cos \theta$

If we take flux entering through the entrance of the net, then $\theta ={180}^{\circ }$. Substituting the values,

$\Rightarrow \varphi =\oint \left(0.003dS\right)\cos {180}^{\circ }$

$\Rightarrow \varphi =-0.003\oint dS$

where, $\oint dS=\pi A^2$ is surface area of the circular cross section.

$\Rightarrow |\varphi |=0.003\times \pi \times \left(R^2\right)$

$\Rightarrow |\varphi |=0.003\times \pi \times (0.11{)}^2$

$\therefore |\varphi |=1.114\times {10}^{-4}{\text{Nm}}^2{\text{C}}^{-1}$

Hence, option (b) is the correct answer.