Question

A $3.0-cm$-diameter circle lies in the $xz$-plane in a region where the electric field is$\overrightarrow{E}=(1500\hat{i}+1500\hat{j}-1500\hat{k})N/C$. What is the electric flux through the circle?

Short answer

The electric flux through the circle is$\mathbf{1}\mathbf{.}\mathbf{07}\mathbf{}\mathbf{}\mathit{N}\mathbf{·}\mathit{m}\mathbf{²}\mathbf{/}\mathit{C}$

Step-by-step solution

Given information and Theory used 

Given :

Diameter of the circle : $3.0-cm$

The electric field is : $\overrightarrow{E}=(1500\hat{i}+1500\hat{j}-1500\hat{k})N/C$

Theory used :

The quantity of electric field passing through a closed surface is known as the Electric flux. Gauss's law indicates that the electric field across a surface is proportional to the angle at which it passes, hence we can determine charge inside the surface using the equation below.

${\Phi }_e=E·A·\cos \theta$

Where $\theta$is the angle formed between the electric field and the normal.

Calculating the electric flux through the circle.

The rectangle is in $xz$-plane, this means the normal of the sheet is in$y$direction which means, the area has component $\hat{j}$. We can get $A$, the area of the flat sheet by

$\overrightarrow{A}=\pi {\left(\frac{d}{2}\right)}^2=\pi {\left(\frac{0.03m}{2}\right)}^2=\underline{7.1\times {10}^{-4}\hat{j}m²}$

When the electric field would be $\overrightarrow{E}=(1500\hat{i}+1500\hat{j}-1500\hat{k})N/C$, the electric flux will be :

$$\begin{gathered} {\Phi }_e=\overrightarrow{E}\overrightarrow{·A} \\ =(1500\hat{i}+1500\hat{j}-1500\hat{k})N/C·(7.1x{10}^{-4}\hat{j})m² \\ =10650\times {10}^{-4}(\hat{i}·\hat{j})+10650\times {10}^{-4}(\hat{j}·\hat{j})-10650\times {10}^{-4}\left(\right(\hat{k}·\hat{j})) \\ =0+10650\times {10}^{-4}-0Nm²/C \\ =\boxed{\mathbf{1}\mathbf{.}\mathbf{07}\mathbf{}\mathbf{}\mathit{N}\mathbf{·}\mathit{m}\mathbf{²}\mathbf{/}\mathit{C}} \end{gathered}$$