Question

Find the electric fluxes ${\Phi }_{1}to{\Phi }_5$through surfaces 1 to 5 in FIGURE P24.29.

Short answer

$$\begin{gathered} {\Phi }_1=\mathbf{-}\mathbf{3200}\mathbf{}\mathit{N}{\mathit{m}}^{\mathbf{2}}\mathbf{/}\mathit{C} \\ {\Phi }_{\mathit{2}}=\mathbf{0} \\ {\Phi }_3=\mathbf{0}\mathbf{} \\ {\Phi }_4=\mathbf{3200}\mathbf{}\mathit{N}{\mathit{m}}^{\mathbf{2}}\mathbf{/}\mathit{C}\mathit{}\mathit{}\mathit{}\mathit{}and \\ {\mathrm{\Phi }}_5=\mathbf{0} \end{gathered}$$

Step-by-step solution

Given information and Theory used 

Given figure :

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.

${\Phi }_e=E·A·\cos \theta$ (1)

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

Finding the electric fluxes for surfaces 1 to 3

The electric field on surface 1 is parallel to the surface's normal but in the opposite direction, resulting in an angle of${\theta }_1=180°$ between the normal and the electric field.

We can get $A_1$, the area of the flat sheet by

$A_1=(4m\times 2m)=8m^2$

We can now insert our $E,A_1,\mathrm{and}{\theta }_1$values into equation (1) to get the electric flux :

$$\begin{gathered} {\Phi }_1=EA\cos \theta \\ =(400N/C)(8m^2)(\cos 180°) \\ =\boxed{\mathbf{-}\mathbf{3200}\mathbf{}\mathit{N}{\mathit{m}}^{\mathbf{2}}\mathbf{/}\mathit{C}} \end{gathered}$$

Surface 2: The electric field is perpendicular to the surface's normal, which is $90°$in this case, and $\cos 90°=0$. As a result, there is no electric flux through surface 2. That is, ${\Phi }_2=\boxed{\mathbf{0}}$

Surface 3: It is the same as the Surface 2. That is,${\Phi }_3=\boxed{\mathbf{0}}$

Finding the electric fluxes for surfaces 4 and 5

Surface 4: The surface's sides are $4m$long, and$\frac{2m}{\sin 30°}=4m$. In the diagram, the angle between the electric field and the normal is displayed, and it may be calculated as ${\theta }_4=90°-30°=60°$.

Surface 4 has an area of :

role="math" localid="1649671479054" $A_4=(4m\times 4m)=16m^2$

We can now put our values of$E,A_4,\mathrm{and}{\theta }_4$ into equation (1) to obtain the electric flux :

role="math" localid="1649671541171" $$\begin{gathered} {\Phi }_4=EA\cos \theta \\ =(400N/C)(16m^2)(\cos 60°) \\ =\boxed{\mathbf{3200}\mathbf{}\mathit{N}{\mathit{m}}^{\mathbf{2}}\mathbf{/}\mathit{C}} \end{gathered}$$

Surface 5: the electric field is perpendicular to the surface's normal, which is $90°$in this case, and $\cos 90°=0$. As a result, there is no electric flux through surface 5. That is,

${\Phi }_5=\boxed{\mathbf{0}}$