At each point on the surface of the cube shown in Fig. 23-31, the electric field is parallel to the z-axis. The length of each edge of the cube is$\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}$. On the top face of the cube the field is $\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{-}\mathbf{34}\hat{\mathbf{k}}\mathbf{}\mathbf{N}\mathbf{/}\mathbf{C}$and on the bottom face it is$\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{+}\mathbf{20}\hat{\mathbf{k}}\mathbf{}\mathbf{N}\mathbf{/}\mathbf{C}$ Determine the net charge contained within the cube.

1. Length of each edge of the cube is $\mathrm{a}=3.0\mathrm{m}$
2. The field at the top face of the cube is role="math" localid="1657341831037" $\overrightarrow{\mathrm{E}}=-34\hat{\mathrm{k}}\mathrm{N}/\mathrm{C}$, and at the bottom face it is role="math" localid="1657341820720" $\overrightarrow{\mathrm{E}}=+20\hat{\mathrm{k}}\mathrm{N}/\mathrm{C}$

The net flux through each cube surface is given by dividing the total flux value through the cube by 6. Thus, the net flux is given using the concept of the Gauss flux theorem.

Formula:

The enclosed charge within the surface,

${\mathrm{q}}_{\mathrm{enc}}={\mathrm{\epsilon }}_0\mathrm{\varphi }={\mathrm{\epsilon }}_0(\mathrm{E}.\mathrm{A})$ (1)

There is no flux through the sides, so we have two “inward” contributions to the flux, one from the top (of magnitude$\left(34\right){\left(3.0\right)}^2$) and one from the bottom (of magnitude$\left(20\right){\left(3.0\right)}^2$). With “inward” flux being negative, the result of the net flux is given using equation (1) such that,

role="math" localid="1657342360739" $\begin{array}{l}\mathrm{\varphi }=-\left((34\mathrm{N}/\mathrm{C}){(3.0\mathrm{m})}^2+(20\mathrm{N}/\mathrm{C}){(3.0\mathrm{m})}^2\right)\\ =-486\mathrm{N}.{\mathrm{m}}^2/\mathrm{C}\end{array}$.

Now, the net charge enclosed within the surface is given using equation (1) such that,

$$\begin{gathered} {\mathrm{q}}_{\mathrm{enc}}=\left(8.85\mathrm{x}{10}^{-12}{\mathrm{C}}^2/\mathrm{N}.{\mathrm{m}}^2\right)(-486\mathrm{N}.{\mathrm{m}}^2/\mathrm{C}) \\ =-4.3\times {10}^{-9}\mathrm{C} \end{gathered}$$

Hence, the value of the charge is $-4.3\times {10}^{-9}\mathrm{C}$.