Question

In a region of space there is an electric field that is in the z-direction and that has magnet $\mathbf{E}\mathbf{=}\mathbf{[}\mathbf{964}\mathbf{\hspace{0.33em}}\mathbf{N}\mathbf{/}\mathbf{(}\mathbf{C}\mathbf{\times }\mathbf{m}\mathbf{\left)}\mathbf{\right]}\mathbf{x}$. Find the flux for this field through a square in the -plane at z=0 and with side length 0.350m. One side of the square is along the +x-axis and another side is along the +y-axis.

Short answer

The electric flux through a surface is$20.666\hspace{0.33em}{\mathrm{Nm}}^2/\mathrm{C}.$

Step-by-step solution

Concept of electric flux and given data

The concept of electric flux, its calculation, and the analogy between the flux of an electric field and that of water. Let us imagine the flow of water with a velocityin a pipe in a fixed direction, say to the right. If we take the cross-sectional plane of the pipe and consider a small unit area given byfrom that plane, the volumetric flow of the liquid crossing that plane normal to the flow can be given as

The magnitude of the electric field in z- direction is$\mathrm{E}=964\hspace{0.33em}\mathrm{N}/\mathrm{C}\cdot \mathrm{m}$ . The side length of the square is $\mathrm{L}=0.35\hspace{0.33em}\mathrm{m}$.

Calculate electric flux

As equation (22.5) mentions, the electric flux through a surface is given by

$$\begin{gathered} {\mathrm{\phi }}_{\mathrm{E}}=\int \hspace{0.33em}\mathrm{EAcos}\left(\mathrm{f}\right) \\ =\int \hspace{0.33em}\overrightarrow{\mathrm{E}}\times \mathrm{d}\overrightarrow{\mathrm{A}} \\ =\mathrm{E}\int \hspace{0.33em}\mathrm{d}\overrightarrow{\mathrm{A}} \\ =\mathrm{E}{\int }_0^{\mathrm{L}}\mathrm{Lxdx} \\ =\mathrm{ELxdx} \\ \hspace{0.33em}=\mathrm{EL}{\left(\frac{{\mathrm{x}}^2}{2}\right)}_0^{\mathrm{L}} \\ =\mathrm{EL}\frac{{\mathrm{L}}^2}{2} \\ =\frac{{\mathrm{EL}}^3}{2} \\ =\frac{964\mathrm{N}/\mathrm{C}\times \mathrm{m}\times {\left(0.35\mathrm{m}\right)}^3}{2} \\ =20.666{\mathrm{Nm}}^2/\mathrm{C} \end{gathered}$$

So, the electric flux through a surface is$20.666{\mathrm{Nm}}^2/\mathrm{C}$