Question

A uniform surface charge of density $\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{nC}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$is distributed over the entire x-yplane. What is the electric flux through a spherical Gaussian surface centered on the origin and having a radius of$\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{cm}$ ?

Short answer

The electric flux through a spherical Gaussian surface centered on the origin and having a radius of 5.0 cm is $7.1{\mathrm{Nm}}^2/\mathrm{C}$.

Step-by-step solution

Listing the given quantities

surface charge of density$8.0\mathrm{nc}/{\mathrm{m}}^2$

the radius of the Gaussian surface is 5.0 cm

Understanding the concept of Gauss Law

Using the statement of Gauss law determine the electric flux through the spherical gauss an surface centered on the origin and having a radius of 5.0 cm

Gauss Law

Formula:

$\mathrm{\varphi }=\frac{{\mathrm{q}}_{\mathrm{enc}}}{{\mathrm{\epsilon }}_0}$

Explanation

From Gauss’s law, we have

$$\begin{gathered} \mathrm{\varphi }=\frac{{\mathrm{q}}_{\mathrm{enc}}}{{\mathrm{\epsilon }}_0} \\ =\frac{{\mathrm{\sigma \pi r}}^2}{{\mathrm{\epsilon }}_0} \\ =\frac{\left(8.0\times {10}^{-9}\mathrm{C}/{\mathrm{m}}^3\right)\mathrm{\pi }{(0.050\mathrm{m})}^2}{(8.85\times {10}^{-12}{\mathrm{C}}^2/\mathrm{N}.{\mathrm{m}}^2)} \\ =7.1{\mathrm{Nm}}^2/\mathrm{C} \end{gathered}$$

Electric flux through a spherical Gaussian surface centered on the origin and having a radius of 5.0 cm is $7.1\mathrm{N}.{\mathrm{m}}^2/\mathrm{C}$