Question

A point charge ${\mathbf{q}}_{\mathbf{1}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{nC}$is located on the x-axis at $\mathbf{x}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{m}$, and a second point charge${\mathbf{q}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{nC}$ is on the y-axis at $\mathbf{Y}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{m}$. What is the total electric flux due to these two-point charges through a spherical surface centered at the origin and with radius (a)$\mathbf{0}\mathbf{.}\mathbf{500}\mathit{m}$ (b) $\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{m}$, (c) $\mathbf{2}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{m}$?

Short answer

(a) The total electric flux due to these two-point charges through a spherical surface centered at the origin and with radius 0.500m is$0\mathrm{N}.{\mathrm{m}}^2/\mathrm{C}$.

(b) The total electric flux due to these two-point charges through a spherical surface centered at the origin and with radius 1.50 m is$-678.0{\mathrm{Nm}}^2/\mathrm{C}$.

(c)The total electric flux due to these two-point charges through a spherical surface centered at the origin and with radius 2.50 m is$-226.0{\mathrm{Nm}}^2/\mathrm{C}$.

Step-by-step solution

Definition of electric flux

The term electric flux may be defined as the application of electric field that may be thought of as the number of electric lines of force that intersect a given area.

The total electric flux due to these two-point charges through a spherical surface centered at the origin and with different radius.

The total electric flux can be calculated by the relation

${\mathrm{F}}_{\mathrm{E}}=\frac{{\mathrm{Q}}_{\mathrm{end}}}{{\mathrm{\epsilon }}_0}$

Where role="math" localid="1664202829532" ${\mathrm{Q}}_{\mathrm{encl}}{\mathrm{\epsilon }}_0$total charged enclosed and electric constant respectively.

For radius 0.500 m.There is no any charge in the radius 0.500m so electric flux is zero.

For radius 1.50m.contain${\mathrm{q}}_2=-6.00\mathrm{nC}$ so the electric flux can be calculated as

$$\begin{gathered} {\mathrm{F}}_{\mathrm{E}}=\frac{-6.00\times {10}^{-9}\mathrm{C}}{8.85\times {10}^{-12}{\mathrm{C}}^{12}/\mathrm{N}.{\mathrm{m}}^2} \\ {\mathrm{F}}_{\mathrm{E}}=-678.{10}^{-12}\mathrm{N}.{\mathrm{m}}^2/\mathrm{C} \end{gathered}$$

Hence, the total electric flux due to these two-point charges through a spherical surface centered at the origin and with radius 0.500m is$0\mathrm{N}.{\mathrm{m}}^2/\mathrm{C}$.

For radius 2.50m contain ${\mathrm{q}}_1=4.00\mathrm{nC},{\mathrm{q}}_2=-6.00\mathrm{nC}$so total charged enclosed is

$$\begin{gathered} {\mathrm{Q}}_{\mathrm{end}}=\left(4.00-6.00\right)\mathrm{nC} \\ {\mathrm{Q}}_{\mathrm{encl}}=-2.00\mathrm{nC} \end{gathered}$$

Hence, the total electric flux due to these two-point charges through a spherical surface centered at the origin and with radius 1.50m is$-6.78.0{\mathrm{Nm}}^2/\mathrm{C}$.

Now the electric flux can be calculated as

$$\begin{gathered} {\mathrm{F}}_{\mathrm{E}}=\frac{-2.00\times {10}^{-9}\mathrm{C}}{8.85\times {10}^{-12}{\mathrm{C}}^2/\mathrm{N}.{\mathrm{m}}^2} \\ {\mathrm{F}}_{\mathrm{E}}=-226.0\mathrm{N}.{\mathrm{m}}^2/{\mathrm{C}}^2 \end{gathered}$$

Hence, the total electric flux due to these two-point charges through a spherical surface centered at the origin and with radius 2.50m is$-226.0{\mathrm{Nm}}^2/\mathrm{C}$.