Question

An unknown charge sits on a conducting solid sphere of radius 10 cm . If the electric field 15 cm from the center of the sphere has the magnitude $\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{}\mathbf{N}\mathbf{/}\mathbf{C}$and is directed radially inward, what is the net charge on the sphere?

Short answer

The net charge on the sphere is $-7.5\times {10}^{-9}\mathrm{C}$.

Step-by-step solution

The given data

a) The radius of the solid sphere,$\mathrm{r}=10\hspace{0.33em}\mathrm{cm}$

b) The magnitude of the electric field,$\left|\mathrm{E}\right|=3\times {10}^3\hspace{0.33em}\mathrm{N}/\mathrm{C}$

c) Distance of the electric field from the center of the sphere,$\mathrm{r}=15\hspace{0.33em}\mathrm{cm}$

Understanding the concept of the electric field

Using the concept of the electric field, we can get the net charge of the sphere by substituting the given values in the given formula.

Formula:

The magnitude of the electric field, $\left|\mathrm{E}\right|=\frac{\mathrm{q}}{4{\mathrm{\pi \epsilon }}_0{\mathrm{r}}^2}$ (1)

Calculation of the net charge on the sphere

The electric field produced by the unknown charge at points outside the sphere is like the field of a point particle with a charge equal to the net charge on the sphere.

Thus, the net charge on the sphere is given by the equation (1) as follows:

$$\begin{gathered} \left|\mathrm{q}\right|=\left(4{\mathrm{\pi \epsilon }}_0{\mathrm{r}}^2\right)\times \mathrm{E} \\ =\frac{{\left(0.15\mathrm{m}\right)}^2\left(3\times {10}^3\mathrm{N}/\mathrm{C}\right)}{9\times {10}^9\mathrm{N}.{\mathrm{m}}^2/{\mathrm{C}}^2} \\ =7.5\times {10}^{-9}\mathrm{C} \end{gathered}$$

The field points inward, toward the sphere center, so the charge is negative.

Hence, the value of the net charge isrole="math" localid="1657345115513" $-7.5\times {10}^{-9}\mathrm{C}$ .