Question

The diagram shows a material formed into a sphere of radius $R$. The diagram shows the magnitude of the electric field $E$ as a function of distance $r$ from the center of the sphere.
Which of the following can be concluded from this graph?

• A. The sphere is made of metal.
• B. The sphere is an insulator with uniform volume charge density.
• C. The sphere is an insulator with volume charge density that is increases as $r$ increases.
• D. The sphere is an insulator with volume charge density that is decreases as $r$ increases.
• E. None of these conclusions is certain based on the graph.

Answer 1

The correct option is D The sphere is an insulator with volume charge density that is decreases as $r$ increases.

The sphere must be an insulator with volume charge density that decreases with increase in $r$.

Let the volume charge density is given by $\rho =\frac{A}{r}$

$\therefore$ Charge enclosed by the Gaussian surface of radius $r$, $q_{enc}={\int }_o^r\rho \times \left(4\pi x^{2}dx\right)$

OR $q_{enc}={\int }_o^r\frac{A}{x}\times 4\pi x^{2}dx=4\pi A\times {\int }_o^{r}xdx$

$⟹$ $q_{enc}=4\pi A\times \frac{r^2}{2}=2\pi A{r}^2$

Using Gauss's law : $\oint \overrightarrow{E}.\overrightarrow{da}=\frac{q_{enc}}{{ϵ}_o}$

$\therefore$ $E\times 4\pi r^2=\frac{2\pi A{r}^2}{{ϵ}_o}$ $⟹E=\frac{A}{2{ϵ}_o}=constant$ for $r<R$