Question

In a spherical distribution, the charge density varies as $\rho \left(r\right)=A/r$ for $a < r < b$ (as shown) where $A$ is a constant. A point charge $Q$ lies at the centre of the sphere at $r=0$ . The electric field in the region $a < r < b$ has a constant magnitude for

• A. $A=0$
• B. $A=Q$
• C. $\text{A} = \frac{\text{Q}}2 \pi \,\text{a}^{2}$
• D. $\text{A} = \frac{\text{Q}}{4 \pi \, \text{a}^{2}}$

Answer 1

Answer: (C)

$Q_{i n}=\int_a^r \frac{A}{r} \times 4 \pi r^2 d r+Q=2 \pi A\left(r^2-a^2\right)+Q$
$\mathrm{E}=\frac{\mathrm{Q}_{\mathrm{in}}}{4 \pi \mathrm{r}^2 \varepsilon_0} \Rightarrow \mathrm{E}=\frac{\frac{\mathrm{A}}{2}-\frac{\mathrm{a}^2 \mathrm{~A}}{2 \mathrm{r}^2}+\frac{\mathrm{Q}}{4 \pi \mathrm{r}^2}}{\varepsilon_0}$
For $E$ = constant,
$-\frac{\mathrm{a}^2 \mathrm{~A}}{2 \mathrm{r}^2}+\frac{\mathrm{Q}}{4 \pi \mathrm{r}^2}=0$
$\Rightarrow \mathrm{A}=\frac{\mathrm{Q}}{2 \pi \mathrm{a}^2}$