Question

In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_A=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as $\rho_A=k r$, where $k$ is positive. In the spherical shell $B$ of outer radius $r_B$, the electrostatic charge density varies as $\rho_B=\frac{2 k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their SI units.

Which of the following statement(s) is(are) correct?

• A. If $r_B=\sqrt{\frac{3}{2}}$, then the electric field is zero everywhere outside $B$.
• B. If $r_B=\frac{3}{2}$, then the electric potential just outside $B$ is $\frac{k}{\epsilon_0}$.
• C. If $r_B=2$, then the total charge of the configuration is $15 \pi k$.
• D. If $r_B=\frac{5}{2}$, then the magnitude of the electric field just outside $B$ is $\frac{13 \pi k}{\epsilon_0}$.

Answer 1

Answer: (B)

$q _1=\int\limits_0^1 kr 4 \pi r ^2 dr =\frac{4 \pi k }{4}=\pi k$
$q _2=\int\limits_1^{ r } \frac{2 k }{ r } 4 \pi r ^2 dr =\frac{8 \pi k \left( r ^2-1^2\right)}{2} $
$q _2=4 \pi k \left[ r ^2-1\right]=4 \pi kr r ^2-4 \pi k $
$q _{ net }= q _1+ q _2 $
$=4 \pi kr ^2-3 \pi k$
$ q _{\text {net }}=\pi k \left[4 r ^2-3\right]$
(A) $E _{\text {net }}=0 \Rightarrow q _{\text {net }}=0 \Rightarrow r =\frac{\sqrt{3}}{2}$
(B) $ V =\frac{ kQ _{ net }}{ r }=\frac{1}{4 \pi \varepsilon_0} \frac{\pi k \left(4 r ^2-3\right)}{ r }$
$ V =\frac{ k }{4 \varepsilon_0}\left[4 r -\frac{3}{ r }\right] $
$ =\frac{ k }{4 \varepsilon_0}\left[4 \times \frac{3}{2}-\frac{3 \times 2}{3}\right]=\frac{ k }{\varepsilon_0}$
(C) $ q _{ net }=\pi k \left[4(2)^2-3\right] $
$ =13 \pi k $
(D) $E _2 =\frac{ kQ }{ r ^2} $
$=\frac{1}{4 \pi \varepsilon_0} \frac{\pi k \left(4 r ^2-3\right)}{ r ^2}$
$ =\frac{ k }{4 \varepsilon_0}\left[\frac{4\left(\frac{5}{2}\right)^2-3}{(5 / 2)^2}\right]$
$=\frac{ k }{25 \varepsilon_0}[25-3]=\frac{22}{25} \frac{ k }{\varepsilon_0}$