Question

The electric field in a certain region is acting radially outward and is given by $E=$ Ar. The charge contained in a sphere of radius '$a$' centred at the origin of the field, will be given by :

• A. $A \,\varepsilon_{0}\, a^{2}$
• B. $4\, \pi\, \varepsilon_{0}\, Aa ^{3}$
• C. $\varepsilon_{0} \,Aa ^{3}$
• D. $4 \,\pi\, \varepsilon_{0}\, Aa ^{2}$

Answer 1

Answer: (B)

$E=A \vec{r}=A r \hat{r}$
$\overrightarrow{\text { Area }}=4 \pi r^{2} \hat{n}$
$\Rightarrow \phi=\vec{E} \cdot \overrightarrow{\text { Area }}=A a \times 4 \pi a^{2}$
also $\phi_{(r=a)}=4 \pi Aa ^{3}$
So, $q_{\text {in }}=\phi \cdot \varepsilon_{0}=4 \pi a^{3} A \in_{0}$