Question

If electric field in a region is radially outward with magnitude $E=Ar$, the charge contained in a sphere of radius $r$ centred at the origin is :

• A. $\frac{1}{4\pi {\epsilon }_0}A{r}^3$
• B. $4\pi {\epsilon }_{0}A{r}^3$
• C. $\frac{1}{4\pi {\epsilon }_0}\frac{A}{r^3}$
• D. $\frac{4\pi {\epsilon }_{0}A}{r^3}$

Answer 1

The correct option is B $4\pi {\epsilon }_{0}A{r}^3$

The electric field at a distance $r$ from the center when charge $q$ of sphere is centered is equal to the field at distance $r$ due to point charge . Thus,
$E=\frac{q}{4\pi {ϵ}_0{r}^2}=Ar\Rightarrow q=4\pi {ϵ}_{0}A{r}^3$