The electric field in a certain region is acting radially outward and is given by $\mathrm{E}=\mathrm{A}\mathrm{r}$. A charge contained in a sphere of radius $\text{a}$ centred at the origin of the field will be given by

• A

  $4\mathrm{\pi }{\mathrm{ϵ}}_0\mathrm{A}{\mathrm{a}}^2$
• B

  $\mathrm{A}{\mathrm{ϵ}}_0{\mathrm{a}}^2$
• Correct answer
  C

  $4\mathrm{\pi }{\mathrm{ϵ}}_0\mathrm{A}{\mathrm{a}}^3$
• D

  ${\mathrm{ϵ}}_0\mathrm{A}{\mathrm{a}}^3$

Solution

We know

${\mathrm{E}}_{\mathrm{Sphere}}=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{\mathrm{q}}{{\mathrm{r}}^2}\mathrm{ }$

$\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{\mathrm{q}}{{\mathrm{a}}^2}=\mathrm{Aa}$

$\mathrm{q}=4{\mathrm{\pi \epsilon }}_0{\mathrm{Aa}}^3$

Hence, the correct answer is option: 'C'.