Question

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

• A. $4\pi {ϵ}_{0}A{a}^3$
• B. ${ϵ}_{0}A{a}^3$
• C. $4\pi {ϵ}_{0}A{a}^2$
• D. $A{ϵ}_0{a}^2$

Answer 1

The correct option is A $4\pi {ϵ}_{0}A{a}^3$

From Gauss's law,

$\oint \overrightarrow{E}\cdot \overrightarrow{dS}=\frac{q_{enc}}{{ϵ}_0}$

At $r=a$, $E=Aa$

$\therefore \oint \overrightarrow{E}\cdot \overrightarrow{dS}=\left(Aa\right)4\pi a^2=\frac{q_{enc}}{{ϵ}_0}$

$\Rightarrow q_{enc}=4\pi {ϵ}_{0}A{a}^3$