Question

The electric field in a region is radially outwards with magnitude $E=\alpha R$. Calculate the charge contained in a sphere of radius $R$ centred at the origin. If the value of charge is $m\times {10}^{-10}C$, then find $m$. (Given $\alpha =100V/m^2$ and $R=0.30m$)

Answer 1

We know that net electric flux can be given by
${\varphi }_{net}=\overrightarrow{E}.\overrightarrow{S}$

Since, the electric field is radially outwards, the angle between electric field and area vector will be $0^{\circ }$
$⟹{\varphi }_{net}=E.S.cos\left(0^{\circ }\right)$
$⟹{\varphi }_{net}=\alpha R4\pi R^2$
$⟹{\varphi }_{net}=\alpha 4\pi R^3$-----(i)

By Gauss law, we know that
${\varphi }_{net}=\frac{Q_{in}}{{ϵ}_0}$
By using (i), we can write
$4\pi \alpha R^3=\frac{Q_{in}}{{ϵ}_0}$
$⟹Q_{in}=4\pi {ϵ}_0\alpha R^3$
$⟹Q_{in}=\frac{1}{9\times {10}^9}\times 100\times (0.3{)}^3$

$\therefore Q_{in}=3\times {10}^{-10}C$
$\Rightarrow m=3$