Question

The electric field in a region is radially outward with magnitude $E=\alpha r$. Calculate the charge contained in a sphere of radius R centered at the origin. Calculate the value of the charge if $\alpha =100V{m}^{-2}$ and R=0.30 m.

• A. $3\times {10}^{-7}C$
• B. $12\times {10}^{-10}C$
• C. $3\times {10}^{-10}C$
• D. $7\times {10}^{-10}C$

Answer 1

The correct option is C $3\times {10}^{-10}C$

Consider a spherical shell of radius x. The electric flux through this surface is

$\varphi ={\int }_s\overrightarrow{E}\cdot d\overrightarrow{S}=E_{r}4\pi r^2$

Therefore, the electric flux through spherical surface of radius R will be

$\varphi =E_{R}4\pi R^2$

When $r=R,E_R=\alpha R$, we have $\varphi =\alpha R4\pi R^2$By Gauss theorem, net electric flux is $\frac{1}{{\epsilon }_0}\times \text{change enclosed}$

$\therefore \alpha R4\pi R^2=\frac{1}{{\epsilon }_0}{Q}_{enclosed}$or $Q_{enclosed}=\left(4\pi {\epsilon }_0\right)\alpha R^3$

Given R=0.30 m, $\alpha =100V{m}^{-2}$

$Q_{enclosed}=\frac{1}{9\times {10}^9}\times 100\times (0.30{)}^3=3\times {10}^{-10}C$