Question

Let there be a spherical symmetric charge distribution with charge density varying as $p\left(r\right)=p_{0}\frac{r}{R}$ upto $r=R$ and $\rho \left(r\right)=0 \, $ for $r>R, \, $ where r is the distance from the origin. The electric field at on a distance $r \, \left(r < R\right)$ from the origin is given by -

• A. $\frac{\rho _{0} r^{2}}{4 \epsilon _{0} R}$
• B. $\frac{\rho _{0} r}{4 \epsilon _{0} R}$
• C. $\frac{\rho _{0} r^{4}}{\epsilon _{0}}$
• D. $\frac{\rho _{0} r^{2}}{\epsilon _{0} R}$

Answer 1

Answer: (A)

$\oint \overset{ \rightarrow }{E} ‍. \, \overset{ \rightarrow }{d s}=\frac{q_{i n}}{\epsilon _{0}}$ ....(1)


$q_{i n}q_{i n}$ =∫0rρ4πr2dr=∫0rρ0rR4πr2dr=4πρ0Rr44
$=\frac{\pi \rho _{0} r^{4}}{R}$ , from (1)
$E.4\pi \, r^{2} \, =\frac{\pi \rho _{0} r^{4}}{\epsilon _{0} R} \, \therefore \, E \, =\frac{\rho _{0} r^{2}}{4 \epsilon _{0} R}$