Let there be a spherically symmetric charge distribution with charge density varying as ρ (r)=ρ

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1. Electric Charges and Fields

hard

Let there be a spherically symmetric charge distribution with charge density varying as $\rho (r)=\;\rho _0\left( {\frac{5}{4} - \frac{r}{R}} \right)$, upto $r = R$ ,and $\rho (r) = 0$ for $r > R$ , where $r$ is the distance from the origin. The electric field at a distance $r(r < R)$ from the origin is given by

$\frac{{{\rho _o}r}}{{3{\varepsilon _0}}}\;\left( {\frac{5}{4} - \frac{r}{R}} \right)\;\;\;\;\;\;$

$\frac{{4\pi {\rho _0r}}}{{3{\varepsilon _0}}}\;\left( {\frac{5}{3} - \frac{r}{R}} \right)$

$\frac{{{\rho _o}r}}{{4{\varepsilon _0}}}\;\left( {\frac{5}{3} - \frac{r}{R}} \right)$

$\frac{{4\pi {\rho _0r}}}{{3{\varepsilon _0}}}\;\left( {\frac{5}{4} - \frac{r}{R}} \right)$

(AIEEE-2010)

Solution

Let us consider a spherical shell of radius $x$ and thickness $dx$

Charge on this shell

$d q=\rho \cdot 4 \pi x^{2} d x=\rho_{0}\left(\frac{5}{4}-\frac{x}{R}\right) .4 \pi x^{2} d x$

$\therefore$ Total charge in the spherical region from centre to $r$ $(r < R)$ is

$q=\int d q=4 \pi \rho_{0} \int_{0}^{r}\left(\frac{5}{4}-\frac{x}{R}\right) x^{2} d x$

$=4 \pi \rho_{0}\left[\frac{5}{4} \cdot \frac{r^{3}}{3}-\frac{1}{R} \cdot \frac{r^{4}}{4}\right]=\pi \rho_{0} r^{3}\left(\frac{5}{3}-\frac{r}{R}\right)$

Electric field at $r, E=\frac{1}{4 \pi \epsilon_{0}} \cdot \frac{q}{r^{2}}$

$=\frac{1}{4 \pi \epsilon_{0}} \cdot \frac{\pi \rho_{0} r^{3}}{r^{2}}\left(\frac{5}{3}-\frac{r}{R}\right)=\frac{\rho_{0} r}{4 \epsilon_{0}}\left(\frac{5}{3}-\frac{r}{R}\right)$

Standard 12

Physics