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A spherically symmetric charge distribution is characterised by a charge density having the following variations
$\rho (r)\, = \,{\rho _0}\left( {1 - \frac{r}{R}} \right)$ for $r < R$
$\rho (r)\,=\,0$ for $r\, \ge \,R$
Where $r$ is the distance from the centre of the charge distribution $\rho _0$ is a constant. The electric field at an internal point $(r < R)$ is
$\frac{{{\rho _0}}}{{4{\varepsilon _0}}}\left( {\frac{r}{3} - \frac{{{r^2}}}{{4R}}} \right)$
$\frac{{{\rho _0}}}{{{\varepsilon _0}}}\left( {\frac{r}{3} - \frac{{{r^2}}}{{4R}}} \right)$
$\frac{{{\rho _0}}}{{3{\varepsilon _0}}}\left( {\frac{r}{3} - \frac{{{r^2}}}{{4R}}} \right)$
$\frac{{{\rho _0}}}{{12{\varepsilon _0}}}\left( {\frac{r}{3} - \frac{{{r^2}}}{{4R}}} \right)$
Solution
Let us consider a spherical shell of radius $x$ and thickness $dx.$
Charge on this shell
$\mathrm{dq}=\rho 4 \pi \mathrm{x}^{2} \mathrm{dx}=\rho_{0}\left(1-\frac{\mathrm{x}}{\mathrm{R}}\right) .4 \pi \mathrm{x}^{2} \mathrm{dx}$
$\therefore$ Total charge in the spherical region from centre to $r(r < R)$
$q=\int d q=4 \pi \rho_{0} \int_{0}^{T}\left(1-\frac{x}{R}\right) x^{2} d x$
$=4 \pi \rho_{0}\left[\frac{x^{3}}{3}-\frac{x^{4}}{4 R}\right]_{0}^{t}$
$=4 \pi \rho_{0}\left[\frac{r^{3}}{3}-\frac{r^{4}}{4 R}\right]$
$=4 \pi \rho_{0} r^{3}\left[\frac{1}{3}-\frac{r}{4 R}\right]$
$\therefore$ Electric field atr, $\mathrm{E}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}}{\mathrm{r}^{2}}$
$ = \frac{1}{{4\pi {\varepsilon _0}}} \cdot \frac{{4\pi {\rho _0}{r^3}}}{{{r^2}}} \cdot \left[ {\frac{1}{3} – \frac{r}{{4R}}} \right]$
$=\frac{\rho_{0}}{\varepsilon_{0}}\left[\frac{\tau}{3}-\frac{\tau^{2}}{4 R}\right]$
