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

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, \(\mathrm{E}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\mathrm{q}}{\mathrm{r}^{2}}\)

\( = \frac{1}{{4\pi {\varepsilon _0}}} \cdot \frac{{4\pi {\rho _0}{r^3}}}{{{r^2}}} \cdot \left[ {\frac{1}{3} - \frac{r}{{4R}}} \right]\)

\(=\frac{\rho_{0}}{\varepsilon_{0}}\left[\frac{\tau}{3}-\frac{\tau^{2}}{4 R}\right]\)