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Let $\rho (r) =\frac{Q}{{\pi {R^4}}}r$ be the charge density distribution for a solid sphere of radius $R$ and total charge $Q$. For a point '$p$' inside the sphere at distance $r_1$ from the centre of the sphere, the magnitude of electric field is
$0$
$\frac{Q}{{4\pi {\varepsilon _0}{r_1}^2}}$
$\;\frac{Q}{{4\pi {\varepsilon _0}{R^4}}}$
$\;\frac{{Q{r_1}^2}}{{3\pi {\varepsilon _0}{R^4}}}$
Solution
Let us consider a spherical shell of thickness $d x$ andradius $x$.
The volume of this spherical shell $=4 \pi r^{2} d r$.
The charge enclosed within shell
$=\frac{Q r}{\pi R^{4}}\left[4 \pi r^{2} d r\right]$
The charge enclosed in a sphere of radius $r_{1}$ is
$=\frac{4 Q}{R^{4}} \int_{0}^{r_{1}} r^{3} d r=\frac{4 Q}{R^{4}}\left[\frac{r^{4}}{4}\right]_{0}^{r}=\frac{Q}{R^{4}} r_{1}^{4}$
$\therefore $ The electric field at point $p$ inside the sphere at a distance $r_{1}$ from the centre of the sphere is
$E=\frac{1}{4 \pi \epsilon_{0}}-\frac{\left[\frac{Q}{R^{4}} r_{1}^{4}\right]}{r_{1}^{2}}=\frac{1}{4 \pi \epsilon_{0}} \frac{Q}{R^{4}} r^{2}$
