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Let $\rho (r)\, = \frac{Q}{{\pi {R^4}}}\,r$ be the volume 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
$0$
$\frac{Q}{{4\pi {\varepsilon _0}r_1^2}}\,$
$\frac{{Q{r_1}}}{{4\pi {\varepsilon _0}{r^4}}}\,$
$\frac{{Qr_{_1}^2}}{{4\pi {\varepsilon _0}{R^4}}}\,$
Solution
$\mathrm{dq}=\frac{\mathrm{Qr}}{\pi \mathrm{R}^{4}} \times 4 \pi \mathrm{r}^{2} \cdot \mathrm{dr}$
${\rm{q}} = \frac{{4{\rm{Q}}}}{{{{\rm{R}}^4}}}\int_0^{{{\rm{r}}_1}} {{{\rm{r}}^3}} \quad {\rm{dr}} = \frac{{{\rm{Qr}}_1^4}}{{{{\rm{R}}^4}}}$
Now the electric field at a distance $\mathrm{r}_{1}$ from the centre (inside)
$\mathrm{E}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{q}}{\mathrm{r}_{1}^{2}}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{Qr}_{1}^{2}}{\mathrm{R}^{4}}$
