Physics - Electrostatics Question with Solution | TestHub

$\rho ={\rho }_0\left(1-\frac{r}{R}\right)$

$dq=\rho dv$

$q_{\mathrm{i}\mathrm{n}}=\int dq=\rho dv$

$\Rightarrow q_{\mathrm{in}}={\int }_0^R{\rho }_0\left(1-\frac{r}{R}\right)4\pi r^{2}dr$

           $=4\pi {\rho }_0{\int }_0^R\left(1-\frac{r}{R}\right)r^{2}dr$

           $=4\pi {\rho }_0{\int }_0^R{r}^{2}dr-\frac{r^3}{R}dr$

           $=4\pi {\rho }_0\left[{{\left[\frac{r^3}{3}\right]}_0}^R-{{\left[\frac{r^4}{4R}\right]}_0}^R\right]$

          $=4\pi {\rho }_0 \left[\frac{R^3}{3}-\frac{R^4}{4R}\right]$

          $=4\pi {\rho }_0\left[\frac{R^3}{12}\right]$

$\Rightarrow q_{\mathrm{in}}=\frac{\pi {\rho }_0{R}^3}{3}$

$E\times 4\pi r^2=\left(\frac{\pi {\rho }_0{R}^3}{3{\epsilon }_0}\right)$

$\Rightarrow E=\frac{{\rho }_0{R}^3}{12{\epsilon }_0{r}^2}$