A non-conducting sphere of radius R has a charge distributed throughout its volume. The charge density $\rho$ varies with distance r from the centre of the sphere as $\rho =\frac{Kr}{R}$,where K is a constant. The electric field at a point P inside the sphere at a distance $r(<R)$ is

We assume the Gaussian surface of radius r. To find the charge q enclosed inside this surface, we consider a thin spherical shell of radius a and thickness $da$

Volume of the shell is $dV=4\pi a^{2}da$. So the charge contained in the shell is
$$\begin{gathered} dq=\rho \times 4\pi a^{2}da \\ =\frac{Ka}{R}\times 4\pi a^{2}da \\ (\because r=a) \end{gathered}$$
Therefore, the total charge enclosed inside the Gaussian surface is
$$\begin{gathered} q={\int }_{a=0}^{a=r}dq=\frac{4\pi K}{R}{\int }_0^r{a}^{3}da \\ =\frac{4\pi K{r}^4}{4R}=\frac{\pi K{r}^4}{R} \end{gathered}$$
Electric flux through the Gaussian surface is
$\varphi =EA=E\times 4\pi r^2$
From Gauss's law,
$\varphi =\frac{q}{{ϵ}_0}$
or
$E\times 4\pi r^2=\frac{\pi K{r}^4}{R{\in }_0}$
$\Rightarrow E=\frac{K{r}^2}{4R{ϵ}_0}$