A large solid sphere has variable but radially symmetrical mass density changing with distance from the centre (r) as $\mathrm{\rho }={\mathrm{\rho }}_0{\mathrm{r}}^{\mathrm{n}}.$ . For what absolute value of n, field will be independent of r? 

We have,

$\mathrm{\rho }={\mathrm{\rho }}_0{\mathrm{r}}^{\mathrm{n}}.$

${\mathrm{g}}_{\mathrm{in}}\times 4{\mathrm{\pi r}}^2=4{\mathrm{\pi Gm}}_{\mathrm{in}}$ (Gauss's law)

${\mathrm{g}}_{\mathrm{in}}=\frac{{\mathrm{Gm}}_{\mathrm{in}}}{{\mathrm{r}}^2} \mathrm{where} {\mathrm{m}}_{\mathrm{in}}$ is the value of mass in the field which is independent of r

$$\begin{gathered} {\mathrm{m}}_{\mathrm{in}}=\int \mathrm{\rho }4{\mathrm{\pi r}}^2\mathrm{dr} \\ =\int {\mathrm{\rho }}_0{\mathrm{r}}^{\mathrm{n}}{\mathrm{\pi r}}^2\mathrm{dr}=4{\mathrm{\pi \rho }}_0\underset{0}{\overset{\mathrm{t}}{\int }}{\mathrm{r}}^{\mathrm{n}+2}\mathrm{dr} \\ =4{\mathrm{\pi \rho }}_0\underset{0}{\overset{\mathrm{t}}{\int }}{\mathrm{r}}^{\mathrm{n}+2}\mathrm{dr}=4{\mathrm{\pi \rho }}_0\frac{{\mathrm{r}}^{\mathrm{n}+3}}{\mathrm{n}+3} \end{gathered}$$

$\Rightarrow {\mathrm{g}}_{\mathrm{in}}=\frac{\mathrm{G}4{\mathrm{\pi \rho }}_0{\mathrm{r}}^{\mathrm{n}+3}}{\left(\mathrm{n}+3\right){\mathrm{r}}^2}=\frac{4{\mathrm{\pi G\rho }}_0}{\mathrm{n}+3}{\mathrm{r}}^{\mathrm{n}+1}$

For g to be independent of r, $\mathrm{n}+1=0\Rightarrow \mathrm{n}=-1$. 

Hence, absolute value of n is 1.