A large solid sphere has variable but radially symmetrical mass density, changing with distance from the centre $\left(r\right) as \rho ={\rho }_0{r}^{-n}.$ . The absolute value of $n$, for which the field will be independent of $r$, is______

Given: $\rho =\frac{{\rho }_0}{r^n}$ Mass enclosed in gaussian surface,

 $$\begin{gathered} M=\int dm \\ M=\int \frac{{\rho }_0}{r^n}dV={\int }_0^r\frac{{\rho }_0}{r^n}4{\mathrm{\pi r}}^2\mathrm{dr}=4{\mathrm{\pi \rho }}_0\frac{{\mathrm{r}}^{3-\mathrm{n}}}{3-n} \end{gathered}$$

Gravitational field at a distance r is given by:

 $$\begin{gathered} \text{E=}\frac{\mathrm{GM}}{{\mathrm{r}}^2} \\ \therefore \mathrm{E}=\mathrm{G}\frac{4{\mathrm{\pi \rho }}_0}{3-\mathrm{n}}{\mathrm{r}}^{3-\mathrm{n}}\left(\frac{1}{{\mathrm{r}}^2}\right) \\ \mathrm{As}, \mathrm{the} \mathrm{field} \mathrm{is} \mathrm{independent} \mathrm{of} \mathrm{r}. \\ \therefore 3-n=2 or n=1 \end{gathered}$$