A uniform sphere has radius ' $R$ ' and mass ' $M$ '. The magnitude of gravitational field at distances '…
- $\frac{\mathrm{R}^2}{\mathrm{r}_1^2 \mathrm{r}_2}$
- $\frac{\mathrm{R}^3}{\mathrm{r}_1 \mathrm{r}_2}$
- $\frac{\mathrm{R}^3}{\mathrm{r}_1^2 \mathrm{r}_2}$
- $\frac{\mathrm{R}^3}{\mathrm{r}_1 \mathrm{r}_2^2}$
Solution
Gravitational Field Ratio Inside and Outside a Uniform Sphere
To determine the ratio $E_1:E_2$ for a gravitational field at distances $r_1 > R$ and $r_2 < R$ from the center of a uniform sphere, the expressions for the gravitational field in each region are applied.
Outside the sphere ($r > R$), the field is $E_{\text{out}} = \frac{GM}{r^2}$, so $E_1 = \frac{GM}{r_1^2}$.
Inside the sphere ($r < R$), only the enclosed mass contributes, with $M_{\text{in}} = M \cdot \frac{r^3}{R^3}$ due to uniform density, giving $E_{\text{in}} = \frac{GM_{\text{in}}}{r^2} = \frac{GMr}{R^3}$, thus $E_2 = \frac{GMr_2}{R^3}$.
The ratio is $\frac{E_1}{E_2} = \frac{\frac{GM}{r_1^2}}{\frac{GMr_2}{R^3}} = \frac{R^3}{r_1^2 r_2}$.
This corresponds to option C.
Asked in: MHT CET 2025 (05 May Shift 2)