The magnitude of the gravitational field at distance `r_(1)` and `r_(2)` from the centre of a uniform sphere of radius `R` and mass `M` are `F_(1)` and `F_(2)` respectively. Then:

Options

A. `(F_(r))/(F_(2))=(r_(1))/(r_(2))` if `r_(1)ltR` and `r_(2)ltR`
B. `(F_(1))/(F_(2))=(r_(2)^(2))/(r_(1)^(2))` if `r_(1)gtR` and `r_(2)gtR`
C. `(F_(1))/(F_(2))=(r_(1)^(3))/(r_(2)^(3))` if `r_(1)ltR` and `r_(2)ltR`
D. `(F_(1))/(F_(2))=(r_(1)^(2))/(r_(2)^(2))` if `r_(1)ltR` and `r_(2)ltR`

Explanation

To solve the problem regarding the gravitational field at distances \( r_1 \) and \( r_2 \) from the center of a uniform sphere of radius \( R \) and mass \( M \), we need to analyze the gravitational field in two scenarios: when the distances are greater than the radius \( R \) and when they are less than \( R \).

### Step-by-Step Solution:

1. **Gravitational Field Outside the Sphere (\( r_1 > R \) and \( r_2 > R \))**:
- For a point outside the sphere, the gravitational field \( F \) at a distance \( r \) from the center is given by:
\[ F = \frac{GM}{r^2} \]
- Therefore, the gravitational fields at distances \( r_1 \) and \( r_2 \) can be expressed as:
\[ F_1 = \frac{GM}{r_1^2} \quad \text{and} \quad F_2 = \frac{GM}{r_2^2} \]

2. **Finding the Ratio \( \frac{F_1}{F_2} \)**:
- To find the ratio of the gravitational fields \( F_1 \) and \( F_2 \):
\[ \frac{F_1}{F_2} = \frac{\frac{GM}{r_1^2}}{\frac{GM}{r_2^2}} = \frac{r_2^2}{r_1^2} \]

3. **Gravitational Field Inside the Sphere (\( r_1 < R \) and \( r_2 < R \))**:
- For a point inside the sphere, the gravitational field \( F \) at a distance \( r \) from the center is given by:
\[ F = \frac{GM_{\text{enclosed}}}{r^2} \]
- The mass enclosed \( M_{\text{enclosed}} \) can be calculated using the density \( \rho \) of the sphere:
\[ M_{\text{enclosed}} = \rho \cdot V = \rho \cdot \frac{4}{3} \pi r^3 \]
- The density \( \rho \) can be expressed as:
\[ \rho = \frac{M}{\frac{4}{3} \pi R^3} \]
- Thus, substituting this into the expression for \( M_{\text{enclosed}} \):
\[ M_{\text{enclosed}} = \frac{M}{\frac{4}{3} \pi R^3} \cdot \frac{4}{3} \pi r^3 = M \cdot \frac{r^3}{R^3} \]
- The gravitational field inside the sphere becomes:
\[ F = \frac{G \cdot M \cdot \frac{r^3}{R^3}}{r^2} = \frac{GM}{R^3} \cdot r \]
- Therefore, the gravitational fields at distances \( r_1 \) and \( r_2 \) inside the sphere are:
\[ F_1 = \frac{GM}{R^3} \cdot r_1 \quad \text{and} \quad F_2 = \frac{GM}{R^3} \cdot r_2 \]

4. **Finding the Ratio \( \frac{F_1}{F_2} \) Inside the Sphere**:
- The ratio of the gravitational fields \( F_1 \) and \( F_2 \) inside the sphere is:
\[ \frac{F_1}{F_2} = \frac{\frac{GM}{R^3} \cdot r_1}{\frac{GM}{R^3} \cdot r_2} = \frac{r_1}{r_2} \]

### Summary of Results:
- For \( r_1 > R \) and \( r_2 > R \):
\[ \frac{F_1}{F_2} = \frac{r_2^2}{r_1^2} \]
- For \( r_1 < R \) and \( r_2 < R \):
\[ \frac{F_1}{F_2} = \frac{r_1}{r_2} \]