The magnitude of the gravitational field at distance `r_1 and `r_2` from the centre of a unifrom sphere of radius R and mass m are `F_1 and F_2` respectively. Then:

Options

A. `F_1/F_2=r_1/r_2` if `r_1 lt R` and `r_2 lt R `
B. `F_1/F_2=r_2^2/r_1^2` if `r_1 gt R` and `r_2 gt R`
C. `F_1/F_2 = r_1/r_2` if `r_1 gt R` and `r_2 gt R`
D. `F_1/F_2=r_1^2/r_2^2` if `r_1 lt R` and `r_2 lt R `

Explanation

To solve the problem, we need to analyze the gravitational field at two different distances \( r_1 \) and \( r_2 \) from the center of a uniform sphere of radius \( R \) and mass \( m \). The gravitational field \( F \) at a distance \( r \) from the center of a sphere can be described using the following principles:

1. **Inside the Sphere (r < R)**: The gravitational field is given by:
\[ F = \frac{Gm}{R^3} r \]
where \( G \) is the gravitational constant, \( m \) is the mass of the sphere, and \( R \) is the radius of the sphere.

2. **Outside the Sphere (r ≥ R)**: The gravitational field behaves like that of a point mass located at the center of the sphere:
\[ F = \frac{Gm}{r^2} \]

### Step-by-Step Solution:

**Step 1: Determine the conditions for \( r_1 \) and \( r_2 \)**

- We need to consider two cases based on the values of \( r_1 \) and \( r_2 \) in relation to \( R \):
- Case 1: Both \( r_1 < R \) and \( r_2 < R \)
- Case 2: Both \( r_1 \geq R \) and \( r_2 \geq R \)

**Step 2: Case 1 - Both \( r_1 \) and \( r_2 \) are less than \( R \)**

- For \( r_1 < R \):
\[ F_1 = \frac{Gm}{R^3} r_1 \]
- For \( r_2 < R \):
\[ F_2 = \frac{Gm}{R^3} r_2 \]
- Now, we can find the ratio \( \frac{F_1}{F_2} \):
\[ \frac{F_1}{F_2} = \frac{\frac{Gm}{R^3} r_1}{\frac{Gm}{R^3} r_2} = \frac{r_1}{r_2} \]

**Step 3: Case 2 - Both \( r_1 \) and \( r_2 \) are greater than or equal to \( R \)**

- For \( r_1 \geq R \):
\[ F_1 = \frac{Gm}{r_1^2} \]
- For \( r_2 \geq R \):
\[ F_2 = \frac{Gm}{r_2^2} \]
- Now, we can find the ratio \( \frac{F_1}{F_2} \):
\[ \frac{F_1}{F_2} = \frac{\frac{Gm}{r_1^2}}{\frac{Gm}{r_2^2}} = \frac{r_2^2}{r_1^2} \]

### Conclusion:

- The correct relationship for the gravitational fields depends on whether both distances are inside or outside the sphere.
- If both \( r_1 \) and \( r_2 \) are less than \( R \), then \( \frac{F_1}{F_2} = \frac{r_1}{r_2} \).
- If both \( r_1 \) and \( r_2 \) are greater than or equal to \( R \), then \( \frac{F_1}{F_2} = \frac{r_2^2}{r_1^2} \).