Two identical spheres each of radius `R` are placed with their centres at a distance `nR`, where `n` is integer greater than `2`. The gravitational force between them will be proportional to

Options

A. `1//R^(4)`
B. `1//R^(2)`
C. `R^(2)`
D. `R^(4)`

Explanation

To solve the problem of finding the gravitational force between two identical spheres, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Parameters:**
- We have two identical spheres, each with a radius \( R \).
- The distance between their centers is \( nR \), where \( n \) is an integer greater than 2.

2. **Use the Gravitational Force Formula:**
- The gravitational force \( F \) between two masses \( M_1 \) and \( M_2 \) separated by a distance \( d \) is given by:
\[ F = \frac{G M_1 M_2}{d^2} \]
- Here, \( G \) is the gravitational constant.

3. **Substitute the Masses:**
- Since the spheres are identical, we can denote their mass as \( M \). Thus, \( M_1 = M_2 = M \).
- The distance \( d \) between the centers of the spheres is \( nR \).

4. **Rewrite the Gravitational Force Expression:**
- Substituting the values into the gravitational force formula, we have:
\[ F = \frac{G M^2}{(nR)^2} \]
- This simplifies to:
\[ F = \frac{G M^2}{n^2 R^2} \]

5. **Express Mass in Terms of Radius:**
- The mass \( M \) of a sphere can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ M = \rho V = \rho \left(\frac{4}{3} \pi R^3\right) \]
- Therefore, we can substitute \( M \) in the force equation:
\[ F = \frac{G \left(\rho \frac{4}{3} \pi R^3\right)^2}{n^2 R^2} \]

6. **Simplify the Expression:**
- Expanding the mass term:
\[ F = \frac{G \left(\rho^2 \frac{16}{9} \pi^2 R^6\right)}{n^2 R^2} \]
- This simplifies to:
\[ F = \frac{16 G \pi^2 \rho^2 R^6}{9 n^2 R^2} \]
- Further simplifying gives:
\[ F = \frac{16 G \pi^2 \rho^2}{9 n^2} R^4 \]

7. **Conclusion:**
- From the final expression, we can conclude that the gravitational force \( F \) is proportional to \( R^4 \):
\[ F \propto R^4 \]

### Final Answer:
The gravitational force between the two identical spheres is proportional to \( R^4 \).