Two identical spheres of radius R made of the same material are kept at a distance d apart. Then the gravitational attraction between them is proportional to

Options

A. `d^(-2)`
B. `d^(2)`
C. `d^(4)`
D. `d`

Explanation

To find the gravitational attraction between two identical spheres of radius R made of the same material and kept at a distance d apart, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Mass of Each Sphere**:
Since the spheres are made of the same material and have the same radius R, we can express their mass (m) using the formula for the volume of a sphere and the density (ρ) of the material:
\[ m = \text{Volume} \times \text{Density} = \frac{4}{3} \pi R^3 \rho \]

2. **Apply the Law of Universal Gravitation**:
The gravitational force (F_g) between two masses (m) separated by a distance (d) is given by Newton's law of universal gravitation:
\[ F_g = \frac{G m_1 m_2}{d^2} \]
Here, both masses are equal (m_1 = m_2 = m), so we can write:
\[ F_g = \frac{G m^2}{d^2} \]

3. **Substitute the Mass Expression**:
Substitute the expression for mass (m) into the gravitational force equation:
\[ F_g = \frac{G \left(\frac{4}{3} \pi R^3 \rho\right)^2}{d^2} \]

4. **Simplify the Expression**:
The expression simplifies to:
\[ F_g = \frac{G \cdot \frac{16}{9} \pi^2 R^6 \rho^2}{d^2} \]

5. **Determine the Proportionality**:
From the final expression, we can see that the gravitational force is proportional to \( \frac{1}{d^2} \):
\[ F_g \propto \frac{1}{d^2} \]
Therefore, we can conclude:
\[ F_g \propto d^{-2} \]

### Conclusion:
The gravitational attraction between the two identical spheres is proportional to \( d^{-2} \).