Let `F_1` be the magnitude of the gravitational force exerted on the Sun by Earth and `F_2` be the magnitude of the force exerted on Earth by the Sun. Then:

Options

A. (a)`F_1` is much greater than `F_2`
B. (b)`F_1` is slightly greater than `F_2`
C. (c)`F_1` is equal to `F_2`
D. (d)`F_1` is slightly less than `F_2`

Explanation

To solve the problem, we need to analyze the gravitational forces exerted between the Earth and the Sun. We will use Newton's law of universal gravitation, which states that the gravitational force between two masses is given by the formula:

\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the distance between the centers of the two masses.

### Step 1: Define the masses and distance
Let:
- \( M_E \) be the mass of the Earth,
- \( M_S \) be the mass of the Sun,
- \( D \) be the distance between the Earth and the Sun.

### Step 2: Write the gravitational force exerted on the Sun by the Earth
According to Newton's law, the gravitational force \( F_1 \) exerted on the Sun by the Earth is given by:

\[ F_1 = \frac{G \cdot M_E \cdot M_S}{D^2} \]

### Step 3: Write the gravitational force exerted on the Earth by the Sun
Similarly, the gravitational force \( F_2 \) exerted on the Earth by the Sun is given by:

\[ F_2 = \frac{G \cdot M_S \cdot M_E}{D^2} \]

### Step 4: Compare the two forces
From the equations derived in Steps 2 and 3, we can see that:

\[ F_1 = \frac{G \cdot M_E \cdot M_S}{D^2} \]
\[ F_2 = \frac{G \cdot M_S \cdot M_E}{D^2} \]

### Step 5: Conclusion
Since both \( F_1 \) and \( F_2 \) are equal, we conclude that:

\[ F_1 = F_2 \]

This means that the magnitude of the gravitational force exerted on the Sun by the Earth is equal to the magnitude of the force exerted on the Earth by the Sun.

### Final Answer
Thus, the correct statement is that \( F_1 \) is equal to \( F_2 \).

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