Let the minimum external work done in shifting a particle from centre of earth to earth's surface be `W_(1)` and that from surface of earth to infinity be `W_(2)` . Then `(W_(1))/(W_(2))` is equal to

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `1:3`

Explanation

To solve the problem, we need to find the ratio \( \frac{W_1}{W_2} \), where \( W_1 \) is the work done in shifting a particle from the center of the Earth to its surface, and \( W_2 \) is the work done in shifting the particle from the surface of the Earth to infinity.

### Step 1: Calculate \( W_1 \)

1. **Potential Energy at the Center of the Earth**:
The gravitational potential energy \( U \) at the center of the Earth is given by:
\[ U_{\text{center}} = -\frac{3}{2} \frac{G M m}{R} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the particle, and \( R \) is the radius of the Earth.

2. **Potential Energy at the Surface of the Earth**:
The gravitational potential energy at the surface is:
\[ U_{\text{surface}} = -\frac{G M m}{R} \]

3. **Calculate \( W_1 \)**:
The work done \( W_1 \) in moving the particle from the center to the surface is the change in potential energy:
\[ W_1 = U_{\text{surface}} - U_{\text{center}} = \left(-\frac{G M m}{R}\right) - \left(-\frac{3}{2} \frac{G M m}{R}\right) \]
Simplifying this gives:
\[ W_1 = -\frac{G M m}{R} + \frac{3}{2} \frac{G M m}{R} = \frac{1}{2} \frac{G M m}{R} \]

### Step 2: Calculate \( W_2 \)

1. **Potential Energy at Infinity**:
The gravitational potential energy at infinity is:
\[ U_{\infty} = 0 \]

2. **Calculate \( W_2 \)**:
The work done \( W_2 \) in moving the particle from the surface to infinity is:
\[ W_2 = U_{\infty} - U_{\text{surface}} = 0 - \left(-\frac{G M m}{R}\right) = \frac{G M m}{R} \]

### Step 3: Calculate the Ratio \( \frac{W_1}{W_2} \)

Now we can find the ratio:
\[ \frac{W_1}{W_2} = \frac{\frac{1}{2} \frac{G M m}{R}}{\frac{G M m}{R}} = \frac{1}{2} \]

### Final Answer
Thus, the ratio \( \frac{W_1}{W_2} \) is:
\[ \frac{W_1}{W_2} = \frac{1}{2} \]