At what depth below the surface of the earth, acceleration due to gravity g will be half its value 1600km above the surface of the earth

Options

A. `4.2xx10^(6)m`
B. `3.91xx10^(6)m`
C. `1.59xx10^(6)m`
D. none of these

Explanation

To solve the problem of finding the depth below the surface of the Earth where the acceleration due to gravity \( g_d \) will be half its value at a height of 1600 km above the surface, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Problem**:
We need to find the depth \( d \) below the Earth's surface where the acceleration due to gravity \( g_d \) is half of the value of gravity at a height of 1600 km above the Earth's surface.

2. **Known Values**:
- Radius of the Earth, \( R = 6400 \) km.
- Acceleration due to gravity at the surface, \( g = 9.8 \, \text{m/s}^2 \).

3. **Calculate Gravity at Height \( h \)**:
The formula for gravitational acceleration at a height \( h \) above the Earth's surface is given by:
\[ g_h = g \left( \frac{R}{R + h} \right)^2 \]
Here, \( h = 1600 \, \text{km} = 1.6 \times 10^6 \, \text{m} \).

4. **Substituting Values**:
We substitute \( R \) and \( h \) into the formula:
\[ g_h = g \left( \frac{6400}{6400 + 1600} \right)^2 = g \left( \frac{6400}{8000} \right)^2 = g \left( \frac{4}{5} \right)^2 \]
\[ g_h = g \left( \frac{16}{25} \right) \]

5. **Finding \( g_d \)**:
According to the problem, \( g_d \) is half of \( g_h \):
\[ g_d = \frac{1}{2} g_h = \frac{1}{2} \left( \frac{16}{25} g \right) = \frac{8}{25} g \]

6. **Using the Formula for Gravity at Depth**:
The formula for gravitational acceleration at a depth \( d \) below the surface of the Earth is:
\[ g_d = g \left( 1 - \frac{d}{R} \right) \]

7. **Setting Up the Equation**:
We set the two expressions for \( g_d \) equal to each other:
\[ \frac{8}{25} g = g \left( 1 - \frac{d}{R} \right) \]

8. **Canceling \( g \)**:
Since \( g \) is non-zero, we can cancel it from both sides:
\[ \frac{8}{25} = 1 - \frac{d}{R} \]

9. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{d}{R} = 1 - \frac{8}{25} = \frac{17}{25} \]

10. **Calculating Depth \( d \)**:
Now, substituting \( R = 6400 \, \text{km} \):
\[ d = R \cdot \frac{17}{25} = 6400 \cdot \frac{17}{25} \]
\[ d = 6400 \cdot 0.68 = 4352 \, \text{km} \]

### Final Answer:
The depth below the surface of the Earth where the acceleration due to gravity is half its value at 1600 km above the surface is approximately **4352 km**.