The acceleration due to gravity at a height `1km` above the earth is the same as at a depth `d` below the surface of earth. Then :

Options

A. `d=1km`
B. `d=3/2km`
C. `d=2km`
D. `d=1/2km`

Explanation

To solve the problem, we need to find the depth \( d \) below the Earth's surface where the acceleration due to gravity is equal to that at a height of 1 km above the Earth's surface.

### Step-by-step Solution:

1. **Understanding the Problem**:
We know that the acceleration due to gravity \( g \) changes with height and depth. We need to find the depth \( d \) such that:
\[ g_h = g_d \]
where \( g_h \) is the acceleration due to gravity at height \( h \) (1 km) and \( g_d \) is the acceleration due to gravity at depth \( d \).

2. **Acceleration due to Gravity at Height**:
The formula for acceleration due to gravity at a height \( h \) above the Earth's surface is given by:
\[ g_h = \frac{g}{(1 + \frac{h}{R})^2} \]
where \( R \) is the radius of the Earth and \( g \) is the acceleration due to gravity at the surface.

3. **Using Binomial Expansion**:
Since \( h = 1 \text{ km} = 1000 \text{ m} \) is much smaller than \( R \) (approximately \( 6.4 \times 10^6 \text{ m} \)), we can use the binomial approximation:
\[ g_h \approx g \left(1 - 2 \frac{h}{R}\right) \]
Thus, substituting \( h = 1000 \text{ m} \):
\[ g_h \approx g \left(1 - 2 \frac{1000}{R}\right) \]

4. **Acceleration due to Gravity at Depth**:
The formula for acceleration due to gravity at a depth \( d \) below the Earth's surface is:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]

5. **Setting the Two Equations Equal**:
Now, we set \( g_h \) equal to \( g_d \):
\[ g \left(1 - 2 \frac{1000}{R}\right) = g \left(1 - \frac{d}{R}\right) \]
Dividing both sides by \( g \) (assuming \( g \neq 0 \)):
\[ 1 - 2 \frac{1000}{R} = 1 - \frac{d}{R} \]

6. **Solving for Depth \( d \)**:
Rearranging the equation gives:
\[ -2 \frac{1000}{R} = -\frac{d}{R} \]
Multiplying through by \( -R \):
\[ 2 \times 1000 = d \]
Thus, we find:
\[ d = 2000 \text{ m} = 2 \text{ km} \]

### Conclusion:
The depth \( d \) below the Earth's surface where the acceleration due to gravity is the same as at a height of 1 km above the surface is \( 2 \text{ km} \).