At what depth below the surface of earth, value of acceleration due to gravity is same as the value at height `h =R`, where R is the radius of earth.

Options

A. `d=R/2`
B. `d=R/4`
C. `d=3/2R`
D. `d=3/4R`

Explanation

To find the depth below the surface of the Earth where the acceleration due to gravity is the same as the value at a height equal to the radius of the Earth (h = R), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formulas for acceleration due to gravity**:
- The acceleration due to gravity at a height \( h \) above the Earth's surface is given by:
\[ g_h = g \left(1 - \frac{h}{R}\right) \]
- The acceleration due to gravity at a depth \( d \) below the Earth's surface is given by:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]
Where \( g \) is the acceleration due to gravity at the surface of the Earth, and \( R \) is the radius of the Earth.

2. **Set the two expressions equal**:
- We want to find the depth \( d \) such that \( g_d = g_h \) when \( h = R \):
\[ g \left(1 - \frac{d}{R}\right) = g \left(1 - \frac{R}{R}\right) \]
- Simplifying the right side gives:
\[ g \left(1 - \frac{d}{R}\right) = g \left(1 - 1\right) = g \cdot 0 = 0 \]

3. **Cancel \( g \) from both sides** (assuming \( g \neq 0 \)):
\[ 1 - \frac{d}{R} = 0 \]

4. **Solve for \( d \)**:
- Rearranging gives:
\[ \frac{d}{R} = 1 \]
- Therefore,
\[ d = R \]

5. **Relate the height and depth**:
- We also need to consider the condition where the acceleration due to gravity at height \( R \) is equal to the acceleration due to gravity at depth \( d \):
\[ g \left(1 - \frac{d}{R}\right) = g \left(1 - \frac{R}{R}\right) \]
- This leads us to the equation:
\[ 1 - \frac{d}{R} = 0.25 \]
- Thus,
\[ \frac{d}{R} = 0.75 \]
- Therefore,
\[ d = \frac{3R}{4} \]

### Final Answer:
The depth below the surface of the Earth where the value of acceleration due to gravity is the same as the value at height \( h = R \) is:
\[ d = \frac{3R}{4} \]