If the change in the value of `g` at a height `h` above the surface of earth is the same as at a depth `d` below it (both `h` and `d` are much smaller than the radius of the earth), then

Options

A. `d=h`
B. `d = 2h`
C. `d = h//2`
D. `d = h^(2)`

Explanation

To solve the problem, we need to establish the relationship between the change in gravitational acceleration \( g \) at a height \( h \) above the Earth's surface and at a depth \( d \) below the Earth's surface.

### Step-by-Step Solution:

1. **Understanding Gravitational Acceleration**:
The gravitational acceleration at the surface of the Earth is denoted as \( g \).

2. **Change in \( g \) at Height \( h \)**:
The formula for gravitational acceleration at a height \( h \) above the Earth's surface is given by:
\[ g_h = g \left(1 - \frac{2h}{R}\right) \]
where \( R \) is the radius of the Earth.

3. **Change in \( g \) at Depth \( d \)**:
The formula for gravitational acceleration at a depth \( d \) below the Earth's surface is given by:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]

4. **Setting the Changes Equal**:
According to the problem, the change in \( g \) at height \( h \) is equal to the change in \( g \) at depth \( d \):
\[ g \left(1 - \frac{2h}{R}\right) = g \left(1 - \frac{d}{R}\right) \]

5. **Cancelling \( g \)**:
Since \( g \) is common on both sides and is not zero, we can cancel it out:
\[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \]

6. **Simplifying the Equation**:
By simplifying, we get:
\[ -\frac{2h}{R} = -\frac{d}{R} \]
This leads to:
\[ 2h = d \]

7. **Final Relationship**:
Thus, the relationship between the height \( h \) and depth \( d \) is:
\[ d = 2h \]

### Summary:
The relationship established is that the depth \( d \) below the Earth's surface is twice the height \( h \) above the Earth's surface, or \( d = 2h \).