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 analyze the change in the acceleration due to gravity (g) at a height (h) above the Earth's surface and at a depth (d) below the Earth's surface. We will use the formulas for gravitational acceleration at these two positions.

### Step-by-Step Solution:

1. **Understanding the Change in g at Height (h)**:
The acceleration due to gravity at a height \( h \) above the Earth's surface can be expressed as:
\[ g_h = g \left(1 - \frac{2h}{R}\right) \]
where \( g \) is the acceleration due to gravity at the surface of the Earth, and \( R \) is the radius of the Earth.

2. **Understanding the Change in g at Depth (d)**:
The acceleration due to gravity at a depth \( d \) below the Earth's surface can be expressed as:
\[ g_d = g \left(1 - \frac{d}{R}\right) \]

3. **Setting the Changes Equal**:
According to the problem, the change in \( g \) at height \( h \) is the same as at depth \( d \). Therefore, we can set the two equations equal to each other:
\[ g \left(1 - \frac{2h}{R}\right) = g \left(1 - \frac{d}{R}\right) \]

4. **Canceling g**:
Since \( g \) is common on both sides, we can cancel it out:
\[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \]

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

6. **Final Result**:
Therefore, we conclude that:
\[ d = 2h \]

### Conclusion:
The depth \( d \) below the Earth's surface is equal to twice the height \( h \) above the Earth's surface.