The change in the value of 'g' at a height 'h' above the surface of the earth is the same as at a depth 'd' below the surface of earth. When both 'd' and 'h' are much smaller then the radius of earth, then which one of the following is correct?

Options

A. `d = (3h)/(2)`
B. `d = (h)/(2)`
C. d= h
D. d =2h

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 g at height and depth and then equate them to find the relationship between h and d.

### Step-by-Step Solution:

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

2. **Understanding the Variation of g with Depth:**
The formula for 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:
- \( g_d \) is the gravity at depth \( d \).

3. **Setting the Equations 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 from Both Sides:**
Since \( g \) is not zero, we can cancel it from both sides:
\[ 1 - \frac{2h}{R} = 1 - \frac{d}{R} \]

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

6. **Final Relationship:**
Rearranging gives us the final relationship:
\[ d = 2h \]

### Conclusion:
The relationship between the depth \( d \) and the height \( h \) is:
\[ d = 2h \]