If `g_e, g_h` and `g_d` be the acceleration due to gravity at earth’s surface, a height h and at depth d respectively . Then:

Options

A. `g_e gt g_h gt g_d`
B. `g_e gt g_h lt g_d`
C. `g_e lt g_h lt g_d`
D. `g_e lt g_h gt g_d `

Explanation

To solve the problem regarding the acceleration due to gravity at different positions relative to the Earth's surface, we need to analyze the formulas for gravitational acceleration at the Earth's surface, at a height \( h \), and at a depth \( d \).

### Step-by-Step Solution:

1. **Understand the Definitions**:
- \( g_e \): Acceleration due to gravity at Earth's surface.
- \( g_h \): Acceleration due to gravity at height \( h \).
- \( g_d \): Acceleration due to gravity at depth \( d \).

2. **Formula for \( g_h \)** (at height \( h \)):
The formula for gravitational acceleration at a height \( h \) above the Earth's surface is given by:
\[ g_h = g_e \left( \frac{R}{R + h} \right)^2 \]
where \( R \) is the radius of the Earth.

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

4. **Comparing \( g_e \), \( g_h \), and \( g_d \)**:
- At the surface, we have \( g_e \).
- At height \( h \), as \( h \) increases, \( g_h \) decreases since the term \( \left( \frac{R}{R + h} \right)^2 \) becomes smaller.
- At depth \( d \), as \( d \) increases, \( g_d \) also decreases because \( \left( 1 - \frac{d}{R} \right) \) becomes smaller.

5. **Assuming \( h = d \)**:
To compare the values, we can set \( h = d \). This gives us:
- \( g_h = g_e \left( \frac{R}{R + h} \right)^2 \)
- \( g_d = g_e \left( 1 - \frac{h}{R} \right) \)

6. **Analyzing the Values**:
- As \( h \) increases, \( g_h \) will always be less than \( g_e \).
- \( g_d \) will also be less than \( g_e \) but will decrease linearly with depth.
- Since both \( g_h \) and \( g_d \) decrease as \( h \) and \( d \) increase, we can conclude that \( g_h < g_d < g_e \) when \( h = d \).

7. **Conclusion**:
Therefore, the order of acceleration due to gravity is:
\[ g_h < g_d < g_e \]

### Final Answer:
The correct relationship is \( g_h < g_d < g_e \).