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, we need to analyze the relationship between the acceleration due to gravity at the Earth's surface (\(g_e\)), at a height \(h\) above the Earth's surface (\(g_h\)), and at a depth \(d\) below the Earth's surface (\(g_d\)).

### Step-by-Step Solution:

1. **Acceleration due to gravity at the Earth's surface (\(g_e\))**:
The acceleration due to gravity at the surface of the Earth is given by:
\[ g_e = \frac{GM}{R^2} \]
where \(G\) is the universal gravitational constant, \(M\) is the mass of the Earth, and \(R\) is the radius of the Earth.

2. **Acceleration due to gravity at a height \(h\) (\(g_h\))**:
The acceleration due to gravity at a height \(h\) above the Earth's surface is given by:
\[ g_h = \frac{g_e}{(1 + \frac{h}{R})^2} \]
This formula shows that as we move away from the Earth's surface, the value of \(g_h\) decreases with the square of the distance from the center of the Earth.

3. **Acceleration due to gravity at a depth \(d\) (\(g_d\))**:
The acceleration due to gravity at a depth \(d\) below the Earth's surface is given by:
\[ g_d = g_e \left(1 - \frac{d}{R}\right) \]
This indicates that as we go deeper into the Earth, the value of \(g_d\) decreases linearly with depth.

4. **Comparing \(g_h\) and \(g_d\)**:
- At the surface (\(d = 0\)), \(g_d = g_e\).
- At height \(h\), \(g_h\) is less than \(g_e\) because it decreases with height.
- At depth \(d\), \(g_d\) is also less than \(g_e\) but decreases linearly.

5. **Conclusion**:
From the above relations, we can summarize:
- \(g_e > g_h\)
- \(g_e > g_d\)
- \(g_h < g_e\) and \(g_d < g_e\)

Thus, the relationship can be expressed as:
\[ g_h < g_d < g_e \]