Which of the following graphs correctly represents the variation of g on the - Earth?

Options

A.
B.
C.
D.

Answer

To solve the question regarding the variation of gravitational acceleration (g) on Earth, we will analyze how g changes with height above the Earth's surface and depth below the Earth's surface.

### Step-by-Step Solution:

1. **Understanding the Variation of g with Height**:
- The formula for gravitational acceleration at a height \( h \) above the Earth's surface is given by:
\[ g' = \frac{g}{(1 + \frac{h}{R})^2} \]
- Here, \( g' \) is the gravitational acceleration at height \( h \), \( g \) is the gravitational acceleration at the Earth's surface, and \( R \) is the radius of the Earth.
- As \( h \) increases, \( g' \) decreases, indicating that gravity decreases with height.

2. **Understanding the Variation of g with Depth**:
- The formula for gravitational acceleration at a depth \( d \) below the Earth's surface is:
\[ g' = g \left(1 - \frac{d}{R}\right) \]
- Here, \( d \) is the depth below the surface.
- As \( d \) increases, \( g' \) decreases, indicating that gravity also decreases with depth.

3. **Analyzing the Behavior at the Center of the Earth**:
- At the center of the Earth, when \( d = R \), the gravitational acceleration \( g' \) becomes:
\[ g' = g \left(1 - \frac{R}{R}\right) = g(1 - 1) = 0 \]
- This means that at the center of the Earth, the gravitational acceleration is zero.

4. **Graphical Representation**:
- The graph of gravitational acceleration \( g' \) versus distance from the center of the Earth would start from \( g \) at the surface, decrease linearly to zero at the center, and then decrease again as height increases above the surface.
- Therefore, the graph will show:
- A linear decrease from \( g \) to \( 0 \) as we go from the surface to the center.
- A decrease in \( g' \) as we move above the surface.

5. **Identifying the Correct Graph**:
- Based on the analysis, the correct graph will show a decrease in gravitational acceleration as we move from the surface to the center and then a further decrease as we move above the surface.

### Conclusion:
The correct graph that represents the variation of gravitational acceleration \( g \) on Earth is option B.