Dependence of intensity of gravitational field `(E)` of earth with distance `(r)` from centre of earth is correctly represented by

Options

A. a)
B.
C. c)
D.

Answer

To determine the dependence of the intensity of the gravitational field \( E \) of the Earth with distance \( r \) from the center of the Earth, we need to analyze the gravitational field both inside and outside the Earth.

### Step-by-Step Solution:

1. **Understanding Gravitational Field Inside the Earth**:
- For a solid sphere (like Earth), when we are inside the Earth (i.e., when \( r < R \), where \( R \) is the radius of the Earth), the gravitational field \( E \) varies linearly with distance \( r \) from the center.
- The formula for the gravitational field inside a uniform solid sphere is given by:
\[ E = \frac{G M r}{R^3} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth.

2. **Understanding Gravitational Field Outside the Earth**:
- When we are outside the Earth (i.e., when \( r > R \)), the gravitational field \( E \) behaves according to the inverse square law:
- The formula for the gravitational field outside the Earth is given by:
\[ E = \frac{G M}{r^2} \]
- This indicates that the gravitational field decreases with the square of the distance from the center of the Earth.

3. **Graphical Representation**:
- Inside the Earth, since \( E \) is directly proportional to \( r \), the graph of \( E \) versus \( r \) will be a straight line passing through the origin.
- Outside the Earth, since \( E \) is inversely proportional to \( r^2 \), the graph will be a curve that decreases sharply as \( r \) increases.

4. **Conclusion**:
- Combining both behaviors, the graph of gravitational field intensity \( E \) with respect to distance \( r \) from the center of the Earth will show a linear increase when \( r < R \) and a rapid decrease when \( r > R \).
- Therefore, the correct representation of the dependence of the intensity of the gravitational field \( E \) with distance \( r \) from the center of the Earth is a graph that shows a straight line for \( r < R \) and a curve for \( r > R \).

### Final Answer:
The correct option that represents the dependence of intensity of gravitational field \( E \) of Earth with distance \( r \) from the center of Earth is **Option A**.