The diagram showing the variation of gravitational potential of earth with distance from the centre of earth is

Options

A.
B.
C.
D.

Explanation

To solve the question regarding the variation of gravitational potential of Earth with distance from the center of Earth, we will analyze the gravitational potential both inside and outside the Earth.

### Step-by-Step Solution:

1. **Understanding Gravitational Potential**:
Gravitational potential (V) is defined as the work done per unit mass to bring a mass from infinity to a point in the gravitational field. It is given by the formula:
\[ V = -\frac{GM}{r} \]
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. **Inside the Earth (r < R)**:
When we are inside the Earth (at a distance \( r \) from the center where \( r < R \)), only the mass enclosed within that radius contributes to the gravitational potential. The gravitational potential inside a uniform sphere is given by:
\[ V = -\frac{GM}{2R} + \frac{GM}{R^3} r^2 \]
where \( R \) is the radius of the Earth. This equation shows that the potential increases quadratically with distance \( r \) from the center.

3. **At the Surface of the Earth (r = R)**:
At the surface of the Earth, we can substitute \( r = R \) into the equation for the potential:
\[ V = -\frac{GM}{R} \]
This is the gravitational potential at the surface.

4. **Outside the Earth (r > R)**:
For distances greater than the radius of the Earth, the Earth can be treated as a point mass located at its center. Thus, the gravitational potential is again given by:
\[ V = -\frac{GM}{r} \]
This shows that the potential decreases inversely with distance \( r \).

5. **Plotting the Graph**:
- For \( r < R \): The potential increases from \(-\frac{3GM}{2R}\) (at the center) to \(-\frac{GM}{R}\) (at the surface).
- For \( r = R \): The value is \(-\frac{GM}{R}\).
- For \( r > R \): The potential decreases towards zero as \( r \) increases.

6. **Conclusion**:
The graph of gravitational potential versus distance from the center of the Earth will show a parabolic increase for \( r < R \) and a hyperbolic decrease for \( r > R \). The correct representation of this variation is depicted in option (T).