By which curve will be variation of gravitational potential of a hollow sphere of radius R with distance be depicted ?

Options

A.
B.
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D.

Explanation

To determine the variation of gravitational potential \( V \) of a hollow sphere of radius \( R \) with distance \( r \), we can analyze the gravitational potential in different regions relative to the hollow sphere.

### Step-by-Step Solution:

1. **Understanding the Hollow Sphere**:
- A hollow sphere is a spherical shell with mass distributed uniformly over its surface. The gravitational field inside a hollow sphere is zero.

2. **Gravitational Potential Inside the Hollow Sphere**:
- Since the gravitational field \( g \) inside the hollow sphere is zero, the gravitational potential \( V \) remains constant throughout the interior of the sphere.
- Let’s denote the gravitational potential at the surface of the sphere (radius \( R \)) as \( V(R) \).

3. **Calculating Gravitational Potential at the Surface**:
- The gravitational potential \( V \) at the surface of the hollow sphere can be calculated using the formula:
\[ V(R) = -\frac{GM}{R} \]
- Here, \( G \) is the universal gravitational constant and \( M \) is the mass of the hollow sphere.

4. **Gravitational Potential Outside the Hollow Sphere**:
- For points outside the hollow sphere (where \( r > R \)), the gravitational potential can be expressed as:
\[ V(r) = -\frac{GM}{r} \]
- This shows that the potential decreases as we move further away from the sphere.

5. **Graphing the Variation of Gravitational Potential**:
- On a graph where the x-axis represents the distance \( r \) and the y-axis represents the gravitational potential \( V \):
- From \( r = 0 \) to \( r = R \), the potential \( V \) remains constant at \( -\frac{GM}{R} \).
- At \( r = R \), the potential starts decreasing as \( -\frac{GM}{r} \) for \( r > R \).
- The graph will show a horizontal line at \( V = -\frac{GM}{R} \) until \( r = R \), and then it will curve downward as \( r \) increases.

6. **Conclusion**:
- The variation of gravitational potential of a hollow sphere with distance is depicted by a graph that is constant for \( r < R \) and decreases for \( r > R \).