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

Options

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Answer

To determine the variation of gravitational potential \( V \) of a hollow sphere with distance \( r \) from its center, 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 of radius \( R \) has mass distributed uniformly over its surface. The gravitational potential is influenced by this mass distribution.

2. **Regions to Consider**:
- There are three distinct regions to consider when analyzing the gravitational potential:
- Inside the hollow sphere (\( r < R \))
- On the surface of the hollow sphere (\( r = R \))
- Outside the hollow sphere (\( r > R \))

3. **Gravitational Potential Inside the Hollow Sphere**:
- For a hollow sphere, the gravitational potential \( V \) inside the sphere (for \( r < R \)) is constant and equal to the potential at the surface. This is because, according to the shell theorem, a uniform spherical shell exerts no gravitational force on a mass located inside it.
- The potential at the surface is given by:
\[ V(R) = -\frac{GM}{R} \]
- Therefore, for \( r < R \):
\[ V(r) = -\frac{GM}{R} \]

4. **Gravitational Potential on the Surface**:
- At the surface of the hollow sphere, the potential is:
\[ V(R) = -\frac{GM}{R} \]

5. **Gravitational Potential Outside the Hollow Sphere**:
- For \( r > R \), the gravitational potential behaves as if all the mass \( M \) of the hollow sphere is concentrated at its center. The potential at a distance \( r \) is given by:
\[ V(r) = -\frac{GM}{r} \]

6. **Graphical Representation**:
- The graph of gravitational potential \( V \) versus distance \( r \) will show:
- A horizontal line at \( V = -\frac{GM}{R} \) for \( r < R \) (constant potential inside the hollow sphere).
- A point at \( V = -\frac{GM}{R} \) at \( r = R \).
- A curve that decreases as \( -\frac{GM}{r} \) for \( r > R \), approaching zero as \( r \) increases.

### Conclusion:
The variation of gravitational potential of a hollow sphere with distance can be depicted as a graph that has a constant value for \( r < R \), a value at \( r = R \), and a decreasing curve for \( r > R \).