Let V and E denote the gravitational potential and gravitational field at a point. It is possile to have

Options

A. `V=0 and E=0`
B. `V=0 and E!=0`
C. `V=0 and E=0`
D. `V!=0 and E!=0`

Answer

To solve the problem, we need to analyze the possible scenarios for gravitational potential (V) and gravitational field (E) at a point in relation to a spherical shell.

### Step-by-Step Solution:

1. **Understanding Gravitational Potential (V) and Gravitational Field (E)**:
- Gravitational potential (V) at a point is defined as the work done in bringing a unit mass from infinity to that point without any acceleration.
- Gravitational field (E) at a point is defined as the force experienced by a unit mass placed at that point.

2. **Case A: V = 0 and E = 0**:
- At infinity, the gravitational potential (V) is defined to be zero. Since the gravitational field (E) is derived from the potential, it is also zero at infinity.
- Therefore, it is possible to have both V = 0 and E = 0 at a point at infinity.
- **Conclusion**: This case is possible.

3. **Case B: V = 0 and E ≠ 0**:
- If we consider a point just outside a spherical shell, the potential (V) is not zero, but the gravitational field (E) is not zero either.
- Hence, it is not possible to have V = 0 while E is not zero.
- **Conclusion**: This case is not possible.

4. **Case C: V ≠ 0 and E = 0**:
- Inside a spherical shell, the gravitational field (E) is zero, but the potential (V) is constant and does not equal zero.
- Thus, it is possible to have V ≠ 0 while E = 0.
- **Conclusion**: This case is possible.

5. **Case D: V ≠ 0 and E ≠ 0**:
- Outside the spherical shell, both the gravitational potential (V) and gravitational field (E) are non-zero.
- Therefore, it is possible to have both V ≠ 0 and E ≠ 0.
- **Conclusion**: This case is possible.

### Final Summary of Cases:
- **Case A**: Possible (V = 0, E = 0)
- **Case B**: Not possible (V = 0, E ≠ 0)
- **Case C**: Possible (V ≠ 0, E = 0)
- **Case D**: Possible (V ≠ 0, E ≠ 0)