Let `V` and `E` be the gravitational potential field. Then select the correct alternative (s)

Options

A. the plot of `E` against `r` (distance from centre) is discontinuous for a spherical shell
B. the plot of `V` against `r` is continous for a spherical shell
C. the plot of `E` against `r` is discontinuous for a solid sphere
D. the plot of `V` against `r` is continuous for a solid sphere

Answer

To solve the problem regarding the gravitational potential field \( V \) and the gravitational field \( E \), we need to analyze the behavior of these fields for both a spherical shell and a solid sphere.

### Step-by-Step Solution:

1. **Understanding Gravitational Field \( E \) and Potential \( V \)**:
- The gravitational field \( E \) is defined as the force per unit mass experienced by a small test mass placed in the field.
- The 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.

2. **Behavior of \( E \) and \( V \) for a Spherical Shell**:
- **Inside the Spherical Shell**:
- The gravitational field \( E \) inside a spherical shell is zero.
- The gravitational potential \( V \) is constant and equal to the potential on the surface of the shell.
- **Outside the Spherical Shell**:
- The gravitational field \( E \) behaves as if all the mass were concentrated at the center, following \( E = \frac{GM}{r^2} \).
- The gravitational potential \( V \) decreases with distance and follows \( V = -\frac{GM}{r} \).

3. **Behavior of \( E \) and \( V \) for a Solid Sphere**:
- **Inside the Solid Sphere**:
- The gravitational field \( E \) increases linearly with distance from the center, given by \( E = \frac{GM}{R^3}r \) where \( r \) is the distance from the center.
- The gravitational potential \( V \) is given by \( V = -\frac{GM}{2R} + \frac{GM}{R^3}r^2 \), which is continuous and has a maximum at the center.
- **Outside the Solid Sphere**:
- Similar to the shell, the gravitational field \( E \) behaves as \( E = \frac{GM}{r^2} \).
- The gravitational potential \( V \) behaves as \( V = -\frac{GM}{r} \).

4. **Analyzing the Plots**:
- For the **spherical shell**:
- \( E \) is discontinuous at the surface (it jumps from 0 to a non-zero value).
- \( V \) is continuous across the surface.
- For the **solid sphere**:
- \( E \) is continuous throughout.
- \( V \) is also continuous throughout.

5. **Selecting the Correct Alternatives**:
- The correct statements based on the analysis are:
- The plot of \( E \) against \( R \) for a spherical shell is discontinuous.
- The plot of \( V \) against \( R \) is continuous for a spherical shell.
- The plot of \( E \) against \( R \) for a solid sphere is continuous.
- The plot of \( V \) against \( R \) is continuous for a solid sphere.

### Conclusion:
The correct alternatives are:
- The plot of \( E \) against \( R \) for a spherical shell is discontinuous.
- The plot of \( V \) against \( R \) is continuous for a spherical shell.
- The plot of \( V \) against \( R \) is continuous for a solid sphere.