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

Options

A. The plot of E against r ( distance form centre) is discontinuous for a spherical shell
B. The plot of V against r is continuous 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 question regarding the gravitational potential field (V) and gravitational field (E) for a spherical shell and a solid sphere, we will analyze the behavior of these fields with respect to distance (r).

### Step-by-Step Solution:

1. **Understanding Gravitational Field (E)**:
- The gravitational field (E) is defined as the force experienced by a unit mass at a point in space due to the gravitational attraction of a mass.
- For a spherical shell, the gravitational field outside the shell behaves like a point mass, while inside the shell, the gravitational field is zero.

2. **Graph of E for a Spherical Shell**:
- Outside the shell (r > R), the gravitational field E is given by the formula:
\[ E = \frac{GM}{r^2} \]
- Inside the shell (r < R), the gravitational field E is:
\[ E = 0 \]
- Therefore, the graph of E vs. r for a spherical shell is discontinuous at r = R, where it jumps from 0 to a positive value.

3. **Understanding Gravitational Potential (V)**:
- The gravitational potential (V) is the work done per unit mass in bringing a mass from infinity to a point in space.
- For a spherical shell, the potential inside the shell is constant and equal to the potential at the surface.

4. **Graph of V for a Spherical Shell**:
- Outside the shell (r > R), the potential is:
\[ V = -\frac{GM}{r} \]
- Inside the shell (r < R), the potential is constant:
\[ V = -\frac{GM}{R} \]
- Thus, the graph of V vs. r for a spherical shell is continuous, with a constant value inside the shell and a decreasing value outside.

5. **Graph of E for a Solid Sphere**:
- For a solid sphere, the gravitational field inside the sphere (r < R) is given by:
\[ E = \frac{GM}{R^3} r \]
- Outside the sphere (r > R), it behaves as:
\[ E = \frac{GM}{r^2} \]
- The graph of E vs. r for a solid sphere is continuous, starting from 0 at the center and increasing to a maximum value at the surface.

6. **Graph of V for a Solid Sphere**:
- For a solid sphere, the potential inside (r < R) is:
\[ V = -\frac{GM}{2R} + \frac{GM}{R^3} r^2 \]
- Outside the sphere (r > R), it is:
\[ V = -\frac{GM}{r} \]
- The graph of V vs. r for a solid sphere is also continuous.

### Conclusion:
Based on the analysis:
- The plot of E against r for a spherical shell is discontinuous.
- The plot of V against r for both the spherical shell and solid sphere is continuous.
- The plot of E against r for a solid sphere is continuous.
- The plot of V against r for a solid sphere is continuous.

### Correct Alternatives:
- A: E against r is discontinuous (True)
- B: V against r is continuous (True)
- C: E against r for solid sphere is discontinuous (False)
- D: V against r for solid sphere is continuous (True)

Thus, the correct options are A, B, and D.