Let V and E denote the gravitational potential and gravitational field at a point. Then, match the following columns
`{:(,"Column-I",,"Column-II"),("(A)",E = 0","V = 0,"(p)","At centre of spherical shell"),("(B)",E ne0","V=0,"(q)","At centre of solid sphere"),("(C)",V ne0","E=0,"(r)","At centre of circular ring"),("(D)",V ne0","E ne0,"(s)","At cetre of two point masses of equal magnitude"),(,,"(t)","None"):}`

Answer

To solve the problem, we need to match the gravitational potential (V) and gravitational field (E) at specific points with the corresponding descriptions in the columns provided. Here’s a step-by-step breakdown:

### Step 1: Understand the Properties of Gravitational Field and Potential
- 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) is the work done per unit mass to bring a mass from infinity to a point in the field.

### Step 2: Analyze Each Case in Column-I
1. **(A) E = 0, V = 0**:
- This occurs at the center of a spherical shell. According to Gauss's law, the gravitational field inside a uniform spherical shell is zero. Since the potential is constant throughout the shell and can be taken as zero at infinity, V = 0 at the center.
- Match: (A) → (p)

2. **(B) E ≠ 0, V = 0**:
- This situation does not occur in classical mechanics. The gravitational field cannot be non-zero while the potential is zero at the same point. Therefore, this option does not match any case.
- Match: (B) → (t)

3. **(C) V ≠ 0, E = 0**:
- This occurs at the center of a solid sphere. Inside a uniform solid sphere, the gravitational field is zero at the center, but the potential is non-zero and is maximum at the center.
- Match: (C) → (q)

4. **(D) V ≠ 0, E ≠ 0**:
- This scenario occurs at the center of a circular ring of mass. At the center of the ring, the gravitational field is zero, but the potential is non-zero.
- Match: (D) → (r)

### Step 3: Compile the Matches
Based on the analysis:
- (A) → (p)
- (B) → (t)
- (C) → (q)
- (D) → (r)

### Final Matches:
- A → p
- B → t
- C → q
- D → r