A positive charge Q is uniformly distributed along a circular ring of radius. A small test charge q is placed at the centre of the ring. Then A. If `qgt0` and is displaced away from the centre in the plane of the ring., it will be pushed bactowards the centre.
B. If `qlt0` and is displced away from the centre in the plane of the ring, it will never return no the centre and will continue moving til it hits the ring.
C. If `qlt0`, it will performe SHM for small displacement along the axis.
D. Q at the centre of the ring is in an unstable equilibrium within the plane of the ring for `qgt0`

Options

A. A,C are true
B. All are true
C. A,D are true
D. B,C are true

Explanation

To solve the problem, we will analyze each statement regarding the behavior of a test charge \( q \) placed at the center of a circular ring with a uniformly distributed positive charge \( Q \).

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a circular ring of radius \( r \) with a positive charge \( Q \) uniformly distributed along its circumference.
- A small test charge \( q \) is placed at the center of the ring.

2. **Analyzing Statement A**:
- **Statement**: If \( q > 0 \) and is displaced away from the center in the plane of the ring, it will be pushed back towards the center.
- **Analysis**: When \( q \) is displaced, the electric field created by the ring will exert a force on \( q \) that points towards the center. This is because the electric field due to a uniformly charged ring is zero at the center, but as soon as \( q \) is moved away, the symmetry is broken, and the net force will act towards the center.
- **Conclusion**: Statement A is **correct**.

3. **Analyzing Statement B**:
- **Statement**: If \( q < 0 \) and is displaced away from the center in the plane of the ring, it will never return to the center and will continue moving until it hits the ring.
- **Analysis**: A negative charge \( q \) will be attracted to the positive charge \( Q \). When displaced, there is no restoring force to bring it back to the center since the electric field at the center is zero. The negative charge will continue to move towards the ring due to the attractive force from the positive charge.
- **Conclusion**: Statement B is **correct**.

4. **Analyzing Statement C**:
- **Statement**: If \( q < 0 \), it will perform SHM for small displacements along the axis.
- **Analysis**: When \( q \) is displaced slightly along the axis of the ring, it experiences a restoring force due to the electric field created by the ring. This restoring force is proportional to the displacement, which is a characteristic of simple harmonic motion (SHM).
- **Conclusion**: Statement C is **correct**.

5. **Analyzing Statement D**:
- **Statement**: \( Q \) at the center of the ring is in an unstable equilibrium within the plane of the ring for \( q > 0 \).
- **Analysis**: If \( q \) is placed at the center, any small displacement will result in a force that pushes it further away from the center due to the repulsive force from the positive charge \( Q \). Thus, the equilibrium is unstable.
- **Conclusion**: Statement D is **correct**.

### Final Conclusion:
All statements A, B, C, and D are correct.