A positively charged thin metal ring of radius R is fixed in the xy plane, with its centre at the origin O. A negatively charged particle P is released from rest at the point `(0,0,z_(0))` where `z_(0)gt0`. Then the motion of P is
a. Periodic for all value of `z_(0)` satisfying `0ltz_(0)ltoo` M
b. Simple harmonic, for all values of `Z_(0)` satisfying `0ltz_(0)leR`
c. Approximately simple harmonic, provided `z_(0)lt lt R`
d. Such that P crosses O and continues to move along the negative z-axis towards `z=-oo`
Choose the correct answer

Options

A. a and b are true
B. c is true
C. a,c,d are true
D. a,b,c,d are true

Answer

To solve the problem, we need to analyze the motion of a negatively charged particle \( P \) released from a point \( (0, 0, z_0) \) in the electric field created by a positively charged thin metal ring fixed in the xy-plane.

### Step 1: Understanding the Electric Field
The electric field \( E \) due to a charged ring at a point along its axis (the z-axis) can be expressed as:
\[ E = \frac{kQz}{(R^2 + z^2)^{3/2}} \]
where:
- \( k \) is Coulomb's constant,
- \( Q \) is the total charge on the ring,
- \( R \) is the radius of the ring,
- \( z \) is the distance from the center of the ring along the z-axis.

### Step 2: Analyzing the Force on the Particle
The force \( F \) acting on the negatively charged particle \( P \) is given by:
\[ F = -qE \]
where \( q \) is the charge of particle \( P \) (which is negative). Therefore, the force acting on \( P \) will be directed towards the ring (the center), as the electric field points away from the positively charged ring.

### Step 3: Motion of the Particle
Since the force is directed towards the center of the ring, the particle will experience a restoring force. This means that if the particle is displaced from its equilibrium position (the center of the ring), it will be pulled back towards that position.

### Step 4: Conditions for Simple Harmonic Motion
For the motion to be classified as simple harmonic motion (SHM), the force must be directly proportional to the negative of the displacement:
\[ F \propto -z \]
This condition holds true when \( z \) (the distance from the center) is small compared to \( R \) (the radius of the ring).

### Step 5: Evaluating the Options
- **Option A:** The motion is periodic for all values of \( z_0 \) satisfying \( 0 < z_0 < \infty \). This is true because the particle will always oscillate back and forth around the center.

- **Option B:** The motion is simple harmonic for all values of \( z_0 \) satisfying \( 0 < z_0 < R \). This is not true, as SHM only occurs for small displacements.

- **Option C:** The motion is approximately simple harmonic provided \( z_0 \ll R \). This is true, as for small displacements, the force behaves like a restoring force.

- **Option D:** The particle crosses the origin and continues moving along the negative z-axis towards \( z = -\infty \). This is incorrect because the particle will oscillate back and forth rather than moving indefinitely in one direction.

### Conclusion
The correct answers are **A** and **C**. The particle exhibits periodic motion for all values of \( z_0 \) and approximately simple harmonic motion when \( z_0 \) is much smaller than \( R \).