Problem 101 - Master Resource Book in JEE Main Physics

Question

A positive charged thin metal ring of radius \(R\) is fixed in the \(x y\)-plane with the centre at the origin \(O . \mathrm{A}\) negatively charged particle \(P\) is released from rest at the point \(\left(0,0, z_{0}\right)\) where, \(z_{0}>0 .\) Then, the motion of \(P\) is (a) periodic, for all values of \(z_{0}\) satisfying \(0 \leq z_{0}<\infty\) (b) simple harmonic, for all values of \(z_{0}\) satisfying \(0 \leq z_{0} \leq R\) (c) approximately simple harmonic, provided \(z_{0}

Step-by-Step Solution

Answer

The answer is (c) approximately simple harmonic, provided \(z_0 < R\).

Step 1: Understand the Problem

We have a positive charged metal ring in the xy-plane with center at origin O and a negatively charged particle P released from a position above the ring at (0, 0, z_0). We need to determine the nature of the motion of P after it is released.

Step 2: Analyze the Electrostatic Force

The electric field at a point on the axis of a ring due to the ring attracts the negatively charged particle towards the ring. The force due to an electric field is given by Coulomb's law: \[ F = qE \]where \( F \) is the force, \( q \) is the charge of the particle, and \( E \) is the electric field.

Step 3: Electric Field on the Axis of a Ring

The electric field at a point along the axis of a uniformly charged ring with total charge \( Q \), radius \( R \), at a distance \( z \) from its center, is given by: \[ E = \frac{kQz}{(z^2 + R^2)^{3/2}} \]where \( k \) is Coulomb's constant. This expression tells us that the field and thus the force on the particle depends on its position \( z \).

Step 4: Consider Simple Harmonic Motion

For motion to be simple harmonic, the force must be directly proportional to the displacement and directed towards the equilibrium position, which occurs when the restoring force is linear with respect to displacement. Given the expression for \( E \), this is only approximately true for small \( z \):\[ F \approx -kQq \frac{z}{R^3} \] when \( z \ll R \), which matches the condition for a harmonic oscillator: \( F = -kz \).

Step 5: Evaluate Each Option

(a) Periodic for all \( z_0 \) implies a general solution for any value on the z-axis, which is not true since the force does not maintain linearity for larger \( z \).(b) Simple harmonic for \( z_0 \leq R \) is not entirely correct, it might have initially SHM-like properties but P can go beyond.(c) Approximately simple harmonic, \( z_0 < R \): since the force is approximately linear for small displacements (\( z \ll R \)), this is a valid description.(d) When \( P \) passes the origin, the force reverses, preventing motion to \(-\infty\), thus not satisfying this option.

Key Concepts

Coulomb's Law

Coulomb's Law is the foundation of the study of electrostatics. It describes the force of interaction between two charged particles. This law tells us that this force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them.

The mathematical representation is:

• Formula: \[ F = k \frac{|q_1 q_2|}{r^2} \]
• Where:
  • \( F \) is the magnitude of the force between the charges,
  • \( k \) is Coulomb's constant (\( 8.9875 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \)),
  • \( q_1 \) and \( q_2 \) are the amounts of the charges,
  • \( r \) is the distance between the centers of the two charges.

This law essentially governs the electrostatic force acting on any charged particle. In the problem, Coulomb's Law helps understand the force that the charged metal ring experiences from the charged particle and vice versa. Coulomb's Law allows us to calculate the force, but it's important to account for the direction, as it determines whether the forces are attractive or repulsive.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and is directed towards that equilibrium. For SHM to occur, the system should be linear related to displacement:

• Restoring force: \[ F = -kx \]
• Where:
  • \( F \) is the restoring force,
  • \( k \) is the proportionality constant known as the spring constant,
  • \( x \) is the displacement from the equilibrium position.

In the context of the exercise, the motion of the charged particle can be approximately described as simple harmonic only for small displacements \( z \) when \( z \ll R \). This approximate linear relationship between force and displacement satisfies the condition for SHM. Larger displacements deviate from this linear property, thus will not exhibit simple harmonic behavior.

Understanding these conditions is crucial as it helps identify under what constraints the charged particle will exhibit simple harmonic motion.

Motion of Charged Particles

The motion of charged particles under electrostatic influence is determined by the forces they experience due to electric fields. In a uniform electric field, a charged particle will experience a constant force along the field's direction.

In this particular problem, the negatively charged particle is released above a positively charged ring. As it descends towards the charge ring, it encounters an electrostatic force due to the electric field emanating from the ring.

• Key Aspects of Motion:
  • The particle accelerates towards the ring initially because it is attracted to the opposite charge.
  • Upon passing through or towards the center of the ring, the direction of the force changes, gradually reducing the speed of the particle and pulling it back.
  • If started at a small height compared to the ring's radius, the motion can be approximated as simple harmonic.
  • For larger distances, the motion no longer remains simple harmonic but will still be periodic due to the nature of Coulomb forces.

The motion of charged particles like this acts as a profound demonstration of electrostatic theory in action, illustrating how forces and fields influence movement.