Problem 99 - Master Resource Book in JEE Main Physics

Question

A positive charge \(Q\) is uniformly distributed along a circular ring of radius \(R\). A small test charge \(q\) is placed at the centre of the ring figure. Then [NCERT Exemplar] (a) If \(q>0\) and is displaced away from the centre in the plane of the ring, it will be pushed back towards the centre (b) If \(q<0\) and is displaced away from the centre in the plane of the ring, it will never return to the centre and will continue moving till it hits the ring (c) \(q<0\), it will perform 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 \(q>0\)

Step-by-Step Solution

Answer

All options (a), (b), (c), and (d) are correct.

Step 1: Understanding the Problem

We need to analyze the forces on a test charge \( q \) placed at the center of a uniformly charged ring with total charge \( Q \) and radius \( R \). The question consists of multiple parts, asking about the behavior of the charge \( q \) depending on its sign and its displacement relative to the ring.

Step 2: Analyze Positive Test Charge in the Plane

If \( q > 0 \), it is initially at the equilibrium in the center due to symmetry (no net force). If displaced in the plane of the ring, each segment of the ring pulls the charge equally from different directions. Still, any displacement off-center would result in stronger force from the nearer side, pushing it back toward the center.

Step 3: Analyze Negative Test Charge in the Plane

For \( q < 0 \), initially at equilibrium at the center, if displaced, it will also experience a net force towards the nearest segment of the ring. The negative charge will be attracted towards the positively charged side, thus, if displaced, it's likely to continue towards the ring until collision.

Step 4: Analyze Negative Charge Along the Axis

If \( q < 0 \) is displaced along the axis of the ring, the forces from all segments balance out in a manner resembling a spring force, guiding it back towards the center. For small displacements along this axis, the negative charge performs simple harmonic motion (SHM) around the center of the ring.

Step 5: Analyze Stability of Equilibrium for Positive Charge

At the center, the positive charge is in equilibrium due to symmetrical forces from the ring. If \( q > 0 \) and displaced slightly in the plane, the forces no longer balance symmetrically, leading the charge away from the center, indicating unstable equilibrium in the plane.

Step 6: Conclude the Correct Options

Summarizing our analyses, (a) is true because \( q > 0 \) experiences a restoring force towards the center, (b) is true because \( q < 0 \) accelerates outward toward the ring, (c) is true since \( q < 0 \) exhibits SHM when displaced along the axis, and (d) is true because the equilibrium is unstable for \( q > 0 \).

Key Concepts

Electric Field

An electric field is a region around a charged object where other charges experience a force. In our scenario, the positively charged ring creates an electric field surrounding it. If we place a small test charge, say \( q \), at the center of this ring, it initially feels no net force because the electric field is symmetrical.

However, if the test charge \( q \) is slightly moved from the center, the electric field will exert different forces from different parts around the ring. This force varies with the position of the charge and its own sign (whether \( q \) is positive or negative), thereby affecting the motion of \( q \).

- When \( q > 0 \), if displaced in the plane, it experiences a stronger electric field from the nearest segment of the ring, pulling it back towards the center.- When \( q < 0 \), if displaced forward in the plane of the ring, it will be attracted towards the ring's positively charged section, which may cause it to move toward the ring continuously until it collides.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic oscillation where a particle moves back and forth over the same path, such as along the axis of a charged ring in our example. SHM occurs when the restoring force is proportional to the displacement and acts towards the mean position.

Here, if \( q < 0 \) is displaced slightly along the ring's central axis, it experiences a force akin to a restoring spring force. This is because the electric field exerts forces that balance out similarly to Hooke's Law, guiding it back and forth through the central point.

For small displacements, this setup leads to SHM where:- The charge oscillates around the central position.- The oscillation's period and frequency depend on the charge's mass and the magnitude of the restoring electric field.SHM is significant because it indicates a predictable, stable displacement behavior that can balance out due to the symmetrical properties of the electric field along the axis.

Equilibrium Stability

The equilibrium of a charge in an electric field represents a state where there is no net force acting on it. This equilibrium can either be stable or unstable, depending upon how the system responds to small displacements.

In the scenario of a charged ring, if \( q > 0 \) is positioned at the center of the ring, it experiences forces from all sides equally — keeping it in equilibrium. However, this equilibrium is unstable in the plane of the ring because any small movement leads to a force imbalance. The charge moves away from the center, as the attraction from one side increases.
- This happens because the electric field does not restore the charge to its initial position but rather pushes it further away.- Consequently, slight disturbances are magnified.In simpler terms, think of balancing a pencil on its tip; if nudged slightly, it will fall over instead of returning to the upright position. For this reason, even though the charge might start in equilibrium, it's sensitive to changes, making it unstable when disturbed.