Problem 64 - Master Resource Book in JEE Main Physics

Question

Two equal negative charges \(-q\) are fixed at the points \((0, a)\) and \((0,-a)\) on the \(Y\)-axis. A positive charge \(Q\) is released from rest at the point \((2 a, 0)\) on the \(X\)-axis. The charge \(Q\) will (a) execute simple harmonic motion about the origin (b) move to the origin and remain at rest (c) move to infinity (d) execute oscillatory but not simple harmonic motion

Step-by-Step Solution

Answer

The charge Q will execute simple harmonic motion about the origin (Option a).

Step 1: Understand the Charge Positions

The negative charges \(-q\) are positioned symmetrically on the \(Y\)-axis at \((0, a)\) and \((0, -a)\). The positive charge \(Q\) is initially placed at \((2a, 0)\) on the \(X\)-axis.

Step 2: Analyze Forces on Charge Q

The forces on the charge \(Q\) due to the two charges \(-q\) will be along the lines connecting \(Q\) and each of the \(-q\) charges. Since the \(-q\) charges are equidistant from the \(X\)-axis, their vertical components of force on \(Q\) will cancel, leaving only horizontal components that add up.

Step 3: Calculate Net Force Direction

The horizontal components of the force on \(Q\) from both \(-q\) charges will point towards the origin because each force component is in the direction towards its respective charge. Therefore, the net force on \(Q\) is towards the origin.

Step 4: Determine the Nature of Charge Q's Motion

The charge \(Q\) will experience a restoring force towards the origin, similar to a spring-mass system. This scenario resembles simple harmonic motion (SHM) since the net force is linear and directed towards an equilibrium point (the origin).

Step 5: Identify the Type of Motion

Due to the linear nature of the restoring force and its direct relation to the displacement from the origin, the motion of charge \(Q\) qualifies as simple harmonic motion (SHM).

Key Concepts

Electrostatics

Electrostatics refers to the study of electric charges at rest. It involves understanding how charged particles interact with each other via electric forces. In this scenario, we have two negative charges, each \(-q\), symmetrically placed along the \(Y\)-axis at points \((0, a)\) and \((0, -a)\). The positive charge \(Q\) is initially positioned on the \(X\)-axis at \( (2a, 0)\).
The key to this exercise lies in comprehending how the configuration of these charges affects their interaction. Each negative charge exerts a force on the positive charge \(Q\). These forces stem from Coulomb's Law, which states that the force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of their distance.
The interaction through electrostatics sets up a system where the forces are crucial in determining the dynamics of the positive charge's motion. When the forces from the negative charges are considered together, their vertical components cancel out due to symmetry, focusing the impact solely along the horizontal direction.

Simple Harmonic Motion

Simple harmonic motion (SHM) occurs when an object moves according to a periodic and oscillatory motion, centered around an equilibrium point. In this exercise, the charge \(Q\) is subjected to forces that pull it towards the origin, akin to the force exerted by a spring. The net force is proportional to its displacement from the origin, meeting the criteria for simple harmonic motion.
SHM in this context means that charge \(Q\) is under the influence of a restoring force proportional to:

• The linear displacement from equilibrium.
• The system's symmetrical nature, offering a consistent force path.

The conditions of symmetry ensure the force \(Q\) experiences due to \(-q\) is both linear and directed towards the origin. As \(Q\) deviates from this point, the forces guide it back in a way that resembles a back-and-forth motion, characteristic of SHM.

Force Analysis

In physics, force analysis involves examining how forces interact within a system to influence an object's movement. Here, the focus is on how the electric forces on charge \(Q\) lead to simple harmonic motion.
First, consider the forces acting due to each negative charge. These forces are aligned along the lines joining each charge \(-q\) and the point where \(Q\) is located. Using vector addition, we observe:

• Since both \(-q\) charges are equidistant from the \(X\)-axis, the vertical force components cancel out.
• This cancellation results in only horizontal force components contributing to the overall net force.

The remaining horizontal forces point towards the origin, acting as a restoring force similar to that found in a mass-spring system. By returning the charge \(Q\) to its initial position with symmetric oscillation around the origin, the net forces uphold the system's stability, confirming simple harmonic motion.