91Ó°ÊÓ

Coulomb's Law is all about understanding how charged particles interact. According to Coulomb's Law, the force between any two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. It is described by the formula:

• \( F = k \frac{|q_1 \, q_2|}{r^2} \)

where:

• \( F \) is the electrostatic force between the charges,
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
• \( r \) is the distance between the centers of the two charges, and
• \( k \) is Coulomb's constant, approximately equal to \( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \).

This law helps us determine the magnitude and direction of the force experienced by a charged particle. In the given scenario, the positive charge \( Q \) experiences forces from the two fixed negative charges \( -q \). Because these negative charges are symmetrically placed above and below the x-axis, their individual forces on \( Q \) combine, leading to a net force toward the origin along the x-axis.

Simple Harmonic Motion (SHM) is a type of oscillatory motion characterized by a restoring force proportional to displacement. In SHM, if you displace an object from its equilibrium position, the system exerts a force to bring it back. This force is described as:

• \( F = -kx \)

Here:

• \( F \) is the restoring force,
• \( k \) is a constant related to the stiffness of the system, and
• \( x \) is the displacement from equilibrium.

In this exercise, the positive charge \( Q \) experiences such a restoring force due to the symmetrical positioning of the fixed charges. When \( Q \) moves along the x-axis, it experiences a force that is always aimed toward the origin, behaving like SHM. The symmetrical forces ensure that, as \( Q \) is displaced from the origin, the system acts to restore it back, explaining why the charge goes into oscillation.

The forces on charged particles are essential for analyzing their motion. In electrostatics, analyzing forces involves examining interactions between multiple charges. Here, the situation is straightforward because of symmetry. Each negative charge \(-q\) exerts a force on the positive charge \(Q\). The forces have both x and y-components:

• The y-components are equal and opposite, thus cancel each other out.
• The x-components add up, giving a net force towards the origin.

This net force causes the positive charge \( Q \) to move towards the origin when released from rest. As \( Q \) passes through the origin, the forces continue directing it toward the opposite side. This results in a continuous back-and-forth motion characteristic of SHM. Understanding the resultant force's direction is crucial in predicting the nature of motion for the charged object.