91Ó°ÊÓ

Coulomb's law is pivotal in understanding how electric charges interact. It states that the electric force between two point charges is directly proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them. Mathematically, it is represented as: \[ F = \frac{k \cdot Q_1 \cdot Q_2}{r^2} \]where:

• \( F \) is the magnitude of the force between the charges.
• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \).
• \( Q_1 \) and \( Q_2 \) are the charges.
• \( r \) is the distance between the charges.

When applied to the exercise, this principle helps calculate the force experienced by particle \( C \) due to charges \( A \) and \( B \) when displaced. This force governs how the particles interact based on relative distance and charge magnitudes.

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. For SHM to occur in our scenario, particle \( C \) must feel a restoring force that pulls it back to its equilibrium position. The condition for SHM is mathematically expressed as:\[ F = -k x \]where \( k \) is the force constant and \( x \) is the displacement from the equilibrium.For particle \( C \), SHM is expected when the restoring force derived from Coulombic interactions is proportional and opposite to its displacement. The force calculated in the problem is linearly dependent on \( x \), thereby satisfying one of the criterias for SHM. This leads to the cyclic motion of the charged particle back and forth about its equilibrium position.

Electric charges are fundamental properties of matter; objects with electric charge experience force when placed in an electromagnetic field. Charge can be positive or negative, and like charges repel each other, whereas opposite charges attract. In this exercise:- Particle \( A \) and \( B \) carried the same charge \( Q \).- Particle \( C \) carried a charge \( q \).The interaction between these charges causes the electric force acting on particle \( C \). The configuration given - \( A \) and \( B \) fixed, while \( C \) is free to move - demonstrates a scenario known as an electric dipole's behavior in a non-uniform field.

The effective force constant, denoted as \( k_{effective} \), is a crucial parameter in determining the dynamics of simple harmonic systems. It acts as an indicator of how "stiff" the system is, influencing the speed and period of oscillation. In the context of the exercise, the effective force constant is derived from the force equilibrium condition \( F = -kx \):\[ k_{effective} = \frac{4k \cdot Q \cdot q}{d^3} \]The higher the value of \( k_{effective} \), the quicker particle \( C \) will oscillate. This parameter provides insight into how the properties like charge magnitudes \( Q \) and \( q \), and separation distance \( d \) affect the rate and nature of oscillations. The time period of oscillation is then inversely related and can be calculated using:\[ T = 2\pi \sqrt{\frac{m}{k_{effective}}} \]This formula highlights the dependency of oscillation time period on mass and the effective force constant, linking back to how mass inertia and interaction strength influence SHM.