91Ó°ÊÓ

To grasp the concept of simple harmonic motion in charged particles, we first need to understand Coulomb's Law. This fundamental principle in electrostatics describes how the force between two stationary, electrically charged particles relates to the distance between them. Mathematically, it's expressed as
\( F = K \frac{{Qq}}{{r^2}} \),
where \(F\) is the force between the charges, \(K\) is Coulomb's constant, \(Q\) and \(q\) are the magnitudes of the charges, and \(r\) is the distance separating them. In this scenario, since the charged particle is affected by the forces from two fixed charges at equal distances, Coulomb's law helps us determine the initial forces acting on the particle before it starts oscillating.

Understanding the role of Newton's Second Law is crucial when exploring the motion of particles. According to this law, the force acting on an object is equal to the mass of the object times its acceleration, formalized as
\( F = m \times a \).
In our exercise, we applied this law to determine the acceleration of the charged particle. The fact that the forces from both fixed charges reinforce at the center and oppose at the ends results in a restoring force that nudges the particle back toward equilibrium. By setting the electrostatic force equal to mass times acceleration, we're setting up the stage for the particle's oscillatory motion.

Hooke's Law provides a link between the forces we've been discussing and simple harmonic motion. This law is commonly associated with springs and states that the force needed to extend or compress a spring by some distance is proportional to that distance, that is
\( F=-kx \),
where \(F\) is the force applied, \(k\) is the spring constant, and \(x\) is the displacement from the spring's equilibrium position. In the context of our charged particle, the restoring force follows a similar pattern, indicating that our particle indeed undergoes simple harmonic motion. Identifying the electric force as analogous to a spring's restoring force allows us to treat the particle's movement like that of a mass on a spring.

The oscillation frequency of an object in simple harmonic motion is a pivotal variable that tells us how many oscillations occur per unit of time. For a mass-spring system, or in our case, a charge moving under conditions analogous to Hooke's Law, the frequency is determined by
\( u = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \),
where \(u\) is the frequency, \(k\) is the effective force constant (analogous to the spring constant in Hooke's Law), and \(m\) is the mass of the oscillating object. By calculating the effective force constant from Coulomb's law and incorporating the mass of our charged particle, we deduced the oscillation frequency for the particle constrained to one-dimensional motion between the two fixed charges.