91Ó°ÊÓ

Coulomb's Law is a fundamental principle in electrostatics that describes the force between two point charges. It states that the force (F) between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it is expressed as \( F = \frac{k_e Q_1 Q_2}{r^2} \), where \( k_e \) is Coulomb's constant (approximately \( 8.99 \times 10^9 \text{N m}^2/\text{C}^2 \)), \( Q_1 \) and \( Q_2 \) are the charges, and \( r \) is the distance between the charges.

• This law explains how charges interact - either repelling or attracting each other.
• When applying Coulomb's Law, always ensure units are consistent for force, charge, and distance.

In our given problem, each fixed charge \( Q \) at positions \( x = a \) and \( x = -a \) exert a repulsive force on the moving charge \( q \) along the x-axis.
Using symmetry and Coulomb’s Law simplifies calculations, especially when evaluating small displacements along the x-axis.

Simple Harmonic Motion (SHM) is a type of periodic motion where the force acting on an object is proportional to the displacement of the object and is directed towards the object's equilibrium position. This can be simplified into the formula \( F = -kx \) where \( k \) is a constant and \( x \) is the displacement.

• Important characteristics of SHM include constant amplitude, constant frequency, and sinusoidal wave forms.
• Analyzing forces in SHM often reveals patterns or symmetries that make complex problems easier to solve.

In our problem, the force calculated between charge \( q \) and the fixed charges \( Q \) is akin to the restoring force in SHM. The net restoring force is proportional to \( -x \), confirming that \( q \) moves in simple harmonic motion along the x-axis.
This harmonic motion makes it possible to predict the charge’s behavior over time.

The Binomial Expansion is an algebraic tool used to simplify calculations involving powers of \((1+z)\), especially when \(|z| \ll 1\). The expansion can be expressed as \((1+z)^n \approx 1 + nz + \frac{n(n-1)}{2}z^2 + \cdots\). The ability to approximate complicated expressions as a series is especially useful in physics.

• It allows us to make approximations where the value of \(z\) is small, simplifying complex expressions.
• High-order terms can often be neglected, making computations more tractable.

In the given exercise, using the Binomial Expansion for approximating \(\frac{1}{(a \pm x)^2}\) simplifies the calculation of forces on the charge \( q \).
By expanding to the first order, we can approximate the expressions more easily, which leads us directly to the conclusion about the behavior of the oscillating charge.

Oscillation Frequency refers to the number of oscillations that happen per unit of time. For an object in simple harmonic motion, frequency is closely related to the systems' physical parameters such as mass and spring constant (or electrostatic forces in this case). The formula used is \( f = \frac{\omega}{2\pi} \), where \( \omega \) is the angular frequency given by \( \sqrt{\frac{k}{m}} \).

• The angular frequency \( \omega \) itself reflects how quickly an object moves through one complete cycle of motion.
• Key factors like mass \( m \) and the effective "spring" constant \( k \) (originating from force terms) determine the oscillation frequency.

In this problem, the frequency of charge \( q \)'s oscillation is derived from the equation of motion for SHM (\( m \ddot{x} = -kx \)).
The expression \( \omega = \sqrt{\frac{4k_e Qq}{ma^3}} \) emerges from matching terms, leading directly to the frequency by dividing \( \omega \) by \( 2\pi \).
Examining these derivations clarifies how physical constraints influence the frequency at which the system oscillates.