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Coulomb's law is a fundamental principle that describes the force between two electric charges. It states that the electric force (F) between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The mathematical form of Coulomb's law is given as: - \( F = k_e \frac{q_1 q_2}{r^2} \) Here, \( k_e \) represents Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges. Coulomb's law is critical for understanding interactions between charged particles. It's similar to Newton's law of gravitation but applies to electrical forces.

To analyze how electric charges interact, we use electric force equations based on Coulomb's law. For a scenario with charges fixed on the x-axis and a third charge free to oscillate, the initial forces can be determined using: - \( F_1 = k_e \frac{Qq}{(a-x)^2} \) - \( F_2 = k_e \frac{Qq}{(a+x)^2} \) These equations calculate the electric force exerted by charges positioned at \( x=a \) and \( x=-a \) on a charge \( q \) at position \( x \). The formula involves substituting the respective distances into Coulomb's law to determine the individual forces.

The binomial expansion is a mathematical tool that simplifies expressions involving powers. It is especially useful when the expressions have small values in the denominators, allowing easier calculations. In this scenario, since we have \(|x| << a\), small terms in expressions can be expanded using: - \((1 \, \pm \, z)^{-2} \approx 1 \, \mp \, 2z\) For the electric forces given earlier: - \((a-x)^{-2} \approx a^{-2}(1 \, + \, 2\frac{x}{a})\) - \((a+x)^{-2} \approx a^{-2}(1 \, - \, 2\frac{x}{a})\) The binomial expansion simplifies the complexity of expressions, making it easier to derive equations necessary for calculating oscillations.

The oscillation frequency of an object describes how fast it moves back and forth in its repetitive movement, a concept tied to simple harmonic motion (SHM). Here, the charge \( q \) oscillates due to the electric forces between it and the fixed charges. We derive the oscillation frequency from the force equation: - The net force simplifies to: \( F_{net} = 4k_e \frac{Qq}{a^3}x \) - For SHM, the force being proportional to displacement \((-kx)\) indicates that the motion follows a periodic path. The effective spring constant \( k \) here involves electrical terms. the angular frequency \( \omega \) of the charge is: - \( \omega = \sqrt{\frac{4k_e Qq}{a^3 m}} \) The actual frequency \( f \) is given by dividing this by \( 2\pi \): - \( f = \frac{1}{2\pi}\sqrt{\frac{4k_e Qq}{a^3 m}} \) This describes how fast \( q \) oscillates about its equilibrium position.