Review problem. A negatively charged particle -q is placed...

Review problem. A negatively charged particle $-q$ is placed at the center of a uniformly charged ring, where the ring has a total positive charge $Q$ as shown in Example 23.8 . The particle, confined to move along the $x$ axis, is displaced a small distance $x$ along the axis (where $x<a$ )and released. Show that the particle oscillates in simple harmonic motion with a frequency given by $$ f=\frac{1}{2 \pi}\left(\frac{k_{e} q Q}{m a^{3}}\right)^{1 / 2} $$

Review problem. A negatively charged particle $-q$ is placed at the center of a uniformly charged ring, where the ring has a total positive charge $Q$ as shown in Example 23.8 . The particle, confined to move along the $x$ axis, is displaced a small distance $x$ along the axis (where $x<a$ )and released. Show that the particle oscillates in simple harmonic motion with a frequency given by
$$
f=\frac{1}{2 \pi}\left(\frac{k_{e} q Q}{m a^{3}}\right)^{1 / 2}
$$

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is characterized by a restoring force that is directly proportional to the displacement from equilibrium, leading to oscillatory behavior with constant frequency. In the context of the charged particle problem, deriving the frequency of oscillation involves equating the linearized force (from the Taylor expansion of the potential) to the mass times acceleration, resulting in the familiar expression for the SHM frequency.

Coulomb's Law

This fundamental law governs the interaction between electric charges, stating that the force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them. It is used to determine the forces at work in systems involving charged particles, such as the force exerted by a charged ring on a nearby charged particle.

Electric Potential from a Symmetric Charge Distribution

When charges are arranged with symmetry, such as a uniformly charged ring, the electric potential along the symmetry axis can be computed by integrating the contributions from all charge elements. This concept is crucial for analyzing how the potential energy of a charged particle changes as it moves, and it helps in deriving the effective force that acts on the particle.

Equilibrium and Stability

A particle in an electric field may be in equilibrium if the net force acting on it is zero. Analyzing the stability of this equilibrium involves determining whether small displacements lead to a restoring force that tends to bring the particle back to equilibrium. This is a key step in understanding when the system can exhibit oscillatory behavior.

Taylor Series Expansion for Small Oscillations

For small displacements around an equilibrium point, the potential energy function can be approximated by its Taylor series expansion. Keeping only the quadratic term leads to a linearized force (restoring force) proportional to the displacement, which is a necessary condition for simple harmonic motion. This method simplifies the analysis of the system's dynamics.

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