An electron is constrained to the central axis of the ring...

An electron is constrained to the central axis of the ring of charge of radius $R$ in Fig. $22-11,$ with $z \ll R .$ Show that the electrostatic force on the electron can cause it to oscillate through the ring center with an angular frequency$$\omega=\sqrt{\frac{e q}{4 \pi \varepsilon_{0} m R^{3}}}$$ where $q$ is the ring's charge and $m$ is the electron's mass.

An electron is constrained to the central axis of the ring of charge of radius $R$ in Fig. $22-11,$ with $z \ll R .$ Show that the electrostatic force on the electron can cause it to oscillate through the ring center with an angular frequency$$\omega=\sqrt{\frac{e q}{4 \pi \varepsilon_{0} m R^{3}}}$$
where $q$ is the ring's charge and $m$ is the electron's mass.

Key Concepts

Electrostatic Force and Coulomb’s Law

This concept describes how charged particles interact via forces that are inversely proportional to the square of the distance between them. Coulomb’s law quantifies the magnitude and direction of the force between point charges, forming the basis to derive the electric field generated by any charge distribution.

Charge Distribution on a Ring

A ring of charge is a classic charge distribution in which the charge is uniformly spread along a circular path. Calculating the electric field along the central axis of the ring involves integrating the contributions from each infinitesimal element, which, due to symmetry, results in a simplified expression for the net field.

Taylor Series Expansion and Small Oscillation Approximation

For small displacements from an equilibrium position, any potential energy function can be approximated by its Taylor series expansion where the leading quadratic term governs the behavior. This approximation allows the complex force expression to be treated as linear, enabling the system to be modeled as a simple harmonic oscillator.

Simple Harmonic Motion in Electrostatic Systems

When the restoring force acting on a displaced charge is proportional to its displacement, the system undergoes simple harmonic motion. This principle is applied by equating the linearized expression for the electrostatic force to the form of Hooke’s law, thereby allowing the determination of the oscillation frequency.

Angular Frequency Determination

Angular frequency is a measure of how rapidly a system oscillates in simple harmonic motion. By equating the effective restoring force constant obtained from the Taylor expansion of the electrostatic potential to the mass of the oscillating particle, one can derive an expression for the angular frequency of the oscillations.

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