Problem

Problem

A particle of mass mm and charge −Q-Q is constrained to move along the axis of a ring of radius aa. The ring carries a uniform charge density +λ+\lambda along its length. Initially, the particle is in the centre of the ring where the force on it is zero. Show that the period of oscillation of the particle when it is displaced slightly from its equilibrium position is given by T=2π2ϵ0ma2λQT = 2\pi \sqrt{\frac{2\epsilon_0 m a^2}{\lambda Q}}
Intuitive explanation

The key is recognizing that a charged particle at the ring's center sits in an unstable equilibrium—any small displacement along the axis triggers a restoring force from the ring's charge. You'll use symmetry to find the electric field along the axis, then show that for small displacements the force is proportional to displacement (simple harmonic motion). Finally, extract the spring constant from the field gradient and apply T=2πm/kT = 2\pi\sqrt{m/k} to get the period.

Intuitive explanation

The key is recognizing that a charged particle at the ring's center sits in an unstable equilibrium—any small displacement along the axis triggers a restoring force from the ring's charge. You'll use symmetry to find the electric field along the axis, then show that for small displacements the force is proportional to displacement (simple harmonic motion). Finally, extract the spring constant from the field gradient and apply T=2πm/kT = 2\pi\sqrt{m/k} to get the period.

Browse the problem library