Imagine a charged ring of a radius with a uniform linear charge density and a single point charge, of opposite charge, moving along the ring's symmetry axis $($the $?-$axis in Figure$).$

a$.)$ ?Show that there is a restoring force on the charge, that is$,$ ?a force trying to keep the charge at $?=0.$

b$.)$ ?Plot the dimensionless force as a function of the scaled coordinate, $/???(?)=\frac{?}{?}.$

c$.)$ ?Show that for small oscillations, a particle with mass $?$ ?and charge q$\text{'}$ ?undergoes simple harmonic motion.

d$.)$ ?Find an expression for the frequency of the oscillation and then evaluate that frequency for an electron in the center of a $2\mathrm{.}0??$ ?diameter ring charged to $1\mathrm{.}0\times 10-13?.$

Transcribed Image Text:

y Segment i with charge R Divide the ring into segments. x 12 Z Oi The field perpendicular to the axis cancels for two diametrically opposed segments. Oi > Ei y Segment i with charge R Divide the ring into segments. x 12 Z Oi The field perpendicular to the axis cancels for two diametrically opposed segments. Oi > Ei