Angular frequency of electron in an electric field

1. The problem statement, all variables and given/known data
An electron is constrained to the central axis of the ring of charge of radius R , Show that the electrostatic force exerted on the electron can cause it to oscillate through the center of the ring with an angular frequency

ω = [itex]\sqrt{\frac{eq}{4π\epsilon_{0}mR^{3}}}[/itex]

where q is the ring’s charge and m is electron’s mass.
2. Relevant equations

Electric field at the axis due to a ring of charge q,
E = [itex]\frac{qz}{4π\epsilon_{0}(z^{2}+R^{2})^{3/2}}[/itex]

where is the distance from the center of the ring

3. The attempt at a solution

Given E, F = qE
[itex]\Rightarrow[/itex] a = F/m
This isn’t simply SHM so
ω ≠ [itex]\sqrt{k/m}[/itex]
So that wouldn’t work
Then I thought if i could find x(t) , I could easily find the time period
So, x(t) = x(t+T)
But a(x) = [itex]\frac{eqz}{4πm\epsilon_{0}(z^{2}+R^{2})^{3/2}}[/itex]
I couldn’t derive anything using the equations of motion , or simple calculus.
So I need some help, not the whole solution but possibly some hints or pointers…

Leave a Reply

Name *
Email *