# 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 = $\sqrt{\frac{eq}{4π\epsilon_{0}mR^{3}}}$

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 = $\frac{qz}{4π\epsilon_{0}(z^{2}+R^{2})^{3/2}}$

where is the distance from the center of the ring

3. The attempt at a solution

Given E, F = qE
$\Rightarrow$ a = F/m
This isn’t simply SHM so
ω ≠ $\sqrt{k/m}$
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) = $\frac{eqz}{4πm\epsilon_{0}(z^{2}+R^{2})^{3/2}}$
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…
Help…

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