# 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…

Help…

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