Angular frequency of electron in an electric field

[amind]

Homework Statement

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.

Homework 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

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...

 

[Tanya Sharma]

But a(x) = [itex]\frac{eqz}{4πm\epsilon_{0}(z^{2}+R^{2})^{3/2}}[/itex]

Take out R from the denominator . Then binomially expand the expression.The condition for small oscillation is z<<R .

 

[rude man]

Hint: it's a very low order expansion ...

 

[amind]

Take out R from the denominator . Then binomially expand the expression.The condition for small oscillation is z<<R .

Oh ! I just didn't see that , thank you.
But what if instead of the electron we take a spherical charged body and where z is not very small

 

[Tanya Sharma]

But what if instead of the electron we take a spherical charged body

I do not know .May be rude man has the answer .

But if I have to make a guess ,then if the spherical body is uniformly charged ,then we may replace it with a point like particle of equivalent charge.

and where z is not very small

Well ,then you will not be able to apply the approximation and the motion will not be simple harmonic.

 

[rude man]

I do not know .May be rude man has the answer .

But if I have to make a guess ,then if the spherical body is uniformly charged ,then we may replace it with a point like particle of equivalent charge.

Well ,then you will not be able to apply the approximation and the motion will not be simple harmonic.

I wouldn't want to tackle the case of a finite-size sphere. I wonder about polarization effects, i.e. asymmetric surface charges since the E field is not uniform over the sphere.

And right, if it's still a point mass but z is not << R then you wind up with a nonlinear diff. eq. which again I would not want to tackle.

 

[amind]

@Tanya and @rude man
I said spherical charged body , so that unlike an electron it is not very small (point size) , okay instead now consider a point charge with charge q' and z is not very small , now what.

I am thinking of making a c++ simulation with unit constants for having a better idea to see what answer it might give

 

[rude man]

@Tanya and @rude man
I said spherical charged body , so that unlike an electron it is not very small (point size) , okay instead now consider a point charge with charge q' and z is not very small , now what.

I am thinking of making a c++ simulation with unit constants for having a better idea to see what answer it might give

As tanya and I said, with a point charge but z not << R you get a nonlinear differential equation which is very difficult to solve in closed form. But you will still get oscillations, only they aren't SHM and the z(t) waveform vs. t will look like a horrible distorted sine wave. This is somewhat like a simple pendulum oscillating with a large angle, say pi/4.

Go ahead and simulate - that is a great idea! Use various z/R, starting with z << R and building up.

 

[amind]

Okay , thanks for your time and help , :)