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Angular frequency of electron in an electric field

  1. Apr 17, 2014 #1
    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...
     
  2. jcsd
  3. Apr 17, 2014 #2
    Take out R from the denominator . Then binomially expand the expression.The condition for small oscillation is z<<R .
     
    Last edited: Apr 17, 2014
  4. Apr 17, 2014 #3

    rude man

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    Hint: it's a very low order expansion ...
     
  5. Apr 17, 2014 #4

    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
     
  6. Apr 17, 2014 #5
    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.
     
  7. Apr 17, 2014 #6

    rude man

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    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.
     
  8. Apr 17, 2014 #7
    @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
     
  9. Apr 17, 2014 #8

    rude man

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    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.
     
  10. Apr 17, 2014 #9
    Okay , thanks for your time and help , :)
     
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