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Oscillation of a Charged Particle

  1. Feb 9, 2017 #1
    1. The problem statement, all variables and given/known data
    At our university we were given this problem: charged ball with mass of ##m = 0.0001 kg## and charge ##Q = -10^{-5} C## is placed on geometric axis of thin torus with inner radius of ##r_{inner} = 0.05 m##, outer radius of ##r_{outer} = 0.1 m## and surface charge density ##\sigma = 10^{-5} C##. Compute oscillation time for small deviation, this is when we only slightly flick the ball from stable state.

    2. Relevant equations
    First, I took a look at this article and a PDF presentation.

    3. The attempt at a solution
    Using the equation for electric field of a charged ring $$ E_z = \frac{Qz}{4 \pi \varepsilon_0 (r^2 + z^2)^{\frac{3}{2}}} $$ I tried obtaining formula for electric field of torus (wider ring) by integration infinitesimal rings from inner to outer radius. Using Symbolab I managed to obtain following equation $$ E_{torus} = \frac{Qz}{4 \pi \varepsilon_0} (\frac{r}{z^3 \sqrt{\frac{r^2}{z^2}+1}})_{r_{inner}} ^ {r_{outer}} $$.

    From here on, I'm lost. Can somebody please help me or at least give me some guidance?
     
  2. jcsd
  3. Feb 9, 2017 #2

    BvU

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    Hi
    means you are looking for something like the ##k## in ##\vec F = -k\vec x##. So if ##|\vec F|## is a more complicated function like what you have found it here, you simply want the linear development around the equilibrium position.
     
  4. Feb 9, 2017 #3

    BvU

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    On 2nd reading I wonder if you are already aware of what I posted and are stuck in working out the electric field expression ?
    Not clear what you are doing there. The exercise geves that the charge is at the surface and unfiormly distributed. Looks like the pdf assumes a thin torus (yours is fat).
     
  5. Feb 9, 2017 #4
    I believe we're on the wrong footing here.

    My task considers thin torus. Due to the fact that such configuration is composed of filamentary thin rings, I tried integration individual contributions (rings) from inner to outer radius.
     
  6. Feb 9, 2017 #5

    BvU

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    Guy here (page 23) calls ##a/\rho = 5## fat. With ##a/\rho = 3##, yours is 67% fatter !
    Surface charge on the inside is twice as close to the axis as surface charge on the outside !

    Or do I have the wrong idea of inner (## r_{\rm inner} = 0.05## m) and outer (## r_{\rm outer} = 0.10## m) radius ?
     
  7. Feb 9, 2017 #6

    vela

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    Is the shape really a torus (donut-shaped)? If it is, the dimensions given seem to contradict the description that it's thin. Or is it supposed to be a flat ring of charge with the given inner and outer radii?
     
  8. Feb 9, 2017 #7

    BvU

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    A washout ?!?

    This is a torus
    upload_2017-2-9_18-20-28.png

    very clear what is meant with a torus in math and physics.

    This is a washer (not a very scientific name, but clear enough)
    upload_2017-2-9_18-24-3.png
     

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  9. Feb 13, 2017 #8
    This is the exact shape I was trying to describe. I'm so sorry for ambiguity in definition, English is not m native tongue.
     
  10. Feb 13, 2017 #9

    BvU

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    Good; makes life easier. Thin disk with a hole in the center it is. I should have been more suspicious when you mentioned a 'wider ring' and called it a torus.
    First expression in post #1 (field of charged ring) looks exactly as the one here , but with ##\ Q\ ## instead of ##\ 2\pi\sigma R' dR' , \ \ ## and -- apart from that -- seems OK to me. This you now want to integrate from ##r_{\rm min}## to ##r_{\rm max}## (instead of from ##0## to ##r_{\rm max}## like here) and there you lose me :

    Using my common sense I got something else. Pretty unpleasant, but the first derivative at ##z=0## is what we are after and that should be fairly decent. (In fact, working out the potential instead of the E-field would have been more economical -- hindsight....)

    One reason I don't trust your result is that it diverges for ##z=0## wich should not happen. Could you check ?
     
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