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The instability of Rutherford's atomic model

  1. Oct 21, 2007 #1
    I understand Rutherford proposed that electrons orbit around a central nucleus. However, since accelerating charges produce electromagnetic radiation, the orbiting electron should lose energy via E&M and spiral into the nucleus.

    But my question is: How do I calculate the time it takes for the electron to spiral into the nucleus, given the rate of energy loss (as a function of acceleration) and the initial electron-nucleus distance?

    The power loss equation is: P = (e^2 a^2 ) / (6 pi epsilon c^3)

    So far I've thought of calculating the initial energy of the system and integrating the power, and then equating the lost energy to the initial energy; however the final energy is negative inifinity, so this doesn't seem to work.

    Algebraic manipulation of circular motion equations didn't get me anywhere either; I'm not really sure how else to proceed now.
  2. jcsd
  3. Oct 22, 2007 #2


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    You need a differential equation for the radius R.
    The P you give is dE/dt.
    Use the Bohr formula for E in terms of R, and use a=v^2/R.
  4. Oct 22, 2007 #3
    I realized a mistake in my earlier analysis; when the electron enters the nucleus, r is not 0 but rather on the order or 10^-14 - this means when the electron enters the nucleus, the electric potential energy does not diverge to negtive inifity like I first thought - so I integrated P from initial r to the nucleus radius and found the total energy loss.

    Then I found the average power loss by dividing the power integral by the interval I integrated over (r final - r initial); for the hydrogen atom I came up with time = 10^-9 which seems about right.

    Does my analysis make sense though? I haven't had much experience with in this particular part of physics and I'm not sure if I just came up with a reasonable answer by a wrong route.
  5. Oct 22, 2007 #4


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    Your approach is probably good for an approximation, but is not correct for getting the desired value.
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