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Jackson electrodynamics exercise

  1. Sep 24, 2009 #1
    1. The problem statement, all variables and given/known data

    An electron moves in a helix : [tex]\vec{r}(t)=v_{z}t \hat{z}+a e^{i\omega_{0}t}(\hat{x}-i\hat{y})[/tex], where [tex]a[/tex] is the radius of the helix and [tex]v_{z}[/tex] the relativistic z-component of the velocity.
    1) Find the position vector of the electron in a system of reference that is moving with velocity [tex]v_{z}\hat{z}[/tex]
    2) Find the central frequency of radiation that the electron emits in the [tex]\hat{z}[/tex] direction in the laboratory reference frame.
    3)Calculate the angular distribution of the power of radiation, [tex]\frac{dP(t')}{d\Omega}[/tex]

    2. Relevant equations

    Jackson 3rd edition, chapter 14 (par. 14.4)

    3. The attempt at a solution

    1) is easy, just a lorentz transformation to find [tex]\vec{r}'(t')[/tex]. It turns out that in the moving frame [tex]\vec{r}'(t')[/tex] has no z-component. So in that frame it actually moves in a circle rather than a helix.

    For 2)I have no idea.

    3)I can maybe calculate [tex]\frac{dP(t')}{d\Omega}[/tex] from equation 14.38 but I am not sure

    Any ideas? Especially for 2)...
  2. jcsd
  3. Sep 24, 2009 #2


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    For 2), what about calculating in the frame from 1), and then transforming to the lab frame?

    For 3), why is the t primed?
  4. Sep 24, 2009 #3

    2) yes but what does "central frequency" means and how do I calculate it?

    3) If you check Jackson (3rd edition page 668), t' refers to the moving particle's own time.
  5. Sep 25, 2009 #4


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    I thought that central frequency would just mean peak frequency. However, after reading Chapter 14, I didn't see the term "central frequency" used once. Maybe I missed it. Or maybe "critical frequency". I don't know. If I had to solve this problem, I would assume peak frequency.
  6. Sep 27, 2009 #5
    You are right, it's just peak frequency
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