Jackson electrodynamics exercise

[qasdc]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

1) is easy, just a lorentz transformation to find $$\vec{r}'(t')$$. It turns out that in the moving frame $$\vec{r}'(t')$$ 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 $$\frac{dP(t')}{d\Omega}$$ from equation 14.38 but I am not sure


Any ideas? Especially for 2)...

 

[turin]

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

For 3), why is the t primed?

 

[qasdc]

turin,

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.

 

[turin]

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.

 

[qasdc]

You are right, it's just peak frequency