# Jackson electrodynamics exercise

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

For 3), why is the t primed?

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
Homework Helper
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.

You are right, it's just peak frequency