E and B fields for charged particle's parabolic motion

[peasqueeze]

I am looking to find a combination of electric and magnetic fields that create something similar to the (d) crests

Currently, I have the cracks flipped clockwise 90deg, so that the crests are concave up. And each crest can be defined by a parabolic function (which are uniform with each new turn).

If I look at one crest, treat it as a particle moving in a concave up parabola I have...

$$\vec{E} = E\hat{y} , \vec{B} = B\hat{z}$$

Now, this is only for a single parabola. I need to make it so that the particle reaches a turning point and goes backwards along the same parabola (to the left now). I think this can be done by making B oscillate between a positive and negative value.

I also need to add a "group velocity" so that a string of these parabolas will create a chain like that seen in (d). In my example, though, the parabolas should be moving downwards.

TO SIMPLIFY:
1. What E and B fields (can be time dependent) will cause a charged particle to oscillate along a parabola
2. What can be done to include a drift/ group velocity so that the parabolas move in one direction.

 

[mfb]

Undulators give particle paths similar to (d): Alternating magnetic fields, the particles follow circle segments in each section. A fine-tuning of the magnetic fields can modify the path a bit more.

 

[peasqueeze]

Ok, so my research is basically modeling self replicating cracks with analog E and B fields. This means that I need an analytical expression (not experimental method).

 

[mfb]

An analytical expression of what?
You can make up magnetic fields (or electric fields) that lead to parabolic paths of particles moving through that field.

Your question is quite vague.