# Electron in solenoid

1. Mar 7, 2006

### coolnessitself

I'm tyring to develop some equations to model the path of an electron beam in a magnetic field. It starts out with a velocity in the z direction but also a tangential velocity, that sets up a helix given by
x(t) = (E/(wB))[cos(wt)-1]
y(t) = (E/(wB))[sin(wt)-wt]
z(t) = vt where v and k and constants and w is the cyclotron frequency, given by (e/m)*B
This part I'm certain about, as I got the correct values of (e/m) from experiment.

now I'm varying the magnetic field with a variac at a high frequency W (I don't want the answer in terms of this because all modifying W does is blur what's visible) so that every time you see a B above, it becomes Bsin(Wt). The electron following this path follows a type of helix like the one above, but it switches directions frequently, so only an arc of the helix (or an arc of a circle if it were projected onto a screen) is created. I'd think it kind of looks like a snake (not sure about this 100%). If you go to this handy site and type in
x=2(cos(.1t*sin(t))-1)
y=2(sin(.1t*sin(t))-.3)
z=.5t
t=[0,20]
you can see an idea of what I think it may resemble in the solenoid. What I see on the screen at the end of the tube looks like this. Since this what I see is off the axis and the plot created by the applet above is on the axis, these equations obviously aren't right.

If I lower the frequency W I don't see anything but a less-blurred arc like this one, so I can't have an answer in terms of W. I'm trying to solve for lowercase w here in order to find (e/m) by the way.
Anyways,
When I look on the screen where the electrons land, I see the arc, and since the frequency W is so high, this is a solid arcing line. I measured the distance r to the arc and the degree measure between either side of the arc, as seen in that pic linked to above.

Now, I've been stumped the last few days as to how to mathematically model this so that I can find the points where the arc starts and ends so I can find w and therefore e/m. If I type in a [const]*sin(t) everywhere there is a w in the original parametric equations it becomes very squiggly and doesn't resemble what I see. Any suggestions on a correct way to model this trajectory?