Model Electron in Solenoid: Path & Frequency Effects

[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 http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/parapath.html" 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 http://plaza.ufl.edu/rosspa/pf.JPG" . 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?

 

[Vailanor]

I'm pretty sure it's something simple that I just need to realize.My thought process is that the electron is traveling in a helical path, so I want to find the equations of the path that describes this helix. Since the magnetic field is changing with time, the helix will not be a perfect spiral but rather an arc. I think I can use the equations given above, but with a modified form of the cyclotron frequency, w = (e/m)*Bsin(Wt). This would give me the position of the electron at any given time. Then I could use the trigonometry to find the distance of the electron from the origin, and the angle between any two points on the arc. From this information I should be able to calculate the length of the arc and the frequency of the cyclotron. I hope this makes sense, and if anyone can provide some insight as to how to go about solving this problem I would really appreciate it. Thanks!

 

[astrokreng]

I would suggest using the equations for a charged particle moving in a magnetic field, known as the Lorentz force equations. These equations take into account the magnetic field strength, the charge and mass of the particle, and its velocity. By solving these equations for your specific setup, you should be able to accurately model the path of the electron beam in the solenoid, including the frequency effects.

Additionally, I would recommend conducting more experiments to gather more data points and accurately measure the distance and angle of the arc on the screen. This will help to improve the accuracy of your model and ultimately provide a more precise value for the cyclotron frequency and (e/m) ratio.

Furthermore, it may be helpful to consult with other scientists or experts in the field to discuss and refine your equations and experimental setup. Collaborating with others can often lead to new insights and solutions to complex problems.

In conclusion, modeling the path of an electron beam in a solenoid can be a challenging task, but with careful experimentation and the use of established equations, it is possible to accurately predict and understand the behavior of the beam.