Electron helix in a magnetic field

[Basil Fawlty]

I thought that a nearly parallel entry path would result in a helix of very small, but constant, radius. I would not expect the electrons to focus at a point, but continue along the infinite helix. What have I missed?

 

[anorlunda]

I moved this thread from a technical forum. No template.

@Basil Fawlty , You must show us your attempt at the solution before our homework helpers are allowed to help. So, please show us your work, or I'll have to tell Sybil :-)

 

[Basil Fawlty]

Well, producing the equations for the helix will not solve this, as the electrons will continue along the helix but not come to a focus.

Assuming magnetic field in the x direction, and v a (very small) initial velocity in the y direction (without loss of generality):

x=ut, y=(mv/eB)sin(eBt/m), z=(mv/eB)(1-cos(eBt/m))

But, of course, there is no limiting case when v tends to zero, so no focus.

I thought the question may be wrong, and 'parallel' should read 'perpendicular', so interchange u and small v in the above equations, but even then the answer provided in the question, which is the circumference of my helix, does not appear obvious.

This question is bugging me, as electrons can be focused on a screen in various devices but only by changing current in coils and so the magnetic field. Is there some physical phenomenon that can make this happen in a uniform field?

 

[Dale]

What is the distance between two such electrons with different v?

 

[Basil Fawlty]

With two different velocities v1 and v2 d(t)= (2m(v1-v2)/eB)sin(eBt/2m) is the simplest form of the distance function. Ah yes, so d(t)=0 when eBt/2m=π, t=2mπ/eB and x= 2πmu/eB
It's not intuitive at all that no matter what the initial relative y velocity component, the two electrons will always focus at a particular point in time and space based on e,m,B,u alone. Plus the question did not help by saying 'negligible interaction'!
Many thanks Dale.

 

[Dale]

I agree with you. It is a confusingly worded question. I am not sure I would call that a focus at all since to me “focus” invokes ray optics and not helical paths converging.