The geometry of acceleration

I made a picture because I'd struggle to get out a question without it. In the picture all things are constant except the strength of the magnetic field. It is at two different values. We see 2 cycloid paths of electrons that starts at rest. The circumference of the large path is exactly twice that of the smaller path.

My question is, would the green path have the higher velocity away from the origin?
Although both paths are equal in length, shouldn't the green path be the fastest? I base this conclusion on the amount of work the electric field does in both cases.

Answers and Replies

Twigg
Science Advisor
Gold Member
I would say the purple path is faster because it's at a lower potential energy on average than the green. At first I agreed with your assessment, but then I noticed these were electron paths, so their potential energy increases in the direction of the electric field (since they're negatively charged). By that logic, the purple path should have a higher average kinetic energy and should be faster based purely on inspection. You could run a simulation or perform the time integral to double check.

I would say the purple path is faster because it's at a lower potential energy on average than the green. At first I agreed with your assessment, but then I noticed these were electron paths, so their potential energy increases in the direction of the electric field (since they're negatively charged). By that logic, the purple path should have a higher average kinetic energy and should be faster based purely on inspection. You could run a simulation or perform the time integral to double check.

I used the diagram to show how often the electrons on the green path are moving faster than the purple path. The yellow bars on the lorentz force line represent times when "green" electrons exceed the top speed of the purple electrons. I think this answers my question because now I realise that the red areas are when electrons in both paths are moving at the same velocity.