Helical motion

If a charged particle such as an electron enters a magnetic field at an angle its path will be a helix because it now has a component along the magnetic field direction (see Figure 1).

The electron makes one orbit of the helix while it travels a distance x along the direction of the field - this is called the pitch of the helix. If the angle that the initial path of the electron makes with field direction is θ we have:

mv²sin²θ/r = Bevsinθ and x = vtcosθ
But t = 2πr/vsinθ giving vsinθ = 2πr/t
and so mvsinθ/r = Be
Combining the two previous equations gives: 2πm/t = Be

Therefore: t = 2πm/Be x = vtcosθ and finally: