# Longest distance travelled by an electron in E&M field

Let's assume we have a cylindrical wire where the currents are following. Since this is not a static case, we would have magnetic and electric field both from the current carrying wire.

Here I have a concern. As I understand that the drift velocity is the average velocity the electron has due to the electric field. Since there is also magnetic field in the current conducting wire, we would have magnetic field in the $$\phi$$ direction (in cylindrical coordinates).

Can I find the maximum distance that an electron can travel perpendicular to the wire?

Do you think this equation would work?

$$m \ddot x = qE+ qvB$$

My intention is to know the longest perpendicular distance travelled by an electron. Can we call this motion as like cycloid properties in electromagnetic field?