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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?

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Electron in metal does not move like electron in vacuum in EM field. In classical theory, electron in conductor experiences more complicated force (as a function of time) than the Lorentz formula with macroscopic field E,B would suggest. As the electron moves through the conductor, it experiences forces due to other particles - electrons, nuclei ( especially if the atomic lattice has imperfections). This manifests as drag.

The magnetic field of the conductor's current pushes the electrons towards the center, but this effect is very small and is usually ignored in DC circuits, and overpowered by skin effect in AC circuits.

On the other hand, the effective attraction of mobile charges due to magnetic force is strong in case of current in plasma, where it leads to so-called pinch effect.

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