On-axis entry into a solenoidal tube ($r_{\max }=2m{v}_{\perp }/(|q|B)$)

A particle entering a uniform axial magnetic field on the tube axis follows a transverse circle through that entry point. Its guiding centre is one Larmor radius from the axis, so the maximum radial displacement from the axis is twice the Larmor radius. For tube radius $a$, clearance requires $v_{\perp }\le |q|Ba/(2m)$ when the transit samples a complete gyration. If the transit is shorter, the actual sampled arc determines clearance instead. The axis is revisited after each full cyclotron period, not after each half-period.

A nonrelativistic particle accepted only at a fixed flight time through an ideal uniform-field tube has fixed parallel speed $v_{\parallel }=l/t_0$. On-axis entry and wall clearance bound the independent perpendicular speed by $|q|Ba/(2m)$, provided a full gyration occurs before exit. Adding parallel and perpendicular kinetic energies gives the displayed ideal energy window, where $K_0=m{v}_{\parallel }^2/2$. The upper endpoint is a grazing limit; strict wall exclusion makes it a supremum.