Charged particle in a uniform magnetic field

A velocity cross magnetic-field force bends perpendicular motion into a circle while doing no work.

The cylindrical vector potential $A_{\theta }=\mu r$ gives a uniform axial magnetic field $2\mu$, while scalar potential $\lambda z^2$ supplies harmonic axial confinement for positive charge and positive $\lambda$. Transverse motion is cyclotron motion about a fixed guiding centre, and axial motion is an independent harmonic oscillation. A circle centred on the cylindrical origin has $\dot{\theta }=-{\Omega }_c$ for positive charge.

A rational ratio of axial and cyclotron frequencies makes a nondegenerate combined orbit close after an integer number of turns. For ratio $p/q$ in lowest terms it closes after $q$ transverse periods and $p$ axial periods. An integer ratio is a sufficient special case. An irrational ratio instead gives quasiperiodic motion when both component amplitudes are nonzero.

For a particle obeying $m\dot{\mathbf{v}}=q\mathbf{v}\times \mathbf{B}$ in a constant uniform magnetic field, the displayed combination is conserved. The derivative of mechanical angular momentum projected onto the field is canceled by the quadratic position term. This is the rotational analogue of the distinction between mechanical and canonical momentum in an electromagnetic field.

In a uniform magnetic field, the Lorentz force leaves the component of velocity parallel to the field constant and rotates the perpendicular component at the nonrelativistic cyclotron frequency $|q|B/m$. The resulting path is a helix around a field line.

The guiding center is the center of a charged particle's circular cyclotron orbit. In a nonuniform field it drifts slowly compared with the rapid cyclotron frequency.

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.

Adding linear drag to cyclotron motion produces an exponentially contracting spiral.