Charged particle in a uniform magnetic field

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

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.

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