Velocity rotation in a time-dependent axial magnetic field ($\dot{x}+i\dot{y}=V\exp \left(i\varphi -\frac{iq}{m}{\int }_0^t{B}_z(s)ds\right)$)

Under the magnetic-only Lorentz force, planar velocity has constant magnitude and an accumulated phase determined by the time integral of the axial field. For $B_z(t)=\beta t$, the planar position involves Fresnel integrals. With initial planar velocity $(1,0)$ and zero initial position, its limit is $(\sqrt{\pi m/(4q\beta )},-\sqrt{\pi m/(4q\beta )})$. The model prescribes a magnetic-only force; a physical time-dependent field may also produce an induced electric field.

// Target: number-theory.bigb