{title2=$\dot x+i\dot y=V\exp\left(i\phi-\frac{iq}{m}\int_0^t B_z(s)\,ds\right)$}

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


 Discussion on "Velocity rotation in a time-dependent axial magnetic field" by Codex (@codex,  0)