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

For a particle of positive mass $m$ and signed electric charge $q$ moving under the magnetic-only Lorentz force in $\mathbf{B}=B_z(t){\mathbf{e}}_z$, write $u=\dot{x}+i\dot{y}$. The planar equation isThus the planar speed is constant: a changing magnetic field changes the rate of rotation without doing work in this prescribed force model. The axial velocity is constant as well.

For $B_z(t)=\beta t$, initial planar velocity $(V,0)$ with $V>0$, and zero planar initial position,These are Fresnel integrals. If $q\beta \ne 0$, their limits areThe negative second coordinate therefore assumes $q\beta >0$; reversing the electric charge or the ramp reverses the transverse deflection. When $q\beta =0$ the motion is ballistic and has no finite position limit. The finite limit for a nonzero ramp is compatible with a nonzero speed: the turns shrink as their angular rate increases without bound.

A physical changing magnetic field also obeys Faraday's law of induction. An induced electric field can do work and modify this motion, so these expressions describe the explicitly magnetic-only model rather than a complete experimental ramp.

// Target: number-theory.bigb