= Velocity rotation in a time-dependent axial magnetic field
{title2=$\dot x+i\dot y=V\exp\left(i\phi-\frac{iq}{m}\int_0^t B_z(s)\,ds\right)$}
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 is
$$
\dot u=-i\frac qm B_z(t)u,\qquad
u(t)=u(0)\exp\left(-i\frac qm\int_0^tB_z(s)\,ds\right).
$$
Thus 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,
$$
x(t)=V\int_0^t\cos\left(\frac{q\beta s^2}{2m}\right)ds,\qquad
y(t)=-V\int_0^t\sin\left(\frac{q\beta s^2}{2m}\right)ds.
$$
These are <Fresnel integrals>. If $q\beta\ne0$, their limits are
$$
(x_\infty,y_\infty)=
V\sqrt{\frac{\pi m}{4|q\beta|}}
\bigl(1,-\operatorname{sgn}(q\beta)\bigr).
$$
The 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
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