Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 4 / 9B / Solution

With $\omega =B/m$, magnetic force does no work, so $|\dot{x}|$ is constant; its zero $z$ component also makes $\dot{z}$ constant. For the stated data,This is a circle of radius $u/\omega$ centred at $(0,-u/\omega )$.

With $\alpha =\mu /m$, the complex velocity is $\dot{x}+i\dot{y}=u{e}^{-(\alpha +i\omega )t}$. HenceThe path spirals into $u(\alpha ,-\omega )/({\alpha }^2+{\omega }^2)$. As $u$ varies, this lies on $y=-(\omega /\alpha )x$, at distancefrom the origin.