NBPhO 2025

A rod of mass $m$ carries a charge $q$; both the charge and the mass are homogeneously distributed over its entire length $l$. The system is in a homogeneous magnetic field of strength $B$, parallel to the $z$-axis, whereas the rod is in the $x$–$y$-plane. Neglect any forces except for the Lorentz force. One end of the rod is painted red, and the other — blue.

i) (2 points) Consider the case when the rod rotates around its centre of mass. What should be the angular speed $\omega$ for the mechanical tension force at the centre of the rod to be zero?

ii) (4 points) Consider now a case when initially the blue end of the rod is at the origin ($x = y = 0$), and the red end at $x = l$. The blue end’s initial speed is zero while the red end’s speed is $v$, parallel to the $y$-axis. It turns out that after a certain time $t$, the red end passes through the origin. Find the smallest possible value for $t$ and express the corresponding value of $v$ in terms of $m$, $q$ and $l$.