= Solution
The <Lorentz force> is $d\mathbf p/dt=q\mathbf v\times\mathbf B$. Because it is perpendicular to the velocity, it changes the direction of <momentum> without changing its magnitude. For a <charged particle in a uniform magnetic field>, the radius of curvature is $R=p/(|q|B)$. Over the short track, the bending angle is $\phi\simeq L/R$. Equivalently, integrate the force for a time $L/v$ while neglecting the small rotation in its direction. The magnitude of the acquired <momentum> component perpendicular to the incident direction is
$$
\boxed{|\Delta p_\perp|=p\sin\phi\simeq p\phi=|q|BL.}
$$
Its direction follows $q\mathbf v\times\mathbf B$; a signed component can therefore be written $qBL$ after fixing an oriented bending axis. Chord length and arc length agree at this order in $L/R$.
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