Past exam of the particle physics course of the University of Oxford / 2022 / 4 / a

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 $\varphi \simeq L/R$. 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 isIts 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 $L/R$.

The sagitta is the maximum distance between the circular track and its chord. The perpendicular bisector of a chord of length $L$ gives a right triangle with hypotenuse $R$ and sides $L/2$ and $R-s$. HenceAt fixed electric charge, magnetic field and lever arm, $p=|q|B{L}^2/(8s)$, so differentiation and propagation of uncertainty giveThus a fixed spatial measurement uncertainty produces a relative momentum uncertainty growing linearly with $p$. If momentum is expressed in $\mathrm{GeV}/c$, $B$ in tesla, $L,s$ in metres and $z=q/e$, restore the usual conversion $p=0.2998|z|BR$.