Quick questions on Relativistic momentum and particle accelerators: HSC Physics Module 7

The correct expression for momentum of a particle of rest mass $m$ moving at velocity $\vec{v}$ is:

The whole job of an accelerator is to push charged particles to extremely high energies for collision experiments. Relativistic momentum dominates the design.

A particle of momentum $p$ in a perpendicular magnetic field $B$ has radius:

A linac uses successive RF cavities to add small kicks to the particle's energy along a straight line. Relativistic momentum determines the spacing of the drift tubes: as $\gamma$ grows, $v$ saturates near $c$ but $p$ keeps increasing, so cavity spacings only need to grow modestly along the line.