A nice feature of the cyclotron described in the preceding problem is that the alternating current frequency applied to the "Dees" is a constant \(\omega=q B / m c\) for nonrelativistic particles, regardless of their energy, so the circulating particles will arrive at the gaps at just the right time. No matter the radius at which a particle orbits, the time it takes to travel between two gap encounters is exactly the same.

(a) Show that this is no longer true for relativistic particles. Find a new expression for \(\omega\) in terms of \(q, B, m, c\), and \(\gamma \equiv\left(1-\beta^{2}\right)^{-1 / 2} \equiv\left(1-v^{2} / c^{2}\right)^{-1 / 2}\).

(b) How might one design an "isochronous cyclotron," in which relativistic protons will still reach the gaps at the correct time, with the same constant-frequency alternating current applied to the Dees?

(c) The TRIUMF isochronous cyclotron has a proton outer orbital radius of \(7.9 \mathrm{~m}\), where the protons have a kinetic energy of \(510 \mathrm{MeV}\). What is the magnetic field strength at the outer orbit?

(d) How fast are these protons moving, expressed as a fraction of the speed of light?