Question: A cyclotron (Fig.) is an apparatus for accelerating charged particles to very high kinetic energies. Rapidly moving charged particles are coaxed into circular orbits by a strong uniform magnetic field that points in a direction perpendicular to the plane of the particles' orbit. During each lap around the cyclotron, the particle is accelerated (twice) by an electric field, boosting its energy and also increasing the radius of its orbit. The National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University is a world leader in the study of rare isotopes and fundamental nuclear physics. The K500 cyclotron at NSCL has a diameter of 3 m and a maximum magnetic field strength of 5T. You can learn more (including the answers to this problem) at An Introduction to the K500 Cyclotron
a. Singly charged argon (m = 40 u) ions are accelerated within the K500 cyclotron. What is the maximum speed to which they can be accelerated? (Express this as a fraction of the speed of light c.)
b. Prove that the time needed for an Ar+ ion to make a full circle within the cyclotron is independent of its velocity. (Thisis true as long as v
c. Calculate this time. Note that the electric field accelerating the charges must flip its polarity twice in this time.
Fig. - A diagram of a small part of the spiral path of particles accelerating inside a cyclotron where the magnetic field B is prependicular to and out of the plane of the page. Accleration occurs at the gap between the Dees.