Let's look first at a cyclotron. A simple cyclotron has a particle injection site just barely off center (red dot) and a band of electric field E = 1.172 x 10' N/C for 10 cm along the diameter, as shown in yellow in the picture the arrows are the direction of the electric field). Everywhere else inside, particles are subject to a constant magnetic field, out of the page and perpendicular to the electric field. When charged particles are injected and both fields are turned on, they follow a spiral trajectory; the magnetic field bends their paths and the electric field adds speed. Assume the cyclotron in this question has a radius of R = 2 m and has a magnetic field of magnitude B = 0.10 T. (a) First, find an expression for the orbital frequency of an alpha particle with some non-zero velocity v inside a cyclotron while the electric field is turned off (alpha particles are bare helium nuclei; you may use Google to look up their mass and charge). You may use r, m, e, v, and B. This value is called the cyclotron frequency. (b) Now, before the electric field is turned on, we inject alpha particles leftward from the injection site located at r = 0.1 m (the red dot below the center of the cyclotron), and they stay orbiting at that radius. What is their initial velocity? Write it out first in symbols, then insert numbers. (c) Now let's turn the electric field on. How many orbits does the alpha particle make before it leaves the cyclotron? Now let's turn to synchrotrons. In a cyclotron, the particle will spiral outward as its velocity increases and eventually end up either exiting or colliding with the inner wall of the device. Synchrotrons are not limited in that way and can accelerate particles to much higher speeds. One reason is that the magnetic field strength in the synchrotron is the tunable parameter, rather than the orbital radius. The particles enter the synchrotron via an accelerating tube where they are subjected to an electric field. Let's see what it would take to accelerate an alpha particle to 0.1c, or one-tenth of the speed of light.