In this problem you will write a computer program to compute and display the motion of a proton inside a cyclotron (see Section 21.1). The first cyclotrons were small desktop devices; we will model a...

In this problem you will write a computer program to compute and display the motion of a proton inside a cyclotron (see Section 21.1). The first cyclotrons were small desktop devices; we will model a small device in order to understand the basic physical phenomena. Consider a cyclotron whose radius is 5 cm, with a gap of 0.5 cm between the dees. External magnets create a uniform magnetic field of 1.0 tesla in the +y direction throughout the device. A proton starts essentially from rest at the center of the cyclotron. The maximum potential difference between the dees is 5000 V. Note that the electric field inside the dees is 0 at all times; Figure 21.132 shows that the metal dees form a nearly closed conducting enclosure.

(a) What is the angular frequency ω of the sinusoidally oscillating potential difference you will need to apply? (b) Compute and display the trajectory of a proton in the cyclotron; display the trail of the proton. You may assume that the proton does not reach a speed sufficiently close to the speed of light to require a relativistically correct calculation, but do check this assumption. To get your program going, you can use a large value of dt (such as 1E10 seconds), but you will need to decrease this value later. (c) Plot the proton's kinetic energy K (in eV) vs. time. Explain the shape of the graph you observe. (d) What happens if you use a different angular frequency for the potential difference? For example, try an angular frequency of . (e) If the proton gains a significant amount of kinetic energy while inside the dees, your dt is too large. Adjust your dt appropriately. What value of dt gives reasonable results? (f) How much time does the proton take to reach the outer edge of the cyclotron? (g) What is K (in eV) when the proton reaches the outer edge of the cyclotron? (h) Given your answer to (g), how many orbits must the proton have made before reaching the outer edge of the cyclotron? (i) What is the final speed of the proton? Were you justified in assuming that v