A company is interested in hiring you to build something for their use. They want to make a cyclotron that will accelerate protons to very high speeds. They would rather not deal with special relativity, so they only want to accelerate protons to a maximum speed of 10% the speed of light (c = speed of light = 3.0 x 10^8 m/s).
They also have other restrictions so you will need to use your physics skills to build a cyclotron based on all their requirements and limitations:
1) They can only establish a magnetic field of maximum magnitude: B = 1.00 T to curve the protons inside of the cyclotron.
2) They already know they want to use a gap spacing between the two halves of the cyclotron of 2.00 cm.
3) They want to avoid having to create a high vacuum for the entire cyclotron, so the maximum voltage they can use to accelerate the protons is restricted as well: if the (voltage/distance) becomes larger than thedielectric breakdown of moderate vacuum, small sparks or lightning bolts will be created in the gap between the two halves (even moderate vacuum cannot keep charges from staying separated if the voltage across thegap becomes too high). The dielectric breakdown for moderate vacuum is 20.5 MV/m. Based on this restriction and the gap spacing of 2.00 cm, there is a maximum voltage you can use to accelerate the protons.
4) The entire size of the cyclotron from one edge to the other (width) can only be just over two feet (25 inches). This puts a restriction on the maximum radius for the protons' motion right before they exit the cyclotron at the far edge after they have accelerated to their maximum speed.
5) The protons will be sent into the center of the cyclotron at an initial speed of 5.0 x 10 m/s before they ever accelerated
Based on this information and these restrictions, answer the following questions:
1. What frequency does the changing voltage (AC voltage) across the gap need to have in order to match up with the protons and ensure they are accelerated each time they encounter the gap (rather than being accelerated and slowed down every other time...)? How come this frequency doesn't have to be changed as the protons' speed continues to increase?
2. Is it even possible to build the cyclotron based on the company's abilities and restrictions? If yes, explain in detail why, showing relevant calculations. If not, explain what restrictions must be changed so that perhaps they can make some adjustments.
3. Once you've perfected your design, how many times will the protons pass through the gap before they exit at the edge of the cyclotron with a speed of 10% the speed of light? (1 point) How long will they actually be inside the cyclotron?
4. Does the initial speed of the protons when they first enter the cyclotron really make a difference? If their initial speed were to change to 500 m/s would you need to make any significant changes to your design? Why or why not?