A cyclotron designed to accelerate protons has an outer radius of 0.570 m. The p

Question

A cyclotron designed to accelerate protons has an outer radius of 0.570 m. The protons are emitted nearly at rest from a source at the center and are accelerated through 600 V each time they cross the gap between the dees. The dees are between the poles of an electromagnet where the field is 0.720 T.

(a) Find the cyclotron frequency for the protons in this cyclotron.
______rad/s
(b) Find the speed at which protons exit the cyclotron.
______m/s
(c) Find their maximum kinetic energy.
______eV
(d) How many revolutions does a proton make in the cyclotron?
______revolutions
(e) For what time interval does one proton accelerate?
______s

Explanation / Answer

'll give you enough information to get started on the problem. But the exact details are not something that I can work out in my head at the moment.

So we have a cyclotron in this problem. You can essentially think of it as a circle split in half by a narrow gap along its diameter (This is why each half is called a Dee). You have a proton cycling around inside the circle because you have a perpendicular mag field B = 0.720 T, and it goes through V = 600V of pot. diff. every time it crosses the gap. Let's denote proton charge by e, its mass by m, and its instantaneous velocity by v.

a) Since our proton is going around in a circle, we know the centripetal force exerted on it has to be F = mv^2/r. But since the proton is also moving in a B field, F = evB. Setting those two equal we have v/r = eB/m. But v/r = omega, the angular frequency. And the actual frequency is just omega/(2pi).

b) ,c) This actually requires some manipulation to come up with an exact answer, however, we know 2 things that can help simplify so we get a very good approximation of the exact answer.
i) If you find the "max" energy that a proton can possibly have in the cyclotron, ie you find the velocity that the proton needs to have to travel in a circle of r=0.570m, you'd notice that the max energy is about a thousand times 700eV, suggesting that the proton needs to travel through the gap about a thousand times.

ii) Notice in the beginning, when the proton has 0 velocity, and you accelerated it across the gap for the first time, the center of motion is not the center of the cyclotron. But as the proton goes over the gap the second time, the acceleration increases the speed of the proton and shifts the center of motion in the direction of the center of the cyclotron. This actually occurs every time the proton crosses the gap, so the center of motion is always within the vicinity of the center of the cyclotron. Thus, when you find the speed at which the proton exit the cyclotron, you can simply approximate the speed as the max speed at which a particle can move in a circle within the cyclotron.

For b, and c, however, when you look at the velocities we are talking about, it's dangerously close to c. Although you are fine if you are working with velocities less than 0.1c, you should still check and see if you actually need the relativistic KE, the relativistic mass, etc.

anyways, if you find the velocity in b), you can find K in c)

d) if you've found K in c), you get this trivially because you gain 700 eV of energy as you cross a gap. You cross the gap twice per rev. And the maximum energy your particle obtains is in part c)

e) You've found the number of revolutions in d), and you've found the frequency at which the particle goes around the cyclotron, you can find the time of acceleration. Notice the frequency in the cyclotron is constant, I hope you understand why that's true.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.