The 1.0-km radius Fermilab Tevatron takes about 20 seconds...

The 1.0-km radius Fermilab Tevatron takes about 20 seconds to bring the energies of the stored protons from $150 \mathrm{GeV}$ to $1.0 \mathrm{TeV}$. The acceleration is done once per turn. Estimate the energy given to the protons on each turn. (You can assume that the speed of the protons is essentially $c$ the whole time.)

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Key Concepts

Relativistic Kinematics

At high energies, particles such as protons travel at speeds very close to the speed of light. This relativistic behavior allows for simplifications in calculations, such as assuming a constant transit speed close to c, which is useful for estimating cycle times and energy gains in accelerators.

Circular Particle Accelerators

Circular particle accelerators use magnetic fields to bend the path of charged particles so that they travel in closed loops. In these systems, particles repeatedly pass through accelerating structures, which allow small increases in energy to accumulate over many cycles until the desired high-energy state is achieved.

Energy Gain per Turn

Energy gain per turn refers to the incremental increase in kinetic energy that a particle receives each time it passes through the accelerator's energy-boosting section. By knowing the overall energy difference and the number of cycles (or turns) over a given period, one can divide the total energy increase by the number of turns to find the energy imparted per cycle.

Transcript

00:01 In this problem, we're looking at how much energy the magnets add to the particle beam on each turn of the tevatron.

00:10 So the tevotron collider is approximately a kilometer radius.

00:16 We're given an initial energy of 150 giga electron volts, and it goes up to 1 trillion electron volts over a span of 20 seconds.

00:32 It tells us to assume the speed is approximately the speed of light.

00:35 So we're not doing a relativistic calculation.

00:39 It kind of gamma terms is kind of swept under the rug inside the energy terms.

00:45 We're dealing in time from the stationary lab reference frame.

00:50 So to find out how much energy is added per cycle or per turn, we have to find the time per turn, then the number of turns.

01:00 And then we can use this energy difference here to find that the energy for each of those turns that's added.

01:11 So one turn that's a length of 2 pi r because that's a circumference.

01:24 It's a circular accelerator.

01:27 So the time for each turn is that distance divided by the speed.

01:36 And we're told to consider the speed as approximately c.

01:43 It's getting closer to c as it gets more energetic, but it's already very close to c at 150 giga electron volts.

01:55 So then, so that's 2 pi 10 to the 3 meters divided by 2 .998 times 10 to the 8 meters percent...

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