Teaching Modern Physics - QuarkNet - Fermilab

Teacher’s Key
Constant Acceleration/Force Problems
1. The Fermilab accelerator has a circular shape, with radius of 1 km. It can take protons
with an initial energy of 120 GeV and accelerate them to 980 GeV in 20 seconds. The
acceleration actually only takes place via an electric force that is only 50 feet along the
orbit. The remainder contains magnetic fields that only bring the protons back around in a
circle for another acceleration phase.
a. Calculate the average power added to the protons.
= (Efinal – Einitial)/time = (980 – 120)/20 = 43 GeV/s = 6.88 × 10 -9 W
Since the protons are relativistic (i.e., traveling at the speed of light at all energies), one
can calculate the number of orbits per second.
b. Calculate the number of orbits in 20 seconds.
d = v t, v = c, → d = (3 × 10 8 m/s)(20 s) = 6 × 10 9 m
Circumference = 2 π r = 6.28 km
(# orbits) = distance/circumference = 9.6 × 10 5 orbits
c. Calculate the increase in energy per orbit.
ΔE/Δorbit = (980 – 120)/ 9.6 × 10 5 = 9 × 10 -4 GeV/orbit = 900 keV/orbit
d. Calculate the average strength of the electric field in the acceleration region. Assume
the electric field is constant in the acceleration region.
If the energy increase in each pass is 900 keV, and the particle is a proton, then the
accelerating voltage is 900 kV.
V = Ed → E = V/d = (900 kV)/(50 ft) = 60 kV/m
1. The first accelerator in the Fermilab accelerator chain is the Cockcroft-Walton. A proton
enters the accelerator with essentially zero energy and exits with a velocity of 4 percent
of the speed of light (thus allowing you to ignore any relativistic effects). If the
acceleration time is 0.16 µs,
a. Calculate the average acceleration of the proton and compare it to the acceleration
due to gravity.
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