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February 2008 - Solutions

February 26, 2008

1. It is stated in “The Discovery Machine” that during a full Moon at high tide, the Geneva area rises by 25 cm and the circumference of the LHC ring increases by 1 mm. As a consequence, the beam energy changes by 0.02%. Check if these numbers are correct by solving this problem.

Discussion of solution Draw a triangle from the center of the Earth to the edges of the LHC ring of diameter d = 27km/π. The radius of the Earth is R ∬ 6 × 103 km, so the angle subtended at the center of the Earth is θㄱ d/R. If the length of the triangle is stretched by h = 25 cm, R ! R + h, then the base increases by s = hθ (by similar triangles), and s is the increase in the diameter of the LHC ring. So, s 25 cm ×(27/πkm)/6 × 103 km 0.036 cm 0.36 mm. The circumference increases by π times this, or about 1 mm.

2. The CMS detector has Resistive Plate Chambers (RPCs) built at Korea University and surrounding the experiment at its outer radii. The RPCs at the ends are about 8 m from the interaction point where the particles are created in collisions. By the time particles from one collision reach these RPCs,

A. another collision has already happened

B. the next collision will not happen for a long time

C. a collision is about happening at this time

Discussion of solution The particles are traveling at very nearly the speed of light 3 × 108 m/s (or almost exactly 1 foot/sec in English units), so it takes 27 ns for the particles the reach the RPCs. The collisiions happen every 25 ns, so a collision is just happening.

3. What is the ratio of the highest mass quark to the lowest mass quark?

A. 1

B. 102

C. 103

D. 104

E. 105

Discussion of solution The u (“up”) quark is about 2 MeV/c2 in mass and the t (“top”) quark is about 171 GeV/c2.

4. What is the ratio of the highest mass charged lepton to the lowest mass charged lepton?

A. 4

B. 40

C. 400

D. 4000

E. 40, 000

Discussion of solution The electron is 0.5 MeV/c2 and the is 1.8 GeV/C2 and the is 1.8GeV/C2.

5. What geometrical object has the highest symmetry?

A. a cylinder

B. a sphere

C. a cube

D. a 3-sided pyramid (tetrahedron)

E. a 4-sided pyramid

Discussion of solution A sphere looks like the same sphere no matter how it is rotated or translated in space. This is an extremely high degree of geometrical symmetry. Laws of nature and equations in physics are also said to be symmetric under an operation if the equations do not change when that operation is performed.

6. What is the energy of the electrons and positrons emerging from the 11.3 km linacs if the superconducting radio-frequency cavities (SCRF) generate an electric field of 35 MV/m, and the SCRF fill about 60% of the linac length ?

A 400 MeV

B 2.5 GeV

C 25 GeV

D 250 GeV

E 400 GeV

Discussion of solution An electron will gain an energy of 35 MeV by falling through (i.e., accelerated by) an electrostatic potential of 35 MV. This is the definition of the “electron volt” as a unit of energy. Only about 60% of the linac is filled with accelerating electric fields, so the energy gain is 35 MV/m ×11.3 km ×0.60 ∼ 240 GeV.

7. How long does it take for a bunch of electrons to travel the length of the linac?

A. 10 ns

B. 40 ns

C. 1 μs

D. 40 μs

E. 1 ms

Discussion of solution The electrons are ultra-relativistic and therefore travel at nearly the speed of light c = 3 × 108 m/s, so the time is 11.3 × 103 m /3 × 108 m/s ㅁ 40μs.