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Homework #1
| Due Date: 2/9/05, in class |
80 Points |
Each problem is worth 10 points
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Derive the relationship between the units of length and time in
natural and SI systems of units, i.e. show that 1 GeV-1 = 0.2 fm or 1
GeV-1 = 6.6x10-25 s. In natural units, are you higher or older?
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Perkins, 1.2.
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In natural system of units, what are the units of
a. Electric charge;
b. Electric field;
c. Voltage;
d. Electric current;
e. Magnetic field;
f. Force?
In each case, derive the relationship express the corresponding SI units in
natural units.
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Show that the interval is indeed a Lorentz-invariant quantity. Show that the product of two arbitrary 4-vectors is Lorentz-invariant.
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Perkins, 1.1.
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Perkins, 1.3.
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HERA is an electron-proton collider with the electron and proton
beam energies of 30 and 820 GeV, respectively.
a. What is the c.o.m. energy at HERA?
b. Find the energy of an electron beam, which would create
the same c.o.m. energy in collisions with stationary
protons.
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The GZK effect. Ultrarelativistic cosmic protons could be
effectively stopped over short distances (on a cosmological scale!) if their
energy is sufficient for the following scattering process on the photons of
Cosmic Microwave Background (CMB):
p + g ® D,
to be allowed kinematically. Consequently, one does not expect to observe any
protons above this threshold
(called the GZK-threshold) to reach the Earth. Find the GZK threshold, given the
mass of the proton and
D of 0.938 GeV and 1.232
GeV, respectively, and the typical energy of a CMB photon of 2.35x10-13
GeV.
Hint: consider a head-on
collision of a proton with the photon, which corresponds to the highest possible
energy released in collision.
In preparation to the next (Monday,
2/7)
lecture, you might find it useful to finish reading Chapter 1 of the
Perkins book. |
