Problem Set #03 (Relativistic Loose Ends and Particles of Light)

Physics 322: Modern Physics
Spring 2015
Problem Set #03 (Relativistic Loose Ends and Particles of Light)
Due Friday, February 6 at 12 noon (in lecture)
ASSUMED READING: Before starting this homework, you should read
Chapter 3 of Harris’ Modern Physics. 1. [Harris 2.79 tweaked] At Earth’s location, the intensity of Sunlight is
1.5 kW/m2. Assuming Earth is spherical with a radius of 6370 km and no
energy escapes Earth, by how much would Earth’s mass increase per day
due to this absorbed energy? HINT: You will want to consider the “cross
2
sectional” area of the Earth facing the Sun (that is, ! A = π rEarth
) rather than
1
2
2
the actual surface area of the Earth facing the Sun, ! A = 4π rEarth
= 2π rEarth
.
2
Clearly explain why for full credit.
2. [Variant of Harris 2.94] A kaon (denoted K0) is an unstable particle of
mass 8.87×10-28 kg. One of the means by which it decays is by
spontaneous creation of two pions, a π+ and a π–. The decay process may
be represented as K0 → π+ + π–. Both the π+ and π– have mass 2.49×10-28
kg. Suppose that a kaon moving in the +x direction decays by this process,
with the π+ moving off at speed 0.9c and the π- at 0.8c.
a. What was the speed of the kaon before the decay?
b. Argue that the two pions created by the decay must be less massive
than the kaon since there must be an increase in the kinetic energy
of the system during this decay. HINT: It may help to imagine you
are in a frame of reference moving with the kaon initially (so that it
is at rest). HINT #2: Or it may help you to examine your answer
to Harris 2.12.
c. Demonstrate that the two pions resulting from the decay of the
kaon can’t be restricted to the x-axis in their motion.
d. (Extra Credit) Recalling that the consequence of (c) is that the
pions resulting from the decay must have some component of their
momenta perpendicular to the x axis (let’s call it a y component), in
what directions do the pions move after the decay?
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Physics 322: Modern Physics
Spring 2015
3. [Harris 2.82 tweaked] A high-explosive material employing chemical
reactions has an explosive yield of 106 J/kg, measured in Joules of energy
released per kilogram of material.
a. By what fraction does its mass change when it explodes?
b. What is the explosive yield of a kilogram of material that produces
energy via nuclear reactions in which its mass decreases by 1 part in
10,000?
4. [Harris 3.3] You are conducting a photoelectric effect experiment by
shining light of 500 nm wavelength at a piece of metal and determining
the stopping potential. If, unbeknownst to you, your 500 nm light source
actually contained a small amount of ultraviolet light, would it throw off
your results by a small amount or by quite a bit? Explain.
5. [Harris 3.9] A low-intensity beam of light is sent toward a narrow single
slit. On the far side, individual flashes are seen sporadically at the
detectors over a broad area that is orders of magnitude wider than the slit
width. What aspects of the experiment suggest a wave nature for light,
and what aspects suggest a particle nature?
6. [Harris 3.11 (tweaked)] For small z, a Taylor’s series expansion reveals
! ez ≈ 1+ z .
a. Use this to show that Planck’s spectral energy density dU
hf
8π V
!
= hfkBT
× 3 f 2 (equation 3-1) df
e
−1 c
agrees with the result of the classical wave theory dU
8π V
!
= kBT 3 f 2 (listed before equation 3-1 in the book)
df
c
in the limit of small frequencies (e.g. low energy or long
wavelength).
b. Show that, whereas the classical formula diverges at high
frequencies (this is the so-called ultraviolet catastrophe of this
theory), Planck’s formula approaches 0. Explain why this is
significant!
– Page 2! of 3! –
Physics 322: Modern Physics
Spring 2015
7. [Harris 3.25] You are an early 20th-Century
Wavelength of
experimental physicist and do not know the value of
Light (nm)
Planck’s constant. By a suitable plot of the following
550
data (shown in the table), and using Einstein's
explanation of the photoelectric effect (KE = hf - φ
500
where h is not known), determine Planck’s constant.
450
NOTE: Just to clarify, if you want to use Excel or some
other plotting package, that is OK, but you better
400
clearly explain what you are doing!
Stopping
Potential (V)
8. [Harris 3.41 tweaked] A stationary muon µ– annihilates with a
stationary antimuon µ+ (same mass, 1.88×10-28 kg, but opposite charge).
The two disappear, replaced by electromagnetic radiation.
a. Clearly explain why is it not possible for a single photon to result?
HINT: Photons have momentum.
b. Suppose two photons result. Describe the possible directions of
motion and the wavelengths.
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0.06
0.286
0.563
0.908