Unformatted text preview: D = 150 ∗ 106 km, and Ts = Quiz 10
5800 K. How does this compare to the observed value
of about 300 K? Fall 2006
Due in section week of March 30, 2008 b) For what wavelength is the power output of the Sun maximum? What about for the
Earth (assuming Te = 300 K)?
c) You The formula in Tipler the Earth is a cold place with the Sun alone. The situation is
Problem 1 should have found thatfor Compton scattering relates to the photon which scatters oﬀ
signiﬁcantly worsened by the fact that much of the
an electron. Suppose, however Greenhouse Eﬀect Sun’s light is about the electron. In
1. Blackbody Radiation and the that we are interested in information reﬂected at the surface
particular, assume why aren’t is traveling down an ice age? In at 10 keV, and it collides While
of the Earth, so an electron we in a permanent xray tube short, the atmosphere. with
a it eV photon headon. Thefor the is reﬂected for incoming T+y comes from the Sun, and
2 is good temperature to assume that most 90 to the e , direction (relative Sun
Call theasurfaceapproximationphotonSun Ts , thatoﬀ atthe ◦Earthheat the distance to the to the
original path).the radius of the Earth Re , and the (after all are maximum see whybothdon’t
from Earth D,of What angle is the electron reﬂected radius of the Sun Rs . to in the visibleas
that most
this gets through the atmosphere to? (You its suppose Modeling we and
worry eyes answer theradiation dispersing cathode fact (Hint: you should understand the
blackbodies, have evolved to take advantage of the rays).that visible light gets through), its
our about ambient following:
derivation of the Compton formula to answer this question See other problem. Earth (which is
not a good approximation per area output for the the heat 4 , ﬁnd a the One suggestion
a)to boost to the rest frame of to assume that most ofSun is σTs leaving naive estimate
Using
is max at that the total power the electrongets then apply compton’s formula for the the for
a much diﬀerent wavelength) and through the atmosphere. If all of photon.)
Sun’s
the temperature of the Earth by doing the following steps:
light gets through the atmosphere, what fraction f of the radiation power coming from
(i.) Calculate the total power leaving the Sun.
Earth must escape the atmosphere for the Earth to have its observed tempurature (again
(ii.) Calculateall of the Sun’sthis that absorbed at the Earth’s surface)?
assuming that the fraction of light is is incident on the Earth.
(iii.) Calculate the total power leaving the Earth if its temperature is Te .
(iv.) Equate the power incident on the Earth from the Sun to the power leaving the Earth
to ﬁnd Te . (They Eﬀect
2. Derivation of Compton should be equal if the Earth is in equilibrium assuming the Earth is
a perfect absorber of the Sun’s light.) Use the values Rs = 7∗105 km, Re = 6400 km,
6
D = wavelength i moving in the How does this compare to an electron at rest
A photon with 150 ∗ 10 km,λand Ts = 5800 K. +ˆdirection collides withthe observed value
x
of about 300 K?
with mass m and scatters in the xyplane. If the angle that the photon is scattered (above the
xaxis) Forθwhat the photon’s ﬁnal power output of fthe Sun maximum? for howabout for the
ˆ b) is and wavelength is the wavelength is λ , derive a formula What the wavelength
Earth changes T = 300 K)?
of the photon(assuming ineterms of m and θ by following the steps below:
c) You should have found that the Earth is a cold place with the Sun alone. The situation is
a) Call φ the angle that the electron scatters below the xaxis, pi the magnitude of the
ˆ
signiﬁcantly worsened by the fact that much of the Sun’s light is reﬂected at the surface
photon’sEarth, so why aren’t we in a permanent ice age? In short,and atmosphere. While of
initial momentum, pf the photon’s ﬁnal momentum, the pe the momentum
of the
the electron after the collision.assume that formulas for momentum conservation (in both
it is a good approximation to Write the most incoming heat comes from the Sun, and
xthat most of this gets through the atmosphere (after all its maximum, ine the visible and
ˆ and y directions) and energy conservation in terms of θ, φ, pi , pf p , and m.
ˆ
our eyes have evolved to take advantage of the fact that visible light gets through), its
not a good approximation to assume that most of the heat leaving the Earth (which is
1
max at a much diﬀerent wavelength) gets through the atmosphere. If all of the Sun’s
light gets through the atmosphere, what fraction f of the radiation power coming from
Earth must escape the atmosphere for the Earth to have its observed tempurature (again
assuming that all of the Sun’s light is absorbed at the Earth’s energy
b) Use your formula for energy conservation to write the electronsurface)? after the collision in terms of pi , pf and m, and your formulas for momentum conservation to write the
electron momentum components, p cos φ and pe sin φ, in terms of pi , pf and pe .
2. Derivation of Compton Eﬀect e
c) You should now have 3 equations so we can in principle use 2 to eliminate the intermediate
A photon with pe in the λi moving in the +ˆdirection collides with an electron prevent
variables φ andwavelength third, which wouldxthen be the desired equation. Toat rest
with mass algebra, the in the xyplane. If the angle that use that
formidablem and scatters following trick may be helpful: the photon is scattered (above the
xaxis) is θ and the photon’s ﬁnal wavelength is λf , derive a formula for how the wavelength
ˆ
of the photon changes)in = E 2 − (pe c)2 = E 2 followingcos φ)2 + (pe c sin φ)2 )
(mc2 2 terms of m and θ by − ((pe c the steps below:
a) Call φ the angle that the electron scatters below the xaxis, pi the magnitude of the
ˆ for the electron and substitute pyour formulas ﬁnal E, pe cosφ, and p e sinφ from partof
photon’s initial momentum, f the photon’s for momentum, and pe the momentum b).
There will be many terms but most either formulas for momentum conservation (inyou are
the electron after the collision. Write the cancel or simplify. Show that what both
left with after the dust settles can conservationas terms of θ, φ, pi , pf , pe , and m.
x and y directions) and energy be written in
ˆ
ˆ
m(pi c − pf c) = pi pf (1 − cos θ)
11 .
d) Finally, write the photon’s initial momemtum pi in terms of λi and its ﬁnal momentum
pf in terms of λf . Show that with these substitutions, Compton’s formula is obtained:
λf − λi = h
(1 − cos θ)
mc Quiz 10
Fall 2006 1. Blackbody Radiation and the Greenhouse Eﬀect
Call the surface temperature for the Sun Ts , that for the Earth Te , the distance to the Sun
from Earth D, the radius of the Earth Re , and the radius of the Sun Rs . Modeling both as
blackbodies, answer the following:
4
a) Using that the total power per area output for the Sun is σTs , ﬁnd a naive estimate for
the temperature of the Earth by doing the following steps: (i.)
(ii.)
(iii.)
(iv.) Calculate the total power leaving the Sun.
Calculate the fraction of this that is incident on the Earth.
Calculate the total power leaving the Earth if its temperature is Te .
Equate the power incident on the Earth from the Sun to the power leaving the Earth
to ﬁnd Te . (They should be equal if the Earth is in equilibrium assuming the Earth is
a perfect absorber of the Sun’s light.) Use the values Rs = 7∗105 km, Re = 6400 km,
D = 150 ∗ 106 km, and Ts = 5800 K. How does this compare to the observed value
of about 300 K? b) For what wavelength is the power output of the Sun maximum? What about for the
Earth (assuming Te = 300 K)?
c) You should have found that the Earth is a cold place with the Sun alone. The situation is
signiﬁcantly worsened by the fact that much of the Sun’s light is reﬂected at the surface
of the Earth, so why aren’t we in a permanent ice age? In short, the atmosphere. While
it is a good approximation to assume that most incoming heat comes from the Sun, and
that most of this gets through the atmosphere (after all its maximum in the visible and
our eyes have evolved to take advantage of the fact that visible light gets through), its
not a good approximation to assume that most of the heat leaving the Earth (which is
max at a much diﬀerent wavelength) gets through the atmosphere. If all of the Sun’s
light gets through the atmosphere, what fraction f of the radiation power coming from
Earth must escape the atmosphere for the Earth to have its observed tempurature (again
assuming that all of the Sun’s light is absorbed at the Earth’s surface)?
2. Derivation of Compton Eﬀect
A photon with wavelength λi moving in the +ˆdirection collides with an electron at rest
x
with mass m and scatters in the xyplane. If the angle that the photon is scattered (above the
xaxis) is θ and the photon’s ﬁnal wavelength is λf , derive a formula for how the wavelength
ˆ
of the photon changes in terms of m and θ by following the steps below:
a) Call φ the angle that the electron scatters below the xaxis, pi the magnitude of the
ˆ
photon’s initial momentum, pf the photon’s ﬁnal momentum, and pe the momentum of
the electron after the collision. Write the formulas for momentum conservation (in both
x and y directions) and energy conservation in terms of θ, φ, pi , pf , pe , and m.
ˆ
ˆ
1 ...
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 Spring '08
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 Quantum Physics, Light, Compton

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