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Blackbody Radiation
The Death of Classical Physics II
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Points to Remember!
X-rays are produced when cathode rays (electrons),
l t d th h l t ti l difference collide
accelerated through a large potential diff llid
with a metal target in an evacuated glass tube.
The charge of an electron is quantized, coming in
integral multiples of a fundamental amount: e = 1.602 x
10-19 C
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Question
Bedouins are desert nomads
who wander the deserts of
Africa and the Middle East.
Why do Bedouins where black
clothing in the desert?
"Bedouin shepherdess. Gaza, Palestine,
[June 1956]." The World of Allah, p. 25.
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Blackbody Radiation
In 1792, Thomas Wedgewood, Charles Darwin’s relative
,
and a renowned maker of china, noted the universal
character of all heated objects to emit radiation. He
observed that all the objects in his ovens became red at
the same temperature, regardless of their size, shape,
or chemical nature.
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Blackbody Radiation
In 1859, Gustav Kirchhoff proved a theorem based on
y y y
thermodynamics that said for any body in thermal
equilibrium with radiation, the intensity of its emitted
power is proportional to the power it absorbs, given by:
power absorbed per unit area
per unit wavelength (eA ≤ 1)
ef = eAI(λ, T)
emitted power total power radiated per unit
area per unit wavelength λ at a
given temperature T.
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Blackbody Radiation
1) An ideal blackbody is an object that absorbs all
l t g ti di ti incident (e 1).
electromagnetic radiation i id t upon it ( A = 1)
2) A blackbody in thermal equilibrium with its
environment must emit as much radiation as it
absorbs.
The
3) Th power (i i ) f h itt d di ti
(intensity) of the emitted radiation
depends only on λ and T and not on the size, shape,
or chemical composition of the body.
ef = I(λ, T)
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Blackbody Radiation
A cavity such as a hollow
p
box or sphere with a small
hole in it can serve as a
blackbody. Any radiation
that enters the hole is
reflected inside the box
until most of it is eventually
absorbed. Only a small
fraction of the incident
radiation will be re-emitted
through the hole.
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Blackbody Radiation
A blackbody emits radiation over a
distribution of frequencies, such that:
λmax
1) Intensity (total power radiated)
intensity/u wavelength
increases with temperature
2) Peak wavelength varies
4000 K
inversely with temperature: the
unit
g p ,
higher the temperature, the
3000 K lower the peak wavelength.
2000 K
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wavelength
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Wien Displacement Law
In 1893, based on empirical observations, Wilhelm Wien
p p general form of I(λ, T) for blackbodies
in proposed a g ( , )
that provided the correct experimental behavior of the
peak wavelength λmax with temperature T:
λmaxT = 2.898 x 10-3 m K
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Emissivity
Emissivity ε is a value between 0 and 1 and is a
body s
measure of the body’s ability to emit and/or absorb
radiation.
0 ≤ ε ≤1
j y ,
Dark colored objects have emissivity near 1, while
light colored objects will have an emissivity closer to
0. A blackbody radiator is a perfect radiator having
emissivity equal to 1.
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Stefan-Boltzmann Law
In 1879, Josef Stefan found empirically and later
g y p
Ludwig Boltzmann found theoretically the power perp
unit area at temperature T emitted by a blackbody is:
Stefan-Boltzmann constant,
σ = 5.6705 x 10-8 W/m2K4
power per unit area
P(T) = εσT4 Stefan-Boltzmann Law
temperature
emissivity (ε = 1 for blackbodies,
ε < 1 for everything else)
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Example
When a small door is opened, a furnace is observed to
p q y
emit maximum peak radiation at a frequency of 1.94 x 10-
14 Hz.
a) What is the peak wavelength emitted (in nm)?
b) What is the temperature of the walls (in K)?
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Example
The wavelength of maximum intensity of the sun’s
radiation is observed to be near 500. nm. Assume the
sun to be a blackbody and calculate
a) the sun’s surface temperature,
b) the power per unit area R(T) emitted from the sun’s
surface, and
c) the energy received by Earth each day from the
sun s radiation.
sun’s radiation
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Blackbody Radiation
Wien’s displacement law only accurately described the
temperature,
peak wavelength at a given temperature but could not
account for the distribution of wavelengths at that
temperature.
Further attempts to understand blackbody radiation
failed to describe the whole spectrum of wavelengths
that were observed.
Theories that fit the higher wavelengths failed to
accurately describe the shorter wavelengths. Theories
that accurately described the shorter wavelengths failed
to encompass the higher wavelengths. 14
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Rayleigh-Jeans Formula
The best classical description of blackbody radiation
p y y g
was developed by Lord Rayleigh and combined
electromagnetism with thermodynamics to show:
2π ck BT
I (λ,T ) = Rayleigh-Jeans formula
λ4
where kB is the Boltzmann constant:
kB = 1.3807 x 10-23 J/K
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The Ultraviolet Catastrophe!
• The Wiens displacement law indicated that peak
g p
wavelength decreases with temperature. The
Rayleigh-Jeans formula predicted a continual
increase in radiated energy with decreasing
wavelength.
• Ergo, as the wavelength approaches 0, the intensity
of radiation approaches infinity!
• This boundless energy at infinitesimally small
wavelengths predicted by the Rayleigh-Jeans formula
was coined “The Ultraviolet Catastrophe.”
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The Ultraviolet Catastrophe!
1200 K
Empirical evidence did not
support the “ultraviolet
catastrophe”.
intensity/u wavelength
Rayleigh-Jeans
formula
unit
experimental
data
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wavelength
Planck’s Radiation Law
In 1900, Max Planck explained the
blackbody radiation spectrum by
y p y
postulating that the radiation was emitted
by oscillating atoms, and furthermore,
that the energy was quantized, coming in
integral multiples of a fundamental
energy hf.
2π c 2 h 1
I (λ,T ) = hc λ k BT
λ 5
e −1
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Planck’s Radiation Law
Planck arrived at his blackbody formula by making two
p
critical assumptions:
1) The energy of each oscillator of frequency f is an
integral number of hf.
n = 1, 2, 3, . . .
Energy of a single
oscillator En = nhf frequency
Planck’s
constant
where h = 6.6261 x 10-34 Js is Planck’s constant 19
Planck’s Radiation Law
Planck arrived at his blackbody formula by making two
critical assumptions:
p
2) Each oscillator can absorb or emit energy in
integral multiples of hf.
∆E = hf
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Points to Remember!
Blackbody radiation
1) The energy of each oscillator of frequency f is an
integral number of hf.
En = nhf
2) Each oscillator can absorb or emit energy in
integral multiples of hf.
∆E = hf
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