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What is the nature of
Part I
The invention of radio?
Hertz proves that light is really an electromagnetic wave.
Waves could be generated in one circuit, and electric pulses with the same
frequency could be induced in an antenna some distance away.
These electromagnetic waves could be reflected, and refracted, focused,
polarized, and made to interfere—just like light!
The mystery:
In 1792, the famous china-maker, Thomas Wedgewood,
noticed that all hot objects became red at the same
temperature independent of size, composition, etc…
Now, for a bit background of thermodynamics….
You may have noticed yourself that things glow (i.e. emit light) when they heat up.
your
stove…
Unlike gases, solids do not have
characteristic lines.
a glassblower
blowing
glass…
the sun
and
other
stars.
Teaser: Why these characteristic
lines? You’ll find out shortly.
Definition of a blackbody…
An object that absorbs all radiation falling on it and therefore appears
...well...black. (It does not reflect.)
emitted power
per unit
frequency
e  J ( , T ) A
fraction of incident
power absorbed per unit
area per unit frequency
a smooth function
that is independent
of the material
Since A=1, a perfect blackbody is also a perfect radiator. This
also follows from the second law of thermodynamics.
The energy radiated by a blackbody is created
in the random thermal motions of the atoms
and electrons in the material itself. Before this
radiation reaches the surface and is emitted, it
has been absorbed and reemitted many times
within the material which “washes out” the
original spectral information. Therefore the
spectral emittance is sculpted into a smooth
function of the wavelength, depending only on
the temperature.
The Second Law of Thermodynamics
Heat will not flow spontaneously from a cold object to a hot object.
A
B
Consider two “blackbody cavities” at equal
temperature with their openings facing each
other. They are radiating into each other.
•If the flux emitted by A were larger than B
then the temperature of A would decrease and
B would increase.
•If the flux emitted by B were larger then the
temperature of B would decrease and the
temperature of A would increase.
The energies emitted by both cavities must be
the same.
Now put a wavelength filter at the opening between the cavities and do the same
mental exercise.
The fluxes for a particular wavelength interval must therefore be the same.
Therefore the flux per unit area and the flux per unit wavelength for a blackbody is a
function of temperature alone, and is independent of the material, size, etc.
Here T is the absolute
temperature (temperature
in Kelvins).
S=
More generally:
As temperature increases, so
does frequency of peak.
a
where a=1 for a blackbody
This is related to the energy
density by:
Wien’s displacement law
These were originally an experimental results.
So what is this function and what does it mean?
It is the job of physicists to ask these questions because a
full understanding can either confirm what we know or
expose some physical laws that were not previously
apparent.
a valiant attempt by Lord Rayleigh
When logic leads you to the wrong conclusion…
Asumption A: An electromagnetic wave in a cavity must have zero electric field at
the wall.
Consequence: you get a resonating cavity with standing modes. The shorter
the wavelength the more ways it can fit into the cavity.
As the frequency gets lower, the number of
modes that will fit into the cavity gets smaller
Assumption B: by the equipartition theorem, each mode has an average thermal
energy of kT.
Equipartition theorem: the idea that the mean energy of the molecules of a gas is equally divided
among the various degrees of freedom of a molecule.
The number of modes per unit frequency was calculated to be:
8v 2
N ( )dv  3 d
c
The spectral energy is therefore calculated to be:
8v 2
u ( )dv  3 kTd
c
energy per unit
volume per unit
frequency
It becomes infinite at short wavelengths!
Starting with logic similar to Rayleigh, you can count the classical modes and the spectral
energy density can be written as
8v 2
u ( )dv  3 E dn
c
where dn is the number of modes of vibration
within a frequency d and E is the average
energy of an oscillator of frequency 
Planks requires: In an oscillator of frequency , the only permitted values
of the energy are:
E  nh  0, h ,2h ,3h ,...
So what does this do to the probability that a particle will be in a particular energy
state? (Does this eliminate the ultraviolet catastrophe?)
E   nh  (probabilit y of nh )
average energy of an oscillator
n
the probability that an oscillator is in a state nhv is
proportional to the Boltzmann factor:
e
E
kT
e
 nhv
kT
 1/ e
Here, kT is the thermal energy that is available.
nh
kT
Maxwell-Boltzmann distribution
As the energy of a state, hv, gets larger
compared to the amount of thermal energy
available, kT, the probability that the state
is occupied trends toward zero.
The ultraviolet catastrophe is avoided!
Average energy of an oscillator:
h
E
e
h
kT
1
To find the spectral energy, multiply by the density of modes:
Planck’s Law
8v 2
hv
u ( , T )  3 h
c e kT  1
8v 2
u ( )dv  3 E dn
c
Does it work, and is it in agreement with some thermodynamic principles that
have been shown to be true, namely Wein’s displacement law, and the StefanBoltzmann law?
Stefan-Boltzmann’s law:


0
u ( )dv  
8v 2
hv
2 5 k 4 4
dv 
T
h
3
3 2
c e kT  1
15h c

0
Wein’s displacement law:
To find the maximum frequency, differentiate and find extrema by setting the result =0
vmax  2.822
Limiting behaviors: hv/kT>>1
1
e
hv
kT
1
Limiting behaviors: hv/kT<<1
e
hv
kT
1
kT
h
1


e
hv
kT
1
 1 1  hv  ...  1
kT

kT
hf
So now we’ve found evidence of the quantization of
energy.
What does that say about the nature of light?
Tune in next time…
Credit: Many of the figures in this lecture came from
http://hyperphysics.phy-astr.gsu.edu/. Check it out.
Star classifications:
Evidence for the “Big Bang”:
1978 Nobel Prize for Penzias and Wilson
The COBE satellite was
developed by NASA's Goddard
Space Flight Center to
measure the diffuse infrared
and microwave radiation from
the early universe to the limits
set by our astrophysical
environment. It was launched
November 18, 1989
2006 Nobel prize to Mather for measuring the CMB blackbody spectrum
(announced 1990)