Download Planck`s Law

BLACKBODY
RADIATION:
PLANCK’S LAW
COLOR and SPECTRAL CLASS
• The light emitted by stars consists of a mixture of
all colors, but our eyes (and brain) perceive such
light as being white or tinged with pastel color.
• In fact, different stars have varying amounts of
each color in their light; this causes stars to have
different colors.
• Most people, however, have never noticed that
stars come in a variety of colors.
• When light from the Sun (or any other star) is
passed through a prism, it is separated into its
component colors -- a continuous spectrum.
When a beam of white light is passed
through a prism, it is broken up into a
rainbow-like spectrum.
COLOR and SPECTRAL CLASS
• If the spectra of different stars are analyzed, it is
found that the intensity of the various colors differs
from star to star.
• Relatively cool stars have their peak intensity in
the red or orange part of the spectrum.
• The hottest stars emit blue light most strongly.
• In other words, the color (or wavelength, ) of the
maximum intensity depends upon the temperature
of the star.
• The star is not necessarily the color of the maximum intensity; in fact, there are no green stars.
Max
Karl
Ernst
Ludwig
Planck
1858 - 1947
Max Planck
1858 - 1947
• In the late 1890’s, Wien and
Rayleigh had unsuccessfully
attempted to formulate an
equation expressing the
intensity of electromagnetic
radiation as a function of
wavelength and the
temperature of the source.
• In 1900, Planck derived the
equation empirically.
• By December of 1900,
Planck had derived the
equation from fundamental
principles.
Planck’s Law
Intensity of Radiation vs. Wavelength
2 hc
1
I(  ) 

5
hc / kT

e
1
2
The intensity (I) of electromagnetic radiation at a
given wavelength () is a complicated function of
the wavelength and the temperature (T).
Planck’s Law
Intensity of Radiation vs. Wavelength
#
#
#
#
nm
400
450
500
550
3000
2.27E+11
4.77E+11
8.18E+11
1.21E+12
3.7415343E-16
#
#
600
650
1.63E+12 7.83E+13 4.81E+14
2.02E+12 7.26E+13 3.96E+14
At 400 nm: 2hc2 / 5
3.65384E+16
#
700
2.36E+12 6.63E+13 3.27E+14
At 400 nm, 3000 K: hc / kT
11.9894485
Planck's Constant (h):
Speed of Light (c):
Pi ():
Boltzmann's Constant (k):
6.6262E-34
2.9978E+08
3.1415927E+00
1.38066E-23
2hc2:
exp(hc / kT)
161046.5126
exp(hc / kT) - 1
161045.5126
I()
2.26883E+11
#
400
450
500
550
600
650
700
3000
0.113441
0.238543
0.40892
0.6073
0.813035
1.007968
1.179203
5800
7.42E+13
8.22E+13
8.45E+13
8.27E+13
5800
0.927449
1.027196
1.055752
1.033287
0.978845
0.907138
0.828339
10000
1.03E+15
8.64E+14
7.14E+14
5.86E+14
10000
1.029757
0.864157
0.713981
0.586331
0.481167
0.395804
0.326925
Planck’s Law
Radiation Intensity vs. Wavelength at 3000oK
(Note Peak in Infrared)
3.50E+12
3.00E+12
2.50E+12
2.00E+12
1.50E+12
3000
1.00E+12
5.00E+11
00
10
0
90
0
80
0
70
0
60
0
50
0
40
0
30
0
20
10
0
0.00E+00
Planck’s Law
Radiation Intensity vs. Wavelength at 6000oK
(Note Peak in Visible)
1.2E+14
1E+14
8E+13
6E+13
6000
4E+13
2E+13
1000
950
900
850
800
750
700
650
600
550
500
450
400
350
300
250
200
150
100
0
Planck’s Law
Radiation Intensity vs. Wavelength at 10000oK
(Note Peak in Ultraviolet)
1.4E+15
1.2E+15
1E+15
8E+14
6E+14
10000
4E+14
2E+14
00
10
0
90
0
80
0
70
0
60
0
50
0
40
0
30
0
20
10
0
0
Planck’s Law
Actual Radiation Intensity vs. Wavelength at
3000, 6000, and 10000oK
1.40E+15
1.20E+15
1.00E+15
8.00E+14
3000
6000
6.00E+14
10000
4.00E+14
2.00E+14
85
0
90
0
95
0
10
00
10
0
15
0
20
0
25
0
30
0
35
0
40
0
45
0
50
0
55
0
60
0
65
0
70
0
75
0
80
0
0.00E+00
Planck’s Law
Intensity of Radiation vs. Wavelength;
Normalized Intensity vs. Wavelength
Pi ():
Boltzmann's Constant (k):
3.1415927E+00
1.38066E-23
2hc 2:
3.7415343E-16
At 400 nm: 2hc 2 / 5
3.65384E+16
At 400 nm, 3000 K: hc / kT
11.9894485
exp(hc / kT)
161046.5126
exp(hc / kT) - 1
161045.5126
I()
2.26883E+11
#
450
500
550
600
650
700
4.77E+11
8.18E+11
1.21E+12
1.63E+12
2.02E+12
2.36E+12
8.22E+13
8.45E+13
8.27E+13
7.83E+13
7.26E+13
6.63E+13
8.64E+14
7.14E+14
5.86E+14
4.81E+14
3.96E+14
3.27E+14
400
450
500
550
600
650
700
3000
0.113441
0.238543
0.40892
0.6073
0.813035
1.007968
1.179203
5800
0.927449
1.027196
1.055752
1.033287
0.978845
0.907138
0.828339
10000
1.029757
0.864157
0.713981
0.586331
0.481167
0.395804
0.326925
Planck’s Law
Normalized Intensity vs. Wavelength
at 3000, 6000, and 10000 oK
1.2
1
0.8
3000
0.6
6000
0.4
1000
0
0.2
90
0
10
00
80
0
70
0
60
0
50
0
40
0
30
0
20
0
10
0
0
Planck’s Law
Normalized Radiation Intensity vs.
Wavelength at Various Temperatures
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
3000
1.78E-14
2.05E-08
1.44E-05
0.000572
0.005623
0.02553
0.072602
0.152667
0.261709
0.388672
0.520343
0.645099
0.75469
0.84446
0.912762
0.960102
0.988322
0.999951
0.997754
4000
6.82E-10
1.45E-05
0.001377
0.016459
0.072761
0.18665
0.345935
0.52148
0.685109
0.818679
0.914758
0.973641
0.999929
0.999935
0.980096
0.946169
0.902917
0.854084
0.802489
5000
2.97E-07
0.000573
0.016459
0.095838
0.262282
0.47775
0.685111
0.846308
0.948629
0.996081
0.999937
0.972733
0.925454
0.866649
0.802491
0.737185
0.673439
0.612884
0.556407
NORMALIZED INTENSITIES
6000
7000
8000
1.45E-05 0.0002056 0.001377
0.005636 0.0255825 0.072758
0.072763 0.1866222 0.345919
0.262286 0.4776782 0.685077
0.52149 0.7562274 0.914714
0.756354 0.9326971 0.999882
0.914775 0.9997807 0.980049
0.990501 0.9864388 0.902875
0.999954 0.9253141 0.802451
0.964589 0.8412268 0.698591
0.902934 0.7500772 0.601214
0.828503 0.6609354 0.51441
0.750203 0.5783871 0.439165
0.67345 0.5043227 0.374956
0.601254 0.4391209 0.320629
0.535067 0.3823763 0.274852
0.475385 0.3333228 0.236329
0.422145 0.2910713 0.203895
0.374981 0.2547345 0.176541
9000
0.00569
0.154473
0.526496
0.854446
1.000009
0.996075
0.911602
0.797008
0.679915
0.572807
0.479948
0.401652
0.336579
0.28286
0.238614
0.202153
0.17204
0.147091
0.126338
10000
0.016462
0.262322
0.685218
0.948777
1.000093
0.925598
0.802616
0.673544
0.556494
0.457
0.375033
0.308499
0.254813
0.211535
0.176578
0.148237
0.125154
0.106255
0.090697
Planck’s Law
Normalized Radiation Intensity vs.
Wavelength at Various Temperatures
1.2
1
3000
0.8
4000
5000
0.6
6000
7000
8000
0.4
10000
0.2
0
100
150 200
250
300
350 400
450
500
550 600
650
700 750
800
850
900 950 1000
Stefan-Boltzmann Law
ET =  T4
where ET = total energy radiated per unit area
over all wavelengths,
and  = 5.67051  10-12 J / cm2 s K4
1.4E+15
1.2E+15
1E+15
8E+14
ET
6E+14
4E+14
2E+14
00
10
0
95
0
90
0
85
0
80
0
75
0
70
0
65
0
60
0
55
0
50
0
45
0
40
0
35
0
30
0
25
0
20
0
15
10
0
0
Wilhelm
Carl
Werner
Otto
Fritz
Franz
Wien
1864 - 1928
Wilhelm Wien
1864 - 1928
• In 1896, Wilhelm Wien
unsuccessfully attempted to
derive what is now known
as Planck’s Law.
• However, he did notice a
relationship between the
temperature of a glowing
object and the wavelength
of its maximum intensity of
emission.
• The result of his
investigation is now known
as Wien’s Displacement
Law.
Wien’s
Displacement
Law:
The peak of the
emission spectrum
of a glowing object
is a function of its
temperature. The
hotter the object,
the shorter the
peak wavelength.
Wien’s Displacement Law
Gives max as f(T), which allows us to calculate
the temperature of a star if we know the
wavelength of its maximum emission, which
is easy to measure from its spectrum.
From Planck’s Law, take dI/dset = 0.
Then, maxT = 2.8979  106 nmK.
Example: max for the Sun = 502 nm.
Therefore, T = 5770K = 5500C.
The three types of Spectra:
Continuous, Emission Line, and Absorption Line
Sodium Absorption Lines:
The sodium vapor “subtracts out” the yellow lines
from the continuous spectrum emitted by the
source.
As an excited hydrogen atom returns to its ground
state, it emits the extra energy in the form of a
photon with a certain wavelength.
Each energy
transition
within an
atom gives
rise to a
photon of a
particular
wavelength.
Solar Spectrum
(Original Drawings by Fraunhofer)
Absorption
lines in a
star’s
spectrum
reveal the
presence of
elements
and
compounds.
Continuous
Spectrum
Absorption
Spectrum
of the Sun
Bright-line
Spectrum
of Sodium
Bright-line
Spectrum
of Hydrogen
Bright-line
Spectrum
of Calcium
Bright-line
Spectrum
of Mercury
Bright-line
Spectrum
of Neon
The “Inverse Square” Law: When light from a point
source travels twice as far, it covers four times the
area, and is therefore only one fourth as bright.
THE END