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Black Black BodyBody RadiationRadiation

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ContentsContentsMotivationMotivationDefinition of a BlackDefinition of a Black--BodyBodyBlackBlack--Body Radation LawsBody Radation Laws

11-- The Planck Law The Planck Law 22-- The Wien Displacement LawThe Wien Displacement Law33-- The StefanThe Stefan--Boltzmann LawBoltzmann Law44-- The RayleighThe Rayleigh--Jeans LawJeans Law

Application for Black Body Application for Black Body ConclusionConclusionSummarySummary

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MotivationMotivation

•• The black body is importance in thermal The black body is importance in thermal radiation theory and practice. radiation theory and practice.

•• The ideal black body notion is importance The ideal black body notion is importance in studying thermal radiation and in studying thermal radiation and electromagnetic radiation transfer in all electromagnetic radiation transfer in all wavelength bands. wavelength bands.

•• The black body is used as a standard with The black body is used as a standard with which the absorption of real bodies is which the absorption of real bodies is compared.compared.

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Definition of a black bodyDefinition of a black bodyA black body is an ideal body which allows the whole of the incident radiation to pass into itself ( without reflecting the energy ) and absorbs within itself this whole incident radiation (without passing on the energy). This propety is valid for radiation corresponding to all wavelengths and to all angels of incidence. Therefore, the black body is an ideal absorber of incident radaition.

Univ. of Oregon web site

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BlackBlack--Body Radiation Laws (1)Body Radiation Laws (1)11-- The RayleighThe Rayleigh--Jeans Law.Jeans Law.* It agrees with experimental * It agrees with experimental

measurements for long measurements for long wavelengths.wavelengths.

* It predicts an energy output that * It predicts an energy output that diverges towards diverges towards infinityinfinity as as wavelengths grow smaller.wavelengths grow smaller.

* The failure has become known * The failure has become known as the as the ultraviolet catastropheultraviolet catastrophe..

4

2),(λπλ ckTTI =

http://www.egglescliffe.org.uk/physics/astronomy/blackbody/Image22c.gif

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This formula also had a This formula also had a problem. The problem was problem. The problem was the term in the the term in the denominator.denominator.For large wavelengths it For large wavelengths it fitted the experimental data fitted the experimental data but it had major problems at but it had major problems at shorter wavelengths.shorter wavelengths.

Ultraviolet CatastrophUltraviolet Catastrophee

λ

4

2),(λπλ ckTTI =

http://theory.uwinnipeg.ca/users/gabor/foundations/quantum/images/slide5.gif

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22-- Planck LawPlanck Law-- We have two forms. As a function We have two forms. As a function

of wavelength.of wavelength.

And as a function of frequency And as a function of frequency

The Planck Law gives a distribution that The Planck Law gives a distribution that peaks at a certain wavelength, the peak peaks at a certain wavelength, the peak shifts to shorter wavelengths for higher shifts to shorter wavelengths for higher temperatures, and the area under the temperatures, and the area under the curve grows rapidly with increasing curve grows rapidly with increasing temperature.temperature.

BlackBlack--Body Radiation Laws (2)Body Radiation Laws (2)

http://scienceworld.wolfram.com/physics/PlanckLaw.html

1

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kTh

echTI ν

νν

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Comparison between Classical and Comparison between Classical and Quantum viewpointQuantum viewpoint

http://upload.wikimedia.org/wikipedia/commons/a/a1/Blackbody-lg.png

There is a good fit at long wavelengths, but at short wavlengths there is a major disagreement. Rayleigh-Jeans ∞, but Black-body 0.

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33-- Wein Displacement LawWein Displacement Law-- It tells us as we heat an object up, its It tells us as we heat an object up, its

color changes from red to orange to color changes from red to orange to white hotwhite hot..

-- You can use this to calculate the You can use this to calculate the temperature of stars.temperature of stars.

The surface temperature of the The surface temperature of the SunSun is is 5778 K, this temperature 5778 K, this temperature corresponds to a peak emission = corresponds to a peak emission = 502 nm = about 5000 Å.502 nm = about 5000 Å.

-- b is a b is a constant of proportionalityconstant of proportionality, , called Wien's displacement constant called Wien's displacement constant and equals 2.897 768 5(51) × 10and equals 2.897 768 5(51) × 10––33 m m K = 2.897768 5(51) × 10K = 2.897768 5(51) × 1066 nmnm K.K.

BlackBlack--Body Radiation Laws (3)Body Radiation Laws (3)

http://www.rumford.com/radiant/images/Wiengraph.gif

Tb

=maxλ

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44-- The StefanThe Stefan--Boltzmann LawBoltzmann Law* Gives the total energy being * Gives the total energy being

emitted at all wavelengths by emitted at all wavelengths by the blackbody (which is the the blackbody (which is the area under the Planck Law area under the Planck Law curve).curve).

* Explains the growth in the * Explains the growth in the height of the curve as the height of the curve as the temperature increases. Notice temperature increases. Notice that this growth is very abrupt.that this growth is very abrupt.

* * Sigma = 5.67 * 10Sigma = 5.67 * 10--88 J sJ s--11 mm--22 KK--44, , Known as the StefanKnown as the Stefan--Boltzmann constant.Boltzmann constant.

BlackBlack--Body Radiation Laws (4)Body Radiation Laws (4)4Tj σ=

http://ceos.cnes.fr:8100/cdrom-98/ceos1/science/dg/fig10.gif

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BlackBlack--Body Radiation Laws (5)Body Radiation Laws (5)

http://en.wikipedia.org/wiki/Image:RWP-comparison.svg

Comparison of Rayleigh-Jeans law with Wien's law and Planck's law, for a body of 8 mK temperature.

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Application for Black BodyApplication for Black Body-- The area of Earth's disk as viewed from space The area of Earth's disk as viewed from space

is, Area = πris, Area = πr22..-- The total energy incident on Earth is, Incident The total energy incident on Earth is, Incident

energy = (πrenergy = (πr22)S)Soo..-- The energy absorbed by the Earth/atmosphere The energy absorbed by the Earth/atmosphere

system, as viewed from space issystem, as viewed from space isAbsorbed energy = (πrAbsorbed energy = (πr22)S)Soo(1 (1 -- A). As we know A). As we know

that bodies must be in radiative equilibrium. that bodies must be in radiative equilibrium. The solar energy striking Earth's disk as The solar energy striking Earth's disk as viewed from space is reviewed from space is re--emitted as thermal emitted as thermal radiation by the surface of the entire globe, radiation by the surface of the entire globe, as described by the Stefanas described by the Stefan--Boltzmann Law, Boltzmann Law, Emitted energy = (4πrEmitted energy = (4πr22)σT)σT44..

-- Set the absorbed energy equal to the emitted Set the absorbed energy equal to the emitted energy:energy:

(πr(πr22)S)Soo(1 (1 -- A) = (4πrA) = (4πr22)σT)σTEE44, Solving for T yields:, Solving for T yields:

TTEE = [S= [Soo(1 (1 -- A)/(4σ)]A)/(4σ)](1/4)(1/4)

= [1370•(1= [1370•(1--0.3)/(4•5.67x100.3)/(4•5.67x10--88)])](1/4)(1/4) = 255 K.= 255 K.

http://soer.justice.tas.gov.au/2003/image/378/atm_structure/o_atm_Radn_Balance-m.jpg

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ConclusionConclusionAs the temperature As the temperature increases, the peak increases, the peak wavelength emitted by the wavelength emitted by the black body decreases.black body decreases.As temperature increases, As temperature increases, the total energy emitted the total energy emitted increases, because the increases, because the total area under the curve total area under the curve increases.increases.The curve gets infinitely The curve gets infinitely close to the xclose to the x--axis but axis but never touches it.never touches it.

http://www.astro.ufl.edu/~oliver/ast3722/lectures/BasicDetectors/BlackBody.gif

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SummarySummary-- A black body is a theoretical object that absorbs 100% of thA black body is a theoretical object that absorbs 100% of the e

radiation that hits it. Therefore it reflects no radiation and radiation that hits it. Therefore it reflects no radiation and appears perfectly black.appears perfectly black.

-- Roughly we can say that the stars radiate like blackbody Roughly we can say that the stars radiate like blackbody radiators. This is important because it means that we can use radiators. This is important because it means that we can use the theory for blackbody radiators to infer things about stars.the theory for blackbody radiators to infer things about stars.

-- At a particular temperature the black body would emit the At a particular temperature the black body would emit the maximum amount of energy possible for that temperature.maximum amount of energy possible for that temperature.

-- Blackbody radiation does not depend on the type of object Blackbody radiation does not depend on the type of object emitting it. Entire spectrum of blackbody radiation depends on emitting it. Entire spectrum of blackbody radiation depends on only one parameter, the temperature, T.only one parameter, the temperature, T.