BLACK BODY RADIATIONBLACK  BODY RADIATION

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Black Body RadiationBlack Body Radiation

Black Body :‐ It absorbs all radiation it receivesBlack Body : It absorbs all radiation it receives

Practical Realization A I th l it ith ll t th hPractical Realization:  An Isothermal cavity with a small aperture through which radiation from out side can be admitted

ρ(υ) dν = energy per unit volume of cavity is contributedρ(υ) dν energy per unit volume of cavity is contributed by the radiation between ν and ν + d ν

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Black Body RadiationBlack Body Radiation

Black Body :‐ It absorbs all radiation it receivesBlack Body : It absorbs all radiation it receives

Practical Realization A I th l it ith ll t th hPractical Realization:  An Isothermal cavity with a small aperture through which radiation from out side can be admitted

ρ(υ) dν = energy per unit volume of cavity is contributed by the radiation between ν and ν + d ν

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Blackbody Radiation Spectrum

Energy densityin the windowυ and υ +d υ

ρ(υ)T=1400 K

T=1800 Kρ(υ)J/m -Hz3

T=1000 K

frequency υ4

Lord Rayleigh- James Jeans LawLord Rayleigh James Jeans LawNumber of independent standing waves in the frequency interval  

G (υ) dυ =  8πυ2 dυ / c3According to the theorem of equipartition 

L = 2(λ/2)g q p

energy , the average energy per degrees of freedom = ½ kT.  Each standing wave originates from oscillating electric charge  in the wall and  2 degrees of freedom is associated ith the

ε = kT2 degrees of freedom  is  associated with the wave . Hence average energy   L =λ/2

ε kT

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Ultrasonic Catastrophe

T=1800 K

( )T=1800 K

ρ(υ)

T=1400 KT=1000 K

frequency υ6

ρ(υ) T=1800 K

υ dependence2

frequency υ7

Planck’s Formula

when hυ << kT

Rayleigh Jeans law

when  hυ << kT

Rayleigh-Jeans law

when  hυ >> kTυ large

8Wien’s law ρ(v) dv = (8πv2/c3 )(hv)e-hv/kT dv

Planck’s Concept of Energy Quantization

14th December, 1900

νhE = νhE =

Max Karl Ernst Ludwig Planck1858‐1947 9

Planck’s postulateAny physical entity whose ``co-ordinate” is oscillating sinusoidally with frequency y can possess only total energies E as integral multiple of

h = Planck’s constant

42Classical

E = 0 10

εn = nhv, n = 0, 1, 2, 3,….r

No. of oscillators with energy εn is proportional to e-εn/ kT

. If there are N oscillators and E be the total energy, the average energy ε = E/N. Let there be No , N1 , N2 ,…Nr

ill h i 0 h 2h h d N Noscillators having energy 0, hv, 2hv ,…,rhv and Nr = No e-rhv/ kT

N = N + N + N + +NN = No + N1 + N2 + … +NrE= 0. No e-0/ kT + No hv. e-hv/ kT +No 2hv.e-2hv/ kT +………+

N rhv e-rhv/ kTNo rhv.e

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N = No (e-0/ kT + e -hv/ kT + e -2hv/ kT +………+ e -rhv/ kT )= N / ( 1- e-hv/ kT ) [ 1+x + x2 + x3 + …= 1 /(1-x) ] No / ( 1 e ) [ 1+x + x + x + … 1 /(1 x) ]

E = No hv. e-hv/ kT (1+ 2e-2hv/ kT +………+ re -(r-1)hv/ kT )

= No hv. e-hv/ kT {1/ (1- e-hv/ kT ) }2 [ 1 + 2x + 3 x2 +.. = 1/(1-x)2 ]

ε = E/N

ε = 

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Planck’s FormulaPlanck s Formula

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Wien’s displacement Law

Wien's displacement law states that the wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature except that each wavelength isas the distribution at any other temperature, except that each wavelength is displaced on the graph.

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