Question

For blackbody radiation obeying Planck's distribution law show by direct computation that the energy fluctuation for radiation in a volume V in the spectral interval is given by Einstein's fluctuation formula:

(u,v + du)

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Answer 1

dw X²r3 ehW/KT-1 Solution for black body radiation obeying blanck's distribution law. weens law Planck's radiation law, energy deniety relation in terms of waretero frequency range is wand w+ dw is, uw, T) dw= w2 hw snergy fluctuation Sinetein fluctuation formula > let us feist teemmarije Eintein” Diginal denevaliny giren ((AE) >=thuvu vrhu uut (3 40²)dv 8 TV2 Einstein's formula fluctuations fromula for a Rystem of orcillators wees derived by lave by wing the planek - Bose détsébution By a simple calculation we obtain To cna, =(n>+ 2072 (0)2 daw ation Ana <ny average <n) + <n>². =<n;=<n>² neember deviation / Energy flucluatim dennity. Mulhu on? hexEv> + <Ev> where <£u) = Muhoin Mu degrees of freedom of the radiation field 2 2 ΔΕν 2 Μν

When + 1- which is (e hulkT1) blanckh formula. we would ou In Rayleigh Mu= no. of modes en volume w in the frequency interval of audth A du. we but ev and Mv in eq? 0 we end up with ( AE v) = (hu) Lehbekt-1 When we weed weens limet formula, have only got the first tam. Jeans lemit only the second terme come- let us take Einstein formula. and but relation one limit for ween formula and R.J. law one by one. the probability of finding exaclly N- photons en vis giren by binomial distribution WINDFNollar(W-1975 WIN)= N photons can be selected from the tolal number of No in a number of No! differentway O) ( ) ! take limits No ao and vo is such way Nolu - AVE (N> remain fiendte ps photon derity, and wiw) followes poisondistribution " then WIN)=re d=N Love thes-wolvo SC logo w because م ۔ a

for blanck's low <E) hut ZT hu KT Tehulkt 1] ANZ-XN2>-<N>= <N). So AE 2 = (hu)(AN)? DER E hule fo when all particles have the energy hu right side term looks exactly first term er o this consideration for partiele - like fluctuation, hun lev for virible light at room temperature KT no.025ev. he have hu ~40» I KT csi KT that is weens formula is good abbroximation and barticle like flucluctuation dominates ief for hu - Rayleigh limit Rayleigh Jeans low - u. Zudy.DE,? mu.DEIZ Ē u² tellows Ho Tott UVEZU.KT (AEU)? mu . this emphase bure ware like fluctuation