486 Introduction to Quantum PhysicsThus, λmax...T=×⋅××=×⋅−−−6 626 075 102 997 925 104 965 115 1 380 658 102897755 10348233JsmsJKmKej.This result is very close to Wien’s experimental value of max.T⋅−2898 103m K for this constant.P40.63(a)Planck’s radiation law predicts maximum intensity at a wavelength maxwe find fromdIdddhcehceehckThcehck Thck Thck Thck Tλλπλ==−RSTUVW=−−−FHGIKJ+−−−−−−−−0211125125122261BBBBBbgafor−−+−=hcekTeehck Thck Thck TBBBB7261510which reduces to51hceehck Thck TBFHGIKJ−=.Define xhc=B.Then we require 55exexx.Numerical solution of this transcendental equation gives x=4965.to four digits. Somax.=hcB, in agreement with Wien’s law.The intensity radiated over all wavelengths is ITdABhc dehck T,025021∞∞zz=+=−B.Again, define xhc=Bso =hcxk TBand dhcxkTdx=−2B.Then, ABhc x k T hcdxhcxkTehcxdxexxx+=−−=−=∞∞2121255 555 2043230ππBBB4.The integral is tabulated as π415, so (in agreement with Stefan’s law)
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