and so on to construct a spectrum in which ε λand α � vary continuously but are

And so on to construct a spectrum in which ε λand

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4.37Consider a closed spherical cavity in which the walls are opaque and all atthe same temperature. The surfaces on the top hemisphere are black andthe surfaces on the bottom hemisphere reflect all the incident radiation atall angles. Prove that in all directionsIλ=Bλ.Downward directed radiation is blackbody radiation for whichIλ=Bλby definition.Since the bottom hemisphere is a perfect reflector, alongeach ray path,Iλ=Iλand since the lower hemisphere is at the sametemperature as the upper hemisphere,Iλ=Iλ=Bλ.4.38(a) Consider the situation described in Exercise 4.35, except that bothplates are gray, one with absorptivityα1and the other with absorptivityα2.Prove thatF01α1=F02α2whereF01andF02are theflux densities of the radiation emitted from thetwo plates.Make use of the fact that the two plates are in radiativeequilibrium at the same temperature but do not make use of Kirchhoff’sLaw.[Hint:Consider the totalflux densitiesF1from plate 1 to plate2 andF2from plate 2 to plate 1.
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4.39Consider the radiation balance of an atmosphere with a large number ofisothermal layers, each of which is transparent to solar radiation and ab-sorbs the fractionαof the longwave radiation incident on it from aboveor below. (a) Show that theflux density of the radiation emitted by thetop-most layer isαF/(2α)whereFis theflux density of the planetaryradiation emitted to space. By applying the Stefan-Boltzmann law (4.12)
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