# 2063 lei de stefan boltzmann em 1884 o matematico

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esverdeada e azulada. 20.6.3 Lei de Stefan-Boltzmann Em 1884, o matem´atico austr´ ıaco Josef Stefan (1835-1893) e seu aluno na ´ epoca, o tamb´ em austr´ ıaco Ludwig Boltzmann (1844-1906), descobriram empiricamente que o fluxo (energia por unidade de ´area, por unidade de tempo) de um corpo negro de temperatura T ´ e dado por: F = 2 π Z π/ 2 0 cos θ sen θ dθ Z 0 B ν ( T ) = σT 4 onde σ = 5 , 67 × 10 - 5 ergs cm - 2 K - 4 s - 1 = 5 , 67 × 10 - 8 W m - 2 K - 4 ´ e a cons- tante de Stefan-Boltzmann. Essa lei pode ser demonstrada considerando que: B ( T ) Z 0 B ν = 2 h c 2 Z 0 ν 3 e kT - 1 205
= 10 7 ergs. Como o raio do Sol ´ e de R = 700 000 km, segue da equa¸c˜ ao (20.26) que a temperatura efetiva do Sol ´ e T ef = 5400 K. A energia que atinge a Terra por unidade de ´area e de tempo, por de- fini¸c˜ ao de fluxo, ´ e de: F = L 4 πr 2 onde r ´ e a distˆancia do Sol `a Terra, de 1 unidade astronˆomica (UA) = 150 milh˜oes de km. Portanto, a potˆ encia luminosa interceptada pela Terra, que tem uma sec¸c˜ ao reta πR 2 , onde R ´ e o raio da Terra, R = 6400 km, ´ e dada por: P = πR 2 F = πR 2

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