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Motion In Straight Line

Question
CBSEENPH11020766

Assuming the sun to have a spherical outer surface of radius r, radiating like a black body at a temperature  to C, the power received by a unit surface (normal to the incident rays) at a distance R from the centre of the sun is :

  • fraction numerator 4 πr squared σt to the power of 4 over denominator straight R squared end fraction
  • fraction numerator straight r squared straight sigma left parenthesis straight t plus 273 right parenthesis to the power of 4 over denominator 4 πR squared end fraction
  • fraction numerator 16 space straight pi squared straight r squared σt to the power of 4 over denominator straight R squared end fraction
  • fraction numerator straight r squared straight sigma left parenthesis straight t plus 273 right parenthesis to the power of 4 over denominator straight R squared end fraction

Solution

D.

fraction numerator straight r squared straight sigma left parenthesis straight t plus 273 right parenthesis to the power of 4 over denominator straight R squared end fraction

From Stefan's law, the rate at which energy is radiated by sun at its surface si 
P = σ x 4 πr2T4

[Sun is a perfectly black body as it emits radiations of all wavelengths and so for it e =1]
The intensity of this power at earth's surface (under the assumption R >>ro) is
straight I space equals space fraction numerator straight P over denominator 4 space πR squared end fraction
space equals space fraction numerator straight sigma space straight x space 4 space πr squared straight T to the power of 4 over denominator 4 πR squared end fraction
space equals space fraction numerator σr squared straight T to the power of 4 over denominator straight R squared end fraction
equals space fraction numerator σr squared left parenthesis straight t space plus 273 right parenthesis squared over denominator straight R squared end fraction