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Question
Compare the rates of emission of heat by a blackbody maintained at 727°C and at 227°C, if the black bodies are surrounded by an enclosure (black) at 27°C. What would be the ratio of their rates of loss of heat?
Solution
Data: T1 = 273 + 727 = 1000 K,
T2 = 273 + 227 = 500 K,
T0 = 273 + 27 = 300 K
(i) The rate of emission of heat, `"dQ"/"dt" = sigma"AT"^4`.
We assume that the surface area A is the same for the two bodies.
`∴ ("dQ"//"dt")_1/("dQ"//"dt")_2 = "T"_1^4/"T"_2^4 = ("T"_1/"T"_2)^4`
`= (1000/500)^4 = 2^4 = 16`
(ii) The rate of loss of heat, `"dQ'"/"dt" = sigma"A" ("T"^4 - "T"_0^4)`
`∴ ("dQ'"//"dt")_1/("dQ'"//"dt")_2 = ("T"_1^4 - "T"_0^4)/("T"_2^4 - "T"_0^4)`
`= (10^12 - 81 xx 10^8)/(625 xx 10^8 - 81 xx 10^8)`
`= ((10000 - 81) xx 10^8)/(544 xx 10^8)`
`= 9919/544 = 18.23`
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