Three bodies `A , B` and `C` have equal surface area and thermal emissivities in the ratio `e_(A) : e_(B) : e_(C) = 1 : (1)/(2): (1)/(4)`. All the three bodies are radiating at same rate. Their wavelengths corresponding to maximum intensity are `alpha_(A), alpha_(B)` and `alpha_(C)` respectively and their temperatures are `T_(A) , T_(B)` and `T_(C)` on kelvin scale , then select the incorrect statement

Options

A. `sqrt(T_(A)T_(C)) = T_(B)`
B. `sqrt(lambda_(A) lambda_(C)) = lambda_(B)`
C. `sqrt(e_(A) T_(A)) sqrt(e_(C)T_(C)) = e_(B)T_(B)`
D. `sqrt(e_(A)lambda_(A)T_(A).e_(B)lambda_(B)T_(B)) = e_(C)lambda_(C)T_(C)`

Answer

D

Explanation

For same power of radiation
`P_(A) = P_(B) = P_(C) implies e_(A)sigmaAT_(A)^(4) = e_(B)sigmaA T_(B)^(4) = e_(C)sigmaAT_(C)^(4)` & `lambda_(A)T_(A) = lambda_(B)T_(B) = lambda_(C)T_(C)(e_(A):e_(B):e_(C) = 1 : (1)/(2):(1)/(4))`
`implies sqrt(T_(A)T_(C)) = T_(B)` or `sqrt(e_(A) T_(A)) sqrt(e_(A)T_(C)) = e_(B)T_(B)` & `sqrt(lambda_(A) lambda_(C)) = lambda_(B)`