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STD 11 - 10.2. transmission of heat — NEET STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceNEETSTD 11 - 10.2. transmission of heat4 Marks
Question
The power radiated by a black body is $P$ and it radiates maximum energy at wavelength,$\lambda_0.$  If the temperature of the black body is now changed so that it radiates maximum energy at wavelength $\frac{3}{4}\lambda_0$, the power radiated by it becomes $nP$. The value of $n$ is
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Answer

From $Wien's$ law, ${\lambda _{\max }}T = constant$

So, ${\lambda _{{{\max }_1}}}{T_1} = {\lambda _{{{\max }_2}}}{T_2}$

$ \Rightarrow {\lambda _0}T = \frac{{3{\lambda _0}}}{4}T' \Rightarrow \frac{{T'}}{T} = \frac{4}{3}$                       $...(i)$

According to $Stefan-Boltzmann\,law$, energy emitted unit time by a blck body is $Ae\sigma T',i.e.,$ power radiated,

$\therefore \,\,P \propto {T^4}$

$So,\frac{{{P_2}}}{{{P_1}}} = {\left( {\frac{{T'}}{T}} \right)^4} \Rightarrow n = {\left( {\frac{4}{3}} \right)^4} = \frac{{256}}{{81}}$

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