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A black body at $200K$ is found to emit maximum energy at a wavelength of $14\mu m$. When its temperature is raised to $1000K$, the wavelength at which maximum energy is emitted is
A. $14\mu m$
B. $70\mu m$
C. $2.8\mu m$
D. $28\mu m$

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Last updated date: 17th Jul 2024
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Answer
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Hint:We can use the Wien displacement law to find the wavelength at maximum energy ${\lambda _m}T = cons\tan t$. Where ${\lambda _m}$is the wavelength of maximum energy and T be the temperature. Blackbody is the surface that absorbs all the radiation falling on it that’s why it has the maximum energy.

Complete step by step answer:
Applying Wien displacement law: $\lambda T = $constant
$\lambda $ is the wavelength of maximum energy
$T$ is the absolute temperature
Using the given values from the question for black body
${\lambda _{m1}}{T_1} = {\lambda _{m2}}T{}_2$
Where we had taken ${\lambda _{m1 = }}14\mu m$, ${T_1} = 200K$, ${T_2} = 1000K$. ${\lambda _{m2}} = $?
Substituting the values
$14 \times 200 = {\lambda _{m2}} \times 1000$
$\Rightarrow{\lambda _{m2}} = \dfrac{{14 \times 200}}{{1000}}$
$\therefore{\lambda _{m2}} = 2.8\mu m$

Hence, the correct answer is C.

Additional information:
Wein displacement was named after the scientist Wilhelm Wien in the year $1893$ which states that black body radiation curve for different wavelengths will peak at different wavelengths that are inversely proportional to temperature. ${\lambda _m} = \dfrac{b}{T}$
Where ${\lambda _m}$ is the wavelength at maximum energy, $T$ is the absolute temperature and $b$ is the constant of proportionality called Wien’s displacement constant having value $2.89771 \ldots \times {10^{ - 3}}mK$

Note:Absolute temperature is the temperature of an object taken on a scale where $0$ is taken as absolute zero. Absolute zero is $0K$ or $ - 273.15^\circ C$ .All objects having absolute zero temperature emit electromagnetic radiation.