The power radiated by a black body is and it radiates maximum energy at wavelength, . If the temperature of the black body is now changed so that it radiates maximum energy at wavelength , the power radiated by it becomes . The value of is

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Solution

Q.
The power radiated by a black body is $P$ and it radiates maximum energy at wavelength, $λ_{0}$ . If the temperature of the black body is now changed so that it radiates maximum energy at wavelength $\frac{3}{4} λ_{0}$ , the power radiated by it becomes $n P$ . The value of $n$ is [2018]

1 $\frac{3}{4}$

2 $\frac{4}{3}$

3 $\frac{256}{81}$

4 $\frac{81}{256}$

Ans.
(3)

From Wien’s law, $λ_{max} T = constant$

So, $λ_{{max}_{1}} T_{1} = λ_{{max}_{2}} T_{2}$

$\Rightarrow λ_{0} T = \frac{3 λ_{0}}{4} T' \Rightarrow \frac{T'}{T} = \frac{4}{3}$

According to Stefan–Boltzmann law, energy emitted per unit time by a black body is $A e σ T^{4}$ , i.e., power radiated.

$∴$ $P \propto T^{4}$

So, $\frac{P'}{P} = {(\frac{T'}{T})}^{4} \Rightarrow n = {(\frac{4}{3})}^{4} = \frac{256}{81}$

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