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

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

$\text{ So, }{\lambda }_{\underset{ }{\max }1}{T}_1={\lambda }_{\underset{ }{\max }2}{T}_2$

$\Rightarrow {\lambda }_{0}T=\frac{3{\lambda }_0}{4}{T}^{\text{'}}\Rightarrow T^{\text{'}}=\frac{4}{3}$

According to Stefan-Boltzmann law, energy emitted unit time by a black body is $Ae\sigma T^4,\text{ i.e., }$power radiated is proportional to the fourth power of temperature.

$\mathrm{i}.\mathrm{e}. \mathrm{P}\propto {\mathrm{T}}^4$

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