Q. A black body emits radiation at the rate $P$ when its temperature is $T$ . At this temperature the wavelength at which the radiation has a maximum intensity is $\lambda _{0}$ . If at another temperature $T^{'}$ the power radiated is $P^{'}$ and wavelength at maximum intensity is $\lambda _{0} / 2$ then
NTA AbhyasNTA Abhyas 2022
Solution:
For a black body, wavelength corresponding to maximum intensity, $\lambda \propto \frac{1}{T}$
Since, wavelength becomes half. So, new temperature, $T'=2T$
The rate of radiation emission, $P \propto T^{4}$
Therefore, $\Rightarrow P \propto \frac{1}{\lambda ^{4}}$
Here, wavelength becomes half. Therefore, new rate of energy emission will be, $P^{'}=16\,P$
Hence, $P^{'}T^{'}=32\,PT$
Since, wavelength becomes half. So, new temperature, $T'=2T$
The rate of radiation emission, $P \propto T^{4}$
Therefore, $\Rightarrow P \propto \frac{1}{\lambda ^{4}}$
Here, wavelength becomes half. Therefore, new rate of energy emission will be, $P^{'}=16\,P$
Hence, $P^{'}T^{'}=32\,PT$