Question

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 $T^{{}^{\prime }}$ the power radiated is $P^{{}^{\prime }}$ and wavelength at maximum intensity is ${\lambda }_0/2$ then

• A. $P^{{}^{\prime }}{T}^{{}^{\prime }}=32PT$
• B. $P^{{}^{\prime }}{T}^{{}^{\prime }}=16PT$
• C. $P^{{}^{\prime }}{T}^{{}^{\prime }}=8PT$
• D. $P^{{}^{\prime }}{T}^{{}^{\prime }}=4PT$

Answer 1

Answer: (A)

For a black body, wavelength corresponding to maximum intensity, $\lambda \propto \frac{1}{T}$
Since, wavelength becomes half. So, new temperature, $T^{\prime }=2T$
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, $P^{{}^{\prime }}=16P$
Hence, $P^{{}^{\prime }}{T}^{{}^{\prime }}=32PT$