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

Answer 1

Answer: (A)

For a black body, wavelength for maximum intensity :
$\lambda \propto \frac{1}{\text{T}}$ and $\text{P}\propto {\text{T}}^4\Rightarrow \text{P}\propto \frac{1}{{\lambda }^4}$
${\text{P}}^{{}^{\prime }}=16P$
$\therefore {\text{P}}^{{}^{\prime }}{\text{T}}^{{}^{\prime }}=32PT$