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 maximum intensity is ${\lambda }_0$. If at another temperature $T^{\prime },$ the power radiated is $P^{\prime }$ and wavelength at maximum intensity is $\frac{{\lambda }_0}{2}$. If $P^{\prime }{T}^{\prime }=nPT,$ then value of $n$ is

Answer 1

For a black body, wavelength for maximum intensity.
Using Wein's displacement law,
$\lambda \propto \frac{1}{T}$
$T^{\prime }=2T$
Using Stefan Boltzmann law
$P\propto T^4\Rightarrow P\propto \frac{1}{{\lambda }^4}$
$\Rightarrow P^{\prime }=16P.\therefore P^{\prime }{T}^{\prime }=32PT$

Why this question ? Caution: Wein's displacement law$({\lambda }_{m}T=\text{constant}),$ here ${\lambda }_m$ corresponds to wavelength of maximum intensity not maximum wavelength possible. Practical application: The Stefan-Boltzmann Law and Wien's Displacement Law are important in designing and selecting sensors because they tell us how much energy a body will produce based on its temperature and where in the electromagnetic spectrum this energy will be maximum.