Question

A black body is at a temperature of $2880\,K$. The energy of radiation emitted by this body with wavelength between $499$ nm and $500$ nm is ${U_1}$ , between $999$ nm and $1000$ nm, is ${U_2}$ and between $1449$ nm and $1500$ nm is ${U_3}$ . Wein’s constant, $b = 2.88 \times {10^6}\,nm\,K$. Then:
A) ${U_1} = 0$
B) ${U_3} = 0$
C) ${U_1} = {U_2}$
D) ${U_2} > {U_1}$

Answer 1

Hint: Wein’s law states that the peak wavelength is inversely proportional to temperature of the black body. The intensity of the radiation emitted by a blackbody is given as a function of wavelength at fixed temperature by Planck’s radiation law. Use this law to compare the energies of the given black body at the given temperature.

Complete step by step solution:
A black body is a physical body capable of absorbing all the incident electromagnetic radiations irrespective of the frequency or angle of incidence. We are given with a black body at some temperature $2880\,K$ . The energy of radiation emitted by this body is measured at three events.
The energy of radiation between wavelength $499$ nm and $500$ nm,${U_1}$
The energy of radiation between wavelength $999$ nm and $1000$ nm, ${U_2}$
The energy of radiation between wavelength $1449$ nm and $1500$ nm, ${U_3}$
Using Stefan-Boltzmann’s law, we know that the energy emitted by a black body $E$ is proportional to the fourth power of its temperature. Thus, we need to find the temperature of the black body at the given wavelength.
$ \Rightarrow E\propto {{\rm T}^4}$
Here, ${\rm T}$ is the temperature of the black body at some given wavelength.
Wein’s law states that the peak wavelength ${\lambda _{\max }}$ is inversely proportional to temperature of the black body
$ \Rightarrow {\lambda _{\max }}\propto \dfrac{1}{T}$
$ \Rightarrow T\propto \dfrac{1}{{{\lambda _{\max }}}}$
Using this relation in Stefan-Boltzmann law, we have:
$E\propto {\left( {\dfrac{1}{{{\lambda _{\max }}}}} \right)^4}$
It is now clear that the energy for radiation is maximum if the wavelength is minimum.
Therefore, the relation between ${U_1},\,{U_2}$ and ${U_3}$ will be
${U_1} > {U_2} > {U_3}$
From this relation, we can say that option A, B and C is incorrect.
The correct option is option C.

Note: A black body radiates a range of wavelengths. Wein’s law is used to find the temperature of distant stars by observing their wavelength and also inverse is possible. For two black bodies to have equal energy of radiation they must be at the same temperature. The temperature used in Stefan-Boltzmann law is absolute temperature in Kelvin scale.