Question

The energy spectrum of a black body exhibits a maximum energy around a wavelength \[{\lambda _o}\] . The temperature of the black body is now changed such that the energy is maximum around a wavelength \[\dfrac{{3{\lambda _o}}}{4}\] .The power radiated by the black body will now increase by a factor of:
(A) \[\dfrac{{256}}{{81}}\]
(B) \[\dfrac{{64}}{{27}}\]
(C) \[\dfrac{{16}}{9}\]
(D) \[\dfrac{4}{3}\]

Answer 1

Hint We first use wien's displacement law to calculate the ratio of temperatures of both the bodies as energy and wavelength present so this law is applicable.
Then we use Stefan’s law as when there is power radiated, we use this law. By the use of Stefan’s law, we will find energy radiated per unit time per unit area i.e. E.
Then we will find the ratio of energy radiated for both the bodies and equate with the other values obtained and find the correct option.

Complete step by step solution
We know that when there is maximum energy and wavelength present, we use the concept of Wein’s displacement law.
According to Wien's displacement law, wavelength corresponding to maximum energy decreases when the temperature of black body increases i.e. \[\lambda T = b\] , where b is constant, \[\lambda \] is the wavelength and T is the temperature.
So, we can write Wien's displacement law for both the temperature i.e. \[{\lambda _1}{T_1} = {\lambda _2}{T_2}\]
So \[\dfrac{{{T_2}}}{{{T_1}}} = \dfrac{{{\lambda _1}}}{{{\lambda _2}}}\] , now we will put the value of wavelength given in the question.
 \[\dfrac{{{T_2}}}{{{T_1}}} = \dfrac{{{\lambda _o}}}{{\dfrac{{3{\lambda _o}}}{4}}}\] , so \[\dfrac{{{T_2}}}{{{T_1}}} = \dfrac{4}{3}\] .
And also, we know when there comes power radiated, we use Stefan’s law:
 Energy radiated per unit time per unit area i.e. \[E = \sigma {T^4}\] , where \[\sigma \] is Stephan-Boltzmann constant. T is the temperature.
So, for first body \[{E_1} = \sigma {T_1}^4\] and for second body \[{E_2} = \sigma {T_2}^4\]
It is said that power radiated by the black body will now increase by a factor in the question. So, we get \[\dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{{T_1}^4}}{{{T_2}^4}}\] ;
 \[\dfrac{{{E_1}}}{{{E_2}}} = {\left( {\dfrac{{{T_1}}}{{{T_2}}}} \right)^4}\] . Now we put the value \[\dfrac{{{T_2}}}{{{T_1}}} = \dfrac{4}{3}\]
So, we get \[\dfrac{{{E_2}}}{{{E_1}}} = {\left( {\dfrac{4}{3}} \right)^4} = \dfrac{{256}}{{81}}\] .

So, the correct option is A

Note Remember the formula of both Wien's displacement law and Stefan's Law and also remember when to apply these laws.
Note that Stefan's Law clarifies that total radiant heat energy emerging from a surface is proportional to the fourth power of its absolute temperature.
Also remember Stefan-Boltzmann law applies only to blackbodies, surfaces that absorb all incident heat radiation