Question

When the temperature of a luminous body is increased then the frequency corresponding to maximum energy of radiation
A. increased
B. decreased
C. remain unchanged
D. First increases and then decreases

Answer 1

Hint: You could recall the Wien’s displacement law and hence find the relation of peak wavelength with temperature. Now you could think of the relation connecting the frequency and the wavelength. By using the above two expressions we could derive an expression relating the frequency with temperature and hence find the answer.

Formula used:
Wien’s displacement law,
${{\lambda }_{peak}}=\dfrac{b}{T}$
Expression for frequency,
$f=\dfrac{v}{\lambda }$

Complete answer:
In the question, we are given a luminous body and we are increasing its temperature. We are asked to find what happens to the frequency corresponding to maximum energy of radiation when the temperature of this luminous body is being increased.
Let us recall Wien’s displacement law that relates the peak wavelength with temperature.
This law basically states that the black body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature. We know that Planck’s radiation law describes the spectral brightness of black-body radiation as a function of wavelength at a given temperature and also the shift in wavelength mentioned above happens as the direct consequence of this law.
So, Wien’s displacement law states that the spectral radiance of blackbody radiation per unit wavelength peaks at a particular wavelength that is given by the expression,
${{\lambda }_{peak}}=\dfrac{b}{T}$ ………………………………… (1)
‘T’ here is the absolute temperature and ‘b’ is the constant of proportionality called Wien’s displacement constant and has the value of,
$b=2.898\times {{10}^{-3}}mK$
From (1), we see that the peak wavelength is inversely proportional to temperature. But we have an expression relating the wavelength of radiation and its frequency that is given by,
$f=\dfrac{v}{\lambda }$ …………………………………. (2)
Where, $v$ is the speed of the radiation, $f$ is the frequency and $\lambda $ is the wavelength.
From (2), frequency is inversely related to wavelength.
So, from (1) and (2), we know that the frequency corresponding to maximum energy of radiation is directly proportional to the temperature of a luminous body. Therefore, as the temperature of a luminous body is increased then the frequency corresponding to maximum energy of radiation is also increased.

So, the correct answer is “Option A”.

Note:
The Wien’s displacement law is actually useful in determining the temperatures of hot radiant objects such as stars or may be any other object whose temperature is far above that of its surroundings. We could also answer the given question using the Stefan-Boltzmann law that is expressed as,
$E=\sigma {{T}^{4}}$
Also, we know that the energy is directly proportional to frequency and hence the answer.