Question

What will be the ratio of temperature of sun and moon if the wavelength of their maximum emission radiations rates are $140 \times 10^{-10}$ m and $4200 \times 10^{-10}$ m respectively?
(A) $1:30$
(B) $30:1$
(C) $42:14$
(D) $14:24$

Answer 1

Hint
We are provided with the wavelength of maximum emission radiation of sun and moon, so use Wien's displacement law to find the ratio of temperature of sun and moon.
Formula used:${\text{T = }}\dfrac{1}{{{\lambda _{{\text{max}}}}}}$
Where, $T$ is the temperature, $\lambda $ is the wavelength.

Complete step by step answer
Given, The wavelength of maximum emission radiation of sun is $140 \times 10^{-10}$ m
The wavelength of maximum emission radiation of moon is $4200 \times 10^{-10}$ m
Wien’s law or Wien’s displacement law deals with the relationship between the temperature of blackbody and the wavelength at which it emits the light. The law states that the absolute temperature of the blackbody is inversely proportional to the maximum wavelength.
$\Rightarrow {\text{Temperature }} \propto {\text{ }}\dfrac{1}{{{\text{wavelength}}{{\text{h}}_{{\text{max}}}}}}$
$\Rightarrow {\text{T = }}\dfrac{1}{{{\lambda _{{\text{max}}}}}}$
Where, $T$ is the temperature, $\lambda $ is the wavelength.
A blackbody is an object that absorbs all radiant energy falling on it at all wavelengths. When a blackbody is at uniform temperature to maintain the thermal equilibrium it emits radiations.
Since the sun absorbs any radiations that hits on it and it is an ideal radiator that emits most radiations. Moon absorbs all the sunlight that falls on it which consists of most types of radiations and emits it back in the form of infra-red. Both the sun and moon acts as a black body.
Since both the sun and moon are blackbodies and we can apply Wien’s law to it.
Applying Wien’s law,
$ \Rightarrow {\text{T = }}\dfrac{1}{{{\lambda _{{\text{max}}}}}}$
$\Rightarrow \dfrac{{{\text{The temperature of sun}}}}{{{\text{The temperature of moon}}}}{\text{ = }}\dfrac{{\dfrac{1}{{{\text{The maximum wavelength of sun}}}}}}{{\dfrac{1}{{{\text{The maximum wavelength of moon}}}}}}$
$ \Rightarrow \dfrac{{{\text{The temperature of sun}}}}{{{\text{The temperature of moon}}}}{\text{ = }}\dfrac{{{\text{The maximum wavelength of moon}}}}{{{\text{The maximum wavelength of sun}}}}$
$ \Rightarrow \dfrac{{{\text{The temperature of sun}}}}{{{\text{The temperature of moon}}}}{\text{ = }}\dfrac{{4200}}{{140}}$
$ \Rightarrow \dfrac{{{\text{The temperature of sun}}}}{{{\text{The temperature of moon}}}}{\text{ = }}\dfrac{{30}}{1}$
The ratio of temperature of sun and moon is $30:1$.
Hence the correct answer is option (A).

Note
Do not forget that the wavelength and temperature is inversely proportional. In this question if frequency is given instead of wavelength, in that case temperature is directly proportional to frequency, as you know that wavelength is inversely proportional to frequency.