Question

Star \[{{S}_{1}}\] emits maximum radiation of wavelength 420 nm and the star \[{{S}_{2}}\] emits maximum radiation of wavelength 560 nm. What is the ratio of the temperature of \[{{S}_{1}}\] and \[{{S}_{2}}\]?
A. \[\dfrac{4}{3}\]
B. \[{{\left( \dfrac{4}{3} \right)}^{\dfrac{1}{4}}}\]
C. \[\dfrac{3}{4}\]
D. \[{{\left( \dfrac{3}{4} \right)}^{\dfrac{1}{2}}}\]

Answer 1

Hint: The emitting radiation problems can be solved by either Stefan’s-Boltzmann law or Wien’s displacement law. Here we are going to find the ratio between the temperatures of the stars with the help of Wien’s displacement law. From the ratio of maximum wavelengths of radiations, we can find out the ratio of temperatures of the stars.

Formula used:
\[{{\lambda }_{\max }}=\dfrac{c}{T}\], where c is the Wien’s constant, T is the temperature and \[{{\lambda }_{\max }}\] is the maximum wavelength.

Complete step by step answer:
We can use Wien’s displacement law to find the ratio of the temperature of two different stars.
According to the Wien’s displacement law, the maximum wavelength of any radiation is inversely proportional to the temperature of the star.
So, for the star \[{{S}_{1}}\],
\[{{T}_{1}}\propto \dfrac{1}{{{\lambda }_{1\max }}}\]……………..(1), where \[{{T}_{1}}\] is the temperature and \[{{\lambda }_{1\max }}\] is the maximum wavelength of radiation.
For the star \[{{S}_{2}}\],
\[{{T}_{2}}\propto \dfrac{1}{{{\lambda }_{2\max }}}\]……………(2), where \[{{T}_{2}}\] is the temperature and \[{{\lambda }_{2\max }}\] is the maximum wavelength of radiation.
Now, we can find out the ratio of temperatures of different stars.
\[\dfrac{{{T}_{2}}}{{{T}_{1}}}=\dfrac{{{\lambda }_{1\max }}}{{{\lambda }_{2\max }}}\]
\[\dfrac{{{T}_{2}}}{{{T}_{1}}}=\dfrac{420\times {{10}^{-9}}}{560\times {{10}^{-9}}}\]
\[\dfrac{{{T}_{2}}}{{{T}_{1}}}=\dfrac{42}{56}\]
\[\dfrac{{{T}_{2}}}{{{T}_{1}}}=\dfrac{3}{4}\]
Here the ratio of the temperature of \[{{S}_{2}}\]and \[{{S}_{1}}\] is \[\dfrac{3}{4}\]. So the ratio of the temperature of the star \[{{S}_{1}}\] and \[{{S}_{2}}\] is \[\dfrac{4}{3}\]. Therefore, the correct answer is option A.

Additional information:
Black body concept has evolved from the study of thermal radiations and electromagnetic radiation. The black body is considered as an ideal radiation absorber and it is used for the comparison with the real physical bodies radiations. There are two laws of black body radiation.
Stefan’s law;
“The energy radiated by a blackbody radiator per second per unit area is proportional to the fourth power of the absolute temperature”. Stefan-Boltzmann law for ideal radiators can be written as,
\[\dfrac{P}{A}=\sigma {{T}^{4}}\], where P is the power, A is the area, T is the temperature and \[\sigma \] is the Stefan-Boltzmann constant.
Wien’s displacement law;
“The wavelength distribution peaks at a value that is inversely proportional to the temperature”. It can be written as,
\[{{\lambda }_{\max }}=\dfrac{c}{T}\], where c is the Wien’s constant, T is the temperature and \[{{\lambda }_{\max }}\] is the maximum wavelength.
It says that objects at a different temperature will emit different spectra of wavelengths. Hotter objects normally emit shorter wavelengths. That’s why it appears as in blue colour. Cooler objects emit most of their radiations at longer wavelengths. Therefore it appears in reddish.

Note: Here, the calculation may get wrong if we calculate the ratio of \[\dfrac{{{T}_{2}}}{{{T}_{1}}}\]. Since in the question they have asked for \[\dfrac{{{T}_{1}}}{{{T}_{2}}}\]. So, we have to find the inverse if we are finding \[\dfrac{{{T}_{2}}}{{{T}_{1}}}\]. Otherwise, you can directly calculate the \[\dfrac{{{T}_{1}}}{{{T}_{2}}}\]. Candidates are advised to cross-check the answer after the calculations since the options contain similar answers. We are dealing with the ratio, so we don’t have to consider the constants. It will consume our time if we are adding constants also.