Question

The maximum energy in the thermal radiation from a hot source occurs at a wavelength of $12\times {{10}^{-5}}cm$. According to Wien's displacement law, the temperature of this source will be $n$ times the temperature of another source for which the wavelength at maximum energy is $6\times {{10}^{-5}}cm$. Then the value of $n$ will be
$\begin{align}
  & A.\dfrac{1}{2} \\
 & B.1 \\
 & C.2 \\
 & D.4 \\
\end{align}$

Answer 1

Hint: The Wien’s displacement law is a law which states the relationship between the emitted radiations and temperature of a black body. The wavelength is selected as the wavelength possessed by the most of the radiation. According to this law, the wavelength will be inversely proportional to the temperature of the black body. Here we can compare both the situations and finally we can find the value of $n$. These all may help you to solve this question.

Complete step by step answer:
As we all know, the wavelength of the light will be inversely proportional to the temperature of the body. This can be expressed mathematically as,
$\lambda \propto \dfrac{1}{T}$
That is we can write this as,
$\dfrac{{{\lambda }_{1}}}{{{\lambda }_{2}}}=\dfrac{{{T}_{2}}}{{{T}_{1}}}$
Where \[{{\lambda }_{1}}\]be the wavelength of the radiation in the first case. This can be written as per the question as,
\[{{\lambda }_{1}}=12\times {{10}^{-5}}cm\]
\[{{\lambda }_{2}}\]be the wavelength of the radiation mentioned in the second case. This can be written as,
\[{{\lambda }_{2}}=6\times {{10}^{-5}}cm\]
\[{{T}_{1}}\] be the temperature at which the maximum wavelength in the first case is radiating.
It can be written as,
\[{{T}_{1}}=nT\]
And \[{{T}_{2}}\]be the temperature at which the maximum wavelength in the second case is radiating,
The value can be expressed as,
\[{{T}_{2}}=T\]
Substituting all these values in the Wien’s displacement law, we can write that,
\[\begin{align}
  & \dfrac{{{\lambda }_{1}}}{{{\lambda }_{2}}}=\dfrac{{{T}_{2}}}{{{T}_{1}}} \\
 & \dfrac{12\times {{10}^{-5}}}{6\times {{10}^{-5}}}=\dfrac{T}{nT} \\
\end{align}\]
From this we will get that,
\[n=\dfrac{1}{2}\]

So, the correct answer is “Option A”.

Note: Wien's displacement law is helpful in determining the temperatures of hot radiant bodies as stars. This is needed for the determination of the temperature of any radiant body which is having a high temperature. Wien’s law is plotted in a graph where intensity is plotted as a function of wavelength.