Question

A black body has a wavelength of $\lambda$ at temperature 2000 K. Its corresponding wavelength at temperature 3000 K will be
A. $\dfrac{2\lambda }{3}$
B. $\dfrac{3\lambda }{2}$
C. $\dfrac{4\lambda }{9}$
D. $\dfrac{9\lambda }{4}$

Answer 1

According to Wien's displacement law the wavelength corresponding to the maximum intensity for the wavelength of maximum peak is inversely proportional to the temperature. Substituting the wavelength and the temperature given in the question we can find the required wavelength.

Complete step by step answer:
According to Wein's displacement law, for a black body radiation curve the wavelength corresponding to the maximum intensity peak is inversely proportional to the temperature.
$\lambda \propto \dfrac{1}{T}$
$\lambda T=b$
Where b is the constant of proportionality.
Given wavelength of the black body at a temperature of 2000 K$=\lambda$.
Let the unknown wavelength corresponding to the temperature of 3,000 K be $\lambda '$.
Therefore we can write,
${{\lambda }_{1}}{{T}_{1}}={{\lambda }_{2}}{{T}_{2}}$
Substituting the given values we get,
\lambda 2000=\lambda '3000 \\\ \therefore\lambda '=\dfrac{2}{3}\lambda \\\
Therefore, we get the wavelength corresponding to the temperature of 3000K is $\dfrac{2}{3}\lambda$.

Note: The most important point to solve this question is to be aware of Wein's law, according to which, ${{\lambda }_{1}}{{T}_{1}}={{\lambda }_{2}}{{T}_{2}}$. Wien's displacement law is one of the most fundamental laws of thermodynamics and it states that the black-body radiation curve will peak at different wavelengths which is inversely proportional to the temperature.
$\lambda \propto \dfrac{1}{T}$
$\lambda T=b$
Where b is the constant of proportionality. Also, the rate of heat transfer by emitted radiation is determined by the Stefan-Boltzmann law of radiation which says that the radiation rate is directly proportional to the fourth power of the temperature. A temperature change is accompanied by a colour change of the radiating body. For example, on continuously heating the iron it changes its colour as temperature increases.