Question

The dimensions of Wien’s constant are
A. \[\left[ M{{L}^{0}}TK \right]\]
B. \[\left[ {{M}^{0}}L{{T}^{0}}K \right]\]
C. \[\left[ {{M}^{0}}{{L}^{0}}TK \right]\]
D. \[\left[ MLTK \right]\]

Answer 1

Hint: Wien’s displacement law has been used to describe the energy radiation of a black body. According to the Wien’s displacement law, \[{{\lambda }_{\max }}=\dfrac{b}{T}\], where \[b\] is the Wien’s constant. From this equation we can find out the dimension of Wien’s constant.

Complete step-by-step answer:
Wien’s displacement constant can be found from the Wien’s displacement law. It states that, spectral radiance of black-body radiation per unit wavelength is inversely proportional to the temperature.
It can be written as,
\[{{\lambda }_{\max }}=\dfrac{b}{T}\], where \[b\] is the Wien’s constant and \[T\] is the temperature.
\[b={{\lambda }_{\max }}T\]
The dimensional formula of wavelength can be written as
\[{{\lambda }_{\max }}\Rightarrow \left[ L \right]\]
The dimensional formula of temperature can be written as
\[T\Rightarrow \left[ K \right]\]
So, the dimensional formula of Wien’s constant is
\[b\Rightarrow \left[ {{M}^{0}}L{{T}^{0}}K \right]\]
So, the correct option is B.

Additional information:
Wien’s displacement law is actually related to the black body. A good approximation of blackbody is a small hole leading to the inside of a hollow object and that can act as a perfect absorber. The nature of the radiation leaving the cavity through the hole depends only on temperature only. Intensity of blackbody radiation increases with temperature. So, the amount of radiation also increases. According to Wien’s displacement law, the peak wavelength decreases with increasing temperature.
For each temperature, there is a wavelength, at which energy radiated is maximum. Increase in temperature increases the energy radiated but decreases the peak wavelength. The area under the curve for a particular temperature gives the total energy emitted by the blackbody. Stefan’s law also used to describe the radiated energy relation with the temperature. It states that the energy radiated by a black body per second per unit area is directly proportional to the fourth power of absolute temperature.

Note: \[{{\lambda }_{\max }}\] is representing the peak wavelength at a particular temperature. Wavelength is a distance between the identical points of adjacent cycles of a wave. So, the dimensional formula of wavelength is L. Dimensional formula of temperature is represented by K. Whereas, the T has been used for the time. Candidates do not make any mistakes by interchanging these quantities.