Question

The unit of Wien’s constant $b$ is
A) ${\text{W}}{{\text{m}}^{ - 2}}{{\text{K}}^{ - 4}}$
B) ${{\text{m}}^{ - 1}}{{\text{K}}^{ - 1}}$
C) ${\text{W}}{{\text{m}}^2}$
D) ${\text{mK}}$

Answer 1

Wien’s displacement law gives the relationship between the peak wavelength of black body radiation and surface temperature of the black body to be inversely proportional to each other. The proportionality constant is referred to as Wien’s constant.

Formula used:
-Wien’s constant is given by, $b = {\lambda _m}T$ where ${\lambda _m}$ is the peak wavelength of the black body radiation and $T$ is the surface temperature of the black body.

Complete step by step answer.
Step 1: Express the Wien’s constant to determine its unit.
Wien observed that the radiation curves of a black body for different temperatures have different peaks of wavelength. Wien’s law states that the product of the peak wavelength ${\lambda _m}$ of the blackbody radiation and the surface temperature $T$ of the black body is a constant value denoted by $b$. This constant is termed as Wien’s constant.
i.e., $b = {\lambda _m}T$ ------- (1)
We know that the S.I. The unit of wavelength is meters and that of the surface temperature is Kelvin. So substituting the units of the peak wavelength ${\lambda _m}$ and the surface temperature $T$ in equation (1) we get, $b = {\text{mK}}$ .
Thus the unit of Wien’s constant is ${\text{mK}}$ and its value is $b = 2 \cdot 89 \times {10^{ - 3}}{\text{mK}}$ .
So the correct option is D.

Hence, option D is correct.

Note: A black body is an ideal body that absorbs all radiations irrespective of the frequency of the radiation and the angle at which it is incident. Wien’s law suggests that the hotter an object is the shorter will be the wavelength of the radiation emitted by the object and so the energy emitted will also be more. This is why the filament in an incandescent bulb appears redder as the temperature of the filament decreases. The wavelength of the radiation becomes longer.