Question

The unit of Wien’s constant b is:
A. \[W{m^{ - 2}}{K^{ - 4}}\]
B. \[{m^{ - 1}}{K^{ - 1}}\]
C. \[W{m^2}\]
D. \[mK\]

Answer 1

Hint - In order to solve this problem obtain the formula in which b is included and you know the unit of all other variables then get the value of b and get the unit of b. doing this will solve your problem and will give you the right answer.

Complete step-by-step answer:
\[\lambda \propto \dfrac{1}{T}\]
\[ \Rightarrow \lambda = \dfrac{b}{T}\] , where b is Wien’s constant of proportionality.
Wien's constant is a physical constant that determines the relationship between the thermodynamic temperature of a black body (an object that radiates electromagnetic energy perfectly) and the frequency at which the radiation intensity becomes maximum.
I.e. \[ \Rightarrow \lambda = \dfrac{b}{T}\] where b is constant of proportionality.
Using the dimensional continuity rule, the usage on L.H.S. will be equal to the value on R.H.S.
I.e. LHS=\[\lambda \] since S.I. unit of \[\lambda \] is m. therefore, its dimension is \[\left[ {{M^0}{L^1}{T^0}} \right]\]
On RHS we have \[\dfrac{b}{T}\] where S.I. unit is K therefore dimensions of T is \[{K^{ - 1}}\]
Because it is a statute, the proportions of L.H.S. will be equal to R.H.S.
i.e. \[\left[ {{M^0}{L^1}{T^0}} \right] = b\left[ {{K^{ - 1}}} \right]\]
\[ \Rightarrow b = \dfrac{{\left[ {{M^0}{L^1}{T^0}} \right]}}{{\left[ {{K^{ - 1}}} \right]}}\]
\[ \Rightarrow b = \left[ {{M^0}{T^0}{L^1}K} \right]\]
Since the dimension of b is \[\left[ {{M^0}{T^0}{L^1}K} \right]\]. The S.I., then. The Wien constant unit is m. K., where m is S.I. The length unit and the K is the S.I. The temperature unit.
Hence, the correct option is D.

Note – In such types of questions of finding units we can simply apply the basic principle of Dimensional analysis, according to which for an equation to be valid it should follow dimensional consistency,i.e. the units on L.H.S. should be equal to the units on R.H.S.