Question

If $A$ represents Boltzmann constant, $B$ represents Planck's constant and $C$ represents speed of light in vacuum, then the quantity having the dimensions of $A^4{B}^{-3}{C}^{-2}$ is

• A. universal gas constant
• B. specific heat capacity
• C. stefan's constant
• D. heat energy

Answer 1

Answer: (C)

Dimension of Boltzmann's constant
$A=\frac{J}{K}=\frac{M{L}^2{T}^{-2}}{K^1}=\left[M{L}^2{T}^{-2}{K}^{-1}\right]$
Dimension of Planck constant
$B=Js=\left[M{L}^2{T}^{-1}\right]$
Dimension of speed of light,
$C=\left[L{T}^{-1}\right]$
Desired dimension $=A^4{B}^{-3}{C}^{-2}$
$\Rightarrow {\left[M{L}^2{T}^{-2}{K}^{-1}\right]}^4{\left[M{L}^2{T}^{-1}\right]}^{-3}{\left[L{T}^{-1}\right]}^{-2}$
$\Rightarrow \left[M^1{L}^0{T}^{-3}{K}^{-4}\right]$
Stefan constant $(\sigma )$ is given by
$E=\sigma T^4$
$\Rightarrow \sigma =\frac{E}{T^4}$
where, $E=$ energy per unit area per unit time
and $T=$ absolute temperature.
$\Rightarrow$ Dimension, $\sigma =\frac{\left[M{L}^2{T}^{-2}\right]}{\left[L^2\right][T]K^4]}$
$\sigma =\left[M{L}^0{T}^{-3}{K}^{-4}\right]$
So, desired dimension is of Stefan constant.