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
$A=\frac{ J }{ K }=\frac{ ML ^{2}\, T ^{-2}}{ K ^{1}}=\left[ ML ^{2} \, T ^{-2} \, K ^{-1}\right]$
Dimension of Planck constant
$B= Js =\left[ ML ^{2} \, T ^{-1}\right]$
Dimension of speed of light,
$C=\left[ LT ^{-1}\right]$
Desired dimension $=A^{4} \, B^{-3}\, C^{-2}$
$\Rightarrow \left[ ML ^{2} \, T ^{-2}\, K ^{-1}\right]^{4}\left[ ML ^{2} \, T ^{-1}\right]^{-3}\left[ LT ^{-1}\right]^{-2}$
$\Rightarrow \left[ M ^{1} \, L ^{0} \, T ^{-3}\, K ^{-4}\right]$
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, $\sigma=\frac{\left[M L^{2}\, T^{-2}\right]}{ \left.\left[L^{2}\right][T] K^{4}\right]}$
$\sigma=\left[ ML ^{0}\, T ^{-3}\, K ^{-4}\right] $
So, desired dimension is of Stefan constant.