If $A$ represents Boltzmann constant, $B$ represents Planck's constant and $C$ represents speed of light in…

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
  1. universal gas constant
  2. specific heat capacity
  3. stefan’s constant
  4. heat energy

Solution

Dimension of Boltzmann's constant
$A=\frac{\mathrm{J}}{\mathrm{K}}=\frac{\mathrm{ML}^2 \mathrm{~T}^{-2}}{\mathrm{~K}^1}=\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]$
Dimension of Planck constant
$B=\mathrm{Js}=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]$
Dimension of speed of light,
$C=\left[\mathrm{LT}^{-1}\right]$
Desired dimension $=A^4 B^{-3} C^{-2}$
$\Rightarrow\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]^4\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^{-3}\left[\mathrm{LT}^{-1}\right]^{-2}$
$\Rightarrow\left[\mathrm{M}^1 \mathrm{~L}^0 \mathrm{~T}^{-3} \mathrm{~K}^{-4}\right]$
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 $\quad T=$ absolute temperature.
$\Rightarrow$ Dimension, $\sigma=\frac{\left[\mathrm{M} \mathrm{L}^2 \mathrm{~T}^{-2}\right]}{\left.\left[\mathrm{L}^2\right][\mathrm{T}] \mathrm{K}^4\right]}$
$\sigma=\left[\mathrm{ML}^0 \mathrm{~T}^{-3} \mathrm{~K}^{-4}\right]$
So, desired dimension is of Stefan constant.

Asked in: AP EAMCET 2018 (24 Apr Shift 1)

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