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Dimensional Analysis

The dimension...

Question

The dimensions of Stefan-Boltzmann constant $σ$ can be written in terms of Planck's constant $h$ , Boltzmann constant $K_{B}$ and the speed of light $c$ as $σ = h^{α} K_{B}$ $^{β} c^{γ}$ . Here,

α = 3, β = 4

and

γ = - 3

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α = 3, β = - 4

and

γ = 2

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α = - 3, β = 4

and

γ = - 2

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α = 2, β = - 3

and

γ = - 1

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Solution

The correct option is A $α = - 3, β = 4$ and $γ = - 2$
By Stefan Boltzmann's Law, energy radiated per unit area per unit time is given by:
$J = σ T^{4}$
Using dimensional analysis:
$[M L^{2} T^{- 2}] [L]^{- 2} [T]^{- 1} = [σ] [K^{4}]$
$[σ] = [M^{1} T^{- 3} K^{- 4}] . . . . . . . . . . . . (i)$

Energy of photon is given by:
$E = h ν$
Using dimensional analysis:
$[M L^{2} T^{- 2}] = [h] [T^{- 1}]$
$[h] = [M^{1} L^{2} T^{- 1}] . . . . . . . . . . (i i)$

Dimensions of speed of light is given by:
$[c] = [L^{1} T^{- 1}]$

Average translational kinetic energy of an ideal gas is given by:
$E = \frac{3}{2} K_{B} T$
Using dimensional analysis:
$[M L^{2} T^{- 2}] = [K_{B}] [K]$
$[K_{B}] = [M^{1} L^{2} T^{- 2} K^{- 1}]$

It is given that:
$σ = h^{α} K_{B}^{β} c^{γ}$
Again, using dimensional analysis:
$[M^{1} T^{- 3} K^{- 4}] = [M^{1} L^{2} T^{- 1}]^{α} [M^{1} L^{2} T^{- 2} K^{- 1}]^{β} [L T^{- 1}]^{γ}$
$= [M^{α + β} L^{2 α + 2 β + γ} T^{- α - 2 β - γ} K^{- β}]$

$α + β = 1$
$2 α + 2 β + γ = 0$
$- α - 2 - γ = - 3 β$
$- β = - 4$

Solving, we get
$β = 4$ , $α = - 3$ , $γ = - 2$

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The dimensions of Stefan-Boltzmann constant σ can be written in terms of Planck's constant h, Boltzmann constant KB and the speed of light c as σ=hαKBβcγ. Here,

The dimensions of Stefan-Boltzmann constant $σ$ can be written in terms of Planck's constant $h$ , Boltzmann constant $K_{B}$ and the speed of light $c$ as $σ = h^{α} K_{B}$ $^{β} c^{γ}$ . Here,