An expression of energy density is given by . Where are constants, is displacement, is Boltzmann constant and is the absolute temperature

Question

An expression of energy density is given by $u = \frac{α}{β} \sin (\frac{α x}{k t})$ . Where $α, β$ are constants, $x$ is displacement, $k$ is Boltzmann constant and $t$ is the absolute temperature. The dimensions of $β$ will be:

Smart Achievers · Accepted Answer

(4)

$\sin (\frac{α x}{k t}) = Dimensionless$

$∴$ $\frac{α [L]}{[M L^{2} T^{- 2}]} = [M^{0} L^{0} T^{0}]$ $\Rightarrow α = [M L T^{- 2}]$

$\frac{α}{β} = \frac{Energy}{Volume} = \frac{[M L^{2} T^{- 2}]}{[L^{3}]}$

$\Rightarrow β = \frac{[M L T^{- 2}] [L^{3}]}{[M L^{2} T^{- 2}]}$

$= [M^{0} L^{2} T^{0}]$

Solution

Q.
An expression of energy density is given by $u = \frac{α}{β} \sin (\frac{α x}{k t})$ . Where $α, β$ are constants, $x$ is displacement, $k$ is Boltzmann constant and $t$ is the absolute temperature. The dimensions of $β$ will be:

1 $[M L^{2} T^{- 2} θ^{- 1}]$

2 $[M^{0} L^{2} T^{- 2}]$

3 $[M^{0} L^{0} T^{0}]$

4 $[M^{0} L^{2} T^{0}]$

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Solution

Q. An expression of energy density is given by u=αβsin(αxkt). Where α,β are constants, x is displacement, k is Boltzmann constant and t is the absolute temperature. The dimensions of β will be:

1 [ML2T-2θ-1]

2 [M0L2T-2]

3 [M0L0T0]

4 [M0L2T0]