The dimensions of Stefan-Boltzmann's constant | vedclass

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(C) The Stefan-Boltzmann constant $\sigma$ has dimensions $[M T^{-3} K^{-4}]$.The dimensions of the constants are:$h = [M L^2 T^{-1}]$$k_B = [M L^2 T^

Vedclass · Accepted Answer

$\alpha=-3, \beta=4$ and $\gamma=-2$

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(C) The Stefan-Boltzmann constant $\sigma$ has dimensions $[M T^{-3} K^{-4}]$.The dimensions of the constants are:$h = [M L^2 T^{-1}]$$k_B = [M L^2 T^{-2} K^{-1}]$$c = [L T^{-1}]$Given $\sigma = h^\alpha k_B^\beta c^\gamma$,we equate the dimensions:$[M T^{-3} K^{-4}] = [M L^2 T^{-1}]^\alpha [M L^2 T^{-2} K^{-1}]^\beta [L T^{-1}]^\gamma$$[M T^{-3} K^{-4}] = [M^{\alpha+\beta} L^{2\alpha+2\beta+\gamma} T^{-\alpha-2\beta-\gamma} K^{-\beta}]$Comparing the exponents of $M, L, T,$ and $K$:$1$) $M: \alpha + \beta = 0$ (Wait,$\sigma$ is energy per unit area per unit time per unit temperature to the fourth power,so $[M T^{-3} K^{-4}]$ is correct. The mass dimension is $1$,so $\alpha + \beta = 1$)$2$) $K: -\beta = -4 \implies \beta = 4$$3$) $M: \alpha + \beta = 1 \implies \alpha + 4 = 1 \implies \alpha = -3$$4$) $T: -\alpha - 2\beta - \gamma = -3 \implies -(-3) - 2(4) - \gamma = -3 \implies 3 - 8 - \gamma = -3 \implies -5 - \gamma = -3 \implies \gamma = -2$Thus,$\alpha = -3, \beta = 4, \gamma = -2$.

The dimensions of Stefan-Boltzmann's constant $\sigma$ can be written in terms of Planck's constant $h$,Boltzmann's constant $k_B$ and the speed of light $c$ as $\sigma=h^\alpha k_B^\beta c^\gamma$. Here,

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