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Question: The dimensions of Stefan’s constant are: A. \([M^{0}LT^{-3}K^{-4}]\) B. \([MLT^{-3}K^{-3}]\) C...

The dimensions of Stefan’s constant are:
A. [M0LT−3K−4][M^{0}LT^{-3}K^{-4}]
B. [MLT−3K−3][MLT^{-3}K^{-3}]
C. [ML2T−3K−4][ML^{2}T^{-3}K^{-4}]
D. [ML0T−3K−4][ML^{0}T^{-3}K^{-4}]

Explanation

Solution

Hint: Stefan-Boltzmann constant is 5.673×10−8kgsec3kelvin45.673\times10^{-8} \dfrac{kg}{sec^{3}kelvin^{4}}, where SI units are kgsec3kelvin4\dfrac{kg}{sec^{3}kelvin^{4}}, this can be used to reduce dimension of the constant. The value of the Stefan–Boltzmann constant is derivable and experimentally tested as well. It is derived from the Boltzmann constant.

Complete step-by-step answer:
Stefan-Boltzmann constant or the Stefan’s constant is denoted by σ , it is a constant of proportionality, used in the Stefan–Boltzmann law of blackbody radiation; which is stated as the total intensity radiated over the wavelength, it is proportional the temperature.

The value of the Stefan–Boltzmann constant is both derivable and experimentable.It is derived from the Boltzmann constant as: σ=2π5kB415h3c2=5.673×10−8Jm−2s−1K−4\sigma=\dfrac{2\pi^{5}{k_{B}}^4}{15h^{3}c^{2}}=5.673\times10^{-8} Jm^{-2}s^{-1}K^{-4}

Where σ\sigma is Stefan-Boltzmann's constant, cc is speed of light, hh is Planck’s constant,kBk_{B} is Boltzmann constant.
The value of Stefan Boltzmann constant is universally accepted and given in SI units as-

σ=5.673×10−8kgsec3kelvin4\sigma=5.673\times10^{-8} \dfrac{kg}{sec^{3}kelvin^{4}}, where SI units are kgsec3kelvin4\dfrac{kg}{sec^{3}kelvin^{4}}, thus the dimensions are.[ML0T−3K−4][ML^{0}T^{-3}K^{-4}]

The value of the Stefan-Boltzmann constant is also written as follows:

In CGS: σ=5.673×10−5ergcm2sec1K4\sigma=5.673\times10^{-5} \dfrac{erg}{cm^{2}sec^{1}K^{4}},
In thermodynamics: σ=11.7×10−8calcm2day1K4\sigma=11.7\times10^{-8} \dfrac{cal}{cm^{2}day^{1}K^{4}},
Furthermore, the Stefan–Boltzmann constant allows for temperature (K) to be converted to units for intensity (W⋅m−2), which is power per unit area.
Hence the answer is, D. [ML0T−3K−4][ML^{0}T^{-3}K^{-4}]

Note:
The Slovenian physicist Josef Stefan formulated the constant in 1879 and derived by Austrian physicist Ludwig Boltzmann in 1884. In Physics, the applications of Stefan Boltzmann constant are many. Some of them are listed below-
(I)It is used to measure the amount of heat radiated from the black body.
(II)It can be used to convert temperature (K) to units for intensity (W.m-2) that is basically Power per unit area.