Question

Stefan’s law, E = σ T⁴ (E = energy radiated per unit area per unit time; T = temperature of black body), and Wien’s displacement law λ_mT b (λ_m = wavelength of maximum emission, T = temperature) are used to define the Stefan’s constant ,σ , and the Wien’s constant, b. If σ , b, along with mass and time, are taken as the ‘base units’ the dimensions of the ‘thermal capacity’ (C_m ) of a body, would be

(1) [b^1/2MT^-3σ^-5]

(2) [b^1/2σ^1/2M^-1T^-2]

(3) [M^3/2bσ^-1/2 T^-7/2]

(4) [M^3/2bσ^-1/2 T^7/2]

Answer 1

(3) [M^3/2bσ^-1/2 T^-7/2]

We have

[σ] = \(\frac{[E]}{[T^4]}=[[\frac{ML^2T^{-2}}{L^2T}]K^{-4}]\)

= [MT^-3K^-4]  ∴K = [σM^-1T³]^-1/4

and [b] = [λ_mT] = [LK]

∴ [L] = [bk^-1]

Now thermal capacity = [C_m] = \(\frac{Heat\,energy}{Change\,in\,Temperature}\)

= \([\frac{ML^2\,T^{-2}}{K}]\) = M [bk^-1]² T^-2 (K^-1)

= MbT^-2 (σM^-1 T³)^-1/2

= [M^1/2 bσ^-1/2T^-7/2]