Dimensions of Physical Constants

To determine the dimensions of the expression $\sigma_s k_B^{-1} b$, we first need the dimensions of each constant involved:

• Stefan-Boltzmann constant ($\sigma_s$): This constant relates to the energy radiated per unit area and time by a black body. Its dimensions are found to be $[M T^{-3} K^{-4}]$.
• Boltzmann constant ($k_B$): This constant relates the average kinetic energy of particles in a gas with the thermodynamic temperature. Its dimensions are $[M L^2 T^{-2} K^{-1}]$.
• Wien's displacement law constant ($b$): This constant relates the peak wavelength of emitted black-body radiation to its temperature. Its dimensions are $[L K]$.

Calculating Combined Dimensions

Now, we combine the dimensions of these constants according to the expression $\sigma_s k_B^{-1} b$. We substitute the dimensions of each term:

$ [\sigma_s k_B^{-1} b] = [\sigma_s] \times [k_B^{-1}] \times [b] $

First, find the dimensions of $k_B^{-1}$:

$ [k_B^{-1}] = \left( [M L^2 T^{-2} K^{-1}] \right)^{-1} = [M^{-1} L^{-2} T^{2} K^{1}] $

Next, substitute all dimensions into the expression:

$ [\sigma_s k_B^{-1} b] = [M T^{-3} K^{-4}] \times [M^{-1} L^{-2} T^{2} K^{1}] \times [L K] $

Combine the powers for each dimension (Mass M, Length L, Time T, Temperature K):

$ [\sigma_s k_B^{-1} b] = [M^{1 + (-1)}] [L^{0 + (-2) + 1}] [T^{-3 + 2}] [K^{-4 + 1 + 1}] $ $ [\sigma_s k_B^{-1} b] = [M^{0}] [L^{-1}] [T^{-1}] [K^{-2}] $

Simplifying, the dimensions are $[L^{-1} T^{-1} K^{-2}]$.