Question

Consider that \(\sigma_s\), \(k_B\), and \(b\) represent Stefan-Boltzmann constant, Boltzmann constant, and Wien's displacement law constant, respectively. The dimension of \(\sigma_s k_B^{-1} b\) is:

• A. \([L^{-1}T^{-1}K^{-4}]\)
• B. \([L^{-1}T^{-1}K^{-2}]\)
• C. \([L^{-1}K^{-2}]\)
• D. \([L^{-1}T^{-1}K^{-3}]\)

Hint:

In dimensional analysis, first write dimensions of each constant separately and then combine powers systematically.

Answer 1

The Correct Option is B

Concept: \[ [\sigma_s] = [MT^{-3}K^{-4}] \] \[ [k_B] = [ML^2T^{-2}K^{-1}] \] \[ [b] = [LK] \]

Step 1: Write the required dimensional expression.
\[ [\sigma_s k_B^{-1} b] = [\sigma_s] [k_B]^{-1} [b] \]

Step 2: Substitute dimensions.
\[ = [MT^{-3}K^{-4}] [M^{-1}L^{-2}T^2K] [LK] \]

Step 3: Simplify powers.
Mass: \[ M^{1-1}=M^0 \] Length: \[ L^{-2+1}=L^{-1} \] Time: \[ T^{-3+2}=T^{-1} \] Temperature: \[ K^{-4+1+1}=K^{-2} \] Hence, \[ [\sigma_s k_B^{-1} b] = [L^{-1}T^{-1}K^{-2}] \] \[ \boxed{\text{Option (B)}} \]