Question

Match the following physical quantities with their respective dimensional formulas.

• A. \( a - i, \quad b - iii, \quad c - iv, \quad d - ii \)
• B. \( a - i, \quad b - ii, \quad c - iv, \quad d - iii \)
• C. \( a - iii, \quad b - ii, \quad c - i, \quad d - iv \)
• D. \( a - ii, \quad b - i, \quad c - iii, \quad d - iv \)

Hint:

To match dimensional formulas, always break down the physical quantity into its fundamental SI units and derive the expression step by step.

Answer 1

The Correct Option is A

Step 1: Understanding the dimensional formulas. 1. Thermal conductivity (\( k \)): It is given by \[ k = \frac{ML^1 T^{-3}}{K} \] So, its dimensional formula is \( MLT^{-3}K^{-1} \) (i). 2. Boltzmann constant (\( k_B \)): It relates energy per temperature per particle, given by \[ k_B = \frac{ML^2 T^{-2}}{K} \] So, its dimensional formula is \( ML^2 T^{-2} K^{-1} \) (iii). 3. Latent heat (\( L \)): It is energy per unit mass \[ L = \frac{ML^2 T^{-2}}{M} \] So, its dimensional formula is \( M^0 L^2 T^{-2} \) (iv). 4. Specific heat (\( C \)): It is heat energy per unit mass per unit temperature, given by \[ C = \frac{ML^2 T^{-2}}{M K} \] So, its dimensional formula is \( M^0 L^2 T^{-2} K^{-1} \) (ii). Thus, the correct matching is: \[ \begin{aligned} & a - i, \quad b - iii, \quad c - iv, \quad d - ii \end{aligned} \]