Match Physical Quantities with Their Dimensional Formulas

To determine the correct matching between the physical quantities in List I and their corresponding dimensional formulas in List II, we derive the dimensions of each quantity step-by-step:

1. P. Boltzmann constant ($k_B$): The average thermal energy $E$ of a gas molecule is related to temperature $T$ by the formula: $E = \frac{3}{2} k_B T \implies k_B = \frac{2E}{3T}$ Taking dimensions on both sides: $[k_B] = \frac{[E]}{[T]} = \frac{[M L^2 T^{-2}]}{[K]} = [M L^2 T^{-2} K^{-1}]$ Hence, P matches with 4.

2. Q. Coefficient of viscosity ($\eta$): According to Newton's law of viscosity, viscous force $F$ is given by: $F = \eta A \frac{dv}{dx}$ Rearranging for $\eta$: $\eta = \frac{F}{A \left(\frac{dv}{dx}\right)}$ Taking dimensions: $[\eta] = \frac{[M L T^{-2}]}{[L^2] \cdot \left[\frac{L T^{-1}}{L}\right]} = \frac{[M L T^{-2}]}{[L^2] [T^{-1}]} = [M L^{-1} T^{-1}]$ Hence, Q matches with 2.

3. R. Planck constant ($h$): The energy $E$ of a photon is related to its frequency $\nu$ by Planck's equation: $E = h \nu \implies h = \frac{E}{\nu}$ Taking dimensions: $[h] = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}]$ Hence, R matches with 1.

4. S. Thermal conductivity ($k$): The rate of heat transfer through a material is given by Fourier's law of heat conduction: $\frac{dQ}{dt} = k A \frac{dT}{dx}$ Rearranging for $k$: $k = \frac{\left(\frac{dQ}{dt}\right)}{A \left(\frac{dT}{dx}\right)}$ Taking dimensions: $[k] = \frac{[M L^2 T^{-3}]}{[L^2] \cdot [K L^{-1}]} = [M L T^{-3} K^{-1}]$ Hence, S matches with 3.

Comparing our results: $\text{P} \rightarrow 4, \quad \text{Q} \rightarrow 2, \quad \text{R} \rightarrow 1, \quad \text{S} \rightarrow 3$

This corresponds to Option C.

Correct Answer: C