Match Physical Quantities with Dimensional Formulas for Constants

To find the correct match between the physical constants in List - I and their corresponding dimensional formulas in List - II, we determine the dimensions of each constant using their fundamental physical equations:

1. Boltzmann Constant ($k_B$): The thermal energy $E$ is related to absolute temperature $T$ by the formula: $E = k_B T \implies k_B = \frac{E}{T}$

   Substituting the dimensional formulas for Energy ($[E] = [M L^2 T^{-2}]$) and Temperature ($[T] = [K]$): $[k_B] = \frac{[M L^2 T^{-2}]}{[K]} = [M L^2 T^{-2} K^{-1}]$

   Thus, A matches with III.

2. Stefan's Constant ($\sigma$): According to Stefan-Boltzmann law, the total radiant energy emitted per unit area per unit time (Emissive power) is proportional to the fourth power of absolute temperature: $\frac{E}{A \cdot t} = \sigma T^4 \implies \sigma = \frac{E}{A \cdot t \cdot T^4}$

   Substituting the dimensions of Energy ($[E] = [M L^2 T^{-2}]$), Area ($[A] = [L^2]$), Time ($[t] = [T]$), and Temperature ($[T] = [K]$): $[\sigma] = \frac{[M L^2 T^{-2}]}{[L^2] [T] [K^4]} = [M L^0 T^{-3} K^{-4}]$

   Thus, B matches with IV.

3. Planck's Constant ($h$): The energy $E$ of a photon is related to its frequency $\nu$ by: $E = h \nu \implies h = \frac{E}{\nu}$

   Substituting the dimensions of Energy ($[E] = [M L^2 T^{-2}]$) and Frequency ($[\nu] = [T^{-1}]$): $[h] = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}]$

   Thus, C matches with II.

4. Gravitational Constant ($G$): According to Newton's law of gravitation, the force between two masses $m_1$ and $m_2$ separated by a distance $r$ is: $F = G \frac{m_1 m_2}{r^2} \implies G = \frac{F r^2}{m_1 m_2}$

   Substituting the dimensions of Force ($[F] = [M L T^{-2}]$), Distance ($[r] = [L]$), and Mass ($[m] = [M]$): $[G] = \frac{[M L T^{-2}] [L^2]}{[M]^2} = [M^{-1} L^3 T^{-2}]$

   Thus, D matches with I.

Comparing with the given choices: $\text{A - III, B - IV, C - II, D - I}$

Hence, the correct option is C.