Match Physical Quantities with Dimensional Formulas in Modern Physics

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

1. Planck's constant ($h$): Using the energy equation for a photon: $E = h\nu \implies h = \frac{E}{\nu}$ where $E$ is energy and $\nu$ is frequency.

   • Dimension of Energy $[E] = [\text{M}\text{L}^2\text{T}^{-2}]$
   • Dimension of Frequency $[\nu] = [\text{T}^{-1}]$

   Thus, the dimension of Planck's constant is: $[h] = \frac{\text{M}\text{L}^2\text{T}^{-2}}{\text{T}^{-1}} = \text{M}\text{L}^2\text{T}^{-1}$ Therefore, A matches with III.

2. Stopping potential ($V_0$): Electric potential is defined as potential energy per unit charge: $V_0 = \frac{W}{q}$ where $W$ is work (energy) and $q$ is electric charge ($q = I \cdot t$).

   • Dimension of Work $[W] = [\text{M}\text{L}^2\text{T}^{-2}]$
   • Dimension of Charge $[q] = [\text{A}\text{T}]$

   Thus, the dimension of stopping potential is: $[V_0] = \frac{\text{M}\text{L}^2\text{T}^{-2}}{\text{A}\text{T}} = \text{M}\text{L}^2\text{T}^{-3}\text{A}^{-1}$ Therefore, B matches with IV.

3. Work function ($\phi$): The work function represents the minimum energy required to liberate an electron from the metal surface. Hence, its dimension is equal to the dimension of energy: $[\phi] = \text{M}\text{L}^2\text{T}^{-2}$ Therefore, C matches with I.

4. Threshold frequency ($\nu_0$): Frequency is defined as the reciprocal of time period: $\nu_0 = \frac{1}{T}$

   Thus, the dimension of threshold frequency is: $[\nu_0] = \text{T}^{-1}$ Therefore, D matches with II.

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

This corresponds to Option A.