JEE PYQ: Units & Measurements - Question ID 715bcd91c9e2 (JEE Main 2026)

Core Formula & Concept:

In dimensional analysis, every physical quantity can be expressed in terms of the fundamental dimensions: mass ($\mathrm{M}$), length ($\mathrm{L}$), time ($\mathrm{T}$), electric current ($\mathrm{A}$), etc. The question requires us to match four physical quantities (Planck’s constant, stopping potential, work function, and threshold frequency) with their correct dimensional formulae from List-II.

Key formulas and their dimensions:

• Planck’s constant ($h$): From the relation $E = h \nu$, where $E$ is energy and $\nu$ is frequency, we get $h = \frac{E}{\nu}$. The dimensions of energy are $\mathrm{ML}^2 \mathrm{T}^{-2}$ and frequency is $\mathrm{T}^{-1}$, so $h$ has dimensions $\frac{\mathrm{ML}^2 \mathrm{T}^{-2}}{\mathrm{T}^{-1}} = \mathrm{ML}^2 \mathrm{T}^{-1}$.
• Stopping potential ($V_0$): This is an electric potential, defined as the work done per unit charge. The dimensions of work are $\mathrm{ML}^2 \mathrm{T}^{-2}$ and charge is $\mathrm{IT}$, so potential has dimensions $\frac{\mathrm{ML}^2 \mathrm{T}^{-2}}{\mathrm{IT}} = \mathrm{ML}^2 \mathrm{T}^{-3} \mathrm{A}^{-1}$.
• Work function ($\phi$): This is the minimum energy required to eject an electron from a metal surface. Energy has dimensions $\mathrm{ML}^2 \mathrm{T}^{-2}$.
• Threshold frequency ($\nu_0$): This is the minimum frequency of light required to eject an electron. Frequency has dimensions $\mathrm{T}^{-1}$.

Step-by-Step Derivation:

1. Planck’s constant (A): From $E = h \nu$, we derive: $h = \frac{E}{\nu} \implies [h] = \frac{\mathrm{ML}^2 \mathrm{T}^{-2}}{\mathrm{T}^{-1}} = \mathrm{ML}^2 \mathrm{T}^{-1}$. This matches III in List-II.


2. Stopping potential (B): Stopping potential is an electric potential, given by $V_0 = \frac{W}{q}$, where $W$ is work and $q$ is charge. The dimensions are: $[V_0] = \frac{\mathrm{ML}^2 \mathrm{T}^{-2}}{\mathrm{IT}} = \mathrm{ML}^2 \mathrm{T}^{-3} \mathrm{A}^{-1}$. This matches IV in List-II.


3. Work function (C): The work function is energy, so its dimensions are: $[\phi] = \mathrm{ML}^2 \mathrm{T}^{-2}$. This matches I in List-II.


4. Threshold frequency (D): Frequency has dimensions: $[\nu_0] = \mathrm{T}^{-1}$. This matches II in List-II.

Thus, the correct matching is: A-III, B-IV, C-I, D-II, which corresponds to Option A.

Common Traps & Exam Tip:

Students often confuse the dimensions of stopping potential with those of electric field or resistance. Remember:

• Stopping potential is a potential (work per unit charge), not an electric field (force per unit charge).
• Planck’s constant is not energy; it is energy per unit frequency.
• Work function is energy, not power (which has dimensions $\mathrm{ML}^2 \mathrm{T}^{-3}$).
• Threshold frequency is a frequency, not angular frequency (which has dimensions $\mathrm{T}^{-1}$ but is often mistakenly associated with $\mathrm{T}^{-2}$).

Always derive dimensions from first principles to avoid such mistakes.