JEE PYQ: Units & Measurements - Question ID d5526a41bbf6 (JEE Main 2025)

In dimensional analysis, every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), time ($T$), and temperature ($K$ or $\Theta$). The key idea is to write the defining formula for each quantity in LIST-I and then express its dimensions in the form $M^a L^b T^c K^d$.

1. Boltzmann constant ($k_B$)
Definition: $k_B = \dfrac{E}{T}$, where $E$ is energy and $T$ is absolute temperature.
Energy has dimensions $ML^2T^{-2}$; temperature has dimension $K$.
Therefore, $[k_B] = \frac{ML^2T^{-2}}{K} = ML^2T^{-2}K^{-1}.$ This matches LIST-II entry III.

2. Coefficient of viscosity ($\eta$)
Definition from Newton’s law of viscosity: $\tau = \eta\,\dfrac{dv}{dx}$.
Shear stress $\tau$ has dimensions $ML^{-1}T^{-2}$; velocity gradient $\dfrac{dv}{dx}$ has dimensions $T^{-1}$.
Therefore, $[\eta] = \frac{ML^{-1}T^{-2}}{T^{-1}} = ML^{-1}T^{-1}.$ This matches LIST-II entry IV.

3. Planck’s constant ($h$)
Definition: $E = h\nu$, where $E$ is energy and $\nu$ is frequency.
Energy has dimensions $ML^2T^{-2}$; frequency has dimension $T^{-1}$.
Therefore, $[h] = \frac{ML^2T^{-2}}{T^{-1}} = ML^2T^{-1}.$ This matches LIST-II entry I.

4. Thermal conductivity ($k$)
Definition from Fourier’s law: $\dfrac{dQ}{dt} = kA\,\dfrac{dT}{dx}$.
Heat flow rate $\dfrac{dQ}{dt}$ has dimensions $ML^2T^{-3}$; area $A$ has dimension $L^2$; temperature gradient $\dfrac{dT}{dx}$ has dimension $KL^{-1}$.
Therefore, $[k] = \frac{ML^2T^{-3}}{L^2 \cdot KL^{-1}} = MLT^{-3}K^{-1}.$ This matches LIST-II entry II.

Matching LIST-I to LIST-II gives: A → III, B → IV, C → I, D → II.

• Students often confuse the dimensions of Boltzmann’s constant with those of the gas constant $R$, forgetting that $R = N_A k_B$ and so $[R] = ML^2T^{-2}K^{-1}$ as well. • Planck’s constant is sometimes mistakenly assigned the dimensions of angular momentum (which is also $ML^2T^{-1}$), but in this question it must be matched to the given options. • Thermal conductivity’s dimensions can be misremembered if one forgets that the temperature gradient introduces a $K^{-1}$ factor.

The correct option is therefore A: A-III, B-IV, C-I, D-II.