JEE PYQ: Units & Measurements - Question ID bf8ebb077c83 (JEE Main 2025)

Core Formula & Concept:

In dimensional analysis, every physical quantity is expressed in terms of the fundamental dimensions: Mass ($M$), Length ($L$), Time ($T$), and Electric current ($A$). The key formulas and concepts for the given quantities are:

• Angular Impulse: Angular impulse is the change in angular momentum. Angular momentum ($L$) = Moment of inertia ($I$) × Angular velocity ($\omega$). Moment of inertia has dimensions $M L^2$, and angular velocity has dimensions $T^{-1}$. Thus, angular impulse (change in angular momentum) has dimensions $M L^2 T^{-1}$.
• Latent Heat: Latent heat ($L$) is the heat energy ($Q$) per unit mass ($m$), i.e., $L = Q/m$. Heat energy has dimensions $M L^2 T^{-2}$ (since $Q = m c \Delta T$, and specific heat $c$ has dimensions $L^2 T^{-2} \Theta^{-1}$ but temperature cancels out in latent heat). Thus, latent heat has dimensions $M^0 L^2 T^{-2}$.
• Electrical Resistivity: Resistivity ($\rho$) is defined by the relation $R = \rho \frac{L}{A}$, where $R$ is resistance, $L$ is length, and $A$ is area. Resistance has dimensions $M L^2 T^{-3} A^{-2}$ (from Ohm’s law $V = IR$, where $V$ is voltage with dimensions $M L^2 T^{-3} A^{-1}$). Thus, resistivity has dimensions $M L^3 T^{-3} A^{-2}$.
• Electromotive Force (emf): emf is the work done per unit charge, i.e., $\mathcal{E} = \frac{W}{q}$. Work has dimensions $M L^2 T^{-2}$, and charge has dimensions $A T$. Thus, emf has dimensions $M L^2 T^{-3} A^{-1}$.

Step-by-Step Derivation:

We derive the dimensions for each quantity in List-I and match them with List-II.

1. (A) Angular Impulse: Angular impulse = Change in angular momentum = $I \omega$. Dimensions of moment of inertia ($I$) = $M L^2$. Dimensions of angular velocity ($\omega$) = $T^{-1}$. Thus, dimensions of angular impulse = $M L^2 \times T^{-1} = M L^2 T^{-1}$. This matches with (III) in List-II.
2. (B) Latent Heat: Latent heat = Heat energy per unit mass = $\frac{Q}{m}$. Dimensions of heat energy ($Q$) = $M L^2 T^{-2}$. Dimensions of mass ($m$) = $M$. Thus, dimensions of latent heat = $\frac{M L^2 T^{-2}}{M} = M^0 L^2 T^{-2}$. This matches with (I) in List-II.
3. (C) Electrical Resistivity: Resistivity ($\rho$) = $R \frac{A}{L}$. Dimensions of resistance ($R$) = $M L^2 T^{-3} A^{-2}$. Dimensions of area ($A$) = $L^2$. Dimensions of length ($L$) = $L$. Thus, dimensions of resistivity = $M L^2 T^{-3} A^{-2} \times \frac{L^2}{L} = M L^3 T^{-3} A^{-2}$. This matches with (IV) in List-II.
4. (D) Electromotive Force: emf = Work done per unit charge = $\frac{W}{q}$. Dimensions of work ($W$) = $M L^2 T^{-2}$. Dimensions of charge ($q$) = $A T$. Thus, dimensions of emf = $\frac{M L^2 T^{-2}}{A T} = M L^2 T^{-3} A^{-1}$. This matches with (II) in List-II.

The correct matching is: (A) - (III), (B) - (I), (C) - (IV), (D) - (II). This corresponds to Option C.

Common Traps & Exam Tip:

Students often make the following mistakes:

• Confusing angular impulse with linear impulse. Linear impulse has dimensions $M L T^{-1}$, while angular impulse has $M L^2 T^{-1}$.
• Misidentifying the dimensions of latent heat as $M L^2 T^{-2}$ (which is heat energy) instead of $M^0 L^2 T^{-2}$ (heat energy per unit mass).
• Forgetting that electrical resistivity involves area in its definition, leading to an extra $L^2$ in its dimensions compared to resistance.
• Mixing up the dimensions of emf and resistance. emf has dimensions $M L^2 T^{-3} A^{-1}$, while resistance has $M L^2 T^{-3} A^{-2}$.

Exam Tip: Always write down the defining formula for each quantity before deriving its dimensions. This avoids confusion between similar-sounding quantities.