JEE PYQ: Units & Measurements - Question ID fefbcadd8790 (JEE Main 2024)

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 used here are:

• Torque ($\tau$): Torque is the cross product of force and the lever arm distance. Since force has dimensions $[M^1 L^1 T^{-2}]$ and distance has $[L^1]$, torque has dimensions: $[\tau] = [F] \cdot [r] = [M^1 L^1 T^{-2}] \cdot [L^1] = [M^1 L^2 T^{-2}]$
• Magnetic Field ($B$): The magnetic field can be defined using the Lorentz force law: $\vec{F} = q (\vec{v} \times \vec{B})$. Rearranging for $B$: $[B] = \frac{[F]}{[q][v]} = \frac{[M^1 L^1 T^{-2}]}{[A^1 T^1] \cdot [L^1 T^{-1}]} = [M^1 T^{-2} A^{-1}]$
• Magnetic Moment ($m$): The magnetic moment of a current loop is given by $m = I \cdot A$, where $I$ is current and $A$ is area. Thus: $[m] = [I] \cdot [A] = [A^1] \cdot [L^2] = [L^2 A^1]$
• Permeability of Free Space ($\mu_0$): From Ampere’s law, the magnetic field due to a long straight wire is $B = \frac{\mu_0 I}{2 \pi r}$. Rearranging for $\mu_0$: $[\mu_0] = \frac{[B] \cdot [r]}{[I]} = \frac{[M^1 T^{-2} A^{-1}] \cdot [L^1]}{[A^1]} = [M^1 L^1 T^{-2} A^{-2}]$

We will match each entry in List I to its corresponding dimensional formula in List II:

1. Torque (A): As derived above, torque has dimensions $[M^1 L^2 T^{-2}]$. This matches IV in List II.
   Match: A → IV
2. Magnetic Field (B): The magnetic field has dimensions $[M^1 T^{-2} A^{-1}]$, which corresponds to III in List II.
   Match: B → III
3. Magnetic Moment (C): The magnetic moment has dimensions $[L^2 A^1]$, matching II in List II.
   Match: C → II
4. Permeability of Free Space (D): Permeability has dimensions $[M^1 L^1 T^{-2} A^{-2}]$, which aligns with I in List II.
   Match: D → I

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

Students often confuse the following:

• Magnetic Field vs. Magnetic Moment: The magnetic field ($B$) has dimensions involving mass ($M$), while the magnetic moment ($m$) does not. Students sometimes mix up their dimensions due to similar terminology.
• Permeability vs. Permittivity: Permeability ($\mu_0$) has dimensions involving current ($A$), while permittivity ($\epsilon_0$) does not. Misremembering their definitions can lead to incorrect matches.
• Torque vs. Work/Energy: Both torque and work have the same dimensions ($[M^1 L^2 T^{-2}]$), but they are distinct physical quantities. The question tests dimensional awareness, not physical interpretation.

Exam Tip: Always derive dimensions from first principles (e.g., force laws, definitions) rather than memorizing them. This reduces errors in matching questions.