JEE PYQ: Units & Measurements - Question ID 7a2aaadd2a71 (JEE Main 2021)

In the SI system, every physical quantity can be expressed in terms of four fundamental dimensions: mass ($M$), length ($L$), time ($T$), and electric current ($A$). The key formulas we use are:

• Magnetic induction (B): From the Lorentz force $\vec{F} = q(\vec{v} \times \vec{B})$, we get $[B] = \frac{[F]}{[q][v]} = \frac{MLT^{-2}}{(AT)(LT^{-1})} = MT^{-2}A^{-1}$.
• Magnetic flux (Φ): $\Phi = \vec{B} \cdot \vec{A}$ gives $[\Phi] = [B][A] = (MT^{-2}A^{-1})(L^{2}) = ML^{2}T^{-2}A^{-1}$.
• Magnetic permeability (μ): From $B = \mu H$ and knowing $[H] = L^{-1}A$, we obtain $[\mu] = \frac{[B]}{[H]} = \frac{MT^{-2}A^{-1}}{L^{-1}A} = MLT^{-2}A^{-2}$.
• Magnetization (M): Magnetization is magnetic moment per unit volume, so $[M] = \frac{[m]}{[V]} = \frac{AL^{2}}{L^{3}} = L^{-1}A$.

Step 1: Magnetic Induction (B)

From $\vec{F} = q(\vec{v} \times \vec{B})$ we isolate $B$: $[B] = \frac{[F]}{[q][v]} = \frac{MLT^{-2}}{(AT)(LT^{-1})} = MT^{-2}A^{-1}$. This matches List-II entry (iii).

Step 2: Magnetic Flux (Φ)

$\Phi = B \cdot A$ gives $[\Phi] = [B][A] = (MT^{-2}A^{-1})(L^{2}) = ML^{2}T^{-2}A^{-1}$. This matches List-II entry (i).

Step 3: Magnetic Permeability (μ)

$B = \mu H$ and $[H] = L^{-1}A$ lead to $[\mu] = \frac{[B]}{[H]} = \frac{MT^{-2}A^{-1}}{L^{-1}A} = MLT^{-2}A^{-2}$. This matches List-II entry (iv).

Step 4: Magnetization (M)

Magnetization is magnetic moment per unit volume: $[M] = \frac{[m]}{[V]} = \frac{AL^{2}}{L^{3}} = L^{-1}A$. This matches List-II entry (ii).

Step 5: Matching to Options

We have: (a) → (iii), (b) → (i), (c) → (iv), (d) → (ii). This corresponds exactly to option D.

Students often confuse the dimensions of magnetic induction ($B$) with those of magnetic flux ($\Phi$), swapping (iii) and (i). Another frequent error is misidentifying magnetization as having the same dimensions as magnetic field strength ($H$), leading to an incorrect match for (d). Always derive each dimension from first principles rather than relying on memory.