JEE PYQ: Units & Measurements - Question ID fa0049b818b2 (JEE Main 2021)

In the chapter Units & Measurements, every physical quantity is expressed in terms of the fundamental dimensions: Mass ($M$), Length ($L$), Time ($T$), and Electric Current ($I$). The key formulas we use are:

• Kinetic energy: $E = \tfrac{1}{2}mv^2 \quad \Rightarrow \quad [E] = [M][L^2T^{-2}] = [ML^2T^{-2}]$.
• Linear momentum: $P = mv \quad \Rightarrow \quad [P] = [M][LT^{-1}] = [MLT^{-1}]$.
• Planck’s constant: $E = h\nu \quad \Rightarrow \quad h = \tfrac{E}{\nu} \quad \Rightarrow \quad [h] = \tfrac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}]$.
• Electric potential: $V = \tfrac{W}{q} = \tfrac{Force \times distance}{current \times time} \quad \Rightarrow \quad [V] = \tfrac{[MLT^{-2}][L]}{[I][T]} = [ML^2I^{-1}T^{-3}]$.

Step 1: Dimension of Planck’s constant $h$

From $E = h\nu$, we get $h = \tfrac{E}{\nu}$. Since $[E] = [ML^2T^{-2}]$ and $[\nu] = [T^{-1}]$, we have $[h] = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}].$ This matches List II entry (ii).

Step 2: Dimension of kinetic energy $E$

$E = \tfrac{1}{2}mv^2$ gives $[E] = [M][L^2T^{-2}] = [ML^2T^{-2}].$ This matches List II entry (iii).

Step 3: Dimension of electric potential $V$

$V = \tfrac{W}{q} = \tfrac{Force \times distance}{current \times time}$. $[V] = \frac{[MLT^{-2}][L]}{[I][T]} = [ML^2I^{-1}T^{-3}].$ This matches List II entry (iv).

Step 4: Dimension of linear momentum $P$

$P = mv$ gives $[P] = [M][LT^{-1}] = [MLT^{-1}].$ This matches List II entry (i).

Step 5: Matching the lists

We obtain:

• (a) $h$ $\to$ (ii)
• (b) $E$ $\to$ (iii)
• (c) $V$ $\to$ (iv)
• (d) $P$ $\to$ (i)

This corresponds exactly to option A.

Students often confuse the dimensions of energy ($[ML^2T^{-2}]$) with those of Planck’s constant ($[ML^2T^{-1}]$), or mistake electric potential ($[ML^2I^{-1}T^{-3}]$) for momentum ($[MLT^{-1}]$). A quick check is to remember:

• Energy has $T^{-2}$ while Planck’s constant has $T^{-1}$.
• Electric potential involves current $I$ in its dimension, momentum does not.

Always derive each dimension from its defining formula rather than memorizing the list.