JEE PYQ: Units & Measurements - Question ID 71153e8f4fab (JEE Main 2021)

In dimensional analysis, every physical equation must satisfy the principle of dimensional homogeneity. This means that the dimensions (fundamental units like mass [M], length [L], time [T], etc.) on both sides of an equation must be identical. If an equation is dimensionally incorrect, it cannot represent a valid physical relationship.

Key formulas and dimensions relevant to this question:

• $t$ = time $\rightarrow$ [T]
• $h$ = height $\rightarrow$ [L]
• $s$ = surface tension $\rightarrow$ [MT^-2]
• $\theta$ = angle $\rightarrow$ dimensionless [1]
• $\rho$ = density $\rightarrow$ [ML^-3]
• $a, r, L$ = radius or length $\rightarrow$ [L]
• $g$ = acceleration due to gravity $\rightarrow$ [LT^-2]
• $v$ = volume $\rightarrow$ [L³]
• $p$ = pressure $\rightarrow$ [ML^-1T^-2]
• $W$ = work done $\rightarrow$ [ML²T^-2]
• $T$ or $\Gamma$ = torque $\rightarrow$ [ML²T^-2]
• $\in$ = permittivity $\rightarrow$ [M^-1L^-3T⁴I²]
• $E$ = electric field $\rightarrow$ [MLT^-3I^-1]
• $J$ = current density $\rightarrow$ [L^-2I]
• $\eta$ = viscosity (not explicitly given, but implied in option A) $\rightarrow$ [ML^-1T^-1]

Option A: $v = \frac{\pi p a^4}{8 \eta L}$

Left-hand side (LHS): $v$ = volume $\rightarrow$ [L³]
Right-hand side (RHS): $\frac{p a^4}{\eta L}$
Substitute dimensions:

• $p$ = [ML^-1T^-2]
• $a^4$ = [L⁴]
• $\eta$ = [ML^-1T^-1]
• $L$ = [L]

Option B: $h = \frac{2 s \cos \theta}{\rho r g}$

LHS: $h$ = [L]
RHS: $\frac{s}{\rho r g}$ (since $\cos \theta$ is dimensionless)
Substitute dimensions:

• $s$ = [MT^-2]
• $\rho$ = [ML^-3]
• $r$ = [L]
• $g$ = [LT^-2]

Option C: $J = \in \frac{\partial E}{\partial t}$

LHS: $J$ = current density $\rightarrow$ [L^-2I]
RHS: $\in \frac{\partial E}{\partial t}$
Substitute dimensions:

• $\in$ = [M^-1L^-3T⁴I²]
• $E$ = [MLT^-3I^-1]
• $\frac{\partial E}{\partial t}$ = [MLT^-4I^-1]

Option D: $W = \Gamma \theta$

LHS: $W$ = work done $\rightarrow$ [ML²T^-2]
RHS: $\Gamma \theta$ (where $\Gamma$ = torque, $\theta$ = angle)
Substitute dimensions:

• $\Gamma$ = [ML²T^-2]
• $\theta$ = dimensionless [1]

Students often overlook the following:

• Ignoring dimensions of constants: In Option A, $\pi$ and 8 are dimensionless, so they don’t affect dimensional analysis. However, students sometimes mistakenly assign dimensions to them.
• Confusing torque and work: Both $W$ and $\Gamma$ have the same dimensions [ML²T^-2], so $W = \Gamma \theta$ is valid. Students may think torque and work are different dimensionally.
• Derivatives in Option C: $\frac{\partial E}{\partial t}$ introduces an extra [T^-1], which students sometimes forget. However, permittivity $\in$ compensates for this, making the dimensions match.
• Angle as dimensionless: $\theta$ and $\cos \theta$ are dimensionless. Students sometimes incorrectly assign dimensions to angles.

Final Answer: Option A is dimensionally incorrect.