JEE PYQ: Units & Measurements - Question ID 633818e326a0 (JEE Main 2025)

In dimensional analysis, every physical quantity is expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), and time ($T$). The key formulas and concepts used here are:

• Coefficient of viscosity ($\eta$): Defined by Newton’s law of viscosity, $F = \eta A \frac{dv}{dx}$, where $F$ is force, $A$ is area, and $\frac{dv}{dx}$ is the velocity gradient. Dimensional formula: $[\eta] = [ML^{-1}T^{-1}]$.
• Intensity of wave ($I$): Intensity is power per unit area. Power has dimensions $[ML^2T^{-3}]$, so intensity: $[I] = [ML^0T^{-3}]$ (since area is $[L^2]$).
• Pressure gradient ($\frac{dP}{dx}$): Pressure is force per unit area, $[P] = [ML^{-1}T^{-2}]$. Gradient means dividing by length, so $[\frac{dP}{dx}] = [ML^{-2}T^{-2}]$.
• Compressibility ($\kappa$): Compressibility is the reciprocal of bulk modulus ($B$), which has dimensions of pressure: $[B] = [ML^{-1}T^{-2}]$. Thus, $[\kappa] = [M^{-1}LT^2]$.

Step 1: Coefficient of viscosity (A)

From Newton’s law of viscosity: $F = \eta A \frac{dv}{dx}$ Dimensional analysis: $[F] = [MLT^{-2}], \quad [A] = [L^2], \quad \left[\frac{dv}{dx}\right] = [T^{-1}]$ Solving for $\eta$: $[\eta] = \frac{[F]}{[A] \left[\frac{dv}{dx}\right]} = \frac{[MLT^{-2}]}{[L^2][T^{-1}]} = [ML^{-1}T^{-1}]$ This matches (IV).

Step 2: Intensity of wave (B)

Intensity is power per unit area: $I = \frac{P}{A}$ Power $P$ has dimensions $[ML^2T^{-3}]$, and area $A$ has $[L^2]$, so: $[I] = \frac{[ML^2T^{-3}]}{[L^2]} = [ML^0T^{-3}]$ This matches (I).

Step 3: Pressure gradient (C)

Pressure $P$ has dimensions $[ML^{-1}T^{-2}]$. Gradient means dividing by length: $\left[\frac{dP}{dx}\right] = \frac{[ML^{-1}T^{-2}]}{[L]} = [ML^{-2}T^{-2}]$ This matches (II).

Step 4: Compressibility (D)

Compressibility $\kappa$ is the reciprocal of bulk modulus $B$: $\kappa = \frac{1}{B}$ Bulk modulus has dimensions of pressure: $[B] = [ML^{-1}T^{-2}]$, so: $[\kappa] = \frac{1}{[ML^{-1}T^{-2}]} = [M^{-1}LT^2]$ This matches (III).

Matching Summary:

• (A) → (IV)
• (B) → (I)
• (C) → (II)
• (D) → (III)

This corresponds to Option C.

Students often confuse:

• Intensity vs. Power: Intensity is power per unit area, not just power. Forgetting to divide by area leads to incorrect dimensions.
• Pressure Gradient vs. Pressure: Pressure gradient is pressure divided by length, not just pressure. Misidentifying this leads to wrong dimensions.
• Compressibility vs. Bulk Modulus: Compressibility is the reciprocal of bulk modulus. Students sometimes forget to invert the dimensions.

Exam Tip: Always write down the defining formula for each quantity before deriving dimensions. This avoids confusion and ensures accuracy.