JEE PYQ: Units & Measurements - Question ID 0a8925b254f0 (JEE Main 2022)

In dimensional analysis, every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($I$), thermodynamic temperature ($\Theta$), amount of substance ($N$), and luminous intensity ($J$). The dimensions of a derived quantity are obtained by expressing it in terms of these base dimensions.

Key formulas and their dimensions relevant to this question:

• Velocity gradient ($G$): $G = \frac{dv}{dx}$, where $v$ is velocity and $x$ is displacement. Dimensions: $\frac{[L T^{-1}]}{[L]} = [T^{-1}]$.
• Decay constant ($\lambda$): In radioactive decay, the decay law is $N = N_0 e^{-\lambda t}$. Since the exponent must be dimensionless, $\lambda t$ must be dimensionless. Thus, dimensions of $\lambda$: $[T^{-1}]$.
• Wien's constant ($b$): From Wien's displacement law: $\lambda_{\text{max}} T = b$. Dimensions: $[L \Theta]$.
• Stefan constant ($\sigma$): From Stefan-Boltzmann law: $E = \sigma T^4$. Dimensions: $\frac{[M T^{-3} \Theta^{-4}] \cdot [L^2]}{[L^2]} = [M T^{-3} \Theta^{-4}]$.
• Angular frequency ($\omega$): $\omega = 2\pi f$, where $f$ is frequency. Dimensions: $[T^{-1}]$.
• Angular momentum ($L$): $L = mvr$, where $m$ is mass, $v$ is velocity, $r$ is radius. Dimensions: $[M] \cdot [L T^{-1}] \cdot [L] = [M L^2 T^{-1}]$.
• Wave number ($k$): $k = \frac{2\pi}{\lambda}$, where $\lambda$ is wavelength. Dimensions: $\frac{1}{[L]} = [L^{-1}]$.
• Avogadro number ($N_A$): Number of entities per mole. Dimensions: $[N^{-1}]$ (since it's a count per mole, and mole is amount of substance).

We compare the dimensions of each pair in the options:

1. Option A: Velocity gradient and decay constant
   Velocity gradient: $[T^{-1}]$
   Decay constant: $[T^{-1}]$
   Dimensions match.
2. Option B: Wien's constant and Stefan constant
   Wien's constant: $[L \Theta]$
   Stefan constant: $[M T^{-3} \Theta^{-4}]$
   Dimensions do not match.
3. Option C: Angular frequency and angular momentum
   Angular frequency: $[T^{-1}]$
   Angular momentum: $[M L^2 T^{-1}]$
   Dimensions do not match.
4. Option D: Wave number and Avogadro number
   Wave number: $[L^{-1}]$
   Avogadro number: $[N^{-1}]$
   Dimensions do not match.

Only Option A has two physical quantities with the same dimensions.

Students often confuse:

• Angular frequency ($\omega$) and angular momentum ($L$): Both have "angular" in their names, but their dimensions differ. $\omega$ is $[T^{-1}]$, while $L$ is $[M L^2 T^{-1}]$.
• Wave number ($k$) and Avogadro number ($N_A$): Both involve "number," but wave number is spatial frequency ($[L^{-1}]$), while Avogadro number is a count per mole ($[N^{-1}]$).
• Decay constant ($\lambda$) and velocity gradient ($dv/dx$): Both have dimensions $[T^{-1}]$, but students may overlook the physical meaning and assume they differ.

Exam Tip: Always write down the dimensional formula for each quantity before comparing. Avoid relying on intuition or similar-sounding names.