JEE PYQ: Units & Measurements - Question ID b0741747278b (JEE Main 2003)

In the chapter Units & Measurements, the central idea is that two physical quantities can be added, subtracted, or compared only if they share the same dimensional formula. The dimensional formula expresses the dependence of a quantity on the fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($I$), thermodynamic temperature ($\Theta$), amount of substance ($N$), and luminous intensity ($J$).

Key formulas and concepts used:

• Work ($W$) and torque ($\tau$) are both defined as force times distance. Their dimensional formula is $[W] = [\tau] = M L^2 T^{-2}$.
• Momentum ($p$) is mass times velocity, so $[p] = M L T^{-1}$. Planck’s constant ($h$) appears in the relation $E = h \nu$, where $E$ is energy ($M L^2 T^{-2}$) and $\nu$ is frequency ($T^{-1}$). Hence $[h] = \frac{[E]}{[\nu]} = M L^2 T^{-1}$.
• Stress ($\sigma$) and Young’s modulus ($Y$) are both force per unit area, so $[\sigma] = [Y] = M L^{-1} T^{-2}$.
• The speed of light in vacuum is given by $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$, so $\left(\mu_0 \varepsilon_0\right)^{-1/2} = c$, whose dimension is $L T^{-1}$.

Step 1: Dimensions of torque and work

Torque $\tau = \vec r \times \vec F$ and work $W = \vec F \cdot \vec d$ both involve force ($M L T^{-2}$) times distance ($L$). Thus $[\tau] = [W] = M L^2 T^{-2}$. They have the same dimensions.

Step 2: Dimensions of momentum and Planck’s constant

Momentum $p = m v$ has dimensions $[p] = M \cdot (L T^{-1}) = M L T^{-1}$. Planck’s constant $h$ satisfies $E = h \nu$, so $[h] = \frac{[E]}{[\nu]} = \frac{M L^2 T^{-2}}{T^{-1}} = M L^2 T^{-1}$. Clearly $[p] \neq [h]$.

Step 3: Dimensions of stress and Young’s modulus

Stress $\sigma = \frac{F}{A}$ and Young’s modulus $Y = \frac{\text{stress}}{\text{strain}}$ both have dimensions $[\sigma] = [Y] = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2}$. They share the same dimensions.

Step 4: Dimensions of speed and $(\mu_0 \varepsilon_0)^{-1/2}$

Speed $v$ has dimension $L T^{-1}$. From Maxwell’s equations, the speed of light $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$, so $\bigl(\mu_0 \varepsilon_0\bigr)^{-1/2} = c$, whose dimension is also $L T^{-1}$. They match.

Conclusion:

The only pair that does not share the same dimensions is momentum and Planck’s constant. Therefore, the correct option is B.

1. Confusing torque and work: Students often think torque (a vector) and work (a scalar) must differ dimensionally. In fact, both are force times distance and share the same dimensions. 2. Planck’s constant misstep: Some forget that $h$ has dimensions of angular momentum ($M L^2 T^{-1}$), not momentum ($M L T^{-1}$). 3. Electromagnetic constants: A common error is to miscalculate the dimensions of $\mu_0$ and $\varepsilon_0$ separately instead of using the known relation $c = (\mu_0 \varepsilon_0)^{-1/2}$.

Exam Tip: Always write down the dimensional formula explicitly for each quantity before comparing. This avoids confusion between scalar and vector quantities that happen to share the same dimensions.