JEE PYQ: Units & Measurements - Question ID be2ea82db078 (JEE Main 2020)

Core Formula & Concept:

In dimensional analysis, every physical quantity can be expressed in terms of fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($I$), thermodynamic temperature ($\Theta$), amount of substance ($N$), and luminous intensity ($J$). The given problem involves determining the dimensions of the quantity $f = \sqrt{\frac{h c^5}{G}}$ where:

• $h$ is Planck’s constant, with dimensions $[h] = M L^2 T^{-1}$.
• $c$ is the speed of light, with dimensions $[c] = L T^{-1}$.
• $G$ is the universal gravitational constant, with dimensions $[G] = M^{-1} L^3 T^{-2}$.

The goal is to find the dimensions of $f$ and match them with one of the given options (Energy, Momentum, Area, or Volume).

Step-by-Step Derivation:

Step 1: Write the expression for $f$ and substitute the dimensions of each constant: $f = \sqrt{\frac{h c^5}{G}}$ Substituting the dimensions: $[f] = \sqrt{\frac{[h] [c]^5}{[G]}}$ Step 2: Substitute the known dimensions:

• $[h] = M L^2 T^{-1}$
• $[c] = L T^{-1} \implies [c]^5 = (L T^{-1})^5 = L^5 T^{-5}$
• $[G] = M^{-1} L^3 T^{-2}$

Thus, $[f] = \sqrt{\frac{(M L^2 T^{-1})(L^5 T^{-5})}{M^{-1} L^3 T^{-2}}}$ Step 3: Simplify the expression inside the square root:

• Numerator: $(M L^2 T^{-1})(L^5 T^{-5}) = M L^{2+5} T^{-1-5} = M L^7 T^{-6}$
• Denominator: $M^{-1} L^3 T^{-2}$
• Fraction: $\frac{M L^7 T^{-6}}{M^{-1} L^3 T^{-2}} = M^{1 - (-1)} L^{7-3} T^{-6 - (-2)} = M^2 L^4 T^{-4}$

So, $[f] = \sqrt{M^2 L^4 T^{-4}}$ Step 4: Take the square root of the dimensions: $[f] = M^{2/2} L^{4/2} T^{-4/2} = M L^2 T^{-2}$ Step 5: Compare with the dimensions of the given options:

• Energy: $[E] = M L^2 T^{-2}$ (e.g., kinetic energy $\frac{1}{2}mv^2$)
• Momentum: $[p] = M L T^{-1}$ (e.g., $mv$)
• Area: $[A] = L^2$
• Volume: $[V] = L^3$

The dimensions of $f$ match exactly with those of Energy.

Common Traps & Exam Tip:

1. Incorrect substitution of dimensions: Students often confuse the dimensions of $G$ (gravitational constant) with those of other constants like the Coulomb constant. Always double-check the dimensions of fundamental constants from memory or the formula sheet.
2. Algebraic errors in exponents: When simplifying $\frac{M L^7 T^{-6}}{M^{-1} L^3 T^{-2}}$, it is easy to make mistakes in subtracting exponents. Remember: $\frac{M^a}{M^b} = M^{a-b}$ and $\frac{T^a}{T^b} = T^{a-b}$.
3. Square root of dimensions: Forgetting to halve the exponents when taking the square root of the dimensions is a common oversight. For example, $\sqrt{M^2 L^4 T^{-4}} = M L^2 T^{-2}$, not $M^2 L^4 T^{-4}$.
4. Mismatching with options: Some students may correctly derive $[f] = M L^2 T^{-2}$ but mistakenly associate it with momentum (which is $M L T^{-1}$) or other options. Always cross-verify with the dimensions of the options.

Exam Tip: When in doubt, recall that energy has the same dimensions as work ($W = F \cdot d$), which is $M L^2 T^{-2}$. This can serve as a quick sanity check.