JEE PYQ: Units & Measurements - Question ID 4f3c5aa1ceb4 (JEE Main 2026)

Core Formula & Concept:

In dimensional analysis, every physical constant can be expressed in terms of the fundamental dimensions: mass ($\mathrm{M}$), length ($\mathrm{L}$), time ($\mathrm{T}$), and temperature ($\mathrm{K}$). The question requires us to match four constants—Boltzmann constant ($k_B$), Stefan’s constant ($\sigma$), Planck’s constant ($h$), and the gravitational constant ($G$)—with their correct dimensional formulae.

Key relations used:

• Boltzmann constant: $k_B = \frac{R}{N_A}$ (gas constant per molecule), and from the ideal-gas law $PV = nRT$, we deduce $[k_B] = \frac{[R]}{[N_A]} = \frac{[PV]}{[nT]}$
• Stefan’s law: $P = \sigma A T^4$, so $[\sigma] = \frac{[P]}{[A][T^4]}$
• Planck’s constant: $E = h\nu$, so $[h] = \frac{[E]}{[\nu]}$
• Newton’s law of gravitation: $F = G\frac{m_1m_2}{r^2}$, so $[G] = \frac{[F][r^2]}{[m_1][m_2]}$

Step-by-Step Derivation:

1. Boltzmann constant ($k_B$)

From the ideal-gas law $PV = nRT$, we have $R = \frac{PV}{nT}.$ Since $n = \frac{N}{N_A}$, $R = \frac{PV\,N_A}{N\,T} = k_B N_A.$ Thus $k_B = \frac{R}{N_A}.$ Dimensionally, $[R] = \frac{[P][V]}{[T]} = \frac{(\mathrm{M\,L^{-1}\,T^{-2}})(\mathrm{L^3})}{\mathrm{K}} = \mathrm{M\,L^2\,T^{-2}\,K^{-1}}.$ Because $N_A$ is dimensionless (count of molecules), $[k_B] = [R] = \mathrm{M\,L^2\,T^{-2}\,K^{-1}}.$ This matches List-II entry III.

2. Stefan’s constant ($\sigma$)

Stefan’s law states $P = \sigma A T^4$, so $\sigma = \frac{P}{A\,T^4}.$ Dimensionally, $[P] = \mathrm{M\,L^2\,T^{-3}},\quad [A] = \mathrm{L^2},\quad [T^4] = \mathrm{K^4}.$ Hence $[\sigma] = \frac{\mathrm{M\,L^2\,T^{-3}}}{\mathrm{L^2\,K^4}} = \mathrm{M\,L^0\,T^{-3}\,K^{-4}}.$ This matches List-II entry IV.

3. Planck’s constant ($h$)

From $E = h\nu$, we get $h = \frac{E}{\nu}.$ Dimensionally, $[E] = \mathrm{M\,L^2\,T^{-2}},\quad [\nu] = \mathrm{T^{-1}}.$ Therefore $[h] = \frac{\mathrm{M\,L^2\,T^{-2}}}{\mathrm{T^{-1}}} = \mathrm{M\,L^2\,T^{-1}}.$ This matches List-II entry II.

4. Gravitational constant ($G$)

Newton’s law $F = G\frac{m_1m_2}{r^2}$ gives $G = \frac{F\,r^2}{m_1m_2}.$ Dimensionally, $[F] = \mathrm{M\,L\,T^{-2}},\quad [r^2] = \mathrm{L^2},\quad [m_1m_2] = \mathrm{M^2}.$ Thus $[G] = \frac{\mathrm{M\,L\,T^{-2}}\cdot\mathrm{L^2}}{\mathrm{M^2}} = \mathrm{M^{-1}\,L^3\,T^{-2}}.$ This matches List-II entry I.

Matching the constants to their dimensions yields:

• A (Boltzmann constant) → III
• B (Stefan’s constant) → IV
• C (Planck’s constant) → II
• D (Gravitational constant) → I

This corresponds to option C.

Common Traps & Exam Tip:

Students often confuse the dimensions of Boltzmann’s constant with those of the gas constant or misapply the temperature exponent in Stefan’s law. A frequent error is swapping the dimensions of $k_B$ and $\sigma$. Always derive each constant from its defining equation rather than memorising the dimensions.