Units and Measurements Part 3

Concept:Each physical quantity has a unique dimensional formula derived from its defining equation.We find the dimensions of spring constant, thermal conductivity, Boltzmann constant, and inductive reactance using their standard relations, then match with the given list.Explanation:A. Spring constant $k$: From Hooke’s law, $F = -k x$. Taking magnitude, $k = \frac{F}{x}$.$[F] = [MLT^{-2}]$, $[x] = [L]$. So $[k] = \frac{[MLT^{-2}]}{[L]} = [ML^{0}T^{-2}]$. This matches List-II, II.B. Thermal conductivity $K$: From heat conduction, $H = \frac{K A \Delta T}{L}$, so $K = \frac{H L}{A \Delta T}$.$[H] = \frac{\text{energy}}{\text{time}} = [ML^{2}T^{-3}]$, $[L] = [L]$, $[A] = [L^{2}]$, $[\Delta T] = [K]$.Thus $[K] = \frac{[ML^{2}T^{-3}][L]}{[L^{2}][K]} = [MLT^{-3}K^{-1}]$. This matches IV.C. Boltzmann constant $k_{\text{B}}$: From kinetic theory, $E = \frac{3}{2} k_{\text{B}} T$. So $k_{\text{B}} = \frac{E}{T}$.$[E] = [ML^{2}T^{-2}]$, $[T] = [K]$. Hence $[k_{\text{B}}] = [ML^{2}T^{-2}K^{-1}]$. This matches I.D. Inductive reactance $X_L$: It behaves like resistance, so $[X_L] = [R]$. From Ohm’s law $V = IR$, $R = \frac{V}{I}$.$V = \frac{\text{work}}{\text{charge}}$, $[\text{work}] = [ML^{2}T^{-2}]$, $[\text{charge}] = [A T]$. So $[V] = [ML^{2}T^{-3}A^{-1}]$.Then $[R] = \frac{[ML^{2}T^{-3}A^{-1}]}{[A]} = [ML^{2}T^{-3}A^{-2}]$. This matches III.Therefore the correct matching is: A→II, B→IV, C→I, D→III.Answer:Option D: A-II, B-IV, C-I, D-III.