Step 1: Understanding the Question
We need to find the dimensional formulas for four fundamental physical constants and match them with the given options. Step 2: Key Formula or Approach
To find the dimension of a physical constant, we use a physical law or equation that contains it and express the constant in terms of other quantities whose dimensions are known. The fundamental dimensions are Mass (M), Length (L), Time (T), and Temperature (K). Step 3: Detailed Explanation A. Boltzmann constant (k):
From the ideal gas law for one molecule, \( \text{Energy} = \frac{3}{2}kT \). We can write \( k = \frac{\text{Energy}}{\text{Temperature}} \).
Dimension of Energy is \( [ML^2T^{-2}] \). Dimension of Temperature is \( [K] \).
\[ [k] = \frac{[ML^2T^{-2}]}{[K]} = [ML^2T^{-2}K^{-1}] \]
This matches III. B. Stefan's constant (\(\sigma\)):
From the Stefan-Boltzmann law, the energy radiated per unit area per unit time (Power/Area) is \( E = \sigma T^4 \). So, \( \sigma = \frac{\text{Power}}{\text{Area} \cdot T^4} \).
Dimension of Power is \( \frac{\text{Energy}}{\text{Time}} = \frac{[ML^2T^{-2}]}{[T]} = [ML^2T^{-3}] \). Dimension of Area is \( [L^2] \).
\[ [\sigma] = \frac{[ML^2T^{-3}]}{[L^2][K^4]} = [ML^0T^{-3}K^{-4}] \]
This matches IV. C. Planck's constant (h):
From the photon energy equation, \( E = hf \), where \(f\) is frequency. So, \( h = E/f \).
Dimension of Energy is \( [ML^2T^{-2}] \). Dimension of frequency is \( [T^{-1}] \).
\[ [h] = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}] \]
This matches II. D. Gravitational constant (G):
From Newton's law of universal gravitation, \( F = G\frac{m_1 m_2}{r^2} \). So, \( G = \frac{Fr^2}{m_1 m_2} \).
Dimension of Force is \( [MLT^{-2}] \). Dimension of distance \(r\) is \( [L] \). Dimension of mass is \( [M] \).
\[ [G] = \frac{[MLT^{-2}][L^2]}{[M][M]} = [M^{-1}L^3T^{-2}] \]
This matches I. Step 4: Final Answer
The correct matching is: A \(\to\) III, B \(\to\) IV, C \(\to\) II, D \(\to\) I. This corresponds to option (C).
