Question

Match the following quantities with their dimensions.

• A. A-Q, B-R, C-S, D-P
• B. A-P, B-Q, C-S, D-R
• C. A-P, B-S, C-R, D-P
• D. A-R, B-Q, C-S, D-P

Hint:

When matching physical quantities with their dimensions, break down the formula for each constant and evaluate the dimension using the basic quantities like mass \(M\), length \(L\), time \(T\), and temperature \(K\).

Answer 1

The Correct Option is D

Step 1: Analyzing the Boltzmann constant.
The dimensional formula of the Boltzmann constant \( K \) is given as:
\[ K = \frac{R}{N_A} \quad \text{(where \( R \) is gas constant and \( N_A \) is Avogadro’s number)} \] Dimension of \( K \) is:
\[ [M L^3 T^{-2}] \] Thus, \( \text{A} \) matches with \( \text{R} \).
Step 2: Analyzing Planck's constant.
The dimensional formula of Planck's constant \( h \) is:
\[ h = \frac{E}{\nu} \quad \text{(where \( E \) is energy and \( \nu \) is frequency)} \] Dimension of \( h \) is:
\[ [M L^2 T^{-1}] \] Thus, \( \text{B} \) matches with \( \text{Q} \).
Step 3: Analyzing Stefan's constant.
The dimensional formula of Stefan's constant \( \sigma \) is:
\[ \sigma = \frac{P}{T^4} \quad \text{(where \( P \) is power and \( T \) is temperature)} \] Dimension of \( \sigma \) is:
\[ [M L^2 T^{-3} K^{-4}] \] Thus, \( \text{C} \) matches with \( \text{S} \).
Step 4: Analyzing Gravitational constant.
The dimensional formula of gravitational constant \( G \) is:
\[ G = \frac{F R^2}{M} \quad \text{(where \( F \) is force, \( R \) is distance, and \( M \) is mass)} \] Dimension of \( G \) is:
\[ [M^{-1} L^3 T^{-2} K^4] \] Thus, \( \text{D} \) matches with \( \text{P} \).
Step 5: Conclusion.
The correct matching of quantities with their dimensions is:
\[ A-R, B-Q, C-S, D-P \] Thus, the correct answer is option (D).
Final Answer: (D) A-R, B-Q, C-S, D-P