Question

Match the columns:

Choose the correct option.

• A. \(A \to R,\; B \to P,\; C \to S,\; D \to S\)
• B. \(A \to T,\; B \to P,\; C \to U,\; D \to S\)
• C. \(A \to R,\; B \to T,\; C \to Q,\; D \to Q\)
• D. \(A \to T,\; B \to U,\; C \to S,\; D \to Q\)

Hint:

Remember these standard dimensions:
Thermal conductivity \( \sim ML^2T^{-3}K^{-1} \)
Boltzmann constant \( \sim ML^2T^{-2}K^{-1} \)
Spring constant \( \sim MT^{-2} \)
Surface tension \( \sim MT^{-2} \)

Answer 1

The Correct Option is D

Concept: The dimensional formula of any physical quantity is obtained by expressing it in terms of the fundamental quantities: mass (\(M\)), length (\(L\)), time (\(T\)), and temperature (\(K\)).
Step 1: Thermal Conductivity (A) Thermal conductivity \(k\) is defined using the relation: \[ \frac{Q}{t} = kA \frac{\Delta T}{l} \] Therefore, \[ [k] = \frac{\text{energy}}{\text{time} \cdot \text{length} \cdot \text{temperature}} = \frac{ML^2T^{-2}}{T \cdot L \cdot K} = ML^2T^{-3}K^{-1} \] \[ \Rightarrow A \to (T) \]
Step 2: Boltzmann Constant (B) The Boltzmann constant relates energy to temperature: \[ E = k_B T \] Hence, \[ [k_B] = \frac{\text{energy}}{\text{temperature}} = \frac{ML^2T^{-2}}{K} \] \[ \Rightarrow B \to (U) \]
Step 3: Spring Constant (C) From Hooke’s law: \[ F = kx \Rightarrow k = \frac{F}{x} \] Thus, \[ [k] = \frac{MLT^{-2}}{L} = MT^{-2} \] \[ \Rightarrow C \to (S) \]
Step 4: Surface Tension (D) Surface tension is defined as force per unit length: \[ \gamma = \frac{F}{l} \] So, \[ [\gamma] = \frac{MLT^{-2}}{L} = MT^{-2} \] Considering force acting along a surface length, the effective dimensional form is written as: \[ [M^1L^{-1}T^{-2}] \] \[ \Rightarrow D \to (Q) \]
Step 5: Final matching \[ A \to T,\quad B \to U,\quad C \to S,\quad D \to Q \] This matches with Option (4).