Step 1: Determining the Spring Constant According to Hooke’s law: \[ F = kx \quad \Rightarrow \quad k = \frac{F}{x} \] The dimensional formula for the spring constant is: \[ [k] = \frac{MLT^{-2}}{L} = MT^{-2} \] Hence, the spring constant corresponds to option (iii). Step 2: Understanding Thermal Conductivity The equation for heat conduction is: \[ Q = kA\frac{(T_1-T_2)}{\ell}t \] The dimensional formula for thermal conductivity is: \[ [k] = \frac{ML^2T^{-2}}{LTK \cdot T} = MLT^{-3}K^{-1} \] Therefore, thermal conductivity matches option (ii). Step 3: Exploring the Boltzmann Constant The definition of the Boltzmann constant is: \[ k_B = \frac{\text{Energy}}{\text{Temperature}} \] The dimensional formula for the Boltzmann constant is: \[ [k_B] = \frac{ML^2T^{-2}}{K} = ML^2T^{-2}K^{-1} \] This gives the Boltzmann constant as option (i). Step 4: Inductance The energy stored in an inductor is given by: \[ U = \frac{1}{2}LI^2 \] The dimensional formula for inductance is: \[ [L] = \frac{ML^2T^{-2}}{A^2} = ML^2T^{-2}A^{-2} \] Thus, inductance corresponds to option (iv). Final Matching: \[ (1) \to (iii), \quad (2) \to (ii), \quad (3) \to (i), \quad (4) \to (iv) \] \[ \boxed{\text{Correct option is (4)}} \]
