Question

A temperature difference can generate e.m.f. in some materials. Let $ S $ be the e.m.f. produced per unit temperature difference between the ends of a wire, $ \sigma $ the electrical conductivity and $ \kappa $ the thermal conductivity of the material of the wire. Taking $ M, L, T, I $ and $ K $ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $ Z = \frac{S^2 \sigma}{\kappa} $ is:

• A. \([M^0L^0T^0I^0K^0]\)
• B. \([M^0L^0T^0I^0K^{-1}]\)
• C. \([M^1L^2T^{-2}I^{-1}K^{-1}]\)
• D. \([M^1L^2T^{-4}I^{-1}K^{-1}]\)

Hint:

Always break compound expressions into steps and simplify dimensions using exponent rules.

Answer 1

The Correct Option is D

Step 1: Dimensional formula of \( S \)
\( S \) is e.m.f. per unit temperature, so: \[ S = \frac{\text{emf}}{\text{temperature}} = \frac{ML^2T^{-3}I^{-1}}{K} = [M^1L^2T^{-3}I^{-1}K^{-1}] \]
Step 2: Dimensional formula of \( \sigma \) (electrical conductivity) \[ \sigma = \frac{1}{\text{resistance} \cdot \text{length}} = \frac{1}{(ML^2T^{-3}I^{-2}) \cdot L} = [M^{-1}L^{-3}T^3I^2] \]
Step 3: Dimensional formula of \( \kappa \) (thermal conductivity) \[ \kappa = \frac{\text{energy}}{\text{time} \cdot \text{length} \cdot \text{temperature}} = \frac{ML^2T^{-2}}{T \cdot L \cdot K} = [M^1L^1T^{-3}K^{-1}] \]
Step 4: Compute \( Z = \frac{S^2 \sigma}{\kappa} \) \[ S^2 = [M^2L^4T^{-6}I^{-2}K^{-2}] \] \[ S^2 \sigma = [M^2L^4T^{-6}I^{-2}K^{-2}] \cdot [M^{-1}L^{-3}T^3I^2] = [M^1L^1T^{-3}K^{-2}] \] \[ Z = \frac{S^2 \sigma}{\kappa} = \frac{[M^1L^1T^{-3}K^{-2}]}{[M^1L^1T^{-3}K^{-1}]} = [M^0L^0T^0K^{-1}] \] Wait — that gives option (2). But this contradicts the earlier derivation. Let's re-evaluate carefully: \[ Z = \frac{S^2 \sigma}{\kappa} = \frac{[M^2L^4T^{-6}I^{-2}K^{-2}] \cdot [M^{-1}L^{-3}T^3I^2]}{[M^1L^1T^{-3}K^{-1}]} \] Numerator: \[ = [M^{2-1}L^{4-3}T^{-6+3}I^{-2+2}K^{-2}] = [M^1L^1T^{-3}K^{-2}] \] Denominator: \[ = [M^1L^1T^{-3}K^{-1}] \] So, \[ Z = [M^{1-1}L^{1-1}T^{-3+3}K^{-2+1}] = [M^0L^0T^0K^{-1}] \] So final result: Correct dimensional formula is: \[ \boxed{[M^0L^0T^0I^0K^{-1}]} \] Updated Correct Answer: (2) \([M^0L^0T^0I^0K^{-1}]\)