Question

Same quantity of ice is filled in each of the two identical metal containers P and Q having the same size and shape but of different materials. In P ice melts completely in time \( t_1 \), whereas in Q the time taken is \( t_2 \). Then the ratio of conductivities of P and Q is:

• A. \( \frac{t_2}{t_1} \)
• B. \( \sqrt{\frac{t_1}{t_2}} \)
• C. \( \frac{t_2}{t_1^2} \)
• D. \( \frac{t_2}{t_2^2} \)

Hint:

The time taken to melt a substance is inversely proportional to the thermal conductivity of the material.

Answer 1

The Correct Option is B

The rate of heat transfer \( Q \) is given by: \[ Q = k A \frac{\Delta T}{d} \] where: - \( k \) is the thermal conductivity, - \( A \) is the surface area, - \( \Delta T \) is the temperature difference, - \( d \) is the thickness of the material. The time to melt the ice is inversely proportional to the thermal conductivity: \[ t \propto \frac{1}{k} \] Thus, the ratio of the times for containers P and Q is related to the square root of the ratio of their thermal conductivities: \[ \frac{t_2}{t_1} = \sqrt{\frac{k_1}{k_2}} \] Thus, the ratio of conductivities is: \[ \frac{k_1}{k_2} = \frac{t_1^2}{t_2} \]