Question

Two hollow metal spheres with radii in the ratio $1:2$ and ratio of thickness $2:3$ are filled with ice. If the ratio of times taken for the ice to completely melt is $2:3$, the ratio of their thermal conductivities is :

• A. 1 : 4
• B. 3 : 8
• C. 1 : 2
• D. 2 : 1

Answer 1

The correct option is C 1 : 2

$\frac{r_1}{r_2}=\frac{1}{2}\frac{l_1}{l_2}=\frac{2}{3}$

$\frac{t_1}{t_2}=\frac{2}{3}$

Let in container1, $Q_1$ heat is required and in container2 $Q_2$ heat is required

Ratio of time taken to melt complete ice=$\frac{2}{3}$

$\Rightarrow \frac{\frac{\frac{4}{3}\pi r_1^3\times {\rho }_{ice}\times L_f}{\frac{△Q\times k_1{A}_1}{l_1}}}{\frac{\frac{4}{3}\pi r_2^3\times {\rho }_{ice}\times L_f}{\frac{△Q\times k_2{A}_2}{l_2}}}=\frac{2}{3}$

$\Rightarrow \frac{\frac{1^3}{\frac{k_1\times 1^2}{2}}}{\frac{8}{\frac{k_2\times 1}{3}}}=\frac{2}{3}\frac{k_2}{k_1}=\frac{2}{1}\frac{k_1}{k_2}=\frac{1}{2}$