Q181 / 200 · WBJEE 2014 · Physics

Same quantity of ice is filled in each of the two metal containers $P$ and $Q$ having the same size, shape and wall thickness but made of different materials. The containers are kept in identical surroundings. The ice in $P$ melts completely in time $t_{1}$ whereas in $Q$ takes a time $t_{2}$. The ratio of thermal conductivities of the materials of $P$ and $Q$ is

  • Correct answer
    A
    $t_{2}: t_{1}$
  • B
    $t_{1}: t_{2}$
  • C
    $t_{1}^{2}: t_{2}^{2}$
  • D
    $t_{2}^{2}: t_{1}^{2}$

Solution

We know that Relation between temperature gradient (TG) and thermal conductivity $(K)$ So, $$ \frac{d \theta}{d t}=-K A \frac{d \theta}{d x}=-K A \times(\mathrm{TG}) $$ Le. $\mathrm{TG} \propto \frac{1}{K}\left(\frac{d \theta}{d t}=\right.$ constant $)$ or $K \times \frac{1}{T}$ $$ \begin{array}{l} K_{1} \times \frac{1}{t_{1}} \\ K_{2} \times \frac{1}{t_{2}} \end{array} $$ From the Eqs. (i) and (ii), we get $$ \frac{K_{1}}{K_{2}}=\frac{1 / t_{1}}{1 / t_{2}} $$ $$ K_{1}: K_{2}=t_{2}: t_{1} $$

Hence, the correct answer is option: 'A'.

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