• question_answer 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 completes is Rune ${{t}_{1}}$ whereas in Q takes a time ${{t}_{2}}$. The ratio of thermal conductivities of the materials of P and Q is : A)  ${{t}_{2}}:{{t}_{1}}$                                    B)  ${{t}_{1}}:{{t}_{2}}$ C)  ${{t}_{1}}^{2}:{{t}_{2}}^{2}$                               D)  ${{t}_{2}}^{2}:{{t}_{1}}^{2}$

Relation between temperature gradient (TG)and thermal conductivity (K) So, $d\frac{\theta }{dt}=-KAd\frac{\theta }{dx}\,=-KA\,\times (TG)$ i.e., $t\propto \,\frac{1}{K}\,\,\,\,\,\,or\,\,\,\,K\propto \,\frac{1}{t}$ ${{K}_{1}}\propto \,\frac{1}{{{t}_{1}}}\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,{{K}_{2}}\propto \,\frac{1}{{{t}_{2}}}$ From eqs. (i) and (ii), we get $\frac{{{K}_{1}}}{{{K}_{2}}}\,=\frac{1/{{t}_{1}}}{1/{{t}_{2}}}$ ${{K}_{1}}:{{K}_{2}}\,={{t}_{2}}:{{t}_{1}}$