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Two spheres of different materials one with double the radius and one-fourth wall thickness of the other, are filled with ice. If the time taken for complete melting ice in the large radius one is $25$ minutes and that for smaller one is $16$ minutes, the ratio of thermal conductivities of the materials of larger sphere to the smaller sphere is
$4:5$
$5:4$
$25:1$
$1:25$
Solution
(d) $Q = \frac{{KA\,(\Delta \theta )t}}{l}$
$Q$ and $\Delta \theta $ are same for both spheres hence
$K \propto \frac{l}{{At}} \propto \frac{l}{{{r^2}t}}$ ==> $\frac{{{K_{{\rm{larger}}}}}}{{{K_{{\rm{smaller}}}}}} = \frac{{{l_l}}}{{{l_s}}} \times {\left( {\frac{{{r_s}}}{{{r_l}}}} \right)^2} \times \frac{{{t_s}}}{{{t_l}}}$. It is given that ${r_l} = 2{r_s},$ ${l_l} = \frac{1}{4}{l_s}$ and ${t_1} = 25$min, ${t_s} = 16$min.
==> $\frac{{{k_{{\rm{larger}}}}}}{{{k_{{\rm{smaller}}}}}} = \left( {\frac{1}{4}} \right){\left( {\frac{1}{2}} \right)^2} \times \frac{{16}}{{25}} = \frac{1}{{25}}$
