Two spheres of different materials one with double the radius and… - Vidyadip

MCQ

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

A
$4: 5$
B
$5: 4$
C
$25: 1$
✓
$1: 25$

✓

Answer

Correct option: D.

$1: 25$

$Q=\frac{K A(\Delta \theta) t}{l}$
$\because Q$ and $\Delta \theta$ are same for both spheres hence$K \propto \frac{l}{A t} \propto \frac{l}{r^2 t}$
$\Rightarrow\frac{K_{\text {larger }}}{K_{\text {smaller }}}=\frac{l_l}{l_s} \times\left(\frac{r_s}{r_l}\right)^2\times \frac{t_s}{t_l}$.
It is given that $r_l=2 r_s, l_l=\frac{1}{4l_s}$ and $(t_1)=25\mathrm{~min}, t_s=16 \mathrm{~min}$.
$\Rightarrow \frac{K_{\text {larger}}} {K_{\text{smaller}}}=\left(\frac{1}{4}\right)\left(\frac{1}{2}\right)^2 \times \frac{16}{25}=\frac{1}{25}$

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