Two rectangular blocks, having identical dimensions, can be arranged either in configuration $I$ or in configuration $II$ as shown in the figure. One of the blocks has thermal conductivity $k$ and the other $2k$. The temperature difference between the ends along the $x-$ axis is the same in both the configurations. It takes $9s$ to transport a certain amount of heat from the hot end to the cold end in the configuration $I$. The time to transport the same amount of heat in the configuration $II$ is .......... $\sec$
Diffcult
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$R_{1}=\frac{\ell}{K A}=2 R$
in configuration $1$
in configuration $2$
$\Delta Q_{1}=\frac{\Delta T}{3 R} t_{1}$
$\mathrm{R}_{2}=\frac{\ell}{2 \mathrm{kA}}=\mathrm{R}$
equivalent $R=3 R$
equivalent $\mathrm{R}=\frac{2}{3} \mathrm{R}$
$ \Delta \mathrm{Q}_{2}=\frac{\Delta \mathrm{T}}{\frac{2 \mathrm{R}}{3} \mathrm{t}_{2}}$
$\Rightarrow \frac{\Delta \mathrm{T}}{3 \mathrm{R}} \mathrm{t}_{1}=\frac{3 \Delta \mathrm{T}}{2 \mathrm{R}} \mathrm{t}_{2} \Rightarrow \mathrm{t}_{2}=\frac{2}{9} \mathrm{t}_{1}=2 \mathrm{sec}$
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