Two cylindrical rods of lengths $l_{1}$ and $l_{2}$ , radii $r_{1}$ and $r_{2}$ have thermal conductivities $k_{1}$ and $k_{2}$ respectively. The ends of the rods are maintained at the same temperature difference. If $l_{1} = 2 l_{2}$ and $r_{1} = \frac{r_{2}}{2}$ , the rates of heat flow in them will be the same if $\frac{k_{1}}{k_{2}}$ is

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Solution

The correct option is D
8

The rate of heat flow in rods A and B are given by
$\frac{Q_{1}}{t} = \frac{k_{1} π r_{1}^{2} Δ θ}{l_{1}}$
and $\frac{Q_{2}}{t} = \frac{k_{2} π r_{2}^{2} Δ θ}{l_{2}}$
Now, $Q_{1} = Q_{2} \Rightarrow \frac{k_{1} r_{1}^{2}}{l_{1}} = \frac{k_{2} r_{2}^{2}}{l_{2}} \Rightarrow \frac{k_{1}}{k_{2}} = \frac{l_{1}}{l_{2}} \times \frac{r_{2}^{2}}{r_{1}^{2}} = 2 \times (2)^{2} = 8$
Hence, the correct choice is (d).

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