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The ratio of thermal conductivity of two rods of different material is $5: 4$. The two rods of same area of cross-section and same thermal resistance will have the lengths in the ratio
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$5: 4$
The thermal resistance of a rod is given by the formula
$\mathrm{R}=\mathrm{L} / \mathrm{kA}$
Given that the two rods have the same thermal resistance and the same cross-sectional area, we can set their thermal resistances equal to each other and solve for the ratio of their lengths
$\mathrm{L} 1 / \mathrm{k} 1 \mathrm{~A}=\mathrm{L} 2 / \mathrm{k} 2 \mathrm{~A}$
Solving for L1/L2 gives:
L1/L2=k1/k2
So, the ratio of the lengths of the two rods is equal to the ratio of their thermal conductivities. Therefore, if the ratio of the thermal conductivities of the two rods is $5: 4$, then the ratio of their lengths will also be $5: 4$
$\mathrm{R}=\mathrm{L} / \mathrm{kA}$
Given that the two rods have the same thermal resistance and the same cross-sectional area, we can set their thermal resistances equal to each other and solve for the ratio of their lengths
$\mathrm{L} 1 / \mathrm{k} 1 \mathrm{~A}=\mathrm{L} 2 / \mathrm{k} 2 \mathrm{~A}$
Solving for L1/L2 gives:
L1/L2=k1/k2
So, the ratio of the lengths of the two rods is equal to the ratio of their thermal conductivities. Therefore, if the ratio of the thermal conductivities of the two rods is $5: 4$, then the ratio of their lengths will also be $5: 4$
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