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The ratio of thermal conductivity of two rods of two different materials is $5:4$ . The two rods of same area of cross-section and same thermal resistance will have the length in the ratio of
(A) $4:5$
(B) $8:10$
(C) $10:8$
(D) $5:4$

Last updated date: 01st Mar 2024
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IVSAT 2024
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Hint: In physics, thermal conductivity is defined as the measurement of the ability of a material to conduct heat or allow it to pass heat energy. It's denoted by $K$ . We will use the general formula of thermal conductivity to find the ratio of lengths of two rods.

Complete step-by-step solution:
We know that the general formula of thermal conductivity is given as:
$K = \dfrac{{Qd}}{{AT}}$ where,
$Q$ Is the amount of heat produced in the rod at given temperature
$d$ Is the length of the rod
$A$ Is the area of the cross section of the rod.
$T$ Is the temperature at which thermal conductivity is measured
$K$ Is called the thermal conductivity of a material.
We have given that the thermal resistances and area of cross section of two rods are same which means the thermal conductivity of both rods depends upon the length only, so
Let ${K_1}$ and ${K_2}$ be the thermal conductivity of two rods and there ratio are given as $\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{5}{4}$
Since, thermal conductivities are directly proportional to lengths of rod so,
$\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{{d_1}}}{{{d_2}}}$
$\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{5}{4} = \dfrac{{{d_1}}}{{{d_2}}}$
$ \Rightarrow \dfrac{{{d_1}}}{{{d_2}}} = \dfrac{5}{4}$
So, the ratio of length of two rods are $5:4$
Hence, the correct option is (D).

Note: It should be remembered that, the SI unit of thermal conductivity is $W{m^{ - 1}}{K^{ - 1}}$ and thermal conductivity decreases with increase in area of a body and also decreases with rise in temperature which means body at low temperature allow much heat to pass through them.

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