Question

The ratio of coefficient of thermal conductivity of two different materials is $5:3$. If the thermal resistance of rods of the same area of these materials is the same, then what is the ratio of length of these rods?
(A) $3:5$
(B) $5:3$
(C) $25:9$
(D) $9:25$

Answer 1

The ratio of the length of the rods can be determined by using the formula of the thermal conductivity which shows the relation between the thermal conductivity, length of the rod, thermal resistance of the rod and the area of the rod. By using this formula, the ratio of the length can be determined.
Useful formula:
The thermal conductivity of the material is given by,
$k = \dfrac{l}{{{R_\theta }A}}$
Where, $k$ is the thermal conductivity of the material, $l$ is the length of the material, ${R_\theta }$ is the thermal resistance of the material and $A$ is the area of the material.

Complete answer:
Given that,
The ratio of coefficient of thermal conductivity of two different materials is $5:3$.
And also given that, the thermal resistance of rods of the same area of these materials is the same, which means ${R_1} = {R_2}$ and ${A_1} = {A_2}$.
Now,
The thermal conductivity of the material is given by,
${k_1} = \dfrac{{{l_1}}}{{{R_1}{A_1}}}\,.................\left( 1 \right)$
Then,
The thermal conductivity of the material is given by,
${k_2} = \dfrac{{{l_2}}}{{{R_2}{A_2}}}\,.................\left( 2 \right)$
By the given information, the thermal resistance of rods of same area of these material is same, then the equation (1) and equation (2) are written as,
${k_1} \propto {l_1}\,...................\left( 3 \right)$ and ${k_2} \propto {l_2}\,...................\left( 4 \right)$
From equation (3) and equation (4), then
$\dfrac{{{k_1}}}{{{k_2}}} = \dfrac{{{l_1}}}{{{l_2}}}$
By substituting the ratio of the thermal conductivity in the above equation, then
$\dfrac{{{l_1}}}{{{l_2}}} = \dfrac{5}{3}$

Hence, the option (B) is the correct answer.

Note:
The thermal conductivity of the material is directly proportional to the length of the material and inversely proportional to the thermal resistance of the material and the area of the material. As the length of the material increases, then the thermal conductivity of the material also increases.