Question

The ratio of the diameters of two metallic rods of the same material is 2:1 and their lengths are in the ratio of 1:4. If the temperature difference between them is equal, then find the ratio of the rate of flow of heat in them.
A. 2:1
B. 4:1
C. 8:1
D. 16:1

Answer 1

Hint:In order to solve this problem we need to understand the rate of heat transfer. The rate of flow of heat is the amount of heat that is transferred per unit of time. Here, using the formula for heat flow we are going to find the solution.

Formula Used:
To find the heat flow the formula is,
\[\dfrac{{dQ}}{{dt}} = - \dfrac{{KA\Delta T}}{L}\]
Where,
A is a cross-sectional area
\[\Delta T\] is the temperature difference between two ends of the metal
L is the length of the metal plate
K is the thermal conductivity

Complete step by step solution:
Here, the heat is flowing through two cylindrical rods of the same material. The diameters of the rods are in the ratio of 2:1 and their lengths are in the ratio of 1:4. If the temperature difference between their ends is the same, then we need to find the ratio of amounts of heat conducted through them per unit of time. The rate of flow of heat is,
\[\dfrac{{dQ}}{{dt}} = - \dfrac{{KA\Delta T}}{L}\]

Since we have two cylindrical rods,
\[\dfrac{{d{Q_1}}}{{dt}} = - \dfrac{{K{A_1}\Delta T}}{{{L_1}}}\]and \[\dfrac{{d{Q_2}}}{{dt}} = - \dfrac{{K{A_2}\Delta T}}{{{L_2}}}\]
Now, if we take the ratios of these two, we get,
\[\dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{{\dfrac{{K{A_1}\Delta T}}{{{L_1}}}}}{{\dfrac{{K{A_2}\Delta T}}{{{L_2}}}}}\]
\[\Rightarrow \dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{{\dfrac{{{A_1}}}{{{L_1}}}}}{{\dfrac{{{A_2}}}{{{L_2}}}}} \\ \]
\[\Rightarrow \dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{{{A_1}}}{{{L_1}}} \times \dfrac{{{L_2}}}{{{A_2}}}\]

We know that area, \[A = \pi {r^2}\]
\[\dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{{\pi {r_1}^2}}{{{L_1}}} \times \dfrac{{{L_2}}}{{\pi {r_2}^2}} \\ \]
\[\Rightarrow \dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{{{r_1}^2}}{{{r_2}^2}} \times \dfrac{{{L_2}}}{{{L_1}}}\]………….. (1)
Here, ratio of diameters is 1:2 that is,
\[\dfrac{{{d_1}}}{{{d_2}}} = \dfrac{2}{1}\] so, \[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{2}{1}\]

The lengths have the ratio of,
\[\dfrac{{{L_1}}}{{{L_2}}} = \dfrac{1}{4}\]
Then, equation (1) will become,
\[\dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{4}{1} \times \dfrac{4}{1} \\ \]
\[\Rightarrow \dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{{16}}{1}\]
That is, \[\dfrac{{d{Q_1}}}{{dt}}:\dfrac{{d{Q_2}}}{{dt}} = 16:1\]
Therefore, the ratio of the rate of flow of heat in them is 16:1

Hence, Option D is the correct answer.

Note: Do not get confused with the formula for thermal conductivity and the rate of heat transfer. Since thermal conductivity and heat transfer are related to each other.