Question

Heat is flowing through two cylindrical rods of the same material. The diameters of the rods are in the ratio of 1:2 and their lengths are in the ratio of 2:1. If the temperature difference between their ends is the same, then find the ratio of amounts of heat conducted through them per unit of time.
A. 1:1
B. 2:1
C. 1:4
D. 1:8

Answer 1

Hint:>In order to solve this problem we need to understand the amount of heat flow in the conductor. It is defined as the transfer of heat down a temperature gradient between two bodies in close physical contact.

Formula Used:
To find the rate of heat flow the formula is,
\[\dfrac{{dQ}}{{dt}} = - \dfrac{{KAT}}{L}\]
Where, A is cross sectional area, \[T\] is temperature and l is length of the cylinder.

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 1:2 and their lengths are in the ratio of 2:1. 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,
\[\dfrac{{dQ}}{{dt}} = - \dfrac{{KAT}}{L}\]
Since we have two cylindrical rods,
\[\dfrac{{d{Q_1}}}{{dt}} = - \dfrac{{K{A_1}T}}{{{L_1}}}\] and \[\dfrac{{d{Q_2}}}{{dt}} = - \dfrac{{K{A_2}T}}{{{L_2}}}\]

Now, if we take the ratios of these two, we get,
\[\dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{{\dfrac{{K{A_1}T}}{{{L_1}}}}}{{\dfrac{{K{A_2}T}}{{{L_2}}}}} \\ \]
\[\Rightarrow \dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{{\dfrac{{{A_1}}}{{{L_1}}}}}{{\dfrac{{{A_2}}}{{{L_2}}}}} \\ \]
\[\Rightarrow \dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{{{A_1}}}{{{L_1}}} \times \dfrac{{{L_2}}}{{{A_2}}} \\ \]
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,
\[\dfrac{{{d_1}}}{{{d_2}}} = \dfrac{1}{2}\] and \[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{1}{2}\]
The lengths have the ratio of,
\[\dfrac{{{l_1}}}{{{l_2}}} = \dfrac{2}{1}\]
Then, equation (1) will become,
\[\dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{1}{4} \times \dfrac{1}{2} \\ \]
\[\therefore \dfrac{{\dfrac{{d{Q_1}}}{{dt}}}}{{\dfrac{{d{Q_2}}}{{dt}}}} = \dfrac{1}{8} \\ \]
That is, \[\dfrac{{d{Q_1}}}{{dt}}:\dfrac{{d{Q_2}}}{{dt}} = 1:8\]
Therefore, the ratio of amounts of heat conducted through them per unit of time is 1:8

Hence, option D is the correct answer.

Note: The rate of conductive heat transfer depends on temperature gradient between the two bodies, the area of contact and the length of the conductor.