Question:
The temperature difference across two cylindrical rods A and B of same material and same mass are 40°C and 60°C respectively. In steady state, if the rates of flow of heat through the rods A and B are in the ratio 3 : 8, the ratio of the lengths of the rods A and B is:
The temperature difference across two cylindrical rods A and B of same material and same mass are 40°C and 60°C respectively. In steady state, if the rates of flow of heat through the rods A and B are in the ratio 3 : 8, the ratio of the lengths of the rods A and B is:
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Updated On: Mar 18, 2026
- 1 : 3
- 5 : 3
- 4 : 3
- 2 : 3
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The Correct Option is C
Approach Solution - 1
To solve this problem, we will use the formula for the rate of heat flow through a cylindrical rod in steady state:
Q/t = (kAΔT)/L
Where:
- Q/t is the rate of heat flow.
- k is the thermal conductivity of the material.
- A is the cross-sectional area of the rod.
- ΔT is the temperature difference across the rod.
- L is the length of the rod.
Since the rods A and B have the same material and mass, their densities and volumes are the same, implying equal cross-sectional areas for equal length rods since they are cylindrical. Thus, the cross-sectional area A can be considered equal for both rods.
For rod A:
QA/t = (kAΔTA)/LA
For rod B:
QB/t = (kAΔTB)/LB
Given:
- ΔTA = 40°C
- ΔTB = 60°C
- QA/t : QB/t = 3 : 8
To find LA/LB}, we set up the ratio using the heat flow formula:
(QA/t) / (QB/t) = (ΔTA * LB) / (ΔTB * LA)
Substituting the given values:
3/8 = (40 * LB) / (60 * LA)
Simplifying the fractions, we get:
3/8 = (2/3) * (LB/LA)
Cross-multiplying to solve for the length ratio:
3 * 3 = 8 * 2 * (LB/LA)
9 = 16 * (LB/LA)
LA/LB = 16/9
Reversing the order for ratios:
LB/LA = 9/16
The ratio of the lengths of rods A to B is therefore:
4 : 3
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Approach Solution -2
We are given the following data:
- Temperature difference across rod A, \( \Delta T_A = 40^\circ C \),
- Temperature difference across rod B, \( \Delta T_B = 60^\circ C \),
- The rates of heat flow through the rods A and B are in the ratio 3 : 8.
In steady state, the rate of heat flow \( Q \) through a rod is given by the formula:
\[
Q = \frac{kA\Delta T}{L},
\]
where:
- \( k \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( \Delta T \) is the temperature difference across the rod,
- \( L \) is the length of the rod.
Since both rods are made of the same material and have the same cross-sectional area, the formula for the rate of heat flow simplifies to:
\[
Q \propto \frac{\Delta T}{L}.
\]
Let the lengths of rods A and B be \( L_A \) and \( L_B \), respectively. The ratio of the rates of heat flow through rods A and B is:
\[
\frac{Q_A}{Q_B} = \frac{\frac{\Delta T_A}{L_A}}{\frac{\Delta T_B}{L_B}} = \frac{\Delta T_A}{\Delta T_B} \times \frac{L_B}{L_A}.
\]
Substitute the given values:
\[
\frac{Q_A}{Q_B} = \frac{40}{60} \times \frac{L_B}{L_A} = \frac{2}{3} \times \frac{L_B}{L_A}.
\]
We are told that the ratio of the heat flow is 3 : 8, so:
\[
\frac{2}{3} \times \frac{L_B}{L_A} = \frac{3}{8}.
\]
Solving for \( \frac{L_B}{L_A} \):
\[
\frac{L_B}{L_A} = \frac{3}{8} \times \frac{3}{2} = \frac{9}{16}.
\]
Thus, the ratio of the lengths of rods A and B is:
\[
\frac{L_A}{L_B} = \frac{4}{3}.
\]
Therefore, the correct answer is option (3).
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