Question

Heat is conducted through a uniform rod ABC of length 1 m keeping the end A at \(100^{\circ}C\). If the temperature at the point B at a distance 0.4 m from the end A is \(80^{\circ}C\) and that at the other end C is \(40^{\circ}C\), then the ratio of heat conducted through AB to that through BC is

• A. \(3:4\)
• B. \(2:1\)
• C. \(1:2\)
• D. \(2:3\)
• E. \(1:1\)

Hint:

For steady state heat conduction, \(\frac{H_{AB}}{H_{BC}} = \frac{\Delta T_{AB}/L_{AB}}{\Delta T_{BC}/L_{BC}}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Rate of heat conduction \(H = \frac{kA(T_1 - T_2)}{L}\). For a uniform rod, \(k\) and \(A\) are constant. So \(H \propto \frac{\Delta T}{L}\).

Step 2: Detailed Explanation:
Length AB = 0.4 m, temperature difference = \(100 - 80 = 20^{\circ}C\)
Length BC = 0.6 m, temperature difference = \(80 - 40 = 40^{\circ}C\)
\(\frac{H_{AB}}{H_{BC}} = \frac{\Delta T_{AB}/L_{AB}}{\Delta T_{BC}/L_{BC}} = \frac{20/0.4}{40/0.6} = \frac{50}{66.67} = \frac{50}{200/3} = 50 \times \frac{3}{200} = \frac{150}{200} = \frac{3}{4}\)
Ratio = \(3:4\)

Step 3: Final Answer:
The ratio is \(3:4\).