Question

Two rods, one of length L and the other of length 2L, are made of the same material and have the same diameter. The two ends of the longer rod are maintained at 100°C. One end of the shorter rod is maintained at 100°C while the other end is insulated. Both the rods are exposed to the same environment at 40°C. The temperature at the insulated end of the shorter rod is measured to be 55°C. The temperature at the mid point of the longer rod would be:

• A. 40°C
• B. 50°C
• C. 55°C
• D. 100°C

Hint:

In heat conduction problems with steady-state conditions, temperature distribution is typically linear in a homogenous material with constant cross-sectional area.

Answer 1

The Correct Option is B

Step 1: Problem Definition. We are analyzing a steady-state heat conduction scenario. The insulated end of the shorter rod is at 55°C, and both rods are in an environment at 40°C. Temperature distribution in both rods will adhere to Fourier's law of heat conduction.

Step 2: Symmetry Application. Given identical material, diameter, and ambient conditions for both rods, a similar linear temperature gradient is anticipated.

The shorter rod has a 55°C temperature at its insulated end and 100°C at its heated end, indicating a 45°C temperature difference over length L.

Step 3: Longer Rod Midpoint Temperature Estimation. For the longer rod, with a heated end at 100°C and the same linear temperature gradient, the temperature at the midpoint is calculated as:

\[ \text{Temperature at midpoint} = 100 - \left(\frac{100 - 40}{2}\right) = 50°C \]

Final Answer: \[ \boxed{50°C} \]