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: Understanding the heat conduction problem.
We are dealing with a steady state heat conduction problem. The temperature at the insulated end of the shorter rod is given as 55°C. Both rods are in the same environment, which is at 40°C. The temperature distribution in both rods will follow the Fourier's law of heat conduction.

Step 2: Apply the symmetry.
Since the two rods are made of the same material and have the same diameter, and both are exposed to the same ambient temperature, we expect a similar linear temperature gradient. The shorter rod has a temperature of 55°C at the insulated end and 100°C at the heated end. Therefore, the temperature difference is 45°C over a length of L.

Step 3: Estimate temperature at the midpoint of the longer rod.
For the longer rod, the temperature at the heated end is 100°C, and the same linear temperature gradient will apply. Hence, the temperature at the midpoint will be: \[ \text{Temperature at midpoint} = 100 - \left(\frac{100 - 40}{2}\right) = 50°C \]

Final Answer: \[ \boxed{50°C} \]