Two ends of rods of length L and radius R of the same material are kept at same temperature. Which of the following rods conducts the maximum heat ?

Question

Accepted Answer

To determine which rod conducts the maximum heat, we need to consider the factors that affect thermal conduction. The rate of heat conduction through a rod is given by Fourier's law of heat conduction, which states that the heat transfer rate (Q) is proportional to the cross-sectional area (A), the temperature difference (ΔT), and inversely proportional to the length (L) of the rod. The formula is:

$$ Q = \frac{k \cdot A \cdot \Delta T}{L} $$

where:
- $ Q $ is the heat transfer rate,
- $ k $ is the thermal conductivity of the material,
- $ A $ is the cross-sectional area of the rod,
- $ \Delta T $ is the temperature difference between the ends of the rod,
- $ L $ is the length of the rod.

Since all rods are made of the same material and are kept at the same temperature difference, $ k $ and $ \Delta T $ are constant for all rods. Therefore, the heat conduction depends on the cross-sectional area and the length of the rod.

1. **Option 4 (L = 100 cm, R = 2.0 cm):**
   - Cross-sectional area, $ A = \pi R^2 = \pi (2.0)^2 = 4\pi $ cm².
   - Length, $ L = 100 $ cm.
   - The ratio $ \frac{A}{L} = \frac{4\pi}{100} = 0.04\pi $.

2. **Option 1 (L = 25 cm, R = 0.5 cm):**
   - Cross-sectional area, $ A = \pi (0.5)^2 = 0.25\pi $ cm².
   - Length, $ L = 25 $ cm.
   - The ratio $ \frac{A}{L} = \frac{0.25\pi}{25} = 0.01\pi $.

3. **Option 2 (L = 50 cm, R = 1.0 cm):**
   - Cross-sectional area, $ A = \pi (1.0)^2 = \pi $ cm².
   - Length, $ L = 50 $ cm.
   - The ratio $ \frac{A}{L} = \frac{\pi}{50} = 0.02\pi $.

4. **Option 3 (L = 75 cm, R = 1.5 cm):**
   - Cross-sectional area, $ A = \pi (1.5)^2 = 2.25\pi $ cm².
   - Length, $ L = 75 $ cm.
   - The ratio $ \frac{A}{L} = \frac{2.25\pi}{75} = 0.03\pi $.

From the calculations above, Option 4 has the highest $ \frac{A}{L} $ ratio, which means it has the greatest capacity to conduct heat. This is because a larger cross-sectional area allows more heat to pass through, and a longer length reduces the rate of heat transfer. However, in this case, the increase in area outweighs the increase in length, resulting in a higher heat conduction rate for Option 4.

Therefore, Option 4 is the correct answer as it provides the maximum heat conduction due to its optimal balance of cross-sectional area and length.