Question

Which of the following circular rods,(given radius r and length l)each made of the same material and whose ends are maintained at the same temperature will conduct most heat?

• A. r = 2r₀ ; I=2l₀
• B. r = 2r₀ ; l=l₀
• C. r = r₀ ; l= l₀
• D. r = r₀ ; l=2l₀

Answer 1

The Correct Option is B

To determine which of the given rods will conduct the most heat, we need to consider the formula for the rate of heat conduction through a rod, which is given by:

Q = \frac{k \cdot A \cdot \Delta T \cdot t}{l}

Where:

• Q is the amount of heat conducted.
• 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.
• t is the time.
• l is the length of the rod.

Since all rods are made from the same material and are maintained at the same temperature difference, k, \Delta T, and t are constant for all rods. The key variables that influence the rate of heat conduction are the cross-sectional area (A = \pi r^2) and the length (l) of the rod.

The heat conduction rate is proportional to \frac{A}{l}. Therefore, for higher heat conduction, A should be maximized and l should be minimized. Now, let's compare the options:

1. Option 1: r = 2r_0 ; l = 2l_0
   • Cross-sectional area: A = \pi (2r_0)^2 = 4\pi r_0^2
   • Length of the rod: 2l_0
   • Ratio \frac{A}{l} = \frac{4\pi r_0^2}{2l_0} = 2\pi \frac{r_0^2}{l_0}
2. Option 2: r = 2r_0 ; l = l_0
   • Cross-sectional area: A = \pi (2r_0)^2 = 4\pi r_0^2
   • Length of the rod: l_0
   • Ratio \frac{A}{l} = \frac{4\pi r_0^2}{l_0} = 4\pi \frac{r_0^2}{l_0}
3. Option 3: r = r_0 ; l = l_0
   • Cross-sectional area: A = \pi r_0^2
   • Length of the rod: l_0
   • Ratio \frac{A}{l} = \frac{\pi r_0^2}{l_0}
4. Option 4: r = r_0 ; l = 2l_0
   • Cross-sectional area: A = \pi r_0^2
   • Length of the rod: 2l_0
   • Ratio \frac{A}{l} = \frac{\pi r_0^2}{2l_0} = 0.5 \pi \frac{r_0^2}{l_0}

Comparing the ratios, we find that Option 2 gives the highest ratio of \frac{A}{l}. Hence, the rod with r = 2r_0 and l = l_0 will conduct the most heat.

Conclusion: The correct answer is r = 2r_0 ; l = l_0.