Which of the following cylindrical rods will conduct most heat, when their ends are maintained at the same steady temperature

Options

A. Length 1 m , radius 1 cm
B. Length 2 m , radius 1 cm
C. Length 2 m , radius 2 cm
D. Length 1 m , radius 2 cm

Answer

To determine which cylindrical rod will conduct the most heat when their ends are maintained at the same steady temperature, we can use the formula for heat conduction through a cylindrical rod. The heat conducted (dQ) is given by:

\[ dQ \propto K \cdot A \cdot \frac{\Delta T}{L} \]

Where:
- \( K \) is the thermal conductivity of the material,
- \( A \) is the cross-sectional area,
- \( \Delta T \) is the temperature difference (which is constant in this case),
- \( L \) is the length of the rod.

### Step 1: Identify the parameters for each rod
For each rod, we need to identify:
- The radius (r)
- The length (L)
- The thermal conductivity (K) of the material (if different for each rod)

### Step 2: Calculate the cross-sectional area (A)
The cross-sectional area \( A \) of a cylindrical rod can be calculated using the formula:

\[ A = \pi r^2 \]

### Step 3: Write the expression for heat conduction (Q)
Using the parameters identified, we can express the heat conducted for each rod as:

\[ Q \propto K \cdot \pi r^2 \cdot \frac{\Delta T}{L} \]

This simplifies to:

\[ Q \propto \frac{K \cdot r^2}{L} \]

### Step 4: Compare the values for each rod
For each rod, substitute the values of \( K \), \( r \), and \( L \) into the expression \( \frac{K \cdot r^2}{L} \) to find which rod has the highest value.

### Step 5: Identify the rod with the maximum heat conduction
The rod with the highest value of \( \frac{K \cdot r^2}{L} \) will be the one that conducts the most heat.

### Example Calculation
Assuming we have the following rods:
- Rod A: \( K = K_1, r = 1 \text{ cm}, L = 1 \text{ m} \)
- Rod B: \( K = K_2, r = 1.5 \text{ cm}, L = 1 \text{ m} \)
- Rod C: \( K = K_3, r = 2 \text{ cm}, L = 1 \text{ m} \)
- Rod D: \( K = K_4, r = 2 \text{ cm}, L = 2 \text{ m} \)

Calculate \( Q \) for each rod:
- For Rod A: \( Q_A \propto \frac{K_1 \cdot (1^2)}{1} \)
- For Rod B: \( Q_B \propto \frac{K_2 \cdot (1.5^2)}{1} \)
- For Rod C: \( Q_C \propto \frac{K_3 \cdot (2^2)}{1} \)
- For Rod D: \( Q_D \propto \frac{K_4 \cdot (2^2)}{2} \)

### Conclusion
After calculating the values, compare them to find which rod has the highest value of \( Q \). In this case, based on the example, Rod C would likely conduct the most heat due to its larger radius, assuming all other factors are equal.