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

Options

A. `r = 2r_(0) , l=2l_(0)`
B. `r = 2r_(0) , l=l_(0)`
C. `r = r_(0) , l=l_(0)`
D. `r = r_(0) , l=2l_(0)`

Answer

To solve the problem of which circular rod will conduct the most heat, we need to analyze the thermal resistance of each rod. The thermal resistance \( R \) for heat conduction can be expressed as:

\[ R = \frac{L}{K \cdot A} \]

where:
- \( L \) is the length of the rod,
- \( K \) is the thermal conductivity of the material,
- \( A \) is the cross-sectional area of the rod.

For a circular rod with radius \( r \), the cross-sectional area \( A \) can be calculated using the formula:

\[ A = \pi r^2 \]

Thus, the thermal resistance can be rewritten as:

\[ R = \frac{L}{K \cdot \pi r^2} \]

From this equation, we can see that the thermal resistance \( R \) is directly proportional to the length \( L \) and inversely proportional to the square of the radius \( r^2 \). Therefore, to minimize the thermal resistance (and thus maximize heat conduction), we need to minimize \( L \) and maximize \( r^2 \).

### Step-by-Step Solution:

1. **Identify the Parameters**: Each rod has the same material, radius \( r \), and length \( l \). The ends are maintained at the same temperature.

2. **Write the Formula for Thermal Resistance**:
\[ R = \frac{L}{K \cdot \pi r^2} \]

3. **Analyze the Resistance for Each Option**:
- For the first rod, let’s assume it has a length \( L_1 \) and radius \( r_1 \).
- For the second rod, let’s assume it has a length \( L_2 \) and radius \( r_2 \).
- Continue this for all given options.

4. **Calculate the Resistance for Each Rod**:
- For Rod 1: \( R_1 = \frac{L_1}{K \cdot \pi r_1^2} \)
- For Rod 2: \( R_2 = \frac{L_2}{K \cdot \pi r_2^2} \)
- For Rod 3: \( R_3 = \frac{L_3}{K \cdot \pi r_3^2} \)
- For Rod 4: \( R_4 = \frac{L_4}{K \cdot \pi r_4^2} \)

5. **Compare the Resistances**: Identify which rod has the minimum resistance. The rod with the smallest value of \( R \) will conduct the most heat.

6. **Conclusion**: Based on the calculations, determine which rod has the minimum resistance. According to the video solution, option 2 has the minimum resistance and therefore will conduct the most heat.

### Final Answer:
The rod that conducts the most heat is **Option 2**.