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?

Options

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

Answer

To determine which of the circular rods will conduct the most heat, we will analyze the thermal resistance of each rod and identify the one with the lowest resistance, as this will allow for the highest heat conduction.

### Step-by-Step Solution:

1. **Understanding Heat Conduction**:
The heat current (I) through a rod can be expressed as:
\[ I = \frac{\Delta T}{R} \]
where \( \Delta T \) is the temperature difference across the ends of the rod and \( R \) is the thermal resistance.

2. **Thermal Resistance Formula**:
The thermal resistance \( R \) of a rod can be calculated using the formula:
\[ R = \frac{L}{kA} \]
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.

3. **Calculating Cross-Sectional Area**:
The cross-sectional area \( A \) of a circular rod with radius \( r \) is given by:
\[ A = \pi r^2 \]

4. **Substituting Values for Each Rod**:
For each rod, we will calculate the resistance \( R \) using the given lengths and radii. Let's denote the rods as \( R_1, R_2, R_3, R_4 \) with their respective lengths and radii.

- **Rod 1**: Length = \( 2L_0 \), Radius = \( r_0 \)
\[ R_1 = \frac{2L_0}{k \pi (r_0)^2} \]

- **Rod 2**: Length = \( L_0 \), Radius = \( 2r_0 \)
\[ R_2 = \frac{L_0}{k \pi (2r_0)^2} = \frac{L_0}{k \pi \cdot 4r_0^2} = \frac{L_0}{4k \pi r_0^2} \]

- **Rod 3**: Length = \( L_0 \), Radius = \( r_0 \)
\[ R_3 = \frac{L_0}{k \pi (r_0)^2} \]

- **Rod 4**: Length = \( 2L_0 \), Radius = \( 2r_0 \)
\[ R_4 = \frac{2L_0}{k \pi (2r_0)^2} = \frac{2L_0}{k \pi \cdot 4r_0^2} = \frac{L_0}{2k \pi r_0^2} \]

5. **Comparing Resistances**:
Now we compare the coefficients of each resistance:
- \( R_1 = \frac{2L_0}{k \pi r_0^2} \) (coefficient = 2)
- \( R_2 = \frac{L_0}{4k \pi r_0^2} \) (coefficient = 1/4)
- \( R_3 = \frac{L_0}{k \pi r_0^2} \) (coefficient = 1)
- \( R_4 = \frac{L_0}{2k \pi r_0^2} \) (coefficient = 1/2)

The smallest coefficient corresponds to \( R_2 \), which means \( R_2 \) has the lowest resistance.

6. **Conclusion**:
Since \( R_2 \) has the minimum resistance, it will conduct the most heat. Therefore, the correct answer is:
\[ \text{Rod 2 conducts the most heat.} \]