Which of the following rods made of same material will conduct more heat in given time when their ends are maintained at the same temperature difference .

Options

A. `l = 1m, r =1cm`
B. `l =2m, r =2cm`
C. `l =3m, r =1cm`
D. `l =100c m,r =2cm`

Explanation

To determine which rod conducts more heat in a given time when their ends are maintained at the same temperature difference, we can use the formula for heat conduction:

\[ Q = \frac{K \cdot A \cdot (T_1 - T_2)}{L} \]

Where:
- \( Q \) is the heat conducted,
- \( K \) is the thermal conductivity of the material,
- \( A \) is the cross-sectional area of the rod,
- \( (T_1 - T_2) \) is the temperature difference across the rod,
- \( L \) is the length of the rod.

Since the material of the rods is the same, the thermal conductivity \( K \) is constant for all rods. Also, the temperature difference \( (T_1 - T_2) \) is the same for all rods. Therefore, we can simplify our analysis to focus on the ratio of the area \( A \) to the length \( L \):

\[ Q \propto \frac{A}{L} \]

### Step-by-Step Solution:

1. **Identify the parameters for each rod**:
- For each rod, we need to determine the cross-sectional area \( A \) and the length \( L \).

2. **Calculate \( A \) and \( L \) for each rod**:
- **Rod 1**:
- Area \( A_1 = 4\pi(1^2) = 4\pi \) cm²
- Length \( L_1 = 1 \) m
- **Rod 2**:
- Area \( A_2 = 4\pi(2^2) = 16\pi \) cm²
- Length \( L_2 = 2 \) m
- **Rod 3**:
- Area \( A_3 = 4\pi(1^2) = 4\pi \) cm²
- Length \( L_3 = 3 \) m
- **Rod 4**:
- Area \( A_4 = 4\pi(2^2) = 16\pi \) cm²
- Length \( L_4 = 0.01 \) m (1 cm)

3. **Calculate \( \frac{A}{L} \) for each rod**:
- **Rod 1**:
\[ \frac{A_1}{L_1} = \frac{4\pi}{1} = 4\pi \]
- **Rod 2**:
\[ \frac{A_2}{L_2} = \frac{16\pi}{2} = 8\pi \]
- **Rod 3**:
\[ \frac{A_3}{L_3} = \frac{4\pi}{3} \approx 1.33\pi \]
- **Rod 4**:
\[ \frac{A_4}{L_4} = \frac{16\pi}{0.01} = 1600\pi \]

4. **Compare the values of \( \frac{A}{L} \)**:
- Rod 1: \( 4\pi \)
- Rod 2: \( 8\pi \)
- Rod 3: \( 1.33\pi \)
- Rod 4: \( 1600\pi \)

5. **Determine which rod conducts the most heat**:
- Since \( Q \propto \frac{A}{L} \), the rod with the highest \( \frac{A}{L} \) ratio will conduct the most heat.
- From the calculations, Rod 4 has the highest \( \frac{A}{L} \) ratio.

### Conclusion:
Rod 4 will conduct the most heat in the given time when their ends are maintained at the same temperature difference.