Four rods with different radii `r` and length `l` are used to connect two reservoirs of heat at different temperatures. Which one will conduct most heat ?

Options

A. r = 2 cm, l = 0.5 m
B. r = 1 cm, l = 0.5 m
C. r = 2 cm, l = 2 m
D. r = 1 cm, l = 1 m

Answer

To determine which of the four rods will conduct the most heat, we can use the formula for the rate of heat transfer through a rod:

\[ \frac{\Delta Q}{\Delta t} = k \cdot A \cdot \frac{\Delta T}{L} \]

Where:
- \(\Delta Q\) is the amount of heat transferred,
- \(\Delta t\) is the time,
- \(k\) is the thermal conductivity of the material (assumed to be the same for all rods),
- \(A\) is the cross-sectional area of the rod,
- \(\Delta T\) is the temperature difference between the two reservoirs,
- \(L\) is the length of the rod.

### Step 1: Identify the Cross-Sectional Area
The cross-sectional area \(A\) of a rod with radius \(r\) is given by:

\[ A = \pi r^2 \]

### Step 2: Substitute the Area into the Heat Transfer Formula
Substituting the area into the heat transfer formula gives:

\[ \frac{\Delta Q}{\Delta t} = k \cdot (\pi r^2) \cdot \frac{\Delta T}{L} \]

### Step 3: Simplify the Expression
Since \(k\), \(\Delta T\), and \(\pi\) are constants for all rods, we can simplify the expression to focus on the ratio:

\[ \frac{\Delta Q}{\Delta t} \propto \frac{r^2}{L} \]

### Step 4: Calculate the Ratio \( \frac{r^2}{L} \) for Each Rod
Now we will calculate \( \frac{r^2}{L} \) for each of the four rods:

1. **Rod 1**: \(r = 2\), \(L = 0.5\)
\[ \frac{r^2}{L} = \frac{2^2}{0.5} = \frac{4}{0.5} = 8 \]

2. **Rod 2**: \(r = 1\), \(L = 0.5\)
\[ \frac{r^2}{L} = \frac{1^2}{0.5} = \frac{1}{0.5} = 2 \]

3. **Rod 3**: \(r = 2\), \(L = 2\)
\[ \frac{r^2}{L} = \frac{2^2}{2} = \frac{4}{2} = 2 \]

4. **Rod 4**: \(r = 1\), \(L = 1\)
\[ \frac{r^2}{L} = \frac{1^2}{1} = \frac{1}{1} = 1 \]

### Step 5: Compare the Ratios
Now we compare the calculated ratios:

- Rod 1: \(8\)
- Rod 2: \(2\)
- Rod 3: \(2\)
- Rod 4: \(1\)

### Conclusion
Rod 1 has the highest ratio of \( \frac{r^2}{L} \), which means it will conduct the most heat. Therefore, the answer is:

**Rod 1 will conduct the most heat.**

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