Two ends of rods of length L and radius R of the same material of kept at the same temperature. Which of the following rods conducts the maximum heat?

Options

A. L=50 cm, R = 1 cm
B. L = 100cm, R = 2 cm
C. L = 25 cm, R = 0.5 cm
D. L = 75 cm , R = 1.5 cm

Explanation

To determine which rod conducts the maximum heat, we can use the formula for the rate of heat transfer through conduction, which is given by:

\[ H = \frac{k \cdot A \cdot (T_1 - T_2)}{L} \]

Where:
- \( H \) is the rate of heat transfer (in watts),
- \( k \) is the thermal conductivity of the material (which is the same for both rods since they are made of the same material),
- \( A \) is the cross-sectional area of the rod,
- \( L \) is the length of the rod,
- \( T_1 \) and \( T_2 \) are the temperatures at the two ends of the rod.

### Step 1: Identify the Variables
Since both rods are made of the same material, we know that \( k \) is constant for both rods. The variables we need to compare are the cross-sectional area \( A \) and the length \( L \).

### Step 2: Calculate the Cross-Sectional Area
The cross-sectional area \( A \) of a rod with radius \( R \) is given by the formula:

\[ A = \pi R^2 \]

### Step 3: Write the Heat Transfer Equations
For the first rod, we can denote:
- Length \( L_1 \) and radius \( R_1 \), then the area \( A_1 = \pi R_1^2 \).
- The heat transfer rate \( H_1 \) can be expressed as:

\[ H_1 = \frac{k \cdot \pi R_1^2 \cdot (T_1 - T_2)}{L_1} \]

For the second rod:
- Length \( L_2 \) and radius \( R_2 \), then the area \( A_2 = \pi R_2^2 \).
- The heat transfer rate \( H_2 \) can be expressed as:

\[ H_2 = \frac{k \cdot \pi R_2^2 \cdot (T_1 - T_2)}{L_2} \]

### Step 4: Compare the Heat Transfer Rates
To determine which rod conducts the maximum heat, we need to compare \( H_1 \) and \( H_2 \):

\[ \frac{H_1}{H_2} = \frac{R_1^2 / L_1}{R_2^2 / L_2} \]

This simplifies to:

\[ H_1 \propto \frac{R_1^2}{L_1} \quad \text{and} \quad H_2 \propto \frac{R_2^2}{L_2} \]

### Step 5: Analyze the Ratios
From the above relationship, we can see that for maximum heat conduction, we want to maximize \( \frac{R^2}{L} \). This means:
- A larger radius \( R \) increases the heat conduction.
- A smaller length \( L \) also increases the heat conduction.

### Step 6: Conclusion
To find out which rod conducts the maximum heat, we need to evaluate the values of \( R \) and \( L \) for each rod. The rod with the highest value of \( \frac{R^2}{L} \) will conduct the maximum heat.