The length of the two rods made up of the same metal and having the same area of cross-section are 0.6 m and 0.8 m respectively. The temperature between the ends of first rod is `90^(@)` C and `60^(@)` C and that for the other rod is 150 and `110^(@)` C. For which rod the rate of conduction will be greater

Options

A. First
B. Second
C. Same for both
D. None of the above

Explanation

To determine which rod has a greater rate of conduction, we will use the formula for heat conduction, which is given by Fourier's law:

\[ \frac{Q}{T} = \frac{K \cdot A \cdot (\Delta T)}{L} \]

Where:
- \( Q \) is the heat transferred,
- \( T \) is the time,
- \( K \) is the thermal conductivity of the material,
- \( A \) is the cross-sectional area,
- \( \Delta T \) is the temperature difference between the ends of the rod,
- \( L \) is the length of the rod.

### Step 1: Identify the parameters for both rods
- **Rod 1:**
- Length \( L_1 = 0.6 \, m \)
- Temperature difference \( \Delta T_1 = 90^\circ C - 60^\circ C = 30^\circ C \)

- **Rod 2:**
- Length \( L_2 = 0.8 \, m \)
- Temperature difference \( \Delta T_2 = 150^\circ C - 110^\circ C = 40^\circ C \)

### Step 2: Calculate the rate of heat transfer for Rod 1
Using the formula for Rod 1:

\[ \frac{Q_1}{T} = \frac{K \cdot A \cdot \Delta T_1}{L_1} \]

Substituting the values:

\[ \frac{Q_1}{T} = \frac{K \cdot A \cdot 30}{0.6} \]

This simplifies to:

\[ \frac{Q_1}{T} = K \cdot A \cdot 50 \]

### Step 3: Calculate the rate of heat transfer for Rod 2
Using the formula for Rod 2:

\[ \frac{Q_2}{T} = \frac{K \cdot A \cdot \Delta T_2}{L_2} \]

Substituting the values:

\[ \frac{Q_2}{T} = \frac{K \cdot A \cdot 40}{0.8} \]

This simplifies to:

\[ \frac{Q_2}{T} = K \cdot A \cdot 50 \]

### Step 4: Compare the rates of heat transfer
From the calculations, we find:

- For Rod 1, \( \frac{Q_1}{T} = K \cdot A \cdot 50 \)
- For Rod 2, \( \frac{Q_2}{T} = K \cdot A \cdot 50 \)

Both rods have the same rate of heat transfer.

### Conclusion
The rate of conduction is the same for both rods.

**Final Answer: Both rods have the same rate of conduction.**

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