The ratio of the areas of cross section of two rods of different materials is 1:2 and the ratio of ther themal conductivities of their mateirals is 4: 3. On keeping equal temperature difference between the ends of theserods, the rate of conduction of heat are equal. The ratio of the lengths of the rods is

Options

A. `2:3`
B. `2:4`
C. `4:2`
D. `4:5`

Answer

To solve the problem step by step, we will use the formula for the rate of heat conduction through a rod, which is given by Fourier's law of heat conduction:

\[ \frac{dQ}{dt} = \frac{k \cdot A \cdot \Delta T}{L} \]

where:
- \( \frac{dQ}{dt} \) is the rate of heat transfer,
- \( k \) is the thermal conductivity,
- \( A \) is the cross-sectional area,
- \( \Delta T \) is the temperature difference across the length of the rod,
- \( L \) is the length of the rod.

### Step 1: Set Up the Ratios

Given:
- The ratio of the areas of cross-section of the two rods \( A_1 : A_2 = 1 : 2 \).
- The ratio of their thermal conductivities \( k_1 : k_2 = 4 : 3 \).
- The temperature difference across both rods is equal, i.e., \( \Delta T_1 = \Delta T_2 \).

### Step 2: Write the Equations for Heat Transfer

For rod 1:
\[ \frac{dQ_1}{dt} = \frac{k_1 \cdot A_1 \cdot \Delta T}{L_1} \]

For rod 2:
\[ \frac{dQ_2}{dt} = \frac{k_2 \cdot A_2 \cdot \Delta T}{L_2} \]

### Step 3: Set the Heat Transfer Rates Equal

Since the rate of heat conduction is equal for both rods:
\[ \frac{dQ_1}{dt} = \frac{dQ_2}{dt} \]

This implies:
\[ \frac{k_1 \cdot A_1 \cdot \Delta T}{L_1} = \frac{k_2 \cdot A_2 \cdot \Delta T}{L_2} \]

### Step 4: Cancel Out the Common Terms

Since \( \Delta T \) is the same for both rods, we can cancel it out:
\[ \frac{k_1 \cdot A_1}{L_1} = \frac{k_2 \cdot A_2}{L_2} \]

### Step 5: Rearrange to Find the Length Ratio

Rearranging gives us:
\[ \frac{L_1}{L_2} = \frac{k_1 \cdot A_1}{k_2 \cdot A_2} \]

### Step 6: Substitute the Ratios

Now, substituting the known ratios:
- \( \frac{k_1}{k_2} = \frac{4}{3} \)
- \( \frac{A_1}{A_2} = \frac{1}{2} \)

So:
\[ \frac{L_1}{L_2} = \frac{4/3 \cdot 1/2} = \frac{4}{6} = \frac{2}{3} \]

### Final Answer

Thus, the ratio of the lengths of the rods is:
\[ \frac{L_1}{L_2} = \frac{2}{3} \]