The ratio of thermal conductivity of two rods of different material is `5 : 4`. The two rods of same area of cross-section and same thermal resistance will have the lengths in the ratio

Options

A. `4 : 5`
B. `9 : 1`
C. `1 : 9`
D. `5 : 4`

Answer

To solve the problem of finding the ratio of lengths of two rods made of different materials with given thermal conductivities, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The ratio of thermal conductivities of the two rods is given as \( K_1 : K_2 = 5 : 4 \).
- The area of cross-section of both rods is the same, i.e., \( A_1 = A_2 \).
- The thermal resistance of both rods is the same, i.e., \( R_1 = R_2 \).

2. **Write the Formula for Thermal Resistance**:
- The thermal resistance \( R \) for a rod is given by the formula:
\[ R = \frac{L}{K \cdot A} \]
- Where \( L \) is the length of the rod, \( K \) is the thermal conductivity, and \( A \) is the area of cross-section.

3. **Set Up the Equations for Both Rods**:
- For rod 1:
\[ R_1 = \frac{L_1}{K_1 \cdot A_1} \]
- For rod 2:
\[ R_2 = \frac{L_2}{K_2 \cdot A_2} \]

4. **Since the Areas are Equal**:
- Since \( A_1 = A_2 \), we can denote them as \( A \). Thus, we can rewrite the equations as:
\[ R_1 = \frac{L_1}{K_1 \cdot A} \]
\[ R_2 = \frac{L_2}{K_2 \cdot A} \]

5. **Equate the Resistances**:
- Since \( R_1 = R_2 \):
\[ \frac{L_1}{K_1 \cdot A} = \frac{L_2}{K_2 \cdot A} \]

6. **Cancel Out the Area**:
- The area \( A \) cancels out from both sides:
\[ \frac{L_1}{K_1} = \frac{L_2}{K_2} \]

7. **Rearranging the Equation**:
- Rearranging gives us:
\[ \frac{L_1}{L_2} = \frac{K_1}{K_2} \]

8. **Substituting the Known Ratio of Conductivities**:
- We know \( \frac{K_1}{K_2} = \frac{5}{4} \):
\[ \frac{L_1}{L_2} = \frac{5}{4} \]

9. **Conclusion**:
- Therefore, the ratio of the lengths of the two rods is:
\[ L_1 : L_2 = 5 : 4 \]

### Final Answer:
The ratio of the lengths of the two rods is \( 5 : 4 \).

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