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The ratio of thermal conductivity of two...

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

A

`4:5`

B

`9:1`

C

`1:9`

D

`5:4`

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To solve the problem, we need to determine the ratio of the lengths of two rods made from different materials, given that their thermal conductivities are in the ratio of 5:4, and they have the same area of cross-section and thermal resistance. ### Step-by-Step Solution: 1. **Understanding Thermal Resistance:** The thermal resistance (R) of a rod is given by the formula: \[ R = \frac{\rho L}{A} \] where: - \( R \) is the thermal resistance, - \( \rho \) is the resistivity (which is inversely related to thermal conductivity \( k \)), - \( L \) is the length of the rod, - \( A \) is the area of cross-section. 2. **Given Ratios:** We know: - The ratio of thermal conductivities \( k_1 : k_2 = 5 : 4 \). - The area of cross-section for both rods is the same, so \( A_1 = A_2 \). 3. **Relating Resistivity and Conductivity:** Since resistivity \( \rho \) is inversely related to thermal conductivity \( k \), we can express the ratio of resistivities as: \[ \frac{\rho_1}{\rho_2} = \frac{k_2}{k_1} = \frac{4}{5} \] 4. **Setting Up the Resistance Equations:** The thermal resistances for the two rods can be expressed as: \[ R_1 = \frac{\rho_1 L_1}{A_1} \quad \text{and} \quad R_2 = \frac{\rho_2 L_2}{A_2} \] Since \( A_1 = A_2 \), we can denote \( A \) as the common area: \[ R_1 = \frac{\rho_1 L_1}{A} \quad \text{and} \quad R_2 = \frac{\rho_2 L_2}{A} \] 5. **Equating Thermal Resistances:** Given that the thermal resistances are equal (\( R_1 = R_2 \)): \[ \frac{\rho_1 L_1}{A} = \frac{\rho_2 L_2}{A} \] Simplifying this gives: \[ \rho_1 L_1 = \rho_2 L_2 \] 6. **Substituting the Resistivity Ratio:** Substitute \( \rho_1 = \frac{4}{5} \rho_2 \) into the equation: \[ \left(\frac{4}{5} \rho_2\right) L_1 = \rho_2 L_2 \] Dividing both sides by \( \rho_2 \) (assuming \( \rho_2 \neq 0 \)): \[ \frac{4}{5} L_1 = L_2 \] 7. **Finding the Length Ratio:** Rearranging gives: \[ L_1 = \frac{5}{4} L_2 \] Therefore, the ratio of lengths \( L_1 : L_2 \) 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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