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Consider two rods of same length and dif...

Consider two rods of same length and different specific heats (`S_1` and `S_2`), conductivities `K_1` and `K_2` and area of cross section (`A_1` and `A_2`) and both having temperature `T_1` and `T_2` at their ends. If the rate of heat loss due to conduction is equal then

A

`K_1A_1=K_2A_2`

B

`(K_1A_1)/S_1=(K_2A_2)/S_2`

C

`K_2A_1=K_1A_2`

D

`(K_2A_1)/S_2=(K_1A_2)/S_2`

Text Solution

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The correct Answer is:
To solve the problem, we need to analyze the heat conduction through two rods with different properties but equal rates of heat loss. ### Step-by-Step Solution: 1. **Define the Parameters:** - Let rod 1 have: - Specific heat: \( S_1 \) - Conductivity: \( K_1 \) - Area of cross-section: \( A_1 \) - Length: \( L \) - Temperature difference: \( T_1 - T_2 \) - Let rod 2 have: - Specific heat: \( S_2 \) - Conductivity: \( K_2 \) - Area of cross-section: \( A_2 \) - Length: \( L \) - Temperature difference: \( T_1 - T_2 \) 2. **Write the Formula for Heat Loss:** The rate of heat loss due to conduction through a rod can be expressed using Fourier's law of heat conduction: \[ \frac{dQ}{dt} = -K \cdot A \cdot \frac{dT}{dx} \] For a rod of length \( L \) with a temperature difference \( \Delta T \) across its ends, this can be simplified to: \[ \frac{dQ}{dt} = \frac{K \cdot A \cdot (T_1 - T_2)}{L} \] 3. **Set Up the Equations for Both Rods:** For rod 1: \[ \frac{dQ_1}{dt} = \frac{K_1 \cdot A_1 \cdot (T_1 - T_2)}{L} \] For rod 2: \[ \frac{dQ_2}{dt} = \frac{K_2 \cdot A_2 \cdot (T_1 - T_2)}{L} \] 4. **Equate the Rates of Heat Loss:** Given that the rate of heat loss due to conduction is equal for both rods, we set the two equations equal to each other: \[ \frac{K_1 \cdot A_1 \cdot (T_1 - T_2)}{L} = \frac{K_2 \cdot A_2 \cdot (T_1 - T_2)}{L} \] 5. **Cancel Common Terms:** Since \( (T_1 - T_2) \) and \( L \) are common in both sides, we can cancel them out: \[ K_1 \cdot A_1 = K_2 \cdot A_2 \] 6. **Final Relationship:** Thus, we arrive at the relationship: \[ K_1 \cdot A_1 = K_2 \cdot A_2 \] ### Conclusion: The relationship between the conductivities and cross-sectional areas of the two rods is given by: \[ K_1 A_1 = K_2 A_2 \]
To solve the problem, we need to analyze the heat conduction through two rods with different properties but equal rates of heat loss. ### Step-by-Step Solution: 1. **Define the Parameters:** - Let rod 1 have: - Specific heat: \( S_1 \) - Conductivity: \( K_1 \) ...
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