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(C) Since the plates do not touch each other, heat exchange occurs through radiation.In a steady state, the heat gained by the middle plate must equal

Vedclass · Accepted Answer

$1.7$

Vedclass · Answer

(C) Since the plates do not touch each other, heat exchange occurs through radiation.In a steady state, the heat gained by the middle plate must equal the heat lost by it.Let $A$ be the area of each plate, $\varepsilon$ be the emissivity, and $\sigma$ be the Stefan-Boltzmann constant. Let $T_1$ be the temperature of the middle plate.The heat received by the middle plate from the outer plates is $Q_{in} = A \varepsilon \sigma T^4 + A \varepsilon \sigma (2T)^4$.The middle plate emits radiation from both of its surfaces, so the heat lost is $Q_{out} = 2 A \varepsilon \sigma T_1^4$.Equating $Q_{in} = Q_{out}$:$A \varepsilon \sigma T^4 + A \varepsilon \sigma (16T^4) = 2 A \varepsilon \sigma T_1^4$$17 T^4 = 2 T_1^4$$T_1^4 = \frac{17}{2} T^4 = 8.5 T^4$$T_1 = (8.5)^{1/4} T \approx 1.71 T$.Thus, the temperature is close to $1.7 T$.

Three large identical plates are kept parallel to each other. The outer two plates are maintained at temperatures $T$ and $2T$, respectively. The temperature of the middle plate in steady state will be close to ........... $T$.

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