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Question

(C) Let the temperature of the middle plate be $T_0$.Since the plates are in a steady state,the heat absorbed by the middle plate must equal the heat

Vedclass · Accepted Answer

$\left(\frac{97}{2}\right)^{\frac{1}{4}} T$

Vedclass · Answer

(C) Let the temperature of the middle plate be $T_0$.Since the plates are in a steady state,the heat absorbed by the middle plate must equal the heat emitted by it.According to Stefan-Boltzmann Law,the power radiated by a black body is $P = \sigma A T^4$.For the middle plate,the heat gained from the third plate (at $3 T$) is $\sigma A (3 T)^4 - \sigma A T_0^4$.The heat lost to the first plate (at $2 T$) is $\sigma A T_0^4 - \sigma A (2 T)^4$.In steady state:$\sigma A (3 T)^4 - \sigma A T_0^4 = \sigma A T_0^4 - \sigma A (2 T)^4$$(3 T)^4 + (2 T)^4 = 2 T_0^4$$81 T^4 + 16 T^4 = 2 T_0^4$$97 T^4 = 2 T_0^4$$T_0^4 = \frac{97}{2} T^4$$T_0 = \left(\frac{97}{2}\right)^{\frac{1}{4}} T$

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