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Stefan-Boltzman's Law

Three very la...

Question

Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures $2 T$ and $3 T$ respectively. The temperature of the middle (i.e. second) plate under steady state condition is?

A

{(\frac{65}{2})}^{1 / 4} T

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B

{(\frac{97}{4})}^{1 / 4} T

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C

{(\frac{97}{2})}^{1 / 4} T

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D

(97)^{1 / 4} T

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Solution

The correct option is C ${(\frac{97}{2})}^{1 / 4} T$
The net heat received by the middle plate from the hotter plate is equal to the net heat given to the cooler plate. Hence
$σ A ({(3 T)}^{4} - T_{o}^{4}) = σ A (T_{o}^{4} - {(2 T)}^{4})$
or
$T_{o}^{4} = \frac{97 T^{4}}{2} \Rightarrow T_{o} = {(\frac{97}{2})}^{\frac{1}{4}} T$

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