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

Options

A. `(65/2)^(1//4) T`
B. `(97/4)^(1//4)T`
C. `(97/2)^(1//4) T`
D. `(97)^(1//4) T`

Answer

C

Explanation

Let `T'` be the temperature of the middle plate (II) and A be area of each plate. Under steady state, the rate of energy received by the middle plate is equal to rate of energy emiited by it.
i.e., `sigmaA(3T)^(4) - sigmaA(T')^(4) = sigmaA(T')^(4) - sigmaA(2T)^(4)`
or `sigmaA[(3T)^(4) - (T')^(4)] = sigmaA[(T')^(4) - (2T)^(4)]`
or `(3T)^(4) - (T')^(4) = (T')^(4) - (2T)^(4)`
or `2(T')^(4) = T^(4)(3^(4)+2^(4)) = T^(4)(81+16) = 97T^(4)`
or `T'^(4) = 97/2 T^(4)` or `T' = (97/2)^(1//4) T`