Three very large plates of same area are kept parrallel 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

there is no process of conduction for the flow of heat between the plates but there is a process of radiation of heat between them. Let `T_(1)` be the temperature of the mid-plate B. At steady state, the rate of flow of heat from plate C to B is equal to the rate of flow of heat from plate B to A.

Using Stefan's law, we have
`sigma A[(3T)^(4) - T_(1)^(4)] = sigma A[T_(1)^(4) -(2T)^(4)]`
or `81T^(4) - T_(1)^(4) = T_(1)^(4)-16 T^(4)`
or `2T_(1)^(4) =97 T^(4)`
or `T_(1) =((97)/(2))^(1//4) T`