Three very large plates of same area are kept parallel (in vertical...

Let the temperature of the middle plate be $T_2$.

In steady state, the net rate of heat received by the middle plate must be zero. The middle plate receives radiation from the first plate on one side and from the third plate on the other side. It also emits radiation from both of its sides.

According to Stefan-Boltzmann law, the net rate of heat transfer per unit area to the second plate is given by:

$\sigma(T_1^4 - T_2^4) + \sigma(T_3^4 - T_2^4) = 0$

$T_1^4 + T_3^4 = 2T_2^4$

Given $T_1 = 2T and T_3 = 3T$, substituting these values:

$(2T)^4 + (3T)^4 = 2T_2^4$

$16T^4 + 81T^4 = 2T_2^4$

$97T^4 = 2T_2^4$

$T_2^4 = \dfrac{97}{2}T^4$

$T_2 = \left(\dfrac{97}{2}\right)^{1/4}T$

Answer: \left(\dfrac{97}{2}\right)^{1/4}T