Three large identical plates are kept parallel to each other. The outer two plates are maintained at temperatures $T$ and $2T$, respectively. The temperature of the middle plate in steady state will be close to

Plates does not touch each other, so exchange of heat takes place in form of radiations.

In steady state, heat gained by middle plate $=$ heat lost by it.

$\Rightarrow$ Heat radiations from plate $A$ $+$ Heat radiations from plate $B=$ Heat radiations emitted from both surfaces 

$\Rightarrow A\epsilon \sigma T^4+A\epsilon \sigma {\left(2T\right)}^4=2A\epsilon \sigma {\left(T_1\right)}^4$

$\Rightarrow T^4+2^4{T}^4=2T_1^4\Rightarrow \frac{17}{2}{T}^4=T_1^4$

$\Rightarrow T_1^4={\left(\frac{17}{2}\right)}^{\frac{1}{4}}·T$ or $T_1=1.7 T$