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

$\Rightarrow A \varepsilon \sigma T^{4}+A \varepsilon \sigma(2 T)^{4}=2 A \varepsilon \sigma\left(T_{1}\right)^{4}$

$\Rightarrow \quad T^{4}+2^{4} T^{4}=2 T_{1}^{4} \Rightarrow \frac{17}{2} T^{4}=T_{1}^{4}$

$\Rightarrow \quad T_{1}^{4}=\left(\frac{17}{2}\right)^{\frac{1}{4}} \cdot T \text { or } T_{1}=1.7 T$