Three parallel plates, approximated as black bodies

Question

Consider three parallel plates, which can be approximated as ideal black bodies. If the left plane is kept at temperature $T$, and the rightmost one at $2T$, what is the temperature of the middle one?

I'm not sure how to approach the problem. I'm looking for a conceptual explanation of this problem, not a worked out solution. Where does one begin when considering black body radiation of this type?

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EDIT: After consulting the problem with my professor, I came to this solution, which he confirmed.

For the plates A, B, and C in order and by the Stefan-Boltzman law, $$ P_A = \sigma T_A^4A \ , P_B = \sigma T_B^4A \ , P_C=\sigma T_C^4A \\ P_B = P_A + P_B \Rightarrow \sigma T_B^4A=\frac{\sigma A (T_A^4 + T_C^4)}{2} \\ \Rightarrow T_B^4=\frac{(T)^4+(2T)^4}{2}=\frac{17}{2}T^4 \\ \therefore T_B = \bigg( \frac{17}{2} \bigg)^{\frac{1}{4}} T $$

Is this correct?

Answer 1

Suppose B receives thermal energy at the rate $P_A$ from the front face of A and $P_C$ from the back face of C. B will reach an equilibrium temperature at which the power $P_B$ it radiates from both its faces equals the sum $P_A+P_C$.

The calculation has to make use of the Stefan-Boltzmann Law of black body radiation, together with the fact that B is emitting from both faces while receiving radiation from only one face each of A and C. You must assume that the temperatures of A and C are controlled so they are not affected by the radiation from B.

Answer 2

Consider the equilibrium condition of these plates ( say A, B and C in order) i.e. when the amount of energy radiated from A to B is the same as that radiated from B to C.

Then think of Stefan-Boltzmann's Law. I hope you will get your answer.