Black body radiation and heat transfer Consider a spherical shell of radius $r_1$ and a concentric spherical shell of radius $r_2>r_1$ in vacuum. The inner sphere is at thermal equilibrium with temperature $T1>T2$, and the outer sphere is at thermal equilibrium with temperature $T_2$.
We should find the power necessary to maintain both, the inner sphere and the outer sphere, in thermal equilibrium.
Now, honestly, I am extremely confused on how to solve it:
For the inner sphere, $P_1 + \sigma 4 \pi r_1^2 T_2^4 = \sigma 4 \pi r_1 ^2 T_1^4$ apparently gives the right answer, the main problem is with the outer sphere:
$P_2 + \sigma 4 \pi r_2 ^2 T_1^4 = \sigma 4 \pi r_2 ^2 T_2^4$ gives the wrong answer. Why?
I mean, I have used the same reasoning for the inner spherical shell, but it only works for one. What could be the problem?
 A: Sorry for deleting/editing this answer a few times. There isn't really enough information to give an answer, and some of the assumptions I made I think were initially wrong.

Consider a spherical shell of radius $r_1$ and a concentric spherical shell of radius $r_2>r_1$ in vacuum. The inner sphere is at thermal equilibrium with temperature $T1>T2$, and the outer sphere is at thermal equilibrium with temperature $T_2$.

After thinking a little more, I think the question is asking about a solid sphere of radius $r_1$ embedded in a thick spherical shell of inner radius $r_1$ and outer radius $r_2$.
(But if the OP has a picture or a diagram that was given as part of the homework question it would be helpful to include it...)

We should find the power necessary to maintain both, the inner sphere and the outer sphere, in thermal equilibrium.

The term "thermal equilibrium" does not make sense here, at least as far as it involves both objects together. I think OP means the entire system is in a "steady state." (Thermal equilibrium of both objects with each other would imply the objects are at the same temperature.)

$P_2 + \sigma 4 \pi r_2 ^2 T_1^4 = \sigma 4 \pi r_2 ^2 T_2^4$ gives the wrong answer. Why?

Assuming both the sphere and the thick spherical shell are black bodies and obey the Stefan–Boltzmann law, OP already found that

$P_1 + \sigma 4 \pi r_1^2 T_2^4 = \sigma 4 \pi r_1 ^2 T_1^4$

This results because we want the energy change in steady state to be zero. But we know the sphere is emitting energy $\delta t \sigma 4 \pi r_1 ^2 T_1^4$ and absorbing energy $\delta t\sigma 4 \pi r_1^2 T_2^4$. And we are told that there is some external power $P_1$ attached to the inner sphere adding energy, such that the total change in energy of the inner sphere is:
$$
\delta E_1 = \delta t \left( 
P_1 + \sigma 4 \pi r_1^2 T_2^4 - \sigma 4 \pi r_1 ^2 T_1^4
\right) \;.
$$
Setting the above energy equation to zero gives OP's initial result:
$$
P_1 = \sigma 4 \pi r_1 ^2 T_1^4 - \sigma 4 \pi r_1^2 T_2^4 
$$

Now, for the outer thick spherical shell...
The outer thick spherical shell also loses energy by emitting it from its outer surface at a rate of:
$$
\sigma 4 \pi r_2^2 T_2^4.
$$
The outer spherical shell also receives energy from its external environment at a rate of:
$$
\sigma 4 \pi r_2^2 T_0^4,
$$
where $T_0$ is the temperature of the environment (e.g., 2.7 K for the cosmic microwave background in our universe).
The outer spherical shell also emits from it's inner surface and receives energy from its inner surface in exactly the opposite values as the inner sphere emits and receives.
The outer spherical shell also has an external power $P_2$ attached.
Therefore the total energy change of the outer thick spherical shell is
$$
\delta E_2 = \delta t 
\left( 
P_2 + \sigma 4 \pi r_2^2 T_0^4 - \sigma 4 \pi r_2 ^2 T_2^4 - (\sigma 4 \pi r_1^2 T_2^4 - \sigma 4 \pi r_1 ^2 T_1^4)
\right)
$$
$$
= \delta t 
\left( 
P_2 + \sigma 4 \pi r_2^2 T_0^4 - \sigma 4 \pi r_2 ^2 T_2^4 + P_1
\right)
$$
Setting the above to zero for steady state gives the result:
$$
P_2 = \sigma 4 \pi r_2 ^2 T_2^4 - \sigma 4 \pi r_2^2 T_0^4 - P_1
$$
