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Exactly. That is why use of the room as the first term in the SB equation is really stupid. The equation should be valid for any configuration. You are making the whole computation extremely complex (and stupid) if you want to switch the roles of T and Tc.
You just never get any smarter....You keep forgetting that the basic premise of the S-B law is that the temperature of T is always greater than that of Tc. You can't switch the roles of T and Tc because T is always warmer...Tc does not emit to T...EVER...energy moves in one direction...that is the problem with your bastardized version of the S-B law..it allows you to set Tc to a temperature higher than that of T. It violates the basic premise of the S-B law.
The whole point is that you DON'T know P. That is what you are trying to compute. You lost track of the whole point of the SB equation.
You don't think you can determine how much energy the sphere is absorbing if you have real time information on the amount and rate of its change in temperature? Really? That's what you think? And if you know how much it is absorbing, you think that then you don't know how much energy is being radiated to it?
Your confusion with a simple problem of a sphere in a hotter room is because you eschew two way energy flow.
I have no confusion...I realize that the equation of the S-B law in question describes nothing more than a simple, one way energy exchange between an object of one temperature and an object of another temperature...you on the other hand, have interpreted it to mean something that is so far removed from what the equation actually says that it is little wonder that you have become hopelessly lost in your crazy model trying to envision some crazy scenario where the emitter becomes the absorber and the absorber becomes the emitter...sorry, not possible.