What is the net energy radiated per unit time by a black body at temperature $T$ with emissivity $e$ and absorptivity $a$, if it’s placed in an environment of temperature $T_o$.Assume that $T$$\gt$$T_o$
Here’s my calculation:
I used the Stefan-Boltzmann law to calculate the $Q$ radiated by object due to its temperature which is $$(Q_{rad})_{obj}=\sigma eAT^4 $$ Where $\sigma$ is Stefan-Boltzmann constant.
The energy is being radiated by the surroundings also so,$$(Q_{rad})_{surr}=\sigma AT_o^4$$So the object absorbs and also emits the energy due the radiation from surroundings which is given by$$Q_{abs}=\sigma aAT_o^4$$ $$Q_{em}=\sigma eAT_o^4$$ Therefore the net energy emitted is given by $$Q_{net}=(Q_{rad})_{obj}-Q_{abs}+Q_{em}$$ which equals to $$Q_{net}=\sigma eAT^4-\sigma aAT_o^4+\sigma eAT_o^4$$
Now if this all is correct, how to simplify it further, because the answer is $Q_{net}=\sigma eA(T^4-T_o^4)$ which I had an attempt to prove.So please guide me.