# Energy radiated/emitted by an object in a black body environment?

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 $$T\gtT_o$$

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.

If we raise the temperature of the object it will radiate more energy than it receives by an amount of $$e\sigma (T^4 - T_0^4)$$. It will still absorb and reflect the same amount of energy, as this is determined solely by the blackbody environment.