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The Stefan-Boltzmann equation states $e=\sigma T^4$, but how do we interpret this?

Is this completely wrong: A body of size $s^2$ generates the radiation/temperature $T^4$ for a given size and a given temperature or something similar?

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The Stefan-Boltzmann law governs the irradiance (radiant power per unit area). The total energy is not $\sigma T^4\!$, but rather the total power $P$ of a body with surface area $A$ and temperature $T$ is given by \begin{equation} P=A\sigma T^4 \end{equation} This result may be surprising, but it is correct. The reason irradiance rises so quickly is because the wavelengths of light decrease with increasing temperature, carrying away more energy with each photon (on average). At the same time, higher temperatures cause the photon emission rate from the surface to increase. It is the increase in the energy of the photons coupled with the increase in photon production rate that gives the $T^4$ dependence.

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The Stefan-Boltzmann law tells us the power per unit area radiated by a black body.

A black body is an idealized object which absorbs all light incident upon it. Therefore, a black body in thermal equilibrium must radiate energy equal to the amount it absorbed, so that it doesn't spontaneously increase it's temperature (which would contradict the assumption that it is in equilibrium!). Assuming the object you're describing is not particularly reflective (shiny), approximating it as a black body is reasonable.

Specifically, it says that irregardless of size or shape, a black body at temperature $T$ will radiate a power per unit area equal to $\frac{P}{A}=\sigma * T^4$. Therefore, to find the total power radiated by an object of surface area $A$, you would simply solve the above equation for $P$

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