According to the formula: $$ \frac{\Delta Q}{\Delta t}=\sigma\epsilon A T^4 $$ What does $T$ refer to in a situation where I am modelling the power of radiation from air of temperature to surface of emissivity $𝜖$?

Is it the temperature difference? And if so, would the equation look like this? $$ \frac{\Delta Q}{\Delta t}=\sigma\epsilon A (T_{2} - T_{1})^4 \hspace{0.5in}or\hspace{0.2in} \frac{\Delta Q}{\Delta t}=\sigma\epsilon A (T_{2}^4 - T_{1}^4) $$

Further, what would the equation look like if the surface had greater temperature than the air?

Edit: the source of the formula is from the IBO Exam Data Booklet (https://ibphysics.org/wp-content/uploads/2016/01/annotated-physics-data-booklet-2016.pdf)

  • $\begingroup$ You should cite the source you're getting this formula from, for clarity. $\endgroup$
    – agaminon
    Commented Nov 21, 2022 at 19:20
  • $\begingroup$ Added. Thanks for reminding me $\endgroup$
    – Rohin
    Commented Nov 22, 2022 at 14:21

1 Answer 1


The term $\sigma\varepsilon A T^4$ is used to model the graybody output power—i.e., the outgoing radiative heat flux from some surface with emissivity $\varepsilon$, area $A$, and temperature $T$, without considering any radiative input.

If that surface entirely faces an environment at temperature $T_\mathrm{env}$, then the net output is typically modeled as $\sigma\varepsilon A(T^4-T_\mathrm{env}^4)$ because that environment itself radiates heat toward the surface. (Alternatively, the net rate of heat gain at the surface can be modeled as $\sigma\varepsilon A(T_\mathrm{env}^4-T^4)$.)

Special care may be needed to treat a surrounding environment of gas only (as in our atmosphere), as some wavelengths may be unabsorbed and some radiation passing essentially transparently to outer space ($T\approx 0\,\mathrm{K}$). This can cause the effective temperature of the atmosphere for radiative heat transfer calculations (or the so-called "sky temperature") to differ from the actual temperature.

All these topics are discussed in detail in introductory heat transfer textbooks, e.g., Incropera & DeWitt's Fundamentals of Heat and Mass Transfer.


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