# Why is the power of $T$ to the fourth in Stefan's law? Any particular reason? [duplicate]

Someone posed a question in front of me that why is the power of t 4 in Stefan's law? please help

## 2 Answers

For what it's worth, thermal radiation into a 1D space (better known as Johnson noise) has a total power proportional to $T^2$. Thermal radiation into a 2D space is proportional to $T^3$, and into 3D space it's proportional to $T^4$. More dimensions means there are more spatial modes that you can fill with photons. This is somewhat related to the fact that the area of a circle is proportional to radius squared, while the volume of a sphere is proportional to radius cubed, and so on. I'm not sure I can give a completely satisfying answer without re-deriving the whole blackbody formula. Maybe somebody else will.

• hi there. thanks a lot for your effort. This question was posed to me by a person who is in school. Can you give a more basic answer (which i can explain to him)? Sep 12 '17 at 18:33

In short, if space had $D$ dimensions we would expect proportionality to $T^{D+1}$. If you want a classical argument, see here; if you want to fix the proportionality constant, see here. (Both of those links present the $D=3$ case, but the generalisation is immediate.)