# The shape of the body in Stefan Boltzmann Law modern use

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## Main Question or Discussion Point

Hello,

It looks like Stefan Boltzmann Law can be used for a lot of different purposes: to calculate the temperature of stars, sun, temperature of the Earth's sky, temperature of particular surface, wall, the radiation emitted by the body by knowing its temperature, and so on.

What confuses me is: in all these modern examples of the use of Stefan Boltzmann Law, there is an initial assumption that the black or gray body which emits the energy is approximated to be a sphere? Is this correct or not?

Is it crucial for the Stefan Boltzmann Law, that the body in question is a sphere, so that it equally emits the energy in all directions?

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mfb
Mentor
The shape does not matter at all, unless the whole object is small compared to the wavelength of radiation.

Thank you mfb.
Would give an example of an object which is not small compared to the wavelength of its radiation?
My apology for the personal ignorance.

mfb
Mentor
Literally everything you can see.

The wavelength range of infrared radiation is micrometers, and visible light is between 0.4 and 0.8 micrometers.

Thank you mfb.

How does a ball of certain dimension relate with the wavelength it radiates?
I still do not understand you, I apologize for that.

mfb
Mentor
Ignore the comment on size. It was a remark about an irrelevant special case. If you take a really tiny dust particle or even individual atoms, it won't follow the law.

Why is that so? Why wouldn't a tiny dust particle which emits energy, follow the law? Because it is not visible by the naked eye?
The Stefan Boltzmann Law is applicable only to the objects which can be seen by a naked eye?

mfb
Mentor
Why wouldn't a tiny dust particle which emits energy, follow the law?
Quantum mechanics. But I really don't think going into more detail there would help. This has nothing to do with the human eye.

Thank you.
But I am still not sure I understand you.
We can apply the Stefan Boltzmann Law to an object of any shape, as long as its size is not smaller than the size of the wavelength of the radiation it emits.

But when we apply the Stefan Boltzmann Law to that object, do we approximate its shape to be a sphere or not?
Forgive me if my questions sound repetitive, but the way I understood your reply:
The shape does not matter at all, unless the whole object is small compared to the wavelength of radiation.
is: Stefan Boltzmann Law can be applied to an object of any shape...

But when we apply the Stefan Boltzmann Law to an object of any shape, do we assume it to be a sphere, or we do not?

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mfb
Mentor
as long as its size is not smaller than the size of the wavelength of the radiation it emits.

But when we apply the Stefan Boltzmann Law to that object, do we approximate its shape to be a sphere or not?
What exactly do you mean by "apply the Stefan Boltzmann Law"? The law does not depend on the shape of the object at all, it is a law about radiation per surface area. If you want to calculate the total amount of radiation emitted by the sun, for example, then you have to take care about the shape (and the size!), but that is a different thing.

What exactly do you mean by "apply the Stefan Boltzmann Law"?
"Apply" as Stefan Boltzman Law can be used to calculate either the temperature of the particular body by knowing the radiation it emits, or the radiation it emits by knowing the its temperature.

The law does not depend on the shape of the object at all, it is a law about radiation per surface area.
What confuses me is that on a lot of examples of Stefan Boltzman Law, the black or gray body is presented as a sphere.
So does that mean that even though the law does not depend on the shape of the object, when it's applied to the object, is it assumed that the shape of the object is a sphere?

mfb
Mentor
"Apply" as Stefan Boltzman Law can be used to calculate either the temperature of the particular body by knowing the radiation it emits, or the radiation it emits by knowing the its temperature.
Then you have to know size and shape of the object to some extent. Clearly larger objects emit more radiation in total.
That is not just the Stefan-Boltzmann law any more, it is the law integrated over the outer surface.
when it's applied to the object, is it assumed that the shape of the object is a sphere?
Not necessarily. You can assume that if you know that the object is roughly spherical, otherwise you cannot.

Not necessarily. You can assume that if you know that the object is roughly spherical, otherwise you cannot.
And the fact that "you cannot" assume that the object's shape is not a sphere, does not prevent one to apply the Stefan-Boltzmann law to that object?

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mfb
Mentor
It does not prevent you from applying it, but you cannot use $4 \pi r^2$ for its surface area.

Thank you mfb.