Questions About Stephan's Law of Thermal Radiation

[Biker]

I took Stephan's law for thermal radiation and I have a couple of questions about it.

1) The law states that the full energy radiated in 1 sec is equal to c T^4 where c is 5.67*10^-8 and T in kelvins and In the book they said if it has surroundings then the net energy emitted would be
q = c A (T^4 -t^4) where t is the temperature of the surroundings. I don't understand how the energy received by the body would be c A t^4.

2) There was an example where a ball with some temperature T and it was connected to a metal cylinder fully covered from the sides by an insulator. When the book tried to calculate the amount of heat emitted by radiation, It subtracted the intersected area between the cylinder and the ball. But, Doesn't it also radiate? It should radiate with net energy. Energy from the radiation of the body minus energy radiated from the cylinder but How do you use stephan's law for non uniform temperature?

 

[sophiecentaur]

There's an 'a' missing in your first formula.

I don't understand how the energy received by the body would be c A t^4.

There is a very basic consideration here. If the two formula were not the same, you could have a steady build up (or fall) in internal energy, which could be the basis for a perpetual motion machine. And that ain't allowed.
If you want to make it more complicated to include non-uniform temperature then you need to consider internal heat transfers and the emissivities for each part of an object's surface. There will still be an equilibrium situation but I would find it tiresome to work it all out.

 

[Biker]

There's an 'a' missing in your first formula.

There is a very basic consideration here. If the two formula were not the same, you could have a steady build up (or fall) in internal energy, which could be the basis for a perpetual motion machine. And that ain't allowed.
If you want to make it more complicated to include non-uniform temperature then you need to consider internal heat transfers and the emissivities for each part of an object's surface. There will still be an equilibrium situation but I would find it tiresome to work it all out.

Could you clarify first part?

I just wonder how would someone calculate it. Does every slice of the conductor radiate? I find it troublesome to use the law because I don't know how it is applicable other than if I have something like a rigid body with some temperature.

 

[sophiecentaur]

Could you clarify first part?

If the factors 'a' were different then at any temperature you could get more energy going in than going out (or vice versa). That would permit perpetual motion - you could connect up a heat engine that worked for free. Nonsense.

 

[Biker]

If the factors 'a' were different then at any temperature you could get more energy going in than going out (or vice versa). That would permit perpetual motion - you could connect up a heat engine that worked for free. Nonsense.

Oh I get what you mean. But I was talking about, Applying this formula to the surrounding such as air. There is some sense about being rigid and definite volume or area. I don't know how it was derived and I can't. So I don't know how to apply it for different situations. Ex, Air and non uniform temperature objects.

 

[sophiecentaur]

Oh I get what you mean. But I was talking about, Applying this formula to the surrounding such as air. There is some sense about being rigid and definite volume or area. I don't know how it was derived and I can't. So I don't know how to apply it for different situations. Ex, Air and non uniform temperature objects.

I think the problem would get very complex. I believe it's included in the design of satellite solar power systems. You'd have to break the object down into identifiable sections and do a thermal balance calculation for each (including interactions).