Where am I going wrong (energy transfer between black bodies)?

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In summary, a black body at temperature T radiates energy at an intensity of R W/m². When a smaller black body (BB2) is placed at equilibrium with BB1, BB2 will emit energy at an intensity of 10⁴R W/m². BB2's temperature must be 10T in order for energy to spontaneously flow from BB1 to BB2.
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Where am I going wrong (energy transfer between black bodies)?
I have a problem with a very basic ‘thought experiment’. I can’t see my mistake(s) - I’m pretty sure there must be at least one! So I’m accepting likely humiliation/embarrassment and asking if anyone can explain where I’m going wrong...

The surface of a black body (BB1) is at temperature T and radiates at R W/m².

An ‘optical’ system collects/redirects/focuses the radiated energy from 1m² of BB1 onto a smaller black body (BB2) of area 10⁻⁴m². We now have R watts directed onto 10⁻⁴m². That’s an incident intensity onto BB2 of 10⁴R W/m².

When BB2 reaches equilibrium, the power it receives (R watts) will be the same as the power it emits. So BB2 will emit R watts from an area of 10⁻⁴m². BB2’s surface is radiating at 10⁴R W/m².

Since a black body's radiated power/unit area is proportional to ##T_{abs}^4## this means BB2’s temperature must be 10T.

So energy is spontaneously flowing from an object at temperature T to one at temperature 10T. Err...
 
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You still only have 1 watt of radiated power to be transferred among bodies.
BB2 can't radiate more than it receives from the 1 m x 1 m of BB1 surface.
 
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You appear to be assuming that all of the energy leaving a 1 square meter surface can be focused down to 1 square cm. It cannot. This is related to the conservation of etendue and view factors. I believe, without calculating it, that at most 1/10000 of the light can be focused that small.
 
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  • #4
Lnewqban said:
You still only have 1 watt of radiated power to be transferred among bodies.
BB2 can't radiate more than it receives from the 1 m x 1 m of BB1 surface.
Not quite with you. The radiated power from 1m² BB1 is equal to the power received by (the much smaller) BB2. This, in turn is the same as the radiated power from BB2. Agreed,

But this means the power/m² radiated by BB2 is 10⁴ times the power/m² originally radiated by BB1. (Because BB2 is so much smaller than BB1.)

From the Stefan-Boltzmann law, this means BB2's temperature must be (⁴√(10⁴) =) 10 times that of BB1.
 
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Dale said:
You appear to be assuming that all of the energy leaving a 1 square meter surface can be focused down to 1 square cm. It cannot. This is related to the conservation of etendue and view factors. I believe, without calculating it, that at most 1/10000 of the light can be focused that small.
Aha! I have indeed made that assumption. I'm not familiar with the conservation of etendue. I'll go and do some reading. Many thanks!

(Minor edit.)
 
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Related to Where am I going wrong (energy transfer between black bodies)?

1. What is the concept of energy transfer between black bodies?

The concept of energy transfer between black bodies is based on the principles of thermodynamics and electromagnetic radiation. Black bodies are theoretical objects that absorb and emit all wavelengths of radiation perfectly, making them ideal for studying energy transfer. When two black bodies are in close proximity, they exchange energy through radiation, which can be described by the Stefan-Boltzmann law.

2. How is energy transferred between black bodies?

Energy is transferred between black bodies through radiation, which is the emission and absorption of electromagnetic waves. The hotter black body emits more radiation than the cooler one, and this energy is absorbed by the cooler body, causing its temperature to increase. This process continues until both bodies reach thermal equilibrium, where the rate of energy transfer is equal in both directions.

3. Why is energy transfer between black bodies important?

Studying energy transfer between black bodies is crucial in understanding the behavior of heat and radiation in various systems. It has practical applications in fields such as astronomy, where black bodies are used to model the behavior of stars and planets. It also plays a role in the development of technologies such as solar panels, which utilize the principles of energy transfer between black bodies.

4. What factors affect energy transfer between black bodies?

The rate of energy transfer between black bodies is influenced by several factors, including the temperature difference between the bodies, the surface area of the bodies, and the properties of the materials that make up the bodies. For example, a larger temperature difference between the bodies will result in a faster rate of energy transfer, while a larger surface area will increase the amount of energy exchanged.

5. How can we improve energy transfer between black bodies?

One way to improve energy transfer between black bodies is by using materials with high thermal conductivity, which allows for more efficient absorption and emission of radiation. Additionally, optimizing the surface area and temperature difference between the bodies can also improve the rate of energy transfer. In some cases, using reflectors or insulators can also help to enhance energy transfer between black bodies.

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