Is the following proof of Kirchoff's law correct?

[neelakash]

The standard book used in our university proves the Kirchoff's law of radiation in an elaborate manner...I understand the proof.But I think using the same principles,this can be proved in a much better way:

Consider an enclosure is at temperature T and the enclosure is filled with temperature radiation emitted by the wall.

Let a body A has been placed inside an enclosure along with a blackbody B of IDENTICAL geometry.The enclosure have wall opaque to all wavelengths and is insulated thermally from the surroundings.In equilibrium state when A and B attain the enclosure temperature T,the total energy aborbed by them will be equal to the total energy emitted by them.

Let emissive power of absorptive power be given by e_λ and a_λ.
If the body A is of abnormal geometry different parts of the body may have different a_λ and e_λ...Suppose an amount of ∑∆U(λ) of radiation falls on the body A in a given time ∆t.As A and B are of identical geometrical shape the radiation falling on the blackbody B is also ∑∆U(λ) in the same time ∆t.

Then for the blackbody we have ∑∆U(λ)=k*∑(e_bλ) where k is a constant and ∑(e_bλ) is the radiation from the blackbody

For the body A,we have ∑a_λ*∆U(λ)=k*∑e_λ where ∑e_λ is the radiation from the body A.

The same proportionality constant has been used because the two bodies are of identical geometrical shapes and radiation emitted in the same time ∆t has been considered.

Hence from the above relations,

[∑a_λ*∆U(λ)/∑∆U(λ)]=[∑e_λ/∑(e_bλ)]

Since this equality should hold for each portion of the spectrum,we must have
a_λ=(e_λ/e_bλ) Which proves Kirchoff's law.

Please give me feedback.Any incorrect area or somewhere the logic should be more solid...

 

[eljose79]

Hello, thank you for sharing your thoughts on proving Kirchoff's law of radiation. Your approach is definitely interesting and can be a useful alternative to the standard book proof. However, there are a few points that I would like to address for better understanding.

Firstly, when you mention that the enclosure is filled with temperature radiation emitted by the wall, it is important to clarify that this radiation is in thermal equilibrium with the wall and not just randomly emitted. This ensures that the radiation inside the enclosure is also at the same temperature as the wall.

Secondly, in your equation for the blackbody, it should be ∑∆U(λ)=k*∑(e_bλ), where k is the Stefan-Boltzmann constant and not a general constant. This is because the total energy emitted by a blackbody is proportional to the fourth power of its temperature, as given by the Stefan-Boltzmann law.

Lastly, the proportionality constant that you have used in the equation for the body A should be the same as the one used for the blackbody B. This is because both bodies are in thermal equilibrium and therefore, the same constant should be used for their respective emissive powers.

Overall, your approach is sound and the logic is solid. However, it would be helpful to clarify these points for better understanding. Thank you for sharing your thoughts and for your contribution to the discussion on Kirchoff's law of radiation.

 

[charng]

I cannot provide a definitive answer on whether or not the proof presented is correct without thoroughly examining the equations and assumptions used. However, I can provide some general feedback on the proof.

Firstly, it is important to note that there are different versions of Kirchoff's law, so it would be helpful to specify which version is being proved in this proof. Additionally, it would be useful to provide a brief explanation of what Kirchoff's law is for readers who may not be familiar with it.

The proof starts off by stating that it is using the same principles as the standard book, but it would be helpful to mention which principles these are. Additionally, it would be beneficial to provide a brief overview of the proof in the beginning so that readers can follow along more easily.

The use of symbols such as ∑ and λ may be confusing for readers who are not familiar with them, so it would be helpful to define them or provide a key for readers to refer to.

One potential issue with the proof is the assumption that the bodies A and B are of identical geometrical shape. While this may be true for some cases, it may not always be the case and could limit the applicability of the proof.

Overall, the proof seems to be using sound logic and equations, but it would be beneficial to provide more context and explanations for readers to follow along more easily. It would also be helpful to provide a clear conclusion that summarizes the proof and clearly states that Kirchoff's law has been proven.