Calculating Spectral Exitance for Planck's Law & Blackbody

AI Thread Summary
The discussion focuses on calculating the average spectral exitance of a blackbody over a finite spectral band using Planck's Law. The original poster successfully computes spectral exitance for a single wavelength and integrates to derive the Stefan-Boltzmann law but seeks to determine the area under the curve for a specific temperature within a limited wavelength range. Responses indicate that the integral for this case lacks an exact solution and suggest using numerical methods or tabulated values of Debye integrals for approximation. The poster acknowledges the need for numerical analysis techniques and expresses a willingness to learn more. Overall, the challenge lies in finding a practical method to calculate spectral exitance without advanced computational tools.
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Hello,

My question is in regards to Planck's Law and a blackbody:

For the single lambda case I can readily find the spectral exitance. Alternately, if I substitute to create an integral in the form of x^3 / (e^x - 1) and integrate over all lambda, I reach Stefan-Boltzmann. No problems there, but I am really interested in finding the (average?) spectral exitance over a small, finite spectral band. Any tips on how to go about this?

Thanks in advance.
 
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i am not quite sure what your question is, but it seems to me you are looking for the answer of this integral:
\int_{b}^a \frac{x^3}{e^x-1} dx,
the above integral has no exact solution... unless a=0, b=infinity, or a=b... the best you could do is use numerical analysis...
 
If you don't have a computer software to give you the result,learn that Debye integral (D_{3}) values are tabulated...

Daniel.
 
Thank you both for your quick responses. To state my question more directly: "What is the area under the curve generated by a blackbody at some given temperature for some finite lambda range?" (Say in the visible region only.)

I can generate an approximate answer by using small increments of area and summing, but I wanted to compare my answer the solution obtained from a definite integral. However, from your answers I believe it is not possible (I do not have any computer software - just pencil and paper!).

I have taken your advice and will study more about numerical analysis and Debye integrals. Thank you again for your help.
 
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