Blackbody emission in 2D coordinates

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The discussion centers on adapting the concept of blackbody emission from 3D to a 2D Cartesian coordinate system. Participants explore how to modify Planck's equation and its units to account for the change in dimensionality, suggesting that one can treat the emission as "per meter of height" to simplify calculations. The idea of using a cylinder's cross-section instead of a sphere is proposed to avoid complications with inverse square intensity decreases. It is emphasized that total radiation should be considered over a solid angle of 2 pi, maintaining uniform distribution in the 2D context. The conversation highlights the challenges of mathematically representing 3D radiation phenomena within a 2D framework.
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The spectral radiance of a blackbody has units of W·sr-1·m-2·Hz-1. How do I deal with these units if I want to think about a 2D problem of radiation in Cartesian coordinates? I assume that instead of a sphere of emission (which would result in artificial decrease in intensity with the inverse square of the distance) I should then approximate emission from the cross-section of a cylinder. What kind of changes to Planck's equation, its units, or some other condition of its application, must be made for this?
 
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You don't need changes, you can think of everything as "per meter of height" (simulating a 3D universe that is completely homogeneous in one dimension).
 
I agree about simulating a 3D universe in which one dimension is homogeneous. However, I do not understand how to treat this mathematically because of the intrinsically 3D nature of radiation. It seems I need to find the radiation over a great circle of a sphere and then integrate in the third dimension to give a cylinder of 1 unit thickness. Otherwise, I'm not sure how to do it.
 
There is no need to make spheres, and I don't see where you try to make them.
The total radiation from a surface will be in a solid angle of 2 pi with a uniform distribution over your single 2D angle.
 

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