Blackbody emission in 2D coordinates

[Hypatio]

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?

 

[mfb]

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).

 

[Hypatio]

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.

 

[mfb]

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.