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.

 

[pmacias]

The spectral radiance of a blackbody in 2D coordinates can be thought of as the intensity of radiation emitted per unit area, per unit solid angle, and per unit frequency. In Cartesian coordinates, this can be visualized as a cross-section of a cylinder, where the intensity of radiation is constant along the length and width of the cylinder.

To account for this 2D emission, modifications would need to be made to Planck's equation. Instead of using the volume of the blackbody, the cross-sectional area of the cylinder would be used to calculate the total energy emitted. Additionally, the units of the equation would need to be adjusted to account for the change in dimensions - instead of meters cubed, the units would be meters squared.

Another important consideration is the solid angle, which represents the portion of the total sphere that is being observed. In 2D coordinates, the solid angle would be proportional to the length of the cylinder, rather than the full 4π steradians in 3D coordinates. This would need to be taken into account when calculating the spectral radiance.

Overall, the changes required for dealing with blackbody emission in 2D coordinates would involve adjusting the dimensions and units used in Planck's equation, as well as taking into account the modified solid angle. With these modifications, it is possible to accurately model and calculate the emission of a blackbody in 2D coordinates.