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- What is the intensity of radiation at the center of a spherical shell of stars?

Given that L is the luminosity of a single star and there are n stars evenly distributed throughout this thin spherical shell of radius r with thickness dr, what is the total intensity from this shell of stars?

My calculations were as follows: Intensity is the power per unit area per steradian of the sky; the power per unit area is ##\frac {nL}{4πr^2}##; the whole sky covers a solid angle of 4π steradians; and so the intensity should be equal to ##\frac{nL}{16π^2r^2} dr##. However, my book says that the total intensity is ##\frac{nL}{4π} dr##. Can anyone help explain my mistake here? I suspect the error has to do with my lack of familiarity with a solid angle, so if that could be explained, that would be very helpful.

My calculations were as follows: Intensity is the power per unit area per steradian of the sky; the power per unit area is ##\frac {nL}{4πr^2}##; the whole sky covers a solid angle of 4π steradians; and so the intensity should be equal to ##\frac{nL}{16π^2r^2} dr##. However, my book says that the total intensity is ##\frac{nL}{4π} dr##. Can anyone help explain my mistake here? I suspect the error has to do with my lack of familiarity with a solid angle, so if that could be explained, that would be very helpful.