How can the irradiance of a point source behave proportionally to 1/R^2, where R is the range of the source yet the irradiance of an extended source be independent of range? It doesn't make any sense to me.
Consider two formulas I found online looking for irradiance of a point source vs extended source:
point source: Irradiance = dpsi/dA = I / R^2, where I is the intensity (W/sr) dpsi is the power and dA is an area element.
extended source: Irradiance = dpsi/dA = pi/4 L (d/f)^2 where d is the diameter of a lens, f its focal length, and L the radiance of the object in W/m^2 sr.
Take an object of uniform radiance (spatially) and approximate it with a point source. As the object gets closer the estimate based on the point source approximation increases quickly where the irradiance calculated from the extended source doesn't change at all. Now, I know the point source is a simplification, but looking at these formulas it sure looks like a bad one!!! What am I missing? Because I know it is used a lot and looking at it now, I'm not sure why.
I have a heavy mathematical background but very little physics. So, go ahead and lay it on me with the math in your answer but give context to the physics.