system
Given the following post, respond as if you're commenting on Reddit:
human
When some physical effect has a strength that is inversely proportional to the distance, why is it not infinite at the center?

I'm asking about all of the inverse x^y laws, really. For example, the brightness of a light source obeys the inverse square law. If halving the distance doubles the brightness, going to the source would make the brightness infinite.

I *suspect* that the reason this absurdity exists is that when you get close enough, you can no longer treat the source as a point; getting close to the center of a mass, the effects of the layers above you cancel out.

I guess what I'm asking is, 1) am I on the right track with the above, and 2) do all sources have physical extent, or is there anything that produces a field or effect that is really a point? (Is there something here to do with quantum uncertainty in position?)
gpt
You are really on track. Point 1) is correct. The power-law behaviour stops when you reach the typical size of the source, and going deeper in the source it's replaced by a smooth behaviour with no singularity. Precisely for the reason you claim. 

Point 2) is a good insight. Yes, there are truly or at least very pointlike sources in the Universe, these are the fundamental particles of the standard model. The divergence is not exactly fixed by quantum mechanics and its "fuzziness" (though *some* problems of the classical theory of particles are fixed by it) but rather by a procedure called renormalization. It's rather complex, but the fundamental starting point is "are we asking the right questions to this physical theory?". And upon reflecting on this question you reach the conclusion that most likely no, we aren't, and once we craft our questions more carefully (which is vaguely what renormalization is) you only get sensible answers, no infinities.