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1

I believe Jordans lemma, https://en.wikipedia.org/wiki/Jordan%27s_lemma, provides that the integral II goes to 0 as $R\rightarrow \infty$.


2

$dl'$ is equivalent to "$d|\mathbf{r}|$", it is essentially a "scalar length measure". The electrodynamics integral you wrote here is a vector-valued integral, so no dotting happens. If you use a linear coordinate system, it may be evaluated as three scalar line integrals, one for each coordinate. Vector valued integrals cannot really be evaluated using a ...


0

A curious result of the physics involved is that the dropped mass oscillating up and down through the hole (say from North Pole to South Pole and back) would be matched exactly by a mass in a polar circular orbit at ground level. Note that the max velocity in the answer above is the same as the circular orbital velocity. If the object were dropped at ...


3

This is a surprisingly simple thing to calculate. It is a well known result that a consequence of the inverse square law is that there is no force inside a symmetrical hollow shell. This means that as the object falls into the hole, it will appear to be attracted by a sphere of decreasing radius - the mass outside "doesn't count." The acceleration of ...


4

The reason you run into problems is you haven't really regularized your loop integral. You're dealing with an ill-defined quantity, and it is natural that you run into contradictions. The way to go is to regularize first (which in essence defines your object of interest), and then manipulate it. You'll then see that some formal operations you use are ...



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