A point mass sits at the origin

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Topology and global structure

Conditions of Validity, Ch Gauss, Stokes and Green proved, and the first inversion of a differential operator

Problem

A point mass sits at the origin, producing the gravitational field g=−GMr2r^\mathbf{g} = -\dfrac{GM}{r^2}\hat{\mathbf{r}} for r>0r>0, with GM=3.0×10−9 m3 s−2GM = 3.0\times10^{-9}\,\mathrm{m^3\,s^{-2}}. Consider the spherical shell 0.50 m<r<2.0 m0.50\,\mathrm{m} < r < 2.0\,\mathrm{m}, which contains no mass. Someone argues: "Every loop drawn inside this shell can be shrunk to a point without leaving the shell, and ∇⋅g=0\nabla\cdot\mathbf{g}=0 throughout the shell, so g\mathbf{g} must be expressible as g=∇×A\mathbf{g}=\nabla\times\mathbf{A} for some vector potential A\mathbf{A} on the shell." ((a)) Compute, by excision, the flux ∮Sg⋅dS\oint_{S}\mathbf{g}\cdot\mathrm{d}\mathbf{S} through a sphere SS of radius c=1.0 mc=1.0\,\mathrm{m} lying inside the shell. ((b)) Explain why this flux is incompatible with g\mathbf{g} being a curl, and identify exactly which hypothesis of the divergence-free converse theorem the shell fails, given that every loop in it does shrink.

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