22 1. Introduction Figure 1.12. Geometry of a spherical shell 1.6.5. Gauss’s law. Gauss’s flux theorem for gravity states that the gravitational flux through a closed surface is proportional to the enclosed mass. Gauss’s theorem can be written in differential form, using the divergence theorem, as ∇ · g = −4πGρ , (1.54) where G is the universal gravitational constant, ρ is the (volumetric) density of the enclosed mass, g = F/m is the gravitational field in- tensity, m is the mass of a test point, and F is the force on this test mass. (a) Use this theorem to determine the force F(r) acting on mass m at point P due to the gravitational attraction of a thin homo- geneous spherical shell of mass M, surface density σ, and radius x. Assume that point P is a distance r from the center of the shell. Be sure to consider the case where point P is inside the shell (r x) as well as outside the shell (r x). (b) Assume that F(r) = −dV/dr, where V (r) is the gravitational potential energy. Integrate the above force (starting at a reference point at infinity) to rederive the potential energy in Exercise 1.6.1.
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