A hypothetical hollow planet is a static

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Newtonian gravitation as a field theory

Relativity and Gravitation, Ch Gravitational field, potential and Gauss's law for gravity

Problem

A hypothetical hollow planet is a static, spherically symmetric shell of uniform density ρ\rho with inner radius aa and outer radius bb (a<ba < b); the cavity r<ar < a is empty and nothing else lies nearby. Use Gauss's law for gravity in integral form, ∮g⃗⋅dA⃗=−4πGMenc\oint \vec g\cdot\mathrm{d}\vec A = -4\pi G M_{\rm enc}, with the outward normal, and take Φ→0\Phi \to 0 at infinity. Work to Newtonian (first) order throughout. ((a)) State the symmetry argument that lets the flux integral be evaluated on a centred sphere, and show that for a≤r≤ba \le r \le b the radial field is gr(r)=−4πGρ3(r−a3r2).g_r(r) = -\frac{4\pi G\rho}{3}\left(r - \frac{a^3}{r^2}\right). ((b)) Show that gr=0g_r = 0 everywhere in the cavity r<ar < a. ((c)) Using gr=−dΦ/drg_r = -\mathrm{d}\Phi/\mathrm{d}r and the known potential at r=br = b, show that the potential throughout the cavity is constant and equal to Φcav=−2πGρ (b2−a2).\Phi_{\rm cav} = -2\pi G\rho\,(b^2 - a^2). ((d)) Evaluate Φcav\Phi_{\rm cav} for ρ=3.00×103 kg m−3\rho = 3.00\times10^{3}\ \mathrm{kg\,m^{-3}}, a=2.00×106 ma = 2.00\times10^{6}\ \mathrm{m} and b=3.00×106 mb = 3.00\times10^{6}\ \mathrm{m}. Take G=6.674×10−11 N m2 kg−2G = 6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}.

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