Paul T. Wu Abstract: Understanding Gauss's flux theorem in electrostatics is fundamental for freshman physics students. Yet, its use in calculating gravitational field strengths often receives less attention in undergraduate curricula. Interestingly, Newton's law of gravitation and Coulomb's law of electrostatics share a deep structural similarity, as both follow the inverse-square law. In Newtonian mechanics, a particle moving under such a law traces an elliptical trajectory, a behavior echoed by electrostatic charges. Both gravitational and electrostatic forces also possess potential energy, with their respective potentials satisfying Laplace's equation, 𝛻 2 𝑈 = 0. When these potentials scale as 𝑈(𝑟) ∝ 𝑟 −1 , the resulting forces vary inversely with the square of the distance, 𝐹(𝑟) = −𝑈̇(𝑟) ∝ 𝑟 −2 . This structural resemblance naturally extends Gauss’s law to both gravitational and electromagnetic contexts. In this paper, we explore the application of Gauss's flux theorem to determine gravitational field strength, using a 2023 postgraduate medical school entrance exam problem from Tsinghua University in Taiwan as a case study; the problem was under appeal at the time. Keywords: Potential field, Laplace’s equation, Poisson’s equation, spherical harmonics, spherically symmetric, Green’s function, Green’s identities, Gauss’s theorem, field strength.
Consider the problem of measuring the intensity of the gravitational field sourced by a point mass, described as follows: The mass density inside a planet is usually not uniform. However, the mass density of the concentric shells of a spherical planet is about the same when the planet is decomposed into concentric spherical layers. As a result, for a spherical planet 𝑋 of radius 𝑅, the mass density can be treated as a function of its distance to the center of the planet 𝜌(𝑟) = 𝜌0 (1 −
𝑟 ). 2𝑅 1
Assuming the gravitational field strength at the planet’s surface is 𝑔𝑟=𝑅 ≝ 𝑔0 . Show that at 𝑟 = 𝑅, 2 the magnitude of the gravitational field is less than half that experienced at planet X's surface, yet remains nonzero, that is, 1
𝑔𝑟=½𝑅 < 𝑔0 and 𝑔𝑟=½𝑅 ≠ 0. 2
If not, prove it otherwise.
A planet’s density usually increases as we go inwards from the crust to core. Suppose a planet called 𝑋 is an oblate spheroid of radius 𝑅 and that its density 𝜌 is modeled by 𝜌(𝑟) = 𝜌0 (1 −
𝑟 ) 2𝑅
where 𝑟 is the radial distance from the center, 𝜌0 is constant. The approximate volume of a spherical shell of radius 𝑟 and thickness 𝑑𝑟 is given by 1