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# 5: Gravitational Field and Potential

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

This chapter deals with the calculation of gravitational fields and potentials in the vicinity of various shapes and sizes of massive bodies. The reader who has studied electrostatics will recognize that this is all just a repeat of what he or she already knows.

• 5.1: Introduction
• 5.2: Gravitational Field
The region around a gravitating body (by which I merely mean a mass, which will attract other masses in its vicinity) is a gravitational field. Although I have used the words “around” and “in its vicinity”, the field in fact extents to infinity. All massive bodies (and by “massive” I mean any body having the property of mass, however little) are surrounded by a gravitational field, and all of us are immersed in a gravitational field.
• 5.3: Newton's Law of Gravitation
Newton noted that the ratio of the centripetal acceleration of the Moon in its orbit around the Earth to the acceleration of an apple falling to the surface of the Earth was inversely as the squares of the distances of Moon and apple from the centre of the Earth. Together with other lines of evidence, this led Newton to propose his universal law of gravitation:
• 5.4: The Gravitational Fields of Various Bodies
• 5.5: Gauss's Theorem
The total normal outward gravitational flux through a closed surface is equal to $$−4 \pi G$$ times the total mass enclosed by the surface.
• 5.6: Calculating Surface Integrals
While the concept of a surface integral sounds easy enough, how do we actually calculate one in practice?
• 5.7: Potential
We have defined only the potential difference between two points. If we wish to define the potential at a point, it is necessary arbitrarily to define the potential at a particular point to be zero. We might, for example define the potential at floor level to be zero, in which case the potential at a height h above the floor is gh ; equally we may elect to define the potential at the level of the laboratory bench top to be zero, where the potential at a height z above the bench top is gz.
• 5.8: The Gravitational Potentials Near Various Bodies
• 5.9: Work Required to Assemble a Uniform Sphere
• 5.10: Nabla, Gradient and Divergence
We are going to meet, in this section, the symbol ∇ . In North America it is generally pronounced “del”, although in the United Kingdom and elsewhere one sometimes hears the alternative pronunciation “nabla”, called after an ancient Assyrian harp-like instrument of approximately that shape.
• 5.11: Legendre Polynomials
In this section we cover just enough about Legendre polynomials to be useful in the following section.
• 5.12: Gravitational Potential of any Massive Body
• 5.13: Pressure at the Centre of a Uniform Sphere
What is the pressure at the centre of a sphere of radius a and of uniform density ρ?

Thumbnail: Gravitational field lines around the Earth. Image used with permission (Public Domain; Sjlegg).