Chapter 15.5, Problem 82E

Potential functions Potential functions arise frequently in physics and engineering. A potential function has the property that a field of interest (for example, an electric field, a gravitational field, or a velocity field) is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 17.)

82. Gravitational potential The gravitational potential associated with two objects of mass M and m is φ = −GMm/r, where G is the gravitational constant. If one of the objects is at the origin and the other object is at P(x, y, z), then r²= x² + y² + z² is the square of the distance between the objects. The gravitational field at P is given by F = − ∇φ where ∇φ is the gradient in three dimensions. Show that the force has a magnitude. |F| = GMm/r^2.Explain why this relationship is called an inverse square law.