Chapter 15.5, Problem 81E

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.)

81. Electric potential due to a point charge The electric field due to a point charge of strength Q at the origin has a potential function φ = kQ/r, where r² = x² + y² + z² is the square of the distance between a variable point P(x, y, z) and the charge, and k > 0 is a physical constant. The electric field is given by E = −∇φ where ∇φ is the gradient in three dimensions.

a. Show that the three-dimensional electric field due to a point charge is given by

$E\left(x,y,z\right)=kQ\langle \frac{x}{r^3},\frac{y}{r^3},\frac{z}{r^3}\rangle$

b. Show that the electric field at a point has a magnitude $\left|E\right|=\frac{kQ}{r^2}$ Explain why this relationship r is called an inverse square law.