Electric Field as the Gradient of Potential

In Section 5.8, it was determined that the electrical potential difference

measured over a path

is given by

We begin by identifying the contribution of an infinitesimal length of the integral to the total integral in Equation 5.14.1. At point

, this is

Although we can proceed using any coordinate system, the following derivation is particularly simple in Cartesian coordinates. In Cartesian coordinates,

We also note that for any scalar function of position, including

, it is true that

Note the above relationship is not specific to electromagnetics; it is simply mathematics. Also note that

and so on for

and

. Making these substitutions into the above equation, we obtain:

This equation may be rearranged as follows:

Comparing the above equation to Equation 5.14.2, we find:

Note that the quantity in square brackets is the gradient operator “

” (Section 4.5). Thus, we may write

which is the relationship we seek.

Using Equation 5.14.8, we can immediately find the electric field at any point

if we can describe

as a function of

. Furthermore, this relationship between

and

has a useful physical interpretation. Recall that the gradient of a scalar field is a vector that points in the direction in which that field increases most quickly. Therefore:

This result should not come as a complete surprise; for example, the reader should already be aware that the electric field points away from regions of net positive charge and toward regions of net negative charge (Sections 2.2 and/or 5.1). What is new here is that both the magnitude and direction of the electric field may be determined given only the potential field, without having to consider the charge that is the physical source of the electrostatic field.