22.5.3 The Electric Field of a Point Charge

We will begin to put the definition of the electric field to full use in the next chapter. For now, to develop the ideas, we will determine the electric field of a single point charge q. FIGURE 22.24a shows charge q and a point in space at which we would like to know the electric field. To do so, we use a second charge q′ as a probe of the electric field.

FIGURE 22.24 Charge q′ is used to probe the electric field of point charge q.

The figure consists of two parts labeled (“a”) and (b) that illustrate the electric field on a positive charge labeled q prime due to a positive point charge labeled q.
FIGURE 22.24 Full Alternative Text

For the moment, assume both charges are positive. The force on q′, which is repulsive and directed straight away from q, is given by Coulomb’s law:

F⃗on q′=(14πϵ0qq′r2, away from q) (22.6)

It’s customary to use 1/4πϵ0 rather than K for field calculations. Equation 22.5 defined the electric field in terms of the force on a probe charge; thus the electric field at this point is

E⃗=F⃗on q′q′=(14πϵ0qr2, away from q) (22.7)

The electric field is shown in FIGURE 22.24b.

NOTE   The expression for the electric field is similar to Coulomb’s law. To distinguish the two, remember that Coulomb’s law has a product of two charges in the numerator. It describes the force between two charges. The electric field has a single charge in the numerator. It is the field of a charge.

If we calculate the field at a sufficient number of points in space, we can draw a field diagram such as the one shown in FIGURE 22.25. Notice that the field vectors all point straight away from charge q. Also notice how quickly the arrows decrease in length due to the inverse-square dependence on r.

FIGURE 22.25 The electric field of a positive point charge.

The figure illustrates the electric field of a positive point charge.
FIGURE 22.25 Full Alternative Text

Keep these three important points in mind when using field diagrams:

  1. The diagram is just a representative sample of electric field vectors. The field exists at all the other points. A well-drawn diagram can tell you fairly well what the field would be like at a neighboring point.

  2. The arrow indicates the direction and the strength of the electric field at the point to which it is attached—that is, at the point where the tail of the vector is placed. In this chapter, we indicate the point at which the electric field is measured with a dot. The length of any vector is significant only relative to the lengths of other vectors.

  3. Although we have to draw a vector across the page, from one point to another, an electric field vector is not a spatial quantity. It does not “stretch” from one point to another. Each vector represents the electric field at one point in space.