18.4 Electric Field: Concept of a Field Revisited

Concept of a Field

A field is a way of conceptualizing and mapping the force that surrounds any object and acts on another object at a distance without apparent physical connection. For example, the gravitational field surrounding the earth (and all other masses) represents the gravitational force that would be experienced if another mass were placed at a given point within the field.

In the same way, the Coulomb force field surrounding any charge extends throughout space. Using Coulomb’s law, $F=k\mathrm{|}{q}_1{q}_2\mathrm{|}/r^2$, its magnitude is given by the equation $F=k\mathrm{|}\mathrm{qQ}\mathrm{|}/r^2$, for a point charge (a particle having a charge $Q$) acting on a test charge $q$ at a distance $r$ (see Figure 18.18). Both the magnitude and direction of the Coulomb force field depend on $Q$ and the test charge $q$.

Figure 18.18 The Coulomb force field due to a positive charge $Q$ is shown acting on two different charges. Both charges are the same distance from $Q$. (a) Since $q_1$ is positive, the force $F_1$ acting on it is repulsive. (b) The charge $q_2$ is negative and greater in magnitude than $q_1$, and so the force $F_2$ acting on it is attractive and stronger than $F_1$. The Coulomb force field is thus not unique at any point in space, because it depends on the test charges $q_1$ and $q_2$ as well as the charge $Q$.

To simplify things, we would prefer to have a field that depends only on $Q$ and not on the test charge $q$. The electric field is defined in such a manner that it represents only the charge creating it and is unique at every point in space. Specifically, the electric field $E$ is defined to be the ratio of the Coulomb force to the test charge:

where $\mathbf{\text{F}}$ is the electrostatic force (or Coulomb force) exerted on a positive test charge $q$. It is understood that $\mathbf{\text{E}}$ is in the same direction as $\mathbf{\text{F}}$. It is also assumed that $q$ is so small that it does not alter the charge distribution creating the electric field. The units of electric field are newtons per coulomb (N/C). If the electric field is known, then the electrostatic force on any charge $q$ is simply obtained by multiplying charge times electric field, or $\mathbf{\text{F}}=q\mathbf{\text{E}}$. Consider the electric field due to a point charge $Q$. According to Coulomb’s law, the force it exerts on a test charge $q$ is $F=k\mathrm{|}\mathrm{qQ}\mathrm{|}/r^2$. Thus the magnitude of the electric field, $E$, for a point charge is

Since the test charge cancels, we see that

The electric field is thus seen to depend only on the charge $Q$ and the distance $r$; it is completely independent of the test charge $q$.