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College Physics 2e

18.4 Electric Field: Concept of a Field Revisited

College Physics 2e18.4 Electric Field: Concept of a Field Revisited

Learning Objectives

By the end of this section, you will be able to:

  • Describe a force field and calculate the strength of an electric field due to a point charge.
  • Calculate the force exerted on a test charge by an electric field.
  • Explain the relationship between electrical force (F) on a test charge and electrical field strength (E).

Contact forces, such as between a baseball and a bat, are explained on the small scale by the interaction of the charges in atoms and molecules in close proximity. They interact through forces that include the Coulomb force. Action at a distance is a force between objects that are not close enough for their atoms to “touch.” That is, they are separated by more than a few atomic diameters.

For example, a charged rubber comb attracts neutral bits of paper from a distance via the Coulomb force. It is very useful to think of an object being surrounded in space by a force field. The force field carries the force to another object (called a test object) some distance away.

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|q1q2|/r2F=k|q1q2|/r2, its magnitude is given by the equation F=k|qQ|/r2F=k|qQ|/r2, for a point charge (a particle having a charge QQ) acting on a test charge qq at a distance rr (see Figure 18.18). Both the magnitude and direction of the Coulomb force field depend on QQ and the test charge qq.

In part a, two charges Q and q one are placed at a distance r. The force vector F one on charge q one is shown by an arrow pointing toward right away from Q. In part b, two charges Q and q two are placed at a distance r. The force vector F two on charge q two is shown by an arrow pointing toward left toward Q.
Figure 18.18 The Coulomb force field due to a positive charge QQ is shown acting on two different charges. Both charges are the same distance from QQ. (a) Since q1q1 is positive, the force F1F1 acting on it is repulsive. (b) The charge q2q2 is negative and greater in magnitude than q1q1, and so the force F2F2 acting on it is attractive and stronger than F1F1. The Coulomb force field is thus not unique at any point in space, because it depends on the test charges q1q1 and q2q2 as well as the charge QQ.

To simplify things, we would prefer to have a field that depends only on QQ and not on the test charge qq. 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 EE is defined to be the ratio of the Coulomb force to the test charge:

E = F q , E = F q ,

where FF is the electrostatic force (or Coulomb force) exerted on a positive test charge qq. It is understood that EE is in the same direction as FF. It is also assumed that qq 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 qq is simply obtained by multiplying charge times electric field, or F = q E F = q E . Consider the electric field due to a point charge QQ. According to Coulomb’s law, the force it exerts on a test charge qq is F=k|qQ|/r2F=k|qQ|/r2. Thus the magnitude of the electric field, EE, for a point charge is

E =| F q | = k | qQ qr 2 | = k |Q| r 2 . E =| F q | = k | qQ qr 2 | = k |Q| r 2 .

Since the test charge cancels, we see that

E = k |Q| r 2 . E = k |Q| r 2 .

The electric field is thus seen to depend only on the charge QQ and the distance rr; it is completely independent of the test charge qq.

Example 18.2

Calculating the Electric Field of a Point Charge

Calculate the strength and direction of the electric field EE due to a point charge of 2.00 nC (nano-Coulombs) at a distance of 5.00 mm from the charge.


We can find the electric field created by a point charge by using the equation E=kQ/r2E=kQ/r2.


Here Q=2.00×109Q=2.00×109 C and r=5.00×103r=5.00×103 m. Entering those values into the above equation gives

E = k Q r 2 = ( 8.99 × 10 9 N m 2 /C 2 ) × ( 2.00 × 10 9 C ) ( 5.00 × 10 3 m ) 2 = 7.19 × 10 5 N/C. E = k Q r 2 = ( 8.99 × 10 9 N m 2 /C 2 ) × ( 2.00 × 10 9 C ) ( 5.00 × 10 3 m ) 2 = 7.19 × 10 5 N/C.


This electric field strength is the same at any point 5.00 mm away from the charge QQ that creates the field. It is positive, meaning that it has a direction pointing away from the charge QQ.

Example 18.3

Calculating the Force Exerted on a Point Charge by an Electric Field

What force does the electric field found in the previous example exert on a point charge of –0.250μC–0.250μC?


Since we know the electric field strength and the charge in the field, the force on that charge can be calculated using the definition of electric field E=F/qE=F/q rearranged to F=qEF=qE.


The magnitude of the force on a charge q=0.250μCq=0.250μC exerted by a field of strength E=7.20×105E=7.20×105 N/C is thus,

F = qE = ( 0.250 × 10 –6 C ) ( 7.20 × 10 5 N/C ) = 0.180 N. F = qE = ( 0.250 × 10 –6 C ) ( 7.20 × 10 5 N/C ) = 0.180 N.

Because qq is negative, the force is directed opposite to the direction of the field.


The force is attractive, as expected for unlike charges. (The field was created by a positive charge and here acts on a negative charge.) The charges in this example are typical of common static electricity, and the modest attractive force obtained is similar to forces experienced in static cling and similar situations.

PhET Explorations

Electric Field of Dreams

Play ball! Add charges to the Field of Dreams and see how they react to the electric field. Turn on a background electric field and adjust the direction and magnitude.

Click to view content.

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