Chapter 21 notes

The Electric Field

Recall that Coulomb’s law is very similar to Newton’s Law of Universal Gravitation.

  • There are more similarities between the interaction between electromagnetically charged particles, and the interaction between masses.
  • Surrounding every mass is a 3-dimensional sphere of influence within which gravitational forces are at work. This is called the gravitational field of the mass, and it exists regardless of the presence of any mass other than the central mass creating the field.

We used the symbol g to represent the strength of this field in Newton's Second Law formula:

F = mg (F = ma!)

We also learned that g near the surface of the earth was considered constant at -9.8 m/s2. Where does this value come from?

    Check out Newton's Universal Law of Gravitation:


    • G is the universal gravitation constant (6.67 x 10-11 Nm2/kg2),
    • m1 is the mass of the earth (5.98 x 1024 kg),
    • m2 is the mass of some other object near the surface of the earth, and
    • d is the separation between the centers of the two masses.

    We can refer to the earth as the central mass in this system.

    Near the surface of the earth, d is just the radius of the earth (6.37 x 106 m). In the Universal Gravitation Law, the values G, m1 and d are constant at the earth's surface, so we can give them a constant value. Solve the equation below for g, and see what you get:

    Surprise! (did you get 9.8 m/s2 ?)

    This value is the gravitational field strength of the earth, at it's surface and has units of N/kg which is the same as m/s2. N/kg can be read as Force per unit mass.

What happens to g if d increases, as in the case where that secondary object gets significantly farther from the surface? g would decrease. That is to say that the strength of the gravitational field weakens.

Similarly, another 'invisible force field' exists around a central charged object. You already know that a second charged particle near any other charge will experience an electric force. This force may be directed away from, or toward this central charged object depending on the signs of the charges. But even if there is NO second charged object, an electric field exists around the central object. A force will become evident as soon as a second charge is brought into the field. The electric field consists of an infinite number of points around a centrally charged object at which a second charge may exist. In the diagram, the electric field looks circular, but consider it to be more spherical, with no outer boundaries.

The electric field around a charged object has a strength that can be calculated and measured. At any particular distance from the object, the strength of the field depends upon the magnitude of the charge upon the object. There is no force until a second charge is brought to that location, but the field strength can be described in terms of force per unit charge, or Newtons of force per Coulomb (N/C) of charge on that second object, if it were there!

  • For example, suppose an object with a charge q1 of -9.0 C produces a field strength E of 8.1 x 1010 N/C at a distance d of 1 meter from the object, and 2.0 x 1010 N/C at a distance of 2.0 meters, and so on. Remember how the inverse square law works.
  • What force would exist between the central object in this field, and a 'test ' charge of +1.0 C at at distance of 1.0 m?
    • The force would be 8.1 x 1010 N.
    • The force would be different at d = 2.0 meters, or if the test charge had a different magnitude.
  • Test charges by convention are always positve.
  • These forces have direction, like any other force. If the two charges are similar, then the force on the test charge is repelling, and its direction is said to be outward from the central charged object creating the field. If the two charges are opposite, the force between them will be attractive, and the direction is inward.
  • Lines are drawn around a charge to indicte the direction of the field. Field direction is drawn based upon a positive test charge being in the field. If the central charged object carries a negative charge, then the field lines are drawn toward the central object, because the positive test charge would experience an attractive force.

  • If the central charged object carries a positive charge, then the field lines are drawn away from it, since a positive test charge in the field would experience a repulsive force.

(incidentally... the two diagrams above are exactly the same size)

If you examine the quantity Kq1/d2 from Coulomb’s law, you will find that by cancelling units, one is left with N/C, the unit we use to describe electric field strength. Furthermore, for any particular central charged object, E is constant. Electric field strength 'E ' can be calculated then, by the equation

E = kq1/d2

where q1 is the charge on the central charged object creating the field and d is the distance in the field from that object. The field strength for any object depends upon d. Compare this to g in Newton's Law of Universal Gravitation Constant.

Since we now know that E = kq1/d2 , substituting E into Coulomb’s law, we find that

Fe = Eq2 and E = Fe/q2

  • This represents a second method of calculating the field strength E when the electric force bewteen the central charged object and a second charge is already known.
  • In this case, q2 is the charge of the second object brought into an existing electric field, such as the 'test ' charge mentioned above.
  • Be carefull: two formulas for E have been presented here, each containing a value of 'q '. The value of q in each formula represents charges on different objects.

Helpful link:

video: 'Electric Forces and Electric Fields ' from the Mechanical Universe Series. (15 min.)