Exploratories

Electric Forces & Fields

Terms to define

electric field -

the alteration of the properties of space around a charged body that will affect a
test charge with a force, F. The closer the test charge is to the body, the greater the force. The direction of the net electric field is defined to be in the direction of the force acting on a positive test charge. The strength of the force acting on the test charge at each point in space is the magnitude of the test charge times the electric field strength. Figure 1(a) illustrates the electric field in the region of a pair of unlike charges and Figure 1(b) illustrates the electric field in the region of a pair of like charges.

Figure 1

lines of force -

imaginary lines used in visualizing fields such as electric fields. The lines
emanate from positive charges and terminate on negative charges. The density of the lines in a region indicates the strength of the field there.

SUPPORTIVE BACKGROUND INFORMATION -

An historical look into the eighteenth and mid-nineteenth centuries allows us to glimpse the beginning of James Clerk Maxwell's electromagnetic wave theory. Coulomb (1738 - 1806) had provided direct experimental evidence that the inverse square law held true for electric charges and the fields surrounding them.
Priestly had proposed this even earlier when he predicted that electrical forces should behave in a way similar to gravitational force. However, it took the experimental genius of Michael Faraday (1791 - 1867) to introduce the concept of the electric field and, thereby, to provide a qualitative description of the behavior of electric forces.

Faraday was basically self-taught with little formal education. Nonetheless he became a remarkable chemist before turning his abilities as an experimentalist to physics. Although Faraday understood little mathematics, this did not inhibit his powerful insight which yielded qualitative descriptions of the electrical field and electric forces. Scientists like Gauss and Maxwell would eventually give a mathematical explanation of Faraday's experimental description of the electric
field. After observing how iron filings arrange themselves around a bar magnet, Faraday called the curved paths of the filings lines of force. He visualized a similar pattern of lines around positively and negatively charged bodies. Just as the lines appeared to originate on one pole of the magnet and terminate on the other, so he imagined that the lines of force of an electrical field would originate on a positive charge and end on a negative charge.

Are Faraday's lines of force real? Faraday believed that physical lines of force actually existed everywhere in space; today we no longer believe that this is true. However, Faraday's visualization of lines of force offers a model for understanding the electric field and has provided the foundation for a quantitative explanation of the phenomenon.

To define the electric field in mathematical terms, imagine a single point charge q in space. When this is the only charge present, there is no force acting on it. But let's imagine a small positive charge q
0, called a test charge, placed at a distance r from q. Now both q and q0 experience the Coulomb force

F = K e q q 0 / r 2 r

(Note: r is a dimensionless unit vector in the direction of r .)

Regrouping the equation yields:

F = q 0 (K e q / r 2 r)

The quantity in the parentheses does not depend on the magnitude of the test charge, but only on its distance from q. The test charge detects the force, but the quantity in brackets exist whether or not q 0 is there to detect it. That quantity is denoted by E and is called the electric field generated by an isolated point charge q.

E = K e q / r 2 r

Electric Fields and Potentials

1. An electric field surrounds an electric charge. The field strength at any place in the field can be found by placing a small positive test charge there. Where the force on the test charge is great, the field strength is great. Electric field strength is directly proportional to the force exerted on a positive test charge. The direction of an electric field at any point is the same as the direction of the force
exerted on the positive test charge. Some electric field lines surrounding a positive charge are shown.

They extend radially from the charge. A vector is sketched at point a to represent the force that would be exerted on a positive test charge there (its direction shows that like charges repel). Other points b,c,d,e and f, are all located at the same distance from the positive charge.

Draw a vector at each point b - f to show the force on the same test charge.

2. The electric field about a negative charge is shown to the right. The field lines point radially inward, in the same direction a positive test charge would be forced. Assume the magnitude of the negative charge is the same as the charge above.

Draw field vectors at each of the points h - m.

3. The pair of equal and opposite charges of questions 1 and 2 is shown below. Their individual fields, drawn uninfluenced by each other, over lap to form a field pattern that can be constructed by vector rules. This is shown at locations a and b, where the two forces combine to a single resultant force. Note that point b overlaps point m, and also points c and l overlap. Note how the size of each vector depends on its distance from the charge (inverse-square law). Every point in the field is the result of both the positive and the negative charges.

a. By vector rules, show the resultant of all the vector pairs shown.

b. Sketch in sample vector resultants at a few other places. Does the pattern that emerges agree with the field pattern shown in Figure a on the next page?

4. Three points, (a,b,c) are indicated on each electric field pattern. Point a in each pattern shows the electric field vector at that point. The vector indicates the magnitude and direction of the force that a positive test charge would experience at that point (a curved field indicates that the force on a nearby test charge would be different in magnitude and direction). Use the vector at points a as a reference and sketch in the electric field vectors for points b and c in each pattern, using colored ink or pencil.

Electric Force & Fields

Everyday Connections

Early scientists initiated the intuitive view of a force being a push or a pull. How can electric charges exert such pushes and pulls without coming in contact with each other? In fact, how does one charge "know" that another is around?

The question of pushes and pulls without actual contact centers on the idea of "action at a distance", a concept that bothered even Isaac Newton in his description of universal gravitation. Michael Faraday first proposed the notion that a charge alters the space around itself and creates a force field.

After viewing the three-dimensional patterns of lines of force presented in the video, try sketching below some patterns yourself in two dimensions.

What is the difference between the electric charge on an object and the associated electric field?

An electric field does not deal with the body itself, but the region surrounding it. It describes the alteration of the space in the region surrounding the body and results in an electric force detected by a test charge entering the region. In other words, the charge is the "thing" that causes the field. An analogy might be the difference between a rose and the pleasant fragrance which surrounds it.

Why do electric lines of force never cross?

The electric field E has a unique direction at any point in space. If two lines crossed, two directions would be indicated for E at the point of intersection and the electric field would not be unique at that point.

How do charges distribute themselves on the surface of a spherical conductor under electrostatic conditions?