1.6 Electric Field Lines

LEARNING OBJECTIVES

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

  • Explain the purpose of an electric field diagram
  • Describe the relationship between a vector diagram and a field line diagram
  • Explain the rules for creating a field diagram and why these rules make physical sense
  • Sketch the field of an arbitrary source charge

Now that we have some experience calculating electric fields, let’s try to gain some insight into the geometry of electric fields. As mentioned earlier, our model is that the charge on an object (the source charge) alters space in the region around it in such a way that when another charged object (the test charge) is placed in that region of space, that test charge experiences an electric force. The concept of electric field lines, and of electric field line diagrams, enables us to visualize the way in which the space is altered, allowing us to visualize the field. The purpose of this section is to enable you to create sketches of this geometry, so we will list the specific steps and rules involved in creating an accurate and useful sketch of an electric field.

It is important to remember that electric fields are three-dimensional. Although in this book we include some pseudo-three-dimensional images, several of the diagrams that you’ll see (both here, and in subsequent chapters) will be two-dimensional projections, or cross-sections. Always keep in mind that in fact, you’re looking at a three-dimensional phenomenon.

Our starting point is the physical fact that the electric field of the source charge causes a test charge in that field to experience a force. By definition, electric field vectors point in the same direction as the electric force that a (hypothetical) positive test charge would experience, if placed in the field (Figure 1.6.1).

(Figure 1.6.1)   \begin{gather*}.\end{gather*}

The electric field is shown as arrows at test points on a grid. In figure a, the field is shown in the x y plane, with x and y measured in meters and ranging from -4 meters to 4 meters. The arrows point away from the origin, and are largest near the origin, decreasing with distance from the origin. In figure b, a three dimensional vector field is shown. The charge is at the center and, again, the arrows are largest near the origin, decreasing with distance from the origin.
Figure 1.6.1 The electric field of a positive point charge. A large number of field vectors are shown. Like all vector arrows, the length of each vector is proportional to the magnitude of the field at each point. (a) Field in two dimensions; (b) field in three dimensions.

We’ve plotted many field vectors in the figure, which are distributed uniformly around the source charge. Since the electric field is a vector, the arrows that we draw correspond at every point in space to both the magnitude and the direction of the field at that point. As always, the length of the arrow that we draw corresponds to the magnitude of the field vector at that point. For a point source charge, the length decreases by the square of the distance from the source charge. In addition, the direction of the field vector is radially away from the source charge, because the direction of the electric field is defined by the direction of the force that a positive test charge would experience in that field. (Again, keep in mind that the actual field is three-dimensional; there are also field lines pointing out of and into the page.)

This diagram is correct, but it becomes less useful as the source charge distribution becomes more complicated. For example, consider the vector field diagram of a dipole (Figure 1.6.2).

(Figure 1.6.2)   \begin{gather*}.\end{gather*}

A vector plot of the electric field due to two sources. The sources are not shown. The field is represented by arrows in an x y graph. Both x and y are in meters and both scales are from -2 meters to 4 meters. Near the origin, the arrows are long and point away from it. Near the point at coordinates 2, 0 the arrows are long and point toward the point. The arrows get smaller as we move farther from those two location and point in intermediate directions.
Figure 1.6.2 The vector field of a dipole. Even with just two identical charges, the vector field diagram becomes difficult to understand.

There is a more useful way to present the same information. Rather than drawing a large number of increasingly smaller vector arrows, we instead connect all of them together, forming continuous lines and curves, as shown in Figure 1.6.3.

(Figure 1.6.3)   \begin{gather*}.\end{gather*}

In part a, electric field lines emanating from a positive charge are shown as straight arrows radiating out from the charge in all directions. In part b, a pair of charges is shown, with one positive and the other negative. The field lines are represented by curved arrows. The arrows start from the positive charge, radiating outward but curving to end at the negative charge. The outer field lines extend beyond the drawing region, but follow the same behavior as those that are within the drawing area.
Figure 1.6.3 (a) The electric field line diagram of a positive point charge. (b) The field line diagram of a dipole. In both diagrams, the magnitude of the field is indicated by the field line density. The field vectors (not shown here) are everywhere tangent to the field lines.

Although it may not be obvious at first glance, these field diagrams convey the same information about the electric field as do the vector diagrams. First, the direction of the field at every point is simply the direction of the field vector at that same point. In other words, at any point in space, the field vector at each point is tangent to the field line at that same point. The arrowhead placed on a field line indicates its direction.

As for the magnitude of the field, that is indicated by the field line density—that is, the number of field lines per unit area passing through a small cross-sectional area perpendicular to the electric field. This field line density is drawn to be proportional to the magnitude of the field at that cross-section. As a result, if the field lines are close together (that is, the field line density is greater), this indicates that the magnitude of the field is large at that point. If the field lines are far apart at the cross-section, this indicates the magnitude of the field is small. Figure 1.6.4 shows the idea.

(Figure 1.6.4)   \begin{gather*}.\end{gather*}

Seven electric field lines are shown, generally going from bottom left to top right. The field lines get closer together toward the top. Two square areas, perpendicular to the field lines, are shaded. All of the field lines pass through each shaded area. The area toward the top is smaller than the area toward the bottom.
Figure 1.6.4 Electric field lines passing through imaginary areas. Since the number of lines passing through each area is the same, but the areas themselves are different, the field line density is different. This indicates different magnitudes of the electric field at these points.

In Figure 1.6.4, the same number of field lines passes through both surfaces (S and S'), but the surface S is larger than surface S'. Therefore, the density of field lines (number of lines per unit area) is larger at the location of S', indicating that the electric field is stronger at the location of S' than at S. The rules for creating an electric field diagram are as follows.

Problem-Solving Strategy: Drawing Electric Field Lines

  1. Electric field lines either originate on positive charges or come in from infinity, and either terminate on negative charges or extend out to infinity.
  2. The number of field lines originating or terminating at a charge is proportional to the magnitude of that charge. A charge of 2q will have twice as many lines as a charge of q.
  3. At every point in space, the field vector at that point is tangent to the field line at that same point.
  4. The field line density at any point in space is proportional to (and therefore is representative of) the magnitude of the field at that point in space.
  5. Field lines can never cross. Since a field line represents the direction of the field at a given point, if two field lines crossed at some point, that would imply that the electric field was pointing in two different directions at a single point. This in turn would suggest that the (net) force on a test charge placed at that point would point in two different directions. Since this is obviously impossible, it follows that field lines must never cross.

Always keep in mind that field lines serve only as a convenient way to visualize the electric field; they are not physical entities. Although the direction and relative intensity of the electric field can be deduced from a set of field lines, the lines can also be misleading. For example, the field lines drawn to represent the electric field in a region must, by necessity, be discrete. However, the actual electric field in that region exists at every point in space.

Field lines for three groups of discrete charges are shown in Figure 1.6.5. Since the charges in parts (a) and (b) have the same magnitude, the same number of field lines are shown starting from or terminating on each charge. In (c), however, we draw three times as many field lines leaving the +3q charge as entering the -q. The field lines that do not terminate at -q emanate outward from the charge configuration, to infinity.

(Figure 1.6.5)   \begin{gather*}.\end{gather*}

Three pairs of charges and their field lines are shown. The charge on the left is positive in each case. In part a, the charge on the right is negative. The field lines are represented by curved arrows starting at the positive charge on the left, curving toward and terminating at the negative charge on the right. Between the charges, the field lines are dense. In part b, the charge on the right is positive. The field lines represented by curved arrows start at each of the positive charges and diverge outward. Between the charges, the field lines are less dense, and there is a black region midway between the charges. In part c, the charge on the right is negative. The field lines start at the positive charge. Some of the lines, those that start closest to the negative charge, curve toward the negative charge and terminate there. Lines that start further from the negative charge curve toward it but then diverge outward. There is an area with very low density of lines to the right of the pair of charges.
Figure 1.6.5 Three typical electric field diagrams. (a) A dipole. (b) Two identical charges. (c) Two charges with opposite signs and different magnitudes. Can you tell from the diagram which charge has the larger magnitude?

The ability to construct an accurate electric field diagram is an important, useful skill; it makes it much easier to estimate, predict, and therefore calculate the electric field of a source charge. The best way to develop this skill is with software that allows you to place source charges and then will draw the net field upon request. We strongly urge you to search the Internet for a program. Once you’ve found one you like, run several simulations to get the essential ideas of field diagram construction. Then practice drawing field diagrams, and checking your predictions with the computer-drawn diagrams.

INTERACTIVE

One example of a field-line drawing program is from the PhET “Charges and Fields” simulation.

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Introduction to Electricity, Magnetism, and Circuits Copyright © 2018 by Daryl Janzen is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.