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OpenStax Staging4 module: m10495 1 Electric Field Lines ∗ Words Numbers This work is produced by OpenStax Staging4 and licensed under the Creative Commons Attribution License 4.0† Abstract By the end of this section, you will be able to: • Explain the purpose of an electric eld diagram • Describe the relationship between a vector diagram and a eld line diagram • Explain the rules for creating a eld diagram and why these rules make physical sense • Sketch the eld of an arbitrary source charge Now that we have some experience calculating electric elds, let's try to gain some insight into the geometry of electric elds. 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) eld lines, and of electric eld line diagrams, enables us to visualize the way in which the space is altered, allowing is placed in that region of space, that test charge experiences an electric force. The concept of electric us to visualize the eld. The purpose of this section is to enable you to create sketches of this geometry, so we will list the specic steps and rules involved in creating an accurate and useful sketch of an electric eld. It is important to remember that electric elds 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 eld of the source charge causes a test charge in that eld to experience a force. By denition, electric eld vectors point in the same direction as the electric force that a (hypothetical) positive test charge would experience, if placed in the eld (Figure 1) ∗ Version 1.5: Sep 13, 2016 3:03 am -0500 † http://creativecommons.org/licenses/by/4.0/ http://legacy-staging4.cnx.org/content/m10495/1.5/ OpenStax Staging4 module: m10495 2 Figure 1: The electric eld of a positive point charge. A large number of eld vectors are shown. Like all vector arrows, the length of each vector is proportional to the magnitude of the eld at each point. (a) Field in two dimensions; (b) eld in three dimensions. We've plotted many eld vectors in the gure, which are distributed uniformly around the source charge. Since the electric eld is a vector, the arrows that we draw correspond at every point in space to both the magnitude and the direction of the eld at that point. As always, the length of the arrow that we draw corresponds to the magnitude of the eld 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 eld vector is radially away from the source charge, because the direction of the electric eld is dened by the direction of the force that a positive test charge would experience in that eld. (Again, keep in mind that the actual eld is three-dimensional; there are also eld 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 eld diagram of a dipole (Figure 2). http://legacy-staging4.cnx.org/content/m10495/1.5/ OpenStax Staging4 module: m10495 3 Figure 2: The vector eld of a dipole. Even with just two identical charges, the vector eld diagram becomes dicult 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 3. http://legacy-staging4.cnx.org/content/m10495/1.5/ OpenStax Staging4 module: m10495 4 Figure 3: (a) The electric eld line diagram of a positive point charge. (b) The eld line diagram of a dipole. In both diagrams, the magnitude of the eld is indicated by the eld line density. The eld vectors (not shown here) are everywhere tangent to the eld lines. Although it may not be obvious at rst glance, these eld diagrams convey the same information about the electric eld as do the vector diagrams. First, the direction of the eld at every point is simply the direction of the eld vector at that same point. In other words, at any point in space, the eld vector at each point is tangent to the eld line at that same point. The arrowhead placed on a eld line indicates its direction. As for the magnitude of the eld, that is indicated by the eld line densitythat is, the number of eld lines per unit area passing through a small cross-sectional area perpendicular to the electric eld. This eld line density is drawn to be proportional to the magnitude of the eld at that cross-section. As a result, if the eld lines are close together (that is, the eld line density is greater), this indicates that the magnitude of the eld is large at that point. If the eld lines are far apart at the cross-section, this indicates the magnitude of the eld is small. Figure 4 shows the idea. http://legacy-staging4.cnx.org/content/m10495/1.5/ OpenStax Staging4 module: m10495 5 Figure 4: Electric eld lines passing through imaginary areas. Since the number of lines passing through each area is the same, but the areas themselves are dierent, the eld line density is dierent. This indicates dierent magnitudes of the electric eld at these points. S In Figure 4, the same number of eld lines passes through both surfaces ( is larger than surface the location of S0, S 0 . and S0, but the surface S Therefore, the density of eld lines (number of lines per unit area) is larger at indicating that the electric eld is stronger at the location of S0 than at S. The rules for creating an electric eld diagram are as follows. note: 1.Electric eld lines either originate on positive charges or come in from innity, and either http://legacy-staging4.cnx.org/content/m10495/1.5/ OpenStax Staging4 module: m10495 6 terminate on negative charges or extend out to innity. 2.The number of eld lines originating or terminating at a charge is proportional to the magnitude of that charge. A charge of 2 q will have twice as many lines as a charge of q. 3.At every point in space, the eld vector at that point is tangent to the eld line at that same point. 4.The eld line density at any point in space is proportional to (and therefore is representative of ) the magnitude of the eld at that point in space. 5.Field lines can never cross. Since a eld line represents the direction of the eld at a given point, if two eld lines crossed at some point, that would imply that the electric eld was pointing in two dierent 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 dierent directions. Since this is obviously impossible, it follows that eld lines must never cross. Always keep in mind that eld lines serve only as a convenient way to visualize the electric eld; they are not physical entities. Although the direction and relative intensity of the electric eld can be deduced from a set of eld lines, the lines can also be misleading. For example, the eld lines drawn to represent the electric eld in a region must, by necessity, be discrete. However, the actual electric eld in that region exists at every point in space. Field lines for three groups of discrete charges are shown in Figure 5. Since the charges in parts (a) and (b) have the same magnitude, the same number of eld lines are shown starting from or terminating on each charge. In (c), however, we draw three times as many eld lines leaving the +3q charge as entering the The eld lines that do not terminate at −q −q . emanate outward from the charge conguration, to innity. Figure 5: Three typical electric eld diagrams. (a) A dipole. (b) Two identical charges. (c) Two charges with opposite signs and dierent magnitudes. Can you tell from the diagram which charge has the larger magnitude? The ability to construct an accurate electric eld diagram is an important, useful skill; it makes it much easier to estimate, predict, and therefore calculate the electric eld 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 eld 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 eld diagram construction. Then practice drawing eld diagrams, and checking your predictions with the computer-drawn diagrams. http://legacy-staging4.cnx.org/content/m10495/1.5/ OpenStax Staging4 module: m10495 note: One example of a eld-line drawing program 7 1 is from the PhET Charges and Fields simulation. 1 Summary Electric eld diagrams assist in visualizing the eld of a source charge. The magnitude of the eld is proportional to the eld line density. Field vectors are everywhere tangent to eld lines. 2 Conceptual Questions Exercise 1 (Solution on p. 10.) If a point charge is released from rest in a uniform electric eld, will it follow a eld line? Will it do so if the electric eld is not uniform? Exercise 2 Under what conditions, if any, will the trajectory of a charged particle not follow a eld line? Exercise 3 (Solution on p. 10.) How would you experimentally distinguish an electric eld from a gravitational eld? Exercise 4 A representation of an electric eld shows 10 eld lines perpendicular to a square plate. How many eld lines should pass perpendicularly through the plate to depict a eld with twice the magnitude? Exercise 5 What is the ratio of the number of electric eld lines leaving a charge 10 (Solution on p. 10.) q and a charge q? 3 Problems Exercise 6 Which of the following electric eld lines are incorrect for point charges? Explain why. 1 https://openstax.org/l/21eldlindrapr http://legacy-staging4.cnx.org/content/m10495/1.5/ OpenStax Staging4 module: m10495 http://legacy-staging4.cnx.org/content/m10495/1.5/ 8 OpenStax Staging4 module: m10495 9 Exercise 7 (Solution on p. 10.) In this exercise, you will practice drawing electric eld lines. Make sure you represent both the magnitude and direction of the electric eld adequately. Note that the number of lines into or out of charges is proportional to the charges. (a) Draw the electric eld lines map for two charges +20 −20 µC µC and µC and +20 µC situated 5 cm from µC and situated 5 cm from each other. (b) Draw the electric eld lines map for two charges +20 each other. (c) Draw the electric eld lines map for two charges +20 −30 µC situated 5 cm from each other. Exercise 8 Draw the electric eld for a system of three particles of charges +1 µC,+2 µC, and −3 µC xed at the corners of an equilateral triangle of side 2 cm. Exercise 9 (Solution on p. 11.) Two charges of equal magnitude but opposite sign make up an electric dipole. A quadrupole consists of two electric dipoles are placed anti-parallel at two edges of a square as shown. Draw the electric eld of the charge distribution. Exercise 10 Suppose the electric eld of an isolated point charge decreased with distance as as 1/r2 . 1/r2+δ rather than Show that it is then impossible to draw continous eld lines so that their number per unit area is proportional to E. http://legacy-staging4.cnx.org/content/m10495/1.5/ OpenStax Staging4 module: m10495 10 Solutions to Exercises in this Module Solution to Exercise (p. 7) yes; no Solution to Exercise (p. 7) At the surface of Earth, the gravitational eld is always directed in toward Earth's center. An electric eld could move a charged particle in a dierent direction than toward the center of Earth. This would indicate an electric eld is present. Solution to Exercise (p. 7) 10 Solution to Exercise (p. 9) http://legacy-staging4.cnx.org/content/m10495/1.5/ OpenStax Staging4 module: m10495 http://legacy-staging4.cnx.org/content/m10495/1.5/ 11 OpenStax Staging4 module: m10495 Solution to Exercise (p. 9) Glossary Denition 1: eld line smooth, usually curved line that indicates the direction of the electric eld Denition 2: eld line density number of eld lines per square meter passing through an imaginary area; its purpose is to indicate the eld strength at dierent points in space http://legacy-staging4.cnx.org/content/m10495/1.5/ 12