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OpenStax-CNX module: m52387 1 ∗ 18.6 Electric Field Lines Bobby Bailey Based on Electric Field Lines: Multiple Charges† by OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0‡ Abstract • Calculate the total force (magnitude and direction) exerted on a test charge from more than one charge • Describe an electric eld diagram of a positive point charge; of a negative point charge with twice the magnitude of positive charge • Draw the electric eld lines between two points of the same charge; between two points of opposite charge. Drawings using lines to represent electric elds around charged objects are very useful in visualizing eld strength and direction. Since the electric eld has both magnitude and direction, it is a vector. Like all vectors, the electric eld can be represented by an arrow that has length proportional to its magnitude and that points in the correct direction. (We have used arrows extensively to represent force vectors, for example.) Figure 1 shows two pictorial representations of the same electric eld created by a positive point charge Q. Figure 1 (b) shows the standard representation using continuous lines. Figure 1 (b) shows numerous individual arrows with each arrow representing the force on a test charge q. Field lines are essentially a map of innitesimal force vectors. ∗ Version 1.1: Dec 21, 2014 8:37 pm -0600 † http://cnx.org/content/m42312/1.7/ ‡ http://creativecommons.org/licenses/by/4.0/ http://cnx.org/content/m52387/1.1/ OpenStax-CNX module: m52387 2 Figure 1: Two equivalent representations of the electric eld due to a positive charge Q. (a) Arrows representing the electric eld's magnitude and direction. (b) In the standard representation, the arrows are replaced by continuous eld lines having the same direction at any point as the electric eld. The closeness of the lines is directly related to the strength of the electric eld. A test charge placed anywhere will feel a force in the direction of the eld line; this force will have a strength proportional to the density of the lines (being greater near the charge, for example). Note that the electric eld is dened for a positive test charge q, so that the eld lines point away from a positive charge and toward a negative charge. (See Figure 2.) The electric eld strength is exactly proportional to the number of eld lines per unit area, since the magnitude of the electric eld for a point charge is E = k|Q|/r and area is proportional to r . This pictorial representation, in which eld lines represent the direction and their closeness (that is, their areal density or the number of lines crossing a unit area) represents strength, is used for all elds: electrostatic, gravitational, magnetic, and others. 2 2 Figure 2: The electric eld surrounding three dierent point charges. (a) A positive charge. (b) A negative charge of equal magnitude. (c) A larger negative charge. In many situations, there are multiple charges. The total electric eld created by multiple charges is the http://cnx.org/content/m52387/1.1/ OpenStax-CNX module: m52387 3 vector sum of the individual elds created by each charge. Figure 3 shows how the electric eld from two point charges can be drawn by nding the total eld at representative points and drawing electric eld lines consistent with those points. While the electric elds from multiple charges are more complex than those of single charges, some simple features are easily noticed. For example, the eld is weaker between like charges, as shown by the lines being farther apart in that region. (This is because the elds from each charge exert opposing forces on any charge placed between them.) (See Figure 3 and Figure 4(a).) Furthermore, at a great distance from two like charges, the eld becomes identical to the eld from a single, larger charge. Figure 4(b) shows the electric eld of two unlike charges. The eld is stronger between the charges. In that region, the elds from each charge are in the same direction, and so their strengths add. The eld of two unlike charges is weak at large distances, because the elds of the individual charges are in opposite directions and so their strengths subtract. At very large distances, the eld of two unlike charges looks like that of a smaller single charge. Figure 3: Two positive point charges q1 and q2 produce the resultant electric eld shown. The eld is calculated at representative points and then smooth eld lines drawn following the rules outlined in the text. http://cnx.org/content/m52387/1.1/ OpenStax-CNX module: m52387 4 Figure 4: (a) Two negative charges produce the elds shown. It is very similar to the eld produced by two positive charges, except that the directions are reversed. The eld is clearly weaker between the charges. The individual forces on a test charge in that region are in opposite directions. (b) Two opposite charges produce the eld shown, which is stronger in the region between the charges. We use electric eld lines to visualize and analyze electric elds (the lines are a pictorial tool, not a physical entity in themselves). The properties of electric eld lines for any charge distribution can be summarized as follows: 1. Field lines must begin on positive charges and terminate on negative charges, or at innity in the hypothetical case of isolated charges. 2. The number of eld lines leaving a positive charge or entering a negative charge is proportional to the magnitude of the charge. 3. The strength of the eld is proportional to the closeness of the eld linesmore precisely, it is proportional to the number of lines per unit area perpendicular to the lines. 4. The direction of the electric eld is tangent to the eld line at any point in space. 5. Field lines can never cross. The last property means that the eld is unique at any point. The eld line represents the direction of the eld; so if they crossed, the eld would have two directions at that location (an impossibility if the eld is http://cnx.org/content/m52387/1.1/ OpenStax-CNX module: m52387 5 unique). 1 Section Summary • • • • • • Drawings of electric eld lines are useful visual tools. The properties of electric eld lines for any charge distribution are that: Field lines must begin on positive charges and terminate on negative charges, or at innity in the hypothetical case of isolated charges. The number of eld lines leaving a positive charge or entering a negative charge is proportional to the magnitude of the charge. The strength of the eld is proportional to the closeness of the eld linesmore precisely, it is proportional to the number of lines per unit area perpendicular to the lines. The direction of the electric eld is tangent to the eld line at any point in space. Field lines can never cross. 2 Conceptual Questions Exercise 1 Compare and contrast the Coulomb force eld and the electric eld. To do this, make a list of ve properties for the Coulomb force eld analogous to the ve properties listed for electric eld lines. Compare each item in your list of Coulomb force eld properties with those of the electric eldare they the same or dierent? (For example, electric eld lines cannot cross. Is the same true for Coulomb eld lines?) 3 Problem Exercises Exercise 2 (a) Sketch the electric eld lines near a point charge +q. (b) Do the same for a point charge . Exercise 3 Sketch the electric eld lines a long distance from the charge distributions shown in Figure 4 (a) and (b) Exercise 4 Figure 5 shows the electric eld lines near two charges q and q . What is the ratio of their magnitudes? (b) Sketch the electric eld lines a long distance from the charges shown in the gure. − − 3.00q 1 http://cnx.org/content/m52387/1.1/ 2 OpenStax-CNX module: m52387 Figure 5: The electric eld near two charges. Exercise 5 Sketch the electric eld lines in the vicinity of two opposite charges, where the negative charge is three times greater in magnitude than the positive. (See Figure 5 for a similar situation). Glossary Denition 1: electric eld a three-dimensional map of the electric force extended out into space from a point charge Denition 2: electric eld lines a series of lines drawn from a point charge representing the magnitude and direction of force exerted by that charge Denition 3: vector a quantity with both magnitude and direction Denition 4: vector addition mathematical combination of two or more vectors, including their magnitudes, directions, and positions http://cnx.org/content/m52387/1.1/ 6