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CHAPTER· 21 Electric Fields ~ High Energy Halo Research into ultra-high-voltage-up to two million voltstransmission of electricity is essential for development of future power technology. At a special test facility, this ultrahigh-voltage test lir::ledemonstrates a man-made corona effect. Why is the corona visible around this power line when the effect is not seen in the electric wires in our homes? Chapter Outline 21.1 CREATING AND MEASURING ElECTRIC FIELDS · The Electric Field · Picturing the Electric Field 21.2 APPLICATIONS OF THE ElECTRIC FIELD · Energy and the Electric Potential · The Electric Potential in a Uniform Field · Millikan's Oil Drop Experiment · Sharing of Charge · Electric Fields Near Conductors · Storing Electric Energy-The Capacitor {Concept Check T he blue glow around the wires in the photo is a modern version of "St. Elmo's Fire." Sailors in the time of Columbus saw these ghostly-colored streamers issuing from their high ships' masts. They recognized them as warning signs of an approaching lightning storm. The glow is indeed related to lightning. The man-made "fire" in the photo is called a corona discharge. Besides causing the eery glow, it causes radio interference and can start a spark jumping from the wire to another conductor, so designers of these experimental power lines try to reduce or eliminate it entirely. The following terms or concepts from earlier chapters are important for a good understanding of this chapter. If you are not familiar with them, you should review them before studying this chapter. · gravitational field, Chapter 8 · gravitational potential energy, Chapter 11 · electrical interactions, Coulomb's law, Chapter 20 425 ObjeCtives 21.1 • define an electric field; explain how to measure it. • distinguish between force and field. • be able to solve problems relating field, force, and charge. • distinguish between electric field and field lines. T Charge and Field Conventions • Positive charges are red. • Negative charges are blue. ~ Electric fields are red. POCKET LAB ELECTRIC FIELDS Use a 5-cm long thread to hang a pith ball (or small plasticfoam ball) from the end of a dangling straw. Rub a 25-cm square piece of plasticfoam with fur to put a large negative charge on the foam. Stand the foam in a vertical orientation.Give the pith ball a positive test charge (use the fur). Notice that the pith ball hangs straight down until you bring it towardthe chargedfoam. The pith ball is now in an electric field. How does the pith ball indicate the "strength" of the field? Move the test charge (pith ball) to investigate the electric field around the foam. Record your results. 426 Electric Fields ...................................... CREATING AND MEASURING ELECTRIC FIELDS he electric force, like the gravitational force, varies inversely as the square of the distance between two objects. Both forces can act at a great distance. How can a force be exerted across what seems to be empty space?In trying to understand the electric force, Michael Faraday (1791-1867) developed the concept of an electric field. According to Faraday, a charge creates an electric field about it in all directions. If a second charge is placed at some point in the field, the second charge interacts with the field at that point. The force it feels is the result of a local interaction. Interaction between particles separated by some distance is no longer required. The Electric Field It is easy to state that a charge produces an electric field. But how can the field be detected and measured? We will describe a method that can be used to measure the field produced by an electric charge q. You must measure the field at a specific location, for example, point A. An electrical field can be observed only because it produces forces on other charges, so a small positive test charge is placed at A. The force exerted on the test charge, q', at this location is measured. According to Coulomb's law, the force is proportional to the test charge. If the size of the charge is doubled, the force is doubled. Thus, the ratio of force to charge is independent of the size of the charge. If we divide the magnitude of the force, F, on the test charge, measured at point A, by the size of the test charge, q', we obtain a vector quantity, Flq'. This quantity does not depend on the test charge, only on the charge q and the location of point A. We call this vector quantity the magnitude of the electric field. The electric field at point A, the location of q', is [E~FO""I q' The direction of the electric field is the direction of the force on the positive test charge. The magnitude of the electric field intensity is measured in newtons per coulomb, N/C. Notice that just as the electric field is the force per unit charge, the gravitational field is the force per unit mass, g = Flm. We said that an electric field should be measured by a small test charge. Why? The test charge exerts a force on q. We want to make sure the force exerted by the test charge doesn't move q to another location, and thus change the force on q' and the electric field we are measuring. So far we have found the field at a single point. The test charge is now moved to another location. The force on it is measured again and the electric field at that location calculated. This process is repeated again and again until every location in space has a measurement of the vector quantity, the electric field, on the test charge associated with it. F. Y. I. Example Problem Calculating the Magnitude of the Electric Field A positive test charge of 4.0 x 10-5 C is placed in an electric field. The force on it is 0.60 N acting at 10°. What is the magnitude and direction of the electric field at the location of the test charge? Unknown: Given: size of test charge, q' = 4.0 X 10-5 C electric field, E F Basic equation: force on test charge, F = 0.60 N at 10° E Robert Van de Graaff devised the high-voltage electrostatic generator in the 1930s. These generators can build up giant potentials that can accelerate particles to high energies. = ----; q Solution: Find the size of the field intensity. E F 0.60N = q' = 4.0 x 10-5 C = 1 .5 x 104N/C The field direction is in the direction of the force since it is a positive test charge, so E = 1.5 X 104 N/C at 10°. If the force were in the opposite direction, the field at the test charge would also be in the opposite direction, that is, E = 1.5 X 104 N/C at 10° + 180° = 190°. Practice Problems 1. A negative charge of 2.0 x 10-8 C experiences a force of 0.060 N to the right in an electric field. What is the field magnitude and direction? 2. A positive test charge of 5.0 x 10-4 C is in an electric field that exerts a force of 2.5 x 10-4 N on it. What is the magnitude of the electric field at the location of the test charge? 3. Suppose the electric field in Practice Problem 2 were caused by a point charge. The test charge is moved to a distance twice as far from the charge. What is the magnitude of the force that the field exerts on the test charge now? ~ 4. You are probing the field of a charge of unknown magnitude and sign. You first map the field with a 1.0 x 10-6 C test charge, then repeat your work with a 2.0 x 10-6 C charge. a. Would you measure the same forces with the two test charges? Explain. b. Would you find the same fields? Explain. The collection of all the force vectors on the test charge is called an electric field. Any charge placed in an electric field experiences a force on it due to the electric field at that location. The strength of the force depends on the magnitude of the field, E, and the size of the charge, q. Thus F = Eq. The direction of the force depends on the direction of the field and the sign of the charge. The total electric field is the vector sum of the fields of individual charges. 21.1 Creating and Measuring Electric Fields 427 FIGURE 21-1. An electric field can be shown by using arrows to represent the direction of the field at various locations. Electric field Field charge e A picture of an electric field can be made by using arrows to represent the field vectors at various locations, Figure 21-1. The length of the arrow shows the magnitude of the field; the direction of the arrow shows its direction. To find the field from two charges, the fields from the individual charges can be added vectorially. Or, a test charge can be used to map out the field due to any collection of charges. Typical electric fields produced by charge collections are shown in Table 21-1. Metal dome Insulator Picturing the Electric Field An alternative picture of an electric field is shown in Figure 21-3. The lines are called electric field lines. The direction of the field at any point is the tangent drawn to the field line at that point. The strength of the electric field is indicated by the spacing between the lines. The field is strong where the lines are close together. It is weaker where the lines are spaced farther apart. Remember that electric fields exist in three dimensions. Our drawings are only two-dimensional models. The direction of the force on the positive test charge near a positive charge is away from the charge. Thus, the field lines extend radially outward like the spokes of a wheel, Figure 21-3a. Near a negative charge the direction of the force on the positive test charge is toward the charge, so the field lines point radially inward, Figure 21-3b. Table 21-1 Approximate Values of Typical Electric Fields FIGURE 21-2. Charge is transferred onto a moving belt at A, and from the belt to the metal dome at B. An electric motor does the work needed to increase the electric potential energy. 428 Electric Fields Field Near charged hard rubber rod In television picture tube Needed to create spark in air At electron orbit in hydrogen atom Value (N/C) 1 1 3 5 x x X X 103 105 106 10" a c b FIGURE 21-3. Lines of force are drawn perpendicularly away from the positive object (a) and perpendicularly into the negative object (b). Electric field lines between like- and oppositely-charged objects are shown in (c). When there are two or more charges, the field is the vector sum of the fields due to the individual charges. The field lines become curved and the pattern complex, Figure 21-3c. Note that field lines always leave a positive charge and enter a negative charge. The Van de Graaff machine is a device that transfers large amounts of charge from one part of the machine to the top metal terminal Figure 21-2. A person touching the terminal is charged electrically. The charges on the person's hairs repel each other, causing the hairs to follow the field lines. Another method of visualizing field lines is to use grass seed in an insulating liquid such as mineral oil. The electric forces cause a separation of charge in each long, thin grass seed. The seeds then turn so they line up along the direction of the electric field. Therefore, the seeds form a pattern of the electric field lines. The patterns in Figure 21-4 were made this way. Field lines do not really exist. They are just a means of providing a model of an electric field. Electric fields, on the other hand, do exist. An electric field is produced by one or more charges and is independent of the existence of the test charge that is used to measure it. The field provides a method of calculating the force on a charged body. It does not explain, however, why charged bodies exert forces on each other. That question is still unanswered. a b c d An electric field points away from positive charges and toward negative charges. Electric fields are real. Electric field lines are imaginary, but help in making a picture of the field. FIGURE 21-4. Lines of force between unlike charges (a, c) and between like charges (b, d) describe the behavior of a positively-charged object in a field. The top photographs are computer tracings of electric field lines. 21.1 Creating and Measuring Electric Fields 429 PH Y SICSLAB :: Voltage Charges, Energy, Purpose To make a model that demonstrates the relationship of charge, energy, and voltage. 4. Tape the 3-V rectangle to the 3" mark on the ruler, the 6-V to the 6" mark, and so on. 5. Let each steel ball represent 1 C of charge. 6. Lift and tape 4 steel balls to the 3-V rectangle, 3 to the 6-V rectangle, and so on. Materials · · · · ball of clay ruler cellophane tape 12 of 3-mm diameter steel balls Procedure 1. Use the clay to support the ruler vertically on the tabletop. (The A" end should be at the table.) 2. Cut a 2 cm x 8 cm rectangular piece of paper and write on it "3 V = 3 J/C". 3. Cut three more rectangles and label them: 6 V = 6 J/C, 9 V = 9 J/C 12 V = 12 JlC Observations and Data 1. The model shows different amounts of charges at different energy levels. Where should steel balls be placed to show a zero energy level? Explain. 2. Make a data table with the columns labeled "charge", "voltage", and "energy". 3. Fill in the data table for your model for each level of the model. Analysis 1. How much energy is required to lift each coulomb of charge from the tabletop to the 9-V level? 2. What is the total potential energy stored in the 9-V level? 3. The total energy of the charges in the 6-V level is not 6 J. Explain this. 4. How much energy would be given off if the charges in the 9-V level fell to the 6-V level? Explain. Applications 1. A 9-V battery is very small. A 12-V car battery is very big. Use your model to help explain why two 9-V batteries would not start your car. 430 Electric Fields ........................ CONCEPT REVIEW 1.1 Suppose you are asked to measure the electric field in space. Answer the questions of this step-by-step procedure. a. How do you detect the field at a point? b. How do you determine the magnitude of the field? c. How do you choose the size of the test charge? d. What do you do next? 1.2 Suppose you are given an electric field, but the charges that produce the field are hidden. If all the field lines point into the hidden region, what can you say about the sign of the charge in that region? 1.3 How does the electric field, E, differ from the force, F, on the test charge? 1.4 Critical Thinking: Figure 21-4b shows the field from two like charges. The top positive charge in Figure 21-3c could be considered a test charge measuring the field due to the two negative charges. Is this positive charge small enough to produce an accurate measure of the field? Explain. 21.2 APPLICATIONS OF THE ELECTRIC FIELD Objectives T he concept of energy is extremely useful in mechanics. The law of conservation of energy allows us to solve problems without knowing the forces in detail. The same is true in our study of electrical interactions. The work done moving a charged particle in an electric field can result in the particle gaining either potential or kinetic energy, or both. In this chapter, since we are investigating charges at rest, we will discuss only changes in potential energy. Energy and the Electric Potential Recall the change in potential energy of a ball when you lift it, Figure 21-5. Both the gravitational force, F, and the gravitational field, g = Flm, point toward Earth. If you lift a ball against the force of gravity, you do work on it, and thus its potential energy is increased. a b Ball -\ • define the electric potential difference in terms of work done moving a unit test charge; distinguish potential from potential difference. • know the units of potential; solve problems involving potential in uniform electric fields. • understand how Millikan used electric fields to find the charge of the electron. · understand how minimizing energy determines sharing of charge; define grounding and relate to charge sharing. • know where charges reside on solid and hollow conductors; recognize the relationship between conductor shape and field strength. · define capacitance; describe the principle of the capacitor; solve capacitor problems. Increase in gravitational potential energy FIGURE 21-5. Work is needed to move an object against the force of gravity (a) and against the electric force (b). In both cases, the potential energy of the object is increased. 21.2 Applications of the Electric Field 431 F. Y. I. "Even if I could be Shakespeare, I think I should still choose to be Faraday." -Aldous Huxley (1925) The situation is similar with two unlike charges. These two attract each other, and so you must do work to pull one away from the other. When you do the work, you store it as potential energy. The larger the test charge, the greater the increase in its potential energy, LlPE. The electric field at the location of the test charge does not depend on the size of the test charge. The field is the force per unit charge, or Flq'. In the same way, we will define a quantity that is the change in potential energy per unit charge. This quantity is called the electric potential difference and is defined by the equation Ll V = LlPElq'. The unit of electric Electric potential difference is change in potential energy divided by the charge. The volt is the unit of electric potential. The reference point of zero level of potential is arbitrary. potential, joule per coulomb, is called potential energy; you give it a positive change in potential energy. Because the test charge is positive, the electric potential difference between A and B is also positive. The electric potential at B is larger than the electric potential at A. If you return the test charge from B back to A, the electric field does work on you. The potential energy of the charge is reduced. Therefore, the electric potential difference between B and A is negative. The work done on you returning the charge is equal and opposite to the work you do moving it away. Therefore, the electric potential difference between A and B is equal and opposite to the difference between Band A. The net change in potential going from A to B to A is zero. Thus, the potential of point A, or any other point, depends only on its position, not on the path taken to get there. potential. The reference, ences in electric potential are = V). Consider the situation shown in Figure 21-5. When you move the test charge from A to B, you do work on it. As a result, you increase its As is the case for any kind of potential electric potential energy can be measured. Only potential differences meaningful. a volt (J/C energy, only differences in The same is true of electric or zero, level is arbitrary. Thus, only differare important. We define the potential differ- ence from point A to point B to be V = VB - VA' Potential differences are measured with a voltmeter. Sometimes the potential difference is simply called the voltage. As described in Chapter 11, the potential energy of a system can be defined to be zero at any convenient reference point. In the same way, the electric potential of any point, for example point A, can be defined to be zero. If VA = 0, then VB = V. If instead, VB = 0, then VA = - V. No matter what reference point is chosen, the value of the potential difference from point A to point B will always be the same. a Low V F F High V b __--I FIGURE 21-6. Electric potential energy is smaller when two unlike charges are closer together (a) and larger when two like charges are closer together (b). 432 Electric Fields + High V Low V FIGURE 21-7. An electric field between parallel plates. We have seen that electric potential increases as a positive test charge is separated from a negative charge. What happens when a positive test charge is separated from a positive charge? There is a repulsive force between these two charges. Potential energy decreases as the two charges are moved farther apart. Therefore, the electric potential is smaller at points farther from the positive charge, Figure 21-6. The Electric Potential in a Uniform Field How does the potential difference depend on the electric field? In the case of the gravitational field, near the surface of Earth the gravitational force and field are relatively constant. A constant electric force and field can be made by placing two large flat conducting plates parallel to each other. One is charged positively and the other negatively. The electric field between the plates is constant except at the edges of the plates. Its direction is from the positive to the negative plate. Figure 21-7 shows grass seeds representing the field between parallel plates. In a constant gravitational field, the change in potential energy when a body of mass m is raised a distance h is given by LlPE = mgh. Remember that g, the gravitational field intensity near Earth, is given by g = Flm. The gravitational potential, or potential energy per unit of mass, is mghlm = gh. What is the potential difference between two points in a uniform electric field? If a positive test charge is moved a distance d, in the direction opposite the electric field direction, the change in potential energy is given by LlPE = + Fd. Thus, the potential difference, the change in potential energy per unit charge, is V = + Fdlq = + (Flq)d. Now, the electric field intensity is the force per unit charge, E = Flq. Therefore, the potential difference, V, between two points a distance d apart in a uniform field, E, is given by V = The electric field between two parallel plates is uniform. Ed. The potential increases in the direction opposite the electric field direction. That is, the potential is higher near the positively-charged plate. By dimensional analysis, the product of the units of E and d is (N/C) . m. This is equivalent to a l/C, the definition of a volt. 21 .2 Appl ications of the Electric Field 433 HELP WANTED VENDING MACHINE REPAIRERS If you have a knack for fixing things, and can learn quickly from our in-house training on specific equipment, we would like to hear from you. You must be able to read electrical diagrams and manuals and use sophisticated test equipment. Vocational school training is a plus, but experience will be considered. We need reliable, bondable, productive people willing to work when needed and interested in advancement. F information contact: ational Automatic Merchan ism Association, 20 North Wacker Drive, Chicago, IL 60606 Example Problem Potential Difference Between Two Parallel Plates Two large, charged parallel plates are 4.0 cm apart. The magnitude of the electric field between them is 625 N/C. a. What is the potential difference between the plates? b. What work is done moving a charge equal to that of one electron from one plate to another? Given: d = 0.040 m E = Solution: a. V b. W = = V, W Unknowns: 625 N/C Basic equations: Ed (625 N/C)(0.040 m) 25 N . mlC = 25 J/C 25 V qV (1.6 X 10-19 C)(25 V) 4.0 X 10-18 CV = 4.0 X V W 10-18 = = Ed qV J Example Problem Electric Field Between Two Parallel Plates A voltmeter measures the potential difference between two large parallel plates to be 60.0 V. The plates are 3.0 cm apart. What is the magnitude of the electric field between them? Unknown: E Basic equation: Given: V = 60.0 V d = 0.030 m Solution: V = Ed, so E V = Ed V = d 60.0 V 0.030 m 60.0 J/C 0.030 m 2.0 x 103 N/C Practice Problems 5. The electric field intensity between two large, charged, parallel metal plates is 8000 N/C. The plates are 0.05 m apart. What is the potential difference between them? 6. A voltmeter reads 500 V when placed across two charged, parallel plates. The plates are 0.020 m apart. What is the electric field between them? 7. What potential difference is applied to two metal plates 0.500 m apart if the electric field between them is 2.50 x 103 N/C? ~ 8. What work is done when 5.0 C is raised in potential by 1.5 V? 434 Electric Fields P hysicsand society MINIMOTORS W hat miracles could a doctor perform with a pump or motor small enough to fit into a blood vessel? Drugs, for example, could be administered precisely where needed in the exact amount required. Tiny rotating knives could remove plaque from arteries. The parts for such machines could not be made by normal methods. The new techniques are often called nanotechnology methods. For more than ten years, scientists have adapted the methods used to make computer chips in order to produce electric motors and gears the diameter of a human hair. At this time, however, the most successful products are sensors. These sensors detect gas or Iiquid pressure and produce an electrical signal. They are already used to improve the efficiency of an auto engine and to monitor blood pressure inside a human. Other sensors can detect the presence of minuscule amounts of poisonous gases. Sensors still in development measure acceleration more sensitively than any existing device. Nanotechnology can also produce actuators. An actuator converts an electrical signal into a mechanical force. One device, the size of a postage stamp, moves two million tiny mirros to project a computer display on the wall. One motor that was built uses electric fields to cause the rotor to spin. The magnetic forces that run ordinary motors have also been used to spin micromotors. The rotors of early motors were as flat as pancakes and couldn't do any real work. Using a technique called L1GA, developed in Germany, scientists at the University of Wisconsin at Madison created metal gears one hundred times thicker than the rotors of the early motors. Electric charges were used to assemble these gears into six-gear transmissions. DEVELOPING A VIEWPOINT The development of nanotechnology is expensive and may not result in products that can be sold for decades, if ever. Who should finance the development? Read further and come to class prepared to discuss the following questions. 1. What appl ications do you foresee in some of the following fields: medicine, health and nutrition, transportation, and consumer electron ics? 2. You are a scientist who wants to develop an application of nanotechnology in health, but you need money to support your project. You go to a government agency to ask for funds. What argument might you use to convince them to use tax money to support your work? SUGGESTED READINGS Heppenheimer, T. A. "Microbots." Discover, March 1989, pp. 78-84. Gannon, Robert. "Micromachine Magic./I Popular Science, March 1989, pp. 88-92. Stewart, Doug. "New Machines Are Smaller Than a Hair, and Do Real Work./I Smithsonian, October 1990, pp. 85-95. Hapgood, Fred. "No Assembly Required." Omni, May 1990, pp. 66-70. 21.2 Applications of the Electric Field 435 FIGURE 21-8. This illustration shows a cross-sectional view of Millikan's apparatus for determining the charge on an electron. Charged Charged plate plate Millikan's Oil Drop Experiment Millikan measured the charge of an electron. A drop is suspended if the electric and gravitational forces are balanced. POCKET LAB UNIT CHARGE Place a small container on a balance and determine the mass of the empty container. (On an electronic balance, push the tare bar.) Place a small handful of 3-mm diameter steel balls in the container. Don't count them! Record the mass. Dump the balls out of the container and repeat the procedure with a different number of balls. Don't count! Record the mass. Repeat this loading and massing process a third time. Look closely at your mass values. Suggest a possible unit mass of each ball. Could your unit mass be too large? Could it be too small? Explain. 436 Electric Fields One important application of the uniform electric field between two parallel plates was the measurement of the charge of an electron. This was made by American physicist Robert A. Millikan (1868-1953) in 1909. Figure 21-8 shows the method used by Millikan to measure the charge carried by a single electron. Fine oil drops were sprayed from an atomizer into the air. These drops were often charged by friction with the atomizer as they were sprayed. Gravity acting on the drops caused them to fall. A few entered the hole in the top plate of the apparatus. A potential difference was placed across the two plates. The resulting electric field between the plates exerted a force on the charged drops. When the top plate was made positive enough, the electric force caused negatively-charged drops to rise. The potential difference between the plates was adjusted to suspend a charged drop between the plates. At this point, the downward force of the weight and the upward force of the electric field were equal in magnitude. The magnitude of the electric field, E, was determined from the potential difference between the plates. A second measurement had to be made to find the weight of the drop, mg, which was too tiny to measure by ordinary methods. To make this measurement, a drop was first suspended. Then the electric field was turned off and the rate of the fall of the drop measured. Because of friction with the air molecules, the oil drop quickly reached terminal velocity. This velocity was related to the mass of the drop by a complex equation. Using the measured terminal velocity to calculate mg, and knowing E, the charge q could be calculated. Millikan found that the drops had a large variety of charges. When he used X rays to ionize the air and add or remove electrons from the drops, he noted, however, that the changes in the charge were always a multiple of -1.6 x 10-19 C. The changes were caused by one or more electrons being added to or removed from the drops. He concluded that the smallest change in charge that could occur was the amount of charge of one electron. Therefore, Millikan said that each electron always carried the same charge, -1.6 x 10-19 C. Millikan's experiment showed that charge is quantized. can have only a charge with a magnitude of the charge of the electron. The presently accepted theory This means that an object that is some integral multiple says that protons are made of matter up of fundamental particles called quarks. The charge on a quark is either + 1/3 or - 2/3 the charge on an electron. A theory of quarks that agrees with other experiments states that quarks can never be isolated. Many experimenters have used an updated Millikan apparatus to look for fractional charges on drops or tiny metal spheres. There have been no reproducible discoveries of fractional charges. Thus, no isolated quark has been discovered; the quark theory remains consistent with experiments. Example Problem Finding the Charge on an Oil Drop An oil drop weighs 1.9 x 10-14 N. It is suspended in an electric field of intensity 4.0 X 104 N/C. a. What is the charge on the oil drop? b. If the upper plate is positive, how many excess electrons does it have? Given: W £ Solution: = mg 1.9 x 10-14 N = 4.0 = x 104 a. Electronic mg us, q = The application of an electric stimulus causes a series of changes in the resting membrane potential (RMP) of nerve and muscle cells. A stimulus that is applied at one point on a cell causes a small region of the cell membrane to become depolarized. This is called a local potential. If large enough, the local potential can cause an action potential, which is a larger depolarization of the RMP. This spreads without changing its magnitude over the entire cell membrane. The drop's mass was found from its terminal velocity. q N/C and gravitational h T Unknown: excess charge, Basic equation: £q = mg . .. . . BIOLOGY CONNECTION E forces balance, so £q mg. 14 1.9 x 10N = 4.0 X 104 N/C = 4.8 x 10-19 The charge of an electron is -1.6 10 19 C. x C (total charge on drop) b. n u m be r of electro n s = -'--,----"'----,------'-'(charge per electron) 4.8 x 10-19 C 1.6 x 10 19 Cle 3 electrons There are three extra electrons because a negatively-charged is attracted toward a positively-charged plate. drop Practice Problems 9. A drop is falling field is off. in a Millikan oil drop apparatus when the electric a. What are the forces on it, regardless of its acceleration? b. If it is falling forces on it? at constant 10. An oil drop weighs 1.9 field of 6.0 x 103 N/C. a. What b. How velocity, x 10-15 what can be said about N. It is suspended the in an electric is the charge on the drop? many excess electrons does it carry? 21.2 Applications of the Electric Field 437 11. A positively-charged oil drop weighs 6.4 x 10-13 N. An electric field of 4.0 x 106 N/C suspends the drop. a. What b. How ~ 12. If three Problem is the charge on the drop? many electrons is the drop missing? more electrons were removed from the drop in Practice 11, what field would be needed to balance the drop? Sharing of Charge Charges move until all parts of a conductor are at the same potential. If a large and a small sphere have the same charge, the large sphere will have a lower potential. All systems, mechanical and electrical, come to equilibrium when the energy of the system is at a minimum. For example, if a ball is put on a hill, it will finally come to rest in a valley where its gravitational potential energy is least. This same principle explains what happens when an insulated, negatively-charged metal sphere, Figure 21-9, touches a second, uncharged sphere. The excess negative charges on sphere A repel each other. Thus the potential of sphere A is high. We can choose the potential of the neutral sphere, B to be zero. When one charge is transferred from sphere A to sphere B, the potential of sphere A is reduced because it has fewer excess charges. The potential of sphere B does not change because no work is done adding the first extra charge to this neutral sphere. As more charges are transferred, however, work must be done on the charge to overcome the growing repulsive force between it and the other charges on sphere B. Therefore, the potential of sphere B increases as the potential of sphere A decreases. Negative charges continue to flow from sphere A to sphere B until the work done adding a charge to sphere B is equal to the work gained in removing a charge from sphere A. The potential of sphere A now equals the potential of sphere B. Thus charges flow until all parts of a conducting body, the two touching spheres in this case, are at the same potential. Consider a large sphere and a small sphere that have the same charge, Figure 21-10. The larger sphere has a larger surface area, so charges can spread farther apart than they can on the smaller sphere. With the charges farther apart, the repulsive force between them is reduced. Therefore, as long as the spheres have the same charge, the Metal spheres of equal size Charged sphere FIGURE 21-9. A charged sphere shares charge equally with a neutral sphere of equal size. 438 Electric Fields -, a b Metal spheres Low V (b) High V Same q of unequal size FIGURE 21-10. A charged sphere gives much of its charge to a larger sphere. Same V different q potential on the larger sphere is lower than the potential on the smaller sphere. If the two spheresare now touched together, charges will move to the sphere with the lower potential; that is, from the smaller to the larger sphere. The result is a greater charge on the larger sphere when two different-sized spheres are at the same potential. Earth is a very large sphere. If a charged body is touched to Earth, almost any amount of charge can flow without changing Earth's potential. When all the excess charge on the body flows to Earth, the body becomes neutral. Touching an object to Earth to eliminate excess charge is called grounding. Moving gasoline trucks can become charged by friction. If that charge were to jump to Earth through gasoline vapor, it could cause an explosion. Instead, a metal wire safely conducts the charge to ground, Figure 21-11. If a computer or other sensitive instrument were not grounded, static charges could accumulate, raising the potential of the computer. A person touching the computer could suddenly lower the potential of the computer. The charges flowing through the computer to the person could damage the equipment or hurt the person. The charges are closer together at sharp points of a conductor. Therefore, the field lines are closer together; the field is stronger, Figure 21-10. If the two spheres are at the same potential, the large one will have greater charge. FIGURE 21-11. fuel truck. Ground wire on a 21.2 Applications of the Electric Field 439 Electric Fields Near Conductors All charges are on the outside of a conductor. Electric fields are largest near sharp points. F. Y. I. Even if the inner surface is pitted or bumpy, giving it a larger surface area than the outer surface, the charge will still be entirely on the outside. The charges on a conductor are spread as far apart as they can be to make the energy of the system as low as possible. The result is that all charges are on the surface of a solid conductor. If the conductor is hollow, excess charges will move to the outer surface. If a closed metal container is charged, there will be no charges on the inside surfaces of the container. In this way, a closed metal container shields the inside from electric fields. For example, people inside a car are protected from the electric fields generated by lightning. On an open coffee can there will be very few charges inside, and none near the bottom. Even though the electric potential is the same at every point of a conductor, the electric field around the outside of it depends on the shape of the body as well as its potential. The charges are closer together at sharp points of a conductor. Therefore, the field lines are closer together; the field is stronger, Figure 21-12. The field there can become so strong that nearby air molecules are separated into electrons and positive ions. As the electrons and ions recombine, energy is released and light is produced. The result is the blue glow of a corona. The electrons and ions are accelerated by the field. If the field is strong enough, when the particles hit other molecules they will produce more ions and electrons. The stream of ions and electrons that results is a plasma, a conductor. The result is a spark, or, in extreme cases, lightning. To reduce corona and sparking, conductors that are highly charged or operate at high potentials are made smooth in shape to reduce the electric fields. On the other hand, lightning rods, Figure 21-13, are pointed so that the electric field will be strong near the end of the rod. Air molecules are pulled apart near the rod, forming the start of a conducting path from the rod to the clouds. As a result of the sharply-pointed shape, charges in the clouds spark to the rod rather than to a chimney or other high point on a house. From the rod, a conductor takes the charges safely to the ground. Storing Electric Energy-The Capacitor When you lift a book, you increase its potential energy. This can be interpreted as "storing" energy in a gravitational field. In a similar way, you can store energy in an electric field. In 1746, the Dutch physician and physicist Pieter Van Musschenbroek invented a device that could a FIGURE 21-12. The electric field around a conducting body depends on the structure and shape (b is hollow) of the body. 440 Electric Fields (c) b c FIGURE 21-13. A lightning rod allows charges from the clouds to be grounded, rather than conducted through a barn. Negatively charged cloud ~ ~ @J Positive ions in atmosphere store electric charge. In honor of the city in which he worked, it was called a Leyden jar. The Leyden jar was used by Benjamin Franklin to store the charge from lightning and in many other experiments. As described previously, as charge is added to an object, the potential between that object and Earth increases. For a given shape and size of the object, the ratio of charge to potential difference, q/V, is a constant. The constant is called the capacitance, C, of the object. For a small sphere far from the ground, even a small amount of added charge will increase the potential difference. Thus, C is small. The larger the sphere, the greater the charge that can be added for the same increase in potential difference, thus the larger the capacitance. Van Musschenbroek found a way of producing a large capacitance in a small device. A device that is designed to have a specific capacitance is called a capacitor. All capacitors are made up of two conductors, separated by an insulator. The two conductors have equal and opposite charges. Capacitors are used in electrical circuits to store electric energy. Capacitors often are made of parallel conducting sheets, or plates, separated by air or another insulator. Commercial capacitors, Figure 21-14, contain strips of aluminum foil separated by thin plastic and are tightly rolled up to save room. The capacitance is independent of the charge on it. Capacitance can be measured by placing charge q on one plate and - q on the other, and measuring the potential difference, V, that results. The capacitance is then found by using the equation C Capacitance is the ratio of charge stored to potential difference. Capacitors store energy in the electric field between charged conductors. .:«. V Capacitance is measured in farads, F, named after Michael Faraday. One farad is one coulomb per volt (CN). Just as one coulomb is a large amount of charge, one farad is an enormous capacitance. Capacitors are usually between 10 picofarads (10 x 10-12 F) and 500 microfarads (500 x 10-6 F). FIGURE 21-14. capacitors. Various types of 21.2 Applications of the Electric Field 441 F. Y. I. Michael Faraday, a self-educated man, was hired by chemist Sir Humphrey Davy as a bottle washer. As time went by, Faraday became an even greater scientist than Davy. Besides his brilliant discoveries in the fields of chemistry and physics, he was also a popular lecturer. A lecture for young people, The Chemical History of a Candle, was published in book form and became a classic. Example Problem Finding the Capacitance from Charge and Potential Difference A sphere has a potential difference between it and Earth of 60.0 V when charged with 3.0 x 10-6 C. What is its capacitance? Given: potential difference, V = 60.0 VB' q = 3.0 X 10-6 C Solution: C = 9.. = 3.0 V = 5.0 X Unknown: capacitance, C . C q asic equation: = V 6 10- C = 5.0 X 10-8 C/V 60.0 V 10-8 F = 0.050 fLF X Practice Problems 13. A 27-fLF capacitor has a potential difference of 25 V across it. What is the charge on the capacitor? 14. Both a 3.3-fLF and a 6.8-fLF capacitor are connected across a 15-V potential difference. Which capacitor has a greater charge? What is it? 15. The same two capacitors are each charged to 2.5 x 10-4 C. Across which is the potential difference larger? What is it? ~ 16. A 2.2-fLF capacitor is first charged so that the potential difference is 6.0 V. How much additional charge is needed to increase the potential difference to 15.0 V? ........................ CONCEPT REVIEW 2.1 If an oil drop with too few electrons is motionless in a Millikan oil drop apparatus, a. what is the direction of the electric field? b. which plate, upper or lower, is positively charged? 2.2 If the charge on a capacitor is changed, what is the effect on a. the capacitance, C? b. the potential difference, V? 2.3 If a large, charged sphere is touched by a smaller, uncharged sphere, as in Figure 21-9, what can be said about a. the potentials of the two spheres? b. the charges on the two spheres? 2.4 Critical Thinking: Suppose we have a large, hollow sphere that has been charged. Through a small hole in the sphere, we insert a small, uncharged sphere into the hollow interior. The two touch. What is the charge on the small sphere? 442 Electric Fields CHAPTER 21 REVIEW· . ------------~~--~~~~~=-~ SUMMARY 21.1 Creating REVIEWING CONCEPTS and Measuring Electric Fields · An electric field exists around any charged object. The field produces forces on other charged bodies. · The electric field intensity is the force per unit charge. The direction of the electric field is the direction of the force on a tiny, positive test charge. · Electric field lines provide a picture of the electric field. They are directed away from positive charges and toward negative charges. 21.2 Applications of the Electric Field · Electric potential difference is the change in potential energy per unit charge in an electric field. Potential differences are measured in volts. · The electric field between two parallel plates is uniform between the plates except near the edges. · Robert Millikan's experiments showed that electric charge is quantized and that the charge carried by an electron is - 1.6 x 10-19 C. · Charges will move in conductors until the electric potential is the same everywhere on the conductor. · A charged object can have its excess charge removed by touching it to Earth or to an object touching Earth. This is called grounding. · Electric fields are strongest near sharply-pointed conductors. · Capacitance is the ratio of the charge on a body to its potential. The capacitance of a body is independent of the charge on the body and the potential difference across it. KEY TERMS electric field electric field lines electric potential difference volt potential difference grounding capacitance capacitor 1. Draw the electric field lines between a. two like charges. b. two unlike charges. 2. What are the two properties a test charge must have? 3. How is the direction of an electric field defined? 4. What are electric field lines? 5. How is the strength of an electric field indicated with electric field lines? 6. What SI unit is used to measure electrical potential energy? electric potential? 7. What will happen to the electric potential energy of a charged particle in an electric field when the particle is released and free to move? 8. Define the volt in terms of the change in potential energy of a charge moving in an electric field. 9. Draw the electric field lines between two parallel plates of opposite charge. 10. Why does a charged object lose its charge when it is touched to the ground? 11. A charged rubber rod placed on a table maintains its charge for some time. Why is the charged rod not grounded immediately? 12. A metal box is charged. Compare the concentration of charge at the corners of the box to the charge concentration on the sides. 13. In your own words, describe a capacitor. APPLYING CONCEPTS 1. What happens to the size of the electric field if the charge on the test charge is halved? 2. Does it require more energy or less energy to move a fixed positive charge through an increased electric field? 3. Figure 21-15 shows three spheres with charges of equal magnitude, but with signs as shown. Spheres y and z are held in place but sphere x is free to move. Initially sphere x is equidistant from spheres y and z. Choose the Chapter 21 Review 443 path that sphere x will follow, assuming other forces are acting. no >tc A B FIGURE 21-15. Use with Applying Concepts 3. 4. What is the unit of potential difference in terms of m, kg, s, and C? 5. What do the electric field lines look like when the electric field has the same strength at all points in a region? 6. When doing a Millikan oil drop experiment, it is best to work with drops that have small charges. Therefore, when the electric field is turned on, should you try to find drops that are moving fast or slow? Explain. 7. If two oil drops can be held motionless in a Millikan oil drop experiment, a. can yoy be sure that the charges are the same? b. the ratios of what two properties of the drops would have to be equal? 8. l)m and Sue are holding hands when they are given a charge while they are standing on an insulating platform. Tim is larger than Sue. Who has the larger amount of charge or do they both have the same amount? 9. Which has a larger capacitance, a t-ern diameter or a to-ern diameter aluminum sphere? 10. How can you store a different amount of charge in a capacitor? PROBLEMS 21.1 Creating and Measuring Electric Fields The charge on an electron is -1.60 x 10-19 C. 1. A positive charge of 1.0 x 10 - 5 C experiences a force of 0.20 N when located at a certain point. What is the electric field intensity at that point? 444 Electric Fields I 2. What charge exists on a test charge that experiences a force of 1.4 x 10-8 N at a point where the electric field intensity is 2.0 x 10-4 N/C? 3. A test charge has a force of 0.20 N on it when it is placed in an electric field intensity of 4.5 x 105 N/C. What is the magnitude of the charge? 4. The electric field in the atmosphere is about 150 N/C, downward. a. What is the direction of the force on a positively-charged particle? b. Find the electric force on a proton with charge +1.6 x 10-19 C. c. Compare the force in b with the force of gravity on the same proton (mass 1.7 x 10-27 kg). ~ 5. Electrons are accelerated by the electric field in a television picture tube, Table 21-1. a. Find the force on an electron. b. If the field is constant, find the acceleration of the electron (mass = 9.11 x 10-31 kg). ~ 6. A lead nucleus has the charge of 82 protons. a. What is the direction and magnitude of the electric field at 1.0 x 10-10 m from the nucleus? b. Use Coulomb's law to find the direction and magnitude of the force exerted on an electron located at this distance. 7. Carefully sketch a. the electric field produced by a + 1.0 I-LC charge. b. the electric field due to a + 2.0 I-LCcharge. Make the number of field lines proportional to the change in charge. 8. Charges X, Y, and Z are all equidistant from each other. X has a + 1.0 I-LC charge, Y a + 2.0 I-LC charge, and Z a small negative charge. a. Draw an arrow showing the force on charge Z. b. Charge Z now has a small positive charge on it. Draw an arrow showing the force on it. 9. A positive test charge of 8.0 x 10-5 C is placed in an electric field of 50.0-N/C intensity. What is the strength of the force exerted on the test charge? 21.2 Applications of the Electric Field 10. If 120 J of work are done to move one coulomb of charge from a positive plate to a negative plate, what voltage difference exists between the plates? 11. How much work is done to transfer 0.15 C of charge through a potential difference of 9.0 V? 12. An electron is moved through a potential difference of 500 V. How much work is done on the electron? 13. A 12-V battery does 1200 J of work transferring charge. How much charge is transferred? ~ 14. A force 0.053 N is required to move a charge of 37 f.LC a distance of 25 cm in an electric field. What is the size of the potential difference between the two points? 15. The electric field intensity between two charged plates is 1.5 x 103 N/C. The plates are 0.080 m apart. What is the potential difference, in volts, between the plates? 16. A voltmeter indicates that the difference in potential between two plates is 50.0 V. The plates are 0.020 m apart. What electric field intensity exists between them? 17. A negatively-charged oil drop weighs 8.5 x 10-15 N. The drop is suspended in an electric field intensity of 5.3 x 103 N/C. a. What is the charge on the drop? b. How many electrons does it carry? ~ 18. In an early set of experiments (1911), Millikan observed that the following measured charges, among others, appeared at different times on a single oil drop. What value of elementary charge can be deduced from these data? f. 18.08 X 10-19 C a. 6.563 x 1O-19 19 C 10g. 19.71 X 10-19 C b. 8.204 x 19 C 10h. 22.89 X 10-19 C c. 11.50 x 19 C i. 26.13 X 10.-19 C 10d. 13.13 x e. 16.48 x 1O-19C 19. A capacitor that is connected to a 45.0-V source contains 90.0 f.LC of charge. What is the capacitor's capacrtance? 20. A 5.4-f.LF capacitor is charged with 2.7 x 10-3 C. What potential difference exists across it? 21. What is the charge in a 15.0-pf capacitor when it is connected across a 75.0-V source? ~ 22. The energy stored in a capacitor with capacitance C, having a potential difference V, is given by W = V2CV2. One application is in the electronic photoflash or strobe light. In such a unit, a capacitor of 10.0 f.LFis charged to 3.00 x 102 V. Find the energy stored. c ~ 23. Suppose it took 30 s to charge the capacitor in the previous problem. a. Find the power required to charge it in this time. b. When this capacitor is discharged through the strobe lamp, it transfers all its energy in 1.0 x 10-4 s. Find the power delivered to the lamp. c. How is such a large amount of power possible? ~ 24. Lasers are used to try to produce controlled fusion reactions that might supply large amounts of electrical energy. The lasers require brief pulses of energy that are stored in large rooms filled with capacitors. One such room has a capacitance of 61 x 10-3 F charged to a potential difference of 10 kV. a. Find the energy stored in the capacitors, given W = V2CV2. b. The capacitors are discharged in 10 ns '(1.0 x 10-8 s). What power is produced? c. If the capacitors are charged by a generator with a power capacity of 1.0 kW, how many seconds will be required to charge the capacitors? USING LAB SKILLS 1. In the Pocket Lab on page 426, does the angle of the thread show the direction of the electric field? Explain. Make a sketch as part of your answer. 2. In the Pocket Lab on page 436, you measured the mass of three groups of balls. Based on these measurements, predict what other values should occur if you continue to measure a random number of balls. THINKING PHYSIC-LY Why are many parts of stereo components covered with metal boxes or containers? Chapter 21 Review 445