Chapter 17 By the end of this chapter, you will be able to: Electric Charge 1. Sketch the distribution of charges for both conducting and insulating objects in various and Electric Field arrangements. 2. Calculate the number of fundamental units of charge in a particular quantity of charge. 3. Determine both the magnitude and direction of the force one charge exerts on another using Coulomb’s law. 4. Determine the net force acting on a charge due to an array of point charges. 5. Relate both the magnitude and direction of the electric field at a point to the force felt by a charge placed at that point. 6. Determine the net electric field at a point due to both an array of point charges and a symmetric charge distribution. In a thundercloud, it is believed that 7. Determine the electric flux through a surface. collisions between ice and slush particles give the ice particles a slight positive 8. Relate the net electric flux through a closed charge. Although the details of this process surface to the amount of charge enclosed by are not understood, the resulting charge the surface. separation can produce enormous electric fields that result in a lightning bolt. W hen you scuff your shoes across a carpet, you can In this chapter, we’ll study how electric charges that are at get zapped by an annoying spark of static electricity. rest in our frame of reference influence each other; we call these That same spark could, in principle, totally destroy an electrostatic interactions. We’ll find that charge has interest- integrated circuit chip in your computer. Fortunately, most mod- ing properties. It is quantized: The total electric charge in a sys- ern electronic devices are designed to prevent such a catastrophe. tem must be an integer multiple of the charge of a single electron. Lightning, the same phenomenon on a vastly larger scale, can Electric charge also obeys a conservation law: Charge can be destroy a lot more than computer chips. All these phenomena in- neither created nor destroyed. However, most of the chapter will volve electric charges and the interactions between such charges. be devoted to the forces that charges produce on other charges. 525 M17_YOUN2788_10_SE_C17_525-561.indd 525 9/23/14 4:59 PM 526 CHAPTER 17 Electric Charge and Electric Field These electrostatic forces are governed by Coulomb’s law and are mediated by electric fields. Electrostatic forces hold atoms, molecules, and our bodies together, but they also are constantly trying to tear apart the nuclei of atoms. We’ll explore all these concepts in this chapter. 17.1 Electric Charge The ancient Greeks discovered as early as 600 B.C. that when they rubbed amber with wool, the amber could attract other objects. Today we say that the amber has acquired a net electric charge, or has become charged. The word electric is derived from the Greek word elektron, meaning “amber.” When you scuff your shoes across a nylon carpet, you become electrically charged, and you can charge a comb by passing it through dry hair. Plastic rods and fur (real or fake) are particularly good for demonstrating electric-charge interactions. In Figure 17.1a, we charge two plastic rods by rubbing them on a piece of fur. We find that the rods repel each other. When we rub glass rods with silk (Figure 17.1b), the glass rods also become charged and repel each other. But a charged plastic rod attracts a charged glass rod (Figure 17.1c, top). Furthermore, the plastic rod and the fur attract each other, and the glass rod and the silk attract each other (Figure 17.1c, bottom). These experiments and many others like them have shown that there are exactly two ▲ Application (no more) kinds of electric charge: the kind on the plastic rod rubbed with fur and the kind run! on the glass rod rubbed with silk. Benjamin Franklin (1706–1790) suggested calling these two kinds of charge negative and positive, respectively, and these names are still used. The person in this vacation snapshot, taken at a scenic overlook in Sequoia National Park, was amused to find her hair standing on Like and unlike charges end. Luckily, she and her companion left the Two positive charges or two negative charges repel each other; a positive and a overlook after taking the photo—and before it was hit by lightning. Just before lightning negative charge attract each other. strikes, strong charges build up in the ground and in the clouds overhead. If you’re standing on charged ground, the charge will spread In Figure 17.1, the plastic rod and the silk have negative charge; the glass rod and the fur onto your body. Because like charges repel, have positive charge. all your hairs tend to get as far from each When we rub a plastic rod with fur (or a glass rod with silk), both objects acquire net other as they can. But the key thing is for you charges, and the net charges of the two objects are always equal in magnitude and opposite to get as far from that spot as you can! in sign. These experiments show that in the charging process we are not creating electric Plain plastic rods neither Plain glass rods neither The fur-rubbed plastic attract nor repel each attract nor repel each rod and the silk- other... other... rubbed glass rod attract each other... – – – – – + + + + + Fur Plastic Silk Glass ... but after being ... but after being ... and the fur and silk rubbed with fur, rubbed with silk, each attracts the rod it the rods repel the rods repel rubbed. each other. each other. + + + + + – – – – – + + + + + – + – + ––– + ++ ++++++ + + (a) Interaction between plastic rods rubbed (b) Interaction between glass rods rubbed (c) Interaction between objects with opposite on fur on silk charges ▲ FigurE 17.1 Experiments illustrating the nature of electric charge. M17_YOUN2788_10_SE_C17_525-561.indd 526 9/23/14 4:59 PM 17.1 Electric Charge 527 charge, but transferring it from one object to another. We now know that the plastic rod ac- quires extra electrons, which have negative charge. These electrons are taken from the fur, which is left with a deficiency of electrons (that is, fewer electrons than positively charged protons) and thus a net positive charge. The total electric charge on both objects does not change. This is an example of conservation of charge; we’ll come back to this important principle later. ConCeptuaL anaLySiS 17.1 The sign of the charge SoLution Since balls 1 and 2 repel, they must be of the same sign, either both positive or both negative. Since balls 2 and 3 repel each Three balls made of different materials are rubbed against different other, they also must be of the same sign. This means that 1 and 3 both types of fabric—silk, polyester, and others. It is found that balls 1 and 2 have the same sign as 2, so all three balls have the same sign. The cor- repel each other and that balls 2 and 3 repel each other. From this result, rect answer is C. we can conclude that A. balls 1 and 3 carry charges of opposite sign. B. balls 1 and 3 carry charges of the same sign; ball 2 carries a charge of the opposite sign. C. all three balls carry charges of the same sign. The physical basis of electric charge Atom When all is said and done, we can’t say what electric charge is; we can only describe its properties and its behavior. However, we can say with certainty that electric charge is one Most of the of the fundamental attributes of the particles of which matter is made. The interactions atom’s volume responsible for the structure and properties of atoms and molecules—and, indeed, of all or- ~10-10 m is occupied sparsely by dinary matter—are primarily electrical interactions between electrically charged particles. electrons. The structure of ordinary matter can be described in terms of three particles: the nega- tively charged electron, the positively charged proton, and the uncharged neutron. The protons and neutrons in an atom make up a small, very dense core called the nucleus, with a diameter on the order of 10-15 m (Figure 17.2). Surrounding the nucleus are the elec- Tiny compared with the trons, which orbit the nucleus out to distances on the order of 10-10 m. If an atom were a Nucleus rneusct loeuf st hceo anttoamin,s tohveer few miles across, its nucleus would be the size of a tennis ball. 99.9% of the atom’s mass. The masses of the individual particles, to the precision that they are currently known, ~10-15 m are as follows: Proton: Positive charge Mass of electron = me = 9.10938291 40 * 10-31 kg, Mass = 1.673 * 10-27 kg Mass of proton = mp = 1.6726217771 742 * 10-27 kg, Neutron: No charge Mass of neutron = mn = 1.6749273511742 * 10-27 kg. Mass = 1.675 * 10-27 kg The numbers in parentheses are the uncertainties in the last two digits. Note that the masses Electron: Negative charge 1 2 of the proton and neutron are nearly equal (within about 0.1%) and that the mass of the Mass = 9.109 * 10-31 kg proton is roughly 2000 times that of the electron. Over 99.9% of the mass of any atom is The charges of the electron and concentrated in its nucleus. proton are equal in magnitude. The negative charge of the electron has (within experimental error) exactly the same ▲ FigurE 17.2 Schematic depiction of the magnitude as the positive charge of the proton. In a neutral atom, the number of electrons structure and components of an atom. equals the number of protons in the nucleus, and the net electric charge (the algebraic sum of all the charges) is exactly zero (Figure 17.3a). The number of protons or electrons in neutral atoms of any element is called the atomic number of the element. When the number of protons in an object equals the number of electrons in the object, the total charge is zero, and the object as a whole is electrically neutral. To give a neutral object an excess negative charge, we may either add negative charges to it or remove posi- tive charges from it. Similarly, we can give an excess positive charge to a neutral body by either adding positive charge or removing negative charge. When we speak of the charge on an object, we always mean its net charge. M17_YOUN2788_10_SE_C17_525-561.indd 527 9/23/14 4:59 PM 528 CHAPTER 17 Electric Charge and Electric Field Protons (+) Neutrons Electrons (-) (a) Neutral lithium atom (Li): (b) Positive lithium ion (Li+): (c) Negative lithium ion (Li−): 3 protons (3+) 3 protons (3+) 3 protons (3+) 4 neutrons 4 neutrons 4 neutrons 3 electrons (3-) 2 electrons (2-) 4 electrons (4-) Electrons equal protons: Fewer electrons than protons: More electrons than protons: Zero net charge Positive net charge Negative net charge ▲ FigurE 17.3 The neutral lithium (Li) atom and positive and negative lithium ions. Nonconducting An ion is an atom that has lost or gained one or more electrons. If one or more electrons are nylon threads removed, the remaining positively charged structure is called a positive ion (Figure 17.3b). A negative ion is an atom that has gained one or more electrons (Figure 17.3c). This gaining or ––Charged losing of electrons is called ionization. plastic rod – Ordinarily, when an ion is formed, the structure of the nucleus is unchanged. In a solid – object such as a carpet or a copper wire, the nuclei of the atoms are not free to move about, Metal Copper – ball wire so a net charge is due to an excess or deficit of electrons. However, in a liquid or a gas, a net electric charge may be due to movements of ions. Thus, a positively charged region in a fluid could represent an excess of positive ions, a deficit of negative ions, or both. The wire conducts charge from the negatively charged plastic rod to the metal ball. 17.2 Conductors and insulators (a) Some materials permit electric charge to move from one region of the material to another; others do not. For example, Figure 17.4 shows a copper wire supported by a nylon thread. A negatively charged plastic rod now repels Suppose you touch one end of the wire to a charged plastic rod and touch the other end to a the ball ... metal ball that is initially uncharged. When you remove the copper wire and bring another – – – charged object near the ball, the ball is attracted or repelled, showing that it has become – – Charged – – electrically charged. Electric charge has been transferred through the copper wire between plastic rod the ball and the surface of the plastic rod. The wire is called a conductor of electricity. If you repeat the experiment, but this time using a rubber band or nylon thread in place of the wire, you find that no charge is (b) transferred to the ball. These materials are called insulators. Conductors permit charge to move through them; insulators do not. Carpet fibers on a dry day are good insulators ... and a positively and allow charge to build up on us as we walk across the carpet. Coating the fibers with charged glass rod an antistatic layer that does not easily transfer electrons to or from our shoes is one attracts the ball. solution to the charge-buildup problem; another is to wind some of the fibers around –– + + conducting cores. + + Most of the materials we call metals are good conductors, and most nonmetals are + Charged insulators. Within a solid metal such as copper, one or more outer electrons in each atom glass rod become detached and can move freely throughout the material, just as the molecules of a gas can move through the spaces between the grains in a bucket of sand. The other elec- (c) trons remain bound to the positively charged nuclei, which themselves are bound in fixed ▲ FigurE 17.4 Charging by conduction. positions within the material. In an insulator, there are no, or at most very few, free elec- A copper wire is a good conductor. (a) The trons, and electric charge cannot move freely through the material. wire conducts charge between the plastic Some materials called semiconductors are intermediate in their properties between rod and the metal ball, giving the ball a good conductors and good insulators. Unlike copper, which is always a good conductor, no negative charge. The charged ball is then matter what you do to it, or rubber, which is always a bad conductor, no matter what you do (b) repelled by a like charge and (c) attracted by an unlike charge. to it, a semiconductor such as silicon can be engineered to have a controllable conductivity. M17_YOUN2788_10_SE_C17_525-561.indd 528 9/23/14 4:59 PM 17.2 Conductors and Insulators 529 This is the basis of the silicon-based transistor, which is the fundamental building block of the modern computer. Finally, we note that, in a liquid or gas, charge can move in the form of positive or negative ions. Ionic solutions are usually good conductors. For example, when ordinary table salt (NaCl) dissolves in water, each sodium (Na) atom loses an electron to become a positively charged sodium ion Na+ , and each chlorine Cl atom gains an electron to become a negatively charged chloride ion Cl- . These charged particles can move freely in the solution and thus conduct c1harg2e from one region of 1the2 fluid to another, providing a mechanism for conductivity. Ionic solutio1ns ar2e the dominant conductivity mechanism in many biological processes. induction When we charge a metal ball by touching it with an electrically charged plastic rod, some ▲ Application of the excess electrons on the rod move from it to the ball, leaving the rod with a smaller good conductor, bad conductor. negative charge. In another technique, called charging by induction, the plastic rod can give another object a charge of opposite sign without losing any of its own charge. Salt water is salty because it contains an abundance of dissolved ions. These ions are Figure 17.5 shows an example of charging by induction. A metal sphere is supported charged and can move freely, so salt water on an insulating stand (step 1). When you bring a negatively charged rod near the sphere, is an excellent conductor of electricity. without actually touching it (step 2), the free electrons on the surface of the sphere are Ordinary tap water contains enough ions to repelled by the excess electrons on the rod, and they shift toward the right, away from the conduct electricity reasonably well—which is rod. They cannot escape from the sphere because the supporting stand and the surrounding why you should never, ever, use an electri- cal device in a bathtub. However, absolutely air are insulators. As a result, negative charge accumulates on the right side of the surface pure distilled water is an insulator because it of the sphere and positive charge (due to the positive nuclei that the electrons left behind) consists of only neutral water molecules. accumulates on the left side. These excess charges are called induced charges. Not all of the free electrons move to the right side of the surface of the sphere. As soon as any induced charge develops, it exerts forces toward the left on the other free electrons. These electrons are repelled from the negative induced charge on the right and attracted toward the positive induced charge on the left. The system reaches an equilibrium state in which the force toward the right on an electron, due to the charged rod, is just balanced by the force toward the left, due to the induced charge. If we remove the charged rod, the free electrons shift back to the left, and the original neutral condition is restored. What happens if, while the plastic rod is nearby, you touch one end of a conducting wire to the right surface of the sphere and the other end to the earth (step 3 in Figure 17.5)? The earth is a conductor, and it is so large that it can act as a practically infinite source of extra electrons or sink of unwanted electrons. Some of the negative charge flows through the wire to the earth. Now suppose you disconnect the wire (step 4) and then remove the rod (step 5); a net positive charge is left on the sphere. The charge on the negatively charged rod has not changed during this process. The earth acquires a negative charge that is equal in magnitude to the induced positive charge remaining on the sphere. Charging by induction would work just as well if the mobile charges in the sphere were positive charges instead of (negatively charged) electrons or even if both positive and nega- tive mobile charges were present (as would be the case if we replaced the sphere with a flask of salt water). In this book, we’ll talk mostly about metallic conductors, in which the mobile Electron Electron Metal deficiency buildup Insulbaatlilng Ncrohedagrag–teivd–el–y– ++++–––– –––– ++++ –––– Wire –––– ++++ Ncheagragteiv ien ++++ stand ground – – – – – – – – Ground 1 Uncharged metal ball 2 Negative charge on rod 3 Wire lets electron buildup 4 Wire removed; ball now 5 Rod removed; positive repels electrons, creating (induced negative charge) has only an electron- charge spreads over zones of negative and flow into ground. deficient region of ball. positive induced charge. positive charge. ▲ FigurE 17.5 Charging a metal ball by induction. M17_YOUN2788_10_SE_C17_525-561.indd 529 9/23/14 4:59 PM 530 CHAPTER 17 Electric Charge and Electric Field Ball with positive charge Metal ball + + + with induced + + charges + + – + + A + –B+ + + Fu – + Fu pull push + + + + + Ball A’s (+) charge pulls on the (–) induced charge and pushes on the (+) induced charge. Because the (–) charge is closer to A, the pull is stronger than the push, so B is attracted to A. ▲ FigurE 17.7 The charge on ball A induces charges in ball B, resulting in a net ▲ FigurE 17.6 A charged plastic comb picks up uncharged attractive force between the balls. bits of paper. charges are negative electrons. However, even in a metal, we can describe conduction as though the moving charges were positive. In terms of transfer of charge in a conductor, a PhET: Balloons and Static Electricity movement of electrons to the left is exactly equivalent to a movement of imaginary positive particles to the right. In fact, when we study electric currents, we will find that, for historical Negatively reasons, currents in wires are described as though the moving charges were positive. charged When excess charge is placed on a solid conductor and is at rest (i.e., an electrostatic comb situation), the excess charge rests entirely on the surface of the conductor. If there were The Molecules with excess charge in the interior of the conductor, there would be electric forces among the comb’s (–) induced charges excess charges that would cause them to move, and the situation couldn’t be electrostatic. charge repels F u the electrons in –+ − F u each molecule in Polarization ––+++––+–++–––+++––+ + t h e s ic d o e im n o dbtfhu ttechh epeud as p pchaheparase,r rgca ref seaslac.i tgiTinhnhgtge Aac e biclahilnalgorg ohenad so onnbo ja en creutt gcea lanenc edtxr ietchr tec nfho ahrrcgoeelsd. Aetvhfeeten br o aynlol oouob enjlee cactgtsra itifhnyas tat atchroeem ncboe tib lciyhn ragur,g nient disn ttgihc eiktms t,hs eervolveuengs h.t h Iyfoo yuuogruh h ratuhirbe, +–+ net positive charge. you can pick up uncharged bits of paper on the comb (Figure 17.6). How is this possible? Paper scrap The interaction between the balloon and the ceiling or between the comb and the paper (insulator) is an induced-charge effect. In step 2 of Figure 17.5, the plastic rod exerts a net attrac- tive force on the sphere, even though the total charge on the sphere is zero, because the Positively + positive charges are closer to the rod than the negative charges are. Figure 17.7 shows this charged + + ++ + ++ effect more clearly. The large ball A has a positive charge; the conducting metal ball B is comb + + ++ + uncharged. When we bring B close to A, the positive charge on A pulls on the electrons in ++ + + B, setting up induced charges. Because the negative induced charge on the surface of B is Fu closer to A than the positive induced charge is, A exerts a net attraction on B. (We’ll study – + −Fu the dependence of electric forces on distance in Section 17.4.) Even in an insulator, the – +–+– electric charges can shift back and forth a little when there is charge nearby. Figure 17.8 ++––++––++––++– – i n d c uh caAerd gc eoc hmalasbrog w ecsirt ehtah atae t(s +at)tract sphaopwers. Aholtwho au gshta tthice cehleacrgtreo nesn ainb ltehse ap acphearr gaerde bpolausntidc tcoo tmhebi rt om poilcekc uuleps uanncdh acragnendo tb mitso voef –+– the paper to the comb. freely through the paper, they can still shift slightly to produce a net charge on one side and the opposite charge on the other. Thus, the comb causes each molecule in the paper to ▲ FigurE 17.8 A charged comb picks up uncharged paper by polarizing the paper’s develop induced charges (an effect called polarization). The net result is that the scrap of molecules. paper shows a slight induced charge—enough to enable the comb to pick it up. 17.3 Conservation and Quantization of Charge As we’ve discussed, an electrically neutral object is an object that has equal numbers of Video Tutor Demo electrons and protons. The object can be given a charge by adding or removing either posi- tive or negative charges. Implicit in this discussion are two very important principles. First is the principle of conservation of charge: M17_YOUN2788_10_SE_C17_525-561.indd 530 9/23/14 4:59 PM 17.4 Coulomb’s Law 531 Conservation of charge The algebraic sum of all the electric charges in any closed system is constant. Charge can be neither created nor destroyed; it can only move from one place or object to another. Conservation of charge is believed to be a universal conservation law; there has never been any experimental evidence for a violation of this principle. Even in high-energy in- teractions in which subatomic particles are created and destroyed, the net charge of all the particles is exactly constant. Second, the magnitude of the charge of the electron or proton is a natural unit of charge. Every amount of observable electric charge is always an integer multiple of this basic unit. Hence we say that charge is quantized. A more familiar example of quantization is money. When you pay cash for an item in a store, you have to do it in 1-cent increments. If grape- fruits are selling three for a dollar, you can’t buy one for 331 cents; you have to pay 34 3 cents. Cash can’t be divided into smaller amounts than 1 cent, and electric charge can’t be divided into smaller amounts than the charge of one electron or proton. Quantitative anaLySiS 17.1 Determine the charge SoLution When identical metal objects come in contact, any net charge they carry is shared equally between them. Thus, when A touch- Three identical metal balls A, B, and C are mounted on insulating rods. Ball A has a charge +q, and balls B and C are initially uncharged (q is es B, each ends up with a charge +q 2. When A then touches C, this the usual symbol for electric charge). Ball A is touched first to ball B charge is shared equally, leaving A and C each with a charge of +q 4. > The correct answer is C. and then separately to ball C. At the end of this experiment, the charge > on ball A is A. +q 2. B. +q 3. C. +q 4. > > > The forces that hold atoms and molecules together are fundamentally electrical in nature. The attraction between electrons and protons holds the electrons in atoms, holds atoms together to form polyatomic molecules, holds molecules together to form solids or liquids, and accounts for phenomena such as surface tension and the stickiness of glue. Within the atom, the electrons repel each other, but they are held in the atom by the attractive force of the protons in the nucleus. But what keeps the positively charged protons together in the tiny nucleus despite their mutual repulsion? They are held by another, even stronger interaction called the nuclear force. (We will learn about the nuclear force in Chapter 30.) 17.4 Coulomb’s Law Charles Augustin de Coulomb (1736–1806) studied the forces between charged particles in detail in 1784 using a torsion balance. The torsion balance, which is depicted in Figure 17.9a, consisted of a small rod that was suspended from its midpoint by a fine wire. On each end of the rod was a charged sphere. When Coulomb brought a third charged sphere near one of the ends of the rod, it caused the rod to rotate slightly about its center of mass. By measuring the direction and magnitude of the angular deflection, Coulomb was able to deduce some of the basic properties of the electric force between charges. This very sensitive technique would ▶ B iO Application Static cling. The genetic code is carried by the “double helix” of DNA, which consists of two DNA strands wound around each other. The two strands stick together by what is essentially static cling. Along each strand, specific molecular groups form dipoles, with a positive or negative end projecting outward. The positive charges on one strand interact precisely with the negative charges on the other, “zipping” the two strands together. Crucially, these interactions are strong enough to keep the strands from coming apart on their own, but weak enough that the cellular machinery can “unzip” the strands for copying. M17_YOUN2788_10_SE_C17_525-561.indd 531 9/23/14 4:59 PM 532 CHAPTER 17 Electric Charge and Electric Field be used 13 years later by Cavendish to study the (much weaker) gravitational force between lead spheres, as we discussed in Section 6.3. Coulomb’s experiments led to the very impor- The negatively tant discovery that the electric force between two point charges (charged bodies that are very charged ball attracts small in comparison with the distance r between them) is proportional to the inverse square the positively charged of the distance between the charges, 1 r2. one; the positive ball The force also depends on the quantity of charge on each object, which we’ll denote by Torsion fiber moves until the elastic > forces in the torsion q or Q. To explore this dependence, Coulomb divided a charge into two equal parts by plac- fiber balance the ing a small charged spherical conductor in contact with an identical but uncharged sphere; electrostatic attraction. by symmetry, the charge is shared equally between the two spheres. (Note the essential role of the principle of conservation of charge in this procedure.) Thus, Coulomb could obtain one-half, one-quarter, and so on, of any initial charge. He found that the forces that two Charged point charges q and q exert on each other are proportional to each charge and therefore pith balls 1 2 –+ + are proportional to the product q1q2 of the two charges. Scale Coulomb’s law The magnitude F of the force that each of two point charges q and q a distance r 1 2 (a) A torsion balance of the type used by apart exerts on the other (Figure 17.9b) is directly proportional to the product of the Coulomb to measure the electric force charges and inversely proportional to the square of the distance between them. The Fu relationship is expressed symbolically as 2 on 1 rLike charges repel. ∙q1 q2∙ q1 F = k r2 . (17.1) Fu 1 on 2 This relationship is called Coulomb’s law. Fu1 on 2 = -Fu2 on 1 . q2 Units: q and q are in coulombs (C); F is in newtons (N). q q 1 2 1 2 F1 on 2 = F12 on 1 =2 k r2 Notes: 0 0 • k is a fundamental constant of nature: k = 8.987551789 * 109 N # m2 C2. r Unlike charges attract. • F represents only the magnitude of the force; the direction is determined >using the fact q that like charges repel and unlike charges attract. 1Fu 2 on 1 • r is the distance between the two charges. Fu 1 on 2 q 2 (b) Interaction of like and unlike charges The SI unit of electric charge is called one coulomb (1 C). For numerical calculations in problems, we’ll often use the approximate value ▲ FigurE 17.9 Schematic depiction of the apparatus Coulomb used to determine the k = 8.99 * 109 N # m2 C2, forces between charged objects that can be treated as point charges. which is in error by about 0.03%. > The forces that two charges exert on each other always act along the line joining the charges. The two forces are always equal in magnitude and opposite in direction, even when the charges are not equal. The forces obey Newton’s third law. As we’ve seen, q and q can be either positive or negative quantities. When the charges 1 2 have the same sign (both positive or both negative), the forces are repulsive; when they are unlike, the forces are attractive. We need the absolute value bars in Equation 17.1 because F is the magnitude of a vector quantity. By definition, F is always positive, but the product q q is negative whenever the two charges have opposite signs. 1 2 The proportionality of the electric force to 1 r2 has been verified with great precision. There is no experimental evidence that the exponent is anything different from precisely 2. > The form of Equation 17.1 is the same as that of the law of gravitation, but electrical and gravitational interactions are two distinct classes of phenomena. The electrical interaction de- pends on electric charges and can be either attractive or repulsive; the gravitational interaction depends on mass and is always attractive (because there is no such thing as negative mass). Strictly speaking, Coulomb’s law, as we have stated it, should be used only for point charges in vacuum. If matter is present in the space between the charges, the net force acting on each charge is altered because charges are induced in the molecules of the inter- vening material. We’ll describe this effect later. As a practical matter, though, we can use M17_YOUN2788_10_SE_C17_525-561.indd 532 9/23/14 4:59 PM 17.4 Coulomb’s Law 533 ▶ Application great balls of fire? Before the invention of the cyclotron, which uses both electric and magnetic fields to accelerate sub- atomic particles, physicists used electric-field generators in atom-smashing experiments. These genera- tors, like the Van de Graaff generators shown here, can accumulate either positive or negative charges on the surface of a metal sphere, thus generating immense electric fields. Charged particles in such an electric field are acted upon by a large electric force, which can be used to accelerate the particles to very high velocities. Coulomb’s law unaltered for point charges in air; at normal atmospheric pressure, the pres- ence of air changes the electric force from its vacuum value by only about 1 part in 2000. In SI units, the constant k in Equation 17.1 is often written as 1 k = , 4pP0 where P0 = 8.854 * 10-12 C2 N # m2 is another constant. This alternative form may appear to complicate matters, but it actually simplifies some of the formulas that we’ll encounter later. When we study> 1electrom2agnetic radiation (in Chapter 23), we’ll show that the numerical value of P0 is closely related to the speed of light. The most fundamental unit of charge is the magnitude of the charge of an electron or a proton, denoted by e: e = 1.60217653 * 10-19 C. The electron has a charge of -e and the proton has a charge of +e. One coulomb represents the total charge carried by about 6 * 1018 protons, or the negative of the total charge of about 6 * 1018 electrons. For comparison, the population of the earth is about 7 * 109 persons, and a cube of copper 1 cm on a side contains about 2.4 * 1024 electrons. In electrostatics problems, charges as large as 1 coulomb are very unusual. Two charges with magnitude 1 C, at a distance 1 m apart, would exert forces of magnitude 9 * 109 N (about a million tons) on each other! A more typical range of magnitudes is 10-9 to 10-6 C. The microcoulomb 1 mC = 10-6 C and the nanocoulomb 1 nC = 10-9 C are often used as practical units of charge. The total charge of all the electrons in a penny is about 1.4 * 105 C. This nu1mber shows that2 we can’t disturb electrica1l neutrality very m2 uch with- out using enormous forces. ConCeptuaL anaLySiS 17.3 Charged spheres in motion SoLution Coulomb’s law states that the magnitude of the force between two charged objects that can be treated as particles is Two small identical balls A and B are held a distance r apart on a friction- less surface; r is large compared with the size of the balls. Ball A has a F = k∙q1q2∙ r2. Is this force somehow divided between the two ob- jects? Does the object with the larger charge exert a stronger force? net charge +q; ball B has a net charge +4q. When both balls are released Shoul1d the for2c>e on each object be calculated separately? No; Newton’s at the same instant, which of the statements about the acceleration of third law gives the answer. Whenever two objects interact, the forces the two balls are correct? (There may be more than one correct choice.) that the two objects exert on each other are equal in magnitude (and A. Both balls accelerate away from each other. opposite in direction). Since the balls experience the same magnitude B. The acceleration of ball B is four times larger than the acceleration of force and have the same mass, by Newton’s second law they have the of ball A. same magnitude of acceleration at any instant. In addition, the two balls C. The acceleration of both balls is constant as they move away from will repel each other because they are both positively charged. As they each other. move apart and r increases, the magnitude of acceleration decreases. D. The acceleration of both balls decreases as they move away from Therefore, both A and D are correct answers. each other. M17_YOUN2788_10_SE_C17_525-561.indd 533 9/23/14 4:59 PM 534 CHAPTER 17 Electric Charge and Electric Field Superposition When two charges exert forces simultaneously on a third charge, the total force acting on that charge is the vector sum of the forces that the two charges would exert individually. This important property, called the principle of superposition, holds for any number of PhET: Electric Field Hockey charges. Coulomb’s law, as we have stated it, describes only the interaction between two point charges, but by using the superposition principle, we can apply it to any collection of charges. Several of the examples that follow illustrate the superposition principle. probLem-SoLving Strategy 17.1 Coulomb’s law Set up 1. As always, consistent units are essential. With the value of k given earlier, distances must be in meters, charges in coulombs, and forces in newtons. If you are given dis- tances in centimeters, inches, or furlongs, don’t forget to convert! When a charge is given in microcoulombs, remember that 1 mC = 10-6 C. SoLve 2. When the forces acting on a charge are caused by two or more other charges, the total force on the charge is the vector sum of the individual forces. If you’re not sure you remember how to do vector addition, you may want to review Sections 1.7 and 1.8. It’s often useful to use components in an x-y coordinate system. As always, it’s es- sential to distinguish among vectors, their magnitudes, and their components (using correct notation!) and to treat vectors properly as such. 3. Some situations involve a continuous distribution of charge along a line or over a sur- face. In this book, we’ll consider only situations for which the vector sum described in step 2 can be evaluated by using vector addition and symmetry considerations. In other cases, methods of integral calculus would be needed. refLeCt 4. Try to think of particular cases where you can guess what the result should be, and compare your intuitive expectations with the results of your calculations. exampLe 17.1 Charge imbalance In this example we will use the quantization of charge to determine the number of excess electrons in an object. (a) A large plastic block has a net charge of -1.0 mC = -1.0 * 10-6 C. How many more electrons than protons are in the block? (b) When rubbed with a silk cloth, a glass rod acquires a net positive charge Video Tutor Solution of 1.0 nC. If the rod contains 1.0 mole of molecules, what fraction of the molecules have been stripped of an electron? Assume that at most one electron is removed from any molecule. SoLution The fraction of all the molecules that are ionized is tSreonts uNpe naenedde dS fooLr va en etP cahrat r(gae) :o fW -e 1w.0an*t t1o0 f-in6 dC t hoen nthuem obbejre cotf. Eelaecch- 6.02N*ion1023 = 66..0225**1100293 = 1.0 * 10-14. electron has charge -e. We divide the total charge by -e: Ne = --11.6.00 ** 1100--61 9C C = 6.2 * 1012 electrons. ohrbaevjfeec Lvtese.Cr yCt no emAam rclyoh naer qgoueba jlie mcatmbsa ocluaonnncttsea ionof f pa a obhsouitugivte e 1 aa0mn-do1 4un neistg aotytfi pvciech acalhr gafeorg,r ebcsuh.ta rtgheedy Part (b): First we find the number N of positive ions needed for a ion Practice Problem: A tiny object contains 5.26 * 1012 protons and tc6ou.0tlae2ls c *ihna 1ar0g m2e3 oo mlfe o1isl.e0 Ac nuvCloeg siaf. dAerasoc ’ihsn nipouanmr th b(aeasr) ,,c 6wh.a0er 2gdei*v i+d1ee0. t2Th3e,h setoo n ttauhlme c brhoeadrr g ocefo onmntao tilhnees- 47..802**1100-81 2C e.lectrons. What is the net charge on the object? Answer: rod by the charge of one ion. Remember that 1 nC = 10-9 C. Thus, Nion = 11..60**1100--199 CC = 6.25 * 109. M17_YOUN2788_10_SE_C17_525-561.indd 534 9/23/14 4:59 PM