Electic Foces: Coulomb s Law All the matte aound you contains chaged paticles, and it is the electic foces between these chaged paticles that detemine the stength of the mateials and the popeties of the substances. ut knowing whethe the chages attact o epel is not enough; we must also know the factos that detemine the magnitude of the electic foce between chages. Investigation 7..1 in the Lab Activities section at the end of this chapte exploes these factos. Often scientists use well-established theoies, pattens, and laws when investigating new phenomena. Fo example, when scientists began, in the eighteenth centuy, to study in a systematic way the electic foce between chages, they hypothesized that the foce would obey an invese squae law, dawing upon thei expeience with gavitation. In fact, in 1785, when the Fench physicist Chales Augustin de Coulomb expeimentally established the quantitative natue of the electic foce between chaged paticles, it was aleady widely expected to be an invese squae law. Coulomb devised a tosion balance simila to that used by Cavendish in his study of gavitational foces but with small chaged sphees in place of Cavendish s masses (Figue 1). Coulomb s appaatus consisted of a silve wie attached to the middle of a light hoizontal insulating od. At one end of the od was a pith ball coveed in gold foil. At the othe end, to balance the od, a pape disk was attached. Coulomb bought an identical stationay ball into contact with the suspended ball. He chaged both balls equally by touching one of them with a chaged object. The two balls then epelled each othe, twisting the wie holding the od until coming to est some distance away. Since Coulomb knew how much foce was equied to twist his wie though any angle, he was able to show that the magnitude of this electic foce,, was invesely popotional to the squae of the distance,, between the centes of his chaged sphees (Figue ): 1 Coulomb also investigated the elationship between the magnitude of the electic foce and the chage on the two sphees. y touching eithe chaged sphee with an identical neutal sphee, he was able to divide its chage in half. y epeatedly touching 7. INVESTIGATION 7..1 Factos Affecting Electic Foce between Chages (p. 37) How would you show that the foce between chaged paticles obeys an invese squae law? Can you devise two o thee diffeent expeiments? (a) (b) + + + + + + + + + + Figue 1 (a) Pat of Coulomb s device (b) The two similaly chaged sphees epel each othe, twisting the wie until the estoing foce fom the wie, which esists the twist, balances the electostatic foce. Figue The electostatic foce of epulsion between two identical sphees at diffeent distances NEL Electic Chages and Electic Fields 37
a chaged sphee with an identical neutal sphee, he was able to educe the chage to a quate, an eighth, a sixteenth, of its oiginal value. Such manipulations evealed that, fo example, halving the chage on one sphee deceased the foce of electostatic epulsion to half its oiginal value, wheeas halving both chages educed the foce to a quate of its initial value. Coulomb concluded that the magnitude of the electic foce is diectly popotional to the poduct of the magnitudes of the chages on each sphee: q 1 q whee q 1 and q ae the espective magnitudes of the chages on the two sphees. Combining these two esults, we have what has become known as Coulomb s law of electic foces: q 1 q q and kq 1 whee k is a popotionality constant, known as Coulomb s constant. Coulomb s law applies when the chages on the two sphees ae vey small, and the two sphees ae small compaed to the distance between them. In this case, the chage distibution on the suface of the sphees will be faily unifom. If the chage on a sphee is unifomly distibuted, then the foce measued between the two sphees is the same as if all the chage on each sphee is concentated at the cente. (This is why we measue fom the cente of each sphee.) It can be assumed that Coulomb s law is extemely accuate when using point chages and easonably accuate when the sphees ae small. The diffeence in accuacy is due to the pesence of the second chaged sphee that causes the chage on the suface of each sphee to edistibute so can no longe be measued fom the cente of the sphees. When the sphees ae small, the chage distibution stays nealy unifom due to the stong epulsive foces between the chages on each sphee. This is only tue if the distance between the sphees is lage compaed to the size of the sphees. Now Coulomb s law may be defined in wods: A q 1 q Figue 3 y Newton s thid law, the electic foce exeted on body A by body is equal in magnitude and opposite in diection to the foce exeted on by A. The foces act along the line connecting the two point chages. The chages will epel if the foces ae alike and attact if unlike. In all cases, Coulomb s law is consistent with Newton s thid law, so when using the equation to find the magnitude of one of the foces, the magnitude of the foce on the othe sphee is also known to be equal in magnitude but opposite in diection (Figue 3). Howeve, to calculate the magnitude of an electic foce quantitatively, in newtons, using Coulomb s law, it is necessay to measue the magnitude of each electic chage, q 1 and q, as well as establish a numeical value fo the Coulomb popotionality constant, k. Electic chage is measued in units called coulombs (SI unit, C). The exact definition of a coulomb of chage depends on the foce acting between conductos though which chaged paticles ae moving. This will be explained fully in Chapte 8, when we lean about these foces. How lage, in pactical tems, is the coulomb? A coulomb is appoxcoulomb (C) the SI unit of electic chage Coulomb s Law The foce between two point chages is invesely popotional to the squae of the distance between the chages and diectly popotional to the poduct of the chages. 38 Chapte 7 NEL
Section 7. imately the amount of electic chage that passes though a standad 60-W light bulb (if connected to diect cuent) in s. In compaison, an electostatic shock that you might eceive fom touching a metallic dooknob afte walking acoss a woollen ug, involves the tansfe of much less than a micocoulomb. Chaging by fiction typically builds up aound 10 nc (10 8 C) fo evey squae centimete of suface aea. The attempt to add moe chage typically esults in a dischage into the ai. Theefoe, stoing even 1 C of chage is difficult. ecause Eath is so lage, it actually stoes a huge chage, oughly 400 000 C, and eleases appoximately 1500 C of chage evey second in stom-fee aeas to the atmosphee. The balance of chage is maintained on Eath by othe objects dumping excess chage though gounding and when lightning stikes Eath. A bolt may tansfe up to 0 C (Figue 4). The value of the popotionality constant k may be detemined using a tosional balance simila to that used by Cavendish. y placing chages of known magnitude a given distance apat and measuing the esulting angle of twist in the suspending wie, we can find a value fo the electic foce causing the twist. Then, using Coulomb s law in the fom FE k q q 1 a ough value fo k can be detemined. Ove the yeas, a geat deal of effot has gone into the design of inticate equipment fo measuing k accuately. To two significant digits, the accepted value fo this constant is Figue 4 The chage tansfe that occus between Eath and the clouds maintains the balance of chage. k 9.0 10 9 Nm /C The magnitude of the electostatic foce between two small, essentially pointlike, chaged objects is 5.0 10 5 N. Calculate the foce fo each of the following situations: (a) The distance between the chages is doubled, while the size of the chages stays the same. (b) The chage on one object is tipled, while the chage on the othe is halved. (c) oth of the changes in (a) and (b) occu simultaneously. Solution SAMPLE poblem 1 F 1 5.0 10 5 N F? (a) Since 1, 1 F F 1 F F 1 1 (5.0 10 5 1 N) F 1. 10 5 N When the distance between the chages is doubled, the magnitude of the foce deceases to 1. 10 5 N. NEL Electic Chages and Electic Fields 39
(b) Since q A q, DID YOU KNOW? Adhesive Tape F F 1 F F 1 q A q q q A1 1 (5.0 10 5 N) F 7.5 10 5 N 1 3 1 When the chage on one object is tipled, and the chage on the othe object is halved, the magnitude of the foce inceases to 7.5 10 5 N. (c) Since q A q q Aq, F F 1 1 q A q The electostatic foce contibutes to the stickiness of adhesive tape. When adhesive tape is attached to anothe mateial, the distance between the chages in the two mateials is vey small. The electons can pass ove the small distance, causing the two objects to have opposite chages and contibuting to the adhesive bond. The small pits ae thee because, when pulling adhesive tape off a suface, pats of the adhesive stay stuck on the mateial. A A1 1 F F 1 q q q q 1 (5.0 10 5 1 N) 3 1 1 F 1.9 10 5 N When the chages and the sepaation fom (a) and (b) change simultaneously, the magnitude of the foce deceases to 1.9 10 5 N. SAMPLE poblem What is the magnitude of the foce of epulsion between two small sphees 1.0 m apat, if each has a chage of 1.0 10 1 C? Solution q 1 q 1.0 10 1 C 1.0 m? 1q = kq (9.0 10 9 Nm /C )(1.0 10 1 C) (1.0 m) 9.0 10 15 N The magnitude of the foce of epulsion is 9.0 10 15 N, a vey small foce. Answe 1. (a) 1. 10 5 N Pactice Undestanding Concepts 1. Two chaged sphees, 10.0 cm apat, attact each othe with a foce of magnitude 3.0 10 6 N. What foce esults fom each of the following changes, consideed sepaately? (a) oth chages ae doubled, while the distance emains the same. 330 Chapte 7 NEL
Section 7. (b) An unchaged, identical sphee is touched to one of the sphees and is then taken fa away. (c) The sepaation is inceased to 30.0 cm.. The magnitude of the foce of electostatic epulsion between two small positively chaged objects, A and, is 3.6 10 5 N when 0.1 m. Find the foce of epulsion if is inceased to (a) 0.4 m, (b) 0.30 m, and (c) 0.36 m. 3. Calculate the foce between chages of 5.0 10 8 C and 1.0 10 7 C if they ae 5.0 cm apat. 4. Calculate the magnitude of the foce a 1.5 10 6 C chage exets on a 3. 10 4 C chage located 1.5 m away. 5. Two oppositely chaged sphees, with a cente-to-cente sepaation of 4.0 cm, attact each othe with a foce of magnitude 1. 10 9 N. The magnitude of the chage on one sphee is twice the magnitude of the chage on the othe. Detemine the magnitude of the chage on each. 6. Two equal unifom spheical chages, each of magnitude 1.1 10 7 C, expeience an electostatic foce of magnitude 4. 10 4 N. How fa apat ae the centes of the two chages? 7. Two identical small sphees of mass.0 g ae fastened to the ends of an insulating thead of length 0.60 m. The sphees ae suspended by a hook in the ceiling fom the cente of the thead. The sphees ae given identical electic chages and hang in static equilibium, with an angle of 30.0 between the sting halves, as shown in Figue 5. Calculate the magnitude of the chage on each sphee. Answes 1. (b) 1.5 10 6 N (c) 3.3 10 7 N. (a) 9.0 10 6 N (b) 5.8 10 6 N (c) 4.0 10 6 N 3. 1.8 10 N 4. 1.9 N 5. 1.0 10 11 C;.0 10 11 C 6. 0.51 m 7. 1. 10 7 C 30.0 Coulomb s Law vesus the Law of Univesal Gavitation Thee ae many similaities between Coulomb s law kq 1 and Newton s law of univesal gavitation F g Gm 1m : q.0 g.0 g Figue 5 Fo question 7 oth ae invese squae laws that ae also popotional to the poduct of anothe quantity; fo gavity it is the poduct of two masses, and fo the electic foce it is the poduct of the two chages. The foces act along the line joining the centes of the masses o chages. The magnitude of the foce is the same as the foce that would be measued if all the mass o chage is concentated at a point at the cente of the sphee. Theefoe, distance in both cases is measued fom the centes of the sphees. In both cases we ae assuming that is longe than the adius of the object. These paallels cannot be viewed as a concidence. Thei existence implies that thee may be othe paallels between electic and gavitational foces. Howeve, the two foces also diffe in some impotant ways: The electic foce can attact o epel, depending on the chages involved, wheeas the gavitational foce can only attact. The univesal gavitational constant, G 6.67 10 11 Nm /kg, is vey small, meaning that in many cases the gavitational foce can be ignoed unless at least one of the masses is vey lage. In contast, Coulomb s constant, k 9.0 10 9 Nm /C, is a vey lage numbe (ove one hunded billion billion times bigge than G), implying that even small chages can esult in noticeable foces. NEL Electic Chages and Electic Fields 331
Just as a mass can be attacted gavitationally by moe than one body at once, so a chage can expeience electic foces fom moe than one body at once. Expeiments have shown that the foce between two chages can be detemined using Coulomb s law independently of the othe chages pesent, and that the net foce on a single chage is the vecto sum of all these independently calculated electic foces acting on it. If all the chages lie on a staight line, we can teat electic foces like scalas, using plus and minus signs to keep tack of diections. If the chages do not lie on a staight line, tigonomety and symmeties ae used. SAMPLE poblem 3 Chaged sphees A and ae fixed in position (Figue 6) and have chages 4.0 10 6 C and.5 10 7 C, espectively. Calculate the net foce on sphee C, whose chage is 6.4 10 6 C. A C + + 0.0 cm 10.0 cm +x Figue 6 Solution q A 4.0 10 6 C A 0.0 cm q.5 10 7 C C 10.0 cm q C +6.4 10 6 C F net? Since all thee chages ae in a staight line, we can take the vecto natue of foce into account by assigning foces to the ight as positive. Sphee C has foces acting on it fom sphees A and. We fist detemine the magnitude of the foce exeted on C by A: kq F CA AqC CA F CA.6 N Theefoe, FCA.6 N [ight]. Next, we detemine the magnitude of the foce exeted on C by : kq F C qc C F C 1.4 N (9.0 10 9 Nm /C )(4.0 10 6 C)(6.4 10 6 C) (0.30 m) (9.0 10 9 Nm /C )(.5 10 7 C)(6.4 10 6 C) (0.10 m) Ou fomulation of Coulomb s law gives only the magnitude of the foce. ut since and C ae dissimila chages, we know that attacts C leftwad, so the diection of the foce exeted on C by is F C 1.4 N [left] 33 Chapte 7 NEL
Section 7. The net foce acting on sphee C is the sum of F CA and F C : F F CA F C.6 N [ight] 1.4 N [left] F 1. N [ight] The net foce acting on sphee C is 1. N [ight]. A + 5 cm SAMPLE poblem 4 Identical sphees A,, C, and D, each with a chage of magnitude 5.0 10 6 C, ae situated at the cones of a squae whose sides ae 5 cm long. Two diagonally opposite chages ae positive, the othe two negative, as shown in Figue 7. Calculate the net foce acting on each of the fou sphees. + D C Solution q A q q C q D 5.0 10 6 C s 5 cm 0.5 m 35 cm 0.35 m F? Each sphee expeiences thee electic foces, one fom each of the two adjacent chages (acting along the sides of the squae) and one fom the moe distant chage (acting along the diagonal). While some of the 1 foces acting will be attactions and some will be epulsions, each of the 1 foces has one of just two possible magnitudes: one magnitude in the case of equal chages 5 cm apat, the othe in the case of equal chages sepaated by the length of the diagonal, 35.4 cm. We begin by detemining these two magnitudes: Figue 7 Fo Sample Poblem 4 1.8 N + A 1.8 N NEL F side (9.0 10 9 Nm /C )(5.0 10 6 C) (0.5 m) F side F diag kq 1 q s kq 1 q (9.0 10 9 Nm /C )(5.0 10 6 C) (0.35 m) F diag 1.8 N Then daw a vecto diagam showing each of these foces with its vecto in the appopiate diection, whethe an attaction o epulsion. The equied diagam is shown in Figue 8. We find that the foces acting on each sphee ae simila, compising in each case an attaction of along two sides of the squae and a epulsion of 1.8 N along the diagonal. The net foce on each sphee is the vecto sum of these thee foces. As an example, we find the thee-vecto sum fo sphee A. Using the ules fo vecto addition, and dawing the vectos tip-to-tail, we obtain the diagam in Figue 9. The desied sum (the dashed vecto) has a magnitude equal to the length of the hypotenuse in the vecto tiangle minus 1.8 N. The magnitude of the hypotenuse is (3.6 ) N.6 (3N) 5.1 N + D C 1.8 N 1.8 N +y +x Figue 8 Fo the solution to Sample Poblem 4 chage A F F = v = 45 F D = F C = 1.8 N Figue 9 The net foce on chage A Electic Chages and Electic Fields 333
Theefoe, F 5.1 N 1.8 N F 3.3 N We can now find the diection of the vecto sum fom the diagam: F 3.3 N [45 down fom ight] The same calculation at each of the othe thee cones poduces the same esult: a net foce of 3.3 N diected inwad along the coesponding diagonal. The same poblem may be eadily solved using components of the foces, in the x and y diections, acting on each sphee. Fo the foce on sphee A: F F F C F D Components in the x diection: F Ax F x F Cx F Dx (1.8 N cos 45 ) 0.3 N Components in the y diection: Theefoe, F Ay F y F Cy F Dy 0 (1.8 N cos 45 ) ().3 N F (F Ax ) ) (F Ay (.3 ) N.3 (N) F 3.3 N v tan 1 F Ay FAx tan 1 1 v 45 F 3.3 N [45 down fom ight] The net foce acting on each chage is 3.3 N towad the cente of the squae. Answes 8. 0.54 N [left];.8 N [ight];.3 N [left] 9. 6. N [outwad, 150 away fom each side] Pactice Undestanding Concepts 8. Thee objects, caying chages of 4.0 10 6 C, 6.0 10 6 C, and 9.0 10 6 C, ae placed in a line, equally spaced fom left to ight by a distance of 0.50 m. Calculate the magnitude and diection of the net foce acting on each. 9. Thee sphees, each with a negative chage of 4.0 10 6 C, ae fixed at the vetices of an equilateal tiangle whose sides ae 0.0 m long. Calculate the magnitude and diection of the net electic foce on each sphee. 334 Chapte 7 NEL
Section 7. SUMMARY Electic Foces: Coulomb s Law Coulomb s law states that the foce between two point chages is invesely popotional to the squae of the distance between the chages and diectly 1q popotional to the poduct of the chages: kq,whee k 9.0 10 9 Nm /C. Coulomb s law applies when the chages on the two sphees ae vey small, and the two sphees ae small compaed to the distance between them. Thee ae similaities and diffeences between Coulomb s law and Newton s law of univesal gavitation: oth ae invese squae laws that ae also popotional to the poduct of quantities that chaacteize the bodies involved; the foces act along the line joining the two centes of the masses o chages; and the magnitude of the foce is accuately given by the foce that would be measued if all the mass o chage is concentated at a point at the cente of the sphee. Howeve, the gavitational foce can only attact while the electic foce can attact o epel. The univesal gavitational constant is vey small, while Coulomb s constant is vey lage. Section 7. Questions Undestanding Concepts 1. (a) Descibe the electic foce between two small chages and compae this foce to the gavitational foce between two small masses. How ae the two foces diffeent? (b) State Coulomb s law and Newton s law of univesal gavitation. In which espect ae these two laws simila? How ae they diffeent? Oganize you answe by copying and completing Figue 10. Newton s law diffeent Figue 10 Venn Diagam common Coulomb s law diffeent. Two identical metal sphees, each with positive chage q, ae sepaated by a cente-to-cente distance. What effect will each of the following changes have on the magnitude of the electic foce exeted on each sphee by the othe? (a) The distance between the two sphees is tipled. (b) The distance between the two sphees is halved. (c) oth chages ae doubled. (d) One of the chages becomes negative. (e) One sphee is touched by an identical neutal sphee, which is then taken fa away and the distance is deceased to. 3 3. Two small sphees of chage 5.0 mc and 4.0 mc ae sepaated by a distance of.0 m. Detemine the magnitude of the foce that each sphee exets on the othe. 4. Two 10.0-kg masses, each with a chage of 1.0 C, ae sepaated by a distance of 0.500 km in intestella space, fa fom othe masses and chages. (a) Calculate the foce of gavity between the two objects. (b) Calculate the electic foce between the two objects. (c) Daw FDs showing all the foces acting on the objects. (d) Calculate the net foce on each object, and use it to find the initial acceleation of each object. (e) Repeat (c) ignoing the gavitational foce. What have you found? NEL Electic Chages and Electic Fields 335
5. Two identically chaged small sphees, of negligible mass, ae sepaated by a cente-to-cente distance of.0 m. The foce between them is 36 N. Calculate the chage on each sphee. 6. Neutal metal sphee A, of mass 0.10 kg, hangs fom an insulating wie.0 m long. An identical metal sphee, with chage q, is bought into contact with sphee A. The sphees epel and settle as shown in Figue 11. Calculate the initial chage on. 1 9. Two small sphees, with chages 1.6 10 5 C and 6.4 10 5 C, ae.0 m apat. The chages have the same sign. Whee, elative to these two sphees, should a thid sphee, of opposite chage 3.0 10 6 C, be placed if the thid sphee is to expeience no net electical foce? Do we eally need to know the chage o sign of the thid object? 10. Two sphees ae attached to two identical spings and sepaated by 8.0 cm, as in Figue 14. When a chage of.5 10 6 C is placed on each sphee, the distance between the sphees doubles. Calculate the foce constant k of the spings. 8.0 cm Figue 14 Figue 11 7. Thee objects with chages 5.0 mc, 6.0 mc, and 7.0 mc ae placed in a line, as in Figue 1. Detemine the magnitude and diection of the net electic foce on each chage. 8. Fou objects, each with a positive chage of 1.0 10 5 C, ae placed at the cones of a 45 hombus with sides of length 1.0 m, as in Figue 13. Calculate the magnitude of the net foce on each chage. 45 A +5.0 mc 6.0 mc +7.0 mc Figue 1 0.80 m 0.40 m Applying Inquiy Skills 11. A chaged sphee is attached to an insulating sping on a hoizontal suface. Assume no chage is lost to the suoundings. An identical sphee is attached to an insulating od. Using only this equipment, design an expeiment to veify Coulomb s law. How will the compession of the sping be elated to the poduct of the chages and the distance between the chages? Making Connections 1. Unde nomal cicumstances, we ae not awae of the electic o gavitational foces between two objects. (a) Explain why this is so fo each foce. (b) Descibe an example fo each in which we ae awae of the foce. Explain why. 13. Assume the electic foce, instead of gavity, holds the Moon in its obit aound Eath. Assume the chage on Eath is q and the chage on the Moon is q. (a) Find q, the magnitude of the chage equied on each to hold the Moon in obit. (See Appendix C fo data.) (b) How stable do you think this obit would be ove long peiods of time? (Will the chages stay constant?) Explain what might happen to the Moon. Figue 13 336 Chapte 7 NEL