The Electric Field. Electric Charge HAPTER Historical Origins

Transcription

1 C HAPTER17 The Electic Field Y This electical dischage was poduced at the tun of the 0th centuy by invento Nikola Tesla, shown in the photo. ou can obseve static electicity in vaious ways. Walk acoss a wool ug with ubbe-soled shoes on a dy day and you body becomes chaged, and then, when you touch a metal dooknob, thee is a sudden, painful spak. Remove clothes fom a dye, and they cling togethe because electic chages have been tansfeed between the clothes. Bush you hai on a vey dy day, and both hai and bush become chaged. The bush then attacts dust o small bits of pape and you hai stands on end. All these phenomena esult fom foces acting between electic chages at est. This chapte and the next deal with electostatics, the study of electic chages at est. This will seve as a foundation fo undestanding electic cuent (chages in motion), with its many applications to moden technology. Afte studying cuent, we shall study magnetism and show its connection with electicity. The culmination of all this will be ou study of the laws of electomagnetism, in which electic and magnetic phenomena ae shown to be intimately connected Electic Chage Histoical Oigins One of the ealiest obsevations of electic attaction was that ambe that has been ubbed with a piece of cloth will attact light objects, such as feathes o staw. This 47

2 48 CHAPTER 17 The Electic Field was known to the Geeks as ealy as 600 B.C. In 1600 A.D. William Gilbet, pesonal physician to Queen Elizabeth I, published the fist systematic study of electicity and magnetism. Gilbet showed that, contay to popula opinion, ambe was not unique in its attactive popety; many othe substances could poduce simila attactive effects afte fictional contact. This attactive foce was obviously a vey geneal popety of natue, and so he gave it a special name electic foce. (Elekton is the Geek wod fo ambe. ) No futhe significant pogess was made in the science of electicity until 1734, when Chales du Fay attempted to account fo his obsevation that the electic foce could be epulsive as well as attactive. Du Fay believed that thee must be two kinds of electical fluid that could flow into a body and electify it. Two bodies chaged with the same kind of fluid would epel each othe, wheeas two bodies chaged with diffeent kinds of fluid would attact each othe. In 1750 Benjamin Fanklin, on the basis of extensive expeiments, sought to eplace the two-fluid theoy by a one-fluid theoy. He believed that an excess of this single fluid was esponsible fo one type of electical chaging and a deficiency of it was esponsible fo the othe type of chaging. The body containing the excess fluid he descibed as positively chaged, and the body with the deficiency he called negatively chaged. Fo example, Fanklin believed that when someone ubs a piece of glass with the bae hand both glass and skin become chaged because some of the electic fluid in the skin is tansfeed to the glass. This movement of the fluid gives the skin a deficiency of the fluid and the glass an excess. Thus he assigned a negative chage to the skin and a positive chage to the glass. Fanklin poposed that when two oppositely chaged bodies ae bought togethe the body with the excess fluid gives it up to the body with the deficiency; both bodies become unchaged and no longe poduce electical foce. Today we know that matte nomally consists of equal quantities of positive and negative electic chages potons and electons and theefoe has no net chage. The potons ae tightly bound in the nuclei of atoms, wheeas the electons, which ae fa less massive than the potons, ae moe weakly bound to atoms and theefoe ae moe easily emoved. The masses of the poton and electon ae m p ]7 kg m e ]31 kg When a body does not contain an equal numbe of potons and electons, it has a net positive o negative chage. Tansfe of electons to o fom a body gives that body a net chage.* If we think of electons as Fanklin s electical fluid, we see that his ideas concening electical chaging wee essentially coect. Fanklin s analysis of the chaging of skin and glass was incoect in one aspect, howeve. Fictional contact between skin and glass esults in the flow of the fluid (electons) fom glass to skin, athe than fom skin to glass as supposed by Fanklin; hence the skin, to which Fanklin assigned a negative chage, has an excess of electons. It follows that the electons must have negative chage. This is just a convention, aising fom Fanklin s abitay assignment of negative chage to the skin. Fig Many of the ealy expeiments with static electicity involved human subjects who wee used to detect the pesence of electic chage by the electic shock it poduced. *Sometimes a net chage is the esult not of electon tansfe but of the tansfe of eithe positively chaged ions (atoms missing one o moe electons) o negatively chaged ions (atoms with one o moe exta electons).

3 Demonstation of Electostatic Foces The following simple expeiment demonstates seveal impotant popeties of electic chage. Fist chage a piece of aluminum foil by ubbing it with plastic wap. Then let the aluminum touch two Ping-Pong balls, each suspended fom a sting. When the balls ae bought close togethe, they expeience a mutual epulsive foce (Fig. 17-a). Next chage a piece of ubbe by ubbing it with wool, and then let the ubbe touch two othe Ping-Pong balls. Like the fist two, these balls expeience a mutual epulsive foce (Fig. 17-b). If one of the balls chaged by ubbe is now bought close to one of the balls chaged by aluminum, a mutual attactive foce is obseved (Fig. 17-c), but if these two balls ae allowed to touch each othe, the electical attaction quickly disappeas (Fig. 17-d). (We assume hee that the balls ae equally chaged.) This behavio can be descibed in tems of excess and deficiency of electons. Electons ae taken away fom the aluminum when it is ubbed, leaving it with a deficiency of electons. Thee ae then moe potons than electons and theefoe a net positive chage. When the aluminum touches two balls, they become positively chaged as some of thei electons flow to the electon-deficient foil. The two positively chaged balls (A) epel each othe. Electons fom the wool ae added to the ubbe when it is ubbed, leaving the ubbe and the balls it touches negatively chaged. The negatively chaged balls (R) epel each othe. When a positively chaged ball (A) is close to a negatively chaged ball (R), they attact. When the two ae bought into contact, electons flow fom R to A, leaving each electically neutal, and they no longe expeience any foce. This expeiment demonstates an impotant geneal popety of electic chages: like chages epel, and unlike chages attact. An electoscope is a device that uses the epulsion of like chages to show the pesence of chage. It consists of a metallic od with two small pieces of gold leaf at the bottom. When chaged, the leaves epel each othe (Fig. 17-3) Electic Chage 49 Fig. 17- Foces between chaged Ping-Pong balls. Fig When chaged, the leaves of an electoscope stand out. Insulatos and Conductos When some mateials ae chaged by contact with a chaged object, the excess o deficiency of electons vey quickly leaves the point of contact and distibutes itself ove the entie suface of the newly chaged mateial. Such mateials ae called conductos. They eadily conduct electic chage fom one point to anothe. Metals ae good conductos, as ae the human body and the eath. Fo othe mateials, called insulatos o dielectics, chage placed on one pat of the suface emains localized. Most nonmetals ae insulatos. Fo example, wood, ubbe, glass, and plastic ae all insulatos.

4 430 CHAPTER 17 The Electic Field (a) (b) (c) Fig A coppe od tansfes chage fom one body to anothe (b), but a plastic od does not (c). Fig The chage distibution in an unchaged conducto. The positively chaged ions ae bound to fixed positions, but some electons ae fee to move thoughout the conducto. Metals ae effective in tanspoting electons fom one body to anothe. Suppose you have a negatively chaged body and want to tansfe some of its chage to a second, initially unchaged body (Fig. 17-4a). If you connect a coppe od between the two, some of the excess electons in the fist body will pass though the od and onto the second body (Fig. 17-4b). If, on the othe hand, you connect the two bodies by an insulato a plastic od, fo example thee will be no tansfe of chage, and the second body will emain unchaged (Fig. 17-4c). It is the micoscopic stuctue of insulatos and conductos that is esponsible fo thei vey diffeent conducting popeties. All the electons in an insulato ae bound to individual atoms, but in a conducto each atom gives up some small numbe of electons so that these electons ae fee to move though the conducto. The emaining electons and the positively chaged nucleus of each atom ae bound togethe to fom a positively chaged ion. The positive ions ae bound togethe in the conducto in a fixed stuctue (Fig. 17-5). Although a conducto may have no net chage, it still contains an enomous esevoi of fee electons that can quickly espond to an electic foce imposed by a neaby chaged body. These fee electons then move though the conducto. Consevation of Chage Although chage moves fom place to place, net chage is neve ceated o destoyed at any point in space. This pinciple is known as consevation of chage and is a geneal law of natue. If electons and potons wee indestuctible paticles, chage consevation would be obvious. Expeiments indicate, howeve, that these paticles ae not indestuctible. They, along with many othe paticles, may be ceated and destoyed. Howeve, in all such expeiments thee is neve ceation o destuction of net chage. Fo example, an electon may inteact with a positon with the esult that the two annihilate each othe, leaving no chage behind. (The positon is a paticle that has exactly the same mass as the electon but caies a positive chage.) Since the electon and positon have opposite chages of equal magnitude, the net chage of the system is zeo both befoe and afte the annihilation, and so the pinciple of chage consevation is satisfied. Units of Chage An obvious and natual choice fo the basic unit of chage is the magnitude of the electon s chage, which we denote by e. Thus we expess the electon s chage as ]e and the poton s chage as 1e: electon chage 5]e poton chage 51e A chaged macoscopic body usually has an excess o deficiency of an enomously lage numbe of electons, and so it is convenient to define a lage unit of chage. The coulomb (abbeviation C) is such a unit. It is an expeimentally detemined unit* that may be expessed as Altenatively, e may be expessed as 1 C/ , o 1 C 5 ( )e (17-1) e ]19 C (17-) *Eq is not the definition of the coulomb. Instead, the coulomb is defined in tems of the ampee, the SI unit of electic cuent, which we shall define in Chapte 0.

5 17- Coulomb s Law 431 The symbol q is used to denote chage. The chage on a body can be expessed as an integal multiple of e: q 56ne (17-3) whee n is the numbe of electons that have been eithe taken fom o added to the body. EXAMPLE 1 Finding the Numbe of Excess Electons on a Chaged Body A body has a net chage of ] ]9 C. How many excess SOLUTION Solving Eq fo n, we have electons ae contained in the body? n 5 } q } 5} ]( ]9 ] C) }] 19 e C Coulomb s Law Two Point Chages In 1785 Chales Coulomb established the fundamental foce law fo two static point chages*: 1. Each of the two chages expeiences a foce that is diected along the line between the two chages; the foce is epulsive fo chages of like sign and attactive fo chages of opposite sign (Fig. 17-6).. The magnitude of the foce is popotional to the poduct of the magnitudes of the two chages and invesely popotional to the squae of the distance between them: uquuq9u F 5 k } (17-4) (a) (b) whee k is a foce constant that expeiment shows to have the value k N-m /C In poblem solving, we shall sometimes ound this off to k N-m /C. Fig (c) Foces between chages. *Coulomb used a tosion balance to pefom his expeiments. This device was imitated by Cavendish yeas late to study gavitational foce. The Cavendish tosion balance was descibed in Chapte 6.

6 43 CHAPTER 17 The Electic Field EXAMPLE Foce Exeted by One Chage on Anothe Two point chages, q C and q9 5]1.00 C, ae located.00 m apat (Fig. 17-7a). What ae the magnitude and diection of the foce that q exets on q9? SOLUTION Since the chages ae of opposite sign, the foce between them is attactive, and so the foce F that q exets on q9 is diected towad q, as shown in Fig. 17-7b. (Of couse thee is also an oppositely diected eaction foce acting on q, but we ae not asked to compute it, and so we have not shown it in the figue.) The magnitude of F is found fom Eq. 17-4: F 5 k} Uq UUq9U } 5 ( N-m /C)} (1.0 0 C) (1.00 C) ( }. 00 m) N (a) (b) Fig This is an enomously lage foce, and thus we would nomally neve obseve such lage concentations of chage. In odinay static chaging of bodies though fiction, the chage might typically be something like 10 ]10 C. Coulomb s Law and Univesal Gavitation Coulomb s law is the second fundamental foce law we have encounteed in ou study of physics. It is simila in fom to the fundamental foce law intoduced in Chapte 6, Newton s law of univesal gavitation. Recall that the gavitational foce between two paticles is popotional to the poduct of thei masses and invesely popotional to the squae of the distance between them (Eq. 6-1): mm9 F 5 G } (17-5) whee G ]11 N-m /kg Thus the Coulomb foce depends on chage in the same way that the gavitational foce depends on mass, and both foces have the same 1/ dependence. Thee ae, of couse, impotant diffeences between Coulomb s law and the gavitational foce law. Although thee is only one kind of mass, thee ae two kinds of chage, positive and negative, and although the gavitational foce is always attactive, the electostatic foce may be attactive o epulsive, depending on the signs of the chages. Anothe obvious diffeence between the two foce laws is that the electical foce constant k is much lage than the gavitational constant G. Thus, as we saw in Example, two bodies.00 m apat, each caying a chage of magnitude 1.00 C, expeience a foce of N, o about half a billion pounds. On the othe hand, two 1.00 kg bodies.00 m apat expeience a mutual gavitational foce of only ]11 N. The lage value of the electic foce constant means that the inteaction of even elatively small chages can poduce significant foces.

7 Supeposition Pinciple When seveal point chages ae pesent in a egion of space, each chage expeiences foces exeted by the othe chages. The esultant foce acting on each chage is the vecto sum of all the Coulomb foces acting on that chage. This pinciple is called the supeposition pinciple and is illustated fo thee positive chages in Fig Coulomb s Law 433 Fig Foce F 1 is exeted by q on q 1, while foce F 31 is exeted by q 3 on q 1. The esultant foce on q 1 is the vecto sum S F 5 F 1 1 F 31. EXAMPLE 3 The Resultant of Two Electostatic Foces on a Chage Find the esultant foce on q 3 in Fig. 17-9a. SOLUTION The fist step is to indicate in a figue the foces F 13 and F 3 exeted on q 3 by q 1 and q espectively (Fig. 17-9b). Notice that F 13 is dawn with the tail of the aow on q 3, since that is the chage this foce acts on, and is diected away fom q 1, since both q 1 and q 3 ae positive and chages of like sign epel. Foce F 3 is diected towad q, since q and q 3 have opposite signs and q theefoe exets an attactive foce on q 3. We have also indicated in the figue the distances fom q 3 to the othe two chages. The next step is to calculate the magnitudes of the foces. Applying Coulomb s law (Eq. 17-4), we find F 13 5 k} Uq 1UUq 3 U } ( ( N-m /C ) ]6 C)( ]6 C) }}} ( ] m) 5 50 N F 3 5 k} Uq UUq 3 U } ( ( N-m /C ) ]6 C)( ]6 C) }}} (3.0 Ïw 3 10 m) N The final step is to find the esultant foce on q 3 by taking the sum of vectos F 13 and F 3, using the usual method of vecto addition by components. (It may be helpful to eview Section 1-5.) S F x 51F 13 F 3 cos N (100 N)(cos 45 ) 5]1 N S F y 51F 3 sin 45 5 (100 N)(sin 45 ) 5 71 N (a) (b) Fig The esultant foce has a magnitude given by )SF) 5 Ï(SwFw x )w w1w (wswfw y )w w 5 Ï(ww1w Nw) w 1w (w7w1w Nw) w 5 74 N This foce is diected at an angle u above the negative x-axis, whee u 5 actan } 7 1 N } N

8 434 Fig One can show the electic field by suspending bits of thead in oil aound electic chages. The theads align with the electic field. CHAPTER The Electic Field The Electic Field The Concept of a Field Thee is anothe way to view the inteaction between chages. Rathe than consideing diectly the foces acting between chages, we can instead intoduce the concept of an electic field. A chage ceates an electic field in the egion of space aound the chage (Fig ). The electic field may then act on othe chages in that egion. Gavitational foce may be egaded in a simila way. The eath sets up a gavitational field in space. When a mass is placed nea the eath, the eath s gavitational field acts on the mass, and the mass expeiences a foce. If you have seen the Sta Tek television seies, you may ecall an often epeated scene that can help you undestand the concept of an electic field. When the staship Entepise is in dange of attack, Captain Kik activates an invisible foce field aound the ship. If any incoming missiles ente that foce field, they expeience a foce, causing them to explode befoe they each the ship. Like the Entepise s foce field, an electic field is an invisible foce field one that exets foce on electic chages enteing the field. The electic field concept is an altenative to Coulomb s law fo viewing the inteaction between chages. Fo example, in the case of two inteacting chages, you can eithe think of the Coulomb foce between the two chages, o think of one chage as the souce of an electic field that exets a foce on the second chage. As long as all chages ae at est, the two appoaches ae equivalent. Intoducing the electic field may seem like an unnecessay complication, since we eplace something faily simple and diect, the Coulomb foce between chages, by a less diect appoach. Howeve, the field appoach tuns out to be absolutely essential in late chaptes, when we need to descibe moving chages. When chages move, thei fields change, but the effect is not immediately communicated to othe chages. Instead, changes in the field popagate though space at the speed of light. The field affecting a given chage at a cetain instant is ceated by anothe chage that poduced the field at an ealie time. Fo example, the motion of electons in a television o adio tansmitting antenna ceates an electic field. The effect of this electic field can be expeienced fo many miles aound the tansmitting antenna. When a television o adio in the vicinity is tuned to that paticula station, the electons in the eceiving antenna expeience a foce that, with the help of the eceive s cicuits, esults in the images and/o sounds poduced (Fig ). The electic field to which the eceiving antenna s electons ae esponding at any instant was poduced ealie by the motion of the tansmitting antenna s electons. Definition of the Electic Field We define the electic field E at a point in space to be the foce pe unit chage that a test chage would expeience if placed at that point. Denoting the test chage by q9, we have F E 5 } (17-6) q9 Fig Electons in a adio eceiving antenna espond to the electic field poduced by electons in a distant tansmitting antenna. Fom this definition it follows that the electic field is a vecto quantity and that the SI unit fo the electic field is the foce unit divided by the chage unit: N/C. It is impotant to undestand that although the test chage q9 entes into the fomal definition of the electic field the pesence of an electic field at a given point in space does not depend on the pesence of a test chage at that point. The electic field is the

9 17-3 The Electic Field 435 foce pe unit chage a test chage expeiences if placed in the field. The electic field is pesent whethe o not thee is a test chage to expeience its effect. The souce of the electic field may be a single chage o any numbe of chages (Fig ). In this section we develop methods fo calculating electic fields. If the electic field at a given point in space is known, we can use the equation defining E to find the foce F on any chage q9 placed at that point.* Solving Eq fo F, we obtain E F 5 q9e (17-7) Fig illustates how an electic field is poduced by souce chages and how this field poduces a foce on othe chages placed in it. Single Souce Chage We shall begin by showing how to detemine the stength and diection of the electic field poduced by a single point chage q. We imagine placing a positive test chage q9 at any point P nea the souce chage q (Fig a). The point P whee we place the test chage is called the field point, defined as that point at which we wish to detemine the electic field. If the souce chage q is positive, q9 is epelled by it. The electic field is defined as the foce pe unit chage on q9 (E 5 F/q9), and so the electic field is in the same diection as the foce on q9; that is, the electic field poduced by a positive souce chage is diected away fom the souce chage (Fig b). If the souce chage is negative, q9 is attacted to it. It follows that the electic field poduced by a negative souce chage is diected towad the souce chage (Fig c). (a) Place chage q9 at point P and it expeiences a foce F = q9e (b) Foce on q9 when q9 is placed at P, assuming q9 is postive Fig Souce chages poduce an electic field E that exets a foce F on a chage q9 placed in the field. E E (a) (b) (c) Fig The electic field poduced by a single point chage at vaious field points. (a) A test chage at P esponds to the field at that point. (b) The electic field of a positive chage is diected away fom the chage. (c) The electic field of a negative chage is towad the chage. *We assume that the pesence of q9 does not affect the distibution of chage that is the souce of the electic field; eithe q9 is so small that it exets a negligibly small foce on the souce chages, o the souce chages ae somehow held in place.

10 436 CHAPTER 17 The Electic Field The magnitude of the foce q exets on q9 is given by Coulomb s law (Eq. 17-4): F 5 k } u quuq9u } The electic field is defined as the foce pe unit chage exeted on a test chage q9, and so we find the magnitude of the electic field poduced by q by dividing both sides of the peceding equation by uq9u: uqu E 5 k } (17-8) This equation gives the magnitude of the electic field poduced by a point chage q at any distance fom the chage. EXAMPLE 4 The Electic Field of a Negative Chage at Vaious Field Points Find the magnitude and diection of the electic field poduced by a souce chage q 5] ]9 C (a) at field point a, located 1.00 m to the ight of q, and (b) at field point b, located.00 m to the left of q. (c) Find the foce on eithe a ]1 C chage o a ] ]11 C chage placed at field point a. SOLUTION (a) Applying Eq we find E 5 } k UqU } 5 ( N-m /C)} 1.0 ] C (1 }.00 m) N/C Since the souce chage is negative, the field is diected towad it, that is, to the left (Fig ). (b) Again applying Eq. 17-8, we find E 5 } k UqU } 5 ( N-m /C)} 1.0 ] C ( }. 00 m) 5.5 N/C At this field point, the field is diected to the ight, which is again towad the negative souce chage. The electic field has the same magnitude at all points that ae the same distance fom the souce chage. The field is thus spheically symmetic. Fig shows the electic field at a and b, as well as at othe selected field points at distances of 1 m and m fom q. A positive souce chage would poduce a simila electic field patten, except that the diection of E would be evesed. Fig (c) If we place a chage q9 in the electic field, accoding to Eq. 17-7, the chage expeiences a foce F 5 q9e. Thus a chage of ]1 C placed at a, whee E N/C to the left, would expeience a foce to the left of magnitude ]1 N. A chage of ] ]11 C placed at a would expeience a foce to the ight (in the ]E diection) of magnitude ]11 N.

11 17-3 The Electic Field 437 EXAMPLE 5 The Gavitational Field of the Eath The gavitational field is defined as the gavitational foce pe unit mass that would be expeienced by a test mass placed at the point whee the field is to be evaluated. Find the magnitude and diection of the gavitational field of the eath at field points that ae at distances R and R fom the cente of the eath, whee R is the eath s adius. Use kg as the mass of the eath and m as its adius. SOLUTION The eath is a spheically symmetic distibution of mass M that exets an attactive foce F on any othe mass m. As seen in Chapte 6, this foce is diected towad the cente of the eath and has magnitude F 5 G } m M } whee G is the gavitational constant ( ]11 N-m /kg ) and is the distance fom m to the cente of the eath. Applying the definition of the gavitational field, which we denote by g, we find F GM g 5 } } 5 } m } The gavitational field equals gavitational acceleation. The units ae eithe N/kg o, equivalently, m/s. At 5 R, we find g 5 } G M R} N/kg ( ]11 N-m /kg )( kg) }}}}} ( m) Fig And at 5 R, we find GM g 5 }} 5 } 1 ( R) 4 } 1 }G M R} 5 }1 }(9.8 N/kg) N/kg Fig shows the eath s gavitational field evaluated at selected field points at distances R and R fom the cente. The field patten is the same as fo a negative chage (Fig ). It follows fom the definition of the gavitational field as foce pe unit mass that if we place a mass m in a gavitational field g the mass expeiences a foce F 5 mg. Thus a 10 kg mass at the suface of the eath, whee g N/kg diected towad the eath s cente, will expeience a foce of magnitude 98 N in the diection of g. At a distance 5 R fom the cente of the eath, whee g 5.5 N/kg diected inwad, the same 10 kg mass would expeience a foce of magnitude 5 N in the diection of g. Goup of Point Souce Chages When the souce of an electic field is a goup of point chages (q 1, q,..., q n ), the electic field is the esultant electic foce pe unit chage on a test chage q9 placed at a field point; that is, E 5 F 1 1 F F n }}} q9 F 1 F Fn }q9 }q9 1 } q9 Each tem in this equation is the field that would be poduced by one chage alone. So the total electic field E is the vecto sum of the fields poduced by the individual chages: E 5 E 1 1 E E n (17-9) Fig illustates the calculation of the electic field. You fist calculate the electic field of individual chages as befoe and then compute the vecto sum of these fields. Fig The field poduced by a goup of point chages is the vecto sum of the single chage fields.

12 438 CHAPTER 17 The Electic Field EXAMPLE 6 The Electic Field of Two Chages Point chages q ]9 C and q 5] ]9 C ae on the x-axis with espective coodinates x 5]5.00 cm and x cm. Find the electic field at thee points: point a at the oigin; point b on the x-axis at x 5]10.0 cm; and point c on the y-axis, 10.0 cm fom each chage. SOLUTION At field point a, chages q 1 and q poduce fields E 1 and E, both diected to the ight, away fom q 1 and towad q (Fig ). The fields have equal magnitudes, since the souce chages have equal magnitudes and ae equal distances away fom a: E 5 E 1 5 k } U ] 9 q1u C } 5 ( N/C)} } (5 ] m ) N/C The total field E at point a is diected to the ight and has magnitude E 5 E 1 1 E 5 ( N/C) N/C At field point b, q 1 poduces a field E 1 to the left (away fom q 1 ) and q poduces a field E to the ight (towad q ): E 1 5 k } U ] 9 q1u C } 5 ( N/C)} } N/C (5 ] m) 1 E 5 k} U ] 9 qu C } 5 ( N/C)} } N/C (1 ] m) The total electic field is diected to the left and has magnitude E 5 E 1 E N/C At point c, E 1 and E ae diected as indicated in Fig The fields have equal magnitudes because the two souce chages ae equidistant fom c: E 5 E 1 5 k } U q } 1U 5 ( N/C)} ] 9 C ( } m) N/C The total field is found by vecto addition of E 1 and E : E 5 E 1 1 E E x 5 E 1x 1 E x 5 (4500 N/C)(cos 60.0 ) 1 (4500 N/C)(cos 60.0 ) N/C The vetical components of E 1 and E cancel, and so the y component of E is 0. Thus the field is diected to the ight and has a magnitude of N/C. Fig

13 17-3 The Electic Field 439 EXAMPLE 7 Acceleation of an Electon in an Electic Field Find the instantaneous acceleation of an electon placed at field point c in the pevious example. SOLUTION Fist we calculate the foce on the electon, using Eq (F 5 q9e 5 ]ee). Since the chage is negative, the diection of the foce is opposite the diection of the electic field. The field at c is diected to the ight, and so the foce on the electon is diected to the left. We find the magnitude of the foce by multiplying the magnitude of the electon s chage times the magnitude of the field at c, found in the pevious example to be N/C: F 5 Uq9UE 5 ee 5 ( ]19 C)( N/C) ]16 N Since this is the only foce acting on the electon, the acceleation of the electon, accoding to Newton s second law, is also diected to the left and has magnitude ] 16 F N a 5 } } 5}9 } m ] kg m/s EXAMPLE 8 The Electic Field of Thee Chages Chages q ]6 C, q 5] ]6 C, and q ]6 C ae located on the y-axis, with espective y coodinates 10 cm, 0, and ]10 cm. Find the electic field at the field point having coodinates x 5 10 cm, y 5 0. SOLUTION Fist we indicate in Fig the diections of E 1, E, and E 3 poduced by q 1, q, and q 3. Next we calculate the magnitude of the fields: E 1 5 k } U q1u ]6 C } 5 ( N-m /C )}} N/C 1 (0.10 Ïw m) E 5 k } U qu } 5 ( N-m /C)} ] 6 C ( } N/C 0.10 m) E 3 5 k } U q3u ]6 C } 5 ( N-m /C )}} N/C 3 (0.10 Ïw m) Finally we calculate the vecto sum, using the calculated magnitudes and the diections indicated in Fig : E x 5 E 1x 1 E x 1 E 3x 5 ( N/C)(cos 45 ) N/C 1 ( N/C)(cos 45 ) N/C E y 5 E 1y 1 E y 1 E 3y Fig ( N/C)(sin 45 ) ( N/C)(sin 45 ) N/C This is a vecto of magnitude N/C, diected 80 above the positive x-axis. In the examples above we identified cetain chages as souces of an electic field and then calculated the magnitude and diection of that field. You should ealize that these souce chages can also espond to a field. Thus each chage both is the souce of an electic field and also expeiences a foce caused by the electic field of all othe chages.

14 440 Fig The electic field poduced by two chaged conductos is made visible by a suspension in oil of bits of thead, which align with the field. CHAPTER The Electic Field Fields Poduced by Continuous Distibutions of Chage Often the souce of an electic field is moe than just a few point chages, and so the methods of the last section ae not sufficient to compute the field. Fo example, a chaged metal plate might have pehaps excess electons on its suface. In computing the electic field poduced by the plate, we cetainly can t compute the fields of electons, point by point. Instead we can think of the excess electons as foming a continuous distibution of chage on the plate s suface and in so doing simplify the computation of the field. In this section we descibe the electic fields poduced by such chage distibutions (Fig ). The fist step in evaluating the electic field poduced at a field point P by a continuous distibution of chage is to divide the chage into tiny elements of chage Dq and then conside the field DE poduced by each such element at P (Fig. 17-0). Since Dq is so small, we can use the fomula fo a point chage to expess the magnitude of the field DE poduced by Dq. Applying Eq uqu E 5 k} to ou point chage Dq, we may wite udqu udeu 5 k } (17-10) Fig Field DE poduced at a field point P by a small chage element Dq (assumed positive), pat of a continuous chage distibution. The field DE is diected away fom Dq if Dq is positive and towad Dq if Dq is negative. We find the total electic field at P by applying the supeposition pinciple, that is, by summing the DE s that wee calculated using Eq : E 5S(DE) (17-11) EXAMPLE 9 Electic Field of a Chaged Ring A thin ing is unifomly chaged with a positive chage Q. A field point P is located on the axis of the ing at a distance fom the edge of the ing (Fig. 17-1). Deive an expession fo the electic field at P. SOLUTION Chage Q is unifomly spead ove the ing, and since each element of chage Dq is the same distance fom the field point, each contibutes a field of the same magnitude DE. Fig shows a pai of chage elements, Dq and Dq9, located on opposite sides of the ing, along with thei espective fields, DE and DE9, at field point P. The components pependicula to the axis, DE and DE9, cancel. This means that the only components contibuting to the total field ae those along the x-axis, DE x and DE9 x. Since the entie ing can be divided into othe pais of chages compaable to Dq and Dq9, all field components pependicula to the x-axis cancel. So only components along the x-axis need to be summed to obtain the total electic field E x : E x 5S (DE x ) Fig The field of a chaged ing. Components of DE and DE9 pependicula to the axis cancel. Each of the n chage elements Dq contibutes an equal component DE x 5DE cos u, whee DE is given by Eq DE 5 k } UD qu }. Thus E x 5S3k} UD qu }(cos u)4 5 nk} D q }(cos u) Since n Dq equals the total chage Q on the ing, we may expess this esult as E x 5 k } Q }(cos u) (17-1)

15 Chage Density 17-4 Fields Poduced by Continuous Distibutions of Chage 441 In the peceding example, chage was distibuted ove a line. Moe often, howeve, we encounte poblems in which chage is distibuted ove eithe a suface aea o a volume. Such distibutions lead to the definitions of suface chage density and volume chage density. Suface chage density, denoted by the Geek lette s (sigma), is defined as chage pe unit aea. Volume chage density, denoted by the Geek lette (ho), is defined as chage pe unit volume. Q s 5 } (17-13) A Q 5 } (17-14) V The units fo s ae C/m, and the units fo ae C/m 3. EXAMPLE 10 Chage Density in a Uanium Nucleus The uanium nucleus has a adius of ]15 m and caies a chage of 19e. Find the chage density within the nucleus. SOLUTION Since the nuclea chage is spead ove the volume of the nucleus, we apply the definition of volume chage density (Eq ) and use the fomula fo the volume of a sphee of adius (V 5 } 4 }p 3 ). 3 5 } Q V } 5 19e 5 19( ]19 C) } }}} } 4 }p 3 } 4 }p( ]15 m) C/m 3 Such a lage chage density can occu within a nucleus because the lage foces of electical epulsion between the potons ae balanced by attactive nuclea foces. Unifomly Chaged Infinite Plane An impotant special case of a continuous chage distibution is the infinite, unifomly chaged plane (Fig. 17-a). The solution of a poblem involving an infinite distibution of chage might seem to be an execise of no physical significance. Howeve, the solution to this poblem is a good appoximation to the field a finite plane of chage poduces at field points that ae close enough to the suface of the plane and fa enough fom the edges to make the plane look infinite (Fig. 17-b). Fo such field points, the missing pat of the infinite plane (fom the edges of the actual plane out to infinity) will contibute a negligible amount to the total electic field. Thus the ealistic poblem of a finite plane of chage can be appoximated by the poblem of an infinite plane, which has a simple solution. (a) (b) Fig. 17- (a) An infinite, unifomly chaged plane. (b) A finite, unifomly chaged plane. Point P is close enough to the plane and fa enough fom the edges that the plane looks infinite.

16 44 CHAPTER 17 The Electic Field It is possible to deive the infinite plane s electic field by applying Eq , but since this equies the use of integal calculus, we shall state the esult hee without poof. (Howeve, a deivation based on Gauss s law is povided in Appendix B.) The electic field of an infinite plane having a unifom suface chage density s is unifom, is diected pependicula to the plane, and has magnitude E 5 pkusu (infinite plane; constant s) (17-15) The field of a positively chaged plane is diected away fom the plane (Fig. 17-3a). The field of a negatively chaged plane is diected towad the plane (Fig. 17-3b). (a) (b) Fig (a) Electic field of a positively chaged infinite plane. (b) Electic field of a negatively chaged infinite plane. EXAMPLE 11 Electic Field of a Chaged Sheet of Photocopy Pape Inside a photocopie, a sheet of copy pape with dimensions of 0 cm by 30 cm has electons emoved fom one side, poducing a unifom positive suface chage. The pupose of this positive chage is to attact the negatively chaged black ink, o tone, fom the photocopie dum, whee the image of the oiginal is fist fomed; when the tone is attacted off the dum, the image is tansfeed onto the copy pape. Find the electic field at a point 1.0 cm fom the suface of the pape and not too close to the edge. SOLUTION The unifomly chaged sheet has a suface chage density s 5 } Q A } 5 ( )e 5 ( )( ]19 C) }} }}} A (0.0 m)(0.30 m) ]7 C/m This chage poduces an electic field that is diected away fom the sheet and has a magnitude given by Eq : E 5 pkus U 5 p( N-m /C )( ]7 C/m ) N/C Isolated Conducting Plate Suppose a net positive chage is placed on a lage, isolated metal plate, say, by emoval of some electons though static chaging. The emaining fee electons in the metal will aange themselves in such a way that the net positive chage will be distibuted equally ove the suface of the plate, with equal chage on the two sides (Fig. 17-4). The electic field at any point is the vecto sum of the fields poduced by the sides, each of which has a positive chage density s. Each side poduces a field that is appoximately the field of an infinite plane, that is, a field of magnitude pks, diected away fom the suface. To eithe side of the plate, the fields E 1 and E poduced by the two suface chages ae in the same diection, and thus the total field to

17 17-4 Fields Poduced by Continuous Distibutions of Chage 443 eithe side is diected away fom the plate and has a magnitude twice that poduced by one suface: E 5 (pks) 5 4pks (17-16) Inside the metal, the diection of E 1 is opposite the diection of E. Thus these fields cancel and the esultant field inside is zeo: E 5 E 1 1 E 5 0 (inside conducto) (17-17) Indeed, the static chage aanges itself on the suface of the plate in such a way as to poduce zeo field inside. If a nonzeo field E wee pesent inside, the fee electons in the metal would expeience a foce F 5 ]ee and would keep moving until the aangement of chage poduced zeo field inside. Popeties of Conductos Seveal featues of the chaged metal plate ae popeties of chaged conductos of any shape. The most impotant geneal popeties of chaged conductos ae: 1. The electic field inside a statically chaged conducto always equals zeo. The fee electons in the conducto keep moving until the chage distibution poduces zeo electic field inside.. Any net chage on a conducto is on the suface. It is possible to pove this popety by applying Gauss s law (Appendix B). Gauss s law shows why a net chage inside a conducto must always poduce a nonzeo field inside, and since thee is no field inside, thee can be no chage. 3. The electic field within a cavity in a conducto is always zeo, if thee is no chage inside the cavity. This popety can be poved by use of Gauss s law and the fact that the electic foce is a consevative foce. This featue of conductos can be utilized to shield a egion of space fom electic fields poduced by chages outside the egion. All that is equied is a closed conducting suface suounding the egion of space to be shielded. An automobile s metal body almost completely suounds the automobile s inteio with a conducto and hence povides good potection against the ca being penetated by electic fields, poduced, fo example, by an electical stom (Fig. 17-5). 4. The electic field just outside the suface of a conducto is pependicula to the suface and has magnitude Fig Side view of an isolated conducting plate, with chage equally divided between the two sides. The two chaged sides poduce fields (E 1 and E ) that cancel inside the plate but not outside. E 5 4pkusu (17-18) whee usu is the magnitude of the suface chage density on the pat of the conducto closest to the field point (Fig. 17-6). This popety is a genealization of the esult fo a conducting plate and, like popeties and 3, can be deived fom Gauss s law (Appendix B). Fig Chage density vaies ove the suface of this conducto. The electic field just outside any pat of the suface is popotional to the chage density thee. Fig The metal body of a ca shields the inteio fom electic fields.

18 444 CHAPTER 17 The Electic Field (a) (b) Fig (a) The electic field aound a chaged conducto is stongest nea a shap point. (b) Bits of thead suspended in oil suound a shaply pointed, chaged conducto. Notice that the theads ae concentated nea the point, indicating that the field is stongest thee. 5. The suface chage density on a conducto is geatest whee the suface is least flat, especially at shap points. This popety can be poved using the concept of electic potential (see Poblem 56, Chapte 17). Accoding to Eq (E 5 4pkusu), the field is stongest whee s is geatest. Theefoe the stongest fields aound a chaged conducto ae located just outside shap points, as indicated in Fig Lightning ods utilize this pinciple. These shaply pointed metal ods, connected to gound, ae sometimes placed on tall buildings. They potect the buildings in two ways. Fist, they tend to pevent the occuence of lightning in the immediate vicinity of the building, since the enhanced field nea a od s shap point will tend to dischage a neaby thundecloud befoe it eaches the high chage concentations necessay fo a lightning dischage. Second, any lightning that does occu in the vicinity is likely to be though the od to the gound, athe than though the building. The Empie State building, which is potected by a lightning-od system, is stuck by lightning evey few weeks Field Lines The diection of an electic field can be gaphically epesented by continuous lines called field lines. The diection of the field at any point is the diection of the tangent to the field line at that point (Fig. 17-8). The magnitude of the electic field can also be indicated by field lines but not by thei length, since the lines ae continuous. Instead, the spacing of the lines indicates the stength of the field. Conside any suface pependicula to the field lines (Fig. 17-9). The lines ae dawn so that the magnitude of E is popotional to the numbe n of field lines pe unit aea though a suface of aea A, pependicula to the field lines: n E ~ (17-19) } A Fig Field lines. Whee the field lines ae close togethe, the numbe of lines pe unit aea is geate and so the electic field is stonge. Whee the field lines ae fathe apat, the electic field is weake (Fig. 17-9). The field line epesentation is useful because it is often possible to daw field lines that ae continuous though most egions of space. Field lines begin and end only at points whee thee is electic chage. This geneal popety of field lines is poved in Appendix B on Gauss s law. Howeve, it is easy to veify the continuity of field lines fo two special cases: the unifomly chaged, infinite plane and the point chage. Fig The same numbe of field lines passes though two sufaces. The numbe of lines pe unit suface aea is geate fo the smalle suface, indicating the field is geate thee.

19 Field Lines fo a Unifomly Chaged, Infinite Plane As descibed in the pevious section, the field of a unifomly chaged, infinite plane is unifom and pependicula to the plane. The field line epesentation of such a field consists of continuous, equally spaced lines extending out in eithe diection fom the plane and pependicula to it (Fig ) Field Lines 445 (a) (b) Fig Field lines fo a unifomly chaged infinite plane. (a) Positively chaged plane. (b) Negatively chaged plane. Accoding to Eq , electic field stength is popotional to the numbe of field lines n pe unit aea A. Thus the numbe of field lines though a pependicula suface is popotional to the poduct of field stength and suface aea: n ~ EA O we may intoduce a constant of popotionality c and expess this esult as n 5 cea (17-0) The constant c is a scale facto, chosen so as to obtain the desied numbe of lines in a dawing. Field Lines fo a Single Point Chage We shall apply Eq to the poblem of epesenting by field lines the electic field of a positive point chage q. Since the field is diected adially outwad fom the chage, we choose as a pependicula suface a spheical suface that has adius and is centeed on the chage (Fig ). We apply Eq. 17-0, using the expession fo the field of a point chage q at a distance and the equation fo the suface aea of a sphee of adius : n 5 cea 5 c 1 k } q } (4p ) Fig This spheical suface is pependicula to the electic field poduced by a point chage at the cente of the sphee. o n 5 4pkqc (17-1) Notice that this expession fo n involves only constants. This means that the numbe of field lines passing though a sphee of any adius is the same, independent of. Thus we can epesent the field by continuous field lines (Fig. 17-3). All the lines that oiginate at q pass though any spheical suface centeed on q. The numbe of lines dawn fom q will depend on the value chosen fo the scale facto c. Fig chage. Field lines of a point

20 446 CHAPTER 17 The Electic Field Field of a Unifomly Chaged Sphee A chaged sphee in which chage is unifomly distibuted thoughout the volume of the sphee poduces an electic field that, fo field points outside the sphee, is the same as though the chage wee concentated at the cente of the sphee; that is, the field outside the sphee is the field of a point chage.* This esult, poved in Appendix B on Gauss s Law, means that the field lines outside a unifom sphee of chage look the same as fo a point chage (Fig. 17-3). Two-Dimensional Dawings of Field Lines It is difficult to indicate a thee-dimensional field line patten by a pespective dawing. Theefoe, we shall often use two-dimensional dawings of field lines. Fo example, Fig shows in two dimensions the field lines of an electic dipole, which consists of two point chages of opposite sign but equal magnitude. When viewing such figues, you should keep in mind that the actual patten of field lines is theedimensional. Fig Field lines of a dipole. EXAMPLE 1 Finding the Stength of a Field Fom the Numbe of Field Lines Suppose that two small sufaces of equal aea ae dawn in Fig one a plane suface passing though point P and pependicula to field lines thee and the othe a spheical suface centeed on the negative chage and passing though point R. Suppose that 9 field lines pass though the plane suface and 7 pass though the spheical suface. If the stength of the electic field at P is 00 N/C, what is it at R? SOLUTION Since field stength is popotional to the numbe of lines pe unit aea and the two suface aeas ae equal, the field stength is popotional to the numbe of lines. Thus the field stength at R is E R 5 } 7 } E 9 P 5 8(00 N/C) N/C *Similaly, the eath s gavitational field is the same fo points outside the eath as though the eath s mass wee concentated at its cente. (See Chapte 6, Fig. 6-6.)

21 HAPTER 17 SUMMARY 447 CElectic chage q, measued in coulombs (C), may be eithe positive o negative. Chages of like sign epel one anothe; chages of opposite sign attact one anothe. Unchaged matte consists of equal numbes of potons (q 51e) and electons (q 5]e), whee e ]19 C Electons may be tansfeed to o fom a body, giving it a net chage popotional to the numbe n of electons tansfeed: q 56ne The magnitude of the mutual attactive o epulsive foce between point chages is popotional to the poduct of the magnitude of the chages and invesely popotional to the squae of the distance between them, accoding to Coulomb s law: F 5 k } uq uuq9u } whee k N-m /C The supeposition pinciple states that the esultant foce poduced by a numbe of chages on a chage q is found by fist calculating the magnitudes and diections of the Coulomb foces each chage exets on q and then finding the vecto sum of these foces. The electic field E is a vecto quantity poduced by a distibution of chage in the egion of space suounding that chage. A point in space whee the field is evaluated is called a field point. A chage q9 in an electic field expeiences a foce: F 5 q9e whee E is the field at the location of q9; E is poduced by othe chages. The field poduced by a single point chage q is diected away fom q if q is positive and towad q if q is negative. The magnitude of the field of a point chage at a distance away fom the chage is uqu E 5 k } The field poduced by a distibution of chage is the vecto sum of the fields poduced by each individual chage: Fo a continuous distibution of chage, we find the electic field by dividing the chage into tiny elements of chage Dq, calculating the field DE of each element Dq as though it wee a point chage, and then applying the supeposition pinciple to find the total field: E 5S(DE) whee udeu 5 k } ud qu } Suface chage density s is defined as chage pe unit aea, and volume chage density is defined as chage pe unit volume: Q s 5 } A Q 5 } V An infinite plane having a unifom chage density s poduces a unifom field, diected pependicula to the plane away fom the plane if s. 0 and towad the plane if s, 0. The magnitude of the field is E 5 pkusu Both the electic field and the net chage inside a statically chaged conducto equal zeo. Any net chage on a conducto is on its suface, poducing a suface chage density s. The field at a field point just outside the conducto is pependicula to the conducto s suface and has a magnitude detemined by s on the pat of the suface nea the field point, accoding to the equation E 5 4pkusu Field lines ae diected lines used to epesent an electic field. They ae continuous at most points in space and ae often cuved. Field lines teminate only at points in space whee thee is electic chage, beginning only on positive chages and ending only on negative chages. The tangent to a field line at any point indicates the diection of the field at that point. The magnitude of the field is indicated by the spacing of the lines the numbe of lines pe coss-sectional aea is popotional to the stength of the field; thus the field is stongest whee the lines ae closest. E 5 E 1 1 E E n

22 448 CHAPTER 17 The Electic Field Summay Questions 1 Is the electic foce between an electon and a poton attactive o epulsive? A positon is the antipaticle of an electon, which means that the positon has the same mass as the electon but it has a positive chage 1e. Will a poton attact o epel a positon? 3 Physicists now know that potons and othe paticles peviously thought to be indivisible ae in fact made up of smalle units called quaks. Potons consist of thee quaks. Thee ae vaious kinds of quaks, but all have electic chage of eithe 6} }e o 3 6}1 }e. 3 (a) If one of the quaks in a poton has a chage of 1} }e, what ae the chages of the othe two quaks? 3 (b) A neuton is an unchaged paticle that, togethe with the poton, makes up most of the mass of an atom. Like the poton, the neuton consists of thee quaks, one of which has an electic chage of 1} }e. What ae the chages of the othe two 3 quaks? 4 The magnitude of the electic foce between two point chages is initially 180 N. What will the foce be if the distance between the two chages is (a) doubled; (b) tipled; (c) halved? 5 The wate molecule has an electic dipole moment. This means that one end of the molecule is positively chaged and the othe end is negatively chaged. These dipole moments esult in electic foces between the molecules. Which of the pais of dipoles shown in Fig expeiences a net attactive foce? 6 A small positively chaged object is bought close to one end of a long metal od that is electically insulated and initially unchaged. The object does not touch the od. Does the od exet a foce on the object? Explain. 7 Two initially unchaged metal sphees ae connected by a coppe wie. A positively chaged object is placed nea one of the sphees but not touching it. What can you do to cause the two sphees to etain a chage even afte the object is moved away? This pocess of chaging without contact is called chaging by induction. 8 Fig shows two potons and one electon. What is the diection of the electic field at R? Fig All fou chages in Fig have the same magnitude. At point P the electic field poduced by these chages is diected towad the left. Chage q 1 is negative. What ae the signs of q, q 3, and q 4? Fig Fig

23 Questions Small pieces of pape ae nea a comb that caies a static chage. Although each piece of pape has no net chage, the comb s field causes the pape to become polaized, as shown in Fig All the positive chage in a piece of pape is pushed down in the diection of E, and all the negative chage is pushed up. By consideing the magnitude and diection of the foces on the chages at the two ends of each piece of pape, detemine whethe this nonunifom field exets a net foce on the pape and, if so, in what diection. 1 Which of the sketches in Fig , if any, can epesent electic field lines? Fig Two unifomly chaged infinite planes ae shown in Fig If a positively chaged balloon of negligible weight is attached by a sting to point P, what will the diection of the sting be when the balloon comes to est? Fig In the Sta Tek television seies, pisones on the staship Entepise wee confined to thei cabins by means of an invisible foce field in an open dooway (Fig ). When they attempted to pass though the dooway, they eceived a painful shock. Suppose the foce field is just a stong electic field. A pisone easons that since his body is unchaged an electic field should not bothe him. What s wong with that easoning? Fig If no othe chage distibution is neaby, is it possible fo the ight side of a flat conducting plate to be chaged while the left side emains unchaged? Explain. 15 Fig shows electic field lines aound a conducto. (a) At which of the fou points is the field stongest? (b) At which point is thee negative chage? Fig Fig

24 450 CHAPTER 17 The Electic Field 16 Suppose you ae diving though a thundestom. Would it be safe to be (a) inside a ca with a metal body; (b) inside a fibeglass ca; (c) outside you ca? 17 Suppose you ae in a cave, deep within the eath. Ae you safe fom electical stoms? 18 Suppose a positive chage of 10e is moved though a small opening in a hollow metal sphee and placed at a point P, which is connected to the inside of the sphee by means of a metal wie (Fig. 17-4). (a) What is the final chage distibution, and how is it achieved? (b) Is thee any limit to the amount of chage that can be tansfeed in this way, if the sphee is in a vacuum, so that thee is no way fo the chage to leak off? 0 Detemine the magnitude and sign of q in Fig Fig Fig A Van de Gaaff geneato opeates on the pinciple descibed in the peceding question. Positive chages ae supplied to the bottom end of a conveyo belt though fictional contact (Fig a). The belt then caies the positive chage to the inside of a metal sphee, whee the chage is tansfeed to the sphee s oute suface. (a) Why must an upwad foce be applied to the positively chaged side of the belt to lift it? (b) When the gil in Fig b touches the metallic suface, he body becomes an extension of the conducto, causing he skin to become positively chaged and he hai to stand on end because of epulsion of like chages. Which of the following foces acts on the end of a hai, balancing the electic foce of epulsion: attactive electic foce, nomal foce, tension in the hai, o weight of the hai? (a) (b) Fig

25 Poblems St. Elmo s fie is a glowing of the ai that occus unde cetain conditions aound a chaged object dischaging into the atmosphee. This kind of dischage, also called a coona dischage, is sometimes obseved aound the wings of an aiplane in flight. Ae you moe likely to see St. Elmo s fie nea the ounded font edge of the aiplane s wing o nea the shap back edge? Answes to Odd-Numbeed Questions 1 attactive; 3 (a) 1} }e, 3 }1 }e; (b) 3 }1 }e, 3 }1 }e; 5 b, c; 7 disconnect the wie; 9,, 1; 11 Although the body has 3 no net chage, thee ae cetainly chages within it; 13 vetical; 15 (a) C; (b) D; 17 yes; 19 (a) to balance the foce of the downwad-diected field acting on the belt s chages as a esult of the chages aleady on the sphee (b) tension in the hai; 1 the shap back edge Poblems (listed by section) 17- Coulomb s Law 1 A coppe wie 90.0 cm long and 1.00 mm in diamete has a mass of 6.35 g. (a) Find the numbe of electons in the wie. (Coppe has an atomic numbe of 9; that is, thee ae 9 potons in the coppe atom. Coppe s atomic mass is 63.5.) (b) Thee is one fee electon pe atom in coppe. Find the numbe of fee electons in the wie. Donna and John have masses of 50.0 kg and 80.0 kg espectively. (a) How many potons ae thee in each peson? (Potons make up oughly 55% of the mass of the human body.) (b) How many electons ae in each peson? (c) Suppose John and Donna stand 5.00 m apat. Calculate the foce exeted on John s potons by (1) Donna s potons and () he electons. (d) What is the esultant foce on his potons? 3 The stuctue of a sodium chloide (table salt) cystal is shown in Fig Each sodium ion Na 1 has a chage 1e and is adjacent to a chloide ion Cl ], which has a chage ]e. The electic foce of attaction between sodium ions and chloine ions holds the cystal togethe. (a) What is the magnitude of the foce between adjacent sodium and chloine ions, ]10 m apat? (b) What is the esultant foce on any ion in the cystal? (c) Suppose you attempt to beak a cubic salt cystal, 1.00 mm on a side, by applying foces F and ]F pependicula to opposite sides of the cube, tying to pull it apat. How geat would F have to be to ovecome the attactive foces of all the ions in a 1.00 mm plane of the cystal? 4 Thee ae extemely lage electical foces of epulsion between the potons in the nucleus of an atom. Howeve, these foces ae nomally not as geat as the stong foce, which is the foce that binds all the potons and neutons in the nucleus togethe. The stong foce has a vey shot ange on the ode of 3 10 ]15 m. When potons o neutons ae sepaated by a distance geate than this, the stong foce does not act. Thus, if fo some eason a nucleus splits in two, o fissions, each fagment can expeience an electical epulsive foce without any othe foce to balance it. Suppose a uanium nucleus (9 potons) splits into two nuclei having 46 potons each. (a) Calculate the epulsive foce between these nuclei just afte the split, when they ae 10 ]14 m apat. (b) Suppose that all the nuclei in one coss section of a 1 mm 3 cube (about nuclei) simultaneously split and expeience a foce pependicula to the coss section. What would be the total foce splitting the cube apat? 5 Find the atio of the magnitudes of the electical and the gavitational foces acting between a poton and an electon sepaated by an abitay distance d. 6 Find the foce on a negative chage that is placed midway between two equal positive chages. All chages have the same magnitude. Fig

26 45 CHAPTER 17 The Electic Field 7 Hanging fom theads ae two chaged balls made of pith (a vey light, spongy mateial that comes fom inside the stems of cetain plants). The balls each have a mass of g and a chage of the same magnitude q and ae attacted towad each othe, as shown in Fig (a) What is the magnitude of the electic foce? (b) What is the magnitude of q? (c) How many electons have been tansfeed to o fom each ball? 10 (a) In Fig , what ae the magnitude and diection of the esultant foce on q 1? (b) What is the esultant foce on the cente of mass of the fou chages? Fig Fig Thee chages, q ]9 C, q C, and q C, ae located on the x-axis at x 1 5 0, x cm, and x cm. Find the esultant foce on q 3. 9 An electon is nea a positive ion of chage 19e and a negative ion of chage ]8e (Fig ). (a) Find the magnitude and diection of the esultant foce on the electon. (b) Find the magnitude and diection of the electon s instantaneous acceleation. 11 In a Catesian coodinate system, the chage q 1 5 ] ]4 C is at the oigin, the chage q ]3 C has coodinates x m, y 5 0, and the chage q 3 5] ]4 C has coodinates x 5 0, y 5]5.00 m. Find the magnitude and diection of the esultant foce on q 1. 1 Given two point chages q 1 and q 5 4q 1, find the position of a thid chage q 3 elative to the othe two chages, such that the esultant foce on q 3 is zeo The Electic Field 13 A chage q 1 on the y-axis poduces a field of magnitude 3 N/C at the oigin, and a chage q on the x-axis poduces a field of magnitude 4 N/C at the oigin. What is the magnitude of the total field at the oigin? 14 Fig shows fields E 1, E, E 3, and E 4 at point P, poduced espectively by chages q 1, q, q 3, and q 4. What is the sign of each chage? Fig Fig

27 Poblems Fig shows positive chages q 1 and q. At which of the five points, A, B, C, D, o G, could you place a negative chage q 3 of the ight magnitude so that the field at point P is the field poduced by q 1 alone? Fig In Fig two electons ae the same distance fom a field point P. At which of the points A, B, C, o D could a poton be placed so that the electic field at P is zeo? 0 Two point chages q 1 and q ae sepaated by 0.0 cm. The electic field at thei midpoint is 600 N/C, diected away fom q 1, which is ]9 C. Find q. 1 A ]9 C chage has coodinates x 5 0, y 5 ].00; a ]9 C chage has coodinates x , y 5 0; and a ] ]9 C chage has coodinates x , y , whee all distances ae in cm. Detemine magnitude and diection fo (a) the electic field at the oigin and (b) the instantaneous acceleation of a poton placed at the oigin. Point chages of ]9 C, ]9 C, and ] ]9 C ae placed at the vetices of an equilateal tiangle. Find the magnitude of the electic field at the cente of the tiangle, which is 10.0 cm fom each vetex. 3 Find the x and y components of the electic field poduced by q 1 and q in Fig at (a) point A; (b) point B. Fig Fig Find the electic field at P in Fig Fig At what distance fom a poton does its electic field have a magnitude of 1 N/C? 19 Points A, B, and C ae at the vetices of an equilateal tiangle. A cetain positive chage q placed at A poduces an electic field of magnitude 100 N/C at C. Suppose a second, identical chage is placed at B. What is the magnitude of the new electic field at C? 4 A Ping-Pong ball that has a mass of.40 g is chaged when electons ae added. The ball is stationay in an electic field. Find the magnitude and diection of the field. 5 A small Styofoam ball weighing ] N is suppoted by a thead in a hoizontal electic field of magnitude N/C. The thead makes an angle of with the vetical. Find the magnitude of the chage on the ball. 6 Millikan measued the electon s chage by obseving tiny chaged oil dops in an electic field. Each dop had a chage imbalance of only a few electons. The stength of the electic field was adjusted so that the electic and gavitational foces on a dop would balance and the dop would be suspended in ai. In this way the chage on the dop could be calculated. The chage was always found to be a small multiple of ]19 C. Find the chage on an oil dop weighing ]14 N and suspended in a downwad field of magnitude N/C.

28 454 CHAPTER 17 The Electic Field 7 A beam of electons is shot into a unifom downwad electic field of magnitude N/C. The electons have an initial velocity of m/s, diected hoizontally. The field acts ove a small egion, 5.00 cm in the hoizontal diection. (a) Find the magnitude and diection of the electic foce exeted on each electon. (b) How does the gavitational foce on an electon compae with the electic foce? (c) How fa has each electon moved in the vetical diection by the time it has emeged fom the field? (d) What is the electon s vetical component of velocity as it emeges fom the field? (e) The electons move an additional 0.0 cm afte leaving the field. Find the total vetical distance that they have been deflected by the field. 8 Duing a thundestom the electic field at a cetain point in the eath s atmosphee is N/C, diected upwad. Find the acceleation of a small piece of ice of mass ]4 g, caying a chage of ]11 C. 35 Two lage paallel planes each cay a unifom distibution of chage of the same magnitude s but of opposite signs (Fig ). Find the electic field at points a, b, and c. Fig Two lage flat dielectic sufaces ae paallel to each othe and cay unifom chage densities (Fig ). Find the electic field at points a, b, and c Continuous Chage Distibutions 9 The eath s suface has a chage density of about ] ]9 C/m. Find the total chage on the suface. 30 The eath s suface chage density (] ]9 C/m ) is oughly balanced by a net positive chage in the lowe 10 km of the eath s atmosphee. What is the aveage volume chage density of this atmospheic chage? 31 A thin gold ing of adius 1.00 cm caies a unifom chage pe unit length of ]14 C/m. Find the electic field on the axis of the ing 1.00 cm fom the cente. 3 A 0.0 cm by 30.0 cm sheet of pape has a unifom suface chage density of ]8 C/m. Find the electic field at a distance fom the pape s suface of (a) 1.00 mm; (b) 1.00 cm; (c) 3.00 cm; (d) 5.00 m. None of the field points is close to the pape s edge. 33 A ectangula slab of dimensions 1.00 m m cm has a unifom chage distibution of ]7 C spead thoughout its volume. Find the electic field at a point just outside the chage distibution, close to the cente of one of the lage faces of the slab. 34 The eath is a good electical conducto and, duing peiods of clea weathe, has a downwad-diected electic field of about 100 N/C at low altitudes. Find the chage density on the eath s suface when its field has this value. Fig Fig shows two lage flat sufaces that have unifom chage densities. Find the electic field at points a and b. Fig

29 Poblems A point chage q is nea a unifomly chaged, lage flat suface of a dielectic (Fig ). Find the electic field at P. Fig Field Lines 39 Fig shows the field that esults when a conducting sphee is placed in a unifom extenal field. (a) Find the field at P, inside the conducto. (b) At which of the points R, S, o T is the field weakest, and at which of these points is it stongest? (c) Find the chage density on the suface of the conducto nea R if the field at that point has magnitude N/C. Fig Additional Poblems 41 Chage q ]9 C is at the oigin and chage q ]9 C has coodinates x cm, y 5 0. Find the coodinates of a field point whee E An aluminum sphee of adius 1.00 m caies a chage of ] ]4 C. The sphee is isolated except fo a paticle of mass ]15 kg and chage ]17 C, which obits the sphee in a cicula obit of adius 3.00 m. Find the peiod of the obit. 43 Electic chage on the eath s suface and in the eath s atmosphee poduces an electic field, which on a clea day (no thundestoms) is diected vetically downwad and has a magnitude of about 100 N/C just above the eath s suface. Vey small negative paticles can be suppoted by the field. Fo lage paticles, the chage that must be caied may be so geat that the field aound the chage causes the suounding ai to conduct the chage away. This occus at fields of about N/C in dy ai. Find the adius of the lagest wate dop that could be suppoted in the eath s electostatic field. Assume that the dop is spheical. Fig A coaxial cable consists of an inne cylindical conducto of adius 1.00 mm inside a hollow cylindical conducto of adius 3.00 mm. Chages on the sufaces of these conductos poduce the field lines shown in Fig The field lines ae equally spaced along the axis of the cable. If the magnitude of the field just outside the inne conducto is 600 N/C, what is the magnitude of the field just inside the oute conducto? Fig Above this field of wheat is an invisible atmospheic electic field. Even though you can t see it, the electic field is just as eal as the wheat field.

30 456 CHAPTER 17 The Electic Field 44 Find the adius of the lagest (spheical) wate dop whose weight could be suppoted by any electostatic field in dy ai. The ai becomes conducting when the total field at any point exceeds N/C. 45 An electon is initially at est just outside a lage coppe suface that has a chage density of ] 6 C/m. How fa has the electon moved in ]9 s if the field is unifom ove this egion? 46 A thundecloud contains a lage concentation of chaged paticles: ionized molecules, chaged dops of wate, bits of ice, and specks of dust. Thee is a concentation of positive chage in the uppe pat of the cloud and of negative chage in the lowe pat.* Suppose that the chage distibution in a cetain cloud can be appoximated by two unifom sphees of chage 1100 C and ]100 C, centeed at points P and Q (Fig ). Find the magnitude and diection of the electic field (a) at P and (b) at the location of an aiplane 1.00 km diectly above P. 47 A point chage q is placed midway between two identical positive point chages of magnitude ]9 C. The esultant foce on each of the thee chages is zeo. Find q. 48 The field just outside a point on the suface of a coppe wie of adius 1.00 mm has magnitude N/C. (a) Find the magnitude of the suface chage density at that point on the wie. (b) If the suface chage density is unifom aound the cicumfeence of the wie, how much chage is on a 1.00 m length of the wie? 49 A ing of adius 0.0 cm has unifom chage density of ]6 C/cm. A 1.00 cm section of the ing on the ight side is emoved. What ae the magnitude and diection of the electic field at the cente of the ing? 50 Two point chages m apat expeience a epulsive foce of N. The sum of the two chages equals ]5 C. Find the values of the two chages. Fig *Negative chages ae peiodically tansfeed to the gound in the fom of lightning stokes. Woldwide electical stoms thoughout the day maintain the negative chage density on the eath s suface. This spoadic downwad flow is balanced by a slow upwad flow of negative chage duing clea weathe.